Computational Geomechanics with special Reference to Earthquake Engineering
0 C Zienkiewicz, Institute for Numerical Methods in Engineering, Swansea, Wales
A H C Chan, University of Birmingham, England
M Pastor, CEDEX* and ETS de Ingenieros de Caminos, Madrid, Spain B A Schrefler, University of Padua, Italy
T Shiomi, Takenaka Corporation, Japan
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Centro de E s t ~ i ~ x f i & I & i d n / e Obras Publicas
JOHN WILEY & SONS
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Library of Congress Cataloging-in-Publication Data Computational geomechanics with special reference to earthquake engineering1 O.C. Zienkiewicz . . . [et al.]. p. cm. Includes bibliographical references and index. ISBN0471-98285-7 1. Earthquake engineering 2. Mathematics. I. Zienkiewicz, O.C. TA705.C625 1998 624.1 ' 7 6 2 6 ~ 2 1 98-8795 CIP British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library ISBN 0-471-98285-7 Typeset in 10/12.25pt Times from the author's disks by Pure Tech India Ltd, Pondicherry Printed and bound in Great Britain by Bookcraft (Bath) Ltd, Midsomer Norton, Somerset This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production.
Contents
Preface
1 Introduction and the Concept of Effective Stress 1.1 Preliminary Remarks 1.2 The Nature of Soils and Other Porous Media: Why a Full Deformation Analysis is the Only Viable Approach for Prediction 1.3 Concepts of Effective Stress in Saturated or Partially Saturated Media 1.3.1 A single fluid present in the pores-historical note 1.3.2 An alternative approach to effective stress 1.3.3 Effective stress in the presence of two (or more) pore fluids. Partially saturated media References
2 Equations Governing the Dynamic, Soil-Pore Fluid, Interaction 2.1 General Remarks on the Presentation 2.2 Fully Saturated Behaviour With A Single Pore Fluid (Water) 2.2.1 Equilibrium and Mass Balance Relationship (u, w and p) 2.2.2 Simplified equation sets (u-p form) 2.2.3 Limits of validity of the various approximations 2.3 Partially Saturated Behaviour with Air Pressure Neglected @, = 0) 2.3.1 Why is inclusion of semi-saturation required in practical analysis? 2.3.2 The modification of equations necessary for partially saturated conditions 2.4 Partially Saturated Behaviour with Air Flow Considered (pa 0) 2.4.1 The governing equations including air flow 2.4.2 The governing equation 2.5 Alternative derivation of the governing equations of sections 2.1-2.4, based on the hybrid mixture theory 2.5.1 Kinematic equations 2.5.2 Microscopic balance equations 2.5.3 Macroscopic balance equations 2.5.4 Constitutive equations 2.5.5 General field equations 2.5.6 Nomenclature 2.6 Concluding Remarks References
>
vi
CONTENTS
3 Finite Element Discretization and Solution of the Governing Equations 3.1 The Procedure of Discretization by the Finite Element Method 3.2 u-p Discretization for a General Geomechanics Finite Element Code 3.2.1 Summary of the general governing equations 3.2.2 Discretization of the governing equation in space 3.2.3 Discretization in time 3.2.4 General applicability of transient solution (consolidation, static solution, drained uncoupled, undrained) Time step length The consolidation equation Static problems-undrained and fully drained behaviour 3.2.5 The Structure of the numerical equations illustrated by their Linear equivalent 3.2.6 Damping matrices 3.3 The u-U Discretization and its Explicit Solution 3.3.1 The governing equation 3.3.2 Discretized equation and the explicit scheme 3.3.3 The structure of the numerical equations in linear equivalent 3.4 Theory: Tensorial Form of the Equations 3.5 Conclusions References
4 Constitutive Relations-Plasticity 4.1 Introduction 4.2 The general Framework of Plasticity 4.2.1 Phenomenological aspects 4.2.2 Generalized plasticity 4.2.3 Classical theory of plasticity 4.3 Critical State Models 4.3.1 Introduction 4.3.2 Critical state models for normally consolidated clay 4.3.3 Extension to sands 4.4 Advanced Models 4.4.1 Introduction 4.4.2 A generalized plasticity model for clays 4.4.3 A generalized plasticity model for sands 4.4.4 Anisotropy 4.5 Modified Densification Model 4.5.1 Densification model for cyclic mobility References
5 Examples for Static, Consolidation and Partially Saturated Dynamic Problems 5.1 Introduction 5.2 Static Problems 5.2.1 Example (a): Unconfined situation-small constraint -Embankment -Footing 5.2.2 Example (b): Problems with medium (intermediate) constraint on deformation 5.2.3 Example (c): Strong constraints-undrained behaviour 5.2.4 Example (d): The effect of the K section of the yield criterion
CONTENTS 5.3 5.4 5.5 5.6 5.7
Isothermal Drainage of Water from a Vertical Column of Sand Modelling of Subsidence due to Pumping from a Phreatic Aquifer Air storage Modelling in an Aquifer Flexible Footing Resting on a Partially Saturated Soil Comparison of Consolidation and Dynamic Results Between Small strain and Finite Deformation Formulation 5.7.1 Consolidation of fully saturated soil column 5.7.2 Consolidation of fully and partially saturated soil column 5.7.3 Consolidation of two-dimensional soil layer under fully and partially saturated conditions 5.7.4 Fully saturated soil column under earthquake loading 5.7.5 Elasto-plastic large-strain behaviour of an initially saturated vertical slope under a gravitational loading and horizontal earthquake followed by a partially saturated consolidation phase 5.8 Conclusions References
6 Validation of Prediction by Centrifuge 6.1 Introduction 6.2 Scaling Laws of Centrifuge Modelling 6.3 Centrifuge Test of a Dyke Similar to a Prototype Retaining Dyke in Venezuela 6.4 The VELACS Project 6.4.1 General analysing procedure 6.4.2 Description of the precise method of determination of each coefficient in the numerical model 6.4.3 Modelling of the laminar box 6.4.4 Parameters identified for the Pastor-Zienkiewicz Mark 111 model 6.5 Comparison with the VELACS Centrifuge Experiment 6.5.1 Description of the models Model No. 1 Model No. 3 Model No. I I 6.5.2 Comparison of experiment and prediction 6.6 Centrifuge test of a Retaining Wall 6.7 Conclusions References
7 Prediction Applications and Back Analysis 7.1 Introduction 7.2 Example 1: Simulation of Port Island Liquefaction-Effect of Multi-dimensional Loading 7.2.1 Introductory remarks 7.2.2 Multi-directional loading observed and its numerical modelling-simulation of liquefaction phenomena observed at Port Island -Conditions and modelling -Results of simulation -Effects of multi-directional loading 7.3 Simulation of Liquefaction Behaviour During Niigita Earthquake to Illustrate the Effect of Initial (shear) Stress
vii
viii
CONTENTS 7.3.1
Influence of initial shear stress -Significance of ISS component to the responses -Theoretical considerations 7.4 Quay Wall Failure and a Countermeasure 7.4.1 Conditions and modelling -Configuration -Soil layers and properties -Input Motion 7.4.2 Results and remarks 7.5 Lower San Fernando D a m Failure 7.6 Mechanism of Liquefaction Failure o n a n Earth D a m (the N Dam) 7.6.1 Objective of the analysis 7.6.2 Input motion 7.6.3 Conditions and modelling -Soil properties -Parameters for liquefaction -Initial stress 7.6.4 Results of calculations 7.6.5 Remarks 7.7 Liquefaction Damage in the Niigata Earthquake of 1964 7.7.1 Results 7.8 Interaction Between Ordinary Soil and Improved Soil Layer 7.8.1 Input motions -Earth pressure due to liquefaction 7.8.2 Safety for seismic loading -External safety -Internal safety 7.8.3 Remarks References
8 Some Special Aspects of Analysis and Formulation: Radiation Boundaries, Adaptive Finite Element Requirement and Incompressible Behaviour 8.1 Introduction 8.2 Input for Earthquake Analysis and Radiation Boundary 8.2.1 Specified earthquake motion: absolute and relative displacements 8.2.2 The radiation boundary condition: formulation of a one-dimensional problem 8.2.3 The radiation boundary condition: treatment of two- dimensional problem 8.2.4 Earthquake input and the radiation boundary condition-concluding remarks 8.3 Adaptive Refinement for Improved Accuracy and the Capture of Localized Phenomena 8.3.1 Introduction to adaptive refinement 8.3.2 Localization and strain softening: possible non-uniqueness of numerical solutions 8.4 Stabilization of Computation for Nearly Incompressible Behaviour with Mixed Interpolation 8.4.1 The problem of incompressible behaviour under undrained conditions 8.4.2 The velocity correction, stabilization process 8.4.3 Examples illustrating the effectiveness of the operator split procedure 8.4.4 The reason for the success of the stabilizing algorithm References
CONTENTS
9 Computer Procedures for Static and Dynamic Saturated Porous Media finite element Analysis 9.1 Introduction 9.2 Outline description of DIANA-SWANDYNE I1 9.3 Description of major routines used in DIANA-SWANDYNE I1 9.3.1 The top level routines 9.3.2 Subroutines for control and material data input 9.3.3 Subroutines for mesh data input 9.3.4 Subroutines called by the main control routine for analysis 9.3.5 Subroutines for the formation of element matrices and residual calculation 9.4 Major service subroutines 9.5 Constitutive model subroutines 9.5.1 Standard constitutive model interfacing subroutine CONSTI 9.5.2 Constitutive models available for general dissemination 9.5.3 Other constitutive models implemented 9.6 System-dependent subroutines References
Appendix 9A Implementing New Models into SM2D
Author Index Subject Index
Preface
Although the concept of effective stress in soils is accepted by all soil mechanicians, practical predictions and engineering calculations are traditionally based on total stress approaches. When the senior author began, in the early seventies, the application of numerical approaches to the field of soil mechanics in general and to soil dynamics in particular, it became clear to him that a realistic prediction of the behaviour of soil masses could only be achieved if the total stress approaches were abandoned. The essential model should consider the coupled interaction of the soil skeleton and of the pore fluid. Indeed, the phenomena of weakening and of 'liquefaction' in soil when subjected to repeated loading such as that which occurs in earthquakes, can only be explained by considering this 'two-phase' action and the quantitative analysis and prediction of real behaviour can only be achieved by sophisticated computation. The simple limit methods often applied in statics are no longer useful. It therefore seems necessary at the present time to present, in a single volume, the basis of such computational approaches because a wider audience of practitioners and engineering students will require the knowledge which hitherto has only been available through scientific publications scattered throughout many journals and conferences. The present book is an attempt to provide a rapid answer to this need. The multiple authorship not only ensures a speedy result, it also introduces members of the research team which, during the last decade, have focused attention on the subject which has developed practical computer codes which are now available to both practitioners and researchers. Since 1975 large number of research workers, both students and colleagues, have participated both at Swansea and elsewhere in laying the foundations of numerical predictions which were based largely on concepts introduced in the early forties by Biot. However, the total stress calculation continues to be used by some engineers for earthquake response analysis, often introduced with the linear approximations. Such simplifications are generally not useful and can lead to erroneous predictions. In recent years, centrifuge experiments have permitted the study of some soil problems involving both statics and dynamics. These provide a useful set of benchmark predictions. Here a validation of the two-phase approach was available and close agreement between computation and experiment was found. A very important landmark was a workshop held at the University of California, Davis, in 1993, which
PREFACE
xi
reported results of the VELACS project (Verification of Liquefaction Analysis by Centrifuge Studies)-sponsored by the National Science Foundation of USA. At this workshop a full vindication of the effective stress, two-phase approaches was clearly available and it is evident that these will be the basis of future engineering computations and prediction of behaviour for important soil problems. The book shows some examples of this validation and also indicates examples of practical application of the procedures described. During numerical studies it became clear that the geomaterial-soil, would often be present in a state of incomplete saturation when part of the void was filled with air. Such partial saturation is responsible for the presence of negative pressures which allow some 'apparent' cohesion to be developed in non-cohesive soils. This phenomenon may be present at the outset of loading or may indeed develop during the dynamic process. We have therefore incorporated its presence in the treatment presented in this book and thus achieved wider applicability for the methods described. Despite the large number of authors, we have endeavoured to present a unified approach and have used the same notation, style and spirit throughout. The first three chapters present the theory of porous media in the saturated and unsaturated states and thus establish general backbone to the problem of soil mechanics. Chapter 4, essential before numerical approximation, deals with the very important matter of the quantitative description of soil behaviour which is necessary for realistic computations. Here, the chapter is necessarily long as it starts from simple plasticity models and continues to the presentation of such topics as generalized plasticity, critical state soil mechanics etc., necessary for an adequate description of the soil behaviour. Indeed, in this chapter we also introduce a simplified model of denszfication which, when added to simple classical plasticity, permits a realistic description of liquefaction and cyclic mobility phenomena consecutively with problems of applications to static or quasistatic problems (Chapter 5 ) , verification of computation by dynamic experiments in centrifuge (Chapter 6) and practical applications to earthquake engineering in Chapter 7. In the last chapter, Chapter 8, we address some rather specialized topics which help in the improvement of general programs but are not absolutely necessary. Here special treatment of incompressibility, radiation damping and adaptive refinement are given. The various solutions of static and dynamic situations shown in this book have been obtained by using the code named SWANDYNE which is available from the authors. Similarly the explicit derivative is also available. A simplified version of SWANDYNE is outlined at the end of the book (Chapter 9) and an executable version can be obtained via the Internet using the URL at http://www.bham.ac.uk/ CivEng/swandyne/index.htm.
Introduction and the Concept of Effective Stress
1.1 PRELIMINARY REMARKS The engineer designing such soil structures as embankments, dams, or building foundations should be able to predict the safety of these against collapse or excessive deformation under the various loading conditions which are deemed possible. On occasion he may have to apply his predictive knowledge to events in natural soil or rock outcrops, subject perhaps to new, man-made conditions. Typical of this is the disastrous collapse of the mountain (Mount Toc) bounding the Vajont reservoir which occurred on October 9th 1963 in Italy (Miiller 1965). Figure 1.1 shows both a sketch indicating the extent of failure and a diagram indicating the cross-section of the encountered ground movement. In theabovecollapse, theevident causeand the 'straw that broke thecamel's back' was the filling and the subsequent drawdown of the reservoir. The phenomenon proceeded essentially in a static (or quasi-static) manner until the last moment when the moving mass of soil acquired the speed of 'an express train' at which point it tumbled into the reservoir, displacing the water dynamically and causing an unprecedented death toll of some 4000 people from the neighbouring town of Longarone. Such static failures which occur, fortunately at a much smaller scale, in many embankments and cuttings are subjects of typical concern to practising engineers. However, dynamic effects such as those frequently caused by earthquakes are more spectacular and much more difficult to predict. We illustrate the dynamic problem by the near collapse of the Lower San Fernando dam near Los Angeles during the 1971 earthquake Figure 1.2, (Seed, 1979, Seed et al., 1975). This failure fortunately did not involve any loss of life as the level to which the dam 'slumped' still contained the reservoir. Had this been but a few feet lower, the overtopping of the dam would indeed have caused a major catastrophe with the flood hitting a densely populated area of Los Angeles. It is evident that the two examples quoted so far involved the interaction of pore water pressure and the soil skeleton. Perhaps the particular feature of this interac
2
INTRODUCTION AND THE CONCEPT OF EFFECTIVE STRESS
Figure 1.1 The Vajont reservoir, failure of Mant Toc in 1963 (Oct. 9th): (a) hypothetical slip plane; (b) downhill end of slide (Miiller, 1965) Plate 1 shows a photo of the slides (front page)
tion, however, escapes immediate attention. This is due to the 'weakening' of the soilfluid composite during the periodic motion such as that which is involved in an earthquake. However, it is this rather than the overall acceleration forces which caused the collapse of the Lower San Fernando dam. What appears to have happened here is that during the motion the interstitial pore pressure increased, thus reducing the interparticle forces in the solid phase of the soil and its strength. Such strength reduction phenomena are mainly evident in essentially non-cohesive materials such as sand and slit. Clays in which negative, capillary pressure provide an apparent cohesion are less liable to such strength reduction.
4
INTRODUCTION AND THE CONCEPT OF EFFECTIVE S T R E S S
This phenomenon is well documented and in some instances the strength can drop to near zero values with the soil then behaving almost like a fluid. This behaviour is known as soil liquefaction and Plate 2 shows a photograph of some buildings in Niigata, Japan taken after the 1964 earthquake. It is clear here that the buildings behaved as if they were floating during the active part of the motion. In this book, we shall discuss the nature and detailed behaviour of the various static, quasi-static and dynamic phenomena which occur in soils and will indicate how a computer based, finite element, analysis can be effective in predicting all these aspects quantitatively.
1.2
THE NATURE OF SOILS AND OTHER POROUS MEDIA: WHY A FULL DEFORMATION ANALYSIS IS THE ONLY VIABLE APPROACH FOR PREDICTION
For single-phase media such as those encountered in structural mechanics, it is possible to predict the ultimate (failure) load of a structure by relatively simple calculations, at least for static problems. Similarly for soil mechanics problems such simple, limit-load calculations, are frequently used under static conditions, but even here, full justification of such procedures is not generally valid. However, for problems of soil dynamics, the use of such simplified procedures is almost never admissible. The reason for this lies in the fact that the behaviour of soil or such a rock-like material as concrete, in which the pores of the solid phase are filled with one fluid, cannot be described by behaviour of a single-phase material. Indeed to some it may be an open question whether such porous materials as shown in Figure 1.3 can be treated at all by the methods of continuum mechanics. Here we illustrate two apparently very different materials. The first has a granular structure of loose, generally uncemented, particles in contact with each other. The second is a solid matrix with pores which are interconnected by narrow passages. From this figure, the answer to the query concerning the possibility of continuum treatment is self-evident. Provided that the dimension of interest and the so called 'infinitesimals' dx, dy, etc. are large enough when compared to the size of the grains and the pores, it is evident that the approximation of a continuum behaviour holds. However, it is equally clear that the intergranular forces will be much affected by the pressures of the fluid-p in single phase (or p l , pz etc. if two or more fluids are present). The strength of the solid, porous, material on which both deformations and failure depend can thus only be determined once such pressures are known. Using the concept of effective stress, which we shall discuss in detail in the next section, it is possible to reduce the soil mechanics problem to that of the behaviour of a single phase, once all the pore pressures are known. Then we can use again the simple, single-phase analysis approaches. Indeed on occasion the limit load procedures are again possible. One such case is that occurring under long-term load conditions in material of appreciable permeability when a steady state drainage pattern has been established and the pore pressures are independent of the material deformation and can be determined by uncoupled calculations.
5
T H E NATURE OF SOILS
Solid / I
Figure 1.3 Various idealized structures of fluid saturated porous solids: (a) a granular material; (b) a perforated solid with interconnecting voids
Such drained behaviour, however, seldom occurs even in problems which we may be tempted to consider as static due to the slow movement of the pore fluid and. theoretically, the infinite time required to reach this asymptotic behaviour. In very finely grained materials such as silts or clays this situation may never be established even as an approximation. Thus, in a general situation, the complete solution of the problem of solid material deformation coupled to a transient fluid flow needs generally to be solved. Here no short-cuts are possible and full coupled a n ~ l ~ v s eofs equations which we shall introduce in Chapter 2 become necessary. We have not mentioned so far the notion of so called undrained behaviour, which is frequently assumed for rapidly loaded soil. Indeed, if all fluid motion is prevented, by zero permeability being implied or by extreme speed of the loading phenomena, the pressures developed in the fluid will be linked in a unique manner to deformation of the solid material and a single-phase behaviour can again be specified. While the artifice of simple undrained behaviour is occasionally useful in static studies, it is not applicable to dynamic phenomena such as those which occur in earthquakes as the pressures developed will, in general, be linked again to the straining (or loading) history and this must always be taken into account. Although in early attempts to deal with earthquake analyses and to predict the damage and response, such undrained analyses were invariably used, adding generally a linearization of the total behaviour and an heuristic assumption linking the pressure development with cycles of loading, the behaviour predictions were poor. Indeed recent comparisons with centrifuge experiments confirmed the inability of such methods to predict either the pressure development or deformations (VELACS Arulanandan & Scott, 1993). For this reason we believe that the only realistic type of analysis is of the type indicated in this book. This was demonstrated in the same VELACS tests to which we shall frequently refer in later chapters. -
6
INTRODUCTION AND T H E CONCEPT OF EFFECTIVE S T R E S S
At this point, perhaps it is useful to interject an observation about possible experimental approaches. The question which could be addressed is whether a scale model study can be made relatively inexpensively in place of elaborate computation. A typical civil engineer may well consider here the analogy with hydraulic models used to solve such problems as spillway flow patterns where the cost of a small-scale model is frequently small compared to equivalent calculations. Unfortunately, many factors conspire to deny in geomechanics a readily accessible model study. Scale models placed on shaking tables cannot adequately model the main force acting on the soil structure, i.e., that of gravity though, of course the dynamic forces are reproducible and scalable. To remedy this defect, centrifuge models have been introduced and, here, though at considerable cost, gravity effects can be well modelled. With suitable fluids substituting water it is indeed also possible to reproduce the seepage timescale and the centrifuge undoubtedly provides a powerful tool for modelling earthquake and consolidation problems in fully saturated materials. Unfortunately, even here a barrier is reached which appears to be insurmountable. As we shall see later under conditions when two fluids, such as air and water for instance, fill the pores, capillary effects occur and these are extremely important. So far no success has been achieved in modelling these and hence studies of structures with free (phreatic) water surface are excluded. This of course eliminates possible practical applications of the centrifuge for dams and embankments in what otherwise is a useful experimental procedure.
1.3
CONCEPTS OF EFFECTIVE STRESS IN SATURATED OR PARTIALLY SA TURA TED MEDIA
1.3.1 A single fluid present in the pores-historical
note
The essential concepts defining the stresses which control strength and constitutive behaviour of a porous material with internal pore pressure of a fluid appear to have been defined, at least qualitatively towards the end of the last century. The work of Lye11 (1871), Boussinesq (1876) and Reynolds (1886) was here of considerable note for problems of soils. Later, similar concepts were used to define the behaviour of concrete in dams (Levy, 1895 and Fillunger, 1913a, 1913b and 1915) and indeed for other soil or rock structures. In all of these approaches the concept of division of the total stress between the part carried by the solid skeleton and the fluid pressure is introduced and the assumption made that the strength and deformation of the skeleton is its intrinsic property and not dependent on the fluid pressure. If we thus define the total stress a by its components aq using indicia1 notation these are determined by summing the appropriate forces in the i-direction on the projection, or cuts, dx, (or dx, dy, dz in conventional notation). The surfaces of cuts are shown, for two kinds of porous material structure, in Figure 1.3 and include the total area of the porous skeleton. In the context of the finite element computation we shall frequently use a vectorial notation for stresses, writing
CONCEPTS OF EFFECTIVE STRESS
7
This notation reduces the components to six rather than nine and has some computational merit. Now if the stress in the solid skeleton is defined as the effective stress d again over the whole cross-sectional area then the hydrostatic stress due to the pore pressure, p acting, only on the pore area should be
where n is the porosity and Sji is the Kronecker delta. The negative sign is introduced as it is a general convention to take tensile components of stress as positive. The above, plausible, argument leads to the following relation between total and effective stress with total stress
or if vectorial notation is used we have a = a'- m n p
(1.4)
where m is a vector written as
The above arguments do not stand the test of experiment as it would appear that, with values of porosity n with a magnitude of 0.1-0.2 it would be possible to damage a specimen of a porous material (such as concrete for instance) by subjecting it to external and internal pressures simultaneously. Further, it would appear from equation (1.3) that the strength of the material would be always influenced by the pressure p. Fillunger introduced the concepts implicit in (1.3) in 1913 but despite conducting experiments in 1915 on the tensile strength of concrete subject to water pressure in the pores, which gave the correct answers, he was not willing to depart from the simple statements made above. It was the work of Terzaghi and Rendulic in 1934 and by Terzaghi in 1936 which finally modified the definition of effective stress to
where n,,.is now called the effective areu coef$cient and is such that
8
INTRODUCTION AND THE CONCEPT OF EFFECTIVE STRESS
Much further experimentation on such porous solids as concrete had to be performed before the above statement was generally accepted. Here the work of Leliavsky (1947) McHenry (1948) Serafim (1954, 1964) made important contributions by experiments and arguments showing that it is more rational to take sections for determining the pore water effect through arbitrary surfaces with minimum contact points. Bishop (1959) and Skempton (1960) analysed the historical perspective and more recently de Boer (1996) and de Boer et (11. (1996) addressed the same problem showing how an acrimonous debate between Fillunger and Terzaghi terminated in the tragic suicide of the former in 1937. The subject of effective stress is as of much interest to the senior author who directed his research to analysis of dams, viz Zienkiewicz (1947, 1963) who found that interpretation of the various experiments was not always convincing. However, the work of Biot (1941, 1955, 1956a, 1956b, 1962) and Biot and Willis (1957) clarified many concepts in the interpretation of effective stress and indeed of the coupled fluid and solid interaction. In the following section we shall present a somewhat different argument leading to equations (1.6) and (1.7).
1.3.2
An alternative approach to effective stress
Let us now consider the effect of the simultaneous application of a total external hydrostatic stress and a pore pressure change; both equal to A p , to any porous material. The above requirement can be written in tensorial notation as requiring that the total stress increment is defined as
or, using the vector notation
In the above, the negative sign is introduced since 'pressures' are generally defined as being positive in compression, while it is convenient to define stress components as positive in tension. It is evident that for the loading mentioned, only a very uniform and small volumetric strain will occur in the skeleton and the material will not suffer any damage provided that the grains of the solid are all made of an identical material. This is simply because all parts of the porous medium solid component will be subjected to an identical compressive stress. However, if the microstructure of the porous medium is composed of different materials, it appears possible that non-uniform, localized stresses, can occur and that local grain damage may be suffered. Experiments performed on many soils and rocks and rock-like materials show, however, that such effects are insignificant. Thus in general the grains and hence the total material will be in a state of pure volumetric strain
9
CONCEPTS OF EFFECTIVE S T R E S S
where K, is the average material bulk modulus of the solid components of the skeleton. Alternatively, adopting a vectorial notation for strain in a manner involved in (1.1)
where E is the vector defining the strains in the manner corresponding to that of stress increment definition. Again, assuming that the material is isotropic, we shall have
Those not familiar with soil mechanics may find the following hypothetical experiment illustrative. A block of porous, sponge-like rubber, is immersed in a fluid to which an increase in pressure of A p is applied as shown in Figure 1.4. If the pores are connected to the fluid, the volumetric strain will be negligible as the solid components of the sponge rubber are virtually incompressible. Frame after load
Frame before load
Negligible deformation (a)
AP
(b) Surface membrane
Figure 1.4 A Porous Material subject to external hydrostatic pressure increases A,. and (a) Internal pressure increment A,; (b) Internal pressure increment of zero
10
INTRODUCTION AND THE CONCEPT OF EFFECTIVE STRESS
If, on the other hand, the block is first encased in a membrane and the interior is allowed to drain freely, then again a purely volumetric strain will be realised but now of a much larger magnitude. The facts mentioned above were established by the very early experiments of Fillunger (1915) and it is surprising that so much discussion of "area coefficients" has since been necessary. From the preceding discussion it is clear that if the material is subject to a simultaneous change of total stress Ao and of the total pore pressure Ap, the resulting strain can always be written incrementally in tensorial notation as
or in vectoral notation
with
The last term in (1.1 la) and (1.1 1b), AEO, is simply the increment of an initial strain such as may be caused by temperature changes etc., while the penultimate term is the strain due to the grain compression already mentioned viz Eq. 1.10 D is a tangent matrix of the solid skeleton implied by the constitutive relation with corresponding compliance coefficient matrix D-' = C. These of course could be matrices of constants, if linear elastic behaviour is assumed, but generally will be defined by an appropriate non-linear relationship of the type which we shall discuss in Chapter 4 and this behaviour can be established by fully drained @ = 0) tests. Although the effects of skeleton deformation due to the effective stress defined by (1.6) with n,,= 1 have been simply added to the uniform volumetric compression, the principle of superposition requiring linear behaviour is not invoked and in this book we shall almost exclusively be concerned with non-linear, irreversible, elasto-plastic and elasto-viscoplastic responses of the skeleton which, however, we assume incremental properties. For assesment of strength of the saturated material the effective stress previously defined with n,,.= 1 is sufficient. However, we note that the deforination relation of (1.1 1) can always be rewritten incorporating the small compressive deformation of the particles as (1.12). It is more logical at this step to replace the finite increment by an infinitesimal one and to invert the relations in (1.1 1) writing these as
11
CONCEPTS OF EFFECTIVE STRESS
where a new 'effective' stress, a", is defined. In the above
and the new form eliminates the need for separate determination of the volumetric strain component. Noting that 6..6.. - 3 !I !I -
or
we can write
or simply
Alternatively, in tensorial form, the same result is obtained as
For isotropic materials, we note that,
which is the tangential bulk modulus of an isotropic elastic material with X and being the Lame's constants. Thus we can write
11
The reader should note that in (1.12) we have written the definition of the effective stress increment which can, of course, be used in a non-incremental state as
12
INTRODUCTION AND T H E CONCEPT OF EFFECTIVE S T R E S S
assuming that all the stresses and pore pressure started from a zero initial state, (for example, material exposed to air is taken as under zero pressure). The above definition corresponds to that of the effective stress used by Biot (1941) but is somewhat more simply derived. In the above, cr is a factor which becomes close to unity when the bulk modulus Ks of the grains is much larger than that of the whole material. In such a case we can write. of course
recovering the common definition used by many in soil mechanics and introduced by von Terzaghi (1936). Now, however, the meaning of a is no longer associated with an effective area. It should have been noted that in some materials such as rocks or concrete it is possible for the ratio K,/K, to be as large as 113 with a = 213 being a fairly common value for determination of deformation. We note that in the preceding discussion the only assumption made which can be questioned, is that of neglecting the local damage due to differing materials in the soil matrix. We have also implicitly assumed that the fluid flow is such that it does not separate the contacts of the soil grains. This assumption is not totally correct in soil liquefaction or flow in soil shearing layer during localization.
1.3.3
Effective stress in the presence of two (or more) porefluids. Pavtially saturated media
The interstitial space, or the pores, may in a practical situation be filled with two or more fluids. We shall, in this section, consider only two fluids with the degree of saturation by each fluid being defined by the proportion of the total pore volume n (porosity) occupied by each fluid. In the context of soil behaviour discussed in this book the fluids will invariably be water and air respectively. Thus we shall refer to only two saturation degrees S, that for water and S, that for air, but the discussion will be valid for any two fluids. It is clear that, if both fluids fill the pores completely, we shall always have
Clearly this relation will be valid for any other pair of fluids e.g. oil and water and indeed the treatment described here is valid for any fluid conditions.
13
CONCEPTS OF EFFECTIVE STRESS
The two fluids may well present different areas of contact with the solid grains of the material in the manner illustrated in Figure 1.5 (a) and (b). The average pressure reducing the interstitial contact and relevant to the definition of effective stress found in the previous section (Equations (1.16) and (1.17)) can thus be taken as I-' = XwPw
where the coefficients xw and
+ XaPa
(1.19)
xa refer to water and air respectively and are such that
The individual pressures pw and pa are again referring to water and air and their difference i.e.
is dependent on the magnitude of surface tension or capillarity and on the degree of saturation ( p , is often referred to therefore, as capillary pressure). Depending on the nature of the material surface the contact surface may take on the shapes shown in Figure 1.5 with
and
Occasionally the contact of one of the phases and the solid may disappear entirely as shown in Figure 1.5a giving isolated air bubbles and making in this limit
Figure 1.5 Two fluids in pores of a granular solid (water and air). (a) air bubble not wetting solid surface (effective pressure p = p w ; (b) both fluids in contact with solid surfaces (effective X, pa pressure p = X, p,
+
14
INTRODUCTION AND THE CONCEPT O F EFFECTIVE STRESS
Whatever the nature of contact, we shall find that a unique relationship between p, and the saturation Sw can be written i.e.
Indeed, the degree of saturation will similarly affect flow parameters such as the permeability k to which we shall make reference in the next chapter, giving
Many studies of such relationship are reported in the literature (Liakopoulos, 1965; Neuman, 1975; Van Genuchten et al., 1977; Narasimhan & Witherspoon 1978; Safai & Pinder, 1979; Lloret & Alonso, 1980; Bear et al. 1984; Alonso et al., 1987). Figure 1.6 shows a typical relationship. The concepts of dealing with the effects of multiple pore pressure by introducing an average pressure and using the standard definition of effective stress (1.19, 1.16 and 1.17) were first introduced by Bishop (1959). Certainly the arguments for thus extending the original concepts are less clear than is the case when only a single fluid is present. However, the results obtained by this extension are quite accurate. We shall therefore use such a definition in the study of partially saturated media. In many cases occurring in practice, the air pressure is close to zero (atmospheric datum) as the pores are interconnected. Alternatively, negative air pressure occurs as cavitation starts and here the datum is the vapour pressure of water. In either case the effect of p, can be easily neglected as the water pressure simply becomes negative from Equation (1.24). Such negative pressures are responsible for the development of certain cohesion by the soil and are essential in the study of free surface conditions occurring in embankments, as we shall see later
Figure 1.6 Typical relations between pore pressure head. h, = p, / X , , Saturation, S, , and relative permeability, k, = k , (S,)/k,(l) (Safai & Pinder 1979). Note that relative permeability decreases very rapidly as saturation decreases
REFERENCES
15
REFERENCES Alonso E. E., Gens A. and Hight D. W. (1987) Special problems of soil: general report. Proc. 9th Euro. Con/.' Int. Soc. Soil Meel?. Found. Eng., Dublin. Arulanandan K. and Scott R . F. (Eds) (1993) Proceedings of the VELACS symposium. 1. A. A. Balkema, Rotterdam. Bear J., Corapcioglu M. Y. and Balakrishna J., (1984) Modeling of centrifugal filtration in unsaturated deformable porous media, A h . Water Resour., 7 , 150-167. Biot M. A. (1941) General theory of three-dimensional consolidation, J. Appl. P h j ~ . 12, , 155164. Biot M. A. (1955) Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys, 26, 182-1 85. Biot M. A. (1956a) Theory of propagation of elastic waves in a fluid-saturated porous solid. part I-low-frequency range, J. Acoust. Soc. Am., 28, No. 2, 168-178. Biot M. A. (1956b) Theory of propagation of elastic waves in a fluid-saturated porous solid. part-11-higher frequency range, J. Acoust. Soe. Am., 28, No. 2, 179-191. Biot M. A. (1962) Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 33, No. 4, 1482-1498. Biot M. A. and Willis P. G. (1957) The elastic coefficients of the theory consolidation, J. Appl. Mech., 24. 59&601. Bishop A. W. (1959) The Principle of Effective Stress, Teknisk Ukeblad, 39, 859-863. De Boer R. (1996) Highlights in the historical development of the porous media theory. Applicd Mec1z~1nic.sReview, 49, 20 1-262. De Boer R., Schiffman R. L. and Gibson R. E. (1996) The origins of the theory of consolidation: the Terzaghi-Fillunger dispute, Gkotechnique, 46, No. 2, 175-1 86. Boussinesq J. (1876) Essai theorique sur I'equilibre d'elasticite des massif pulverulents. Mem. savants h a n g e r s , Acad. Belgique, 40, 1-180. Fillunger P. (1913a) Der Ayftrieh in Tcrlsperrcv~,0sterr. Wochenschrift offentlichen Baudienst. 532-556. Fillunger P. (1 9 13b) Der Aujirieh in Tul~sperren,Osterr. Wochenschrift offentlichen Baudienst. 567-570. Fillunger P. (1915) Versuch iiher die Zugfestigkeit he; crllseitigem Wasserdruck, 0sterr. Wochenschrift offentl. Baudienst, H29, 443448. Leliavsky S. (1947) Experiments on effective area in gravity dams, Trans. Am. Soc. Civil Engrs., 112, 444. Levy M. M. (1895) Quelques Considerations sur la construction des grandes barrages, C o w ptes Rendus De L'Academie Des Sciences Serie I-Mathematique. 288. Liakopoulos A. C. (1965) Trczn.sient,flow through unsrrturated porous media, D. Eng. Dissertation, University of California, Berkeley, USA. Lloret A. and Alonso E. E. (1980) Consolidation of unsaturated soils including swelling and collapse behaviour, GPotechnique, 30, 449447. Lyell C. (1871) Student's elements of geology, London. McHenry D. (1948) The effect of uplift pressure on the shearing strength of concrete-R.48. International Congress Large Dams, 3rd, Stockholm, Vol. I. Miiller L. (1965) The Rock slide in the Vajont Valley, Fels Mecharzik, 2, 148-212. Narasimhan T. N. and Witherspoon P. A. (1978) Numerical model for saturated-unsaturated flow in deformable porous media 3. Applications, Water Resources Res., 14. 1017-1034. Neuman S. P. (1975) Galerkin approach to saturated-unsaturated flow in porous media in Finite Elernents in jluids, Wiley, London. Reynolds 0. (1886) Experiments showing dilatancy, a property of granular material. Proc. R. Inst., 11, 354363.
16
INTRODUCTION AND T H E CONCEPT OF EFFECTIVE S T R E S S
Safai N. M. and Pinder G. F. (1979) Vertical and horizontal land deformation in a desaturating porous medium, Adv. Water Resources 2, 19-25. Seed H. B. (1979) Consideration in the earthquake resistant design of earth and rockfill dams, GCotechnique, 29, No. 3, 215-263. Seed H. B., Idriss I. M., Lee K. L. and Makdisi F. I. (1975) Dynamic analysis of the slide in the Lower San Fernando dam during the earthquake of February 9, 1971, J. Geotech. Eng. Div., ASCE, 101, No. 9, 889-911. Serafim J. L. (1954) A subpresseo nos barreyens-Publ. 55, Laboratorio Nacional de Engenheria Civil, Lisbon. Serafim J. L. (1964) The 'uplift area' in plain concrete in the elastic range-C. 17, Int. Congr. Large Dams, 8th, Edinburgh, Vol. V. Skempton A. W. (1960) Effective Stress in Soils, Concretes, and Rocks, Proc. Con$ Pore Pressures and Suction in Soils, 4-16. Terzaghi K . von (1936) The shearing resistance of saturated soils, Proc. 1st ICSMFE, 1 , 5 4 5 6 . Terzaghi K . von and Rendulic L. (1934) Die wirksame Flachenporositat des Betons, Z. Ost. Ing.-u. Archit Ver., 86, No. 112, 1-9. Van Genuchten M. T., Pinder G. F. and Saukin W. P. (1977) Modeling of leachate and soil interactions in an aquifer - EPA-60019-77-026, Proc. 3rd Annual Municipal Solid Waste Res. Symp., 95-103. Zienkiewicz 0.C. (1947) The stress distribution in gravity dams, J. Inst. Civ. Eng., 27,244271. Zienkiewicz 0. C. (1963) Stress analysis of hydraulic structures including pore pressures effects, Water Power, 15, 104108.
Equations Governing the Dynamic, Soil-Pore Fluid, Interaction
2.1 GENERAL REMARKS ON THE PRESENTATION In this chapter we shall introduce the reader to the equations which govern both the static and dynamic phenomena in soils containing pore fluids. We shall divide the presentation into three Sections, Section 2.2 will deal with soil, or indeed any other porous medium, saturated with a single fluid. This, most common, problem contains all the essential features of soil behaviour and the equations embrace and explain the vast majority of problems encountered in practice. We shall show here how the dynamic equations, which are essential for the study of earthquakes, reduce to those governing the quasi-static situations of consolidating soils and indeed to purely static problems without modification. This feature will be used when discretization is introduced and computer codes are derived, since a single code will be capable of dealing with most phenomena encountered in soil and rock mechanics. The limitations of the approximating simplification are discussed in Section 2.2 by using a simple linearized example and deriving conclusions on the basis of an available analytical solution. The same discussion will show the domain of the validity of the assumptions of undrained and fully drained behaviour. In the same section we shall introduce a simplification which is valid for the treatment of most low-frequency phenomena-and this simplified form will be used in the subsequent Section 2.3 dealing with partially saturated soil in which the air pressure is assumed constant and also finally in Section 2.4 dealing with simultaneous water and air flow in the pores. The notation used throughout this chapter will generally be of standard, tensorial form. Thus: ui will be the displacement of the solid matrix with i = 1,2 in 2 dimensions or i = 1.3 in 3 dimensions
18 EQUATIONS GOVERNING T H E DYNAMIC, SOIL-PORE FLUID, INTERACTION
Alternatively, the form
will also be used for the same quantity in vectorial notation. Similarly, we shall use w, and v, or w and v to denote the velocities of water and air relative to the solid components. These velocities are calculated on the basis of dividing the appropriate flow by the total cross-sectional area of the solidfluid composite. As mentioned in the previous chapter, a,and a; refer to the appropriate total and effective stresses, with a and a"being the vectorial alternatives. Similarly, E,, or E refers to the strain components. Further pa, p, and P = XwPw f X & will stand for air and water pressure and the 'effective' pressure defined in the effective stress concept in Equation (1.11) of Chapter 1 when two fluids are present. Sa, S, are the relative degrees of saturation and k, and k , are the permeabilities for air and water flow. Other symbols will be added and defined in the text as the need arises. The derivation of the equations in this chapter follow a physical approach which establishes clearly the interactions involved in the manner presented by Zienkiewicz and Shiomi (1984), Zienkiewicz (1982), Zienkiewicz et (11. (1990a and 1990b) etc. This is a slightly different approach from that used in the earlier presentations of Biot (1941, 1955, 1956a, 1956b and 1962) and Biot & Willis (1957) but we believe it is slightly easier to follow as it explores the physical meaning of each term. Later it became fashionable to derive the equations in the forms of so called mixture theories (see Green & Adkin (1960), Green (1969) and Bowen (1976)). The equations derived were subsequently recast in varying forms. Here an important step forward was introduced by Morland (1972) who used extensively the concept of volume fractions. Derski (1978) introduced a different derivation of coupled equations and Kowalski (1979) compared the various parameters occurring in Derski's equations with those of Biot's equations. A full discussion of the development of the theory is given in the paper by de Boer (1996). For completeness, we shall include such mixture derivations of the equations in Section 2.5 If correctly used, the mixture theory establishes of course identical equations but in the author's view, introduces some arbitrariness in the selection of various parameters. It seems that despite much sophistication of the various sets of coupled equations, most authors limited their works to conventional, linear elastic, behaviour of the solid. Indeed, de Boer and Kowalski (1983) found it necessary to develop a special plasticity theory for porous, saturated solids. In the equations of Zienkiewicz (1982), and Zienkiewicz et ul. (1990a) any non-linear behaviour can be specified for the skeleton and therefore realistic models can be incorporated. Indeed we shall find that such models are essential if practical conclusions are to be drawn from this work.
FULLY S A T U R A T E D BEHA VIOUR W I T H A SINGLE PORE FLUID ( W A T E R )
19
2.2 FULLY SATURATED BEHA VIOUR WITH A SINGLE PORE FL UZD ( WATER) 2.2.1
Equilibrium and mass balance relationship (u, w and p )
We recall first the effective stress and constitutive relationships as defined in equations (1.16), of the previous chapter which we repeat below.
This effective stress is conveniently used as it can be directly established from the total strains developed. However, it should be remembered here this stress definition was derived in the first chapter as a corollary of using the effective stress defined as below
which is responsible for the major part of the deformation and certainly for failure. In soils, the difference between the two effective stresses is negligible as cu z 1. However, for such materials as concrete or rock the value of o in the first expression can be as low as 0.5 but experiments on tensile strength show that the second definition of effective stress is there very much more closely applicable as shown by Leliavsky (1947), Serafim (1954) etc. For soil mechanics problems, to which we will devote most of the examples, o = 1 will be assumed. Constitutive relationships will still, however, be written in the general form using an incremental definition
du" = D(de - deO)
(2.3b)
The vectorial notation used here follows that corresponding to stress components given in (I. I). We thus define the strains as
In the above, D is the 'tangent matrix' and deOis the increment of the thermal or similar autogeneous strain and of the grain compression mp/3Ks. The latter is generally neglected in soil problems.
20 EOUATIONS GOVERNING THE DYNAMIC. SOIL-PORE FLUID. INTERACTION
If large strains are encountered, this definition needs to be modified and we must write
where the last two terms account for simple rotation (via the definition in (2.6b) ) of existing stress components and are known as the Zaremba (1903a, 1903b)-Jaumann (1905) stress changes. We omit here the corresponding vectorial notation as this is not easy to implement. The large strain rotation components are small for small displacement computation and can frequently be neglected. Thus in the derivations that follow we shall do so-though their inclusion presents no additional computational difficulties and they are included in the computer program. The strain and rotation increments of the soil matrix can be determined in terms of displacement increments du; as
and
The comma in the suffix denotes differentiation with respect to the appropriate coordinate specified. Thus
If vectorial notation is used, as is often the case in the finite element analysis, the so called engineering strains are used in which (with the repeated index of du;,; not summed.)
However, the shear strain increments will be written as dyij = 2 d ~ i= j du,,
+ du,,;
FULLY SATURATED BEHA VIOUR WITH A SINGLE PORE FLUID ( W A T E R )
dy,, =
ddu, ax
21
+ ddu, ay
-
We shall usually write the process of strain computation using matrix notation as d~ = Sdu where u
=
[u,u?,u,]
(2.8) T
(2.9)
And for two dimensions the strain matrix is defined as:
with corresponding changes for three dimensions (as shown in Zienkiewicz and Taylor 1989). We can now write the overall equilibrium or momentum balance relation for the soil-fluid 'mixture' as
where w;
dw; = --etc. dt
In about wi (or w) is the average (Darcy) velocity of the percolating water. The underlined terms in the above equation represent the fluid acceleration relative to the solid and the convective terms of this acceleration. This acceleration is generally small and we shall frequently omit it. In derivations of the above equation we consider the solid skeleton and the fluid embraced by the usual control volume: dx . dy . dz. Further, pf is the density of the fluid, b is the body force per unit mass (generally gravity) vector and p is the density of the total composite i.e.
where ps is the density of the solid particles and n is the porosity (i.e. the volume of pores in a unit volume of the soil). The second equilibrium equation ensures momentum balance of the fluid. If again we consider the same unit control volume as that assumed in deriving
22 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
(2.11) (and we further assume that this moves with the solid phase) we can write
In the above we consider only the balance of the fluid momentum and R represents the viscous drag forces which, assuming the Darcy seepage law, can be written as
Note that the underlined terms in (2.12) represent again the convective fluid acceleration and are generally small. Also note that, throughout this book, the permeability k is used with dimensions of [length]3[time]/[mass]which is different from the usual soil mechanics convention k' which has the dimension of velocity, i.e., [length]/[time]. Their values are related by k = kl/p;g' where p[ and g' are the fluid density and gravitational acceleration at which the permeability is measured. The final equation is one accounting for mass balance of the flow. Here we balance the flow divergence w;,~by the augmented storage in the pores of a unit volume of soil occurring in time dt. This storage is composed of several components given below in order of importance: (i) the increased volume due to a change in strain i.e.: Siideii= d~~~= mTdc (ii) the additional volume stored by compression of void fluid due to fluid pressure increase: ndp/Kf (iii) the additional volume stored by the compression of grains by the fluid pressure increase: (1 - n)dp/Ks and (iv) the change in volume of the solid phase due to a change in the intergranular effective contact stress (4,= og + Sup): - Bqd~,/K, = - 5 (drii + Ks
g)
Here KT is the average bulk modulus of the solid skeleton and E;; the total volumetric strain. Adding all the above contributions together with a source term and a second-order term due to the change in fluid density in the process we can finally write the flow conservation equation
FULLY SATURA TED BEHA VIOUR WITH A SINGLE PORE FLUID ( W A T E R )
23
This can be rewritten using the definition of cu given in Chapter 1 (1.15b) as
or in vectorial form
where
In (2.16) the last two (underlined) terms are those corresponding to a change of density and rate of volume expansion of the solid in the case of thermal changes and are negligible in general. We shall omit them from further consideration here. Equations (2.1 I), (2.13) and (2.16) together with appropriate constitutive relations specified in the manner of (2.3) define the behaviour of the solid together with its pore pressure in both static and dynamic conditions. The unknown variables in this system are: The pressure of fluid (water), p = p, The velocities of fluid flow wi or w The displacements of the solid matrix u; or u. The boundary condition imposed on these variables will complete the problem. These boundary conditions are: (1) For the total momentum balance on the part of the boundary I?, we specify the total traction t ; ( t )(or in terms of the total stress avn, (a. G) with n; being the ith component of the normal at the boundary and G is the appropriate vectorial equivalence) while for I?,, the displacement ui(u),is given. (2) For the fluid phase, again the boundary is divided into two parts r, on which the values of p are specified and r,,,where the normal outflow w, is prescribed (for instance, a zero value for the normal outward velocity on an impermeable boundary). Summarising, for the overall assembly, we can thus write
24 EOUATIONS GOVERNING THE DYNAMIC. SOIL-PORE FLUID. INTERACTION
and
Further
and
It is of interest to note, as shown by Zienkiewicz (1982), that some typical soil constants are implied in the formulation. For instance, we note from (2.16) that for undrained behaviour when w;,; = 0 i.e. with no net outflow, we have (neglecting the last two terms which are of second order).
and
If the pressure change dp is considered as a fraction of the mean total stress change mTda/3 or d0;;/3 we obtain the so called B soil parameter (Skempton (1954)) as
Using the assumption that the material is isotropic so that
FULLY SATURATED BEHAVIOUR WITH A SINGLE PORE FLUID ( W A T E R )
25
where KT is (as defined in Eq. (1.10) the bulk modulus of the solid phase and p is once again Lame's constant. B has, of course, a value approaching unity for soil but can be considerably lower for concrete or rock. Further, for unsaturated soils, as will be seen from the next section, the value will be much lower (Terzaghi, 1925, Lambe and Whitman, 1969 and Craig, 1992).
2.2.2 Simplified equation sets (u-p form) The governing equation set (2.1 1) (2.13) and (2.16) together with the auxiliary definition system can of course be used directly in numerical solution as shown by Zienkiewicz and Shiomi (1984). This system is suitable for explicit time stepping computation as shown by Sandhu & Wilson (1969) and Ghaboussi & Wilson (1972) and more recently by Chan et al. (1991). However, in implicit computation, where large algebraic equation systems arise, it is convenient to reduce the number of variables by neglecting the apparently small (underlined) terms of equations (2.11) and (2.13). These contain the variable w;(w)which now can be eliminated from the system. The first equation of the reduced system becomes (from (2.1 1))
sTu- pii + pb = 0
(2.20b)
The second equation is obtained by coupling (2.13) and (2.16) using the definition (2.14) and thus eliminating the variable w; (w). We now have, omitting density changes
26 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
This reduced equation system is precisely the same as that used conventionally in the study of consolidation if the dynamic terms are omitted or even of static problems if the steady state is reached and all the time derivatives are reduced to zero. Thus the formulation conveniently merges with procedures used for such analyses. However, some loss of accuracy will be evident for problems in which high-frequency oscillations are important. As we shall show in the next section, these are of little importance for earthquake analyses. In eliminating the variable w;(w) we have neglected several terms but have achieved an elimination of two or three variable sets depending on whether the two or three-dimensional problem is considered. However, another possibility exists for obtaining a reduced equation set without neglecting any terms provided that the fluid (i.e. water in this case) is compressible. With such compressibility assumed, Equation (2.16) can be integrated in time provided that we introduce the water displacement u ? ( u R ) in place of the velocity w;(w). We define
where the division by the porosity n is introduced to approximate the true rather than the averaged fluid displacement. We now can rewrite (2.16) after integration with respect to time as
and thus we can eliminate p from (2.11) and (2.13). The resulting system which is fully discussed in Zienkiewicz and Shiomi (1984) is not written down here as we shall derive this alternative form in Chapter 3 using the total displacement of water U = uR u as the variable. It presents a very convenient basis for using a fully explicit temporal scheme of integration (see Chan et al. 1991) but it is not applicable for long-term studies leading to steady state conditions, as the water displacement U then increases indefinitely. It is fortunate that the inaccuracies of the u-p version are pronounced only in highfrequency, short-duration, phenomena, since for such problems we can conveniently use explicit temporal integration. Here a very small time increment can be used for the short time period considered (See Chapter 3). Table 2.1 summarises the various forms of governing equations used.
+
FULLY SATURATED BEHAVIOUR WITH A SINGLE PORE FLUID ( W A T E R )
27
Table 2.1 Comparative sets of coupled equations governing deformation and flow u - w - p equations (exact) ((2.11) (2.13) and (2.16))
sTg- pii - pf [w + wvTw] + pb = 0
u - p approximation for dynamics of lower frequencies. Exact for consolidation ((2.20), (2.21))
sTu- pii + pb = 0 u - U, only convective terms neglected (3.72)
sTu+ a Q ( a - n ) v ( v T u )+ a e n V ( v T u ) - (1 - n)pii - p i J + pb = 0
+
( a - n) QV(VTu) n Q v ( V T u ) - k-' ( n u - nu) - pf U + pf b
=0
In all the above a" = u + amTp and d u " = D ~ =E DSdu
2.2.3 Limits of validity of the various approximations It is, of course, important to know the degree of approximation involved in various differential equation systems. Thus it is of interest to know under what circumstances undrained conditions can be assumed, to define the behaviour of the material and when the simplified equation system discussed in the previous section is applicable, without introducing serious error. An attempt to answer these problems was made by Zienkiewicz et al. (1980). The basis was the consideration of a one-dimensional set of linearized equations of the full systems (2.1 I), (2.13), (2.16) and of the approximations (2.20) and (2.21). The limiting case in which w;,, = 0 (representing undrained conditions) was also considered. For all these conditions the exact solution of the equation is possible. We consider thus that the only physical variation is in the vertical, X I ,direction (XI= x) and then we have
where D is called the one-dimensional constrained modulus, E is Young's modulus and u is Poisson's ratio of the linear elastic soil matrix, also
28 EOUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
The differential equations are, in place of (2.11):
In place of (2.13):
and in place of (2.16):
with
Taking K,
--+
cc
For a periodic applied surface load
a periodic solution arises after the dissipation of the initial transient in the form = jjell"' P = ~ e ~ etc. l''~ and a system of ordinary linear differential equations is obtained in the frequency domain which can be readily solved by standard procedure. The boundary conditions imposed are as shown in Figure 2.1. Thus at s=Lu=Ow=Oandatx=O~,=qp=O. In Figure 2.1 (taken from Zienkiewicz et al. 1980) we show a comparison of various numerical results obtained by the various approximations: (i) exact equation (Biot's, labelled B) (ii) the u - p equation approximation (labelled Z) (iii) the undrained assumption (w = 0) and (iv) the consolidation equation obtained by omitting all acceleration terms, labelled C. The reader will note that the results are plotted against two non-dimensional coefficients:
FULLY SATURATED BEHA VIOUR WITH A SINGLE PORE FLUID ( W A T E R )
29
where k' and k are the two definitions of permeability discussed earlier. In the above
where L is a typical length such as the length of the one-dimensional soil column under consideration, g is the gravitational acceleration,
is the speed of sound, T is the natural vibration period and T is the period of excitation. The second non-dimensional parameter is defined as
In the study, the following values were assumed:
= p f / p = 0.333, n(porosity) = 0.333, and
Figure 2.2 Summarizes the conclusions by indicating three zones in which various approximations are sufficiently accurate. We note that, for instance, fully undrained behaviour is applicable when II, < and when 111> lo2 the drainage is so free that fully drained condition can be safely assumed. To apply this table in practical cases, some numerical values are necessary. Consider, for instance, the problem of the earthquake response of a dam in which the typical length is characterized by the height L = 50 metres, subject to an earthquake in which the important frequencies lie in the range
Thus, with the wave speed taken as
2
we have T = = 0.1s the parameter 112is therefore in the range 3.9 < II~< 39
30 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
-
Figure 2.1 The soil column -variation of pore pressure with depth for various values of rl and B (Biot theory) - - - - z (u-p approximation theory) c (Consolidation "2 theory) (Solution (C) is independent of xz).Reproduced from Zienkiewicz (1980) by permission of the Institution of Civil Engineers
PARTIALLY SATURATED BEHAVIOUR WITH AIR PRESSURE NEGLECTED
31
Drained (influence of z, negligible)
Figure 2.2 Zones of sufficient accuracy for various approximations: Zone 1, B = Z = C, slow phenomena (w and u can be neglected) Zone 2, B = Z # C, moderate speed (w can be neglected) Zone 111 B # Z # C, fast phenomena (w cannot be neglected only full Biot equation valid). Definition as in Figure 2.1. Reproduced from Zienkiewicz(1980) by permission of the Institution of Civil Engineers AI = k p ~ , Z /=~2~k 2 pT/~f2 A2 = J L ~ / V , Z= k=&lpg, k - kinematic permeability, f = 2L/ V, , V: = ( D + k f / n ) / p -- Pkf /pin -- kf / p l (speed of sound in water), P = pf / p , n 0.33, P 0.33
-
-
and Ill is dependent on the permeability k with the range defined by
According to Figure 2.2 we can, with reasonable confidence: (i) assume fully undrained behaviour when I I l = 97k' < lop2 or the permeability k' < rnls. (This is a very low value inapplicable for most materials used in dam construction). (ii) We can assume u-p approximation as being valid when k' < m/s to reproduce the complete frequency range. However, when k' < lo-' m/s periods of less then 0.5 s are still well modelled. We shall, therefore, typically use the u-p formulation appropriately in what follows reserving the full form for explicit transients where shocks and very high frequency are involved.
2.3 PARTZALL Y SA TURA TED BEHA VZOUR WITH AIR PRESSURE NEGLECTED (pa = 0) 2.3.1
Why is inclusion of partial saturation required in practical analysis?
In the previous, fully saturated, analysis, we have considered both the water pore pressure and the solid displacement as problem variables. In the general case of
32 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
non-linear nature which is characteristic for the problems of soil mechanics both the effective stresses and pressures will have to be determined incrementally as the solution process (or computation) progresses step by step. In many soils we shall encounter a process of 'densification' implied in the constitutive soil behaviour. This means that the history of straining (associated generally with shear strain) induces the solid matrix to contract (or the material to densify). Such densification usually will cause the pore pressure to increase, leading finally to a decrease of contact stresses in the soil particles to near-zero values when complete liquefaction occurs. Indeed generally failure will occur prior to the liquefaction limit. However, the reverse may occur where the soil 'dilation' during the deformation history is imposed. This will imply development of negative pressures which may reach substantial magnitudes. Such negative pressures cannot exist in reality without the presence of separation surfaces in the fluid which is contained in the pores, and consequent capillary effects. Voids will therefore open up during the process in the fluid which is essentially incapable of sustaining tension. This opening of voids will probably occur when zero pressure (or corresponding vapour pressure of water) is reached. Alternatively air will come out of solution-or indeed ingress from the free water surface if this is open to the atmosphere. The pressure will not then be vapour pressure but simply atmospheric. We have shown in Chapter 1 that, once voids open, a unique relationship exists between the degree of saturation Swand the pore pressures pw (see Figure 1.6). Using this relation, which can be expressed by formula or simply a graph we can modify the equation used in Section 2.2 to deal with the problem of partial saturation without introducing any additional variables assuming that the air throughout is at constant (atmospheric) pressure. Note that both phenomena of densification and dilation will be familar to anybody taking a walk on a sandy beach after the tide has receded leaving the sand semi-saturated. First note how when the foot is placed on the damp sand, the material appears to dry in the vicinity of the applied pressure. This obviously is the dilation effect. However if the pressure is not removed but reapplied several times the sand 'densgies' and becomes quickly almost fluid. Clearly liquefaction has occurred. It is surprising how much one can learn by keeping one's eyes open! The presence of negative water pressures will, of course, increase the strength of the soil and thus have a beneficial effect. This is particularly true above the free water surface or the so called phreatic line. Usually one is tempted to assume simply a zero pressure throughout that zone but for non-cohesive materials this means almost instantaneous failure under any dynamic load. The presence of negative pressure in the pores assures some cohesion (of the same kind which allows castles to be built on the beach provided that the sand is damp). This cohesion is essential to assure the structural integrity of many embankments and dams.
2.3.2 The modification of equations necessary for partially saturated conditions The necessary modification of equations (2.20) and (2.21) will be derived below, noting that generally we shall consider partial saturation only in the slower phenomena for which u-p approximation is permissible.
PARTIALLY SA TURA TED BEHA VIOUR WITH AIR PRESSURE NEGLECTED
33
Before proceeding, we must note that the effective stress definition is modified and the effective pressure now becomes (viz ch. 1 sect. 1.3.3)
with the effective stress still defined by (2.1). Equation (2.20) remains unaltered in form whether or not the material is saturated but the overall density p is slightly different now. Thus in place of (2.7) we can write
neglecting the weight of air. The correction is obviously small and its effect insignificant. However (2.21) will now appear in a modified form which we shall derive here. First, the water momentum equilibrium, Equation (2.8), will be considered. We note that its form remains unchanged but with the variable p being replaced by p,. We thus have
As before, we have neglected the relative acceleration of the fluid to the solid Equation (2.14), defining the permeabilities, remains unchanged as
However, in general, only scalar, i.e. isotropic, permeability will be used here
where I is the identity matrix. The value of k is, however, dependent strongly on S, and we note that:
Such typical dependence is again shown in Figure 1.7 of Chapter 1. Finally, the conservation Equation (2.16) has to be restructured though the reader will recognize similarities. The mass balance will once again consider the divergence of fluid flow w;.; to be augmented by terms previously derived (and some additional ones). These are
34 EOUATIONS GOVERNING THE D YNAMZC. SOIL-PORE FLUID, INTERACTION
(i) Increased pore volume due to change of strain assuming no change of saturation: Giideii = d ~ ; ; (ii) An additional volume stored by compression of the fluid due to fluid pressure increase: nSwdpw/Kf (iii) change of volume of the solid phase due to fluid pressure increase: (1 - n)xwdpw/Ks (iv) change of volume of solid phase due to change of intergranular contact stress: -KT/K,(~E;; xwdpw/K,)
+
(v) and a new term taking into account the change of saturation: ndSw Adding to the above, as in Section 2.2, the terms involving density changes, on thermal expansion, the conservation equation now becomes:
Now, however, place
(? is different from that given in Equation (2.17) and we have in its
which of course must be identical with (2.17) when Sw= 1 and xW= 1 i.e. when we have full saturation. The above modification is mainly due to an additional term to those defining the increased storage in (2.17). This term is due to the changes of the degree of saturation and is simply:
but here we introduce a new parameter Cs defined as
The final elimination of w in a manner identical to that used when deriving (2.21) gives, (neglecting density variation):
PARTIALLY SATURATED BEHAVIOUR WITH AIR PRESSURE NEGLECTED
( k j - p , , , - swpfiij
P + S,pfbj)),;+aEii + Q* + SO = 0 -
35
(2.33~)
The small changes required here in the solution process are such that we shall find it useful to construct the computer program for the partially saturated form, with the fully saturated form being a special case. In the time-stepping computation, we still always assume that the parameters S,, k , and C, change slowly and hence we will compute these at the start of the time interval keeping them, subsequently, constant. Previously, we mentioned several typical cases where pressure can become negative and hence saturation drops below unity. One frequently encountered example is that of the flow occuring in the capillary zone during steady state seepage. The solution of the problem can of course be obtained from the general equations simply by neglecting all acceleration and fixing the solid displacements at zero (or constant) values. If we consider a typical dam or a water retaining embankment shown in Figure 2.3 we note that, on all the surfaces exposed to air, we have apparently incompatible boundary conditions. These are: p, = 0 and w, = 0 (i.e. net zero inflow) Clearly both conditions cannot be simultaneously satisfied and it is readily concluded that only the second is true above the area where the flow emerges. Of course when the flow leaves the free surface, the reverse is true.
Figure 2.3 A partially saturated dam. Initial steady-state solution. Only saturation (a) and pressure contours (b) are shown. Contour interval in (b) is 75 kPa. The Phreatic line is the boundary of the fully saturated zone in (a)
36 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
Figure 2.4 Test example of partially-saturated flow experiment by Liakopoulos (1965). (a) configuration of test (uniform inflow interrupted at t = 0) (b) pressures with - - -, computed; , recorded; (c) data (linear elastic analysis, E = 3000 kPa)
Computation will easily show that negative pressures develop near the surface and that, therefore, a partially saturated zone with very low permeability must exist. The result of such a computation is shown in Figure 2.3 and indeed it will be found that very little flow occurs above the zero-pressure contour. This contour is in fact the well known phreatic line and the partially saturated material procedure has indeed been used frequently purely as a numerical device for its determination (see Desai, 1977a and b, Desai and Li 1983 etc.). Another example is given in Figure 2.4. Here a numerical solution of Zienkiewicz et a[., (1990b) is given for a problem for which experimental data are available from Liakopolous (1965). In the practical code used for earthquake analysis we shall use this partially saturated flow to calculate a wide range of soil mechanics phenomena. However, for completeness in Section 2.4 we shall show how the effects of air movement can be incorporated into the analysis.
2.4 2.4.1
PARTIALLY SATURATED BEHA VIOUR WITH AIR FLOW CONSIDERED (pa > 0) The governing equations including air flow
This part of the chapter is introduced for completeness-though the effects of the air pressure are insignificant in most problems. However, in some cases of consolidation and confined materials, the air pressures play an important role and it is useful to have means for their prediction. Further, the procedures introduced are readily
PARTIALLY SATURATED BEHA VIOUR WITH AIR FLOW CONSIDERED
37
applicable to other pore-fluid mixtures. For instance, the simultaneous presence of water and oil is important in some areas of geomechanics and coupled problems are of importance in the treatment of oil reservoirs. The procedures used in the analysis follow precisely the same lines as introduced here. In particular, the treatment following the physical approach used in this chapter has been introduced by Simoni and Schrefler (1991), Li et al. (1990) for flow of water and oil and Schrefler and Zhan (1993) for flow of water with air. The alternative approach of using the mixture theory in these problems was outlined by Li and Zienkiewicz (1992) and Schrefler (1995). Some simple considerations will allow the basic equations for the dynamic of the soil containing two pore fluids to be derived.
2.4.2
The governing equation
The dynamics of the total mixture can, just as in Section 2.3, be written in precisely the same form as that for a single fluid phase (see (2.1 1)). For completeness we repeat that equation here (now, however, a priori omitting the small convective terms)
However, just as in Equation (2.25) we have to write
noting that
For definition of effective stress we use again (2.24) now, however, without equating the air pressure to zero i.e. writing P = xwpw
+ (1 - xw)pa
(2.36)
For the flow of water and air we can write the Darcy equations separately, noting that
for water as in (2.27) and for the flow of air:
38 EQUATIONS GOVERNING T H E DYNAMIC, SOIL-PORE FLUID, INTERACTION
Here we introduced appropriate terms for coefficients of permeability for water and air, while assuming isotropy. A new variable v now defines the air velocity. The approximate momentum conservation equations (see 2.13) can be rewritten in a similar manner using isotropy but omitting acceleration terms for simplicity. We therefore have for water
and for air
Finally, the mass balance equations for both water and air have to be written. These are derived in a manner identical to that used for equation (2.30). Thus for water we have
n . w;.; aSW&+ Sw-pw Kw
a-n Pw +xwpw+ nSw + nSw- + So = 0
n VT w+crSwmi.+Sw--t),+Kw
a-n Pw PW So KS xwpW+ nSw nSw-
+
Ks
Pw
+
+
=0
(2.41~) (2.41b)
and for air
n v;.; aSaii,+ S, -pa Ka
+
VTv
a-n Ks
Pa + -~ a p a+ nSa + nSaPa + i0= 0
n a-n + crSami.+ S, -pa +xada + nSa + nSaPa + so = 0 Ka Ks -
Pa
(2.42~)
(2.42b)
Now, in addition to the solid phase displacement u;(u),we have to consider the water presure pw and the air presure pa as independent variables. However, we note that now (see (1.16))
and that the relation betweenp, and Sw is unique and of the type shown in Figure 1.6 of Chapter 1. p, now defines Sw and from the fact that
Sw
+ S, = 1
the air saturation can also be found.
and
xW+ xa = I
(2.44)
39
THEORY
We have now the complete equation system necessary for dealing with the flow of air and water (or any other two fluids) coupled with the solid phase deformation.
ALTERNA TIVE DERIVA TION OF THE GOVERNING EQUA TION (OF SECTION 2.2-2.4) BASED ON THE HYBRID MIXTURE THEORY
2.5
It has already been indicated in Section 2.1 that the governing equations can alternatively be derived using mixture theories. The classical mixture theories (see Green (1969) and Bowen (1980,1982), Morland (1972)) start from the macro-mechanical level, i.e. the level of interest for our computations, while the so called hybrid mixture theories (viz. Whitaker (1977) Hassanizadeh and Gray (1979a, 197913, 1980, 1990) start from micro-mechanical level. The equations at macro-mechanical level are then obtained by spatial averaging procedures. Further there exists a macroscopic thermodynamical approach to Biot's theory proposed by Coussy (1995). All these theories lead to a similar form of the balance equations. This was in particular shown by de Boer et al. (199 1) for the mixture theory, the hybrid mixture theories and the classical Biot's theory. Where the theories differ, is in the constitutive equations, usually obtained from the entropy inequality. This is shown here in particular for the effective stress principle, because this was extensively discussed in Chapter 1. Let's consider first the fully saturated case. For instance, Runesson (1978) shows that the principle of effective stress follows from the mixture theory under the assumption of incompressible grains. This means that cu in equation (1.16a) or (2.1a)
+
g!! = g.. r, 11
VP
(2.45)
has to be equal to one, which results in
Only in this case the two formulations given by Biot's theory and the mixture theory coincide. In the hybrid mixture theories, the concept of the solid pressure ps i.e. the pressure acting on the solid grains is introduced (Hassanizadeh and Gray, 1990). From the application of the second principle of thermodynamics, it appears that
where E is the time rate of change of the deformations, k is a coefficient ad 8, and 8, the absolute temperatures of solid and water respectively. Under assumption of local thermodynamic equilibrium, it follows that
and the effective stress principle in the form of (2.46) can be derived following Hassanizadeh and Gray (1990). Equation (2.48) holds also under non-equilibrium
40 EQUATIONS GOVERNING T H E DYNAMIC, SOIL-PORE FLUID, INTERACTION
conditions, it has, however, to be assumed that the solid grains remain incompressible, the same assumption as with the mixture theory. Let us consider now the partially saturated case. There are again several conflicting expressions in literature. The first expression for partially saturated soils was developed by Bishop (1959) and may be written as follows, see Equations (2.36, 2.24 and 1.19)
where X, is the Bishop parameter, usually a function of the degree of saturation, see (1.19) (1.20) (1.22a,b). The same expression, but with Bishop's parameter equal to the water degree of saturation was derived by Lewis and Schrefler (1987) using volume averaging. Hassanizadeh and Gray (1990) find for the partially saturated case that
which considering thermodynamic equilibrium conditions or non-equilibrium conditions but incompressible solid grains reduces to
Taking into account that
the solid pressure (2.51) coincides with that of equation (2.49) if xW=Sw but this is of course not often the case. Finally, Coussy (1995), using the Clausius-Duhem inequality, obtains
where
is the capillary pressure increment. This equation has hence an incremental form and differs substantially from the previous ones. The practical implication of these different formulations for slow phenomena have been investigated in detail by Schrefler and Gawin (1996). It was concluded that in practical soil mechanics situations the resulting differences are small and appear usually after long lasting variations of the moisture content. Only several cycles of drying and wetting would produce significant differences. The formulations of the effective stress principle in finite form coincide if the Bishop parameter x = Sw and the solid grains are incompressible, i.e. a = 1 this ofcourse limits the applicability. This assumption is now made and the governing
41
THEORY
equations are derived again, using the hybrid mixture theory, as has been done by Schrefler (1995) and Lewis and Schrefler (1998). Isothermal conditions are assumed to hold, as throughout this book. For the full nonisothermal case the interested reader is referred to Lewis and Schrefler (1998). We first recall briefly the kinematics of the system.
2.5.1
Kinematic equations
As indicated in Chapter 1, a multi-phase medium can be described as the superposition of all 7r phases, 7r = 1,2, . . . K , i.e. in the current configuration each spatial point x is simultaneously occupied by material points X" of all phases. The state of motion of each phase is, however, described independently. In a Lagrangian or material description of motion, the position of each material point x" at time f is a function of its placement in a chosen reference configuration, X" and of the current time t x; = x;
(x;, x;,x;,t )
To keep this mapping continuous and bijective at all times, the Jacobian of this transformation must not equal zero and must be strictly positive, since it is equal to the determinant of the deformation gradient tensor FT
where U" is the right stretch tensor, V" the left stretch tensor, and the skew-symmetric tensor R" gives the rigid body rotation. The differentiation with respect to the appropriate co-ordinates of the reference or actual configuration are respectively denoted by comma or slash, i.e.
Because of the non-singularity of the Lagrangian relationship (2.55), its inverse can be written and the Eulerian or spatial description of motion follows
The material time derivative of any differentiable function f"(x, t ) given in its spatial description and referred to a moving particle of the 7r phase is
If superscript a is used for phase.
the time derivative is taken as moving with the a
42 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
2.5.2 Micvoscopic balance equations In the hybrid mixture theories, the microscopic situation of any .ir phase is first described by the classical equations of continuum mechanics. At the interfaces to other constituents, the material properties and thermodynamic quantities may present step discontinuities. For a thermodynamic property @ the conservation equation within the .ir phase may be written as
where i is the local value of the velocity field of the .ir phase in a fixed point in space, i is the flux vector associated with a,g the external supply of @ and G is the net production of Q. The relevant thermodynamic properties @ are mass, momentum, energy and entropy. The values assumed by i, g and G are given in Table 2.2 (Hassanizadeh and Gray (1980, 1990) and Schrefler (1995)). The constituents are assumed to be microscopically non-polar, hence the angular momentum balance equation has been omitted. This equation shows, however, that the stress tensor is symmetric.
2.5.3 Macvoscopic balance equations For isothermal conditions, as here assumed, the macroscopic balance equations for mass, linear momentum and angular momentum are then obtained by systematically applying the averaging procedures to the microscopic balance equations (2.60) as outlined in Hassanizadeh and Gray (1979a, 1979b, 1980). The balance equations have here been specialized for a deforming porous material, where flow of water and moist air (mixture of dry and vapour) is taking place (see Schrefler, 1995). The local thermodynamic equilibrium hypothesis is assumed to hold because the timescale of the modelled phenomena is substantially larger than the relaxation time required to reach equilibrium locally. The temperatures of each constituent at a generic point are hence equal. Further, the constituents are assumed to be immiscible and chemically non-reacting. All fluids are assumed to be in contact with the solid phase. As throughout this book, stress is defined as positive tension for the solid phase, while pore pressure is defined as positive compression for the fluids. In the averaging procedure the volume fractions q" appear which are identified as follows: for the solid phase qs = 1 - n: for water qw = nS,; and for air qa = nS,. Table 2.2 Thermodynamic properties for the microscopic mass balance equations Quantity Mass Momentum Energy Entropy
P
i
1 i E
0 t "I tmr - q cP
X
+ 0.5r.r
G
g 0 g g.i+h
0 0 0
S
P
43
THEORY
The averaged macroscopic mass balance equations are given next. For the solid phase this equation reads
where u is the mass averaged solid phase velocity and p" is the intrinsic phase averaged density. The intrinsic phase averaged density p" is the density of the .ir phase averaged over the part of the control volume (Representative Elementary Volume, REV) occupied by the .rr phase. The phase averaged density p,, on the contrary, is the density of the .ir phase averaged over the total control volume. The relationship between the two densities is given by
For water we have
where nSwpwew(p)= -m is the quantity of water per unit time and volume, lost through evaporation and vw the mass averaged water velocity. For air, this equation reads
where va is the mass averaged air velocity. The linear momentum balance equation for the fluid phases is t;,,
+ p"(b;
-
a;)
+ p" Len(pi;) + i;]
=0
where t" is the partial stress tensor, p"b" the external momentum supply due to gravity, p"a" the volume density of the inertia force, p"eT(pr) the sum of the momentum supply due to averaged mass supply and the intrinsic momentum supply due to a change of density and referred to the deviation i" of the velocity of constituent .ir from its mass averaged velocity, and accounts for the exchange of momentum due to mechanical interaction with other phases. pke"(pr) is assumed to be different from zero only for fluid phases. For the solid phase the linear momentum balance equation is hence
The average angular momentum balance equation shows that for non-polar media the partial stress tensor is symmetric ti: = t; at macroscopic level also and the sum of the coupling vectors of angular momentum between the phases vanishes.
44 EOUA TIONS GOVERNING THE DYNAMIC. SOIL-PORE FLUID. INTERACTION
2.5.4
Constitutive equations
Constitutive models are here selected which are based on quantities currently measurable in laboratory or field experiments and which have been extensively validated. Most of them have been obtained from the entropy inequality, see Hassanizadeh and Gray (1980, 1990). It can be shown that the stress tensor in the fluid is
where p, is the fluid pressure, and in the solid phase is
r..S!I = a - - (1 - n)pshii 1
11
(2.68)
with p, = pwSw + p a s a in the case of thermodynamic equilibrium or for incompressible solid grains, (2.51). The sum of (2.67), written for air and water and of (2.68) gives the total stress a, acting on a unit area of the volume fraction mixture
This is the form of the effective stress principle employed in the following, as already explained. Moist air in the pore system is assumed to be a perfect mixture of two ideal gases, dry air and water vapour, with T = ga and r = g w respectively. The equation of a perfect gas is hence valid
where M , is the molar mass of constituent T , R the universal gas constant, and Q the common absolute temperature. Further, Dalton's law applies and yields the molar mass of moisture
Water is usually present in the pores as a condensed liquid, separated from its vapour by a concave meniscus because of surface tension. The capillary pressure is defined as pc = pg - pw, see Equation (2.54). The momentum exchange term of the linear momentum balance equation for fluids has the form
where v"5s the velocity of the r phase relative to the solid. It is assumed that R" is invertible, its inverse being ( R " ) - ' = and KT is defined by the following relation
45
THEOR Y
k,
K$ = -(p", 7 " ) T ) PT
where p, is the dynamic viscosity, k the intrinsic permeability and T the temperature above some datum. In the case of more than one fluid flowing, the intrinsic permeability is modified as
where k'" is the relative permeability, a function of the degree of saturation. For the water density, the following holds,
where Kw is the bulk modulus of water.
2.5.5 General field equations The macroscopic balance laws are now transformed and the constitutive equations introduced, to obtain the general field equations. The linear momentum balance equation for the fluid phases is obtained first. In Equation (2.65) the fluid acceleration is expressed, taking into account Equation (2.59), and introducing the relative fluid acceleration a"" Further, Equations (2.67) and (2.72) are introduced. The terms are dependent on the gradient of the fluid velocity. The effects of phase change are neglected and a vector identity for the divergence of the stress tensor is used. Finally, equations (2.73) and (2.74) are included, yielding
The linear momentum balance equation for the solid phase is obtained in a similar way, taking into account equations (2.68) instead of (2.67). By summing this momentum balance equation with equation (2.76) written for water and air, and by taking into account the definition of total stress (2.69) assuming continuity of stress at the fluid-solid interfaces and by introducing the averaged density of the multiphase medium,
we obtain the linear momentum balance equation for the whole multi-phase medium
46 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID, INTERACTION
The mass balance equations are derived next. The macroscopic mass balance equation for the solid phase (2.61), after differentiation and dividing by pS, is obtained as
This equation is used in the subsequent mass balance equations to eliminate the material time derivative of the porosity. For incompressible grains, as assumed here, DSPS~r - 0. For compressible grains, see equation (2.89) and related remarks. The mass balance equation for water (2.63) is transformed as follows. First Equation (2.75), the material time derivative of the water density with respect to the moving solid phase and the relative velocity vWS,are introduced. Then the derivatives are carried out, the quantity of water lost through evaporation is neglected and the material time derivative of the porosity is expressed through Equation (2.79), yielding
The mass balance equation for air is derived in a similar way
To obtain the equations of Section 2.4.1, further simplifications are needed, which are introduced next. An updated Lagrangian framework is used, where the reference configuration is the last converged configuration of the solid phase. Further, the strain increments within each time step are small. Because of this we can neglect the convective terms in all the balance equations. Neglecting, in the linear momentum balance equation (2.78) the relative accelerations of the fluid phases with respect to the solid phase, yields the equilibrium equation (2.28a)
The linear momentum balance equation for fluids (2.76) by omitting all acceleration terms, as in Chapter 2, can be written for water
where qWv? = W ; and
kwii
k,krw
=!Jw
and for air
NOMENCLA TURE
47
where
The phase densities appearing in Chapter 2 are intrinsic phase averaged densities as indicated above. The mass balance equation for water is obtained from Equation (2.80), taking into account the reference system chosen, dividing by pw, developing the divergence term of the relative velocity and neglecting the gradient of water density. This yields
where the first of Equations (2.84) has been taken into account. This coincides with Equation (2.35a) for incompressible grains except for the source term and the secondorder term due to the change in fluid density. This last one could be introduced in the constitutive relationship (2.75) Similarly, the mass balance equation for air becomes
where again the first of equations (2.86) has been taken into account and the gradient of the water density has been neglected. Similar remarks as for the water mass balance equation apply. In particular, the constitutive relationships for moist air, Equations (2.70) and (2.71), have been used. Finally, if for the solid phase the following constitutive relationship is used (see Lewis and Schrefler (1998))
where Ks is the bulk modulus of the grain material, then the mass balance equations are obtained in the same form as in Chapter 2 (with xW= S,). However, this is not in agreement with what was assumed for the effective stress.
2.5.6
(Nomenclature) for Section 2.5
As this section does not follow the notations used in the use of the book, we summarise below for purposes of nomenclature a" mass averaged acceleration of .rr phase aTS acceleration relative to the solid.
48 EQUATIONS GOVERNING THE DYNAMIC. SOIL-PORE FLUID. INTERACTION b external momentum supply material time derivative E specific intrinsic energy F" deformation gradient tensor f" differentiable function
6
fji fi
aflaxi aflaxi
g external momentum supply related to gravitational effects G net production of thermodynamic property P h intrinsic heat source i flux vector associated with thermodynamic property P k constitutive coefficient k intrinsic permeability tensor kr" relative permeability of .ir phase KT permeability tensor Kw water bulk modulus m mass rate of water evaporation M , molar mass of constituent K n porosity p, capillary pressure p, dry air pressure p, vapour pressure pa air pressure p, solid pressure pw pressure of liquid water
p, macroscopic pressure of the .ir phase i local value of the velocity field R universal gas constant R constitutive tensor R" rigid body rotation tensor S intrinsic entropy source Sw degree of water saturation Sa degree of air saturation t current time t, microscopic stress tensor t" partial stress tensor 2" exchange of momentum due to mechanical interaction of the .rr phase with other phases u mass averaged solid phase velocity Un right stretch tensor vm velocity of the .rr phase with respect to the solid phase vw mass averaged water velocity va mass averaged air velocity
CONCLUDING REMARKS
49
v volume averaged water relative velocity w volume averaged air relative velocity V" left stretch tensor x" material point X" reference configuration E linear strain tensor @ entropy flux p increase of entropy 7" volume fraction of the T phase xw Bishop parameter X specific entropy p" dynamic viscosity 0, absolute temperature of constituent T p averaged density of the multi-phase medium p" intrinsic phase averaged density of the .rr phase p, phase averaged density of the T phase p microscopic density a' effective stress tensor 9 generic thermodynamic property or variable Super or subscripts ga = dry air gw = vapour a = air w = water s = solid
2.6
CONCLUDING REMARKS
The equations derived in this chapter together with appropriately defined constitutive laws allow (almost) all geomechanical phenomena to be studied. In the next chapter, we shall discuss in some detail approximation by the finite element method leading to their solution.
REFERENCES Biot M. A. (1941) General theory of three-dimensionalconsolidation, J. Appl. Phys., 12,155-164. Biot M . A. (1955) Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 182-185. Biot M. A. (1956a) Theory of propagation of elastic waves in a fluid-saturated porous solid, part I-low-frequency range, J. Acoust. Soc. Am., 28, No. 2, 168-178. Biot M. A. (1956b) Theory of propagation of elastic waves in a fluid-saturated porous solid, part 11-low-frequency range, J. Acoust. Soc. Am., 28, No. 2, 179-191.
50 EQUA TIONS GOVERNING T H E DYNAMIC, SOIL-PORE FLUID, INTERACTION Biot M. A. (1962) Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33, No. 4, 1482-1498. Biot M. A. and Willis P. G. (1957) The elastic coefficients of the theory consolidation, J. Appl. Mech., 24, 59&6O 1. Bishop A.W. (1959) The principle of~ffectivestress, Teknisk Ukeblad, 39, 859-863. Bowen R. M. (1976) Theory of Mixtures in Continuum Physics, Academic Press, New York. Bowen R.M. (1980) Incompressible porous media models by use of the theory of mixtures, Int. J. Eng. Sci. 18, 1129-1148. Bowen R.M. (1982) Compressible porous media models by use of theories of mixtures, Int. J. Eng. Sci. 20, 697-735. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D. (1991) A Fully Explicit u-w Scheme for Dynamic Soil and Pore Fluid Interaction, APCOM Hong Kong, 11-13 Dec., 1, 881-887. Coussy 0 . (1995) Mechanics of Porous Media, J . Wiley & Sons, Chichester. Craig R. F. (1992) Soil Mechanics (5th edn), Chapman & Hall, London. De Boer R. (1996) Highlights in the historical development of the porous media theory, Appl. Mech. Rev., 49, 201-262. De Boer R., Ehlers W., Kowalski S. and Plischka J. (1991) Porous Media, a Survey of Different Approaches, Forschungsbericht aus dem Fachbereich Bauwesen, 54, UniversitaetGesamthochschule Essen. De Boer R. and Kowalski S. J. (1983) A plasticity theory for fluid saturated porous solids, Int. J. Eng. Sci., 21, 1343-1357. Derski W. (1978) Equations of motion for a fluid saturated porous solid, BUN. Acud. Polish Sci. Tech., 26, 11-16. Desai C. S. (1977a) Discussion-Finite element, residual schemes for unconfined flow, Int. J. Nurn. Meth. Eng., 11, 80-8 I. Desai C. S. (1977b) Finite Element, residual schemes for unconfined flow, Int. J. Nurn. Meth. Eng. 10, 1415-1418. Desai C. S. and Li G. C. (1983) A Residual Flow Procedure and Application for Free Surface Flow in Porous Media, Adv. Water Res., 6 , 27-35. Ehlers W. (1989) Poroese Medien, Forschungsbericht aus dem Fachbereich Bauwesen, 47, Universitaet-Gesamthochschule Essen. Ghaboussi J. and Wilson E. L. (1972) Variational formulation of dynamics of fluid saturated porous elastic solids, ASCE E M , 98, No. EM4, 947-963. Green A. E. (1969) On basic equations for mixtures, Quart. J. Mech. Appl. Math., 22,428438. Green A. E. and Adkin J. E. (1960) Large Elastic Deformations and Nonlinear Continuum Mechanics, Oxford University Press, London. Hassanizadeh M. and Gray W.G., (1979 a). General conservation equations for multiphase systems: 1 Averaging procedure, Adv. Wcrter Res., 2, 131-144. Hassanizadeh M. and Gray W.G. (1979 b) General conservation equations for multiphase systems: 2 Mass, momenta. energy and entropy equations, Adv. Water Res., 2, 191-203. Hassanizadeh M. and Gray W.G. (1980) General conservation equations for multiphase systems: 3 Constitutive theory for porous media flow, Adv. Water Res., 3, 2 5 4 0 . Hassanizadeh M. and Gray W.G. (1990) Mechanics and Thermodynamics of multiphase flow in porous media including interphase transport, Adv. Water Res., 13, 169-186. Jauman G. (1905) Die Grundlagen der Bewegungslehre von einem modernen Standpunkte aus, Leipzig. Kowalski S. J. (1979) Comparison of Biot's equation of motion for a fluid saturated porous solid with those of Derski, Bull. Acad. Polish Sci. Tech.. 27, 455461.
REFERENCES
51
Lambe T. W. and Whitman R. V. (1969) Soil Mechanics, (SI Version), John Wiley & Sons, New York. Leliavsky S. (1947) Experiments on effective area in gravity dams, Trans. Am. Soc. Civil Engrs., 112, 444. Lewis R.W. and Schrefler B.A. (1987) The Finite Element Method in the Deformation rind Consolidation ofPorous Media, J . Wiley & Sons, Chichester. Lewis R.W. and Schrefler B.A. (1998) The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, J . Wiley & Sons, Chichester. Li X. K. and Zienkiewicz 0. C. (1992) Multiphase flow in deforming porous-media and finiteelement solutions, Comp. Struct., 45, No. 2, 21 1-227. Li X. K., Zienkiewicz 0. C. and Xie Y. M. (1990) A numerical model for immiscible 2-phase fluid-flow in a porous medium and its time domain solution, Int. J. Num. Meth. Eng.. 30, NO. 6, 1195-1212. Liakopoulos A. C. (1965) Transient flow through unsaturated porous media, D.Eng. Dissertation, University of California, Berkeley, USA. Morland L. W. (1972) A simple constitutive theory for fluid saturated porous solids, J. Geophys. Res., 77, 890-900. Runesson K. (1978) On non-linear consolidation of soft clay, Ph.D. Thesis, Chalmers University of Technology, Goeteborg. Sandhu R. S. and Wilson E. L. (1969) Finite element analysis of flow in saturated porous elastic media, ASCE EM, 95, 641-652. Schrefler B. A. (1995) Finite Elements in Environmental Engineering: Coupled thermo-hydromechanical process in porous media involving pollutant transport, Archives of Computer Methods in Engineering, 2, 1-54. Schrefler B.A. and Gawin D. (1996) The effective stress principle: incremental or finite form?. Int. J. Nurn. Anal. Meth. Geom., 20, 785-814. Schrefler B. A. and Zhan X. (1993) A fully coupled model for waterflow and airflow in deformable porous media, Water Resources Res., 29, No. 1, 155-167. Serafim J. L. (1954) A subpresseo nos barreyens - Publ. 55, Laboratorio Nacional de Engenheria Civil, Lisbon. Simoni L. and Schrefler B. A. (1991) A staggered finite element solution for water and gas flow in deforming porous media, Commun. Appl. Num. Meth.. 7, 213-223. Skempton A. W. (1954) The pore pressure coefficients A and B, Gkotechnique. 4. 143-147. Terzaghi K. von (1925) Erdbaumechanik auf bodenphysikalischer Grundlage, Deuticke. Vienna. Whitaker S., (1977). Simultaneous heat mass and momentum transfer in porous media: a theory of drying, Advances in Heat Transfer, 13, Academic Press. New York. Zaremba S. (1903a) Le priniple des mouvements relatifs et les equations de la mecanique physique. Reponse a M. Natanson, Bull. Int. Acad. Sci. Cracovie, 614-621. Zaremba S. (1903b) Sur une generalisation de la theorie classique de la viscosite, Bull. Int. Acad. Sci. Crucovie., 380-403. Zienkiewicz 0. C. (1982) Field equations for porous media under dynamic loads in Num. Meth. in Geomech., D. Reidel, Boston U.S.A. Zienkiewicz 0 . C. and Shiomi T. (1984) Dynamic Behaviour of saturated porous media: The generalized Biot formulation and its numerical solution, Int. J. Nurn. Anal. Geotnech.. 8, 71-96. Zienkiewicz 0. C. and Taylor R. L. (1989) The Finite Element Method - Volume I : Baslc Formulation and Linear Problems (4th edn), McGraw-Hill Book Company, London. Zienkiewicz 0. C., Chang C. T. and Bettess P. (1980) Drained, undrained, consolidating and dynamic behaviour assumptions in soils, GPotechnique, 30, No. 4. 385-395.
52 EQUATIONS GOVERNING THE DYNAMIC, SOIL-PORE FLUID. INTERACTION Zienkiewicz 0. C., Chan A. H. C . , Pastor M., Paul D. K. and Shiomi T. (1990a) Static and Dynamic Behaviour of Geomaterials - A rational approach to quantitative solutions, Part I-Fully Saturated Problems, Proc. Roy. Soc. Lond., A429, 285-309. Zienkiewicz 0. C., Xie Y. M., Schrefler B. A,, Ledesma A. and Bicanic N. (1990b) Static and Dynamic behaviour of soils: a rational approach to quantitative solutions, Part 11: Semisaturated problems, Proc. Roy. Soc. Lond., A429, 310-323.
Finite Element Discretization and Solution of the Governing Equations
3.1
THE PROCEDURE OF DISCRETIZATION BY THE FINITE ELEMENT METHOD
The general procedures of the Finite Element discretization of equations are described in detail in various texts. Here we shall use throughout the notation and methodology introduced by Zienkiewicz & Taylor (1989 and 1991) which is the most recent (fourth) edition of the first text for the finite element method published in 1967. In the application to the problems governed by the equations of the previous chapter we shall typically be solving partial differential equations which can be written as
where A, B are matrices of constants and L is an operator involving spatial differentials such as d l d x , d l d y , etc., which can be, and frequently are, non-linear. The dot notation implies time differentiation so that
a@
-=& at
d2@ dl, z & etc.
--
In all of the above, @ is a vector of dependent variables (say representing the displacements u and the pressure p). The finite element solution of the problem will always proceed in the following pattern: (i) The unknown functions Q, are 'discretized' or approximated by a finite set of parameters 6,and shape function Nk which are specified in spatial dimensions. Thus we shall write
54
FINITE ELEMENT DISCRETIZATION AND SOLUTION
where the shape functions are specified in terms of the spatial coordinates i.e.
and &;
= 6;(t)
(3.4b)
are usually the values of the unknown function at some discrete spatial points called nodes which remain as variables in time. (ii) Inserting the value of the approximating function 6 into the differential equations we obtain a residual which is not identically equal to zero but for which we can write a set of weighted residual equations in the form
which on integration will always reduce to a form
Where M , C and P are matrices or vectors corresponding in size to the full set of numerical parameters 6k.A very suitable choice for the weighting function Wj is to take them being the same as the shape function Ni:
Indeed this choice is optimal for accuracy in so called self-adjoint equations as shown in the basic texts and is known as the Galerkin process. If time variation occurs, i.e. if the parameters Giare time dependent, equation (3.6) which is now an ordinary differential equation requires solution in the time domain. This can be, once again, achieved by discretization in time and use of finite elements there although many alternative approximations (such as the use of finite differences or other integration schemes) are possible. Usually, the parameters & represent simply the values of at specified pointscalled nodes and the shape functions are derived on a polynomial basis of interpolating between the nodal values for elements into which the space is assumed divided. Typical finite elements involving linear and quadratic interpolation are shown in Figure 3.1. The present chapter will be divided into two sections. In Section 3.2 we shall consider solution of the approximation based on the u-p form in which the dependent variables are the displacement of the soil matrix and the
+"
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
55
Figure 3.1 Some Typical two dimensional elements for linear and quadratic interpolations
from nodal values
pore pressure characterized by a single fluid, i.e. water. However, we shall allow incomplete saturation to exist assuming that the air pressure is zero. The formulation thus embraces all the features of the u-p approximation of sections 2.2.2, 2.3.1 and 2.3.2 and is the basis of a code capable of solving all lowfrequency dynamic problems, consolidation problems and static drained or undrained problems of soil mechanics. Only two dimensions will be considered here and in the examples which follow, but extension to three dimensions is obvious. The code based on the formulation contained in this part of the chapter is named SWANDYNE (indicating its Swansea origin) and its outline was presented in literature by Zienkiewicz et al. (1990a). Section 3.3 of this chapter is intended for the solution of dynamic problems where high-frequency effects predominate. The variable here will be u and U i.e. displacement vectors of the solid and of the pore fluid (water). The code based on this form is fully explicit and it is named GLADYS-2E as it was developed in Glasgow (see Chan et al., 1991) following the work described by Zienkiewicz and Shiomi (1984). The time-step limitations of such explicit codes are severe and the code is therefore limited to relatively short time durations. On the other hand, the implicit form in Section 3.2. allows much longer periods to be studied. Indeed, with suitable accuracy control such codes can be used both for earthquake phenomena limited to hundreds of seconds or consolidation problem with a duration of hundreds of days.
3.2 3.2.1
U-P DZSCRE TZZA TZON FOR A GENERAL GEOMECHANZCS FZNZTE ELEMENT CODE Summary of the general governing equations
We will report here the basic governing equation derived in the previous chapter. However, we shall limit ourselves to the use of the condensed, vectorial form of these which is convenient for finite element discretization. The tensorial form of the equations can be found in Section 3.4.
56
FINITE ELEMENT DISCRETIZATION AND SOLUTION
The overall equilibrium or momentum balance equation is given by (2.11) and is here copied for completeness as
In the above and in all following equations, the relative fluid acceleration terms are omitted as only the u-p form is being considered. The strain matrix S is defined in two dimensions as (see (2.10))
{ : z } = Sdu Here u is the displacement vector, p the total density of the mixture (see (2.19))
generally taken as constant and a is the total stress with components
The effective stress is defined as in (2.1)
where a again is a constant usually taken for soils as
and p the effective pressure defined by (2.24) with pa P
= XwPw
= 0.
(3.14)
The effective stress a" is computed from an appropriate constitutive law generally defined as 'increments' by (2.2)
where D is the tangent matrix dependent on the state variables and history and corresponds to thermal and creep strains.
EO
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
57
The main variables of the problem are thus u and p,. The effective stresses are determined at any stage by a sum of all previous increments and the value of p, determines the parameters S, (saturation) and X , (effective area). On occasion the approximation
can be used. An additional equation is supplied by the mass conservation coupled with fluid momentum balance. This is conveniently given by (2.27) which can be written as
with k = k ( S , ) . The contribution of the solid acceleration is neglected in this equation. Its inclusion in the equation will render the final equation system non-symmetric (see Leung, 1984) and the effect of this omission has been investigated in Chan (1988) who found it to be insignificant. However, it has been included in the force term of the computer code SWANDYNE-I1 (Chan, 1995) although it is neglected in the left hand side of the final algebraic equation when the symmetric solution procedure is used. The above set defines the complete equation system for solution of the problem defined providing necessary boundary conditions have been specified as in (2.18) and (2.19) i.e.
and
Assuming isotropic permeability, above becomes
where q, is the influx i.e. having an opposite sign to the outflow w,. The total boundary I? is the union of its components i.e.
58
FINITE ELEMENT DISCRETIZATION AND SOLUTION
3.2.2 Discretization of the governing equation in space The spatial discretization involving the variables u and p is achieved by suitable shape (or basis) functions, writing
We assume here that the expansion is such that the strong boundary conditions (3.18) are satisfied on ru and r, automatically by a suitable prescription of the (nodal) parameters. As in most other finite element formulation, the natural boundary condition will be obtained by integrating the weighted equation by parts. To obtain the first equation discretized in space we premultiply (3.8) by (NU)=and integrate the first term by parts (see for details Zienkiewicz and Taylor, 1989 or other texts) giving:
where the matrix B is given as
and the 'load vector'f ( I ) , equal in number of components to that of vector u contains all the effects of body forces, and prescribed boundary tractions i.e.
At this stage it is convenient to introduce the effective stress (see (3.12)) now defined to allow for effects of incomplete saturation as
The discrete, ordinary differential equation now becomes
where
A GENERAL GEOMECHANZCS FINITE ELEMENT CODE
59
is the MASS MATRIX of the system and
is the coupling matrix-linking servation, and
equation (3.23) and those describing the fluid con-
The computation of the effective stress proceeds incrementally as already indicated in the usual way and now (3.15) can be written in discrete form:
where of course D is evaluated from appropriate state and history parameters. )~ integrating by Finally we discretize equation (3.17) by premultiplying by ( N ~and parts as necessary. This gives the ordinary differential equation
where the various matrices are as defined below
where Q* is defined as in (2.30~)i.e.
where C s , Sw, Cw and k depend onp,.
FINITE ELEMENT DISCRETIZATION AND SOLUTION
60
3.2.3 Discretisation in time To complete the numerical solution, it is necessary to integrate the ordinary differential equations (3.23), (3.27) and (3.28) in time by one of the many available schemes. Although there are many multi-step methods available (see e.g. Wood, 1990), they are inconvenient as most of them are not self-starting and it is more difficult to incorporate restart facilities which are required frequently in practical analyses. On the other hand, the single-step methods handle each step separately and there is no particular change in the algorithm for such restart requirements. Two similar, but distinct, families of single-step methods evolved separately. One is based on the finite element and weighted residual concept in the time domain and the other based on a generalization of the Newmark or finite difference approach. The former is known as the SSpj - Single Step pth order scheme for jth order differential equation ( p 2 j ) . This was introduced by Zienkiewicz et al. (1980b, 1984) and extensively investigated by Wood (1984a, 1984b, 1985a, 1985b). The SSpj scheme has been used successfully in SWANDYNE-I (Chan, 1988). The later method, which was adopted in SWANDYNE-I1 (Chan, 1995) was an extension to the original work of Newmark (1959) (see also Whitman 1953) and is called Beta-m method by Katona (1985) and renamed the Generalized Newmark (GNpj) method by Katona and Zienkiewicz (1985). Both methods have similar or identical stability characteristics. For the SSpj, no initial condition e.g. acceleration in dynamical problems, or higher time derivatives are required. On the other hand, however, all quantities in the GNpj method are defined at a discrete time station thus making transfer of such quantities between the two equations easier to handle. Here we shall use the later (GNpj) method, exclusively, due to its simplicity. In all time stepping schemes we shall write a recurrence relation linking a known value 4, (which can either be the displacement or the pore water pressure), and its &,+, , . . ., which derivatives $, &, . . . at time station t, with the values of are valid at time t, A t and are the unknowns. Before treating the ordinary differential equation system (3.23), (3.27) and (3.28), we shall illustrate the time stepping scheme on the simple example of (3.6) by adding a forcing term:
+
+,+,,
From the initial conditions, we have the known values of a,, 6,. We assume that the above equation has to be satisfied at each discrete time i.e. t, and t,+l, we can thus write:
and
From the first equation, the value of the acceleration at time t, can be found and this solution is required if the initial conditions are different from zero.
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
61
The link between the successive values is provided by a truncated series expansion taken in the simplest case as GN22 as Eq(3.34) is a second-order differential equation j and the minimum order of the scheme required is then two: as (p > j)
Alternatively, a higher order scheme can be chosen such as GN32 and we shall have:
In this case, an extra set of equations is required to obtain the value of the highest time derivatives. This is provided by differentiating (3.35) and (3.36).
and
In the above equations, the only unknown is the incremental value of the highest derivative and this can be readily solved for. Returning to the set of ordinary differential equations, we are considering here i.e. in (3.23), (3.27) and (3.28) and writing (3.23) and (3.28) at the time station t,+l, we have:
assuming that (3.27) is satisfied. Using GN22 for the displacement parameters u and G N l l for the pore pressure parameter p"', we write:
62
FINITE ELEMENT DISCRETIZATION AND SOLUTION
u,+1
= u,
u,+1 = u,,
+ Au, +&at
+ p,au,at
and
where Au, and Ap, are as yet undetermined quantities. The parameters PI,P2 and PI are usually chosen in the range of 0 to 1. For P2 = 0 and PI = 0, we shall have an explicit form if both the mass and damping matrices are diagonal. If the damping matrix is non-diagonal, an explicit scheme can still be achieved with PI = 0 thus eliminating the contribution of the damping matrix. The well known central difference scheme is recovered from (3.41) if PI = 1/2,P2 = 0 and this form with an explicit u and implicit p scheme has been considered in detail by Zienkiewicz et al. (1982) and Leung (1984). However, such schemes are only conditionally stable and for unconditional stability of the recurrence scheme we require 1
P2
> PI > 2
-
and
1
81 2 5
The optimal choice of these values is a matter of computational convenience, the discussion of which can be found in literature. In practice, if the higher order accurate 'trapezoidal' scheme is chosen with P2 = PI = 112 and PI = 112, numerical oscillation may occur if no physical damping is present. Usually some algorithmic (numerical) damping is introduced by using such values as P2 = 0.605 P2 = 0.515
PI = 0.6 and PI = 0.6 or PI = 0.51 and PI = 0.51.
Dewoolkar (1996), using the computer program SWANDYNE I1 in the modelling of a free-standing retaining wall, reported that the first set of parameters led to excessive algorithmic damping as compared to the physical centrifuge results, therefore the second set was used and gave very good comparisons. However, in cases involving soil, the physical damping (viscous or hysterestic) is much more significant than the algorithmic damping introduced by the time stepping parameters and the use of either sets of parameters leads to similar results. Inserting the relationships (3.43) into equations (3.41) and (3.42) yields a general non-linear equation set in which only Au, and Ap,, remain as unknowns. This set can be written as
63
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
w!lj
+
+
= QT+~,LI~&A& ~ , , + ~ p ~ A t A cs ,y+ ~ A ~ ," F1j!
=0
(3.44b)
where @j1and F ~ J can , be evaluated explicitly from the information available at time t, and
In this A u i must be evaluated by integrating (3.27) as the solution proceeds. The values of u,+l and p,+, at the time t,+l are evaluated by equation (3.43). The equation will generally need to be solved by a convergent, iterative process using some form of Newton Raphson procedure typically written as I J{as *a ~ np}, l l = n+ 1
where 1 is the iteration number and {~i,,}~+{sa&}~+~ AP, sap,
The Jacobian matrix can be written as
J=
where
which are the well known expressions for tangent stiffness matrix. The underlined term corresponds to the 'initial stress' matrix evaluated in the current configuration as a result of stress rotation defined in (2.5). Two points should be made here: (a) that in the linear case a single 'iteration' solves the problem exactly (b) that the matrix can be made symmetric by a simple scalar multiplication of the second row (providing KT is itself symmetric). In practice it is found that the use of various approximations of the matrix J is advantageous such as, for instance, the use of 'secant' updates (see for instance Crisfield (1979), Matthies and Strang (1979) and Zienkiewicz and Taylor (1991)
64
FINITE ELEMENT DISCRETIZATION AND SOLUTION
A particularly economical computation form is given by choosing /32 = 0 and representing matrix M in a diagonal form. This explicit procedure was first used by Leung (1984) and Zienkiewicz et al. (1980). It is, however, only conditionally stable and is efficient only for phenomena of short duration. We shall return to such explicit processes in Section 3.3. The iterative procedure allows the determination of the effect of terms neglected in the u-p approximation and hence an assessment of the accuracy. The process of the time-domain solution of (3.44) can be amended to that of successive separate solution of the time equations for variables Au, and Ap, respectively, using an approximation for the remaining variable. Such staggered procedures, if stable, can be extremely economical as shown by Park and Felippa (1983) but the particular system of equations here presented needs stabilization. This was first achieved by Park (1983) and later a more effective form was introduced by Zienkiewicz et al. (1988). Special cases of solution are incorporated in the general solution scheme presented here without any modification and indeed without loss of computational efficiency. Thus for static or quasi-static problems it is merely necessary to put M = 0, and immediately the transient consolidation equation is available. Here time is still real and we have omitted only the inertia effects (although with implicit schemes this apriori assumption is not necessary and inertia effects will simply appear as negligible without any substantial increase of computation). In pure statics the time variable is still retained but is then purely an artificial variable allowing load incrementation. In static or dynamic undrained analysis the permeability (and compressibility) matrices are set to zero, i.e. H . f(*)= 0, and usually S = 0 resulting in a zero-matrix diagonal term in the jacobian matrix of Equation (3.47). The matrix to be solved in such a limiting case is identical to that used frequently in the solution of problems of incompressible elasticity or fluid mechanics and in such studies places limitations on the approximating functions Nu and NP used in (3.19) if the Babuska-Brezzi (Babuska, 1971 and 1973, Brezzi, 1974) convergence conditions or their equivalent (Zienkiewicz et al. 1986) are to be satisfied. Until now we have not referred to any particular element form, and indeed a wide choice is available to the user if the limiting (undrained) condition is never imposed. Due to the presence of first derivatives in space in all the equations it is necessary to use 'Co-continuous' interpolation functions (Zienkiewicz and Taylor, 1989) and Figure 3.2 shows some elements incorporated in the formulation. The form of most of the elements used satisfies the necessary convergence criteria of the undrained limit (Zienkiewicz, 1984). Though the bi-linear u and p quadrilateral does not, it is, however, useful when the permeability is sufficiently large. We shall return to this problem in Chapter 8 where a modification is introduced allowing the same interpolations to be used for both u and p. In a later chapter we shall discuss a possible amendment to the code permitting the use of identical u-p interpolation even in incompressible cases. We note that all computations start from known values of u and p"' possibly obtained as the result of static computations by the same program in a manner which will be explained in the next section. The incremental computation allows the various parameters, dependent on the solution history, to be updated.
65
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
Quadratic for u
Linear for p
a (ii)
Biquadratic for u
Bilinear forp
D)0 (ii)
Linear for u
Linear forp
(ii)
b Linear (with cubic bubble) for u
Linear f o r p
Figure 3.2 Elements used for coupled analysis, displacement (u) and pressure @) formulation (a) (i), quadratic for u; (ii), linear for p; (b) (i), biquadratic for u; (ii), bilinear for p: (c) (i) linear for u; (ii), linear for p: (d)(i), linear (with cubic bubble) for u; (ii), linear for Element (c) is not fully acceptable at incompressible-undrained limits.
Thus for known Au increment the 0'' are evaluated by using an appropriate tangent matrix D and an appropriate stress integration scheme. Further we note that if p,,. 0 (full saturation) then we have
>
and the permeability remains at its saturated value
However, when negative pressures are reached i.e. when p < 0, the values of S,, , x,,,, k , have to be determined from appropriate formulae or graphs. The reader will observe that when p,,. 2 0 we regain the fully saturated behaviour described in Section 2.2 of the previous chapter.
66
FINITE ELEMENT DISCRETIZATION AND SOLUTION
3.2.4 General applicability of transient solution (consolidation, static solution, drained uncoupled, undrained) Time step length As explained in the previous section, the computation always proceeds in an incremental manner-and in the u - p form in general the explicit time stepping is not used as its limitation is very serious. Invariably the algorithm is applied here to the unconditionally stable, implicit, form and the equation system given by the Jacobian of (3.46) with variables Au and Ap needs to be solved at each time step. With unconditional stability of the implicit scheme, the only limitation on the length of the time step is the accuracy achievable. Clearly in the dynamic, earthquake problem short time steps will generally be used to follow the time characteristic of the input motion. In the examples that we shall give later we shall frequently use simply the time interval At = 0.02 s which is the interval used usually in earthquake records. However, once the input motion has ceased and its record no longer has to be followed, a much longer length of time step could be adopted. Indeed after the passage of the earthquake, the remaining motion is caused by something resembling a consolidation process which has a slower response allowing longer time steps to be used. The length of the time step based on accuracy considerations was first discussed in Zienkiewicz et al. (1984), Zienkiewicz and Shiomi (1984), and later by Zienkiewicz and Xie (1991), Bergan and Mollener (1985). The simplest process is that which considers the expansion for such a variable as u given by (3.43) and its comparison with a Taylor series expansion. Clearly for a scalar variable u the error term is given by the first omitted terms of the Taylor expansion i.e. in scalar values
Using an approximation of this third derivative shown below
we have
For a vector variable u we must consider its L2 norm i.e. llul12 = &% etc.
(3.51)
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
67
and we can limit the error to
This limit was re-established later by Zienkiewicz and Xie (1991) who replace the leading coefficient of (3.52) as a result of a more detailed analysis by
Whatever the form of error estimator adopted, the essence of the procedure is identical and this is given by establishing a priori some limits or tolerance which must not be exceeded, and modifying the time steps accordingly. In the above we have considered only the error in one of the variables i.e. u but in general this suffices for quite a reasonable error control. The tolerance is conveniently chosen as some percentage 7 of the maximum value of norm llul12 recorded. Thus we write
with some minimum specified. The time step can always be adjusted during the process of computation noting, however, not to change the length of the time step by more than a factor of 2 or 112 otherwise unacceptable oscillations may arise.
The consolidation equation \
In the standard treatment of consolidation equation (see for instance Lewis and Schrefler, 1998) the acceleration terms are generally omitted a priori. However, there is no disadvantage in writing the full dynamic formulation for solving such a problem. The procedure simply reduces the multiplier of the mass matrix M to a negligible value without influencing in any way the numerical stability, provided of course that an implicit integration scheme is used.
Static problems--undrained and fully drained behaviour Steady state, (static) conditions will only be reached under the extremes of undrained or fully drained behaviour. This can be deduced by rewriting the two, discrete, governing equations (3.23) and (3.28) omitting terms involving time derivatives. The equations now become:
68
FINITE ELEMENT DISCRETIZA TION AND SOLUTION
and
with the effective stresses given by (3.27) and are defined incrementally as dd'
= D(de - de") = D(Bdu
-
de")
(3.57)
First we observe that the equations are uncoupled and that the second of these i.e. (3.56) can be solved independently of the first for the water pressures. Indeed in this solution the negative pressure zones and hence the partially saturated regions can be readily determined following the procedures outlined in the previous chapter. With p,,, determined as
the first equation (3.55) coupled with the appropriate constitutive law (3.57) can be solved once the history of the load applied has been specified. The solution so obtained is of course the well known, drained, behaviour. The case of undrained behaviour is somewhat more complex. We note that with k = 0 i.e. with totally impermeable behaviour
H = 0 and f ( 2 )
=0
(3.59)
But on re-examing equation (3.28) we find that it becomes
which on integration establishes a unique relationship between u and p,, which is not time dependent
assuming that the initial condition of u = 0 and p,,. = 0 coincides. Equation (3.61) now has to be solved together with (3.55). If S = 0 i.e. no compressibility is admitted we have the problem already discussed in the previous section in which only certain u - p,, interpolations are permissible (as shown in Figure 3.2). However, if S # 0 p,, can be eliminated directly and the solution concerns only the variable u. Solving (3.61) for p,, which can only be done provided that some fluid compressibility is available giving S # 0, then (3.55) and the constitutive law are sufficient to obtain the unique undrained condition. The existence of the two steady states is well known and what we have indicated here is a process by which the various matrices given in the original computer program can be used to obtain either of the steady state solutions. However, this does require an
69
A GENERAL GEOMECHANICS FINITE ELEMENT CODE
alternative to the original computer program. However, it is possible to obtain such steady states by the code, using the previous time-stepping procedure. Two types of undrained conditions exist: (a) when k = 0 throughout; (b) k # 0 but the complete boundary is impermeable. Both cases can be computed with no difficulties. Provided that the boundary conditions are consistent with the existence of drained and undrained steady state conditions the time-stepping process will, in due course, converge with
However, this process may be time consuming even if large time steps At are used. A simpler procedure is to use the GNOO scheme with
Equations (3.41) and (3.61) now become, for the undrained problem,
If the material behaviour is linearly elastic, then the equation can be solved directly yielding the two unknowns u,,, and p;+, and if the material is non-linear, an iteration scheme such as the Newton Raphson, Quasi-Newton, Tangential Matrix or the Initial Matrix method can be adopted. With a systematic change of the external loading, problems such as the load-displacement curve of a non-linear soil and pore-fluid system can be traced.
3.2.5 The Structure of the numerical equations, illustrated by their linear equivalent If complete saturation is assumed together with a linear form of the constitutive law we can write the effective stress simply as a" = DBu
(3.62)
70
FINITE ELEMENT DISCRETIZATION AND SOLUTION
We can now reduce the governing u-p equations (3.23) and (3.28) to the form given below
and
where p = p,,. and K
=
1,
B~DB~O
is the well known elastic stiffness matrix which is always symmetric in form. S and H are again symmetric matrices defined in (3.31) and (3.30) and Q is as defined in (3.29) The overall system can be written in the terms of the variable set [ii,plT as
Once again the uncoupled nature of the problem under drained condition is evident (by dropping the time derivatives) giving
in which p can be separately determined by solving the second equation. For undrained behaviour we can integrate the second equation when H = 0 and obtain an anti-symmetric system which can be made symmetric by multiplying the second set equation by minus unity (Zienkiewicz and Taylor, 1985)
It is interesting to observe that in the steady state we have a matrix which, in the absence of fluid compressibility, results in
which only can have a unique solution when the number of ii variables nu is greater than the number of p variables n,. This is one of the requirements of the patch test of Zienkiewicz et al. (1985), Zienkiewicz et al. (1986) and Zienkiewicz and Taylor (1989) and of the Babuska-Brezzi (Babuska, 1973 and Brezzi, 1974) condition.
THE u-U DISCRETIZATION AND ITS EXPLICIT SOLUTION
71
3.2.6 Damping matrices In general, when dynamic problems are encountered in soils (or other geomaterials) the damping introduced by the plastic behaviour of the material and the viscous effects of the fluid flow are sufficient to damp out any non-physical or numerical oscillation. However, if the solutions of the problems are in the low-strain range when the plastic hysteresis is small or when, to simplify the procedures, purely elastic behaviour is assumed, it may be necessary to add system damping matrices of the form Cu to the dynamic equations of the solid phase, i.e., changing (3.23) to
Indeed such damping matrices have a physical significance and are always introduced in earthquake analyses or similar problems of structural dynamics. With the lack of any special information about the nature of damping it is usual to assume the so called 'Rayleigh damping' in which
where cr and p are coefficients determined by experience (see for instance Clough and Penzien, 1975 or 1993). In the above, M is the same mass matrix as given in (3.24) and K is some representative stiffness matrix of the form given in (3.47).
3.3 3.3.1
THE U-UDZSCRETZZA TZON AND ITS EXPLICIT SOL UTZON The governing equation
We shall now return to the original equations of Chapter 2 Section 2.2 without the introduction of the approximation used in Section 3.2.2. Thus (2.1 l), (2.13) and (2.16) are repeated below as:
omitting now only the convective acceleration, density variation and source terms, i.e., the terms underlined in the above equations. Here for brevity the equations are now given only in tensorial notation.
72
FINITE ELEMENT DISCRETIZATION AND SOLUTION
If instead of using the relative velocity w, we introduce the relative displacement
uR,in the manner suggested in (2.22) of Chapter 2 giving
we can integrate, (3.72~)in time and find a direct expression for the pressure provided that the compressibility, l / Q , is different from zero (see (2.17)). Thus we have
Inserting this into (3.72a and b), we obtain the following system rJ,'.
'J
a;,
= 01. .+ ,$..
+ a Q ( ' ~ ~ k+knu&),;
-
(3.75)
VP
pui
-
pf.n ijiR + pbi
=0
(3.76a)
and
The vector form of the above equation is presented in Table 2.1 of Chapter 2. The discretization of the above leads to the equation originally used by Ghaboussi and Wilson (1972). However, this system is inconvenient as second derivatives in time of both variables occur in each of the equations and thus completely diagonal matrices cannot be obtained by mass lumping. A very simple modification can be made here as suggested by Zienkiewicz and Shiomi (1984), which leads to complete matrix decoupling of higher derivatives and which is therefore ideally suited for explicit schemes. Now in place of relative displacements of water, we shall use its total displacement defined as
thus
where the relative displacements are divided by the porosity n, equation (3.73), to approximate the average true displacement. Starting from (3.76), after some algebraic manipulations we have
and
THE u-U DISCRETIZATION AND ITS EXPLICIT SOLUTION
73
This is conveniently rewritten as
and
Multiplying (3.79b) by n and subtracting from (3.79a) we find that the first equation is now free from the acceleration of the fluid displacement and becomes
The second equation (3.79b) is multiplied by n so that symmetry is preserved in the discretized equations.
In the final equation system (3.80) only ii, occurs in the first equation and only the second, thus leading to a convenient diagonal form in discretization.
a in
3.3.2 Discvetized equation and the explicit scheme Approximating the two variables in terms of finite element interpolation, we have, (now including the vector notation)
and
where Nu and N u are the appropriate shape functions. Weighting the first equation (3.80a) with NUTand the second (3.80b) with N ~ 'on~ , integrating by parts over the whole domain, we have the following equation system:
74
3.3.3
FINITE ELEMENT DISCRETIZATION AND SOLUTION
The structure of the numerical equations in linear equivalent (viz. 3.2.5)
Considering each term in the equation system (3.82) in turn: (i) Stiffness Matrix K
where ni are the directions of the normal at the boundary. Linearizing the constitutive equation we have
and making use of the symmetry of shear strains, we have
D ~ i j k l= D ~ i j l k we have
THE U-UDISCRETIZATION AND ITS EXPLICIT SOLUTION
(iii) Stiffness Matrix K2
Note that
(iv) Solid Mass Matrix Ms
(v) Solid Body Force
(vi) Damping Matrix C2
(vii) Damping Matrix Cl
75
76
FINITE ELEMENT DISCRETIZATION AND SOLUTION
(vii) Stiffness Matrix K2 Transpose
(viii) Stiffness Matrix K3
Noting again that
(ix) Damping Matrix C3
(x) Damping Matrix C2 Transpose
THE U-U DISCRETIZATION AND ITS EXPLICIT SOLUTION
77
(xi) Fluid Mass Matrix My N;(-npf
oi)dfl = ,
=
-MflLULi
=
I
~;np~N;dfl6~,
-M~U
(xii) Fluid Body Force
Collecting all the terms and displaying in matrix form which was first presented by Shiomi (1983) and Zienkiewicz and Shiomi (1984):
It is of interest to observe that the mass matrix of the system does not couple variables ti and U and here can be easilty diagonalized. With full symmetry of the system, we can use the generalized Newmark procedure, now applied in terms of combined displacement variable:
for which the whole problem can be written in a simple form of
We recall here the form outlined in Section 3.1. Although, of course, implicit schemes can once again be used here-the explicit operation is of main interest because lumping (by, for instance, nodal integration) allows a very cheap forward marching scheme. This is obtained by putting the following values for y in the original Newmark scheme or P2 for the Generalized Newmark scheme:
for
if
P2
= 0 it disappears
78
FINITE ELEMENT DISCRETIZATION AND SOLUTION
Although the mass matrices can be diagonalized rather simply, if the Newmark parameter p or the Generalized Newmark parameter PI is non-zero then the inclusion of the damping matrix may destroy this diagonal structure. The following two cases can be identified. If anisotropic permeability is used with cross-coupling terms i.e. kv # 0 for i # j, then the damping matrix will assume a block diagonal structure of 4 x 4 for twodimensions and 6 x 6 for three-dimensions if the variables u and U are numbered next to each other. On the other hand, if the cross-coupling terms are all zero, then the block diagonal structure will consist of 2 x 2 matrices whether it is two-dimension or threedimension. In any case, if Rayleigh damping with the full stiffness matrix is used then the diagonal structure will be destroyed, (see Chan et al., 1991), hence only the diagonal contribution of the stiffness matrix is included in the solution matrix while the full matrix is retained in the right hand side during the calculation of the residuals. Having dealt with the problem raised by the Rayleigh damping, the off-diagonal term on the solution matrix can also be removed by further lumping (see, for instance, Simon et al., 1986). One of the major constraints on an explicit scheme is the limitation of the explicit time step. However, for many non-linear and transient problems, the use of a small time step is a positive advantage as the non-linearity may impose a large number of iterations in typical implicit schemes. In the case when the fluid bulk modulus Kfis much bigger than the bulk modulus of the soil matrix KT the critical time step is found to be:
where e is the minimum length between nodes and C1 is a constant depending on the type of element used (and such other factors as porosity). Therefore by a suitable reduction in the value of Kf, an incrase in the critical time step length is possible and this was shown in Chan et al. (1991). Generally, the results are not adversely affected until the value of Kfbecomes comparable to the bulk modulus of soil matrix K T .
3.4
THEORY: TENSORZAL FORM OF THE EQUA TZONS
The equation numbers given here correspond to the ones given earlier in the text.
Noting that the engineering shear strain dy,,. is defined as: dy,. = 2 d ~ , . Equation (3.10) is scalar
79
TENSORIAL FORM OF THE EQUATIONS =
(5 - (5..
(5"
IJ
(3.11b)
v
= 0.. + ~16.. 11
(3.12b)
!IP
Equation (3.13) is scalar. Equation (3.14) is scalar.
Equation (3.16) is scalar
and n;w;
= n;kii(-p
,,.,
,,+ S,,.p/-6,)= w,
on I? = I?,,.
and assuming isotropic permeability
The summation range for the upper case indices will depend on the number of nodes with solid displacement and pore water pressure degrees-of-freedom (dof) respectively.
Applying Green's identity to the internal force term (first term on the left-hand side)
80
FINITE ELEMENT DISCRETIZATION AND SOLUTION
Rearranging NZjordR
+[
I,
N;pN;dR]GLi
=
1
NipbidR +
0
1
rr
N;iidT.
(3.20b and 3.22b)
The definition of the B matrix in equation (3.21) is not needed in tensorial form. 0..lJ-
MKLsL;+
1,
o'llJ.
N;, ,$dR
1 (~ do; = D i j ~ i(N;,kd",
1,
N; (kl(-pw,
-
ax6.. d'
-
~ ~ , ~fi) f I0 r -
+ NtidiiKk)
-
=
(3.23b)
d&li)
(3.27b)
, + Sivpfbj),; + a&,;+ P< + i0)dR = 0 Q
Neglecting source term and integrating by part the first part of the first term -
Ira
N$nikvp.,,jdT.
+
1,
+
+
I
b , , k u p M . , , N$(kgS,vpfb,),i Niazi,; + N g f i dR = 0
Inserting the shape functions
Q* .
CONCLUSIONS
81
equation (3.33) is scalar.
3.5
CONCLUSIONS
In this chapter, the governing equations introduced in Chapter 2 are discretized in space and time using various implicit and explicit algorithms. They are now ready for implementation into computer codes. In Chapters 5-7, we shall show some applications for static, quasi-static and dynamic examples to illustrate the practical applications of the method and to validate and verify the schemes and constitutive models used.
REFERENCES Babuska I. (1971) Error Bounds for finite element methods, Num. Math., 16, 322-333. Babuska I. (1973) The finite element method with Lagrange Multipliers, Num. Math., 20, 179-192. Bellman R. (1960) Introduction to Matrix Analysis (1st edn), Mcgraw-Hill Book Company, London. Bergan P. G . and Mollener (1985) An automatic time-stepping algorithm for dynamic problems, Comp. Meth. Appl. Mech. Eng. 49, 299-318. Brezzi F. (1974) On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, R. A. I. R. 0 . Anal. Numtr., 8, No. R-2, 129-151. Chan A. H. C. (1988) A unqied Finite Element Solution to Static and Dynamic Geomechanics problems, Ph.D. Dissertation, University College of Swansea, Wales. Chan A. H. C. (1995) User Manual for DIANA SWANDYNE-11, School of Civil Engineering, University of Birmingham, December, Birmingham. Chan A. H. C., Famiyesin 0 . 0. and Muir Wood D. (1991) A Fully Explicit u-w Schemes for Dynamic Soil and Pore Fluid Interaction, APCOCM Hong Kong, 11-13 Dec., 1, 881-887. Clough R. W. and Penzien J. (1975) Dynamics of Structures, McGraw-Hill, New York. Clough R. W. and Penzien J. (1993) Dynamics of Structures (2nd edn), McGraw-Hill, Inc., New York.
82
FINITE ELEMENT DISCRETIZATION AND SOLUTION
Crisfield M. A. (1979) A faster modified Newton-Raphson Iteration, Comp. Meth. Appl. Mech. Eng., 20, 267-278. Crisfield M. A. (1991) Non-linear Finite Elenzent Analysis of Solid.7 and Structures, 1, John Wiley & Sons, Chichester. Crisfield M. A. (1997) Non-linear Finite Element Analysis of Solids and Structures, 2. John Wiley & Sons, Chichester. Dahlquist G. (1956) Convergence and stability in the numerical integration of ordinary differential equations, Math. Scnnd., 4, 33-53. Dahlquist G. (1959) Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Teknisku Hogskolans Handlingar, 130. Dahlquist G. (1978) On accuracy and unconditional stability of linear multistep methods for second order differential equations, BIT, 18, 133-1 36. Dewoolkar M. M. (1996) A study of seismic effects on centiliver-retaining ~valls with saturated backfill. PhD Thesis, Dept of Civil Engineering, University of Colorado, Boulder, USA. Gantmacher F. R. (1954a) Applications of the theory of matrices: Second part of ' A Theory o f Matrices', translated and revised by Brenner J. L. et al., Interscience Publishers, Inc. Gantmacher F. R. (1954b) The Theory of Matrices-English translation by Hirsch K. A, 1, Chelsea Publishing Company. Gantmacher F. R. (1954~)The Theory of'Matrices-English translation by Hirsch K. A. 2, Chelsea Publishing Company. Ghaboussi J. and Wilson E. L. (1972) Variational formulation of dynamics of fluid saturated porous elastic solids, ASCE E M , 98, No. EM4, 947-963. Hurwitz A. (1895) ~ b e die r Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen realen Teilen besitzt, Math. Ann., 46, 273-384. Hurwitz A. (1933) ~ b e die r Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen realen Teilen besitzt, Mathematische Werke, Basel, 2, 533-545. Katona M. G. (1985) A general family of single-step methods for numerical time integration of structural dynamic equations, NUMETA 85, 1, 213-225. Katona M. G. and Zienkiewicz 0.C. (1985) A unified set of single step algorithms Part 3: The Beta-m method, a generalisation of the Newmark scheme, Int. J. Num. Meth. Eng., 21, 1345-1 359. Lambert J. D. (1973) Computational Methods in Ordinary Differential Equutions, John Wiley and Sons Ltd., Chichester. Leung K. H. (1984) Earthquake response of saturated soils and liquefaction, Ph.D. Dissertation, University College of Swansea, Wales. Lewis R. W. and Schrefler B. A. (1998) The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, John Wiley & Sons, Chichester. Li X. K. and Zienkiewicz 0. C. (1992) Multiphase flow in deforming porous media and finiteelement solutions, Comp. Struct., 45, No. 2, 21 1-227. Li X., Zienkiewicz 0. C. and Xie Y. M. (1990) A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution, Int. J. Num. Meth. Eng., 30, 1195-1212. Matthies H. and Strang G. (1979) The solution of nonlinear Finite Element equations, Int. J. Num. Meth. Eng., 14, 1613-1626. Newmark N. M. (1959) A method of computation for structural dynamics, Proc. ASCE, 8, 67-94. Park K. C. (1983) Stabilization of partitioned solution procedure for pore fluid-soil interaction analysis, Int. J. Num. Meth. Eng., 19, 1669-1673. Park K. C. and Felippa C. A. (1983) 'Partitioned analysis of coupled systems', Chapter 3 in Computational Methods,for Transient Analysis, Elsevier Science Publishers B. V.
83
REFERENCES
Paul D. K. (1982) Efficient dynamic solutions for single and coupled multipled field problems. Ph.D. Dissertation, University College of Swansea, Wales. Routh E. J. (1877) Stability of a given state of motion-the advanced part of a treatise on the dynamics of a system of rigid bodies (6th edn), Dover, London. Routh E. J. (1930) Stability of a given state of motion-the advanced part of a treatise on the dynamics of a system of rigid bodies, Dover, London. Routh E. J. (1955) Stability of a given state of motion-the advanced part of a treatise on the dynamics of a system of rigid bodies, Dover, New York. Schrefler B. A. (1995) Finite Element in Environmental Engineering: Coupled thermo-hydromechanical process in porous media involving pollutant transport, Archives of Computational Methods in Engineering, 2, 1-54. Schrefler B. A. and Zhan X. (1993) A fully coupled model for waterflow and airflow in deformable porous media, Water Resources Res., 29, No. 1, 155-167. Shiomi T. (1983) Nonlinear behaviour of soils in earthquake-C/Ph/73/83, Ph.D. Dissertation, Univ. Coll. of Swansea, Wales. Simon B. R., Wu J. S. S., Zienkiewicz 0 . C. and Paul D. K. (1986) Evaluation of u-w and u-p FEM for the response of saturated porous media using I-dimensional models, Int. J. Nunz. Anal. Geomech., 10, 461482. Whitman R. V. (1953) After Marcusion (1995): an example ofprofessional modesty, The Earth. Engineers and Education, MIT, 200-202. Wood W. L. (1984a) A further look at Newmark, Houbolt, etc.. time-stepping formulae, Int. J. Num. Meth. Eng., 20, 1009-1017. Wood W. L. (198413) A unified set of single step algorithms Part 2: Theory, Int. J. Num. Meth. Eng., 20, 2303-2309. Wood W. L. (1985a) Addendum to 'a unified set of single step algorithms, Part 2: Theory', Int. J. Num. Meth. Eng., 21, 1165. Wood W. L. (1985b) A unified set of single-step algorithms Part 4: Backward error analysis applied to the solution of the dynamic vibration equation-Numerical Analysis Report 6185, Department of Mathematics, University of Reading, Reading. Wood W. L. (1990) Practical Time-stepping Schemes, Clarendon Press, Oxford. Zienkiewicz 0. C. (1977) The Finite Element Method (3rd edn), McGraw-Hill Book (UK) Ltd, London. Zienkiewicz 0 . C., Hinton E., Leung K. H. and Taylor R. L. (1980a) Staggered, Time Marching Schemes in Dynamic Soil Analysis and Selective Explicit Extrapolation Algorithms in Proceedings of the Second Symposium on Innovative Numerical Analysis for the Engineering Sciences, University of Virginia Press. Zienkiewicz 0 . C., Wood W. L. and Taylor R. L. (1980b) An alternative single-step algorithm for dynamic problems, Earthquake Engineering & Structural Dynamics, 8, 31-40. Zienkiewicz 0 . C., Leung K. H., Hinton E. and Chang C. T. (1982) Liquefaction and permanent deformation under dynamic conditions Numerical solution and Constitutive relations, Chapter 5 in Soil Mechanics- Transient and Cyclic loads, John Wiley, Chichester. Zienkiewicz 0 . C. (1984) 'Coupled Problems and their Numerical Solution,' Chapter 1 in Numerical Methods in Coupled Systems, John Wiley and Sons Ltd., Chichester. Zienkiewicz 0 . C. and Shiomi T. (1984) Dynamic Behaviour of saturated porous media: The generalized Biot formulation and its numerical solution, Int. J. Num. Anal. Geomech.. 8. 71-96. Zienkiewicz 0 . C., Wood W. L., Hine N. W. and Taylor R. L. (1984) A unified set of single step algorithms Part I: General formulation and applications, Int. J. Num. Meth. Eng., 20. 1529-1552. Zienkiewicz 0 . C. (1985) The coupled problems of soil-pore fluid-external fluid interaction: Basis for a general geomechanics code, ICONMIG 5, 1731-1 740. -
84
FINITE ELEMENT DISCRETIZATION AND SOLUTION
Zienkiewicz 0. C. and Taylor R. L. (1985) Coupled Problems - A simple time-stepping procedure, Comm. Appl. Num. Meth., 1, 233-239. Zienkiewicz 0. C., Taylor R. L., Simo J. C. and Chan A. H. C. (1986a) The Patch Test - A condition for assessing F. E. M. Convergence, In?. J. Num. Meth. Engrg., 22, 39-62. Zienkiewicz 0. C., Qu S., Taylor R. L. and Nakazawa S. (1986b) The Patch Test for Mixed Formulation, Int. J. Num. Meth. Eng., 23, 1873-1883. Zienkiewicz 0. C., Paul D. K. and Chan A. H. C. (1988) Unconditionally stable staggered solution procedure for soil pore fluid interaction problems, Int. J. Num. Meth. Eng., 26, NO 5, 1039-1055. Zienkiewicz 0. C., Chan A. H. C., Pastor M., Paul D. K. and Shiomi T. (1990a) Static and dynamic behaviour of geomaterials: A rational approach to quantitative solutions, Part I: Fully saturated problems, Proc. Roy. Soc. Lond., A429, 285-309. Zienkiewicz 0. C., Xie Y. M., Schrefler B. A,, Ledesma A. and Bicanic N. (1990b) Static and dynamic behaviour of soils: a rational approach to quantitative solutions, Part 11: Semisaturated problems, Proc. Roy. Soc. Lond., A429, 3 10-323. Zienkiewicz 0.C. and Taylor R. L. (1989) The Finite Element Method - Volume I : Basic Formulation and Linear Problems (4th edn), McGraw-Hill Book Company, London. Zienkiewicz 0. C. and Taylor R. L. (1991) The Finite Element Method - Volume 2: Solid and Fluid Mechanics, Dynamics and Non-linearity (4th edn), McGraw-Hill Book Company, London. Zienkiewicz 0.C. and Xie Y. M. (1991) A Simple error estimator for adaptative time stepping procedure in dynamic analysis, International Journal for Earthquake and Structural Dynamics, 20, 871-887.
Constitutive Relations Plasticity
4.1 INTRODUCTION The purpose of constitutive models is to capture some of the main features of the mechanical behaviour of solids under given conditions of temperature, velocity of load application, level of strain, nature of stress conditions, etc. Roughly speaking, models can be classified into two main groups: (i) Micromechanical or physical models, based on the behaviour of grains or particles, and (ii) Macromechanical or phenomenological models. Most of the models used in computer codes are of the second class. All materials present a response which depends on time to a greater or lesser degree. For instance, a specimen of soft clay subjected to a constant vertical stress shows a vertical deformation which increases monotonically with time. However, most of the geomaterials under normal engineering conditions present a mechanical behaviour which depends more on the level of stress, pore pressure, past history, direction of load increment and material structure than on time. In fact, the major part of the time dependence observed is generally connected with the pore water flow. For these, plasticity based theories provide a consistent framework in which the behaviour can be accurately understood and predicted. It can be said that history of plasticity began in 1773 with the work of Coulomb in soils, applied later by Poncelet and Rankine to practical soil mechanics problems. It was not until almost a century later, in 1864, when Tresca, based on experimental results on punching and extrusion tests, proposed a yield criterion dependent on the maximum shear stress. Later, St. Venant in 1870 introduced the concept of
86
CONSTITUTIVE RELATIONS
-
PLASTICITY
isotropic flow rule, which was generalized by Levy to three-dimensional conditions. The principal axes of stress and the increment of strain were assumed to be the same. The next significant step had to wait until the beginning of the new century, when von Mises and Huber, independently, proposed a new yield criterion which, with the flow equations derived by Levy, became known as the Levy-Von Mises equations. There, no distinction between total, elastic and plastic parts of the strain increment was implied. The decomposition between elastic and plastic parts was introduced for plane stress conditions by Prandtl in 1924 and later, in 1930 by Reuss for general conditions of stress. Reuss proposed a flow rule for the plastic component. The idea of plastic potential was suggested by von Mises in a work presented in 1928, where the normal to the yield surface was used to provide the direction of plastic flow. Hardening plasticity was studied by Melan, who in 1938 generalized previously established concepts of plasticity to account for this effect. However, the framework of what is known today as Classical Plasticity was established in 1949 by Drucker, who introduced many of the concepts of modern plasticity such as loading surface, loading and unloading, neutral loading, consistency and uniqueness. Since then, much development has taken place, motivated by the development of both faster computers and numerical methods for boundary value problems. There exist today a great variety of models able to deal with most of the observed features of the mechanical behaviour of materials. The approach which will be followed here will, first of all, introduce a general framework within which most models can be cast (section 4.2), then deal with Classical Plasticity formulation in Section 4.3 following with the description of advanced models capable of showing liquefaction phenomena in Section 4.4.
4.2
4.2.1
THE GENERAL FRAMEWORK OF PLASTICITY Phenomenological aspects
The uniaxial behaviour of materials shows that irreversible strain develops in a way which depends on the type of material. In the case of metals such as mild steel, the observed behaviour in tension is schematized in Figure 4.1, where it can be seen that the response is elastic and linear until a point A is reached, from which plastic or irreversible strain upon unloading appears. If the specimen is subjected to an increasing strain, the stress does not change until point E. Along the plateau ABDE, the material behaviour is known as perfectly plastic. If the specimen is unloaded, both loading and unloading follow the same path, without irreversible deformation. The level of stress at which plastic strains appears does not change, and the material does not harden. Once a certain level of strain has been reached (E), the stress again increases. If we unload at some point F and then reload again, the material is able to resist a higher load until new plastic strains develop (hardening). Finally, a maximum load is reached from which the stress decreases until the material fails. In the case of soft soils such as saturated clays, the stress-strain curve is different, as plastic strain are present from very early stages of the test (Fig. 4.2).
87
THE GENERAL FRAMEWORK OF PLASTICITY
Stress
t
0
Strain
Figure 4.1
Stress
Behaviour of mild steel
I
0
Strain
Figure 4.2 Behaviour of soft clay
Finally, some geomaterials such as concrete present degradation due to damage caused by the loading process to the structure of the material (Fig. 4.3). Loading and unloading shows clearly how the aparent elastic modulus of the material degrades as the test progresses. Full understanding of this behaviour needs to take into account this process of degradation, and theories such as Damage Mechanics provide a suitable framework. However, plastic models can be developed to reproduce the observed behaviour with an acceptable degree of accuracy.
4.2.2
Generalized plasticity
In the following, boldface characters will be used for tensors, uppercase (such as D) denoting fourth-order tensors Dijkl,and lower case (such as a) for second-order tensors a,-.
88
CONSTITUTIVE RELATIONS
0
-
PLASTICITY
Strain
Figure 4.3 Behaviour of materials with damage
It is convenient to use a vector-matrix representation of tensorial magnitudes in numerical computations; fourth-order tensors corresponding to matrices and secondorder tensors to vectors. In Chapters 2-3 we have introduced this notation. We shall here indicate the small alteration necessary to return to the matrix notation used in the previous chapters where a and D are vectors and matrices respectively. The convention for products and its matrix equivalence is:
(a) Double dot denotes contracted product in last two indexes
(b) Tensor products are expressed by
The behaviour of geomaterials depends on effective stresses as shown in Chapter 1, which are denoted by a dash. However, in the first part of this chapter, devoted to the Introduction to Elastoplastic Constitutive Equations, we will not use the dash when referring to stress for the sake of simplicity.
Basic theory If the response of the material does not depend on the velocity at which the stress varies the relationship between the increments of stress and strain can be written as
THE GENERAL FRAMEWORK OF PLASTICITY
89
where Q, is a function of the increment of the stress tensor d u and variables describing the 'state' (or history) of the material. This is a general relation embracing most nonlinear, rate-independent constitutive laws. An inverse form is
As the material response does not depend on time, Xde
= @(Ada)
where X E %+ is a positive scalar (Darve 1990). Consequently, is a homogeneous function of degree 1, which can be written as
from which the increments of stress and strain are related by
where
is a fourth-order tensor, homogeneous, of degree zero in d u . Before continuing, some basic properties of C will be described. We will consider a uniaxial loading-unloading-reloading test schematized in Figure 4.4 where the constitutive tensor C is a scalar, the inverse of the slope at the point considered. As can be seen, the slope depends on the stress level, being smaller at higher stresses. However, if we compare the slopes at points A , , A2 and A3, they are not the same, and C depends on past history (stresses, strains, modification of material microstructure, etc.) Taking a closer look at point C, it can be seen that, for a given point, different slopes are obtained in 'loading' and 'unloading', which implies a dependence on the direction of stress increment. This dependence is only on the direction, as C is a homogeneous function of degree zero on d u . Therefore, in this simple one-dimensional case, it is possible to write for loading
90
CONSTITUTIVE RELATIONS
/
Figure 4.4
-
PLASTICITY
Strain
General stress-strain behaviour
and for unloading
We observe that if we consider an infinitesimal cycle with d a followed by - d a , the total change of strain is not zero as
This kind of constitutive law has been defined by Darve (1990) as incrementally non-linear. There are several alternatives to introduce the dependence on the direction of the stress increment, among which it is worth mentioning the multilinear laws proposed by Darve and co-workers in Grenoble (Darve and Labanieh, 1982), or the hypoplastic laws of Dafalias (1986) or Kolymbas (1991). However, the simplest consists of defining in the stress space a normalized direction n for any given state of stress a such that all possible increments of stress are separated into two classes, loading and unloading deL = CL : d a for n : d a > 0 (loading) dev = Cv : d a for n : d a < 0 (unloading)
(4.10)
Neutral loading corresponds to the limit case for which
This is the starting point of the Generalized Theory of Plasticity, introduced by Zienkiewicz and Mroz (Mroz and Zienkiewicz, 1985 and Zienkiewicz and Mroz, 1985) and later extended by Pastor and Zienkiewicz (Zienkiewicz, Leung and Pastor 1985, Pastor Zienkiewicz and Leung 1985, Pastor, Zienkiewicz and Chan 1990).
THE GENERAL FRAMEWORK OF PLASTICITY
91
Introduction of this direction discriminating between loading and unloading defines a set of surfaces which is equivalent to those used in Classical Plasticity as will be shown later, but these surfaces need never be explicitly defined. Continuity between loading and unloading states requires that constitutive tensors for loading and unloading are of the form
and
where n , ~and ngU are arbitrary tensors of unit norm and H L j u two scalar functions defined as loading and unloading plastic moduli. It can be very easily verified that both laws predict the same strain increment under neutral loading where both expressions are valid and hence non-uniqueness is avoided. As for such loading, the increments of strain using the expressions for loading and unloading are
and
It follows that material behaviour under neutral loading is reversible, and it can therefore be regarded as elastic. Indeed, the tensor Ce characterizes elastic material behaviour, and it can be very easily verified that for any infinitesimal cycle of stress (do, -du) where d u corresponds to neutral loading conditions, the accumulated strain is zero. This suggests that the strain increment can be decomposed into two parts
where
and
We note that irreversible plastic deformations have been introduced without the need for specifying any yield or plastic potential surfaces, nor hardening rules. All
92
CONSTITUTIVE RELATIONS
-
PLASTICITY
that is necessary to specify are two scalar functions HLluand three directions, ngLlu and n. To account for softening behaviour of the material, i.e., when HL is negative, definitions of loading and unloading have to be modified as follows: dcL = CL : d u for n : d u e > 0 (loading) dew = C u : d u for n : d u e < 0 (unloading)
(4.17)
where du' is given by
We note here (and in what follows) that in matrix notation the product forms are written simply as deL = CL.du for; nT.du' > 0 etc.
Inversion of the constitutive tensor Implementation of a constitutive model into finite element codes requires on many occasions an inversion of the constitutive tensor in order to express the increment of stress as a function of the strain increment. This inversion can only be automatically performed when the plastic modulus H i s different from zero. Should this not be the case, inversion would have to be carried out according to the procedure described below (Zienkiewicz and Mroz, 1985). First of all, a scalar X is introduced
and the increment of strain is written as
Both sides of above equation are now multiplied by n: D' n : D' : de
=
(n : D') : (C' : d u )
+ (n : D')
: dXngLlu
(4.21)
from which we obtain
where we have taken into account that the product De : C' is the fourth-order identity tensor. Substituting now
93
THE GENERAL FRAMEWORK OF PLASTICITY
we obtain n : D'
: d~ =
+
(HLIU n : D'
: ngLlu)dX
(4.24)
and
If we now multiply by D' both sides of d~ = Cr : d u
+ dXngLlu
we have, du
= D' : d~ - dXD' : ~ , L , u
Substitution of the value of dX, gives du
= D' : d~ -
(n : Dr : d ~ ) (D' : ngLIU) HL/U n : D' : ~ , L / u
+
which can be written as du
= Del' : d~
If we make use of the vectorial formulation to represent tensors, the above expression can be written as
4.2.3
Classical theory of plasticity
Formulation as a particular case of generalized plasticity theory Classical Plasticity Theory can be considered as a particular case of the Generalized Theory described above by a suitable choice of the plastic modulus, directions n and ngLlu and the elastic constitutive tensor.
94
CONSTITUTIVE RELATIONS
-
PLASTICITY
A yield surface is first introduced as
where we have assumed that there is a set of scalar internal variables K accounting for the material state and characterizing the size (and shape) of the yield surface. Sometimes, as will be discussed later, f depends also on the tensor variable a , as in the case of kinematic and anisotropic hardening models, for instance. Here we will restrict the discussion to the isotropic case stated above. In the interior of the yield surface, there is no plastic deformation, and, consequently, the plastic modulus is H = m. The loading-unloading direction is given by the normal to the surface
where
The direction of plastic flow is similarly derived from a plastic potential surface g(u) = 0 passing through the stress point considered,
Both surfaces can coincide, and the flow rule is then said to be associative, or can be different in which case there is a non-associative flow rule. Therefore, the material behaviour predicted by Classical Plasticity models presents a sharp transition from the elastic to the elastoplastic regime, with a discontinuity in the derivative of stress-strain curves. The plastic modulus is obtained through application of the so called 'consistency condition', i.e., the requirement that during yield the stress point should always remain on the yield surface. A certain 'hardening law' has to be introduced, relating d~ to either incremental plastic work or to the increment of plastic strain.
Yield and failure surfaces Following experimental evidence, plasticity theories postulate that irreversible or plastic strain appears whenever the stress reaches a surface f (av,K ) = 0. For all
THE GENERAL FRAMEWORK OF PLASTICITY
95
stress states in the interior of this surface, material behaviour is elastic and K is constant, the material cannot sustain a higher stress and failure takes place. This is the reason why the yield surface is also known as the failure surface. Care should be taken, however, as in the case of materials with hardening, these surfaces can be different. The scalar K usually characterizes the size of the surface. This is, of course, a simplification, and more complex descriptions are available, such as
f (au,K ) < 0. If
If the material is isotropic, the representation theorems of scalar functions of tensor variables allows a simpler expression for f
which can be further simplified to
where Y ( K ) is generally some measure of strength. II is the first invariant of the stress tensor, 11 = 01
+ + 02
03
= u,,
JZ and J3 the second and third invariants of the deviatoric stress tensor s,
and a , , 02, a3 the three principal stresses.
(4.38)
96
CONSTITUTIVE RELATIONS
-
PLASTICITY
At this stage it is convenient to define also the Lode's angle 19often used instead of J 3 .
with
Hardening, softening and failure It is important to distinguish between the yield surface, inside which behaviour of the material is elastic, and the failure surface, where failure takes place. To illustrate this, consider the example given in Figure 4.5 where a specimen of soft clay is being loaded from an initial state PI to failure at P3.There, yield surfaces are the ellipsesf - ( t i ) = 0. The parameter ti in this case is associated with the (negative) plastic volumetric strain, i.e. dti
=
-d~,,
(4.43)
and in Figure 4.5 we show the yield surface in the space of two stress invariants, the second or deviatoric invariant and the first or mean, hydrostatic stress invariant. With each of these is associated an appropriate strain component -n,, being the component in the direction of decreasing volumetric strain if plasticity is assumed associated. Thus the three stages of loading P I ,P2 and P3 correspond to increasing values of ti as shown in Figure 4.5 (b). It has to be noticed that plastic strain appears from the beginning of the test, as the initial stress is on the yield surface. If, for instance, we unload at P2, there will exist a permanent deformation even when the stress has come back to the original state. Deviatoric stress
Deviatoric stress
Axial strain (b)
Figure 4.5 Typical hardening behaviour of clay. (a) Yield surfaces (b) Stress-strain curve showing permanent strain upon unloading
97
THE GENERAL FRAMEWORK OF PLASTICITY f
t Deviatoric stress
Deviatoric stress
-Hydrostatic stress
Axial strain
(a)
('J)
Figure 4.6 Ideal Plasticity ( K = constant). (a) Stress path
Deviatoric stress
(b) Stress-strain curve
Deviatoric stress
*
/
Axial strain
-Hydrostatic stress (a)
Figure 4.7 Softening behaviour. (a) Stress path
(b)
(b) Stress-strain curve
The process of increasing the size of the yield surface in this case is known as hardening. Comparing the conditions at PI and Pz, the elastic domain is bigger in the latter, and the material is harder in this sense. Notice that slopes of the stress-strain curves contradict this definition, as the incremental response of the material is harder in the first case. Hardening is not a common feature of all materials. Indeed, in the case shown in Figure 4.6, the size did not change and failure takes place as soon as the yield surface is reached. In another loading case, the size of the yield surface may decrease, as shown in Figure 4.7, and softening behaviour occurs.
Some frequently used failure and yield criteria. Pressure independent criteria : von Mises-Huber yield criterion von Mises yield criterion assumes that plastic strain appears whenever the second invariant of the stress tensor reaches a critical value Y * ,
98
CONSTITUTIVE RELATIONS - PLASTICITY
where Y(K) is generally the tensile strength. Alternative expressions are (i) In the principal stress space
(ii) In general stress conditions
Taking into account that the condition J2 = constant, corresponds to stress states a1 ,az, a3 such that the distance to the hydrostatic axis a1 = a2 = a3 is constant, von Mises criterion is represented in principal stress axes as a cylinder of radius = &!Y which is schematized in Figure 4.8(a). In the same figure we show a plane perpendicular to the hydrostatic axis, which is referred to as the II plane. Its intersection with the von Mises cylinder is a circle, which is shown in Fig.4.8(b). A simple method of determining the constant Y is to perform a tension test a2 = a3 = 0 and to determine the instant at which plastic strain develops. If the value of limiting tensile stress is a y then we obtain
a
from which
Figure 4.8
ll plane
von Mises - Huber yield criterion. (a) In the principal stress space (b) Section by
99
THE GENERAL FRAMEWORK OF PLASTICITY
Figure 4.9
Von Mises criterion for plane stress conditions
In plane stress conditions, a3 stress axes is
= 0,
and the expression of the criterion in principal
which corresponds in the al, a2 axes to an ellipse with principal axes at 45" (Figure 4.9).
Tresca criterion The Tresca criterion, proposed in 1864, is based on the assumption that plastic straining of a material appears when the maximum shear strain reaches a critical value Y. This condition, expressed in terms of the principal stresses reads (urnax - ffrnin)
=
Y
(4.50)
Substituting now the maximum and minimum principal stresses by their values in terms of the invariants I , , J2 and Lode's angle 8
Noting that
100
CONSTITUTIVE RELATIONS
Figure 4.10
-
PLASTICITY
Tresca Yield criterion. (a) In principal stress axes (b) In the Il plane
we can write finally
When plotted in the space of principal stresses, the Tresca yield criterion is a hexagonal prism, with its axis coincident with the hydrostatic axis a , = 02 = a3 (Figure 4.10a). The section by the II-plane is a regular hexagon as can be seen in Figure 4.10(b). Finally, the plane stress condition a2 = 0 is represented by
which are shown in Figure 4.1 1.
Pressure dependent criteria : Mohr-Coulomb surface In 1773 Coulomb proposed the law T
= C.
-
a,, tan q5
(4.55)
to describe the conditions under which failure takes place in soils. He assumed that failure occurs on a plane on which the shear stress T , and the normal stress a,7 (compression negative) fulfill the above condition. Although it is not advisable to think of it as a yield surface, it has been used frequently in engineering practice, and most finite element codes include it.
T H E GENERAL FRAMEWORK OF PLASTICITY
In terms of principal stresses or invariants, we will write
and
which can be obtained from geommetrical considerations (Figure 4.12.) This results in
Figure 4.11 Tresca criterion for plane stress conditions
Figure 4.12 Mohr-Coulomb law
101
102
CONSTITUTIVE RELATIONS
Figure 4.13
-
PLASTICITY
Mohr-Coulomb yield surface
and
From above, using the relationships between principal stresses and invariants, it is easy to obtain
The Mohr-Coulomb criterion is represented in the space of principal stresses as a hexagonal pyramid, which has been depicted in Figure 4.13. Drucker-Prager criterion The Drucker-Prager criterion is an attempt to create a smooth approximation to the Mohr-Coulomb surface in the same manner as von Mises approximates Tresca. The surface is written as
and. when plotted in the space of principal stresses, consists of a cone in which the axis is coincident with the hydrostatic axis. The section of this cone through the II plane is a circle and when plotted in the mean hydrostatic pressure-deviatoric stress plane, the intersection with it consists of two lines with identical slope (compression and extension). Therefore, the friction angles corresponding to compression and extension are different, and, in fact, given the parameter a and a value of Lode's angle 19,if the intersections of the DruckerPrager cone and the Mohr-Coulomb surfaces are to coincide for a certain value of Lode's angle 8,the relationship between the friction angle and a is
THE GENERAL FRAMEWORK OF PLASTICITY
103
A similar relationship between cohesion and the parameter Y can easily be obtained as
These relationships have to be taken into account when trying to use the DruckerPrager criterion for plane strain conditions and what is known from experiments is cohesion and angle of friction. Under cylindrical triaxial conditions, i.e., uT = ( u l ,uz = u3)the angles of friction in compression and extension are different, and can be obtained using the above relationship with 0 = 7r/6 and 0 = -7r/6 ac =
2sin4 cos 2 - 1sin 2 sin4
JS
from where
and
In a similar way,
The values of Y for compression and extension are
and
104
CONSTITUTIVE R E L A T I O N S PLASTICITY
Finally, it can be seen that, for a given value of a, the relationship between angles of friction in extension and compression is
It is left to the reader as an exercise to demonstrate that there is a value of sin& for which the friction angle in extension reaches 7r/2. It is interesting to mention that 'rounded' Mohr-Coulomb surfaces have been proposed in the past (Zienkiewicz and Pande, 1977), in which the slope M is assumed to vary as:
In this way, the yield surface is smooth. and coincides with the Mohr-Coulomb original surface in both triaxial compression and extension conditions.
Consistency condition for strain hardening materials If we assume that the material hardening is of 'strain' type, there will exist a law relating the increments of K and E
Assuming that yielding occurs on the yield surface given by
and hence
This can be rewritten using (4.71) as
If. in the above expression, we substitute dd' it gives
105
THE GENERAL FRAMEWORK OF PLASTICITY
This expression can be further developed to
from which the plastic modulus is finally obtained as
Using the alternative vector notation, the above expression is written as
where d t i / d ~is~ a) square matrix. Local failure conditions or continuing deformation at a constant stress state can happen whenever H L = 0, which corresponds to:
for which either of the following conditions have to be fulfilled
dn dg = 0 with (b)--. ~ E Pd u
dn # 0 ddl
--
Computational aspects Most of the expressions given in the above sections simplify in the case of isotropic materials, as all necessary items can be defined in terms of stress invariants. The yield surface, for instance, can be expressed as
106
CONSTITUTIVE RELATIONS
-
PLASTICITY
or in terms of another alternative set of invariants. The constitutive tensor DePwhich appears in non-linear finite element computations can be expressed, as it was shown above, as a function of directions n and n,, and the scalar H, all of them dependent on the invariants. However, what it is needed in computations is the general three-dimensional form. Therefore, the expressions have to be transformed from the space of invariants to the general 3D space. In the case of the Classical Theory of Plasticity, the constitutive tensor DeP is written in vector notation as
where we have introduced
having dropped, for simplicity, sub-indexes L/ U referring to loading and unloading. This is precisely the Generalized Plasticity expression (4.28) with
n
af
=-
du
and n,
=
ag
A simple way to obtain either gradient in terms of invariants I , , J2 and 0 is the following
where
3d3
J3
sin 38 = - -2 5,312 Differentiating this expression, we arrive at
THE GENERAL FRAMEWORK OF PLASTICITY
107
and, from here,
Taking into account now that
we have, finally,
which can be written in a more compact way as:
+
n = C~nl C m
+C m
where
and
It can be seen that the set of constants { C ; )depends on the yield criterion chosen, being independent of the vectors ni. Next, we will obtain the explicit form of nl, n2 and n3.
108
Vector
CONSTITUTIVE RELATIONS nl
is given by
nl
811 da,, - d(aikSki) =6/ .! 6kri--6. - r/ au a ~ ! ~doji
= = - -
or.
d --{a\-
aa,
+a,. + a,)
from which, in vector notation,
To obtain vector nl we will use tensor notation
Taking into account that
we arrive at
and
=1
-
PLASTICITY
THE GENERAL FRAMEWORK OF PLASTICITY
109
from which it follows that
Therefore,
Vector n3 is given by
where 1 J3 = j~ i j ~ j k ~ k i
After some algebra, the final expression for n3 is
Constants CI , C2, and C3 depend on the yield criterion chosen. In the case of the von Mises yield criterion, which can be written as f=J2-y2=0
we find
(4.106)
110
CONSTITUTIVE RELATIONS
-
PLASTICITY
Sometimes, an alternative expression using the 1D yield stress is used
and then C2 is
4.3 4.3.1
0; =
a
a.
CRITICAL STA TE MODELS Intvoduction
Constitutive modelling of soil behaviour is a keystone in the process of predicting the behaviour of a geostructure. No finite element code will provide results of better quality than that of the constitutive equation implemented in it. Today, there are a great variety of models, able to deal with situations ranging from simple monotonic stress paths to cyclic loading, rotation of principal stress axes and anisotropy. This has been made possible because of extensive work developed in laboratories throughout the world. In the past years, coordination of effort between different groups has increased, and, as an important result, bench-mark tests on some selected reference materials have been made available to constitutive modellers. Here, the initiatives of laboratories such as 3S in Grenoble, CERMES in Paris (both part of GRECO and GEO networks in France) and Case Western University in Cleveland (USA), have to be mentioned. Most of the proposed benchmark tests deal with key aspects of granular and cohesive soil behaviour: 1. Isotropic consolidation. Loading, unloading and reloading. Memory effects. 2. Shear behaviour in axisymmetric triaxial tests: Drained and undrained tests Effects of density and confining pressure Liquefaction of loose sands Memory effects: overconsolidation. 3. Unloading, reloading and cyclic loading: Densification Pore pressure build-up. Liquefaction and cyclic mobility.
4. Three-dimensional effects
5. Anisotropy: Material Induced by loading.
CRITICAL STATE MODELS
111
These progressively more sophisticated tests have helped to develop constitutive models of increasing complexity. There exists, however, a dramatic gap between these recently developed constitutive models and those used in day-to-day engineering practice. Several factors have contributed to it: 0
0 0
Industrial Finite Element codes do not implement constitutive models suitable to realistic geotechnical analysis. To calibrate more advanced models, the engineer needs to be acquainted with them. Special laboratory tests are frequently required to obtain material parameters which cannot be obtained by direct observation from raw data.
The last section was devoted to the introduction of elastoplastic constitutive models in the framework of Generalized Plasticity, and it was shown there how Classical Plasticity models can be considered as particular cases of the theory. The simple models of Tresca and von Mises present severe limitations when applied to geomaterials in general and soils in particular. Drucker and Prager proposed, in 1952, an elasticperfectly plastic constitutive model with an associated flow rule which could be applied to limit analysis problems. However, this model is not able to describe plastic deformations inside the yield surface cone, as occur in common engineering situations. Moreover, the associated behaviour is not valid, as it would predict large dilatancy at failure. Later, in 1957, Drucker, Gibson and Henkel introduced an elastoplastic model including two fundamental ingredients, a closed yield surface which consisted of a cone and a circular cap, and a hardening law dependent on density, paving the way to modern plasticity. At the same time, extensive research on the basic properties of soils in triaxial conditions was carried out at Cambridge University (Henkel 1956, Henkel, 1960 and Parry, 1960), and these ideas were further elaborated, arriving not only at practical expressions describing volumetric hardening, but also at the concept of a line in the ( e , p', q) space where the residual states lie. This line was referred to as the Critical State Line and is one of the basic ingredients of Critical State Theory introduced by the Cambridge group (Roscoe, Schofield and Thurairajah, 1963; Roscoe, Schofield and Wroth, 1958; Schofield and Wroth, 1958 and Roscoe and Burland, 1968). The purpose of this Part is to describe classical elastoplastic models for soils, together with their limitations, which made it necessary to introduce new concepts.
4.3.2
Cvitical state models fov novmally consolidated clays
Hydrostatic loading: isotropic compression tests One of the basic features of soil behaviour is the importance of its density on its behaviour. Both cohesive and granular soils exhibit changes of density caused by 0
a change in effective confining pressure p' and
0
changes of arrangement of grains in the structure induced by shearing of the material.
112
CONSTITUTIVE RELATIONS
Figure 4.14
-
PLASTICITY
Hydrostatic compression stress path
The simplest case in which the first mechanism occurs is hydrostatic loading of a soil specimen, in which the confining pressure is varied. The process is carried out slowly enough to prevent the development of intersticial pore pressure (drained conditions). If the initial state of stress is
and the specimen is loaded according to
the stress path will consist on a segment of straight line along the hydrostatic axis
d,= d!= d', or along the axis q = 0 if we are using the plane (p',q ) (Figure 4.14). In above, we have considered compressions as negative. The change of volume can be described either by the volumetric strain,
where we have used the minus sign for consistency with the definition of p', or the change of voids ratio e
We will consider now a loading-unloading-reloading process 1-2-3-4-5-6-7-8, in which hydrostatic pressure p1 is increased from p', to p',, and then the specimen is unloaded to pi and reloaded again to pi. This cycle is followed by a loading branch 4-5, with a final pressure of p', (Figure 4.15a)
113
CRITICAL STATE MODELS
Figure 4.15 Hydrostatic compression test on a normally consolidated clay. (a) Experimental results (b) Idealized behaviour
It can be seen that unloading and reloading branches differ, although volumetric strain developing in the branch 2-3-4 can be considered to be reversible. However, irreversible plastic deformation occurs from 1 to 2 and from 4 to 5. This behaviour is typical of soft clays, and can be sketched as shown in Figure 4.15(b). If time effects can be neglected, the response of a soft cohesive soil can be idealized in the In($) e plot as a line of slope X (Points 1-2-5-8) -
or, alternatively X dp' d ~ ?=, -1+ep1
This line is often referred to as the 'Normal Consolidation Line', and is one of the basic ingredients of modern plasticity models for soils. The parameter X depends on the type of soil, and it can be related to the Plasticity Index PI by the empirical relation (Atkinson and Bransby, 1978)
It is important to note that, if a Mohr-Coulomb or a Drucker-Prager yield surface had been used, no plastic deformation would have been produced, and, therefore, this is a severe limitation of all finite element codes implementing, as unique options, yield criteria which are open in the hydrostatic axis. If such plastic deformation needs to be reproduced, closed yield criteria should be used instead. Figure 4.16 illustrates this fact. If the stress increases from point 1 to 2, both states are inside yield
114
CONSTITUTIVE RELATIONS
Figure 4.16
-
PLASTICITY
Open and closed yield surfaces
surfacefi, and therefore, no plastic strain is produced in the process. The solution is to use yield surfaces intersecting the hydrostatic axis q = 0. Loading from 1 to 2 expands the yield surface, which can be assumed to harden as the soil densifies. The observed volumetric strain can be decomposed into elastic and plastic parts according to d ~ ,=, det
A dp' + d ~ := ----1 + e p'
-
During unloading from 2 to 3, and subsequent reloading to 4 the behaviour will be purely elastic d ~ ,=, de;,
dp' ltey' K
= --
where K is a new constant characterizing the elastic volumetric response. It can be related to the bulk modulus by
K,. =
1+e K
Once the stress point reaches the yield surface again, a plastic strain develops, the yield surface expands and the soil continues hardening. A simple law relating the size of the yield surface, which will be denoted by p, to the plastic volumetric strain is obtained as
from where
CRITICAL STATE MODELS
115
The subscript 'c' refers to consolidation, and the process 1-2-5 is referred to as 'isotropic consolidation'. If a soil specimen which has been subjected in the past to a consolidation pressure of p: = p i is tested at a lower pressure pl, it will be possible to observe in the curve e - lnp' a change of slope such as depicted in Figure 4.15(a) in the branch 3-4-5. This soil is referred to as 'overconsolidated', while soils at the normal consolidation line are called 'normally consolidated'. Both concepts can be easily understood in the framework of plasticity, as overconsolidated soils are characterized by the stress state being inside the yield surface at the initial state. The 'Overconsolidation Ratio', or OCR, is a parameter measuring the degree of overconsolidation,
Of course, these definitions apply only to simple hydrostatic loading conditions. but can be generalized to more complex situations where the OCR will be the ratio of the measures of two stress states.
Triaxial test So far, we have only considered stress paths where no shear strain is induced, unless the soil is anisotropic. It was seen that isotropic compression results in densification and hardening of soil, and it was mentioned that another mechanism causing densification was shear. Here we will concentrate in shear behaviour of normally consolidated clays subjected to symmetric or cylindrical triaxial stress conditions
which hereafter will be referred to as 'triaxial'. The stress conditions applied in a triaxial cell are sketched in Figure 4.17, and consist of a cell pressure a3 applied through a fluid, usually water, and a vertical additional load (a,- a3)referred to as a 'deviator' applied with a ram. The triaxial test is commonly used in the laboratory to determine soil properties as the desired stress paths can be reproduced quite accurately. There are several problems like membrane penetration, inhomogeneities caused by the development of narrow zones where the strain localizes, known as 'shear bands', and friction with the upper and lower rigid caps. In addition to these, it is worth mentioning those problems related to accurate measurement of vertical loads, changes in specimen cross-section, homogenization of pore pressures inside the specimen, and measurements of axial, radial and volumetric strain.
116
CONSTITUTIVE RELATIONS - PLASTICITY
Two main types of tests are currently used: (i) Consolidated drained; and (ii) Consolidated undrained. In the first case, a saturated soil specimen is brought to an initial state of hydrostatic stress I T u1= (ah,a; , a,,) = -pl(l, 1,l)
where the pore pressure is zero. The load is applied slowly to avoid pore pressure build-up, and once the initial conditions are reached, the vertical load applied through the ram is increased. The stress path is depicted in Figure 4.18. As pore pressures are zero (drained conditions), the total and effective stress coincide.
Figure 4.17 Triaxial stress conditions
Figure 4.18 Consolidated drained stress path
117
CRITICAL STATE MODELS
In this figure, it can be seen that both the hydrostatic pressurep' and the deviatoric stress are increasing. The stress path can be studied either in the space of principal stresses, or, alternatively, in the spaces ( I ; ,J 2 , 19) or (p',q, 8). The last is very convenient, as these invariants are given by
and
with
=
1
(u', - .;)I
from which we have q
=
-
(u', - u i )
(4.127)
which is precisely the stress induced by the vertical load applied through the ram. In above, both stresses are negative, and we have supposed that the absolute value of d, is higher than that of a',. The measures of strain are those work-associated top' and q
Concerning Lode's angle, it is kept constant during the test, provided that (d,-a',) does not change sign during the test. This fact occurs when applying compression-extension cycles. The stress path in the (p', q) plane is shown in Figure 4.19, where it can be seen that, due to its positive slope, and for a given absolute value of the deviator q, the angle AOB is smaller than AOB'. Therefore, if the failure surface is of MohrCoulomb type, M,. 2 Me, the soil will fail earlier in extension. The slope of the stress path can easily be obtained as
118
CONSTITUTIVE RELATIONS
-
PLASTICITY
and
from where, taking into account that
d' is constant,
ir Figure 4.19 Consolidated drained stress path in the p'
Figure 4.20
-q
plane
Typical results of CD tests on normally consolidated clays
CRITICAL STATE MODELS
Figure 4.21
119
Consolidated undrained tests on normally consolidated clay
The results obtained in compression tests on normally consolidated clays are similar to those depicted in Figure 4.20. The main features are the following: 0
There is a tendency of the soil to compact as the test proceeds, caused by the increase of p' and a rearrangement of soil particles.
0
Failure takes place at a certain value of stress ratio q = M for tests performed at different confining pressures.
0
Soil strength and compaction depend on confining pressure, and increase with it.
A second type of triaxial test is the Consolidated Undrained (CU) test, where, after consolidation, the drainage valve is closed to prevent dissipation of pore pressure. The test has to be carried out slowly enough for the pore pressure to be homogeneous through the specimen. During the test, measurements of pore pressure, axial strain, vertical stress and cell pressure are taken to monitorize the stress path. Figure 4.21 shows typical results obtained in drained consolidated clays. It can be seen how the effective stress path bends towards the origin as a consequence of pore pressure increase caused by the tendency of soil to compact. Again the failure takes place at a line of slope M,., which coincides with that obtained in drained tests. This test has been classically used to characterize soil behaviour under 'fast' loading, where pore pressures do not have time to dissipate, in short-term stability analysis. Of course, this is a simplification, and a complete coupled analysis should be performed instead. It has to be noticed that soil strength is lower in undrained than in drained conditions because of the generated pore pressures. Undrained behaviour of normally consolidated clays provides an interesting illustration of the shortcomings of the Mohr-Coulomb criterion when used as a yield surface. Figure 4.22 compares the predicted behaviour of such a model with that observed in the laboratory. It can be seen how the model overestimates soil strength because it
120
CONSTITUTIVE RELATIONS
-
PLASTICITY
.......... Predicted
Experiment
p'o
Figure 4.22 Predicted (Mohr-Coulomb) and observed behaviour in CU tests
cannot predict the pore pressures caused by plastic volumetric strain which develop during the test. In the Mohr-Coulomb model, no plastic strain is produced until the yield surface is reached. In addition to that, if the flow rule is associative, dilation and negative pore pressures will develop at failure, and the stress path will turn to the right following the yield surface (Bl-C). This process will be endless, and the deviatoric stress will keep increasing continuously. In reality, the process is stopped by cavitation of the pore fluid.
Critical state models It can be said that modern plasticity models for soils are based both on the pioneering work of Drucker, Gibson and Henkel(1957), who first introduced the ideas of volumetric hardening and a closed yield surface, and on the theoretical and experimental work of researchers from the University of Cambridge, who provided the framework of Critical State Soil Mechanics, in which elastoplasticmodels for soilscould bedeveloped. The basic ingredients of Critical State Soil Mechanics are the following: There exists a line in the (e, In p') plane in which all stress paths in normally consolidated clays lie, which is referred to as the 'Normal Consolidation Line' (NCL). This line was depicted in Figure 4.15(b) (1-2-5-8). The interest of this line is that it provides a volumetric hardening rule which can be generalized to general stress conditions (Roscoe, Schofield and Thurairajah, 1963). There exists a line in the space (e, In p', q) where all residual states lie, independently of the type of test and initial conditions (Parry, 1960). The projection of this line on to the (e, lnp') plane is parallel to the NCL, and divides initial states into 'wet' and 'dry', depending on whether they lie in the space between both lines or not (Figure 4.23). At this line, shear deformation takes place without change of volume. The stress paths resulting either from consolidated drained and undrained tests lie on a unique state surface referred to as the 'Roscoe Surface'. This fact was found
121
CRITICAL STATE MODELS
experimentally by Henkel (1960), who plotted the water content contours obtained in drained tests and found that undrained tests paths followed these lines as well (Figure 4.24). This fact is not directly applicable by elastoplastic models, as these isolines are not yield surfaces corresponding to constant values of the hardening parameter. In fact, during undrained paths the soil hardens as plastic volumetric strain is produced, while the sum of the plastic and elastic increments of volumetric strain is kept constant. What is useful, however, is that it gives a hint of the kind of yield surface. From here, simple elastoplastic models can be derived. The first step, as mentioned above, is to assume as a hardening rule
where pb is a parameter characterizing the size of the yield surface. Next, the yield surface is determined. Roscoe, Schofield and Thurairajah (1963) assumed that incremental plastic work
Figure 4.23
Normal consolidation and critical state lines
.......... w=ct. from CD tests CU tests
,4
Figure 4.24
Constant water content lines as obtained from C D and CU tests (sketched)
122
CONSTITUTIVE RELATIONS
-
PLASTICITY
was given by
6 WP = Mp'd&r from where
Using the above expression, dilatancy d,
= d&,P/d&,P is
obtained as
where it is interesting to note that dilatancy is zero at the Critical State Line. The normal to the plastic potential surface is proportional to
and 7
Therefore, dilatancy is given by
If we take into account that, along the surface
we obtain
CRITICAL STATE MODELS
123
which can be integrated to obtain the plastic potential
where p:. is the abscissa at which the surface intersects the hydrostatic axis q = 0. This surface has been depicted in Figure 4.25, where it can be seen that the normal to the surface at p' = p', is not directed along the axis. Therefore, the normal will not be uniquely defined in three-dimensional stress conditions, although it can be assumed that the surface is rounded off in the proximity of the axis so that the normal is directed along it. If the flow rule is assumed to be associated, the yield surface coincides with g. This model was further elaborated by Burland (1965), who suggested an ellipse as a yield surface. The work dissipation was given by
PC
Figure 4.25 Yield and Plastic potential surfaces of Cam clay Model
Figure 4.26 Yield surface of modified Cam clay model.
124
CONSTITUTIVE RELATIONS
-
PLASTICITY
from which dilatancy is obtained as
with
The yield surface can easily be obtained by integration of the above and is given by
which is depicted in Figure 4.26
4.3.3
Extension to sands
The models described so far are able to predict with reasonable accuracy the behaviour of normally consolidated clays. They depart from reality (i) when applied to overconsolidated soils, as it is not possible to reproduce inelastic strain which develops inside the yield surface, and (ii) when applied to sands. The behaviour of granular soils depends mainly on density, and two extreme classes of behaviour can be identified. Sands at very dense states can be prepared in the laboratory by vibration and not by compaction, as occurs in the case of cohesive soils. When sheared in triaxial conditions, the behaviour is similar to that depicted in Figure 4.27. In the first part of the test the sand contracts, reaching a minimum void ratio e, and, from there, it dilates. Concerning the deviatoric stress, it increases until a peak is reached, and then it softens. Finally, it stabilizes at residual conditions, where the plastic flow takes place at constant volume. Therefore, a critical state exists for sands. This fact had been first established by Rowe (1962). The results sketched in the figure follow the ideas of Taylor (1948), who suggested that the moment at which the stress ratio (deviatoric stress in the plot) reaches the value at which residual conditions will take place later, the volumetric strain presents a peak. From the point of view of elastoplasticity, it can be assumed that dilatancy is always zero at the line 7 = Mg,
p
Axial strain
Figure 4.27
.
* Dense
Axial strain
Triaxial tests on dense and loose sand (schematized)
125
CRITICAL STATE MODELS (4 A
2 3
1
'
t
Axial strain
Axial strain
Figure 4.28
Behaviour of dense sand as predicted by critical state models
either before reaching the critical state or at that point. In fact, several investigators proposed a separate denomination of this line referring to it as the 'Characteristic State Line' (Habib and Luong, 1978) or the Line of Phase Transformation' (Ishihara, Tatsuoka and Yasuda, 1975). An important difficulty encountered is that the specimen is no longer homogeneous long before residual conditions, as the strain concentrates in the shear bands. Therefore, the observed softening is rather of a structural than a material nature. If such behaviour is modelled by a basic critical state model such as described in the preceding section, the best choice is to assume the sand as overconsolidated, as shown in Figure 4.28. The behaviour is elastic from 1 to 2, where the yield surface is reached. As the soil dilates, it softens, and the stress path follows from 2 to 3, where it stabilizes at critical state conditions. Under undrained conditions, the results present more important discrepancies. Dense sands separate at the beginning of the test from the undrained stress path of an elastic material, which is a vertical segment on the @', q) plane (Figure 4.29(a)). If such a process is reproduced with a critical state model, the results will be similar to those shown in Figure 4.29 (b). It can be seen there that the difference from the observed behaviour is important. as the strength is underestimated. On the other side of the density spectrum, very loose sands present liquefaction under undrained conditions. The phenomenon consists of a sudden drop of resistance of the soil, which behaves as a viscous fluid (Figure 4.30). It is important to note that the behaviour in the descending branch corresponds to increasing values of the stress ratio, and, therefore, it is not sound to assume this behaviour as softening. The separation from the vertical of the stress path shows that plasticity is present from the beginning of the test.
126
CONSTITUTIVE RELATIONS
-
PLASTICITY
Axial strain
(4
Axial strain (b)
Figure 4.29 Undrained behaviour of dense sand in CU triaxial test. (a) Experimental results (b) Predicted
Such behaviour cannot be described by the models presented above, especially the dramatic loss of resistance. If the material is assumed normally consolidated, the results will be similar to those shown in Figure 4.21. All these limitations motivated further research to extend the range of application of CS models. The three fundamental ingredients were: (i) hardening laws depending on deviatoric and volumetric plastic strain; (ii) non-associative plastic flow rules; and (iii) plastic deformations existing throughout the process.
127
CRITICAL STATE MODELS
I
Axial strain
I Axial strain
Figure 4.30
Liquefaction of very loose sand
The first is required to cross the line 77 = M,, as the plastic modulus is zero there otherwise. Deviatoric hardening was introduced by Nova (1977) and Wilde (1977), who assumed a hardening parameter of the type
Y=E~+D< where
(4.149)
< is the accumulated deviatoric shear strain
The size of the yield surface was made to depend on Y, and the plastic modulus in triaxial conditions was found to be proportional to ag -+
ap'
D-ag
ay.
Therefore, the Critical State Line at which
can be crossed, without the plastic modulus is zero. Failure will occur when
128
CONSTITUTIVE RELATIONS
-
PLASTICITY
which happens at a stress ratio higher than M,. Once there, if D is kept constant, the path will not return to the CSL, and failure will take place with dilation. Another possibility, proposed by Nova (1982), consists of making them drop D to zero, which results in a discontinuity of slope but with the desired result of coming back to CS conditions. Finally, a hardening law with saturation can be assumed to hold for D
This law was suggested by Wilde (1977) and applied to a bounding surface model by Pastor, Zienkiewicz and Leung (1985). If a negative value of D is assumed, then the plastic modulus becomes zero at a stress ratio lower than critical, and liquefaction-like behaviour can be modelled in the softening regime. As discussed above, it is more sound to assume this process to be of the hardening type, as the stress ratio is continuously increasing. The second ingredient is a non-associative flow rule, as suggested by Poorooshasb, Holubec and Sherbourne (1966 and 1967), Nova and Wood (1979), Nova (1982), Zienkiewicz, Humpheson and Lewis (1975), and Pastor, Zienkiewicz and Leung (1985). The plastic potential and flow rules can be determined from experiment, as shown in Nova and Wood (1978), where surfaces were defined by different analytical expressions valid for different ranges of the stress ratio. Pastor, Zienkiewicz and Leung (1985) used as plastic potential a simplification of that proposed by Nova and Wood (1979), assuming now that a single expression was valid for the full range of stress ratios
where pi is the abscissa at which it intersects the p' axis. This surface can be obtained from the dilatancy rule
dg = (1
CSL
,
+a). (Mg
-
rl)
CSL
.
..'
n
Figure 4.31
Plastic potential and yield surfaces for (a) loose sands (b) dense sands
ADVANCED MODELS
129
As yield surfaces, they proposed curves belonging to the same family
where M f # M, in general. An interesting fact reported by them is that the ratio M, / M, depends on the relative density and indeed it can be assumed to be the same as D,. Figure 4.31 (a) and (b) show plastic potential and yield surfaces for very loose and dense sands.
4.4
4.4.1
ADVANCED MODELS Zntvoduction
So far, we have discussed in the previous sections some classical plasticity models for soils which have proven to reproduce accurately enough the behaviour of soil under monotonic loading. They incorporated as basic ingredients a plastic potential and a yield surface, the latter being allowed to expand or contract depending on whether the material was hardening or softening. However, material remained elastic within the yield surface, where no plastic deformation can develop. The immediate consequence is that these models are unable to reproduce either the behaviour of overconsolidated soils, or phenomena occuring during cyclic loading, such as pore pressure generation in fast processes or densification. Indeed, both phenomena are related, as the latter is a direct consequence of the tendency of soil to compact. In fact, it can be shown that variation of pore pressure and densification in undrained conditions are related by the expression
where n is the porosity, Kf the bulk modulus of pore fluid and KT that of the soil skeleton. As these are key aspects which reproduce failure of the soil caused by liquefaction or cyclic mobility phenomena (Martin et ul., 1975), it motivated an important effort of research, which proceeded along two main lines. The first consisted in developing new cathegories of models embracing classical plasticity as a particular case, and the second was based on introducing the volumetric deformation which is produced by cyclic shearing of a soil as an 'autogenous volumetric strain', from which suitable densification laws were produced. Here, we could mention the work of Bazant and Krizek (1976) and Cuellar et 01. (1977) who developed densification laws in the context of the endochronic theory, and that of Zienkiewicz and co-workers at Swansea University, who developed simple densification models and implemented them into coupled numerical models (Zienkiewicz er al., 1978 and Zienkiewicz et a/., 1982). These will be described next.
130
CONSTITUTIVE RELATIONS
-
PLASTICITY
Densification models Phenomena such as liquefaction can be thought of being caused by: (i) an accumulation of pore pressure with the number of cycles; and (ii) liquefaction in the last cycle as it occurs in monotonic loading. This interpretation has motivated the development of the so called densification models, where simple elastoplastic behaviour of soil and the accumulation of pore pressure were taken into account by two different mechanisms. The constitutive equation was written as da'
=
D,,.(d€
-
dq)
(4.159)
where Dep accounted for the elastoplastic behaviour and d ~ for " the densification caused by cyclic loading. In general a non-associative Mohr-Coulomb model with zero dilatancy is assumed here for the elastoplastic behaviour. In the model proposed by Zienkiewicz et nl. (1978), the accumulated deviatoric strain was quantified by a variable defined as
<
where eii is the deviatoric strain. The densification is described by a law
where A and B are constants and
K
a parameter defined as
Here. y is a third parameter of the model, and B is given by
where I@/ is the amplitude of the stress cycles (deviatoric) and a,,,o the initial value of the mean effective stress at the beginning of the cyclic loading process. The 'densification' model differs from other elasto-plastic models described in this text by a rather arbitrary separation of effects. For this reason we defer the discussion of its most recent forms to the end of this chapter. We must, however, remark that it is capable of modelling with good accuracy the liquefaction phenomena encountered in earthquakes and because of its simplicity it deserves wider use.
Kinematic hardening models The second approach consisted in extending the theory of plasticity beyond the limits imposed in the classical formulation. The first successful theory was the multi-surface
ADVANCED MODELS
131
kinematic-hardening model proposed by Mroz in 1967, where a set of 'loading surfaces' within an outer 'boundary' surface was postulated. Since then, further developments and improvements have taken place (Mroz, Norris and Zienkiewicz, 1978; Prevost, 1977; Hirai, 1987). The number of surfaces allows us to keep track of loading events such as the maximum stress level reached, or points at which stress has reversed. Large intensity loading events erase lower intensity events. An elastic domain may also be postulated, corresponding to the volume enclosed by the inner surface. As the stress is increased from an initial state, the surfaces reached by the stress path translate until a new loading surface is attained. This movement must comply with a rule which ensures that surfaces never intersect each other. An improvement of this 'multi-surface model' was provided by Mroz, Norris and Zienkiewicz (198 l), who introduced an infinite number of nested loading surfaces, making the hardening modulus depend on the ratio of the sizes of active loading and outer or consolidation surfaces. In this way, the field of plastic modulii was made continuous in the whole domain enclosed by the outer surface. Memory of loading events was kept through the position and size of the surfaces at which stress reversal took place. The elastic domain was assumed to shrink to a point. It can be seen that both the 'multisurface' and the 'infinite number of surfaces' models are able to reproduce most of the basic features of soils under cyclic loading, such as memory of past events and plastic deformation during unloading. Mroz and Norris (1982) showed an application of the model to the cyclic behaviour of normally consolidated and overconsolidated clays, and found that the model was able to predict final states lying on an 'equilibrium line', as observed by Sangrey, Henkel and Esrig (1969). Other elastoplastic, kinematic or anisotropic hardening models have been shown to perform well in modelling liquefaction and other cyclic loading phenomena (Ghaboussi and Momen, 1982; Hirai, 1987; Aubry, et al. 1982). However, the price to pay in numerical computations is high, and simplified versions were sought.
Bounding surface models and generalized plasticity If the number of surfaces is reduced to two, i.e., the outer or consolidation and the inner or yield, a field of hardening modulii can still be described by prescribing the variation between both surfaces. This model was independently proposed by Krieg (1975) and Dafalias and Popov (1975), and evolved to what is known today as 'Bounding Surface Theory' (Dafalias and Herrmann, 1982; Dafalias, 1986; Wang Dafalias and Shen 1990; Kaliakin and Dafalias 1989 and Bardet, 1989). A similar approach, the 'subloading surface model' was proposed by Hashiguchi and Ueno (1977) and Hashiguchi, Imamura and Ueno (1989). On the bounding surface, plastic strain develops according to classical plasticity theory, with directions n and n, given by the normals to the bounding and plastic potential surfaces, and the plastic modulus obtained through application of the consistency condition describing material hardening or softening properties. In the case of loading processes beginning at the bounding surface, the results coincide with those of classical plasticity. However, for loading processes inside it, such as may
132
CONSTITUTIVE RELATIONS
I
-
PLASTICITY
Bounding surface
Figure 4.32 Bounding surface interpolation
occur in cyclic loading, the difference is that bounding surface models are able to introduce plastic deformations by using some interpolation rules relating the stress point P(C in Figure 4.32) to an image of it on the BS, PBs(Bin Figure 4.32). Simple interpolation rules were proposed by Dafalias and Herrmann (1982), and by Zienkiewicz, Leung and Pastor (1985). There, to obtain the image point PBS,a line was drawn passing through the origin and point P, its intersection with the bounding surface being taken as the image point. Directions n and n, in P were assumed to be those at Pss and the plastic modulus was interpolated according to a simple law
where 6 is the distance from the origin to the stress point P, and So the distance between the origin and the image point PBs, y being a parameter of the model (Figure 4.32). The main shortcoming of early BS models was their inability to reproduce plastic deformations which develop when unloading, and it was overcome within the more general framework of generalized plasticity (Pastor, Zienkiewicz and Leung, 1985). Here, the model was of bounding surface type for loading, but plastic deformations during unloading were introduced within the more general framework of generalized plasticity. A further step was given by Pastor, Zienkiewicz and Chan, introducing a full generalized plasticity model in Pastor and Zienkiewicz (1986) and Pastor, Zienkiewicz and Chan (1990), which was applied by the authors to reproduce the behaviour of both cohesive and frictional soils under monotonic and cyclic loading.
Hypoplasticity and incrementally non-linear models One of the basic characteristics of the elastoplastic tensors Dep and Cep is their dependence on the direction of loading u = du/lldull, which is taken into account
ADVANCED MODELS
133
in a simple way by introducing a direction n for each mechanism of deformation considered. Alternatively, it is possible to provide general expressions for the constitutive tensors satisfying all necessary requirements, such as dependence on u. Hypoplasticity is, in this sense, the most promising framework in which this goal can be achieved. Among all the constitutive models of this kind it is worth mentioning those proposed by Darve and Labanieh (1982); Dafalias, (1986); and Kolymbas, (1991). One of the first models which can be considered of this kind was introduced in Grenoble by Darve and Labanieh (1982), and since the early days of the theory it has been considerably improved (Desrues and Chambon 1993). They are referred to as incrementally non linear models, as they are based on the assumption that the incremental non-linearity in the constitutive tensor may be approximated by suitable interpolation laws once material behaviour along different stress paths has been established. Darve and Labanieh (1982) suggested that these paths could correspond to positive and negative directions along principal stress axes 1,2 and 3. Consequently, six values of C were required for the interpolation along a particular direction. The constitutive tensor was assumed to be given by
were u is the unit tensor along direction of loading increment. If constitutive tensors along positive and negative directions of the principal directions are N+ and N-, a simple incrementally non-linear law could be given by
where
The model proved to reproduce well the behaviour of soils under both monotonic and cyclic loading (Darve, Flavigny and Rojas 1985). Dafalias and co-workers presented extensions of the bounding surface model within the framework of hypoplasticity (Dafalias, 1986; Wang, Dafalias and Shen, 1990). Hypoplastic models have been introduced also in Karlsruhe by Kolymbas and coworkers (Wu and Kolymbas, 1990; Kolymbas and Wu, 1993), producing general expressions for the constitutive tensor.
CONSTITUTIVE RELATIONS
134
4.4.2
-
PLASTICITY
A Generalized plasticity model for clays
Normally consolidated clays The simplest case of normally consolidated clay under virgin loading will be considered first. We will begin by assuming that a residual critical state exists in the space e - p' - q where e is the voids ratio and p', q the effective confining pressure and a measure of deviatoric stress. To obtain a flow rule, it is possible to use values of dilatancy measured in laboratory tests d" ~ lL d~ ' , d -
where
E,.
d& dt,
and de, are strain measures work-conjugated top' and q SW
=
d : de = (p'q).
(2:)
defined as
If the ratio between the plastic increments of volumetric and deviatoric strain is assumed to be the same then the ratio of total (elastic plus plastic) increments observed in laboratory tests, direction n, can be immediately obtained. To this end, experiments carried out by Balasubramanian and Chaudhry (1978) using constant p'lq stress paths suggest that dilatancy can be approximated by a straight line in the p'-q plane (Figure 4.33). Dilatancy is therefore expressed as
where M, is the slope of the critical state line in the p'-q plane, cr a material constant and 7) is the so called stress ratio defined as
Direction n, is now given by
135
ADVANCED MODELS
0.5
I .O
Figure 4.33 Dilatancy of soft Bangkok clay (after Balasubramanian and Chaudry)
with
This law can also be used to describe dilatancy of granular materials, as it was suggested in Pastor, Zienkiewicz and Leung (1985) according to test results reported by Frossard (1983). Concerning direction n, we will assume that the flow rule is associative, following Atkinson and Richardson, who performed experiments in three cohesive soils and found little discrepancies from plastic potential and yield surfaces (Atkinson and Richardson, 1985). Therefore, we will have
with
136
CONSTITUTIVE R E L A T I O N S PLASTICITY
and
In what follows, we will drop the subindex 'g' referring to the plastic potential surface as it coincides with the yield surface. To obtain the plastic modulus for virgin loading, we will consider an isotropic consolidation test of a normally consolidated specimen, for which the increments of volumetric elastic and total strain are given by K dp' d ~ ;= , -l+epl
and
x dp' de,. = -l+epl from which the plastic volumetric strain increment is
~EP= . (AI + eK ) dp' p' --
Comparing now the above equation to the general expression for the plastic strain increment 1 d~~ = -n.(n HL
: da')
(4.183)
which particularizes to
for the stress path considered here, it can be concluded that the plastic modulus H L is given by
The parameters X and K are the slopes of the normal consolidation and elastic unloading lines in the (e, In p') plane, and Ho is a material constant. To generalize this expression of the plastic modulus to other conditions than isotropic compression paths, we will make the assumption that plastic modulus depends on the mobilized stress ratio, decreasing as the later increases until reaching a value of zero at the critical line (7 = q/pl = M).
137
ADVANCED MODELS
Therefore,
where f ( q ) is such that
and
A suitable form was proposed in Pastor, Zienkiewicz and Chan (1990)
+
where do = (1 a ) M and p can be taken as two for most clays. So far we have analyzed only the behaviour in the triaxial plane, but above expressions can be immediately generalized to all three-dimensional conditions by assuming that M depends on Lode's angle 0 according to a suitable law. Below we define a smoothed version of Mohr's criterion widely used in practice (Zienkiewicz and Pande, 1977)
where M,. is the slope of the critical state line obtained in standard compression triaxial tests and 0 Lode's angle
Elastic constants are assumed to depend on p' according to the laws
and K 1 dEt = -dp' 1 ep'
+
138
CONSTITUTIVE RELATIONS
-
PLASTICITY
where
The model presented so far concerns normally consolidated clays under virgin loading. In order to assess model performance, their predictions will be compared against a full set of tests carried out by Balasubramanian and Chaudhry in 1978 for soft Bangkok clay. The proposed model has five parameters, i.e., two elastic constants, the slope of the critical state line M, the constant a characterizing dilatancy and Ho, which appears in the plastic modulus. T o determine them the following procedure may be followed: The elastic constants can be easily determined from unloading-reloading tests. Here, they were found from the constant p' tests reported. The slope M of the critical state line on the (p' - q) plane is found from drained, undrained or constant p' tests. The parameter a controlling dilatancy can be found from dilatancy plots, and it is given by
Finaly, constant Ho can be found as a function of A, K and e as described above.
0
Experiments
- predicted
1r: ,\y--
Figure 4.34 Constant p' test on Bangkok clay (After Balasubramanian and Chaudhry)
::(': 100.
. :sl
Experimants
I:
.
Predicted
p'Wa) 100.
200.
300.
400.
0.10
0.20
Figure 4.35 Consolidated undrained tests on Bangkok clay (After Balasubramanian and Chaudhry)
ADVANCED MODELS
139
Figure 4.36 Consolidated drained tests on Bangkok clay (After Balasubramanian and Chaudhry 1978)
The results obtained with the proposed model are shown in Figure 4.34, 4.35 and 4.36. First of all, Figure 4.34 shows the constant p' tests together with the experimental results. Next, the model is applied to simulate the consolidated drained behaviour of Bangkok clay (Figure 4.35), and, finally, we present the results obtained for Consolidated undrained tests (Figure 4.36)
Overconsolidated clays The model described in the previous section can be extended to describe the behaviour of overconsolidated clays. To this end, we will introduce a function accounting for memory of past history, which will consist of storing the past event of maximum intensity. The mobilized stress function proposed in Pastor, Zienkiewicz and Chan (1990) is
which will be used in the plastic modulus
where f (7) has been given in the previous section and CMAX is the maximum value previously reached by the mobilized stress function. In the above, we have introduced a deviatoric strain hardening function g(<) (Wilde, 1977),
where
140
CONSTITUTIVE RELATIONS
-
PLASTICITY
and
Therefore, two additional parameters y and 00 are needed to extend the range of application of the model to overconsolidated clays. It should be noted that for first or virgin loading of clays above expressions reduce to those previously proposed for normally consolidated clays. Figures 4.37 and 4.38 show the behaviour of normally and heavily consolidated Weald clay reported by Henkel (1956), together with the model predictions. It is important to note that in overconsolidated clays the peak value of the stress ratio may be higher than M, then decreasing to reach it as a residual state. For cyclic loading it is possible to obtain quite satisfactory results for clays using simple elastic unloading and thus avoid the introduction of additional parameters. We show in Figure 4.39 the performance with the above assumption for cyclic tests with constant stress amplitude carried out by Taylor and Bacchus (1969). Finally, Table 4.1 gives the parameters used in the simulations described above. Bangkok clay
Weald clay
Taylor and Bacchus
Figure 4.37 Behaviour of normally consolidated Weald clay (after Henkel 1956)
141
ADVANCED MODELS
r z Axial strain
Figure 4.38 Behaviour of overconsolidated Weald clay (OCR=24)
10 Number of cycles
20
Figure 4.39 Behaviour of clay under two-way strain-controlled triaxial loading (data from Taylor and Bacchus 1969)
4.4.3 A generalized plasticity model for sands Monotonic loading Following experimental results reported by Frossard in drained triaxial tests, dilatancy can be approximated by a linear function of the stress ratio 77 similar to that used in the preceding section for normally consolidated clays.
using M, instead of M as we will assume that flow rule is non-associated. Dilatancy is zero at the line
142
CONSTITUTIVE RELATIONS
-
PLASTICITY
which coincides with the projection of the critical state line on the plane (pl-q). This line has also been referred to as the 'characteristic state line' (Habib and Luong, 1978) or the 'line of phase transformation' (Ishihara, Tatsuoka and Yasuda, 1975) and plays an important role in modelling sand behaviour as will be shown later. It has to be noted that this line is not the critical state line, which will be reached at residual conditions. Whether the critical state line existed or not has been a matter of discussion during past years, due to the difficulty of obtaining homogeneous specimens at failure after shear bands have developed. However, recent experiments carried out at Grenoble by Desrues have shown that inside the shear band a critical void ratio is reached. During a test, this line can be crossed a first time, with the specimen still far from the residual state. If shearing continues, the stress path will finally approach the critical state line. Therefore, the condition 77 = M, represents two different states at which dilatancy is zero, the 'characteristic state' and the critical state. The direction of plastic flow n , ~can be determined in the triaxial space by similar procedures used in cohesive soils, giving
with
So far, the behaviour of granular and cohesive soils coincides. However, use of non-associative flow rules is necessary for modelling of unstable behaviour within the hardening region, and the direction n should be specified as different from n , ~ . We do this by writing
with
ADVANCED MODELS
143
where
Again, both My and M, depend on Lode's angle in the manner suggested in Zienkiewicz and Pande (1977). It has to be remarked that both directions have been defined without reference to any yield or plastic potential surfaces, though, of course, these can be established u posteriori In fact, it is possible to integrate above directions to obtain both plastic potential and yield surfaces
where the size of both surfaces is characterized by the integration constants pi, andpk. Both surfaces were depicted in Figures 4.31(a) and (b) for a medium-loose sand, together with experimental data obtained from accoustic emission (Tanimoto and Tanaka, 1986). Similar yield surfaces were proposed by Nova (1982). To derive a suitable expression for the plastic modulus HL. it is necessary to take into account several well established experimental facts: (i) residual conditions take place at the critical state line
(ii) failure does not necessarily occur when this line is first crossed. (iii) The frictional nature of material response requires the establishment of a boundary separating impossible states from those which are permissible. A convenient law was introduced in Pastor and Zienkiewicz (1986) in the form
where
together with
144
CONSTITUTIVE RELATIONS
-
PLASTICITY
limit the possible states, and where
are of a similar form to the expressions proposed for clays. To illustrate the predictive capability of the proposed model, we will consider next several sets of experiments reported in the literature (Castro, 1969; Taylor, 1948 and Saada and Bianchini, 1989) and which cover the basic features of granular soil behaviour under monotonic loading. (i) Very loose sands exhibit liquefaction under undrained shearing, as has been described in the preceding chapter. Considering the qualitative results shown there, it is important to remember that the material densifies during the whole process, which is shown by a continuous increase in pore water pressure, suggesting that the soil is hardening. This seems to contradict the fact that a peak exists, and the material can be thought of as softening. However, in a frictional material, strength has to be analyzed in terms of mobilized stress ratios rather than deviatoric stress, and no peak is presented by this parameter. This behaviour can be considered unstable in the sense of Drucker (1956, 1959)
having thus
If such a feature is to be modelled with a positive plastic modulus, the associated plasticity has to be abandoned, choosing
Figures 4.40(a), (b) and (c) show, respectively, the stress paths, deviatoric stress vs. axial strain and pore pressures obtained by Castro, together with the model predictions, which agree well with them. (ii) At the other end of the density range, peaks exist in deviatoric stress during drained shear of very dense sands, this effect developing progressively as density is increased. The factor H, is introduced in the expression giving a plastic modulus to account for: crossing of the Characteristic State line (7 = M,) without immediately producing failure, reproduction of softening residual conditions taking place at the critical state line.
145
ADVANCED MODELS (a)
p'
-
392 KPa
200 400 600 Mean e r l t c b v c confinmg pressure p' (kPa )
1
2
3
4
5
6
7
Shear stram e, (36)
x
9
Figure 4.40 Undrained behaviour of Banding sand (data from Castro 1969) Computed results shown by solid line. (a) p'-q plot (b) Deviatoric stress vs. shear strain plot (c) Pore pressure vs. shear strain plot
146
CONSTITUTIVE RELATIONS - PLASTICITY
0
0
Figure 4.41 1948)
j
10 I5 Axial strain (Of;)
5
10 15 Axial strain ("h)
20
0
5 10 IS 20 A x d strain (",I)
10
0
5
10 15 Axial strain ('14)
20
Drained behaviour of dense and loose sand (experimental data from Taylor,
To illustrate the role of the plastic modulus in the transition from softening to hardening regimes, let us consider a drained triaxial test (Figure 4.41). During a first part of the path, both H I ,and H, are positive and decrease in a monotonous way. At 17 = M,, i.e., when crossing the characteristic state line, H,, becomes zero, while H, is still positive. If the process continues, a moment arrives at rl, where
with
If the test is run under displacement control, the deviatoric stress does not change for an infinitesimal variation of the strain dp'
= dq = 0
ADVANCED MODELS
147
Meanwhile, H, has decreased, and consequently, the plastic modulus becomes negative. The soil has entered the softening regime, and from this moment the deviatoric stress will present a descending branch. The deviatoric strain hardening function H , will vanish as deformation progresses, reaching a final asymptotic value of zero at 17 = M,, this time at the critical state line. During the softening process,
and
It can be seen that there is no need on this occasion for non-associativeness to ensure the existence of peaks as H is negative, and, in fact, very dense sands may exhibit the limiting associative behaviour with
The ratio M f / M g seems to be dependent on relative density, and in Pastor, Zienkiewicz and Leung (1985), a suitable relation was proposed as
where D, is the relative density. Figures. 4.41 and 4.42(a) and (b) show model predictions for dense and loose sand response in drained conditions (Taylor, 1948; Saada and Bianchini, 1989). Care should be taken when analyzing the results of tests in general, and in the case of dense sands in particular, as failure localizes along narrow zones referred to as shear bands. From the moment of their inception, the specimen is no longer homogeneous, and the experimental results correspond to a boundary value problem rather than a homogeneous body. However, we have to stress following facts: Even if the specimen is not homogeneous, softening must exist for the sample to exhibit a peak. 0
The overall response is governed by the ratio between the width of the shear band and the length of specimen. This effect is similar to what can be observed in numerical computations and which has been referred to as mesh-dependence.
0
Experimental evidence seems to indicate the existence of a residual critical state. To obtain it, it is simpler to use loose rather than very dense specimens.
148
CONSTITUTIVE RELATIONS - PLASTICITY I
"
r)
2 -50 Hollow cylinder Hostun sand conf.press.207kPA
- Experiment o
Prediction
I
I
-3 -2 Axial strain (%) Test HH3 Hollow cylinder Hostun sand conf.press.207kPA
-
& in in
z2 -100
u
-1
-
- Experiment
'tiu
Prediction
0
-
.m
8
Axial strain (%)
a
-150
0
0
0
"
Axial strain (%)
Axial strain (%)
\
(a)
(b)
Drained behaviour of Hostun sand (Experiments from Saada and Bianchini 1989). (a) Compression test (b) Extension test
Figure 4.42
(iii) Undrained shearing of medium-loose to dense sands shows the intermediate characteristics to be discussed. Once the characteristic state line is reached, an upturn in the stress path is produced as the soil changes from contractive to dilative behaviour. If the material is isotropic, determination of the CSL position can easily be performed from a point at which the undrained stress path has a vertical tangent in (p' - q) space, as then,
and
dp' = 0 Figures 4.40(a), (b) and (c) show how relative density influences the undrained behaviour of sand, together with predictions of the proposed model.
149
ADVANCED MODELS
At this point, a model has been produced such that: (i) it reproduces the most salient features of sand under monotonic shearing; (ii) it is very simple, as no surfaces are involved and consistency conditions do not have to be fulfilled; (iii) it is computationally efficient in FE codes, as the stress point does not have to be brought back to the yield surface and tangent moduli are easily established.
Three-dimensional behaviour So far we have considered the triaxial response of soil (compression and extension). However, proposed relations have been made dependent not only on I; and J2 (or, alternatively, p' and q), but also on the third invariant or equivalently on Lode's angle 6. Hence, the soil response can be generalized to any path out of the triaxial plane. Denoting triaxial stress parameters p', q and 19by a*
and the Cartesian stress tensor by a , invariance of the contracted product u : n results in (Chan, Zienkiewicz and Pastor, 1988) d a :n
= d a * : n*
then, substituting in the above du* d a * =-: da du results in da* n=n*--du and similarly * du*
n,
=n,
da
Finally, the increment of plastic strain is given by
(4.226)
150
CONSTITUTIVE RELATIONS
-
PLASTICITY
and the constitutive tensor CeP in Cartesian coordinates can be written as
This of course, allows any specified stress or strain path to be followed, as illustrated in the following examples. Figure 4.43 shows the predicted behaviour forp' and 0 constant tests against results obtained in the hollow cylinder device (Pastor, Zienkiewicz and Chan, 1989). The model was calibrated with conventional triaxial compression and extension tests only. The hollow cylinder device is able to perform tests with rotation of principal stress axes by combining axial and radial stresses with a variable shear stress r. If, from a triaxial state (al, o2 = 03, r = 0) the shear stress is increased leaving the specimen to drain freely, q will increase while p' is kept constant, and, in addition to this, the principal stress axis will change. In Figure 4.44 (Pastor, Zienkiewicz and Chan, 1989) it is shown how the proposed model can reproduce this special path. In this case, the effect of increasing q prevails over rotation of principal stress axes. However, a pure rotation of principal stress directions will not induce any response from the material, as the model is defined in terms of stress and strain invariants.
Figure 4.43
Constant b tests on Reid sand (experiments from Saada and Bianchini 1989)
151
ADVANCED MODELS
500 -
PREDICTED EXPERIMENTAL - - - - - - - - - -
Shearing of a sand with rotation of principal stress axes (experimentsfrom Saada and Bianchini, 1989)
Figure 4.44
Experiments with pure rotation paths show a plastic volumetric strain under drained conditions and pore pressure generation under undrained loading. Generalization of the proposed model for these phenomena can be made by considering several mechanisms, as it was proposed in Pastor. Zienkiewicz and Chan ( 1 990) using models able to introduce strain and load-induced anisotropy (Pastor, 1991 ).
Unloading and cyclic loading A first step towards modelling cyclic behaviour of sands is to understand what happens when unloading and reloading. Concerning the former, the response is characterized as isotropic and elastic in most classical plasticity models, which is not always very accurate. In fact, it can be observed from experiments that higher pore pressures than those correspondent to elastic unloading appear. Figure 4.45 depicts results obtained by Ishihara and Okada (1982) on undrained shearing of loose sands under reversal of stress. Isotropic elastic unloading is characterized by zero volumetric plastic strain and, as under undrained conditions the volume is constant, the volumetric elastic strain should be also zero and, therefore, p' should not change (a variation of p' causes a change in volumetric elastic strain). Instead of unloading along a vertical line the stress path turns towards the origin which indicates higher pore pressures than isotropic elastic. This phenomenon depends on the stress ratio 7, from which unloading takes place, its importance increasing with it. Two possible explanations are possible: (i) Either the material structure has changed after having crossed the characteristic state line, and the new distribution of contacts makes the specimen anisotropic (Bahda, 1997). or
152
CONSTITUTIVE RELATIONS
-
EXPERIMENTAL
PLASTICITY -
PREDICTED
0.60
1
I Mean confining pressure
Mean confinin pressure P' ( W c m ) -
B
w
n
a
-1.0 o 1.0 2.0 AXIAL STRAIN (96)
AXIAL STRAIN (%)
Figure 4.45 Undrained behaviour of loose sand under reversal of stress (experiments after lshihara and Okada, 1982)
(ii) Plastic deformations develop during unloading. If we assume that plastic strains appear upon unloading, and that they are of a contractive nature, a simple expression for the plastic modulus fulfilling these requirements was proposed in Pastor, Zienkiewicz and Chan (1990)
1
=H
(5) 1 1 '
for
>I
and extends the range of the model so far proposed hierarchically. To determine the direction of plastic flow produced upon unloading, we note that irreversible strains are of a contractive (densifying) nature. The direction n , ~can thus be provided by
ADVANCED MODELS
153
where
and
Concerning reloading, it is necessary, as it was done in the previous section for clays, to take into account the history of past events. Here, we will modify the plastic modulus introducing a discrete memory factor HDMas
where
C was defined above as
and y is a new material constant. Finally, plastic modulus is given by
Figure 4.45 shows a prediction of this model extension for the experimental results of Ishihara and Okada. It is possible now to model cyclic phenomena as liquefaction and cyclic mobility which appear in loose and medium sands under cyclic loading, and which are responsible of catastrophic failure of structures subjected to earthquakes. Both phenomena are largely caused by the overall tendency of medium and loose sands to densify when subjected to drained cyclic shearing. If the load is applied fast enough or the permeability is relatively small, this mechanism causes progressive pore pressure build-up leading to failure. In the case of very loose sands, liquefaction takes place following a series of cycles in which the stress path migrates towards lower confining pressures. Figure 4.46 shows the results obtained by Castro (1969) in his pioneering work. Denser sands do not exhibit liquefaction but cyclic mobility. Failure here is progressive since the stress path approaches the characteristic state line by its shift caused by pore pressure build up. Deformations during unloading cause the stress path to turn towards the origin, and strains produced during the next loading branch are of higher amplitude. Figure 4.47 shows both the experimental results obtained by Tatsuoka on Fuji river sand.
154
CONSTITUTIVE RELATIONS Experimental
-
PLASTICITY
Experimental
(a) 2
1
(bi
0 pt (kg I cm-)
5 Axial stram (%)
10
Predicted
Predicted
2 3 p' (kg 1 cm-)
0
4
5
10
Axial stram (96)
Figure 4.46 Liquefaction of loose banding sand under cyclic loading (predictions and experimental data from Castro, 1969)
,
(a)
(b)
Prcdicted
Predicted
OE$l
g
,
m o
m
5
Y,
0-
u
,
-I
(c)
p'1(kg I cm-)
1
2
2.5
I 10.0
i? j
(4
Experimental
Axial strain (%) 0
Exper~mental
-
-5
"
0
M
5
Y,
u
5
I
I , p' (kg 1 cm-)
-I
2
2.5
-10.0
0 Axial stram (%)
Figure 4.47 Cyclic mobility of loose Niigata sand (predictions and experimental data from Tatsuoka, 1972)
155
ADVANCED MODELS
Table 4.2 Model parameters used in simulations
Fig.4.41(loose) Fig.4.42
Fig.4.45
Fig.4.46
Fig.4.47
Figs.4.43 and 4.44
Table 4.2 above gives the model parameters used in the preceding simulations. The model can be further elaborated as shown in Pastor, Zienkiewicz and Chan (1987) and Pastor, Zienkiewicz, Xu and Peraire (1993) by improving the way in which the history ofpast events is taken into account. To thisend, two elements are introduced: (i) a surface defining the maximum level of stress reached, and (ii) the point at which last reversal took place. Directions n and n,, and the plastic modulus HL depend on the relative position of the stress C with respect to the point at which the load was reversed, B, and an image point, D, defined on the same mobilized stress surface as B. To obtain the values of H L , n and n,, suitable interpolation rules are used. In particular, n is interpolated from -n to n using a linear law. The direction of plastic flow is obtained again by defining a suitable dilatancy at C, d g which is interpolated from an initial value dgoto
The initial value of the dilatancy at the reversal point d8 is given by
156
CONSTITUTIVE RELATIONS
Figure 4.48
-
PLASTICITY
Interpolation rule
where the constant C,(O < C, < 1) varies with the density, being close to zero for medium-loose sands. The plastic modulus is interpolated between an initial value HUoand its final value at the image point on the mobilized stress surface H D . The initial value can be assumed to be infinite to decrease a possible accumulation of plastic strain under very low amplitude cycles.
where f is an interpolation function depending on the relative position of the points B, C and D and which is 1 when C and D coincide. Concerning the rule to obtain the image stress point D, there are several alternative possibilities. For instance, it can be obtained as the intersection of the straight line joining the reversal and the stress point with the mobilized stress surface, as depicted in Figure 4.48. This interpolation law provides a smooth transition between unloading to reloading. In fact, unloading may be considered as a new loading process. It is important to remark that direction of plastic flow and unit vector n will not be functions of the stress state only, but of the past history as well. Finally, the influence of sand densification under cyclic loading can be taken into account by introducing into the plastic modulus a factor H,/
Figure 4.49 shows the densification of a loose sand under cyclic loading, and it can be observed how the volumetric and deviatoric plastic strain produced decreases with the number of cycles. It should be mentioned here that since the simple models we have described here were proposed, several improvements and modifications have been introduced,
157
ADVANCED MODELS
Stress ratio
- 1 .0
1 .0
0
2.0
Shear strain (%)
Figure 4.49
Densification of a loose sand (predicted)
particularly at CERMES (Paris) (Saitta, 1994). where research recently finished has succeeded to include state parameters describing, in a consistent way, the behaviour of sand under different conditions of confining pressure and relative density (Bahda, 1997; Bahda, Pastor and Saitta, 1997).
4.4.4
Anisotropy
Introductory remarks Anisotropy in geomaterials is caused either by the arrangement of particles such as occur in natural deposits in which the grains may have their major axes on the bedding planes, or by the spatial distribution of contacts and contact forces. In the first case, it is found that the strength is higher when tested along the deposition direction. This effect can introduce important errors if not taken into account. For instance, if the number of cycles to liquefaction is determined using a standard triaxial testing machine to evaluate the liquefaction potential of a natural sand
158
CONSTITUTIVE RELATIONS
-
PLASTICITY
deposit, the value obtained will be greater, and, therefore, the strength will be underestimated. Several theories have been proposed within the framework of plasticity to describe both initial and stress-induced anisotropy. Basically, anisotropy has been approached most of the times by changing the position, orientation and shape of isotropic yield, loading or plastic potential surfaces, in such a way that those changes were dependent on tensors such as stress or strain and not only on their invariants. Initial or fabric anisotropy could be reproduced as well by introducing initial movements and distortions on the surfaces. Combination of kinematic and anisotropic hardening laws proposed by Mroz (1967), have provided a suitable way to model anisotropic behaviour of soils (Mroz, Norris and Zienkiewicz, 1979; Hashiguchi, 1980; Hirai, 1987; Ghaboussi and Momen, 1982; Liang and Shaw, 1991; Cambou and Lanier, 1988 and di Prisco, Lanier and Nova, 1993). Surfaces can be allowed to expand following isotropic hardening rules and to translate and rotate under anisotropic hardening laws. If only initial or fabric anisotropy is to be considered, a simple way to introduce anisotropic surfaces is to define a modified second invariant of deviatoric stress tensor
where Aijkl is a fourth-order tensor characterizing material anisotropy. This method was initially suggested by Hill (1950) and extended by Nova and Sacchi (1982) and Nova (1986) to soils and soft rocks. If J i is substituted by J; in any isotropic plasticity model (such as the Cam-Clay, for instance), one finds yield surfaces which have been distorded and rotated. This effect can be introduced also by directly formulating the surfaces on the stress space (Banerjee and Youssif, 1986) or by deriving them from modified anisotropic flow rules, as proposed by Anandarajah and Dafalias (1986). Baker and Desai (1984) suggested to include the effect of stress induced anisotropy via joint stress and stress invariants. Surfaces were made dependent on
in addition to invariants of stress and strain tensors. This approach is based on the representation theorem of scalar functions depending upon two symmetric second-order tensors u and P in this case). All theories mentioned above are able to introduce the anisotropic response of geomaterials even when there is a single mechanism of deformation. Multi-mechanism theories can also describe anisotropic behaviour, provided that they are not formulated in terms of the three stress invariants only. Multi-laminate models as introduced by Pande and Sharma (1983) consider that deformation is caused by dilation and slip taking
159
ADVANCED MODELS
place at all possible contact planes within the material. Of all possible active planes only a reduced number of sampling planes is considered. The overall response is obtained by a process of numerical integration extended to sampling planes. Alternatively, if attention is focused only on planes normal to XY, YZ and ZX, and their responses are grouped together, one finally arrives at a three-mechanism model (Aubry, Hujeux, Lassoudiere and Meimon, 1982 and Matsuoka, 1974). Both multi-laminate and multi-mechanism models of the type described above can produce plastic strain under pure rotation of principal stress axes (Pande and Sharma, 1983; Pastor, Zienkiewicz and Chan, 1990). Alternatively, the behaviour of the material can be assumed to be caused by superposition of responses to variations in 4,o;and 4 , and can then be generalized to more general stress conditions. This has been proposed by Darve and Labanieh (1982) and applied to complex stress paths, including anisotropy effects. Finally, it should be mentioned that material fabric plays a paramount role on geomaterials anisotropic response and it is in turn modified by the deformation process. The fabric may be approximated by a second-order tensor, which can be incorporated into the constitutive equations. An interesting way has been recently proposed by Pietruszczak and Krucinski (1989), and consists on adding two components to obtain the increment of plastic strain. First one corresponds to an isotropic hardening mechanism, and the second accounts for deviations of isotropy, which are made dependent on fabric tensor.
Proposed approach It has been mentioned above that material structure or fabric has to be incorporated in the constitutive equations to account for both initial and induced anisotrpy. Here, it will be assumed that fabric can be described by a second-order structure tensor A, which will determine its type of symmetry. If Q is a rotation or reflection tensor, the class of symmetry will be defined by the set of operators Q which fulfill
For instance, transversely isotropic materials will be described by A invariant under Q given by
Q
=
(a
0 0 cos0 -sin0 s i n O:oc
)
where it has been assumed that the plane of isotropy is XY. The structure tensor will have the form
160
CONSTITUTIVE RELATIONS
-
PLASTICITY
and it can be easily checked that the invariance relation A = Q~ . A . Q is verified. If the initial 'structure' of the material is described by A', A will vary along the loading process, according to
where dA will depend on the plastic strain
Now, the structure tensor can be used to define a fourth-order anisotropy tensor B from which a modified second invariant Ji can be derived as suggested by Hill (1950); Nova and Sacchi (1982) and Nova (1986). Following Cowin (1985), B can be expressed as a combination of terms listed below.
(i) 6 @ 6 (ii) 6 @ A A t 3 6 (iii) 6 t3 A2 A' (iv) A t3 A (v) ABA'
(vi) A'
@
@
6
A'BA
A'
where compact notation has been used. If a transverse isotropic material is considered, the tensor B referred to principal axes is given by
where rows and columns include components of the tensor with their components ordered as
and, therefore, component 1133 is located at first row and third column.
ADVANCED MODELS
161
In the above,
It can be seen that the resulting anisotropy tensor depends only on five constants, and that the form proposed by Nova and Sacchi (1982) is a particular case of the above expression in which BI and B4 have been made one and zero respectively. So far, only JS has been extended to account for anisotropy. However, geomaterial behaviour is also dependent on first and third invariants, and anisotropy should also be reflected on them. New invariants fl and J-; can be introduced in a similar manner, by defining B' and B"' which are tensors of orders two and six. The first anisotropy tensor B' would be dependent on 6 and A, and B'" on double tensorial products of S, A and such as S @ S @ S, A @ S 8 6 , etc. Therefore, the extended set of invariants is given by
Finally, constitutive laws derived for isotropic materials in terms of I,'.J: and J ; can be generalized to anisotropic situations by substituting them by modified forms given above. An interesting particular case is obtained when A is taken as
Then, the constitutive law can be seen to be dependent on joint stress-strain invariants, as proposed by Baker and Desai (1984). As mentioned above, the result of substituting stress invariants by their modified forms can be viewed as introducing a rotation and a distortion of yield and plastic potential surfaces, and indeed some kinematic hardening models in which a back stress is introduced can be considered as particular cases of the theory outlined above. They present, however, the advantage of being simpler to develop. If the three tensors introduced to produce the modified invariants are defined as:
162
CONSTITUTIVE RELATIONS
-
PLASTICITY
it can be checked that this choice corresponds to a pure rotation of the yield and plastic potential surfaces. In the case of generalized plasticity, where surfaces are not introduced, it will be a rotation of directions n and n , ~ , " (Pastor, 1991). The model described in Pastor (1991) and Pastor, Zienkiewicz, Xu and Peraire (1993) introduced a rotation characterized by a new direction of the plastic potential and yield surface axes given by a unit tensor a given by:
The first modified invariant was obtained as
which is proportional to the projection of stress tensor along a. To obtain the second invariant J:, a modified deviatoric stress was defined as
q, g) can be Alternatively, a modified form of the classical set of invariants (j', obtained. Care should be taken when defining a new Lode's angle as the trace of the new modified deviatoric stress is not zero. A possible solution consists of defining a new deviatoric stress t as
from which
and
The modified invariants
ADVANCED MODELS
163
and e can now be introduced into the generalized plasticity model described in the preceding section, to describe the anisotropic behaviour of sand.
A generalized plasticity model for the anisotropic behaviour of sand Sand deposits exhibit anisotropic response caused by the alignment of sand grains on horizontal planes. This initial or inherent anistropy may be modified by subsequent strains developed as the material is loaded. If a specimen of such material is brought to failure, grains will be reorganized as deformation increases, changing the initial structure. It has been shown above how the material response can be described by providing suitable expressions for tensors n,~,", and plastic modulus H L j u . Following this approach, simple models have been derived for isotropic materials (Zienkiewicz. Leung and Pastor, 1985; Pastor, Zienkiewicz and Chan, 1985 and 1990) in terms of invariants I ; , J i and Jj or p', q and 0. It will be shown next how to obtain a simple generalized plasticity model for anisotropic sands. Concerning the plastic flow rule, two experimental facts will be recalled: If experimental data obtained on granular soils with initial anisotropy such as is given in Yamada and Ishihara (1979) are analyzed, it can be found that the zero volumetric incremental strain surface may be described by the same simple relation proposed in the preceding section for isotropic sand
where M, depends only on Lode's angle 0 and q' is the stress ratio. The tests performed by Yamada and Ishihara (1979) consisted of proportional, radial paths performed at constant p' and 0. A detailed description of both the testing procedures and the results obtained is given there. The samples exhibited a strong anisotropy as grains were arranged such that their long axes were horizontal. Therefore, different behaviour was observed along paths such as ZC and YC which have the same value of Lode's angle. However, no such dependence was found for M,, which will be assumed to be independent of anisotropy. 0
A similar analysis may be carried out to study how soil dilatancy, is affected by soil anisotropy. Miura and Toki (1984) analyzed the experiments of Yamada and Ishihara (1979) and concluded that soil anisotropy did not greatly affect the dilatancy behaviour of sand, confirming the linear relation between dilatancy and stress ratio proposed above for isotropic soils.
It was found that the parameter cu depended on 0, and the following relationship was proposed
164
CONSTITUTIVE R E L A T I O N S PLASTICITY
where subindex 'c' refers to values obtained for 6' = 30". So far, it has not been necessary to introduce the modified invariants described above, and expressions giving MJH) and dilatancy holds for both isotropic and anisotropic materials. The modified invariants @'. q, 6 ) will be used, however, in the definitions of direction n and plastic modulus HLI('directly. Microstructure tensor A was assumed to have an initial value A' corresponding to initial fabric. Therefore, it should reflect material symmetries. Naturally deposited sands exhibiting transverse isotropy will have
with axis X I coinciding with direction of deposition, while isotropic materials will be characterized by
As the material is loaded, deformation will produce rotation of grains and rearrangement of microstructure. Therefore, A will change, and dA will be a function of the increment of strain. General expressions for this functions have been suggested by Pietruszczak and Krucinski (1989). Here we will assume that dA may be expressed as
where CCloand CClIare two material parameters and E is the accumulated deviatoric strain. If CLloand C,,, are taken as zero and one respectively, and no initial anisotropy exists, A will coincide with EP, and the modified invariants will be functions of both the stress and mixed stress-strain invariants. To show the performance of the proposed model, Figure 4.50 reproduces both the results obtained by Yamada and Ishihara (1979) and the model predictions (Pastor, 1991). The tests were run on the Fuji River, with specimens constructed by pluviation of sand through water to simulate the natural deposition process. This resulted on a highly anisotropic structure which was modified by subsequent loading. Cubic samples of sand were tested on a true triaxial apparatus along different paths. Those depicted in Figure 4.50 correspond to compression and extension along the vertical. It is interesting to notice how the anisotropic structure induced by the deposition process resulted in a higher overall stiffness for the specimen tested in com-
165
MODIFIED DENSIFICATION M O D E L
3
4 (kglcm-)
Figure 4.50
...
.
Predicted
-Experiment
Anisotropic behaviour of Fuji River sand in triaxial compression and extension
pression along the vertical, and how the situation was reversed when loaded in extension.
4.5 4.5.1
MODIFIED DENSZFICA TZON MODEL Densification model for cyclic mobility
Introduction The densification model is firstly introduced with the classical Mohr-Coulomb criteria for shear strength (Zienkiewicz rt ul, 1978). That model is suited to simulate undrained tests in which the excess pore pressure is built up due to the constant cyclic load, until the full liquefaction takes place or the shear stress reaches the phase transformation line. It cannot, however, simulate the cyclic mobility behaviour after the stress point reaches the phase transformation line. The most important application is the soil layer problem. After the stress reaches the failure line of the Mohr-Coulomb densification model in the middle of the soil layer, the wave does not transfer upward and no response acceleration is observed. This can happen if the soil layer consists of very loose sand but the dense soil layer resists and transfers the
166
CONSTITUTIVE RELATIONS
-
PLASTICITY
upward wave propagation. The modification of the densification model for this kind of phenomena is introduced here. The definition and simulation of a triaxial test is presented.
Modification of the densification model. In order to simulate the cyclic mobility, equation (4.161) and (4.162) are generalized as introduced by Shiomi and Tsukuni (1998). Equations (4.161) can be re-written by the following functions if K ( K ) represents the limitation of the autogeneous volumetric strain at a large damage parameter k,
K is a damage quantity that causes irreversible dilatancy. The damage quantity is dependent on the stress ratio and the accumulated shear train <. Then (4.162) can also be re-written using the following functions. The incremental formula can be as follows.
where 71 is the stress ratio against the current mean stress and H is the stress ratio against the initial mean stress. Most experimental data refers to 0 and very few data are available for the relation between the dilatancy and rl. But if the dilatancy depends on the stress ratio, it should theoretically be 77. Cyclic mobility can be included in the function h(r1). The empirical formulation can be used for this purpose. In order to simulate a triaxial undrained test, the above function can be modified further to avoid numerical difficulty. The dependence of ti is included in the function h; that is h ( 7 , K ) . Therefore, the function K ( K )is defined as
and h ( q ) is replaced by h(r1,K ) . This is because the damage, parameter k defined by equation (4.269) becomes very large when liquefaction takes place and causes numerical difficulty. Now the damage parameter ti has a linear relationship to the dilatancy. r is independent of the stress ratio until the reversal stress ratio is over the phase transfer line qR.Thereafter, C3 takes a constant value as follows, Table 4.3
Definition of function Before cyclic mobility
r After cyclic mobility
where a, b are constants, h(71,~ ) , defined 3 as follows
4q3 K )
=Y,P(~)IXK)
167
MODIFIED DENSIFICATION MODEL
where A,, B,,, y,,are the model parameters of the densification model. 'n' is the index for the strain region, n = 1 before cyclic mobility, n = 2, after cyclic mobility, and n = 3 when K > k,. as shown in Table 4.5 a ( 7 ) is is the differential of the stress path shown in Figure 4.51 and is a function of the stress ratio q(= ~ / a : , )It . is defined as an empirical function and described in the table as { ( a l711, , ( a Zq2), , . . . (a,,,G ) ) .The different function is used before and after the liquefaction as show in Table 4.4.
0.0
0.1
Figure 4.51 Table 4.4
0.2
0.3
0.4
0.5
0.6
o:,, (MPa) Model of stress path for dilation behaviour
Definition of a ( q ) Function a ( q ) Before cyclic mobility After cyclic mobility
For loading
Defined by table (e.g. Table 4.7)
For unloading
lrll 0.1 q'?
lrll 1 -% 11,
-
lrll
'11
where 1 7 ~= tan& and q, = tan$, , & is the phase transfer line and $f is the failure line. P(K) depends on the damage parameter. Table 4.5
Definition of
P ( K ) (a) for K < K,.
and (b) for
K
2 K,.
(a)
Before cyclic mobility (n= 1 ) After cyclic mobility (n=2)
Loading
I - C~L(O) C ~ L ( O ) +I + B l K
cru
+ 1 +1B-Z (Czc K KO) -
Unloading
1.0
I CZL C2Lt1+BI(~-~U) -
168
CONSTITUTIVE RELATIONS
Figure 4.52
Table 4.5 (b)
Relationship between the damage parameter
Contu' for k
K
-
PLASTICITY
and P(K)(CZI.= 0.5)
> kc After cyclic mobility (n=3)
Loading Unloading
Here KO is the damage parameter when cyclic mobility takes place. The coefficients C7L and C2L' are coefficients for the loading process and the unloading process respectively. An example of P ( K ) is shown in Figure 4.52. The function P ( K ) reaches a certain value when the damage parameter ti becomes large.
Simulation of a triaxial test The following simulation of a triaxial test shows the capability of the modified densification model. Tables 4.6 to 4.8 show the parameters of the model. There are many coefficients but these coefficients can be determined from the undrained cyclic test data. Figure 4.53 shows the stress path of the experimental and numerical simulation. Both results show the cyclic mobility behaviour. Figure 4.54 shows the stress-strain curve. Strains become large after the cyclic mobility starts and loses
169
MODIFIED DENSIFICATION MODEL
stiffness at low shear stress. This is well represented. Furthermore the enlargement of the strain after cyclic mobility takes place also shows good agreement, as shown Figure 4.55. Table 4.6 Parameters of modified densification model c
4 4
0.OMPa 41.8 deg. 25.0 deg.
yl AI B1
4.0 x lo7 1.0 x lo-' 2.0
Table 4.7 Function
Table 4.8
25.0 y3 0.016 A3 8.0 B3
y2 A? B2
100.0 CZL 0.60 0.004 Czo 4.0 8.0 n, 0.05
u
b G
471)
Function C4L
Experiment (q = 0.147)
Calculation
(
=
0.147)
0.01 5
Vi
-g2 ez:2 2 rc m
0.000
m
-0.015 0.00
0.02
0.04 0.06 0.08 Effective mean stress o',(MPa)
Figure 4.53 Stress path of simulation
0.10
2.50 0.50 90Mpa
170
CONSTITUTIVE RELATIONS
-
PLASTICITY
Experiment (q=O.147)
Calculation (q=0.147) 0.015
n
-z2 r_ P.
2 V)
0.000
z
-0.0 15 -4
Figure 4.54
-2
0 Shear strain y(%)
2
4
Stress-strain curve of a triaxial test (experiment and simulation)
10 Number of cycles N
100
Figure 4.55 Enlargement of shear strain due to accumulation of damage (experiment and simulation)
REFERENCES
171
REFERENCES Anandarajah, A. and Dafalias. Y. F. (1986) Bounding surface plasticity 111: Application to anisotropic cohesive soils J. Eng. Mech. ASCE, 112, 12, 1292-1 3 18. Atkinson, J. H. and Bransby, P. L. (1978) The Mechanics qfSoi1.s; An Introduction to Criticcrl State Soil Mechanics. McGraw Hill, London. Atkinson, J. H. and Richardson, D . (1985) Elasticity and normality in soil: experimental examinations, GPotecliniqlre,35. 443449. Aubry, D. Hujeux, J. C. Lassoudiere. F. and Meimon, Y. (1982) A double memory model with multiple mechanisms for cyclic soil behaviour, Int. Synzp. Num. Models in Geomechanics. R. Dungar, G . N. Pande and J. A. Studer, (Eds) Balkema, Rotterdam. Bahda. F. (1997) Etude du comportement du sable au triaxial, PhD Thesis, ENPC-CERMES Paris. Bahda, F. Pastor, M. and Saitta, A. (1997) A double hardening model based on generalized plasticity and state parameters for cyclic loading of sands, G . N. Pande and S. Pietruszczak. Baker, R. and Desai, C. S. (1984) Induced anisotropy during plastic straining, Int. J. Num. Anal. Methods Geon~ech.,8, 167-185. Balasubramanian, A. S. and Chaudhry, A. R. (1978) Deformation and strength characteristics of soft Bangkok clay, J. Geotech. En& Div. ASCE, 104, NO.GT9, 1153-1 167. Banerjee, P. K. and Yousif. N. B. (1986) A plasticity model for the mechanical behaviour of anisotropically consolidated clay, Int. J. Num. Anal. Methods Geomech, 10, 521-541. Bardet, J. P. (1989) Prediction of deformations of Hostun and Reid Bedford sands with a simple bounding surface plasticity model, in Constitutive Equations ,for Grunulcw Noncohesive Soils, A. Saada and G . Bianchini (Eds), 131-148, Balkema. Baiant. Z. P. and Krizek, R. J. (1976) Endochronic constitutive law and liquefaction of sand. Proc. Am. Soc. Civ. Eng. EM 102, 225-38. Burland, J. B. (1965) Correspondence on 'The yielding and dilatation of clay'. Gkotechnique 15. 21 1-214. Cambou, B. and Lanier J. (1988) Induced anisotropy in cohesionless soil: experiments and modelling, Computers and Geotechnics 6, 291-3 1 1. Castro, G . (1969) Liquefirction of'scmds, Ph.D. Thesis, Harvard Univ., Harvard Soil Mech. Series no. 8 1. Chan, A. H. C. Zienkiewicz, 0. C. and Pastor, M . (1988) Transformation of incremental plasticity relation from defining space to general cartesian stress space, Comrnrm. Appl. Num. Meth., 4, 577-580. Cowin, S. C. (1985) The relationship between the elasticity tensor and the fabric tensor, Mech. Materials, 4, 137-1 47. Coulomb, C. A. (1773) MPm. M~rth.et Phys. 7 , 343. Cuellar, V.. Baiant, Z. P., Krizek, R. J. and Silver, M. L. (1977) Densification and hysteresis of sand under cyclic shear, J. Geotech. Eng. Div.. ASCE, 103, 9 18. Dafalias, Y. F. (1986) Bounding surface plasticity. I: Mathematical foundation and hypoplasticity, J. Eng. Mech. ASCE, 112, 966-987. Dafalias, Y. F. and Herrmann, L. R. (1982) Bounding Surface Formulation of Soil Plasticity. in Soil Mechanics-Transient and Cyclic L O N ~GP., N . Pande and 0. C. Zienkiewicz (Eds). Ch. 10, 253-282, Wiley. Dafalias, Y. F . and Popov, E. P. (1975) A model of non-linearly hardening materials for complex loadings, Acta Mech. 21, 173-192. Darve, F. (Ed.) (1990) Geonirrter-iuls. Constitutive Equations and Modelling. Elsevier Applied Science. Darve. F. and Labanieh, S. (1982) Incremental constitutive law for sands and clays: simulation of monotonic and cyclic tests, Int. J. Numer. Anal. Meth. Geomechs. 6, 243-275.
172
CONSTITUTIVE RELATIONS
-
PLASTICITY
Darve, F. Flavigny, E. and Rojas, E. (1986) A class of incrementally non-linear constitutive relations and applications to clays, Computers cmd Geotechnics, 2, 43-66. Desrues, J. and Chambon, R. (1993) A new rate type constitutive model for geomaterials: CloE. D. Kolymbas (Ed.), in Modern Approuches to Plrsticity, 309-324, Elsevier. di Prisco, C. Nova, R. and Lanier, J. (1993) A mixed isotropic kinematic hardening constitutive law for sand in D. Kolymbas (ed.), Mo~lernApprouches to Plusticity, pp. 83-124. Balkema. Drucker, D. C. (1956) On uniqueness in the theory of plasticity, Quurt. Appl. Muth.. 14, 35-42. Drucker, D . C., (1959) A definition of unstable inelastic material, J. Appl. Mech., 26, 101-106. Drucker, D. C. and Prager, W. (1952) Soil Mechanics and Plastic Analysis or Limit Design, Quurt. Appl. Math. 10, 157-165. Drucker, D . C. Gibson, R. E. and Henkel, D. J. (1957) Soil Mechanics and Workhardening Theories of Plasticity, Trans. ASCE 122, 338-346. Frossard, E. (1983) Une equation d'ecoulement simple pour les materiaux granulaires, GPotechnique, 33, 1, 21-29. Ghaboussi, J. and Momer, H. (1982) Modelling and analysis of cyclic behaviour of sands, Soil Mecl~anics-Transientund Cyclic Loudk, pp. 313-342 G . N. Pande and 0. C. Zienkiewicz (Eds). Wiley. Habib. P. and Luong, M. P. (1978) Sols pulvurulents sous chargement cyclique, Materiuu.~ and Structures Sous Churgernent Cyclique, Ass. Amicale des lngenieurs Anciens Eleves de I'Ecole Nationale des Ponts et Chaussees (Palaiseau. 28-29 Sept. 1978). pp. 49-79. Hashiguchi, K. (1980) Constitutive equations of elastoplastic materials with elastic-plastic transition. J. Appl. Mech. A S M E , 47, 266-272. Hashiguchi, K. and Ueno, M. (1977) Elastoplastic Constitutive laws of granular materials, Constitutive Equations of'Soils, 9th. Int. Congr. Soil Mech. Found. Engng., S. Murayama and A. N. Schofield (Eds), pp. 73-82, JSSMFE. Hashiguchi, K. Imamura, T. and Ueno. M. (1989) Prediction of deformation behaviour of sands by the subloading surface model, in Constitutive Equntionsfbr Grunulur Non-cohesive Soils, A. Saada and G . Bianchini (Eds). 131-148, Balkema. Henkel. D . J. (1956) The effect of overconsolidation on the behaviour of clays during shear, Georechnique 6, 139-1 50. Henkel, D. J. (1960) The shear strength of saturated remoulded clay, Proc. Research Conference on Sheur Strmgth of' Cohesive Soils, Boulder, Colorado, pp. 533-540. Hill, R. (1950) The Mathemuticul Theory of Plusticity. Oxford, Clarendon Press. Hirai, H . (1987) An elastoplastic constitutive model for cyclic behaviour of sands, Int. J. Nun?. A n d . Metk. Geomech. 11, 503-520. Huber, M. T. (1904) C:~.sopismo technicxe, Lemberg, 22, 81. Ishihara, K. and Okada, S. (1982) Effects of large preshearing on cyclic behaviour of sand, Soils und Foundations, 22, 109- 125. Ishihara, K. Tatsuoka, F. and Yasuda, S. (1975) Undrained deformation and liquefaction of sand under cyclic stress, Soils and Foundutions, 15, 29-44. Kaliakin. V. N. and Dafalias, Y. F. (1989) Simplifications to the bounding surface model for cohesive soils, Int. J. Num. Anal. Merh. Geomech. 13. 91-100. Kolymbas, D. (1991) An outline of hypoplasticity, Archive ofApplied Mechanics, 61, 143-151. Kolymbas, D. and Wu, W. (1993) Introduction to Hypoplasticity, Kolymbas, D. (Ed.) in Modern Approaches to Plasticity. 21 3-224, Elsevier. Krieg, R. D . (1975) A practical two-surface plasticity theory, J. Appl. Mech.. Trans. A S M E , E42, 641-646. Levy, M. (1871) J. Muth. Pures et App. 16, 369. Liang, R. L. and Shaw, H. L. (1991) Anisotropic Plasticity Model for Sands, J. Geotech. Eng. ASCE, 117, N0.6, 913-933.
REFERENCES
173
Martin, G. R. Finn, W. D. L. and Seed, H. B. (1975) Fundamentals of liquefaction under cyclic loading, Proc. Am. Soc. Civ. Eng., GT 5, 423-38. Matsuoka, H. (1974) Stress-strain relationships of sands based on the mobilized plane, Soils and Foundutions, 14, No2, pp. 47-61. Melan, E. (1938) Ingenieur-Archiv, 9, 116. Miura, S. and Toki, S. (1984) Elastoplastic stress-strain relationship for loose sands with anisotropic fabric under three-dimensional stress conditions, Soils and Foundations, 24, 43-57. Mroz, Z. (1967) On the description of anisotropic work-hardening, J. Mech. Phys. Solids. 15. 163-175. Mroz, Z. and Norris, V. A. (1982) Elastoplastic and viscoplastic constitutive models for soils with applications to cyclic loading, in Soil Mechanics-Transient and Cyclic Loads, G. N . Pande a n d 0.C. Zicr~kicwiiz<Eds(,pp. 'I 7~+-21' 7, J ' U ~ W V q am' SVCIJ-. Mroz Z. and Zienkiewicz 0. C. (1985) Uniform formulation of constitutive equations for clay and sand, Mech. Eng. Mater. Ch. 22, pp. 415450 C. S. Desai and R. H. Gallaher (Eds), John Wiley and Sons. Mroz, Z. Norris, V. A. and Zienkiewicz, 0.C. (1978) An anisotropic hardening model for soils and its application to cyclic loading, Int. J. Num. Anal. Meth. Geomech., 2, 203-221. Mroz, Z. Norris, V. A. and Zienkiewicz, 0 . C. (1979) Application of an anisotropic hardening model in the analysis of elastic-plastic deformation of soils, Geotechnique, 29, 1-34. Mroz, Z. Norris. V. A. and Zienkiewicz, 0. C. (1981) An anisotropic critical state model for soils subjected to cyclic loading, Giotechnique, 31, 451469. Nova, R. (1977) On the hardening of soils, Arch. Mech. Stos., 29, 3, 445458. Nova, R. (1982) A Constitutive Model for Soil under Monotonic and Cyclic Loading, in Soil Mechanics-Trunsient und Cyclic L o ~ d sG. , N. Pande and 0. C. Zienkiewicz (Eds) pp. 343374, John Wiley and Sons. Nova, R. (1986) An extended cam clay model for soft anisotropic rocks, Camp. and Geotech.. 2, 69-88. Nova R . and Sacchi G. (1982) A model of the stress-strain relationship of orthotropic geological media, J. Mech. Theor. Appl., 1, 6, pp. 927-949. Nova, R. and Wood, D. M. (1978) an experimental program to define yield function for sand, Soils and Foundutions, 18, 77-86. Nova, R . and Wood, D. M. (1979) A constitutive model for sand, Int. J. Num. Anal. Meth. Geonzech. 3, 255-278. Pande, G. N. and Sharma, K. G. (1983) Multi-laminate model for clays: a numerical evaluation of the influence of principal stress axes, Int. J. Num. Anal. Meth. Geomech., 7, 397418. Parry, R. H. G. (1960) Triaxial compression and extension tests on remoulded saturated clay. Gbotechnique, 10, 166-1 80. Pastor, M. (1991) Modelling of Anisotropic Soil Behaviour, Computers and Geotechnics, 11. 173-208. Pastor, M. and Zienkiewicz, 0 . C. (1986) A generalized plasticity hierarchical model for sand under monotonic and cyclic loading, Proc. 2nd. Int. Con$ Numerical Models in Geomechanics, Ghent (Belgium), 31st March-4 April, Pande, G. N. and Van Impe, W. F. (eds). M. Jackson and Son Publ. pp. 131-150. Pastor, M. Zienkiewicz, 0 . C. and Chan, A. H. C. (1987) A Generalized Plasticity Continuous Loading model for Geomaterials, in Numerical Methods in Engineering: Theory and Appro.ximation, 0 . C . Zienkiewicz, G . N. Pande and J. Middleton (Eds), Martinus Nijhoff Publ. Pastor, M. Zienkiewicz, 0. C. and Chan, A. H. C. (1989) Generalized plasticity model for three-dimensional sand behaviour, in Constitutive Equations for Granular Non-Cohesive Soils, A. Saada and G. Bianchini (Eds), Balkema, 535-549.
174
CONSTITUTIVE RELATIONS
-
PLASTICITY
Pastor, M. Zienkiewicz, 0 . C. and Chan, A. H. C. (1990) Generalized Plasticity and the modelling of soil behaviour, Int. J. Num. Anul. Meth. Geomech., 14, 151-190. Pastor, M. Zienkiewicz, 0 . C. and Leung, K. H. (1985) Simple model for transient soil loading in earthquake analysis. I1 Non-associative models for sands, Int. J. Num. Anal. Meth. Geotnech.. 9, 477498. Pastor, M. Zienkiewicz, 0. C. Xu Guang-Dou and Peraire, J. (1993) Modelling of sand behaviour: cyclic loading, anisotropy and localization, in Modern Approaches to Plasticity D. Kolymbas (Ed.), pp. 469492, Elsevier. Pietruszczak. S. and Krucinski, S. (1989) Description of anisotropic response of clays using tensorial measure of structural disorder, Mech. Muter., 8, 237-249. Poorooshasb, H. B. Holubec, I. and Sherbourne, A. N. (1966) Yielding and flow of sand in triaxial compression (Parts I1 and HI), Can. Geotech. J. 3, pp. 179-190. Poorooshasb, H. B. Holubec. I. and Sherbourne, A. N. (1967) Yielding and flow of sand in triaxial compression (Parts I1 and III), Can. Geotech. J. 4, pp. 376-397. Prandtl L. (1924) Proc. 1st Int. Cong. Appl. Mech., pp. 43, Delft. Prevost, J. H. (1977) Mathematical modelling of monotonic and cyclic undrained clay behaviour, Int. J. Num. Anal. Meth. Geomech., 1, 195-216. Reuss. A. (1930) Zeits. Ang. Math. Mech. 10, 266. Roscoe, K. H. and Burland, J. B. (1968) On the generalized stress-strain behaviour of wet clay, in Engineering Plasticity, J. Heyman and F. A. Leckie (eds.), pp. 535-610, Cambridge University Press. Roscoe, K. H. Schofield, A. N. and Thurairajah. A. (1963) Yielding of clays in states wetter than critical. GPotechnique, 13, 21 1-240. Roscoe, K. H. Schofield, A. N. and Wroth, C. P. (1958) On the yielding of soils, GPotechnique, 8, 22-53. Rowe, P. W. (1962) The stress-dilatancy relation for static equilibrium of an assembly of particles in contact, Proc. Roy. Soc., A269, 500-527. Saada, A. and Bianchini, G . (eds) (1989), Constitutive Equations,for Granular Non-Cohesive Soils, Balkema. de Saint-Venant, B. (1870) Cot7lptes Rendus Acad Sci. Paris, 70, 473. Saitta. A. (1994) Modelisaton elastoplastique d u comportement mecanique des sols. application a la liquefaction des sables et a la sollicitation d'expansion de cavite, P h D Thesis, ENPC-CERMES Paris. Sangrey, D. A. Henkel, D. J. and Esrig, M. I. (1969) The effective stress response of a saturated clay soil to repeated loading, Can. Geotech. J., 6, 241-252. Schofield. A. N. and Wroth, C. P. (1958) Critical State Soil Mechanics, McGraw-Hill. Shiomi. T . and Tsukuni, S., (1998) Applicntion of Root's dynamic equation to seismic liquification problem, Poromechanics, Thimus et.al, (eds.) Balkema, 505-510. Smith. G . F. (1971) On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J. Eng. S c i , 9, 899-916. Tanimoto, K. and Tanaka, Y. (1986) Yielding of soils as determined from acoustic emission, Soils and Foundations, 26, 69-80. Taylor, D . W. (1948) Fundammtuls of'Soil Mechunics, Wiley. Taylor, P. W. and Bacchus, D. R. (1969) Dynamic cyclic strain tests on a clay, Proc. 7th. Int. Cot$ on Soil Mech. and Found. Eng. I, 401409, Mexico. Tresca, H. (1864) Cornptes Rendus Accrd. Sci. Paris, 59, 754. von Mises, R. (1913) Gottinger Nachrichten, math.-phys. Klasse, 582. Wang, Z. L. Dafalias, Y. F. and Shen. C. K. (1990) Bounding surface hypoplasticity model for sand, J. Eng. Mecl~.ASCE, 116, 983-1 001. Wilde, P. (1977) Two-invariant dependent model of granular media, Arch. Mech. Stos, 29, 799-809.
175
REFERENCES
Wineman, A. S. and Pipkin, A. C. (1964) Material symmetry restrictions on constitutive equations, Arch. Rat. Mech. And., 17, 184-214. W.Wu and Kolymbas, D. (1990) Numerical testing of the stability criterion for hypoplastic constitutive equations, Mech. Muter., 29, 195-201. Yamada, Y. and Ishihara, K. (1979) Anisotropic deformation characteristics of sand under three-dimensional stress conditions, Soils und Foundutions, 19, 2. pp. 79-94. Zienkiewicz, 0. C. and Mroz Z. (1985) Generalized Plasticity formulation and application to Geomechanics, Mrcli. Etg. Muter. Desai, C. S. and Gallagher, R. H. (Eds), Ch. 33. pp. 655-680 John Wiley and Sons. Zienkiewicz, 0. C. and Pande, G . N. (1977) Some useful forms of isotropic yield surfaces for soil and rock mechanics, in Finite Eletnmts in Geomechanics, G. Gudehus (Ed.), Wiley. Chapter 5, pp. 179-190. Zienkiewicz, 0. C. Chang, C. T. and Hinton E. (1978) Non-linear seismic response and liquefaction, Int. J. Num. A n d . Metlzods in Geomech, 2, 381404. Zienkiewicz, 0. C. Humpheson C. and Lewis, R. W. (1975) Associated and non-associated viscoplasticity and plasticity in soil mechanics, G&technique, 25, 671-689. Zienkiewicz, 0. C. Leung, K . H. Hinton, E. and Chang, C. T. (1982) Liquefaction and permanent deformation under dynaniic conditions Numerical solution and constitutive relations, in Soil Mpchunic..~ Tvunsimt und Cyclic Lourls. G . N. Pande and 0. C. Zienkiewicz (Eds). pp. 71-104, John Wiley and Sons. Zienkiewicz, 0.C. Leung, K. H. and Pastor, M. (1985) Simple model for transient soil loading in earthquake analysis. 1 Basic model and its application, Int. J. Nutn. A t d Metl~. Geotnerh. 9, 453-476. -
-
Examples for Static, Consolidation and Partially Saturated Dynamic Problems
5.1 INTRODUCTION In this chapter we deal with the solution of static and quasi- static problems in which dynamic (inertial) effects are negligible. In the latter category the phenomena associated with consolidation are typical and both fully and partially saturated cases will be presented. Since finite deformation and partial saturated examples are only given in this chapter, examples involving hypothetical dynamic behaviour using finite deformation and under partially saturated condition respectively are also given. In both classes of problems we are concerned with the deformation and movement of the soil or of its associated foundation. An excessive amount of the latter is a measure of failure generally or loss of servicability where the stress distribution or its magnitude of pore pressures are indicative of the state of the material stressed. It is the dejormution and displucement which are observable and must be determined. For these the knowledge of the constitutive relation discussed in the previous chapter is of paramount importance, but the simplest constitutive law, which answers the question posed by the engineers and which provides the determination of failure, is to be used at all times. Failure is sometimes associated with continuing displacement without a load increment. This is a definition which is often accepted. However, on occasion a finite displacement can be specified as failure by the engineer and knowledge of displacements is important even if these are not excessive. In the first three chapters of this book we have formulated the dynamic problem and its solution with the time dependence being retained in the final discretized equations. It is clear (as indicated earlier) (see Equation (3.66)) that the problems of consolidation can be directly solved by the code based on our formulation, as with slow motion the dynamic effects become automatically negligible. Less computational effort is involved if the acceleration terms are neglected and the G N l l scheme is used for both the skeleton displacement and pore pressure. However, it is not
178
CONSOLIDATION AND P A R T I A L L Y SATURATED D Y N A M I C PROBLEMS
obvious that static problems can be directly dealt with by the general program though it has been customary to create special programs for such analyses. In Section 3.2.4 we have shown how the static problem can be dealt with by the dynamic code without any loss of efficiency when an appropriate time stepping scheme, i.e. the GNOO scheme, is employed. In this chapter, we shall introduce first in Section 5.2 some typical static problems using a non-associative Mohr-Coulomb material model. Particular attention is paid to the effect of the plastic dilatancy effect. Since fully saturated behaviour is but a special case in our formulation, Sections 5.35.6 are devoted to various problems concerning consolidation under partially saturated conditions. First, the analysis of an isothermal drainage of water from a vertical column of sand is compared with experimental results in Section 5.3. It is followed by the modelling of subsidence due to pumping from a phreatic aquifer in Section 5.4. The effect of air flow is considered in Section 5.5 which is the numerical modelling of air storage in an aquifer. Then we return to the theme of bearing capacity in Section 5.6 by the modelling of a flexible footing resting on a partially saturated soil. Lastly, in Section 5.7, a comparison of consolidation and dynamic results for various fully saturated and partially saturated problems using both small strain and finite deformation formulation, is given. The comparison begins with the consolidation of a fully saturated soil column in Section 5.7.1. It is followed by the comparison between the consolidation of a fully saturated and a partially saturated soil column in Section 5.7.2. A two-dimensional consolidation example is introduced in Section 5.7.3 for a soil layer consolidating under fully saturated and partially saturated conditions, respectively. The comparison of the small strain and finite deformation formulation under earthquake loading is given for a one-dimensional soil column example in Section 5.7.4. Lastly in Section 5.7.5, the elasto-plastic largestrain behaviour of an initially saturated vertical slope subjected to gravitational load and horizontal earthquake acceleration, followed by a consolidation phase, is given. In the next section, we shall begin with some typical static problems.
5.2 STA TIC PROBLEMS In this section we shall present some typical problems of static analysis both of the fully drained and totally undrained kind (the latter usually involving fairly rapid load application during which seepage can be considered negligible). In this section we shall endeavor to show the reader that numerical solutions, though costly by comparison with the simple limit methods which are widely and often successfully used in geomechanics, are necessary and provide otherwise unavailable information. The examples chosen will show that: (a) For a relatively unconstrained situation, such as that involved in the stability of embankments and some foundation problems, the failure loads obtained by numerical computation are not very dependent on the plastic flow rules and again match well the predictions by conventional (limit-based) analysis.
STATIC PROBLEMS
179
(b) For situations in which non-homogeneous material distribution or geometry provides an appreciable amount of confinement, limit-based methods do not predict the failure loads satisfactorly. (c) Drained and undrained behaviour can be studied effectively by a single specification of material properties using the effective-stress concept, and finally: (d) The solution can be quite sensitive to the form of the yield section assumed in the T-plane In all the examples we shall specify the material properties by a yield surface of Mohr-Coulomb type with a straight-line tangent and in which 'yield' is assumed to occur when the shear stress T exceeds a given cohesion c' and a friction angle o':
where a', refers to the normal tensile effective stress. The superscript' implies that these quantities are effective stress parameters. For the dry problem, the effective stress will be the same as the total stress n. In this section, as effective parameters are always used, there is no confusion if the superscript' is dropped.
5.2.1 Example (a): Unconfined situation-small constvaint In this set of examples we shall study the importance of associative and non-associative flow rules and will find that here the choice of the flow rule is of almost negligible importance as far as failure loads are concerned.
Embankment The first example is that of an undrained soil in an embankment in which the angle is steeper than that of the internal friction and in which cohesion has been added. The computation was carried out by Zienkiewicz et a1 (1975) from which Figure 5.1 is taken. In this problem the collapse is achieved by progressively reducing the cohesion. The value of the cohesion at the collapse point is identified as that obtained by slip circle analysis (despite the coarse mesh employed). Further, the difference in the collapse load between the assumptions of associative and non-associative behaviour is very small.
Footing This example, also taken from the same reference as the previous one, shows the collapse of a footing exerting a uniform load on the soil stratum. Here no exact closed
180
5
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
5
-8.0
r
10
Cohesion : k ~ / m ' 20 25 30
15 I
Cohesion : k ~ / m '
Collapse cohesion by slip circle
-
.2-13.0 K
b . 0
-
1
I
y
- 0
-
I
Non-associative flow rule (7,b=0,@=20)
m *
I
.8 - 5 . 0 -
II
-8
I
-6.0 - s ~ oconverged t
Material propenles E=Zx 1 0 J k ~ l r n L v=0.25 @=2O y-20 k ~ l r n '
STATIC PROBLEMS
181
Figure 5.1 Embankment deformation flow patterns and maximum effective shear strain contours (gravity constant; progressive decrease of cohesion) (a) Associative 0 = 4 = 20" (b) Non-associative 0 = 0 4 = 20". Reproduced from Zienkiewicz (1975) by permission of the Institution of Civil Engineers
form solution exists, but for collapse, a mechanism suggested by Prandtl (1920) and Terzaghi (1943) gives loads which are compared with the numerical ones in Figure 5.2. Once again very close results were obtained for the limit loads by both associated and non-associated plasticity models. The two previous examples suggest that numerical analysis adds little to the solution of the problem from the point of view of practical geomechanics. Indeed, the main addition appears to be only the increased cost of analysis and little additional information has been gained. This, however, is just not true as we shall illustrate in the following sections with examples in which some degree of constraint exists.
5.2.2 Example (b): Problems with medium (intermediate) constraint on deformation The first example here is that of a heterogeneous embankment illustrated in Figure 5.3. Here the different plastic displacements of each layer are such as to cause a collapsed load for which simple geomechanic methods is not sufficient. In the table attached to the figure we show that a limit approach gives safety factors ranging from 1.09 to 1.54 while numerical solutions show little influence on the flow rule employed giving 1.165 for both associative and non-associative flows. The second example illustrates however a case where differences between associated and non-associated rules becomes appreciable. This case is that of an Axisymetric triaxial sample loaded between two rough platens as shown in Figure 5.4. Different degrees of dilatancy now affect the load appreciably.
182
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
5.2.3 Example ( c ) : Strong constraints-undrained behaviour Probably the highest degree of constraint on the deformation of the soil skeleton is the behaviour occuring during undrained deformation. In such conditions the total volumetric strain is controlled by the compressibility of the pore fluid and this is generally small, resulting in almost no overall change in volume during deformation. Prescribed load 4
Material properties
0
Plane strain 32 quadratic elements
Applied footing pressure (Iblin') 40 60 80 100 120 140 160 180
20
Terzaghi
\wNot Prandtl
Nan-associated flow' rule ( q j
=
0")
!on\ I
Mohr-Coulomb material
:C
=
I0 lblih c$
.'\
JP"(~P'~
= 20"
Critical State 4 model I I
Figure 5.2 (a) Strip load on a foundation of a weightless c-4 material; ideal plasticity with associated and non-associated (non-dilatant) flow rules and strain-hardening plasticity. Mesh and load-deformation behaviour
STATIC PROBLEMS
183
(ii)
Figure 5.2(b) Relative plastic velocities at collapse (drained behaviour) (i) Associated Mohr-Coulomb (dilatant) (ii) Non-associated Mohr-Coulomb (zero dilatancy) (iii) Strain dependent critical state model. Reproduced from Zienkiewicz (1974) by permission of John Wiley & Sons Limited
Such behaviour has two consequences: (i) If the compressive modulus (Bulk modulus) of the skeleton is small compared with that of the fluid, any changes of mean total stress Aa, are compensated by equal and opposite changes of pore pressure Ap and the mean effective stress remains unchanged i.e. Ad, = 0.
184
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
;,,
O r i g i n : & r Granular l fill material ) = 35 Granular material 0 Organic silt
10 20
=
) = 33
;25 lbln)
) = 20 = 125 1wrt2
y=llolb/f?
0
20
40
60
80
100
120
140
Method
Safety factor
Reference
Whitman-Bailey Bishop Fellenius Sarma Morgenstern-Price Bell Associated Mohr-Coulomb Non-associated Mohr-Coulomb
1.24-1.26 1.33 1.09 1.542 1.557 1.49 1.165 1.165
Whitman-Bailey (1967) Whitman-Bailey (1967) Whitman-Bailey (1967) Sarma (1973) Sarma (1973) Be11 (1968)
Figure 5.3 Layered embankment problem (a) geometry and material properties (b) finite element mesh (53 quadratic isoparameric elements) (c) relative shear strain rate contours at collapse (d) table of safety factors using various methods. Reproduced from Zienkiewicz (1975) by permission of the Institution of Civil Engineers
Thus if the material is initially at the point of yielding as shown in Point A of Figure 5.5 any deviatoric stress changes will occur without changes of the mean stress and the material will behave like a von Mises (or Tresca) solid with respect to total stress changes providing no volumetric stress occurs during such straining. (ii) If the plastic yielding is such that deviatoric strain changes of the skeleton must be accompanied by volumetric strain changes (as for instance required by associative laws) then the only way to achieve overall straining without volume change is to compress (or expand) the fluid. With the volume expansion required by associated
STATIC PROBLEMS
1 85 ;Prescribed load 4
Material properties
C = 10 m / m 2
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 4
Displacement v:m x 10
@=45
Figure 5.4 Axisymmetric sample between rough platens. Effect of degree of dilatancy 9 = 0 (zero dilatancy) and 9 = $J = 45" (fully associated flow rule). Reproduced from Zienkiewicz (1978) by permission of John Wiley & Sons Limited
Figure 5.5 The Mohr-Coulomb trace in the mean stress - deviatoric stress plane. Changes of total stress from point A. Reproduced from Zienkiewicz (1974) by permission of John Wiley & Sons Limited
flow rule, the material will develop negative pore pressures during plastic straining and will gain strength continuously. Figure 5.6 shows the same footing problem as that in Figure 5.2 but now solved (on the same mesh) introducing the undrained assumption. It will be observed that for a non-associative behaviour with no dilatancy in failure, or for a critical state model where failure occurs with no volume changes, the failure loads are almost
186
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS Applied footing pressure (lblin')
-
Pressure (lblinL)
IV up to = 3500 Iblin
but no collapse load found
I Elastic ideal plastic (Mohr Coulomb) associative (dilatant) I1 Elastic ideal plastic (Mohr Coulomb) non associative (zerodilatancy) 111 Elastic strain dependent plastic (criticalstate) IV Elastic ideal plastic total analysis of non Mises
Figure 5.6 Load deformation characteristics (undrained conditions) for plane footing. Reproduced from Zienkiewicz (1974) by permission of John Wiley & Sons Limited
identical with those of total stress analysis with Tresca assumption. However the non-associative, expanding material allows no overall failure. It must be remarked that in the present case the failure loads are governed entirely by the cohesion existing in the material as otherwise the strength would be simply zero when gravity were absent giving an entirely different starting point (A) in Figure 5.5 for the material at different response.
5.2.4
Example (d): The effect of the T section of the yield criterion
In all of the previous examples, we have assumed that the full Mohr-Coulomb relation is employed which requires the yield (andlor flow) surface giving, in the absence of friction; the Tresca yield surface rather than the better known and simpler von Mises one. However, with this type of definition, difficulties arise at corners whlch need special treatment (see Zienkiewicz and Taylor, 1991) and many simplifications have been suggested in the literature. In Figure 5.7, we show the 7r plane section of the Mohr-Coulomb surface for 4 = 20" and various approximations of it,
187
STATIC PROBLEMS
1" ,-
om press ion cone
s-Zienkiewicz-Pande ompromise envelope Extension cone Mohr-Coulomb
Figure 5.7 The -/r plane section of the Mohr-Coulomb surface with 4 = 20" and various smooth approximations. Reproduced from Zienkiewicz (1978) by permission of John Wiley & Sons Limited
.LoEompression cone
/
i
/
1
Compramise cone
d
E = 30000 Iblin2 n = 0.3 c = 10 lbIin.2 f = 20" 32 Parabolic elements used Soil condition - drained
Figure 5.8 Load-deformation curves for ideal associated plasticity for various forms of the Mohr-Coulomb approximation (solution on the same mesh as in Figure 5.2). Reproduced from Zienkiewicz (1978) by permission of John Wiley & Sons Limited
188
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
which appear feasible. As discussed in Chapter 4 many possibilities arise. One is to use 'smoothing' and avoid corners in manner described by Zienkiewicz and Pande (1977) and Gudehus (1978). Another is to replace the surface by a circle as is done in the well known Drucker-Prager surface (1952) The effect of such an approximation is surprising as shown in Figure 5.8 where the same foundation problem as that of Figure 5.2 is used as a test bed. It is of interest to note that the full Mohr-Coulomb and the extension cone in the Drucker-Prager give close representation, although the smoothing appears to cause about 20'%,difference in the collapse load!! The simple set of examples presented was computed directly before the advent of the dynamic code. We have, however, recomputed several typical cases using the method suggested in this chapter and have found an identical failure load without extra effort. Some difference in displacement response is obtained due to the difference in plasticity formulation (viscoplasticity formulation had been used in these examples) and a non-linear iterative scheme has also been used.
5.3 ISOTHERMAL DRAINAGE OF WATER FROM A VERTICAL COLUMN OF SAND It is very difficult to choose some appropriate tests to validate the two-phase flow partially saturated model because of the lack of any analytical solutions for this type of coupled problem where deformations of the solid skeleton are studied, together with the saturated-unsaturated flow of mass transfer. There are also very few documented laboratory experiments available. One of these is the experimental work conducted by Liakopoulos (1965), on the isothermal drainage of water from a vertical column of sand. This test was also used by Narasimhan and Witherspoon (1978), Schrefler and Simoni (1988), Zienkiewicz et al. (1990b), as well as by Schefler and Zhan (1993), to check their numerical models. The same example was solved in Gawin et al. (1995), but only for onephase flow. Here (Gawin and Schrefler (1996)) attention is paid to the effects of two-phase flow, as compared to the previous solution (Gawin et al. 1995). In this example, isoparametric Lagrangian elements are used, the same interpolation is used for pressures and displacement fields. Furthermore, linear elastic material behaviour is assumed. In the experiment, a column of one metre high Perspex was packed with Del Monte sand and calibrated to measure the moisture tension at several locations along the column. Before the start of the experiment i.e. at t < 0, water was continuously added from the top and was allowed to drain freely at the bottom through a filter, until uniform flow conditions were established. At the start of the experiment i.e. t = 0, the water supply was stopped and the tensiometer readings were recorded. The porosity n = 29.7594, and the hydraulic properties of Del Monte sand had been measured by Liakopoulos and reported in his PhD dissertation (l965), by a seperate set of experiments. During the numerical simulation, Liakopoulos' saturation-capillary pressure and relative permeability of water-capillary pressure relationships were approximated using the following equations:
I S O T H E R M A L DRAINAGE OF W A T E R
189
where p, is given in the unit of Pa and k,, is the relative permeability for the water phase. For the purpose of numerical modelling, the column was divided, in turn, into 10 and 20 four, eight and nine-noded isoparametric finite elements of equal size and different time steps in the time domain with (At = IOs, Is or 0.5s) gave practically the same results. At the beginning, besides uniform flow conditions i.e. unit vertical gradient of the potential andp, = 0 at the top surface, static equilibrium was also assumed. The boundary conditions were as follows: For the lateral surface, the horizontal displacement and the fluid outflow is zero. For the top surface, p , =pa,, where p,,, means atmospheric pressure. For the bottom surface, pa = pa,, , p , = 0 for r > 150s, while waterp, was assumed to change linearly from the initial value to zero for t < 150s and the base is fixed in both displacement directions. Liakopoulos did not measure the mechanical properties of the soil, so the Young Modulus of the soil was assumed to be E = 1.3MPa, Poisson's ratio v = 0.4 and Biot's constant cr = I, similar to Gawin er al. (1995), Schrefler and Zhan (1993) and Schrefler and Simoni (1988). The calculations were performed for one-phase flow with gas pressure fixed at the atmospheric pressure in the partly saturated zone, as well as for two-phase flow. For the latter case, the switching between saturated and unsaturated solution was performed at p, = 2kPa i.e. S = 0.998, which corresponds to bubbling pressure of the analysed sand (Liakopoulos, 1965), as well as at p, = OPa, in order to analyse the effect of the 'switching' value on the solution obtained. The relative permeability of the gas phase was assumed to follow the relationship proposed by Brooks and Corey (1966):
where S, is the equivalent value of saturation with the additional lower limit of k,, 2 0.0001. The resulting profiles of water pressure for the two-phase flow with switching at pc2kPa shown in Figure 5.9 as solid curves with different symbols at different time stations and for one-phase flow in dash-dot curves are compared with the experimental results of Liakopoulos (1965) in fine lines. The solution from one-phase flow showed better agreement and this is because in Schrefler and Simoni (1988), the Young's modulus value has been manually 'fitted' for the case. The profiles of vertical displacements, water saturation and capillary pressure for two-phase flow with switching pressure atp, = 2kPa (solid curves) andp, = OPa (dashdot curves) are compared with the results for one-phase flow (fine curves) in Figure 5.10.
190
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
Figure 5.9 Comparison of the numerical solutions for water pressure (two-phase flow with switching at p, = 2kPa indicated by solid curves and one-phase flow solution indicated by dash-dot curves) with experimental results of Liakopoulos (1965) in fine lines. Reproduced from Gawin and Schrefler (1996) by permission of MCB University Press Ltd
There are some noticable differences between one and two-phase flow solutions for vertical displacements as shown in Figure 5.10, although their final values are similar. The differences between the one and two-phase flow solutions are more appreciable for saturation of water and capillary pressure as shown in Figures 5.1 1 and 5.12. In the lower zone, where no gas flow occurs, the differences are small, but higher up there is a qualitative change in the soil behaviour caused by the presence of gas under pressure (see also Figure 5.13). The solution for two-phase flow with switching atp, = 2kPa is similar to the one-phase flow solution at the bottom of the column. The one-phase flow solution tends to the two-phase flow solution with switching at p, = OPa in the upper part of the sand column, while in the middle region it forms some characteristic constant-value zones corresponding to switching values of capillary pressure or saturation. In general, the gradients of saturation and capillary pressure are higher for the two-phase solution in the upper zone where gas flow occurs. This is qualitatively in accordance with the solution obtained by Schrefler and Zhan (1993). The profiles of gas pressure for two-phase flow with switching at p, = 2kPa are given as solid curves in Figure 5.13 while at pc = OPa, as dash-dot curves. The discernable differences are caused by the different assumptions about gas flow i.e. no gas flow for zones where degree of saturation is greater than 0.998 in the first case. There is a small difference in the pressure amplitudes, nevertheless the qualitative similarity of the gas pressure profiles in the zone, where gas flow occurs, is obvious. This example shows that the modelling of the transition from fully saturated to partially saturated condition and vice versa is very sensitive to the procedure adopted.
ISOTHERMAL DRAINAGE OF WATER
191
Vertical displacement (m)
0.0045
5 0 0.2 0.4 0.6 0.8 1 Height (m)
Figure 5.10 Comparison of the two-phase flow solutions with switching at p, = 2kPa (solid curves) and p, = OPa (dash-dot curves) with one-phase flow solution (fine curves) made for vertical displacements. Reproduced from Gawin and Schrefler (1996) by permission of MCB University Press Ltd Saturation (-)
Figure 5.11 Comparison of two-phase flow solutions with switching at pc = 2kPa (solid curves) and at p, = OPa (dash-dot curves) with one-phase flow solution (fine curves) made for degree of saturation of water. Reproduced from Gawin and Schrefler (1996) by permission of MCB University Press Ltd
192
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS Capillary pressure (Pa) lo O
O
0 0.2 Height (m)
0
0.4
0.6
0
0.8
1
Figure 5.12 Comparison of the two-phase flow solution switching at p, = 2kPa (solid curves) and p, = 0 Pa (dash-dot curves) with one-phase flow solution (fine curves) made for capillary pressure). Reproduced from Gawin and Schrefler (1996) by permission of MCB University Press Ltd
Figure 5.13 Comparison of the gas pressure profiles (the two-phase flow solution) for switching at p, = 2kPa (solid curves) and at p, = 0 Pa (dash-dot curves). Reproduced from Gawin and Schrefler (1996) by permission of MCB University Press Ltd
193
MODELLING OF SUBSIDENCE
5.4
MODELLING OF SUBSIDENCE DUE TO PUMPING FROM A PHREA TIC AQUIFER
This example (Gawin et a1 (1995)) deals with subsidence of saturated-unsaturated land due to pumping from an axisymmetric aquifer, which was solved previously by Safai and Pinder (1979) and then by Meroi (1993). An aquifer of 10m depth sited on an impervious layer was subjected to pumpage of 20m3/h within a height of 2.5m from the bottom. At the beginning a fully saturated state with unit vertical gradient of the potential was assumed, as well as a mechanical equilibrium state. The boundary conditions were as follows: For the bottom surface, the vertical displacement is fixed. At the top surface, the pressure of the air is fixed at atmospheric pressure p,,,. For the inner lateral surface (radius 0.3m), an outflow rate of 1.179kgmp'sp' within the height of 2.5m from the bottom is assumed and the horizontal displacement at this surface is assumed to be zero. For the outer lateral surface (radius 100.3m), full saturation of water with unit vertical gradient of potential was assumed. The process is assumed to be isothermal. The mechanical and hydraulic properties of the soil were assumed to be the same as in Safai and Pinder (1979) and Meroi (1993) i.e. E = 22MPa, v=0.1. k' = 2 x 1 0 - ~ m / sand n = 0.2. The relationships between capillary pressure, satura-
0
10
20
30
40
50 60 Distance (m)
70
80
90
100
Figure 5.14 Resulting profiles of saturation (full curves) compared with solution of Safai and Pinder (1979) in broken curves. Reproduced from Gawin and Schrefler (1995) by permission of John Wiley & Sons Limited
194
CONSOLIDATION AND PARTIALLY SA TURATED D YNAMZC PROBLEMS
Distance (m)
Distance (m)
Figure 5.15 Resulting saturation and vertical displacement profiles (full curves) compared with solution of Meroi (1993) in broken curves. Reproduced from Gawin and Schrefler (1995) by permission of John Wiley & Sons Ltd
tion and relative permeability of water followed those proposed by Safai and Pinder (1979). The relative permeability of gas proposed by Brooks and Corey (1966) as given in (5.3) and (5.4) were again applied in the partially saturated zone. For numerical purposes, the aquifer was simulated by 50 eight-noded isoparametric elements (five in height and ten in radial direction) using the same mesh as in Meroi (1993). A three-by-three integration scheme was applied. Temporal discretization was performed with an initial step size of 1 min for the first 10 hours, 10 min for the next 20 hours and then 1 hour for the rest of the 28 days-the total required time of analysis.
195
AIR STORAGE MODELLING
The resulting profiles of water saturated on the top surface of the aquifer are compared in Figure 5.14 with the results of Safai and Pinder (1979) in broken curves, showing relatively good agreement. It has to be mentioned that the model of Safai and Pinder (1979) neglects the fluid accumulation due to changes in the degree of saturation, which explains at least part of the differences in the early stage of the pumping phase. The profiles of water saturation on the upper surface and vertical displacements for time stations of 10 min, 3h, 10h, 30h and 28 days are compared in Figure 5.15 with the results of Meroi (1993) in broken curves. In Meroi (1993), only one phase flow is considered with the gas pressure fvted at atmospheric pressure in the partially saturated zone, which could explain the difference in time transident behaviours. The final values at the centre matched very well.
5.5
AIR STORAGE MODELLING IN AN AQUIFER
With the same geometry as in the column drainage test example of Section 5.4, air storage in an aquifer is simulated by the partially saturated model (Schrefler and Zhan (1993)). Meiri and Karadi (1982) simulated this problem by a one-dimensional finite element model, with a rigid soil skeleton. In this simulation the skeleton is considered deformable, with Young's modulus E = 692kN/m2. The porous medium system was assumed initially fully saturated with a reference permeability of 5 x lop"m2, a porosity of 0.2 and initial aquifer pressure of 5.066 M N / or ~ 50 ~ Patm, under isothermal conditions at 149°C (or 300°F). Initial conditions are no displacement and a constant water pressure of 5.066 M N / at ~ ~ all points. The boundary conditions are as follows: lateral surface, q, = 0, q, = 0, uh = 0 bottom surface, q, = O,p, = p,,f, u h 7 u, = 0 top surface, q, = 2.44 x 10p4kg/s/m- (air injection), q,
= 0.
The water and air viscosities are selected as 0.3mNs/m2 and 2 4 p ~ s / m 2respect, ively. The water density for standard conditions is 100kg/m3 and the air density, 1.22kg/m3. The following relationships between the relative permeabilities of water and air, the capillary pressure and water saturation, proposed by Brooks and Corey (1966), are employed in the simulation: 2lli
k,,
=
ST
196
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
---Hrs0010 20
-t30
---*-40 --m--
50 60
--+-.70
-+-80
-0.2
4.1
0.0
0. I
0.2
49.8
49.9
50.0
50.1
50.2
Vertical displacement (m)
Pressure of water (atm)
Pressure of gas (atm)
Saturation of water (Dotted lines. results from Meiri & Karadi)
Figure 5.16 Profiles of vertical displacement, water pressure, air pressure and saturation of air storage modelling in an aquifer. Reproduced with permission from Schrefler and Zhan (1 993) American Geophysical Union
197
AIR STORAGE MODELLING
I
I
I
I
I
I
0.4
0.5
0.6
0.7
0.8
0.9
I
1.0
Saturation Figure 5.17 Profile of saturation of air storage modelling in an aquifer with double Young's modulus for Figure 5.16. Reproduced with permission from Schrefler and Zhan (1993) American Geophysical Union
(c
where S, = (Sw- Swc)/(l- Swc)is the effective saturation, Swcis the irreducible saturation, X is the pore size distribution index and p b is the bubbling pressure. The values for Swc, X and p b are given as 0.2, 3.0, and 1 . 6 8 k ~ / m respectively, ~, corresponding to sand with an intrinsic permeability of 5 x 10-I3m2. The vertical displacement, water pressure, air pressure, and saturation versus depth of the column are shown in Figure 5.16 for different time instances. Water and air pressure have a minimum at the interface of the two fluids. The column was first contracting and then finally expaning vertically with continuing air inflow from the top. The saturations are compared with the result reported in Meiri and Karadi (1982), indicated with dashed lines. The effects of a deformable skeleton assumption can be seen clearly. In general, at the onset of air flow the desaturation is greater in the rigid skeleton model while later on, the situation is reversed, i.e., the deformable skeleton model has greater desaturations. In order to point out the influence of the solid skeleton deformation, a second simulation was performed with a double Young's modulus. The new saturation profiles are shown in Figure 5.17. While the propfile of saturation after 80 hours of pumping is roughly the same with both Young's moduli, the profiles of shorter time spans are different. These simulations confirmed that the model proposed can reproduce this quasi-static case and that the main features of the model, i.e., air flow and solid deformation, have their importance.
198
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
5.6
FLEXIBLE FOOTING RESTING O N A PARTIALL Y SA TURA TED SOIL
In this example (Schrefler and Zhan (1993)), a flexible footing resting on a partially saturated soil is modelled. The discretized cross-section (19 x 12 m) is shown in Figure 5.18. The boundary conditions are as follow: for t < 0, lateral surface, q, = 0, q, = 0, u h = 0; bottom surface, p, = 0, (pa = p w =p,,r = 101.325 kPa), i.e., S, = 1 from S , = S,(p,) and u h = u, = 0; capillary pressure increasing versus top surface such that on the top surface S, = 0.75; for t 2 0, the top surface outside the footing, S, = 1 and the top surface under the footing f,,,, = 405.3 kPa (4 p,,, acting downward): all other conditions are as for t < 0. The wetting process is here simulated for a time span of 10's. This problem was first solved by Lloret et 01. (1987), using two separate computer programs, one for the twophase flow field and one for deformation analysis. From a computational point of view, this represents some sort of matrix partitioning and was applied to the original fully coupled system of equations. Since no iterations were performed between the two programs during a time step, it is not clear how far the coupling was preserved through this procedure. Furthermore, the approach used in Lloret rt LEI.(1987) is based on state surfaces while here the capillary pressure relationship by Brooks and Corey (1966) is used, together with a modified effective-stress principle and the fully coupled system of equations is solved as such. Hence the results of the two approaches are not expected to be the same. The deformation after 100000s is shown in Figure 5.19. The contour lines of the water degree of saturation at the same time is shown in Figure 5.20 and water pressure and gas pressure contour lines, respectively, in Figures 5.21 and 5.22. Finally, the change in the vertical displacement with time of three surface points A, B and C of Figure 5.18 is shown in Figure 5.23 and, for comparison, the results for fully saturated conditions are also indicated.
Figure 5.18 Discretized cross-section for flexible footing resting on a partially saturated soil. Reproduced with permission from Schrefler and Zhan (1993) ci,"~American Geophysical Union
FLEXIBLE FOOTING RESTING
199
Figure 5.19 Deformation shape at t
= 100000s. Reproduced with permission from Schrefler and Zhan (1993) 0 American Geophysical Union
r = 100000s. Reproduced with permission from Schrefler and Zhan (1993) (0American Geophysical Union
Figure 5.20 Water saturation at
First the soil is compacting, with much larger vertical displacements below the footing, and with increasing time the cross-section swells. The constitutive model used in Schrefler and Zhan (1993) required some refinement in order to simulate collapse upon first wetting and rewetting of the soil, in the presence of a mean compressive total stress, as is shown have happened a t point A.
200
CONSOLIDATION AND PARTIALLY SATURATED D YNAMZC PROBLEMS
Figure 5.21 Water pressure (atmospheres) at t = 100000s. Reproduced with permission from Schrefler and Zhan (1993) American Geophysical Union
/c~
Figure 5.22 Air pressure (atmospheres) at t = 100000s. Reproduced with permission from Schrefler and Zhan (1993) 0American Geophysical Union
201
COMPARISON OF CONSOLIDATION AND D Y N A M I C RESULTS
I o4
I o5
I o6
1
108 I o7 Time (seconds)
Figure 5.23 Change in vertical displacement with time at three different points as indicated in Figure 5.18. Reproduced with permission from Schrefler and Zhan (1993) 0American Geophysical Union
5.7
COMPARISON OF CONSOLIDATION AND DYNAMIC RESULTS BETWEEN SMALL STRAIN AND FINITE DEFORMA TION FORMULA TION
A large-deformation model with a number of simplifications has been developed by Meroi et ul. (1995). Although the finite deformation solution has not been described in this book, it would be of interest to compare the small strain formulation
202
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
introduced in Chapters 2 and 3 and to investigate the effect of finite deformation. The large-deformation model has been extensively validated with respect to solutions reported in the literature. For the purpose of validation, a number of one-phase static and dynamic problems (Bathe and Ozdemir, 1976; Bathe et al. 1975; Heyliger and Reddy, 1988; Shantaram et al. 1976) were first solved and then consolidation solutions of fully saturated soils were obtained (Carter et al. 1979; Kim et al. 1993; Lewis and Schrefler, 1987; Meijer, 1984; Monte and Krizek, 1976; Prevost 1981 and 1984). These studies have been reported in Meroi (1993) and they showed good agreement with the corresponding results in the literature. Five examples have been presented in Meroi et ul. (1995) to test the new features offered by the large-deformation approach. All the examples contained comparison with small deformation developed in Chapter 2 and 3. All of them are reproduced in this section.
5.7.1
Consolidation of fully saturated soil column
This example is concerned with the consolidation of a one- dimensional ten metre deep ground, fully saturated by water, infinitely extended in the horizontal direction, and subjected to a step load applied at the top level, with drainage allowed only through the top surface. The problem, which is one-dimensional since each vertical section can be considered as a plane of symmetry, is modelled as a saturated soil column under plane strain condition. The boundary conditions for the displacement field are such that: all nodes are horizontally constrained and the bottom level is fixed with no vertical movement. Atmospheric pressure is assumed at the top level and impermeable boundaries are imposed at the lateral and bottom surfaces. For comparison with Prevost's (1981, 1982 and 1984) results of one-dimensional elastic consolidation, different load levels were considered, reaching a maximum intensity for the uniformly distributed load equal to the Young's modulus of the ground, E. An initial porosity of 0.3, a specific permeability of 0.01 mls, an elastic modulus of lGPa and a zero Poisson ratio are adopted, in accordance with Prevost (1982). A backward difference integrator was used because of its superior efficiency in consolidation analyses reported in Prevost (1982). With the finite-deformation formulation, the theoretical relationship between the applied load and the final displacement is logarithmic for this example and it is represented in Figure 5.24 by the dot-dash curve. The computational results (curve a) are in very good agreement with the theoretical behaviour (curve c). The dashed curve (curve d) represents the results of linear analysis while the dotted curve (curve e) the solution obtained by Prevost (1982). At high load levels, due to the assumption of constant elastic modulus, a near-zero void ratio may occur computationally, and at this stage the soil behaviour should be described by the elastic characteristics of the compacted grain itself: a relationship between the elastic modulus of soil and its void ratio has been given in Monte and Krizek (1976). It is assumed here, for the sake of simplicity, that the soil elastic modulus becomes ten times as large as the initial one when the porosity approaches zero. The results obtained under this assumption are represented by curve b.
COMPARISON OF CONSOLIDATION AND D YN'4MIC RESULTS
203
Figure 5.24 Vertical settlement versus load level in one-dimensional elastic consolidation problem with fully saturated conditions: curves a and b, computational results with the finite deformation approach, for constant and variable elastic behaviour. respectively; curve c. theoretical solution for the finite deformation regime, curve d: linear analysis response; curve e. Prevost solution. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
In Figure 5.25, the maximum vertical displacement versus the normalized time T, ( T , = c,t/h2 = c,t, where c, is the coefficient of consolidation and c, the time factor (Lewis and Schrefler, 1987)) for different load levels: curves a, b and c refer to load levels equal to 0.2,0.4 and 0.6 times the elastic modulus of the soil matrix, respectively. A further improvement to the finite deformation model consists of the introduction of the dependence of the absolute permeability on the void ratio. In the present analysis, permeability is assumed to be a linear function of the void ratio, varying from the initial value to zero when porosity becomes zero. Figure 5.26, in which time is given in the logarithmic scale, shows the influence on computational results of such a relationship. Consolidation with constant and variable permeability is described, respectively, by curves a and c for the load level equal to 0.3E and by curves b and d for a load level of 0.5E. The significant increase in time (in logarithmic scale on the figure) necessary to reach full consolidation can clearly be seen.
5.7.2
Consolidation of fully and partially saturated soil column
This example refers to the work by Advani et ul. (1993) and Kim et at. (1993). but partial saturation conditions were also considered. A one-dimensional column of 7 metres high was modelled in accordance with Kim et al. (1993) using plane strain
204
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
elements. The material properties are: porosity of 0.5, dynamic permeability of 4 x 10-1 lm4/Ns, a n elastic modulus of 6GPa and a Poisson ratio of 0.4. The resulting time factor c, = 1.049 x lo-'. The uniformly distributed load applied in the vertical direction reaches a limiting value of 1OGPa. In Figure 5.27, the vertical settlement, normalized with respect to the corresponding values of Terzaghi's theory, are plotted against normalized time for the case of fully saturated conditions for small deformation (curve a) and large deformations (curve b). The results are in substantial agreement
Figure 5.25 Vertical settlement versus normalized time in the one-dimensional consolidation problem with constant permeability and elastic modulus under fully saturated conditions and finite-deformation assumptions. Curves a, b and c refer to load levels equal to 0.2, 0.4 and 0.6E. Reproduced from Meroi (1995) by permission of John Wiley & Sons Ltd
Figure 5.26 Vertical settlement vs. normalized time in the one-dimensional elastic consolidation problem with fully saturated conditions. Curves a and c refer to a load level of 0.3E while curves b and d to 0.5E with c and d having variable permeability. Reproduced from Meroi (1995) by permission of John Wiley & Sons Ltd
COMPARISON OF CONSOLIDATION AND DYNAMIC RESULTS
205
with the ones presented in Kim et al. (1993).The large deformation analysis was also performed for the case of permeability linearly dependent on the void ratio (curve c). In the same figure, the large deformation results for three different homogeneous partially saturated initial conditions were also plotted. These partially saturated initial conditions were imposed by assigning the initial capillary pressure distribution corresponding to the degree of saturation via the capillary pressure relationship in Figure 5.28. While the small deformation analysis with air pressure equal to the atmospheric pressure gives the final results in one time step (curve d), in the large deformation analysis a real consolidation process has taken place.
Figure 5.27 For the model drawn, vertical settlement, normalized with respect to the corresponding values of Terzaghi's theory, vs. normalized time: fully (curve a, smalldeformation analysis; curves b and c, large-deformation analyses with constant and variable permeability, respectively) and partially saturated conditions (curves d for small deformation analysis). Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
Figure 5.28 Saturation and relative permeability vs. hydraulic head. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
206
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
Figure 5.29 Time transient water pressure. normalized with respect to applied load. for the fully saturated case at the top. Curve a, small-deformation approach; curves b and c, largedeformation cases with constant and variable permeability, respectively. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
The dissipation of water pressure for the fully saturated case in the top element is shown in Figure 5.29 for small deformation as curve a, and for large deformation with constant and variable permeability as curves b and c
5.7.3
Consolidation of two-dimensional soil layer under fully andpartially saturated conditions
This example is also taken from Kim et ul. (1993) and consists of a two-dimensional, plane strain analysis of an 8m deep soil layer loaded by a uniform pressure of 6GPa by a 16m wide foundation (=2b). Because of symmetry, only half of the model is considered, which extends for 48m in the horizontal direction. All the material data are the same as in the previous section, apart from the Poisson ratio which is assumed to be zero in accordance with Kim et al. (1993), the resulting time factor is c, = 0.375 x The boundary conditions for displacements are taken as: the bottom level is fixed, while laterally only vertical displacements are allowed. Both rigid and flexible footings are considered. In order to model the rigid footing, a multi-point constraint technique, i.e. tied-nodes technique, is employed. Pressure is assumed to be zero at the top surface including the underside of the foundation in accordance with Kim et al. (1993). Since a net flow is allowed through the foundation, it is assumed to be permeable. Besides the fully saturated case, a partially saturated model is also investigated with an initial saturation of 92%).The initial condition is obtained by imposing a suction in accordance with the relationship given in Figure 5.28. In the case of a rigid foundation under partially saturated conditions, the foundation is now considered as impervious.
COMPARISON OF CONSOLIDATION AND D Y N A M I C RESULTS
207
Displacements of the top node at the centre line, normalized according to
w, = Ew/2bq are shown in Figure 5.30 for the rigid foundation and in Figure 5.31 for the flexible one. In both figures, curves a refer to small-deformation analysis introduced in Chapters 2 and 3, for the fully saturated case, and in perfect agreement with results proposed by Kim et 01. (1993) and with the analytical solution of Booker (1974). Curves c refer to large-deformation analysis. In the case of a rigid foundation, the result is also given by Kim et 01. (1993) and is in good agreement with the one reported here. Consolidation with the linear dependence of the permeability with the void ratio is indicated by curve d. Curves e describe the behaviour in the case of an initially uniform partial saturation of 92% and large deformations. For permeable flexible foundation, the behaviour is almost time independent, while for a rigid impermeable foundation, some consolidation takes place and the time for transient behaviour is longer than the fully saturated case, because the foundation impermeability forced the water not only to follow a longer path, but also one with a smaller relative permeability. Figure 5.32 represents the deformed mesh in the case of a flexible foundation. giving consolidation patterns at T, equal to 0.01, 0.1 and 0.55. For the same dimensionless times, Figures 5.33 and 5.34 give the results for a rigid foundation with full and partial saturation, respectively. The comparison of the different deformed shapes allows one to appreciate the influence of the different fluid pressures. In particular, a swelling close to the foundation during the first stage of the analysis can be observed under fully saturated conditions.
Figure 5.30 Model description and normalized settlement of the top node at the centre line versus normalized time for the rigid footing. Curves a and b, small-deformation regime for fully and partially saturated initial corlditions, respectively; curves c and d, finite deformation analysis from initial fully saturated conditions with constant and variable permeability, respectively; curve e, finite deformation result from initial partially saturated conditions. Reproduced from Meroi (1995) by permission of John Wiley & Sons Ltd
208
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
Figure 5.31 Normalized settlement of the top node at the centre line for the flexible footing case versus normalized time. Curves a and b, small-deformation analysis for fully and partially saturated initial conditions; curves c and d, finite deformation analysis from initial fully saturated conditions with constant and variable permeability, respectively; curve e, finite deformation results from initial partially saturated conditions. Reproduced from Meroi (1995) by permission of John Wiley & Sons Ltd
Figure 5.32 Deformed mesh for flexible footing, the fully saturated case: the consolidation pattern is given at dimensionless time, T, equal to 0.01, 0.1 and 0.55. Reproduced from Meroi (1995) by permission of John Wiley & Sons Ltd
COMPARISON OF CONSOLIDATION AND DYNAMIC RESULTS
209
Figure 5.33 Deformed mesh for rigid footing, the fully saturated case: the consolidation pattern is given at dimensionless time, T, equal to 0.01, 0.1 and 0.55. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
Figure 5.34 Deformed mesh for rigid footing, initial partial saturation of 92%: the consolidation pattern is given at dimensionless time, T, equal to 0.01, 0.1 and 0.55. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
5.7.4
Fully saturated soil column under earthquake loading
This example is used to test the code's capability of modelling large deformation with plasticity. This example of liquefaction performed by Zienkiewicz et al. (1990b) and Xie (1990) was used for this testing purpose and the Pastor-Zienkiewicz (1986) model as described in Chapter 4 is adopted for the sand. It can be seen that the results were not significantly affected by the use of geometric non-linearity because the fluid pressure can rise even without the large deformation during a cycle of earthquake
210
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
loading. The first 10 seconds of the N-S component of the El-Centro 1940 earthquake is taken as the horizontal base acceleration input during the consolidating phase of the sandy soil under a uniformly applied load of 600kPa. The geometry and mechanical characteristics of the model are the same as those given by Zienkiewicz et 01. (1990). and, in particular, a n initial elastic modulus of 4.5 MPa is assumed. Figure 5.35 shows the increase in pore pressure, both for large deformation (upper of the twin curves) and for the small- deformation approach during the first 15 seconds for
Figure 5.35 Pore pressure versus time in the generation phase at three reference points of the drawn model, both for large-deformation (upper one of the twin curves) and for the smalldeformation approach. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
Figure 5.36 Horizontal displacements versus time at points A and D for both largedeformation (upper of the twin curves) and for the small-deformation approach. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
COMPARISON OF CONSOLIDATION AND D Y N A M I C RESULTS
211
three reference points on the finite-element model. The horizontal displacements obtained using the two different approaches at points A and Dare given in Figure 5.36. Pore pressure and vertical displacements are shown in Figures 5.37 and 5.38 respectively. For any pair of curves, the one with smaller final value belongs to the large-deformation approach. It can be noted that the amount of vertical displacement induced by the earthquakes is not significant in comparison to the one induced by the surface applied load.
5.7.5
Elasto-plastic large-strain behaviour of an initially saturated vertical slope under a gravitational loading and horizontal earthquake followed by a partially saturated consolidation phase
This last example considers the elasto-plastic large-strain behaviour of a 9.15m vertical slope (Chen and Mizuno, 1990) subjected to gravitational loading during the first 25 seconds of El-Centro N-S component of horizontal acceleration and during the following consolidating phase. The fully saturated domain after excavation is modelled by linear triangular finite elements. Figure 5.39 shows the final pressure distribution plotted over the deformed mesh. To impose initial, fully saturated conditions, with the piezometric level at the top nodes of the domain, an undrained analysis is carried out with horizontal constraints at the vertical slope nodes. Then the seismic excitation is applied at the bottom nodes, the above mentioned horizontal constraints are released and the soil is allowed to desaturate.
0
0
1000 2000 3000 4000 5000 6000
t (s)
0
Figure 5.37 Pore pressure versus time in the dissipation phase for the given points. both for the large (the one with the smallest final value of the twin curves) and for the smalldeformation approach. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
212
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS
Figure 5.38 Vertical displacement versus time in the consolidation phase at the given points, both for the large (the one with smallest final value of the twin curves) and for the smalldeformation approach. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited PRESSURE
TIME: 2570.0
Figure 5.39 Final pressure distribution over the deformed mesh. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
At the left and right vertical sides of the domain, horizontal displacements are fixed and a hydrostatic pressure distribution with atmospheric value at the two corresponding top nodes is assigned. The material characteristics of soil-saturation relationships included, are assumed with reference to the Pastor-Zienkiewicz (1986) model of the clay core of the San Fernando dam, in accordanced with Zienkiewicz rt al. (1990).
214
CONSOLIDATION AND PARTIALLY SATURATED DYNAMIC PROBLEMS 5
10
100
15
20
25
1000
time (s)
time (s)
Figure 5.41 Vertical (v) and horizontal (h) displacements versus time during the seismic and consolidation phases of the top point of the slope. Reproduced from Meroi (1995) by permission of John Wiley & Sons Limited
215
CONCLUSIONS
The saturation distributions are represented over the corresponding deformed configurations in the area close to the slope at different times in Figure 5.40, while in Figure 5.41 the time history of vertical and horizontal displacements of the top node of the slope is reported.
5.8 CONCLUSIONS The formulation developed in Chapter 2 and discretized in Chapter 3 is used to analyse various one-dimensional and two-dimensional problems with fully saturated and partially saturated soil respectively. The results obtained from static and consolidation analysis are highly satisfactory and compared well with available analytical and experimental solutions. However, it would be useful for practical purposes for the formulation to be validated using model experiments and this will be introduced in the next chapter.
REFERENCES Advani S. H., Lee T. S., Lee J. H . W. and Kim C. S. (1993) Hygrothermomechanical evaluation of porous media under finite deformation. Part I-Finite element formulations. hlt. J. N u r ~ Meth. . Eng., 36, 147-160. Bathe K . J. and Ozdemir H. (1976) Elastic-plastic large deformation static and dynamic analysis, Coinp. Struct., 6, 81-92. Bathe K. J., Ramm E. and Wilson E. L. (1975) Finite element formulations for large-deformation dynamic analysis, Int. J. Nzm~.Meth. Eng.. 9. 353-386. Booker J. R. (1974) The consolidation of a finite layer subject to surface loading, Int. J. Solids & Structures, 10, 1053-1065. Brooks R. N. and Corey A. T. (1966) Properties of porous media affecting fluid flow. ASCE IR, 92. No. IR2, 61-68. Carter J. P., Booker J. R. and Small J. C. (1979) The analysis of finite elasto-plastic consolidation, Int. J. Num. Anal. Geomech., 3, 107-129. Chen W. F. and Mizuno E. (1990) Nonlinecrr Ancrlysis in Soil Mechcinics-Tlzeory and Implement~ztion, Elsevier, Amsterdam. Drucker D. C. and Prager W., (1952) Soil Mechanics and Plastic Analysis or Limit Design. Quart. Appl. Math. 10, 157-165. Gawin D. and Schrefler B. A. (1996) Thermo-hydro-mechanical analysis of partially saturated porous materials, Eng. Cornput., 13, No. 7, 113-143. Gawin D., Baggio P. and Schrefler B. A. (1995) Coupled heat, water and gas flow in deformable porous media, Int. J. Nurn. Meth. Fluids, 20, 969-987. Heyliger P. R. and Reddy J. N. (1988) On a mixed finite element model for large deformation analysis of elastic solids, Int. J. Non-linenr Mrch. 23, 131-145. Kim C. S., Lee T. S., Advani S. H. and Lee J. H. W. (1993) Hygrothermomechanical evaluation of porous media under finite defromation: part I1 model validations and field simulations, Int. J. Nurn. Meth. Eng., 36, 161-179. Lewis R. W. and Schrefler B. A. (1987) The Finite Elenzent Method in the Drfbrmation crnd Consolidution of Porous Mrdiu, John Wiley & Sons, Chichester. Liakopoulos A. C. (1965) Transient flow through unsaturated porous media. D. Eng. Dissertation, University of California, Berkeley, USA. -
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CONSOLIDATION A N D P A R T I A L L Y S A T U R A T E D D Y N A M I C P R O B L E M S
Lloret A., Gens A., Batlle F. and Alonso E. E. (1987) Flow and deformation analysis of partially saturated soils in Grouncln~rterejfkcts in Geotrc~liniculEngineering. A. A. Balkema, Rotterdam, Netherlands. Meijer K. L. (1984) Comparison of finite and infinitesimal strain consolidation by numerical experiments, Int. J. Nurn. A n d . Geoniecll., 8, 53 1-548. Meiri D. and Karadi G . M. (1982) Sin~ulationof air storage aquifer by finite element model, Int. J. NZUII. A n d . G e m ~ ~ r h6,. .339-351. Meroi E. (1993) Comportamento non lineare per geometria di mezzi porosi parzialmente saturi. Ph.D. Thesis, Istituto di Scienza e Tecnica delle Costruzioni, Universiti di Padova. Meroi E. A , , Schrefler B. A. and Zienkiewicz 0 . C. (1995) Large strain static and dynamic semisaturated soil behaviour, bit. J. Nuni. Anrrl. Geotnecli., 19, 81-106. Monte J. L. and Krizek R. J. (1976) One-dimensional mathematical model for large-strain consolidation. GPotechnique. 26, 495-5 10. Narasimhan T. N. and Witherspoon P. A. (1978) Numerical model for saturated-unsaturated flow in deformable porous media 3. Applications, Wuter Rrsources Rex., 14, 1017-1034. Pastor M. and Zienkiewicz 0 . C. (1986) A generalised plasticity hierarchical model for sand under monotonic and cyclic loading, NUMOG II, Ghent, April. 13 1-1 50. Prandtl L. (1921) ~ b e die r Eindringungsfestigkeit plastisher Baustoffe und die Festigkeit von Schneiden. Zeitschr~ft,fCr Ange~c~rmclte M(rthernntik und Mechnrlik, 1, No. 1 , 15-20. Prevost J. H. (1981) Consolidation of anelastic porous media, A S C E E M , Vol. 107, 169-186. Prevost J. H. (1982) Nonlinear transient phenomena in saturated porous media, Conzp. Merh. Appl. Mech. Eng., 30, 3-18. of'EnginrrrPrevost J. H. (1984) Non-linear transient phenomena in soil media, in Mec~l~crnics ing Materirr1.s. Wiley, Chichester, Chapter 26. Safai N. M. and Pinder G. F. (1979) Vertical and horizontal land deformation in a desaturating porous medium, A h . Wtrter Resourc,es., 2, 19-25. Schrefler B. A. and Simoni L. (1988) A Unified approach to the analysis of saturated unsaturated elastoplastic porous media in Nurnericcrl Met1iod.s in Geon~echcinicr,Innsbruck, Balkema, Rotterdam. Schrefler B. A. and Zhan X. (1993) A fully coupled model for waterflow and airflow in deformable porous media, Wuter Resources R e x , 29, No. 1, 155-167. Schrefler B. A,, Zhan X. Y. and Simoni L. (1995) A coupled model for water flow, airflow and heat flow in deformable porous media, Int. J. Nurw Metl~.Heut Fluid Flolc., 5, 53 1-547. Shantaram D., Owen D. R. J. and Zienkiewicz 0 . C. (1976) Dynamic transient behaviour of two-and three-dimensional structures including plasticity, large deformation effects and fluid interaction, Errrtl~qunkeEngitic~ering& Structurcil Djworvics. 4, 561-578. Soil Mecl~tmics,John Wiley and Sons, Inc.. New York. Terzaghi K. (1943) Tl~c~oreticcil Xie Y. M. (1990) Finite element solution and adaptive analysis for static and dynamic problems of saturated-unsaturated porous media, Ph.D. Dissertation, Univ. Coll. of Swansea. Wales. Zienkiewicz 0. C. and Pande G . N. (1977) Time-dependent multi- laminate model of rocks a . Geornech., 1, numerical study of deformation and failure of rock masses, Int. J. N u t ~ Anrrl. 219-247. Zienkiewicz 0 . C. and Taylor R. L. (1991) The Fitiite E/(wer~tMcthorl-Volutne 2 Solid rrnti Fluid M e c I ~ ~ n i cDynuniic~ .~, ~ n Nun-line(rritj' d (4th erln), McGrew-Hill Book Con~pany, London. Zienkiewicz 0 . C., Humpheson C. and Lewis R. W. (1975) Associated and non-associated viscoplasticity and plasticity in soil mechanics, GPotechniyue, 25, 671-689. Zienkiewicz 0. C., Xie Y. M., Schrefler B. A,, Ledesma A. and Bicanic N. (1990) Static and Dynamic behaviour of soils: a rational approach to quantitative solutions, Part 11: Semisaturated problems, Proc. Roj*.Soc. Lond., A429, 310-323. -
Validation of Prediction by Centrifuge
6.1 INTRODUCTION In the previous chapter (Chapter 5 ) , we presented several examples of the application of the full formulation for various static and consolidation problems. This effectively tested the limit behaviour of various constitutive models and also the interaction of slow drainage with deformation during the consolidation process. The problem did not, however, stretch predictive capacity: in the first case of limit behaviour it gave answers which were quite well known in general, and in the second case, the departure from linear, elastic, behaviour during consolidation was small. To test fully the possibilities offered by the formulation and the models presented we should seek examples where: (1) the repeated loading generates substantial pore pressures and possibly liquefaction; (2) the problem is such that non-linear, nearfailure, stresses are present and at least partial inelastic permanent deformation results. Clearly, the study of earthquake response presents the greatest challenge here but it is hardly possible to measure it on a site where both conditions 1 and 2 above will occur within a reasonable time span, or indeed ever. Further, it would be almost impossible to achieve the so called 'Class A' prediction. so beloved by soil mechanicians, in which the computation precedes the actual event. Even if an earthquake of the desired magnitude with the desired effect did happen, its precise detail of input would not be available before it happened. What is often possible is to reconstruct catatrophic events, particularly if some idea of the input motion is available through measurements within reasonable proximity. In the next chapter, we shall show such reconstructions known frequently as backanalysis. These in the soil mechanics problems are, of course, Class C predictions and therefore mistrusted by some as of course the soil parameters could be adjusted to achieve the already known measured results. For this reason it is desirable to attempt scale model tests of earthquake events for which both Class A and Class B predictions are possible (The last one being computed
218
VALIDATION OF PREDICTION BY CENTRIFUGE
simultaneously with the measurement on the model). Two possible scale models of environmental conditions exist: (a) the shaking table; and (b) the centrifuge. The shaking table has been used with great success in modelling the dynamic effect of structures but, unfortunately, it is less successful for soil mechanics problems. The reason here is that for typical soil problems gravity is the most important external force and this is obviously not modelled correctly in a scaled model in which densities are kept constant and the linear din~ensionsreduced. For this reason, the centrifuge was invented and this device permits a very considerable increase in gravitational acceleration. The scale model in the centrifuge is usually rather small and thus the whole frame of the test has to be rotated at high speed, producing a fairly uniform field of up to 100g in the model area. For this reason, we shall draw our comparisons in the following sections entirely from the centrifuge and here, as we mentioned, both Class A and B type predictions will be possible. In Section 6.2, the basic theory of centrifuge modelling in geotechnical applications is introduced with particular attention to the use of an alternative fluid in order to achieve dynamic compatibility of diffusion and inertial behaviour. It is precisely this substitution of pore pressure which rendered centrifuge testing unsuitable for the modelling of prototype dynamic events under partially saturated conditions and here lies one of the limitations of the procedure. For the interest of the reader, we recommend a study of various publications of Professor Schofield and others concerning the physics of centrifuge modelling. In the first section describing the centrifuge test, we shall concentrate on a model of a dyke (Venter 1985, 1987) performed on the Cambridge geotechnical centrifuge. Here comparisons of computations are done simultaneously with models and perhaps this section should be classified under Class B verifications. In this section and indeed in the later section (6.5) where we describe a somewhat similar test done on an embankment wall at Colorado by Dewoolkar, the same remarks apply. In both cases, we did not use scaling as the centrifuge itself and the artificial earthquake itself, were considered to be the 'prototype'. Section 6.4 represents results of a very major study undertaken in USA under the name of VELACS (VERification of Liquefaction Analysis by Centrifuge Studies - Arulanandan and Scott, 1993). This study was funded by the National Science Foundation, USA and involved some twenty laboratories in various parts of the world performing numerical predictions which later were to be compared to several centrifuge studies in the USA and Cambridge Geotechnical centrifuge, UK. In this example, all the predictions are of Class A type as results had to be presented to the organisers before the centrifuge tests were attempted. Only in one or two cases were the results obtained later and these are specially marked as Class B. During this study, very many alternative situations were investigated and it will be seen later that excellent comparisons were obtained. It is necessary, however, to remark that to date in no case has it been possible to perform centrifuge models with a free fluid surface and such structures as dams, retaining embankments with different levels of water at different sides, etc., cannot be modelled because of the restriction on the partially saturated conditions. For this particular case, the only results which are available will be presented by back-analysis in the following chapter.
SCALING L A W S OF CENTRIFUGE MODELLING
219
6.2 SCALING LA WS OF CENTRIFUGE MODELLING In this section, a brief description and derivation of the centrifuge scaling laws for models are described. This is included to aid the readers in the interpretation of the centrifuge results and their comparison with the numerical results. Furthermore, it is the purpose of this section to explain the concept of dynamic compatibility in centrifuge modelling which led to the use of a different pore fluid from water in saturated centrifuge model tests. As we consider in the numerical analysis that the centrifuge experiment is a prototype itself, issues concerning far-field boundary conditions and particle size will not be dealt with. For readers interested in the centrifuge modelling of dynamic events, the following publications can be referred to (Schofield, 1980 and 1981; Lee & Schofield, 1988; Schofield and Zeng, 1992; Steedman and Zeng, 1995). Assuming an N (typically between 50 and 200) scale model is introduced in the centrifuge, the linear length is scaled by:
where the superscript M denotes the model scale and P denotes the prototype it intends to model. We can write the mixture equilibrium equation (see (2.11) neglecting only convective terms) for model and prototype respectively.
If we assume that the density of the mixture and fluid, together with the stress statet are maintained the same in both the model and the prototype, this would require the acceleration to be scaled by 1/N times with:
and
Comparing with the linear dimension in (6.1), one would conclude that time is also scaled by N times:
t The maintanence of the same stress level is important for soil behaviour as the stress: strain behaviour of soils is highly stress level and strain-history dependent.
220
VALIDATION OF PREDICTION BY CENTRIFUGE
Therefore, in the model, a dynamic event will happen N times as fast as its corresponding prototype, thus pushing up the frequency of the dynamic events by N times. However, for most practical earthquake events, the frequency and wavelengths of the wave within the pores should still be within the range of laminar Darcy's flow. The scaling relationship is then applied for the fluid mass conservation equation (see (2.16))
which implies that pore pressure is the same in the model and the prototype if the compressibilities and the void ratio remain the same. Also for any source term its rate must also be the same. If any substitute fluid is used, its compressibility should not be too different from that of water, though the bulk modulus of water does vary because of the amount of air dissolved in it. So far, there is no problem in the scaling, however, a problem arises when the fluid flow equation (see (2.13) and (2.14) again neglecting only the convective terms) is considered:
Considering the equation for the model and prototype separately:
and
As the velocity is required to be the same in the model and the prototype if the scaling of the displacement, acceleration and time is as described in Equations (6.1, 6.4, 6.5 and 6.6) we require the permeability to be scaled by I/N:
This cannot be readily achieved if the same solid material, pore fluid and porosity are used. One of the solutions to this problem is therefore to use a different fluid. For instance, in the Cambridge test of the dyke, silicon oil with the same density as water is used. The viscosity is chosen to be N centi-stokes or cs (Dow Corning Limited, 1985) because water has a viscosity of 1 cs. This reduces the permeability by N times
CENTRIFUGE TEST OF A DYKE
22 1
and the above relation is retained. However, there is now a possibility that the bulk modulus of the fluid and its damping characteristics are different from those of water. In the VELACS exercise, in order to avoid problems in the interpretation of the centrifuge tests, water was chosen as the pore fluid and consolidation was therefore found to be applied more rapidly than the corresponding prototype. Other substitute fluids used include Metolose (Dewoolkar, 1996). However, for semi-saturated material, the use of another liquid will in all probability lead to a different value for surface tension as well as different drying and wetting characteristics. Although the use of the same fluid in experimental modelling of pollutant transport in a semi-saturated environment has been reported (Cooke. 1991 and 1993; Cooke and Mitchell, 1991a and b) and the scaling of the capillary was reported to be modelled correctly (Hellawell, 1994; Culligan-Hensley and Savvidou, 1995), it would be difficult at this stage to extend such tests to dynamic events.
6.3
CENTRIFUGE TEST OF A DYKE SIMILAR TO A PROTOTYPE RETAINING DYKE IN VENEZUELA
The test represents a fully dynamic analysis with transient behaviour. The physical model is a centrifuge experiment performed by Venter (1985) on the Cambridge Geotechnical Centrifuge. The principle of the centrifuge has been briefly explained in Section 6.2. Simply, the centrifuge reproduces similar stresses and strain history as experienced by the prototype on the scaled model. If the behaviour of soil is controlled mainly by the stress state and its strain history then the centrifuge model is able to predict the generic behaviour of the prototype under earthquake conditions. The layout of the experiment is given in Figure 6.1. The model is built in a strong box with a dyke lying on a sand bed. An oil reservoir is created behind the dyke to provide seepage conditions. Silicon oil is used so that the diffusion equation and the dynamic equation can have the same timescale under the centrifuge condition. This is done by using silicon oil (Dow Corning Limited, 1985) of viscosity of 80 centistokes (80 times the viscosity of the water). Also shown in Figure 6.1 are the measurement devices, which include 11 PPTs - pore pressure transducers, 1 LVDT - Linear Voltage Displacement Transducer, 7 ACCs accelerometers and 3 TSTs-total stress transducers. With the triaxial test results (Venter, 1986) on the Leighton-Buzzard sand used in the centrifuge experiment, soil parameters for the Pastor - Zienkiewicz Mark-111 model are identified. They are listed in Table 6.1. The Finite Element idealization and the boundary conditions are given in Figure 6.2. Also shown in the figure are the positions of 10 pore pressure transducers, 4 accelerometers and the LVDT presented for displacement comparison purposes. The comparison is done for as many points as possible so that an overall picture, together with appropriate mechanisms can be obtained. The test was done at 78g and the material data for the Finite Element analysis are given in Table 6.2. The first study is performed with 4-4 element, i.e. 4noded for soil displacement (u) and 4-noded for pore pressure (p).The computer code used was SWANDYNE-I (Chan, 1988) using SSpj time stepping scheme. For each pair of graphs presented in this section, the left-hand side one is a measured value from experiment, while the right hand side is the computed value. -
222 Table 6.1
VALIDATION O F PREDICTION BY CENTRIFUGE Soil model data ---
Test
14
Predicted
Adjusted Po(kPa) Kevo KevoIP, Keso KesoIP, Mg Mf
* As suggested in original paper (Pastor and Zienkiewicz
* * emin= 0 . 6 5 ( D ~= 100%) em,, = 1.025(DR = 0%)
1986)
Table 6.2 Material data for finite element analysis = 1908kgm-' Bulk density (average soil-pore fluid) = 980kgm-' Density of the pore fluid = 1.092 x 109Pa Bulk modulus of the pore fluid Biot - alpha = 1 .OOOO = 0.444444 Porosity (n) Initial void ratio (e) = 0.80 Permeability of the sand bulk = 2.1 x 10-6ms-' = 2.1 x 10-4ms-' Permeability of the sand drain g-acceleration at lg = 9.81ms-* g-acceleration for the centrifuge test (78g) = 765.18rn~-~) Gauss Point for all cases (Gauss-Legendre) = 2 x 2 No incremental strain subdivision is performed Initial stress method is used for non-linear iterations Convergence criteria for the non-linear iterations: Residual force norm versus current external force norm (for each phase) 5 0.1% Properties of the Rigid block: Density = 2 0 0 0 k ~ m -Mass ~ per unit width: 18.2kg per metre Moment of Inertia per unit width: 0.069503 kgm2 per metre Size of time step = 0.00015 sec (data point spacing of the measurements) Total number of time steps: 1024 0's for the SSpj scheme = 0.5,0.5 Bilinear interpolation for both u and p
223
CENTRIFUGE TEST OF A DYKE
117
I Sand nuxturc 2. Concrete dyke 3. Retaming wall 4. 0 1 1 overflow 5 Oil sea 6. Coarse sand drains
Baw wlth prcscr~bed earthquake movement
7
R i g d boundry
Model d ~ m m r l o nIn rnm
Pore prcssure measurement
@ PPT2626 x
,
-
Figure 6.1
PPT2628 x(69)
Acceleratmn measurement
PPTZR4h ~(90)
- -
I
Sections through model KVV03 showing dimensions a n d transducer locations
Water level k = 2 . I x lo-
k = 2 . 1 x 10/
/ n=nuh
Impervious
u=v=O ii
,
Shaking at bottom
0 Measured
Location Computed location
Computed deformation shape at the end of the earthquake (10 magnification)
Figure 6.2 Finite element idealization
224
VALIDATION OF PREDICTION B Y CENTRIFUGE
I
0.00
I
0.02
0.04
0.06 0.08 Seconds
A
0.10
0.12
0.14
0.08 0 10 Seconds Measured
0.12
0.14
.
Input motion
(ii)
0.00
(iii)
0.0002
-
0.0000
-.
0.02
0.04
DOF: 2
0.06
OF NODE: 26
-g -0.0004 -E -0.0006 .--0.0002
-5 -0.0008 -
2
-0.OOlO
-
-0.0012
-
-0.0014
I
0.0b
0.02
0.04
0.06
0.08 0.10 0.12 Computed Vertical d~splacementof the dyke
,
1
0.14
Figure 6.3(a) Comparison with centrifuge results; (4-4, Gamdm=2) (tip) input motion (iii) vertical displacement of the dyke
(ii) &
225
CENTRIFUGE TEST OF A DYKE
The results are given in Figures 6.3(a)-6.3(h). The input motion was taken from the accelerometer ACC1244 attached to the box. Also shown in Fig. 6.3(a) is the vertical displacement at the crest of the dyke. The comparison is excellent. Besides predicting correctly the final displacement, the rising time and the shape of the rise are also predicted. The reader is reminded that this was the first set of soil parameters obtained directly from the soil model tester, no parametric study has been performed i.e. a prediction is stated as one of Class B-1. The pore pressure transducer A gives good agreement too as shown in Figure 6.3(b) although the value is slightly lower. More oscillation is seen in B; however. the predicted trend is still correct. The agreement of C is remarkable although the size of oscillation is larger, nevertheless, the result is very good. The other graph in Figure 6.3(c) gives the comparison of pore pressure transducer D. The mean value is almost the same as the measured value, although oscillation is more pronounced. E predicts lower pore pressure rise and F is quite satisfactory, so does G. H, I and J are slightly worse but the overall prediction on the pore pressure rise is very good. This oscillation found in the excess pore pressure could be due to the proximity of the rigid boundary condition either at the bottom of the container or near the underside of the dyke. Let us consider the transducers in two groups: (1) Far-field where the influence of the structure is less at locations A, C, E, F, G and H.
EXCESS PORE PRESSURE AT POINT A DEVICE TYPE 6 P R E S U K t 1KANSDUCER D E V I C E h U M B b R 2851
DOb I
OFNODE 198
EXCESS PORE PRESSURE AT POINT B DFVICF TYPE h PRESSURE TRANSDUCER DFVICF N U M B E R : 2846 DOF I
OF NODE: 14
40000 40000 30000 30000 6 20000 h\~~;~~+~+d@+ ~0000 10000 g loo00 0 5 0-1 0000 ;-loo00 -20000 Q 20000 -30000 -30000 I 0 0.00 0.02' 0.04 0.06 0.08 0.1'0 0.i2 0 . K 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Seconds (ii) Mrusr~red Cornputrd
Figure 6.3(b) Comparison with centrifuge results: (4-4, Gamdm=2) (top) excess pore pressure at point A (bottom) excess pore pressure at point B
226
VALIDA TION OF PREDICTION B Y CENTRIFUGE EXCESS PORE PRESSURE AT POINT C 80000
70000
Devlce type pressure tranducer device numher 2848
1
700001 De\ I
Xi1000
Doll l
of node: 98
Doll l
of node 138
70000)
C type: ~
pressure tranducer deb ~ c number e 2338
Figure 6.3(c) Comparison with centrifuge results: (top) excess pore pressure at point C; (bottom) excess pore pressure at point D
(2) Near-structure where the influence of the structure is more pronounced at locations B, D, I and J. It can be observed that comparatively less oscillations are found at locations which are far-field. Even if there is substantial oscillation such as location H, the dominant frequency is more akin to the input frequency (approximately 120Hz). However, for the near-structure locations, the oscillations are much more pronounced and the frequency is more akin to the frequency of the structure and of a higher frequency nature (approximately 240Hz). One possible reason for this oscillation is the proximity of the impermeable solid boundary and the fact the average relative fluid acceleration has been neglected in the u-p formulation. As the average relative fluid acceleration is neglected, any volume change near an impermeable boundary will behave in an undrained manner and lead to a large rate of change in pore water pressure via the fluid continuity equation. Similar behaviour has been found in the proximity of a retaining wall in (Dewoolkar, 1996). When the same analysis is repeated using the fully explicit u-w formulation (Chan et al, 1995),significantly less oscillation is observed in, for instance, point I-PPT2628 (Figure 6.4) when compared with Figure 6.3(f). Initially, all the accelerometers show good agreement with experimental results. As the soil weakens, the value on the higher level (L and M) departs from the experimental value. The predictions of N and 0 are reasonable. This may be due to the
CENTRIFUGE TEST OF A DYKE 100000
IDewce type pressure tranducer dewce number 2847
I
10000
227
Device lype pressure rranducer dewce number 2855
100000
j
I
Dof I
of node 140
I
of node: 196
-
0
0 00
I
I
0 02
0 04
I
0.06
I
0 08
I
I
I
0.10
0 12
0.14
Sccondc .hlemreed (11,
Figure 6.3(d) Comparison with centrifuge results: (top) excess pore pressure at point E; (bottom) excess pore pressure at point F
0 !I0
0 O?
0 01
0 00
(ii)
0 ox 0 10 Seconds
0 1?
0 11
Aleom~d
Figure 6.3(e) Comparison with centrifuge results: (top) excess pore pressure at point G; (bottom) excess pore pressure at point H
228
VALIDATION OF PREDICTION BY CENTRIFUGE Device type pressure tranducer d e v m number: 2628
40000
- Deblce type: pressure tranducer d e v m number 30000 < zoo00 40000
-
10000
-
of node- 23 30000
$ 20000
5
loo00
2
g
0 -
I
000
of node: 17
2626
-: loo00 0
Dof I
40000 -1
I
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001
I
I
000
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(ii)
I
008 010 Sccmds
I
01:
10000
I
011
O(i0
I
I
002
001
I
000
r y y I
008
'
l
010
I 012
I
011
i-ornpu~d
Figure 6.3(f) Comparison with centrifuge results: (top) excess pore pressure at point I; (bottom) excess pore pressure at point J
Dcwcc type: I accelcromctcr dcvlcc numbcr- 1928 300
0'00
0 02
0.04
0.06
0 08
0.10
0 I?
300
7
Dof I
of node. I39
Duf I
o f node: 142
0.14
SECONDS
(i)
wed
Debice type: I acceleronleter device nunihcr. 1583
-'y
300 zoo
1
3004
002
(ii)
004
0.06
0.08
0.10
012
014
SECONDS .&!LUSW~
Figure 6.3(g) Comparison with centrifuge results: (top) acceleration at point L; (bottom) acceleration at point M
229
CENTRIFUGE TEST OF A DYKE 300 200
p I00 ; 0
;-100 5
zoo
{
-300 0.00
(102
ii)
(104
006 008 SCCONDS
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,
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Figure 6.3(h) Comparison with centrifuge results: (top) acceleration at point N; (bottom) acceleration at point 0 Excess pl-essure (kPe)
-20.0
;
0.00
I
0.04
I
0.08 Time (seconds)
I
1
0.12
0.16
Excess prcssurc (kPa) 60.0
40.0
T m e (seconds)
ib)
Figure 6.4 Numerical results of excess pore pressure at point I (PPT2628) using fully explicit GLADYS-2E with 3-3 element: (a) with 44 elements; (b) with 96 elements
VALIDA TION OF PREDICTION BY CENTRIFUGE
230
amount of shear wave energy being transmitted from the bottom to the top. As the soil weakens, less shear wave should be transmitted. However, due to the oscillatory nature of the pore water pressure, the mean effective stress is not reduced as much as in the physical experiment. Therefore the shear modulus is not reduced adequately leading to the excessive transfer of shear wave energy. Despite a number of shortcomings, the results represent an excellent comparison with the experimental results accounting for possible experimental errors. Nevertheless, these results represent a Class B-1 prediction and a set of more convincing Class A predictions are given in the next and subsequent sections.
6.4
T H E VELACS PROJECT
Although many verification exercises have been performed by the authors (Chan, 1988) and (Zienkiewicz et NI., 1990) and others using reported centrifuge results-a more systematic study became recently possible through the project VELACS (VErification of Liquefaction Analysis by Centrifuge Studies-Arulanandan and Scott, 1993) funded by the National Science Foundation, USA. A numerical prediction of several postulated tests was requested from 'predicting participants' before the experiments are performed and results obtained for 'centrifuge experiment participants'. The numerical predictions were kept in sealed condition by a third party (Thompson and Lambe, 1994) and these were not made available to the 'centrifuge experiment participants'. This double blind policy was introduced to minimize possible 'cheating' and thus enhance the credibility of the results. Such 'Class A' predictions were submitted by twenty 'predicting participants' by 30th September, 1992 (Table 6.3) when apparently the centrifuge tests were first commenced by seven universities (University of California, Davis; California Institute of Technology; Cambridge University; University of Colorado, Boulder; Massachusetts Institute of Technology; Princeton University; and Rensselaer Polytechnic Institute). However, some of the specified centrifuge tests could not be carried out-and additional computations ('Class B') mostly because of the difference in the prescribed and actual input earthquake motion were requested-without however supplying other experimental results. It is instructive for the readers to note that, except for the simpliest model No. 1 which represents a level soil layer, all numerical predictors used computer codes based directly or indirectly on the Biot theory and approximation form introduced in this book (Smith, 1994). Nine centrifugal models (see Figure 6.5 taken from Arulanandan and Scott (1993)) were selected for the verification: Model No.1-Horizontally Model No.2-Sloping Model No.3-A 0
layered loose sand in laminar box
loose sand layer in laminar box
sand layer one side dense, the other side loose
Model No.4a-Stratified
soil layers in laminar box
Model No.4b-Stratified
soil layers in rigid box
231
THE VELACS PROJECT
I Model No. 7
1
Laminar box
Laminar box
A
E n d Dr = 40%
Sand: Dr = 40%
1
I(
I
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t
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1 Model No. .
n
~ o d eNo. l 4
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II
Laminar box
1 Sand: Dr = 60%]
1 Model No. t
~ o d e No. l 4
n
Rigid Box
1 Sand: Dr
= 60%
1
*
R i g ~ dBox ;,
3r
I 1 Model No.
'
Rigid Box
n
~ o d eNo. l 1 Rigid Box
n
7.6m silt
1 ~ o d eNo. l I
Internurional Conference on the Verification of N~inlericalProcedure for the Annlvsis ($Soil Liquefaction Problems 17 - 20 October 1993 Orpnized by The University of Californaia, Davis and The California Institute of Technolog?.
Figure 6.5 Centrifuge model configurations for class A: Predictions-VELACS (Arulanandan and Scott. 1993)
Project
232
VALIDATION OF PREDICTION BY CENTRIFUGE
m Q x
X
x
X
X
X
X
m
X
X
m
<
X PI
0
x x x
x
m
x
Q
x x x x
X
x
x
X
x x x
X
x
d
X
G
--
233
THE VELACS PROJECT 0
Model No.6-A
submerged embankment in rigid box
Model No.7-A
submerged sand embankment with slit core
Model No.11-A
gravity quaywall with sand backfill
Model No.12-A
structure embedded in stratified soil layers
Most of the tests have been performed at more than one centrifuge for centrifuge validation purpose and considerable scatter of results can be found between the centrifuge results. Three of the authors led three different groups of predictions using computer codes SWANDYNE-4 (implicit u-p with partial saturation), SWANDYNE-I1 (implicit u-p with full saturation) and MuDIAN (implicit u-U with fullsaturation, to be introduced in Chapter 7) based on their work in University College of Swansea. The SWANSEA group led by the senior author of this book, Zienkiewicz, together with one of the authors, Pastor, presented eight predictions for model Nos. 1,2,3.4a. 4b, 6, 7 and 11 (Zienkiewicz et d . , 1993a-h). The second author, Chan, presented seven predictions for model Nos. 1 , 2, 3, 4a, 4b, 7 and 11 (Chan et ul., 1993a-g) and another of the authors, Shiomi, together with the senior author, presented four predictions for model Nos 1, 2, 11 and 12 (Shiomi et (I/., 1993a-d). Lastly, Chan (Chan rt ul., 1994) and Shiomi (Sture ct (I/.,1994) were also involved in the process of overviewing numerical predictions. Most of the predictive results can be classified as good or excellent. A selection of these will be presented in the subsequent sections in this chapter with a brief description of the experimental set-up and conlparisons with numerical results on the same scale. But before going into the detail of the prediction, the following sections are devoted to the analysing procedure using SWANDYNE-11.
6.4.1 General analysing procedure This analysis procedure is applicable to all the predictions performed by the authors: The information about the centrifuge model is gone through in detail and key data noted. A finite element mesh is generated using a pre-processor. Time required for the subsequent dynamic and consolidation analysis are also taken into consideration so that a reasonable mesh is chosen. The appropriate boundary conditions are then applied at the boundaries of the model. Tied nodes are used to model the laminar box behaviour (see Section 6.4.3). The hydrostatic pressure, which is assumed to be constant throughout the analysis, is prescribed at the fluid nodes concerned and the pressure on the solid phase is also applied.
234
VALIDATION OF PREDICTION BY CENTRIFUGE
The appropriate permeability and gravitational acceleration are then included. The models are modelled at the model scale so the appropriate acceleration level is the centrifugal acceleration imposed. A static analysis was performed to determine the initial stress state of the model. A KO value of 0.4 is assumed. In order to avoid tensile stress and high stress ratio, Mohr Coulomb elasto-perfectly plastic model is used for the initial analysis with a reduced frictional angle of 25.. (vii) The output of the static analysis is considered carefully to check if the initial pore pressure distribution is reasonable and also if the stress state is acceptable. (viii) A no-earthquake dynamic run is then performed to check if the initial stress state is in the correct equilibrium condition. If it is not, a new static initial analysis is performed with modified parameters to obtain equilibrium. When the initial stress state is acceptable, a linear elastic analysis is performed to note the basic behaviour of the finite element mesh. Then a non-linear analysis is performed for the earthquake stage with the supplied horizontal and vertical earthquake with proper scaling. The dynamic analyses were performed using a Generalized Newmark (Katona and Zienkiewicz, 1985) scheme with non-linear iterations using initial linear elastic tangential global matrix. The constitutive model used is the Pastor-Zienkiewicz (1986) mark-111 model. The parameters used are described in Section 6.4.4. The time step used is usually equal to a simple multiple of the earthquake spacing. The choice of the time step depends on the number of the stations in the earthquake input and the frequency of the input. The void ratio i.e. permeability and other geometric properties were kept constant during the analyses. Rayleigh damping of (minimum) 5'%,is applied at 100Hz which is the dominant frequency in the earthquake-like motion input. (xi) The earthquake phase of the analysis is then plotted to check for any anomaly. (xii) The consolidation then follows the dynamic analysis. Usually a larger time step is used for the consolidation analysis, a gradual change in time step is used to avoid numerical instability. The full dynamic equation is used for the consolidation stage of the analysis with the appropriate mass matrix. (xiii) The results are first plotted using a simple post- processing program to check its validity. If the result does not seem reasonable, the dynamic analysis is repeated with another set of numerical parameters, iteration schemes etc. until a reasonable and numerically stable result is obtained. (xiv) Various plots are then performed for the final report. Since total quantities e.g. pore pressure and displacement are used in the program, post-processing is required to obtain the excess pore water pressure and relative displacement required by the specification.
THE VELACS PROJECT
235
(xv) Other post-processing e.g. excess pore water pressure ratio, spectral analysis and response spectrum are calculated for reporting purposes.
6.4.2
Description of the precise method of determination of each coefficient in the numerical model
The determination of each coefficient of the Pastor-Zienkiewicz mark I11 model follows the procedure outlined in Section 5.5 of (Chan, 1988) and is being reproduced in this section. As drained monotonic, undrained monotonic and undrained cyclic tests are the most widely available tests in common engineering applications, they are chosen in the parametric determination process. During earthquake and other rapid loading, the undrained test is more relevant. The tests should be done with samples having relative density around the intended relative density. In this section, the way to identify each of the parameters required by the model is illustrated. One undrained monotonic and cyclic test is taken from each of the loose sands (Dr = 40U%),dense sand (Dr = 60%) and silt experimental data sets, respectively. The comparison of the constitutive model and the physical undrained triaxial tests has been given in Chan er al. (1992a and 1992b). These results are produced using a soil model subroutine for DIANA-SWANDYNE I1 interfaced with a soil model testing program SM2D and the experimental results are also plotted on the same graph for the monotonic test. In the following sections, each parameter will be discussed in turn: Mg (dimensionless): can be estimated from the graph plotting stress ratio versus the shear strain or axial strain. Mg is approximately equal to the maximum value of stress ratio that the test reaches. It can also be estimated from the q versus p' plot with a tangent drawn from the origin to the residual stress path in an undrained triaxial test. Mg corresponds to the maximum slope obtained by this method. Mg can also be obtained from the drained test using the intercept of dilatancy versus stress ratio plot. In this exercise, the stress ratio plot was used. Mf (dimensionless): can be determinated by matching the shape of the stress path in the q versusp' plot in an undrained triaxial test. Alternatively, it can be obtained by matching the critical stress ratio that the behaviour of the soil changes from contractive to dilative behaviour in the case of dense sand. The ~ serve as a good starting point for the evaluation of its value of D R . M can value. In this exercise. the critical stress ratio is used. a g (dimensionless): can be obtained from the slope of the graph between the dilatancy and stress ratio over the Mg graph. However, this value is usually taken as 0.45 and it is also used in this exercise. of (dimensionless): is usually taken to be the same as a g so that the loading locus and plastic potential are having the same shape. KevOc (dimensions of stress): represents the value of the bulk modulus at the mean effective stressplO. It can be obtained by matching the initial slope of the
236
VALIDATION OF PREDICTION BY CENTRIFUGE
mean effective stress p' or pore pressure versus axial strain plot in an undrained test. Its value can be adjusted so that a better match of the curve of pore pressure versus axial strain can be obtained. In the VELACS exercise, this was done so that the end point in the predicted curves stayed close to the experimental data. (vi) KesOc (dimensions of stress): represents the value of three times the shear modulus at the mean effective stress p'O. It can be obtained by matching the initial slope of deviatoric stress q versus axial strain plot in an undrained test. Its value can be adjusted so that a better match of the curve of q versus axial strain can be obtained. In the VELACS exercise, this was done so that the end point in the predicted curves stayed close to the experimental data. (vii) $o (dimensionless): is usually taken as 4.2 and this value is taken here. (viii) fiI (dimensionless): is usually taken as 0.2 and this value is taken here. (ix) Ho (dimensionless): is determined by fitting the curves in p' or q versus the axial strain plot. It can be found by matching the shape of the q versus p' plot for undrained tests also. (x) Huo (dimensionless): is determined by matching the initial slope of the first unloading curve. (xi)
- y (dimensionless): ~ ~ is determined by matching the rate of change of the slope of the first unloading curve or by matching the number of cycles in a series of loading and unloading. The second method is used in this exercise.
(xii)
(dimensionless): is determined by matching the slope of the first reloading curve or by matching the number of cycles in a series of loading and unloading. The second method is used in this exercise.
(xiii) pb (dimensions of stress): is the initial mean effective stress of the undrained triaxial test.
6.4.3 Modelling of the laminar box The way that the boundary conditions were incorporated in the numerical method has been given in the relevant prediction papers. The laminar box (Hushmand et al., 1988) is treated with tied node facility. The horizontal and vertical nodal displacements at the two ends of the soil are restrained to have the same value. The interface between structure and soil is assumed to be perfectly bonded. No change of boundary condition is made during the analyses. Only rigid block and four noded linear isoparametric elements for both the displacement (u) and pore pressure (p) are used in the analyses.
COMPARISON WITH THE VELACS
6.4.4
237
Parameters identified for the Pastor-Zienkiewicz Mark I11 model
There are quite a number of parameters in the Pastor-Zienkiewicz mark 111 model which require definition. Three CIUC (Isotropically Consolidated followed by Undrained Compression test) experimental results starting from 40kPa were chosen to identify the parameters. The experimental results were taken from (Arulmoli et al., 1992) which provided the standard soil model test results for the numerical predictors. The 40kPa ones were chosen because they are close to the mean effective stress value at the middle of the centrifuge model. The permeability of silt is also calculated for this level of mean effective stress. The set used in the Class A prediction for (Chan et a1 1993a - is given below: (9 Loose sand: (Dr = 40%) Experiment CIUC4051 was used. The parameters obtained are as follows: Mg = 1.15 Mf = 1.03 af = cug = 0.45 KevOc = 770kPa KesOc = 1155kPa, the elastic modulus is proportional to the mean effective stress, Bo = 4.2 PI = 0.2 p:, = 4kPa HO = 600 HUO= 4000 kPa Y H = ~ 2yDM= 0. (ii) Dense sand: (Dr = 60%) Experiment CIUC6012 was used. The parameters obtained are as follows: Mg = 1.32, Mf = 1.30, a f = a g = 0.45. KevOc = 2000kPa, KesOc = 2600kPa, the elastic modulus is proportional to the mean effective stress Po = 4.2, P1 = 0.2 pk = 4kPa, Ho = 750, Huo = 40000kPa, YH,, = 2, ?DM = 4 (iii) Silt: Experiment CIUCBS13 was used. The parameters obtained are as follows: Mg = 1.15, Mf = 0.50, af = ag= 0.45, KevOc = 400kPa, KesOc = 1520kPa, the elastic modulus is proportional to the mean effective stress, Po = 4.2. pl = 0.2 pk = 4kPa, Ho = 900, Huo = 100000kPa, TH, = 2,yDM= 8.
6.5
COMPARISON WITH THE VELACS CENTRIFUGE EXPERIMENT
6.5.1 Description of the models Model No. 1 Taboada and Dobry (1993), Stadler et a1 (1993), Ishihara 1994) A water-saturated uniform layer of loose sand (Dr=40%), 10m (in prototype scale) thick, in a laminar box, was mainly subjected to horizontal base motion. The test was instrumented as shown in Figure 6.6(a). Centrifuge test resuts for model No.1 were carried out by three universities. RPI is the primary experimenter and UC Davis and the University of Colorado conducted duplicate tests.
Model No. 3 (Scott et a1 1993) A water-saturated layer of sand deposited at l l m (in prototype scale) thick, in a laminar box was subjected to base shaking. The test was instrumented as shown in
238
VALIDATION OF PREDICTION BY CENTRIFUGE
Figure 6.7(a). Although the model contained sand deposited at both 40% and 70% relative densities, laboratory data were available only for 40% and 60% relative
1-
I
LVOTI LVOTS
Concentrated masses (aluminum alloy rectangular rings) F.E mesh
Instruments for model No. I M
1
RPI Test- 2
(Sccond) Time
ID1
0.6
0.6
0.4
0.4
5
0.2
5 k 0.0
E -
00
2 -0.2
3 -0.2
-0 4
-0.4
5
0.2
5
-0 6
(Second) Tune
101
-0.6 0
2
4
6
8
12 14 16 18 20 0 2 4 6 (Second) T~me Expcrirnental and predicted horizontal ~ c e l c r a t i o nA H 3 10
(Second) Time
8
10 12 (Second) Time
(Second) Time
14
16
18
20
239
COMPARISON WITH T H E VELACS
densities. The predictors were expected to infer properties of sand at 70%)relative density based on the properties at 40'% and 60% relative densities. In our case, the material parameters for relative density 60% is used. The centrifuge results for this model were carried out by CIT as primary experimenter and UC Davis and RPI as duplicate experimenters. 0.6 0.4
0.4
.sc2 0.2 + 0.0
.g
g -0.2
y
C
0.2
+E
0
0.0
U
-0.4
-0.2
414 noded elements
-0.4
-0.6 0
2
4
6
(Second) Time 8 10 12 14 16 18 20 (Second) Time Expennental and Predicted horizontal acceleration AH5 (KNM:)
(KNM')
2 1003 Z 80'2 a 60E 40203 0 0
2 100 $ 802 60-
RPI-Test 2 P5
k g
r/]
8
Prediction P5
4020-
X
1
W
I
O
<
2
-
1
I
l
l
0 10 20 30 40 50 60 70 80 90 '00 l b 20 30 40 50 60 70 80 90 100 (Second) (Second) Ti]% Ti'"e Experimentel and predicted excess pore-pressure (KNM~I (KNM') 2 1100 2 100 2 80 Prediction RPI-Test 2 802 60 60 2 P6 P6 a 40 40 m
5
%
k g
-
20
m
-
0
X
W
W
O
I
0
(KNM')
2 100 80 I 2 60 -
i L
I
I
I
I
I
I
I
I
0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 (Second) (Second) Time Time Experimental and predicted excess pore-pressure P6 (KNM') Prediction
RPI-Test 2 P7 m
g 3
20 0
g
X I W ; I I I I l I 1 I 0 10 20 30 40 50 60 70 80 90 I00 (Second) Time
20 O
0
10 20 30 40 50 60 70 80 90 100 (Second) Time
Experimental and prcdicted excess pore-pressure P7
VALIDATION OF PREDICTION BY CENTRIFUGE
240
Model No. 11 (Madabhushi and Ferg 1994) A soil and water retaining wall was subjected to base shaking. A surcharge was added on the backfill of the retaining wall. The relative density of the sand in this model was approximately 60%. The centrifuge test of this model was carried out by Cambridge University only and was instrumented as shown in Figure 6.8(a). No repeat test was conducted.
6.5.2 Comparison of experiment and prediction Overall, predictions of the SWANDYNE program compared well with experiment. The close agreement of the predicted pore pressures and displacements with those measured in centrifuge tests affirms the reliability of the computation procedure in the SWANDYNE program when used with carefully calibrated material properties and model parameters. The use of fully coupled equations for the soil-pore fluid interaction and of a simple soil model based on the generalized plasticity for the soil skeleton, both introduced in detail in this volume, forms a consistent and powerful prediction procedure. This rigorous analysis procedure promises to be a reliable tool for practical problems in engineering. However, the following problems have to be addressed.
RPI-Tcst 1
Prediction
-p
-0.1
C;
-0.2
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 I
0
I
I
I
I
I
I
I
I
I
I
l O P 3 0 4 1 3 W W 7 C I ~ W l 0 0
(second)
(second) Time Experiment$ and predicted vertid displaccmmt LVDTl
Figure 6.6 VELACS Centrifuge Model No. 1: instrumentation, finite element mesh and comparison of experimental and predicted results
241
COMPARISON WITH T H E VELACS
(a) Although the predicted horizontal acceleration on the surface of the soil layer is much higher than the experimental results, it is obvious that the experimental results of the surface acceleration is not necessarily reliable after soil liquefaction. We also have recognized that the numerical result of the surface acceleration obtained by using mixed 8-4 noded elements, probably gives a much higher peak value of acceleration. This can be improved by using 4 4 noded elements. There is no large difference in the acceleration at the middle points between prediction and experiment. The prediction also showed that the peak acceleration on the surface decreased
Instruments for model No. 3
F. E, mesh 2
(KNIM')
(KNIM
(second) TlME
(second) TIME
50
0
100
150
200
250
200
250
"h (second) Tme
2
(KNIM )
Prediction
M 20 Y
0 0
50
100
I50 (second) Tme
Experimental and ~redictedexcess pore-pressure P8 (loose sand)
Experimental and predicted excess pore-pressure P9 (dense sand)
Figure 6.7 VELACS Centrifuge Model No. 3: instrumentation, finite element mesh and comparison of experimental and predicted results
VALIDATION OF PREDICTION BY CENTRIFUGE
242
-&- - f " !-lTI
.
.
CIT-Test
E
k20
i: 3
0 0
50
100
I50
200
250
(second) Time
(KNlM )
0 0
50
100
I50
200
250
(second) Time Expenmental and predicted excess pore-pressure PB (loose sand)
0
50
100
150
200
I50
200
250
(second) Ttme
"
)
;
(second) T~me
0
I00
50
I50
200
250
(sccond) Tme Ex~nmentaland predtctcd cxcess pare-prcssurc P9 (dense sand)
250
(second) Time
(m)
I00
1
":TI 9
SO
IKNIM )
ti 20
w
0
(KN M
(second) Time
lKN,h4-)
g 3
0
50
100
150
200
250
(second) Time
(m)
(second) Time
Prediction
4.3 1
5
44 0
50
100
150
200
250
(second) Time Experimental and predicted vertical displacement 1 5 (loose sand)
0
50
100
I50
200
250
(second) Time Exper~mentaland pred~ctedvert~caldisplacement L6 (dense sand)
Figure 6.7 (cont.) VELACS Centrifuge Model No. 3: instrumentation, finite element mesh and comparison of experimental and predicted results
243
COMPARISON WITH T H E VELACS
gradually when liquefaction occurred. Similar phenomena can be found in the centrifuge tests and earthquake history.
(b) The predicted time histories of pore-pressure agreed closely with those recorded in the centrifuge tests. In model No. 1, the experimental results showed slightly faster pressure dissipation; however, the differences were not large. The experimental
0
2
4
I
10
8
6
-Initial boundary - - . - - Uoundaty after sarthquate t s s ~
(m)
c
0.5
localion
0.4
-
(25.0.1 2.0.7.0) In (1,VDTI) horizontal
0.3
2 0.2
LVDT 1
Prsdiction
0.1 0.0 0
5
10
15
20
25
30
35
40
Figure 6.8 VELACS Centrifuge Model No. 11: instrumentation, finite element mesh and comparison of experimental and predicted results
244
VALIDATION OF PREDICTION BY CENTRIFUGE
results of model No. 3 by RPI and CIT gave obviously different peak values of porepressure and made comparison very difficult. The predicted post-earthquake consolidation was consistent with the values measured in the centrifuge test. (c) The prediction of displacement showed similarities to those measured during centrifuge experiment. The horizontal displacement of the retaining wall in model No. 11 was well predicted. The predicted values of surface settlement in model No. 1 were lower than those recorded in the centrifuge test, but the difference was not too marked. In model No. 3, although the prediction of surface settlement in the loose sand agreed closely with that measured in the centrifuge tests, the numerical analysis
0.75" AC4 ELEVATION
Figure 6.9 (a)Schematic model configuration of test M M D l (Dewoolkar 1996); (b) Finite element mesh with slip element, in heavy line (Dewoolkar, 1996)
245
CENTRIFUGE T E S T OF A RETAINING W A L L
did not predict more settlement in dense sand than in the loose sand which was recorded in all experiments for this model, despite a later repeat of the experiments which showed that different results could also be obtained (Dobry, 1996). Almost all numerical predictors failed to achieve this qualitative difference. The mechanism involves a failed dense sand wedge moving into the liquefied loose sand due to the lack of support. It is of interest to carry out further research into this phenomenon. More research is also needed on the modelling and calibration of data for the behaviour of dense and and silt under cyclic loading.
6.6 CENTRIFUGE TEST OF A RETAINING WALL The schematic model configuration of test MMDl in Figure 6.9 (a) has been taken from (Dewoolkar, 1996). A layer of slip element can be found behind the retaining wall in Figure 6.9 (b). Without this layer of slip element, a consistent stress state cannot be found as the initial condition for the dynamic analyses. Good agreement with theoretical and experimental results has been found for the static analysis in Figure 6.10. The experimental results are striffer than the theoretical and numerical results because of the stiffening of the retaining wall by the weld at the root of it. Selected results for the horizontal accelerations at the tip of the wall, horizontal accelerations within the soil and dynamic bending strains in the wall can be found in Figure 6.1 1. All of them showed good agreement with experimental results. The same is true for the horizontal displacement at the tip of the wall and long-term excess pore pressure trace in Figure 6.12. In all comparisons, the program SWANDYNE introduced by the authors was used. Although the numerical modelling was performed at MMDI (SWANDYNE 11) Mesh-2
MMDI (SWANDYNE 11) Mesh-2
-0.1
0
0.1
0.2 0.3 0.4 Deflection (inch)
0.5
0.6
(a)
-.
. Numerical
-10 0 10 20 Total lateral earth pressure (psi) ('J)
---- ---
: Theoretical
+ : Experimental
Figure 6.10 (a) Static deflection of the wall (Dewoolkar, 1996); (b) Total lateral earth pressure on the wall (Dewoolkar, 1996)
VALIDATION OF PREDICTION BY CENTRIFUGE
246 (a)
MMDI (DI) (model scale)
MMDI SWANDYNE 11 . . . . . . ., . . . - .; . . . . . . . . . . . . . .:..AC9 ..... '
..
-40
1
0
0.1
0.2 0.3 Time (s)
0.4
I
0.5 Time (s) MMDI SWANDYNE 11
MMDI (DI) (model scale)
(b)
........:........ ;..............
1 ....
4 0
-
0
01
02
03
04
05
0
..)
. . . . . . . . . . . . . . . . :.. . . . . . .:...... .
0.1
T ~ m e(s)
g
0"
Ol
0.2 03 T ~ m e(s)
0.3
0.4
0.5
MMDI SWANDYNE I1
;-500 0
0.2
Time (s)
MMDI (DI) (model scale)
(c)
2 -500
: AC8 ...........
04
05
c
0"
o
0.1
0.2 0.3 Time (s)
0.4
0.5
Figure 6.11 Comparison with centrifuge test MMDl (Dewoolkar, 1996): (a) Horizontal wall accelerations at the top; (b) Horizontal soil accelerations; (c) Dynamic bending strains of the wall
different stages of the centrifuge, all the material parameters used are derived from the VELACS exercise as given in Section 6.4.4. Only the amount of damping has been varied to investigate the effect of different levels of damping.
6.7 CONCL USZONS In this chapter, the u-p formulation derived in Chapter 3 has been used to compare with various centrifuge tests performed on the Cambridge Geotechnical Centrifuge
247
REFERENCES (a)
-
MMDI (DI)
-0.2
(model scale)
MMDI SWANDYNE I I
1 0
0.1
0.2 0.3 Time (s)
0.4
0.5
T ~ m (s) e (a)
MMDl (Dl)
(b)
= 15
MMDI SWANDYNE I I
(model scale)
-a .A
PP6
2 I0
2 2
a 5
2
-
a
3 0 0
-5
.
0
05
1
15 2 2 5 Tune (s)
3
35
4
0
0.5
1
.
1
.
1.5 2 2.5 Time (s)
3
3.5
4
Figure 6.12 Comparison with centrifuge test MMDI (Dewoolkar, 1996): (a) Dynamic wall deflections; (b) Long-term excess pore pressure
and the VELACS project. Very good and excellent agreements have been obtained thus validating the formulation and the computer code for various types of analysis under saturated condition. In the next chapter, we are going to show numerical predictions of practical examples and the use of the numerical procedure in design.
REFERENCES Arulanandan K. and Scott R. F. Eds. (1993) Proceedings of VELACS symposium, A. A. Balkema, Rotterdam. Arulanandan K. and Zeng X. ( I 993a) Experimental results of Model No. 1 in Verificution of Numerical Proceclures,for the Ana1~vsi.sqf Soil Liquefimion Problems, (eds.) Arulanandan K . and Scott R. F., UC Davis, 17-20 Oct, A. A. Balkema, Rotterdam, 19-24. Arulanandan K . and Zeng X. (1993a) Experimental results of Model No. 1 in Verification o f Numerical Procerlures,for the Ana1ysi.r ?fSoil LiqutIfaction Problems, (eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct, A. A. Balkema, Rotterdam, 295-300. ArulmoIi K., Muraleetharan M. M. H. and Fruth L. S. (1992) VELACS Laboratory testing program - Soil data report, Earth Technology Corporation, project No. 90-0562, Irvine. California. Chan A. H. C. (1988) A unified finite element solution to static and dynamic geomechanics problems, Ph.D. Dissertation, University College of Swansea, Wales.
248
VALIDATION OF PREDICTION B Y CENTRIFUGE
Chan A. H. C., Famiyesin 0 . 0. R. and Muir Wood D . (1992a) Report No. CE-GE92-23-0: Numerical Simulation Report for the VELACS Project - General Description, Department of Civil Engineering, Glasgow University, Glasgow. Chan A. H. C.. Famiyesin 0 . 0 . R. and Muir Wood D. (1992b) Report No. CE-GE92-23-1: Numerical Simulation Report for the VELACS Project-Class A prediction of RPI model, Department of Civil Engineering. Glasgow University, Glasgow. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D . (1993a) Numerical Prediction for Model No. 1 in Verification qf Numericrrl Procetlure.s~forthe Anulysis if Soil Liqurfirction Problems, (eds.) Arulanandan K. and Scott R. F., UC Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam. 87-108. Chan A. H. C.. Famiyesin 0. 0. and Muir Wood D . (1993b) Numerical Prediction for Model No. 3 in firificution of Nun~ericrilProcec/~rrc~.s,fi,r the Anulysi.~of Soil Liqu+ction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam. 343-362. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D. ( 1 9 9 3 ~Numerical ) Prediction for Model No. 3 in Verifi'cntion ofArun~ericcrlProceduresfor tile Anrrl~vsisoJ'Soil Liquejkction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam. 489-510. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D. (1993d) Numerical Prediction for Model procedure.^ ,fbr the Ancr1j~si.sof' Soil Liqwfirction ProNo. 4a in Vmficution of' N~in~eric,uI blenu, (eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam, 623-630. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D. (1993e) Numerical Prediction for Model No. 4b in Ver~ficutionof' Nurnericnl Proceclures f i ~ rtlw Ancrlysis q f Soil Liquclfirction Problenis. (eds.) Arulanandan K. and Scott R . F., U C Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam, 71 1-720. Chan A. H. C., Famiyesin 0. 0 . and Muir Wood D. (19930 Numerical Prediction for Model No. 7. in Verification of Nun~eric,ulProcedures ,for the Analj,.r.is @Soil LiquLlfirction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct, 1, A. A. Balkema, Rotterdam, 835-850. Chan A. H. C., Famiyesin 0 . 0 . and Muir Wood D. (19938) Numerical Prediction for Model No. 11, In Ver~ficntion of Nun~c~riccrlProcedures ,for the Anrrl~~sisof Soil Liquefnction Prohlenis, (eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct.. 1, 909-93 1. Chan A . H. C., Siddharthan R. and Ito K. (1994) Overview of the Numerical Predictions for VELACS Model No. 3 in Ver~ficcirionof Nutnericul Procedures ,for the unulysis of soil liquefuction yrobletns, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct, A. A. Balkema, Rotterdam, 1443-1456. Chan A. H. C.. Famiyesin 0. 0. and Muir Wood D. (1995) User Manual for GLADYS-2E, School of Civil Engineering, University of Birmingham, December, Birmingham. Cooke B. (1991) Centrifuge modelling of flow and contaiminant transport through partially saturated soils, Ph.D. Dissertation, Queen's University, Kingston, Ontario, Canada. Cooke B. (1993) Physical modelling of contaminant transport in the unsaturated zone in Geotecl~niculMunugement of Waste and Contaminution, Balkema, Rotterdam. Cooke B. and Mitchell R . J. (1991a) Evuluution of' Contcrn~incrntTrunsport in Purtiully Srrtur n r d Soil in Centrifuge '91, Balkema. Rotterdam. Cooke B. and Mitchell R. J. (1991b) Physical Modelling of dissolved contaiminant transport in an unsaturated sand, Cunudiun Geotecl~nicalJournal, 28, No. 6, 829-833. Culligan-Hensley P. J. and Savvidou C. (1995) Environmental geomechanics and transport process in Geotechniccrl Centrifuge Technology. (ed.) Taylor R. N. Blackie Academic & Professional, London. Chapter 8, 196263.
REFERENCES
249
Dewoolkar M . M. (1996) PhD Thesis, University of Colorado, Boulder. Dobry (1996) Private communication Dow Corning Limited (1985) Dow Corning 200 Fluid in Information about Silicone Fluids Bulletin: 22-069D-01, Dow Corning Data Sheet. Hellawell E. E. (1994) Modelling transport processes in soil due to hydraulic, density and electrical gradients, Ph.D. Dissertation, University of Cambridge. Hushmand B., Scott R . F. and Crouse C. B. (1988) Centrifuge Liquefaction tests in a laminar box, CI.otec/iriiq~w,38, No. 2, 253-262. lshihara K . (1994) Review of the predictions for model 1 in the VELACS program in Verificution of Nurnc~riculProcedures,fi,r the Anulysis qf Soil Liquefirctiori Probleriis, (eds.) Arulanandan K . and Scott R. F., UC Davis, 17-20 Oct, A. A. Balkema, Rotterdam. 1353-1368. Katona M. G . and Zienkiewicz 0 . C. (1985) A unified set of single step algorithms Part 3: The Beta-m method, a generalisation of the Newmark scheme, Int. J. Num. Merh. Eng.. 21. 1345-1 359. Lee F. H. and Schofield A. N. (1988) Centrifuge modelling of sand embankments and islands in earthquakes, GPotechniqzre, 38, No. I , 45-58. Madabhushi S. P. G. and Zeng X. (1994) An analysis of the seismic behaviour of quay walls in Veri/icution of'Numericc11 Procedurc~s.fOrthe Aticil~vsisq f Soil Liqwfirction Problems, (eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct, 2, A. A. Balkema, Rotterdam. 1593-1 606. Pastor M. and Zienkiewicz 0 . C. (1986) A generalised plasticity hierarchical model for sand under monotonic and cyclic loading, NUMOC II, Ghent. April, 131-150. Schofield A. N. (1980) Cambridge Geotechnical Centrifuge Operations - 20th Rankine Lecture, GPotechiiyue, 30, No. 3. 227-268. Schofield A. N. (1981) Dynamic and Earthquake geotechnical centrifuge modelling, Proc. Itit. Conf: on Recent Advclncm iri Geotechnicrrl Engineering r~ndSoil Dj.nriniic.s, University of Missouri-Rolla, Rolla, MO, USA, 3, 108 1-1 100. Schofield A. N. and Zeng X. (1992) Design and performance of an equivalent-shear-beam (ESB) container for earthquake centrifuge modelling - CUEDID-soilslTR24.5, Cambridge University Engineering Department. Cambridge. England. Scott R . F., Hushmand B. and Rashidi H. (1993) Model No. 3 primary test description and test results in Verificrition qf Nuniericul Prucmiurrs jbr rlie Anc11y.sis qf' Soil Liqu
250
VALIDATION OF PREDICTION BY CENTRIFUGE
Stadler A. T.. KO Hon-Y. and Sture S. (1993) Experimental results of Model No. 1 in Ver~fi'cutionqf Numerical Procedures for the Ancr1~~si.sqf' Soil Liquefuction Problems, (eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct, A. A. Balkema, Rotterdam, 25-44. Steedman R. S. and Zeng X. (1995) Dynamics in Geotechnicirl Cmtr~f'ugeTecl~nology,(ed.) Taylor R. N., Blackie Academic & Professional, London Chapter 7, 168-195. Sture S., Law H. K., Shiomi T. and Iai S. (1994) VELACS: Overview of numerical predictions for model No. 12, in Veriji'cation c?f'NumericcrlProceclure.s,fbr the Analysis qf'Soil Liquefuction Problems, (eds.) Arulanandan K. and Scott R . F., U C Davis, 17-20 Oct., 2, A. A. Balkema, Rotterdam, 1635-1646. Taboada V. M. and Dobry R. (1993) Experimental results of Model No. 1 at RPI in Verification of Nurnericd Procedures,fbr rlze Ancrlysis qf' Soil Liqugfuction Problen~s,(eds.) Arulanandan K . and Scott R. F., UC Davis. 17-20 Oct, 1, A. A. Balkema, Rotterdam, 3-18. Thompson P. Y. and Lambe P. C. (1994) Project VELACS management and coordination in Verif~cationof Numerical Procedures ,fbr the Andysis of Soil Liquefirction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct, 2, A. A. Balkema, Rotterdam, 1267-1280. Towhata I. (1994) Review of prediction 'A' on model I1 in Ver;fi'crrtionqfNun~ericcr1Proceduresfor the Anabsis o f s o i l Liyuefuction Problems (eds.) Arulanandan K. and Scott R . F., UC Davis, 17-20 Oct. 1, A. A. Balkema, Rotterdam, 1607-1612. Venter K. V. (1985) KVV03 data report: Revised data report of a centrifuge model test and two triaxial tests, Andrew N. Schofield and Associates, Cambridge, England. Venter K. V. (1986) Triaxial data report: Report on seven triaxial tests, Andrew N. Schofield and Associates, Cambridge, England. Venter K. V. (1987) Modelling the response of sand to cyclic loads, Ph.D. Dissertation, Cambridge University Engineering Department. Zienkiewicz 0. C., Chan A. H. C., Pastor M., Paul D. K. and Shiomi T. (1990) Static and Dynamic Behaviour of Geomaterials - A rational approach to quantitative solutions, Part I - Fully Saturated Problems. Proc. Roy. Soc. Lond.. A429, 285-309. Zienkiewicz 0 . C., Huang M. and Pastor M. (1993a) Numerical Prediction for Model No. 1, in Verrfication of Numerical Procedures for the Analysis qf' Soil Liquefuction Problen~s,(eds.) Arulanandan K . and Scott R. F., U C Davis, 17-20 Oct., 1, 259-276. Zienkiewicz 0. C., Huang M. and Pastor M. (1993b) Numerical Prediction for Model No. 2, in Verifi'cation of Numericcrl Procedures for the Anal~vsiso j Soil Liquejuction Problems, (eds.) Arulanandan K . and Scott R . F., U C Davis, 17-20 Oct., 1,423-434. Zienkiewicz 0 . C., Huang M. and Pastor M. ( 1 9 9 3 ~Numerical ) Prediction for Model No. 3, in Verification of Numerical Procedures,for the Analysis qfSoi1 Liquefirction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct., 1, 583-591. Zienkiewicz 0. C., Huang M. and Pastor M. (1993d) Numerical Prediction for Model No. 4a, in Veriji'cation ofNumerica1 Procedures,for the Analysis qf'Soil Liyuefuction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct., 1, 675-680. Zienkiewicz 0 . C., Huang M. and Pastor M. (1993e) Numerical Prediction for Model No. 4b. in Verrfication of Numericrrl Procedures jbr the Anu1,vsis of' Soil Liquejuction Proble~ns, (eds.), Arulanandan K. and Scott R. F., UC Davis, 17-20 Oct., 1, 731-736. Zienkiewicz 0 . C., Huang M. and Pastor M. (1993f) Numerical Prediction for Model No. 6, in Verification of Numerical Procedures,for the Ancrlysis qj'Soil Liqwfuction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct., 1, 777-782. Zienkiewicz 0 . C., Huang M. and Pastor M. (19938) Numerical Prediction for Model No. 7, in Verif~cationof' Numerical Proceclures,fbr the Anulysis of'Soil Liyuefuction Problems, (eds.) Arulanandan K. and Scott R. F., U C Davis, 17-20 Oct., 1, 873-880.
REFERENCES
251
Zienkiewicz 0 . C., Huang M. and Pastor M. (1993h) Numerical Prediction for Model No. 11, in Verification of Numerical Proceduresfor the Analysis of Soil Liquefaction Problems, (eds.) Arulanandan K . and Scott R. F., UC Davis, 17-20 Oct., 1, 997-1006.
Prediction Applications and Back Analysis
7.1 INTRODUCTION In this chapter we will introduce case histories of the two-phase liquefaction analysis (effective stress dynamic analysis) for real engineering problems. Seven examples are introduced. Soil layer liquefaction problems I: simulation of recorded data at Kobe Port Island, to illustrate the effect of multi-directional loading (1995). Soil layer liquefaction problems 11: simulation of liquefaction behaviour during Niigata earthquake (1964), to illustrate the effect of initial shear stress. Quay wall failure in Kobe City (1995). Dam failure I: Lower San Fernando Dam, to illustrate the effect of pore water migration on stability of dams (1971). Dam failure 11: investigation of liquefaction failure mechanism for an earth dam at Hokkaido (1993). Soil structure interaction problem: building tilted due to Niigata earthquake (1964). Effect of counter measures using the deep soil mixing method-a prediction.
design
The most difficult part of liquefaction analysis is the determination of the soil parameters. It is important that soil parameters are determined from the site investigation data including laboratory test data. For example, the cohesion and friction angle can be determined directly from the drained tri- axial test data. But parameters of liquefaction are not so easy for simple constitutive models. For example, dilatancy
254
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
parameters are dependent on the shape of the potential surface, yield surface, difference of failure line from phase transformation line, hardening parameters of shear behaviour, and so on. For this reason, the densification model mentioned in Chapter 4 is a useful model for design engineers. In this chapter the densification model is used in most of the examples. As the original densification is not capable of representing cyclic mobility phenomena, a modification for the original model is presented in Section 4.5 of Chapter 4. Examples (1)-(7) are all back analyses of liquefaction induced by various earthquakes. In most of these cases, only records of permanent deformations are observed and no measurements are available for acceleration or pore water pressures. Example (1) is not only typical of soil layer problems but is the only case in which accelerations were recorded at four positions in depth during the Hyogoken-Nanbu Earthquake in 1995. The earthquake heavily damaged Kobe City and over five thousand people died. The example is also a very rare case in which the input motion is available at the deep base support layer and this was recorded at the damage site. Thus the input motion at the site can be estimated from the measurements in the vicinity. This contrasts with the Niigata Kawagishi-cho problem in example (2), where the input motion was estimated from the data recorded at a location over 100 km distant. Example (3) is a quay-wall problem, typical of liquefaction. The quay wall was heavily damaged during the same earthquake as in example (1) i.e. the HyogokenNanbu Earthquake of 1995. The foundation behind the quay wall was not damaged because of the counter measure installed. Dams are another type of structure for which liquefaction is important. In this chapter, two dams are analysed and reported in examples (4) a rock-fill dam and (5) an earth-fill dam. Both show a similar failure mechanism but they are included here as they showed different failure patterns. The lower San Fernando dam example, (4), is included to illustrate the effect of pore water migration on the stability of dams, and the earth dam at Hokkaido is included to show an investigation of the liquefaction failure mechanism using a numerical method. Example (6) is included for its interest in three-dimensional analysis. The example is taken from an apartment block heavily damaged during the Niigata Earthquake of 1964. Example (7) is included as a case of dynamic earth pressure action and shows how safety can be increased. Here, dynamic earth pressure acted on the embedded structure when liquefaction occurred and this also presents important problems.
7.2 EXAMPLE 1: SIMULA TZON OF PORT ISLAND LIQUEFACTION - EFFECT OF MULTI-DIMENSIONAL LOADING 7.2.1 Zntroductory remarks Although a real earthquake inevitably consists of multi-directional components, most analyses are done using only one or two-dimensional models. The aim of this
EFFECT OF MULTI-DIMENSIONA L LOADING
255
section is to investigate the importance of multi-directional loading (MDL) in engineering practice. The case studies are made on the liquefaction events that occurred during the Hyogoken-Nanbu Earthquake of 1995. Two-phase dynamic equations used in the examples are those that have been derived in Chapters 2 and 3. It was found that the uni-directional loading along the principal axis of the earthquake orbit agreed well with results from the horizontally multi-directional loading for the maximum response acceleration, except for the process of pore water pressure build-up and other details of response. It was also found that the effect of vertical loading is not significant; however, the effect of initial (static) shear stress (ISS) was, and this will be discussed in Section 7.3. MDL and initial stress obviously play important roles in the geotechnical numerical analysis if material non-linearity occurs. In reality an earthquake, inevitably, has multi-directional movement and initial static stresses with a shear component will always be found in such soil structures as dams, dykes and in natural ground. Most studies of liquefaction analysis in the past have considered only one horizontal component of the earthquake, while very few studies have been made on the behaviour of the ground subjected to multi-directional earthquake loading. and even less attention has been paid to the initial stress condition. Both are important for precise numerical prediction and studies of these are reported in the individual sections. Firstly MDL will be discussed and followed by ISS in the next section. The effect of MDL has been studied experimentally by several researchers. Settlement of a thin dry sand layer was studied by Pyke (1973) for the case of a sand layer shaken on the shaking table in one or two horizontal directions. Circular and random motions were applied for the two-component tests. Their conclusion is that the settlements caused by combined horizontal motion are almost equivalent to the sum of settlements caused by components acting separately. While vertical accelerations of less than lg cause no settlement if acting alone, vertical accelerations superimposed on the horizontal accelerations could cause a marked increase in the settlements. The effect of MDL on the liquefaction strength (stress ratio to induce a certain shear strain, e.g. 3'%, under a given number of cyclic loading; e.g. 15 or 20) has been studied at the end of the 1970's (Seed et al., 1978 and Ishihara and Yamazaki, 1980). Seed et ul. attributed the settlement on the shaking table test to the liquefaction strength, and Ishihara and Yamazaki obtained directly the liquefaction strength with a two-directional simple shear test apparatus under undrained conditions. They found that the cyclic stress ratio dropped approximately 25%35'%, causing 3% strain in a direction depending upon the pattern of the two components for loading. This liquefaction strength is also influenced by the loading irregularity (Ishihara and Nagase, 1988). The volumetric strain due to consolidation following liquefaction also differed due to MDL (Nagase and Ishihara. 1988). From the numerical analysis point of view, there has to be an influence of MDL on the liquefaction induced by earthquakes since negative dilatancy, which causes liquefaction, depends on the accumulated shear strain. The accumulated shear strain is the sum, though not simply the arithmetic sum, of the six components of the strain (three deviatoric strains and three shear strains). Ghaboussi and Dikmen (1981) first
256
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
studied the effect of MDL on a soil layer problem with their proposed numerical method using the fully coupled Biot's equation with u-U formulation (Zienkiewicz and Shiomi, 1984) for the dynamic equation and a non-linear material model. The material model was based on the hyperbolic stress-strain relation for shear (Kondner and Zelasko, 1963), and the effective stress-path approach for dilatancy (Ishihara et al.. 1975). In this model, the decrease of effective stress is a function of both components of the horizontal shear strains. A hypothetical, horizontally layered ground, subjected to the El Centro Earthquake was solved as a case study. Twodimensional analysis showed a marked difference in the build-up behaviour of pore water pressure and some differences in the surface velocity spectra. Even when the amplitude of the acceleration is increased to the resultant peak accelerations from the two directions, the results were different from the results of the two-dimensional analysis. As an alternative, Fukutake et al. (1995) suggested the use of input motions that are 1.3 times larger than the earthquake and of a stronger direction between north-south (NS) or east-west (EW), although this conclusion is considered premature. Therefore there is a need to survey the three-dimensional behaviour of the level ground subjected to MDL, together with other variations such as soil properties and input motions. This section investigates the effects of MDL on a real site. The site is a typical soil 'column' on the Port Island in Kobe City, and earthquake motions for the studies were recorded during the Hyogoken-Nanbu earthquake on January 17th in 1995, where the liquefaction phenomenon has been observed throughout the island and settlement was about 20 cm (estimated from the relative gap between buildings supported by piles and the ground surface after the earthquake). At first, back analysis of the observed data is explained and then the parameter study for M D L is reported.
7.2.2 Multi-directional loading observed and its numerical modellingsimulation of liquefaction phenomena observed at Povt Island During the Hyogoken-Nanbu Earthquake, liquefaction took place along most of the seaside of Kobe City. Sand boiling and flushing water due to liquefaction occurred in many places including Port Island where an array of seismometers was set at four depths (Kansai-Kyogikai 1995). Figure 7.1 shows the orbit of the records at Port Island for the Hyogoken-Nanbu earthquake in 1995. Two-to-four very large amplitudes can be seen from the figures. The maximum acceleration at the surface (GL O.Om) was 314gal (3.14msp2)for the NS and 288gal for the EW direction. They are about half of the value record at GL-83.8m. The diagram at the bottom left of Figure 7.1 shows the orbit at GL-83.8m during zero to five seconds. Several large amplitudes were clearly seen. The direction, when the maximum amplitude occurred, was about 20 degrees from north to west. This direction is considered as the principal axis of the earthquake components. The time history of the direction was the North-South direction (Figure 7.2). The diagram at the bottom right of Figure 7.1 shows the orbit of the NS (north-south)-UD (up down) motion. At GL-83.8m. the UD component was not considered significant.
257
EFFECT OF MULTI-DIMENSIONAL LOADING -obscrved
G.L. 0.0m
-observed
G.L. -83.8m
.--
-800
400
0
400 EW Ace. (gal)
-observed
800
-800
G.L. -83.81~1
-u
800
G.L. -83.8m
800
--m "
400
400
-
M
m
9 in
0 400 EW Ace. (gal)
-observed
800
--w
-400
0
Y n
0 -
3
Z
400
1100
-800 -800
Figure 7.1
400
0 400 EW Acc. (gal)
800
-
-800 -800
400
0 400 NS Ace. (gal)
800
Orbit of an observed earthquake record (after Kansai-Kyogikai 1995)
Conditions and modelling The Effect of MDL was studied by simulating the above records using a column of finite elements. Case studies performed are shown in Table 7.1. The column of soil used in the numerical modelling is shown in Figure 7.3. The record at GL-83.8m was introduced as the input motion. Four cases are studied. Case 1 simulates the observed record by incorporating all three directions of the earthquake motion. Case 2 was studied to investigate the influence of vertical input motion on the liquefaction phenomena. Cases 3-5 are for the comparison between two and three-dimensional modelling. Table 7.2 shows the material properties of the ground layer at Port Island. The other properties are calculated using the data shown in the table and the soil properties at a similar site. For example, friction angle and liquefaction strength were calculated through the N value of standard penetration tests (Tokimatsu and Yoshimi, 1983).
258
PREDICTION APPLICATIONS A N D BACK ANALYSIS
--g -
Time (Sec) 800
s
.4-
-2
0
U
'
. - - - - - - - - - - - - - - - - - - - - - - - - - - -
J
-800 0
5
10
15 Time (Sec)
20
25
30
Principal direction 0
.---.-----------------------
-800
0
i
5
10
15 Time (Sec)
20
25
30
Figure 7.2 Acceleration of NS, EW and principal direction
Table 7.1
Cases studied
Case
Analysis type
Input motion
1 2 3 4 5
3 D analysis 3 D analysis 2 D analysis 2 D analysis 2 D analysis
NS, EW and U D NS and EW only NS and UD only EW and U D only Principal axis and U D only
Results of simulation Figure 7.4 shows the time history of response accelerations in the NS direction overlaying the observed accelerations. These agreed well initially, however, some difference in the response is found at the surface after 5 seconds. The observed acceleration shows a very long period wave, while the calculated acceleration has a component of higher frequencies. Figure 7.5 shows the time history of the excess pore pressure in Cases 1 (NS+EW+UD) and 2 (NS+EW). The layer between GL -12.6m and -14.0m was fully liquefied at about 6 seconds and kept liquefied. But the built-up speed
259
EFFECT O F MULTI-DIMENSIONAL LOADING Table 7.2 Soil layer and material properties Layer no. Depth (m) 0.0 to -5.0 -5.0 to -12.6 - 12.6 to - 16.8 -16.8 to -19.0 -19.0 to -27.0 -27.0 to -32.8 -32.8 to -50.0 -50.0 to -61.0 -61.0 to -79.0 -79.0 and deeper
Soil type
V, (mls)
V, (mls)
Average N-value
Density (kNim3)
Man-made fill Man-made fill Man-made fill Man-made fill Silty c!ay Layers of gravelly sand and silt Silty clay Diluvium sand
Figure 7.3 Analytical model
of the excess pore pressure above the layer was slowed down after 6 seconds. This is because the input motion transferred from the bottom to the surface was significantly reduced by the sudden loss of the strength at the layer between GL - 12.6m and - 14.0m. On the other hand, at a deeper level of GL -15.4m, pore pressure continued to build up and almost reached full liquefaction. Therefore, contrary to popular belief, a tendency for liquefaction at a deeper layer was found in this large impact-type earthquake. In order to investigate the effect of vertical input motion, Case (2) (NS+EW only) was then analysed. The excess pore pressure response using the updown component (UD) has a very high frequency shown by the thin solid line in Figure 7.5. Nevertheless, excess pore pressure response without UD motion (bold
260
PREDICTION APPLICATIONS AND BACK ANALYSIS
600
G.L. 0 . 0 m
.
600
0
- - - - simulation
-observed
5
10 Time(s)
20
15
G.L. 1 6 . 8 m
- - - - simulation
-observed ...
0
5
10 Time(s)
15
20 G.L. -32.8m
- - - - simulation
0
5
10 Time(s)
15
20 G.L. 8 3 . 8 m
....simulation o b s e r v e d
0
5
Figure 7.4
10 Time(s)
15
20
Response acceleration of the NS direction
solid line) passes through nearly the mean value of the excess pore pressure obtained with UD motion. This implied that vertical input motion induced only compressible wave in the water but did not affect the response of the soil skeleton. Therefore the result of Case 1 is found to be very similar to Case 2 when the high frequency is filtered out. Next the results of unidirectional loading (Figure 7.6 and Figure 7.7) and multidirectional loading (Figure 7.5) were compared. The response obtained with unidirectional EW direction loading showed a different response while the response obtained with unidirectional NS direction loading was close to the one obtained with multidirectional loading. This is because the NS direction was very close to the principal direction of the earthquake motion. The build-up of pore water pressure could have been dominated by the NS component of the earthquake. In order to find out the effect of the direction of input motion, response with unidirectional loading
261
EFFECT OF M U L T t - D I M E N S I O N A L LOADING
0
5
10
I5 Time (See)
20
25
30
GL-12.6111-14.0m
Time (See)
G.L.-15.4m -1 6.8m
Time (Sec)
Figure 7.5
Excess pore pressure ratio (NS + EW
+ UD)
in the principal direction was then calculated and shown in Figure 7.8. In this case. the results were very similar to those obtained using the NS component alone.
Effects of Multi-directional loading Maximum accelerations for uni-and multi-directional loading are plotted in Figure 7.9. Results of the NS direction for Case 1 (NS EW UD) and Case 3 (NS + UD) showed similar behaviour. Response of Case 5 with input motion in the principal axis was similar to the response of Case 1 (NS + EW) at the same (principal) direction. Maximum vertical acceleration was hardly influenced by the horizontal excitation. This tendency was found the same as the results of Ghaboussi and Dikmen (1981).
+
+
262
PREDICTION APPLlCATIONS AND BACK A N A L Y S I S
0
5
I0
15 Time (Sec)
20
25 G.L.-9.0m I I .Om
30
0
5
10
15 Time (Sec)
20 25 G.L.-12.6m 1 4 . 0 m
30
0
5
10
15 Time (Sec)
20
30
25
Figure 7.6 Excess pore pressure ratio (NS)
7.3 SIMULA TION OF LIQUEFACTION BEHA VIOUR DURING NZIGATA EARTHQUAKE T O ILLUSTRATE THE EFFECT OF INITIAL (SHEAR) STRESS In addition to MDL, ISS also induces significant effects to the results of the liquefaction analysis. ISS due to self-weight is usually assumed through KO (earth pressure coefficient, at rest) or static analysis due to self-weight, but the real value could not be obtained by the current measurement. No reliable means for measuring initial stress in situ is sufficiently well developed at the current time. Only a few studies have been done for in situ KO (Hatanaka and Uchida, 1995). Consequently there are very few reports in numerical analyses to investigate the effect of ISS. However, they could be modelled implicitly by adjusting the shear strength of the material if only shear resistance is important, and also in static problems. But it is not
T H E EFFECT OF INITIAL ( S H E A R ) S T R E S S
263
0
5
10
15 Time (Sec)
20
25
30
0
5
10
15 Time (Sec)
20
25
30
-
.0 44
-
.
1.0
-- -
---------
G.L.-12.6m 1 4 . 0 m
-
-
-:
-
- -
---------
44
0
0
5
5
10
10
15 Time (Sec)
20
15 Time (Sec)
20
25
30
G.L.-15.4m 1 6 . 8 m
25
30
Figure 7.7 Excess pore pressure ratio (EW)
straightforward, since not only shear strength but also the dilatancy characteristic is affected by ISS. A layered soil ground was analysed with two different initial conditions, i.e. 'with ISS' and 'without ISS' conditions. The earth pressure at rest KOis 1.O in the case of the condition 'without ISS' and 0.5 for the case 'with ISS'. The mean stresses were kept the same at the same depth since the liquefaction strength has long been assumed to be the same for the same mean stress in the soil column problem (Yoshimi, 1991). This means that the vertical stress changes to maintain the mean stress. This assumption was made to simulate the ordinary calculation procedure for the liquefaction safety factor. The analysis of these examples was very significant. The same results were obtained for most of the constitutive models in which criteria depend on the maximum shear stress in three-dimensional space. However, there are some analysis codes which do not detect any significance for this problem. This difference in response is obviously caused by the different constitutive equation used in different codes.
264
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
0 1.0
5
10
15 Timc (Sec)
20
25 G.L.-9.0111-11 .Om
r
30
1
0
5
10
15 Time (Sec)
20 25 G.L.-12.6m 1 4 . 0 m
30
0
5
10
15 Time (Sec)
20 25 G.L.-15.4m -16.8111
30
0
5
10
15 Time (Sec)
20
30
25
Figure 7.8 Excess pore pressure ratio (principal direction)
7.3.1 Influence of initial shear stress Initial stress is calculated or n~odelleddifferently depending upon the analysis codes and/or analysts (Shiomi, rr (11. 1991) and (Shiomi and Shigeno, 1993). However, it is very difficult to evaluate the initial stress since no one knows how the ground was made and no measurement is available even for the ratio of the lateral earth pressure to the vertical stress. Therefore, several calculations should be made to find the appropriate initial stress which is chosen according to the engineer's experience. However different choice could give completely different the results. The difference in the results is mainly caused by the initial shear stress (ISS) component of the initial stress. We are going to comment on this using an example of a one-dimensional layer problem and show how big the difference is.
265
T H E EFFECT OF INITIAL ( S H E A R ) S T R E S S ( N S + EW + UD)
-NS
.--o-.NS i , m.j
EW (NS + EW + UD)
---A-.EW.i, d.j
A
N S (observed)
0
-4-
200
400
EW (observed)
600
800
0
+UD (NS + EW + UD) ( p r i n c i p a l axis + UD) UD (observed)
Max. Acc. (gal)
200 400 600 Max. Acc. (gal)
(a)
(b)
ROO
Figure 7.9 Maximum response acceleration (a) horizontal component; (b) vertical
component
Significance of ISS component to the responses Response acceleration The influence of ISS is significant. Figure 7.10 shows the maximum response acceleration and displacement. The response near the surface of 'with ISS', i.e. KO = 0.5, is larger for acceleration and smaller for displacement than those of 'without ISS'. This means that the stiffness of the soil material for the case 'with ISS' is reduced more than the case 'without ISS'. The existence of ISS places the stress closer to the failure line so the material with ISS becomes weaker and this is the main reason for the smaller acceleration and larger displacement in the case with ISS. The difference is about 301%for acceleration and 20% for displacement. Excess pore water pressure Figure 7.11 shows the time history of the liquefaction ratio. The pore water pressure of 'with ISS' built up quickly and higher. Both cases reach full liquefaction, but the excess pore water pressure is about 10%less in the case of 'without ISS'. This tendency was almost the same as the laboratory element tests
266
PREDICTION APPLICATIONS AND BACK ANALYSIS
[I;.,.:., --,.
0
- . - .--
1 2 Max. disp. (cm)
0
40 80 120 Max. acc. (gal)
....
-KO = 0.5
3
K O = 1.0
Figure 7.10 Profile of maximum response acceleration and displacement
0
2
4
6
8
Time (s)
Figure 7.11 Time history of pore pressure ratio
10
OUA Y W A L L FAILURE AND A COUNTERMEASURE
267
by Vaid and Finn (1979) and Hyodo et ul. (1988) except for the final pore water pressure. The liquefaction ratio is 75% for the 'with ISS' and 40%)for the 'without ISS' at 3.5 seconds. Figure 7.1 1 may give a clue as to why the liquefaction strength curve gives the same value as for the same mean stress. The liquefaction strength curves were determined by the number of cycles taken for the samples to reach final liquefaction. The criteria for the final liquefaction is either 5% shear strain in deformation or pore pressure reaching 95% of the initial vertical stress. The pore water pressure ratios reached 1.0 at almost the same time, at about 8 seconds.
Theoretical considerations ISS can therefore be classified into two types. Type I, ISS is seen in the case where an external force is applied in a perpendicular direction to ISS. In this case, the incremental shear stress has mainly the effect of rotating the principal stress with the increment in the equivalent stress being small. For example, the soil beneath a structure has almost no shear stress in the horizontal direction; however, deviatoric shear stress (vertical stress minus horizontal stress) is relatively large. The maximum shear stress acts for the direction of approximately 45 degrees. An external force due to an earthquake produces a large horizontal shear stress and the principal stress direction is rotated. Type 11, ISS is the case where an external force is applied to the direction parallel to the maximum shear direction of ISS. In this case the incremental equivalent shear stress is equal to the external incremental shear stress. For example. a slope such as that of a dam has ISS close to the horizontal direction. These two types of ISS might work differently. There is no substantial evidence, however, that ISS should affect liquefaction differently. A soil-column-type dynamic effective analysis can obviously be classified into Type I. It was found that the existence of ISS creates a slower build-up of pore water pressure, as indicated by the experiments. In the example problem, the upper soil layers with ISS were weakened more than in the case without ISS. It should be noticed that ground layer analyses frequently neglect ISS. Constitutive models based on a typical elasto-plastic theory, which uses the flow rule, produce different results for the conditions 'with ISS' and 'without ISS.' The constitutive models developed as an extension of a one-dimensional shear soil column model often ignored ISS (deviatoric stress due to the difference of vertical stress and horizontal stress) since most of the models use the hyperbolic stress-strain model for the shear behaviour and their stresses always start at zero. That means that ISS is not involved in the formulation.
7.4 QUA Y WALL FAILURE AND A COUNTERMEASURE Although quay walls surrounding a Hotel at Kobe were greatly damaged due to liquefaction during the Hyogoken-Nanbu Earthquake of 1995 (Figure 7.12), the foundations of the building had no damage. This proved the effect of the
PREDICTION APPLXCATIONS A N D B A C K A N A L Y S I S
268
Figure 7.12
Damaged quay wall along the building (after Suzuki ct rrl., (1995))
countermeasure for the foundations. The countermeasure was lattice walls, which are made of the stiffened ground by mixing cement to reduce the shear movement of the ground (Suzuki et al, 1995). The lattice-shaped stiffened ground walls (LSSGW) were built surrounding the piles. At the design stage, prediction analysis for the structural design stage was conducted to confirm the effects of the LSSGW.
7.4.1
Conditions and modelling
Configuration Two-dimensional analysis was conducted. The numerical model of the foundation and building is shown in Figure 7.13. The building was built on a pier shaped ground,
269
OUA Y WALL FAILURE AND A COUNTERMEASURE
34m
a -10 3m -13 4m -
-15 8m -
-26 1m -32 7m (a) Sectmn
plle
.1
I
.I
.I.
J,
J,
J,
L
m 8 5 ' 8 ' ~ 2 ~ 7 6 ' 8 2 ' fT8,5'6
( b ) Pldne
Figure 7.13 Numerical model of foundation and building
which projected into the sea. The foundation was surrounded by quay walls. Deformation due to the earthquake in the shorter section of the building was anticipated larger than that in the longer section so that the numerical study was made in the shorter section. The foundation of the building was made using a repeated pattern of piles and LSSGW shown in Figure 7.13. LSSGW and ground with piles were modelled into separate groups. These two groups were connected at a corner of the lattice. The building was considered as elastic and Mohr-Coulomb criteria was used for the LSSGW.
Soil layers and properties The zone where liquefaction was anticipated was from G L (ground level) 0.0 m to - 13.4m. The layers from G L - 13.4m to -32.7 m were soft blue clay or silty soil. The piles were supported at the layer G L -32.7 m. The constitutive model used here was the M-C (Mohr-Coulomb) Densification model. Its soil properties (shown in
270
PREDICTION APPLICATIONS AND BACK ANALYSIS
Table 7.3 Soil properties Layer No.
Porosity
Depth -(m)
Friction angle (deg)
Shear modulus (MW 34.00 34.00 59.00 79.00 38.00 141.00 92.00
Table 7.4 Parameters of liquefaction ( R l s and densification model) Layer No.
RIS
r
A
B
2 3 4 5
0.1534 0.1495 0.1574 0.1997
5.0 5.0 5.0 5.0
0.02100 0.02100 0.02000 0.01200
1 .OO 1.OO 1.OO 1 .OO
Table 7.3) were obtained by a site investigation conducted during the design stage. The liquefaction strength R I 5 is the only data available for the liquefaction parameter. This is largely because safety against liquefaction is normally judged by the liquefaction strength. R l s or RZ0 is the shear stress ratio against the confining pressure when liquefaction takes place after 15 or 20 cycles of the undrain triaxial test. Parameters r, A and B in Table 7.4 are for the densification model and are determined to satisfy R 1 5 .Figure 7.14 shows an example of a fitted curve for the liquefaction strength.
log N
Figure 7.14 Liquefaction strength of laboratory test and calculation
O U A Y W A L L FAILURE AND A COUNTERMEASURE
271
Time (s)
Figure 7.15
Predicted earthquake for Meriken Oriental Hotel
Input motion Input motions for design were E l Centro of 1940, EW component, Taft of 1952, NS component and some other records of large earthquakes which occurred in Japan. An artificial earthquake generated from a design spectrum is also often used. Here an earthquake predicted at the site from an earthquake recorded at the Kobe Ocean Weather Station during the Hyogoken-Nanbu Earthquake of 1995, was used.
7.4.2 Results and remarks Figure 7.16 shows the permanent deformation as the quay walls of both sides tilted and the back fill was extensively damaged. However, there was no damage in the foundation zone. That was the intention of reinforcement of the soil foundation. Large pore water pressure built up behind the quay wall and caused the failure of the quay wall during the Hyogoken-Nanbu Earthquake (Figures 7.16 and 7.17). Relatively large pore water pressure built up around the toe of the LSSGW but did not cause any damage. The latter pore water pressure build-up can be easily anticipated because stiffness of LSSGW became much greater than the support layer below the LSSGW so that a large shear strain is induced around there. It is necessary for discontinuity to be avoided between the layers of LSSGW and the under-soil layer. Figure 7.18 shows the profile of the pore water pressure at the end of the earthquake. The figure for Line 1 shows the results for the centre of the model. The figure for Line 4 shows the result for the quay wall. The pore water pressure within the LSSGW was less than 30'% of the vertical component of the initial effective stress shown by the dotted line. The pore water pressure behind the quay wall reached the full liquefaction state. This result matches the investigation after the earthquake.
272
PREDICTION APPLICATIONS A N D BACK ANALYSIS
Figure 7.16 Permanent deformation of quay wall
Figure 7.17 Contour of pore water pressure
Figure 7.19 shows the maximum shear stresses which occurred in the soil layers and in the LSSGW itself. These stresses indicated that most of the shear forces acting in the ground were shared by the LSSGW. Therefore, soil strains of the inner zone of the LSSGW were 0.19% at most of those which took place at the bottom of the LSSGW zone. These small strains were due to the stiff LSSGW and could produce no liquefaction. The strains of the outside of the LSSGW, i.e. back-fill of the quay
273
Q U A Y W A L L FAILURE AND A COUNTERMEASURE 0 -2 -4 E
-6
5
g
-8
-I0 -12 -1 4
JA -
'
? W Pore Pressure
( x I 0 kPa)
5
-5
0
5 10 15 20 25 Pore Pressure
( x 10 kPa)
0
5 10 15 20 Pore Pressure
( x I0 kPa)
Pore Pressure ( x I OkPa)
Figure 7.18 Pore water pressure and initial effective stress (see Figure 7.13 for lines 1 to 4)
Figure 7.19 Shear stress of lattice shaped stiffened ground wall and ground
wall, were very large (more than 10%) since the quay would be able to move freely towards the sea, so that the liquefaction took place and then more, larger, strains were induced.
274
7.5
PREDICTION APPLICATIONS AND BACK ANALYSIS
LOWER SAN FERNANDO DAM FAILURE
The failure of the lower San Fernando earth dam in 1971 with nearly catastrophic consequences, is typical of what can occur in a poorly consolidated soil structure affected by shaking, resulting from an earthquake. Zienkiewicz and Xie (1991) have reported some results of numerical simulation of the failure. Here, the details are presented. In particular, the effect of cohesion resulting from negative pore pressure and the influence of such parameters as permeability and relative density on the dynamic response of the dam, are illustrated. Although full comparative measurements are not available, the reconstruction of the event by Seed et ul. (1975) and Seed (1979) is remarkable in attempting to explain why the failure occurred apparently some 60-90 seconds after the start of the earthquake, which was recorded to last some 14 seconds. The actual collapsed dam and a 'reconstructed' cross-section are shown in Figure 7.20 following Seed (1979). The hypothesis made here was that the important pressure build-up occurring as a result of cyclic loading which manifested itself first in the central portions of the dam, 'migrates' in the post earthquake period to regions closer to the 'heel' of the dam where it triggers the failure.
Cross-section through embankment after earthquake
Reconstructed cross-section Figure 7.20 Failure and reconstruction of original conditions of the Lower San Fernando dam (Seed. 1979)
275
LOWER S A N FERNANDO D A M FAILURE
-
Rigid boundary
Base excitation )le boundary
Impermeable boundary
Figure 7.21 Idealization of San Fernando dam for analysis: (a) material zones (see Table 7.5): (b) displacement discretization and boundary conditions; and (c) pore pressure discretization and boundary conditions
We show in Figure 7.21 the material idealization, finite element meshes and boundary conditions used in the present computations. First an initial, elastic static analysis is carried out by the full program considering a semi-saturated condition and assuming the gravity and external water pressure to be applied without dynamic effects. Figure 7.22 shows such an initial steady-state solution for the saturation and the pore pressure distribution, indicating clearly the 'phreatic' line and the suction pressures developing above.
Figure 7.22 Initial steady-state solution: (a) Pressure (kPa); and (b) Saturation contours
276 Table 7.5
PREDICTION APPLICATIONS AND BACK A N A L Y S I S Material properties used in the Lower San Fernando dam analysis
Material zone p,(kg/m3) pf ( k g / m 3 ) K,(Pa)
K f (Pa)
1 2a 2b 3
2 2 2 2
2090.0 2020.0 2020.0 2020.0
980.0, 980.0 980.0 980.0
10'' 10" lo1' lo1'
v
x 109.2857 x 10' 0.2857 x lo9 0.2857 x 10' 0.2287
Porosity k (mis) 0.375 0.375 0.375 0.375
lop3 lo-' lo-' lo-'
Starting with the above computed effective stress and pressure distribution, a full non-linear dynamic computation is carried out for the period of the earthquake and continued for a further time of 200 seconds. The material properties assumed to describe the various zones of the dam using the constitutive model described in Chapter 4 and also found in Pastor and Zienkiewicz (1986) and Pastor er al. (1988, 1990) are summarized in Table 7.5. Figure 7.23 shows the displaced form of the dam at various times. The displacements at some characteristic points and the development and decay of excess pore pressures are shown in Figure 7.24 and Figure 7.25. It is noted that deformations are increasing for a considerable period after the end of the earthquake. This undoubtedly is aided by the redistribution of pore pressures. Near the upstream surface the pore pressures continue to rise well after the passage of the earthquake. This indeed was conjectured by Seed (1979). It is also noted that the pattern of deformation is very similar to that which occurred in the actual case showing large movements near the upstream base and indicating the motion along the failure plane. The suction pressures which developed above the phreatic line give a substantial cohesion there. Indeed, preliminary computation indicates that, without such cohesion, an almost immediate local failure develops in the dry material upon shaking. The amount of cohesion depends on the S,- pw (or the S,-h,) curve in the following equation. Since p, is assumed as zero, it is seen that the cohesion is of the value Swp, in the otherwise cohensionless granular soil since the effective stress is defined as
When we reduce the parameter b of Van Genuchten's formula by a factor of 100, the pore pressure distribution of static solution will be almost the same as in Figure 7.22(b) but the saturation in the semi-saturated zones will be close to 1.
277
LOWER S A N FERNANDO DAM FAILURE
90s
(iii)
Figure 7.23 Deformed shapes of the dam at various times: ( i ) 15 s (end of earthquake): (ii) 30 s; (iii) 90 s: (iv) 200 s.
Table 7.6 Coefficients of saturation function
For sand' For San Fernando I
h
,?(cm-')
1
u(cm-I)
h
c1
0.0689 0.0842
0.01740 0.00700
2.5 2.0
0.0667 0.0500
5.0 4.0
1 .OO 0.90
After Van Genuchten rt rrl. (1977)
The higher value of S, results in stronger cohesion in the upper part of the dam. The dotted lines in Figure 7.24a are the results of computation occurred because of the now increased cohesion in the upper regions of the dam. If the permeability of the dam material is sufficiently high, it may be impossible for an earthquake to cause any build-up of pore pressures in the embankment, since the
278
PREDICTION APPLICATIONS AND BACK ANALYSIS
Figure 7.24 Horizontal (left) and vertical (right) displacements: (a) at the crest (dashed line represents the result of computation with increased 'cohesion'); (b) at point E; (c) at point H; (d) at point I (see Figure 7.21a).
L O W E R S A N FERNANDO D A M FAILURE
279
Figure 7.25 Excess pore pressure at points A to H (see Figure 7.21a)
pore pressure can dissipate by drainage as rapidly as the earthquake can generate them by shaking. Figure 7.26 shows the results which indicate a rapid dissipation of pore pressures and much reduced permanent deformations.
280
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
Figure 7.26 Results of analysis with increased permeabilities: (a) deformed shape of the dam after 15 s; (b) deformed shape of the dam at 200 s; (c) horizontal displacement on the crest; (d) ~~ertical displacement on the crest; (e) excess pore pressure at point A; excess pore pressure at point D (see Figure 7.21a)
In an additional analysis the relative density of the dam material is assumed lower, which implies in the present constitutive model that the ratio Mf/M, is considered equivalent to the relative density Dr [Pastor et a1 19851. With M, values in Table 7.5 fixed. Mf values are now reduced to 1.24,0.453,0.604 and 0.906 for material zones 1, Za, 2b and 3, respectively. In this case significantly larger displacements are recorded at the early stages of the earthquake shaking as shown in Figure 7.27.
28 1
MECHANISM OF LIQUEFACTION FAILURE 5s
(ii)
10s
(iii)
Figure 7.27 Results of analysis with softer materials, showing deformed shapes at: (i) 5s; (ii) 10s; (iii) 15s; (iv) 200s.
7.6 MECHANISM OF LIQUEFACTION FAILURE ON AN EARTH DAM ( T H E N DAM) 7.6.1 Objective of the analysis In this section, the failure mechanism of an old irrigation dam was studied. The dam collapsed completely during the Nihonkai Nanseibu Earthquake of 1993, due to liquefaction. We shall, however, refer to the dam as the N dam for simplicity. Sand boils due to its liquefaction were found at the foot of the dyke. It was decided that the dam had to be decommissioned and a section of the dam was removed soon after the damage was found. The dam section could then be examined easily. Although the top of the dam had large cracks, no clear slip-line was found in the section of the dam. Any trial to find the failure mechanism by conventional engineering methods such as the sliding analysis, were not successful. The liquefaction analysis (Shiomi et al.. 1996) was then conducted using the dynamic effective stress approach using MuDIAN (Multiphase Dynamic Interaction Analysis: Shiomi et al., 1993). The analysis explained reasonably well the observation, which was made after the failure.
282
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
The modelling of the constitutive relationship and the initial stress condition plays an important role in the analysis and is reported here in detail. Also in this study, the failure pattern of the earth dam is evaluated. In the dam analysis, the treatment of the initial shear stress condition is a difficult issue. At first a static analysis was conducted for the self-weight of the dam. Then liquefaction analysis was conducted with the densification model, which was proposed by Zienkiewicz et a1 (1978). The liquefaction parameters were estimated from a liquefaction strength curve, which was only one test data set, for its liquefaction properties. The dam quickly liquefied and failed using those parameters. To find the reason for this we surveyed the influence of mesh, soil parameters and initial stress conditions. Finally, a reasonable simulation result was obtained by controlling the influence of the initial shear stress condition. There we found that we could not alter the initial shear stress itself with respect to the shear failure criterion, because of the presence of the initial shear stress, due to the sloping of the dam side. This can neither be neglected nor reduced. However, its influence on the dilatancy then came into question. Therefore, this relation was further investigated.
7.6.2 Input motion There was no earthquake acceleration record at the N Dam, so the record at a dam nearby was used for input motion. The dam was about 130km from the epicentre. Figure 7.28 and Figure 7.29 show the response spectra and time history of the recorded earthquake respectively. The N Dam is located about 70km from the epicentre so the maximum acceleration could have been as large as 200-250ga1, according to the attenuation curve of distance from the epicentre. As the actual maximum acceleration could not accurately be determined, three different values of the maximum acceleration for the input motion to the base of the analysis model were attempted and they were 134ga1, 200gal and 250gal.
0.1
1
Time (Sec)
Figure 7.28 Response spectra of input motion
283
MECHANISM OF LIQUEFACTION FAILURE
MAX. 134.8 (23.211gal)
-1501
0
'
'
'
'
'
'
'
13 Time (Sec)
'
'
'
I
26
Figure 7.29 Time history of input motion
7.6.3 Conditions and modelling The N-dam is an old earth dam built in the 1920's. The configuration and structure of the dam are shown in Figure 7.30. The height of the dam from the bottom of the reservoir was about 10.0m. The core of the dam (dark area in Figure 7.30(a)) was in the centre of the dam and 'shielded' the water flow. Therefore the two-phase region (soil skeleton + water) was assumed only at the up-streamside of the core. The first natural frequency was 2.3 Hz for the horizontal mode and the second was 3.5Hz for the vertical mode.
Soil properties The soil parameters used in the analysis are shown in Table 7.7. The various values of the shear moduli G were evaluated through the N values of the standard penetration test. Cohesion c and friction angle 4 were obtained from the drained tests. Densities y
Random IV (silty sand)
'
Sedimentation (a)
Effective stress analysis
Figure 7.30 Configuration of the dam: (a) distribution of soil properties; (b) FEM mesh and
boundary condition
284
PREDICTION APPLICATIONS AND BACK ANALYSIS
were obtained by the physical tests. Permeability was determined through 20% particle size D20 by Creager's approach (Creager et al., (1944)).
Parameters for liquefaction Material parameters for liquefaction are determined from the liquefaction strength curve as in the previous example. The liquefaction strength used here was obtained from the N-value and laboratory test. Figure 7.31 shows the liquefaction strength curve used for this analysis. The solid dot shows the data obtained by the undrained tri-axial test. The other marks shows the liquefaction strength calculated from the Nvalue of the standard penetration test. The solid line shows the liquefaction strength calculated by the assumed soil parameter for the densification. The parameters of the densification models are listed in Table 7.7. Table 7.7 Soil properties
Name
N-Value y(kN/m3) G (MPa) Porosity Permeability c(kPa) (mJs)
Core I Core I1 Random I Random I1 Random 111 Random IV Random V Sand Sand & gravel Sand & gravel Sandy rock I Sandy rock I1 Sandy rock 111
Number of cycles N
Figure 7.31
Liquefaction strength
4 (deg.)
285
M E C H A N I S M OF LIQUEFACTION FAILURE
Table 7.8 Model's parameter for densification model Soil type
N-value
F C ('%I)
Djtr (mm) Average u,
RIzo
A
B
r.
Random I1 Random IV Gravel
3.4 4.7 20.6
32.2 29.3 0
0.28 0.17 2
0.1692 0.2239 0.2998
0.01 1 0.007 0.004
1
8 6 3
-50.65 -86.18 -59.78
1 1
Initial stress Initial stress was calculated for self-weight assuming that Poisson's ratio was 113. Figure 7.32 shows the initial maximum shear stress. The initial mean effective stress and shear stress are important, since both influence the build-up of excess pore water pressure. With the densification model, the initial shear stress causes a rapid build-up of pore water pressure, which does not agree with ordinary engineering experience. Therefore, the initial shear stress was ignored in the densification model modifying (4.162) to the following equation in this analysis.
where Oo is the initial shear stress ratio. The dilatancy of the sand may not depend on the initial shear stress if the material pass a long period after the construction of the dam.
7.6.4 Results of calculations Among the three case studies, the case where the input motion was 200gal showed good agreement to the failure pattern observed. So the actual magnitude that hit the
Figure 7.32
Contour of maximum shear stress
286
PREDICTION APPLICATIONS AND BACK ANALYSIS
Figure 7.33 Crack observed after the earthquake
dam could be said to be about 200gal. Figure 7.33 shows the damage to the dam, where three cracks were observed. The largest depth of the crack C1 was 1.Om-1.2m. The cracks C2 and C3 are shown on the surface of the upper stream. Figure 7.34 shows the progress of the deformation of the dam in the case of 200gal. At about 10 seconds, the surface of the upper stream showed movement toward the toe of the dam. This deformation might have caused the observed cracks C2 and C3 DISP. SCALE :1 - 1
1.9117
At 10 seconds
At 20 seconds
At 26 seconds
Figure 7.34 Transition of the deformation of the dam
LIQUEFACTION D A M A G E IN T H E NIIGA T A EARTHQUAKE OF 1964
287
Figure 7.35 Contour of excess pore water pressure at 26 seconds
at the surface of the upper stream as shown in Figure 7.33. During the first 20 seconds the deformation of the upper stream surface became enlarged continuously. Then the top part of the upper stream began to settle 0.84m due to the liquefaction near the dam core. This might have caused crack C 1. Excess pore water pressure was very high at 26 seconds as shown in Figure 7.35. This caused the overall land slide and the largest crack, C1.
7.6.5 Remavks The liquefaction analysis made clear the failure mechanism of the dam. Since sand region has liquefied and dyke sank under self-weight no significant slip line developed. This mechanism was explained by a rather simple densification model with a simple modification.
7.7 LIQUEFACTION DAMAGE IN THE NZZGA TA EARTHQUAKE OF 1964 An apartment built directly on a sand foundation was tilted substantially due to the Niigata Earthquake in 1964. Until that time, it was believed that a sand layer was stronger than clay, so that a direct foundation could be used. But a sand foundation lost its strength due to liquefaction and then the apartment tilted. The largest tilt was about 60 degrees. The analytical model is shown in Figure 7.36. The objective of this analysis is to evaluate the three-dimensional effect on liquefaction analysis. The apartment was a wall structure but the five-story building is modelled by beam elements adapting the natural frequency equivalent. Half of the problem is modelled considering the symmetric shape. The side boundaries of the excitation direction are modelled by the periodic boundary. The side boundary perpendicular to the excitation and the bottom boundary are modelled by a viscous damper. The constitutive model used here is that of the Pastor and Zienkiewicz generalized plasticity model.
288
PREDICTION APPLICATIONS AND BACK ANALYSIS
Figure 7.36
Analytical model
The input motion used is the recorded data at Akita-Kencho for the Niigata Earthquake of 1964. The time history is shown in Figure 7.37. Table 7.9 shows the soil-layer model and material parameters. The liquefaction strength calculated by the soil parameters is shown in Figure 7.38. All layers have the possibility of liquefaction but layers 7-9 are stronger than the upper layers. Figure 7.39 shows the result of the undrained tri-axial test by changing the applied stress ratio. The N-value is the number of loading cycles for the strain amplitude, 5% is calculated. Figure 7.40 shows the initial stress of ground under and far from the building. The stress contour under the building is higher due to the overburden load.
7.7.1 Results Figure 7.41(a) shows the deformation due to earthquake. Layers 5 and 6 show permanent horizontal deformation for both sections A and B and the ground surface settled. Figure 7.41(b) shows the bird's-eye view of the deformation. Parts of the ground element are eliminated to show the inside. The excess pore pressure beneath the building is large due to the over-burden load. This can be seen in Figure 7.42 by comparing the results at sections A and B.
289
LIOUEFACTION DAMAGE IN T H E NIIGA T A EARTHQUAKE OF 1964
.""
0
1
2
3
4
5
7
6
8
9
Time (Sec)
Figure 7.37 Recorded data at Akita-Kencho for the Niigata Earthquake of 1964
0
Lab. Test. Layer 7.8.9
A
Lab. Test, Layer 5.6 Lab. Test. Layer 3.4 Simulation. Layer 7.8.9 Simulation, Layer 5.6 Sirnulation. Layer 3.4
- --------
10
Number of cycles N
Figure 7.38 Liquefaction strength
10
PREDICTION APPLICATIONS A N D BACK ANALYSIS
290
-
.
0
20
40 60 Mean stress (kPa) N= 5
80
100
0
20
40 60 Mean stress (kPa) N=20
80
100
( I ) Layer 3.4 (GL-2.0-5.Om)
Mean stress (Wa)
Mean strcss (kPa)
Mean stress (Wa)
Mean stress (kPa) N= 5
(3) Layer 7.8.9 (GL-8.0
N=5
- I4.Orn)
Figure 7.39 Stress path of element simulation
Table 7.9 Material parameters Depth (m)
porosity
Poisson's ratio
G (kPa)
Permeability (rnls)
v,(~P.,)
Cohesion (kPa)
Q
(deg.)
LIQUEFACTION DAMAGE IN THE NIIGATA EARTHQUAKE OF 1964
291
For the PZ model Depth (m)
MI
tr,
M,
ru,
8"
3,
Ho
Hue
7"
YDM
GL-2.0 -5.0 GL-5.0 -8.0 GL-8.0-14.0
0.39 0.50 0.58
0.46 0.54 0.62
1.1 1 1.11 1.11
0.35 0.35 0.35
1.0 1.0 1.0
0.12 0.02 0.0062
2.8 2.7 6.5
2.8 5.7 2.5
2.6 3.6 5.7
2.6 3.6 5.7
--
(a) Section A
(b) Section A
Figure 7.40 Initial stress condition predicted by static analysis
292
PREDICTION APPLICATIONS AND BACK A N A L Y S I S
DISP. SCALE : H 5.47E-02m
(a) Section A
Figure 7.41
DISP. SCALE :
5.47E-02
Deformation of the ground and building
INTERACTION BETWEEN ORDINARY SOIL
293
Figure 7.41 (corlt.)
7.8 INTERACTION BET WEEN ORDINARY SOIL AND IMPROVED SOIL LA YER It is very important to prevent liquefaction for key facilities in the industry such as thermal power plants and in order to estimate correctly their damage if subjected to a major earthquake. This example demonstrates how the dynamic effective analysis can be used to evaluate the effectiveness of the newly developed soil reinforcing method for foundations. This reinforcing method uses a deep-soil mixing method. The area where power stations are built is often by the sea due to the water cooling requirements and the earth from these areas is inevitably made up of filled sand. This method improves the strength of the soil in the ground by mixing it with cement and makes the stiffness usually over 10 times greater than that of the original soil layers. Thus the foundation itself will not liquefy but should also be able to share the horizontal seismic force, which is transferred from the bottom of the superstructures. According to common engineering practice, the static prediction method is required to ensure seismic safety of the stiffened ground. In order to do this, the major mechanism should be extracted requiring investigation of the stress for both the inside and outside of the improved soil (Kishion et al., 1998). The summary is reported here.
294
PREDICTION APPLICATIONS A N D BACK ANALYSIS
( a ) Section A
( b )Section B Section A Gl
B
c
0
E F
C H
0.00 10.0 20.0 30.0 40.0 50.0 83.0 70.0 &Pa)
Figure 7.42 Vertical stress immediately after the earthquake stops (Pore Pressure)
In Figure 7.43 numerical models for a cross-section of a power plant is shown. The building is about 320m long and 65m wide and modelled as a simple shear beam with stiffness equivalent to the whole structure like some form of super-element. The dark
295
INTERACTION BETWEEN ORDINARY SOIL
Figure 7.43 Configuration of the numerical model and mesh (shallow layer model)
hatched zone is the improved soil which is supported by a strong support layer with shear wave velocity of over 400mIs in which condition the soil layer will not liquefy. The surrounding area is the sandy soil layer where liquefaction is anticipated if a major earthquake strikes. The liquefaction analysis is conducted for a probable major earthquake artificially generated for the Tokyo Bay area. The maximum velocity is 50 mls. The material properties of the building are summarized in Table 7.10. The soil property is determined by soil sampling at a real site as shown in Table 7.1 1. Density and porosity are obtained by the physical property test. The N-value is obtained by the standard penetration test, the shear velocities ( V s )by the wave velocity investigation at the field, drained cohesion and friction angles ( c and 4) and liquefaction strength, by laboratory tests; permeability from particle size DZo(diameter at which 20'51 of the soil is finer) according to Creager's experimental data (Creager et al. 1944). In this analysis, all necessary soil tests mentioned above have been completed but in most cases, only limited data will be available for the prediction analyses, since they will be carried out at a very early stage of construction. At this stage, the details of the project will not be finalized. Engineers will be forced to interpolate between the limited amount of data, occasionally they need to extrapolate. Table 7.10 Equivalent beam model of a building: (a) stiffness; (b) lumped mass at node (a) Element No.
1
2
3
4
5
6
Height (m) Long dir. Short dir.
29.1(34.4 92.4 3.53
24.0(29.1 57.45 8.51
18.5(24.0 165. 9.54
13.0(18.5 74.1 11.2
7.0(13.0 108. 17.4
0.0(7.0 153. 25.8
(sectional area, unit x 10-Jm'jm (b)
Node. No.
1
2
3
4
5
6
7
8
Height (m) Long dir. Short dir.
34.5 42.6 7.82
29.1 187. 34.34
24.0 88.0 16.1
18.5 46.56 8.54
13.0 241. 44.3
7.0 223. 40.8
0.0 1910. 363.
-5.9 2680. 529.
(unit: Mglm)
Table 7.11 Soil properties of DMS problem Soil type
depth (m) N-value
Fine sand
SFs
Silty sand Finesand Fine sand Silty clay Fine sand
SFs As1 As2 Dcl Dsl
-1.40 2.70 4.40 5.30 -6.80 -8.80 -1 1.90 -13.05 -14.60 -19.80
11 11 7 3 3 3 27
density Share ( ~ ~ / m wave ~ ) velocity (mlsec)
Poisson's Friction ratio Angle (degree)
Cohesion (kPa)
Permeability (cmisec)
Rl2O ud/2uo1
Soil particle density
Porosity
INTERACTION BETWEEN O R D I N A R Y SOIL
297
Determination of liquefaction strength The direct soil properties for liquefaction behaviour obtained in ordinary engineering practice are the liquefaction strength curve, which shows the limit stress ratio against the number of cycles of loading (Figure 7.44). However, this curve does not give the information about how fast the excess pore pressure is built up. Therefore the parameters of the constitutive model are not determined only by the liquefaction strength curve but also by the engineer's experience.
7.8.1 Input motions The input motion used in the study was an artificial earthquake determined in 1992 as a design earthquake for the Tokyo Bay area. This earthquake can be used as input
Figure 7.44 Liquefaction strength of soil layers
PREDICTION APPLICATIONS AND BACK ANALYSIS
298
Max : 3 10.64 (17.72sec)
I
400 1
Time (Sec) (b) Acceleration spectra
Figure 7.45
Rinkai 1992 artificial earthquake
motion at a depth of bearing layer such as the Edogawa Layer (V, = 300-500 mls). To determine the seismic intensity in the Tokyo Bay area, several earthquakes were used such as El Centro, 1940, NS, Taft, 1952, EW, and artificial earthquakes recommended by Japan Architectural Centre. Figure 7.45 shows the time history of the Rinkai 1992, an artificial earthquake. A flat spectrum between 0.2 and 0.6 Hz at maximum is assumed.
Earth pressure due to liquefaction Seismic forces acting on the side of the improved soil ground is necessary for static safety analysis as well as seismic intensity. Among them the pressure acting to the
Earth pressure (kPa) (a)
Figure 7.46 Pressure at the side boundary of the improved soil ground: (a) maximum distribution; (b) comparison with Westergaard's
299
INTERACTION BETWEEN ORDINARY SOIL
side boundary of the improved soil ground was not well known in the case when the surrounding soil layers were liquefied. Therefore, a liquefaction analysis was made. Figure 7.46 (a) shows the pressure of the skeleton and the water at each level from left to right. The maximum total pressure from both sides is shown in Figure 7.46(b). The results obtained by the effective stress FE analysis agreed well with Westergaard's formula for dynamic water pressure.
7.8.2 Safety for seismic loading External safety Evaluations for the external safety of the improved soil ground were carried out for; (1) the possibility of the improved soil ground slipping at the boundary surface to the support soil layer; (2) a sub-grade reaction at the bearing layer; (3) an overturning of the improved soil ground. Safety factors for the slip failure of the examples were over 1.5 as shown in Table 7.12. The safety factor is calculated against the strength of the support soil layer. Safety factors for the sub-grade reaction were about three times larger than the compression. Using the strength of the improved soil ground. which has a value only for cohesion and zero for the friction angle according to the current design rule. Therefore, there will be a limitation for deeper foundations since the deviatonic stress becomes higher as the layer becomes deeper. A guideline for depth limitation was obtained from this numerical research, Table 7.12 Results of FE analysis and simple calculation (design procedure) Check items
-x 5 -2 V]
--
3
x .U a 0
-t:
M
.
-0
3
Maximum contact pressure (kPa) Subgrade reaction
2?
- m0
Subgrade reaction for ISG (kPa) Slip [safety factor]
d
V)
B
(kPa) Horizontal shera force
E
w
(kPa) Vertical shera force (kPa)
Ground kind
Building type
Side
Analysis
Design
Analysis Design
Shallow layer
Lighter
Long Short Short Short Long Short Short Short Long Short Short Short Long Short Short Short Long Short Short Short Long Short Short Short
221 239 293 513 1.69 1.91 2.01 1.99 221 239 290 646 247 310 330 310 81 78 92 186 147 105 1 16 142
224 269 326 688 1.57 1.43 1.44 2.10 224 269 326 688
0.98 0.89 0.90 0.75
0.99 0.89 0.89 0.94
-
-
Heavier Deep layer Heavier Shallow Lighter layer Heavier Deep layer Heavier Shallow Lighter layer Heavier Deep layer Heavier Shallow Lighter layer Heavier Deep layer Heavier Shallow Lighter layer Heavier Deep layer Heavier Shallow Lighter layer Heavier Deep layer Heavier
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
176 205 24 1 253
0.83 0.51 0.48 0.56
300
PREDICTION APPLICATIONS AND BACK ANALYSIS
3 0
2 0
1 0 0 10 Horizontal distance from centre (m)
20
30
Figure 7.47 Stress distribution of the vertical component
but further research is necessary for the limitation taking into account the dependency of the confining pressure for the strength of the improved soil ground. The possibility of overturning was very small since the side ground supported the improved soil.
Internal safety In order to check the safety of the improved soil ground, the maximum stresses at the local point were checked against the strength for normal stress. The maximum stress naturally occurs at the bottom edge of the building or of the improved soil ground as shown in Figure 7.48. There, very little moment was seen. Contour lines for the maximum value at a time of the shear stress (T,,,.) under the building shown in Figure 7.48 is almost horizontal. This means that the failure line can be developed along the line. Distributions of the maximum value of the shear stress along the assumed horizontal slip line for the depth from GL-6.35m to G L - 12.0m. are shown in Figure 7.49. The shear stress close to the foundation has sharp stress concentration at the edge of the building and low stress at the centre of the building. Shear stresses at the side area of the improved soil ground are very low due to liquefaction.
Figure 7.48 Contour of the shear stress
40
301
INTERACTION BETWEEN O R D I N A R Y SOIL
Horizontal distance from centre (m)
Figure 7.49 Shear stress along the horizontal line
Distance from left end of improved soil ground (ni)
Shear stress r,, (kPa)
Figure 7.50 Shear stress distribution along the vertical line
Various types of possible shear failure mechanisms were also checked. The footing type of failure was not triggered by the seismic force in this problem. The failure line may be horizontal near the bottom and/or vertical near the edge. They were both examined using the shear stress (r,.) calculated by the liquefaction analysis. Figure 7.50 (a) shows the distribution along the horizontal section for the maximum value at the time of the average shear stress. As in the ordinary slip-line analysis approach, the accumulated shear stress per unit length along the vertical or horizontal section was compared with the shear strength of the material. In the vertical section, the values were comparable and less than the failure strength. In the horizontal direction, the result of the calculation was much lower than the material strength. From this study it is concluded that the failure mechanism for the safety check is adequate for horizontal section but overestimated the vertical one.
7.8.3 Remarks The safety of the improved soil ground by the deep-soil mixing method for very large structures was examined by dynamic effective stress FE analyses. The analyses concluded that the following mechanisms should be checked and are sufficient for
302
PREDICTION APPLICATIONS A N D BACK A N A L Y S I S
the safety evaluation of the improved soil ground: (1) sub-grade reaction of the improved soil ground to the bearing layer; (2) slip failure at the bottom; (3) maximum contact pressure at the bottom of the improved soil ground; and (4) average shear stress along the vertical line for internal safety. Following these conclusions a design procedure based on the seismic intensity method was proposed. The results of the analyses also helped to determine the seismic intensity and other external forces such as the earth pressure to the side boundary of the improved soil ground.
REFERENCES Bazant Z.P. and Krizek R.J. (1976). Endochronic constitutive law for liquefaction of sand, Proc. ASCE. J. of Mech. Div., 102 (EM4), 701-722. Creager W.P., Justin J.D. and Hinds J. (1944) Engineering for dams, Vol. 111: Earth, Rockfill, Steel and Timber Dams, John Wiley and Sons, 649. Fukutake K. and Ohtsuki A. and Fujikawa S. (1995), Applicability of 2D analysis and merit of 3D analysis in liquefaction phenomena, Proc. Symp. On Three-dimensionul Evaluation o f Ground Failure, Tokyo, 229-236 (in Japanese). Ghaboussi J. and Dikmen S.U. (1981), Liquefaction analysis for multidirectional shaking, J. Geotechnical Engineering Division, Proc. qf ASCE, 197, GT5, 605-627. Hatanaka M. and Uchida A. (1995). Simple method for the determination of the KO value in sandy soil, First In t. Conference on Eurthquake Geotechnical Engineering, I, 309-3 14. Hyodo M., Murata N., Yasufuku and Fuji (1988), Effect of initial shear stress on development of undrained cyclic residual shear strain in saturated sand, Proc. Symp. on the use of' laboratory test for undrained cyclic behaviour of'soil and in-situ test, 199-204. Ishihara K. (1989). Dynamic behaviour of ground and earth structure Numerical method and problem specification, Proc. Symp. on hehaviour of ground and earth structure during earthquake held by Japanese Societj, of Soil and Foundation Engineering in Tokyo, 50-63, (in Japanese) Ishihara K. and Tatsuoka F., Yasuda S. (1975), Undrained deformation and liquefaction of sand under cyclic stresses, Soils and Founclations, 15(1), 29-44. Ishihara K. and Yamazaki F. (1980), Cyclic simple shear tests on saturated sand in multidirectional loading, Soil and Foundations, 20(1), 45-59. Ishihara K. and Nagase H. (1988) Multi-directional irregular loading tests on sand, Soil Dynamics and Earthquake Engineering, 7 , 4 Oct 1988, 201-212. Kansai - Kyogikan, Kansai - Kyogikan report, 1995 Kishino Y., Narikawa M., Masuda A,, Nakamura N., Shiomi T. and Suzuki Y. (1998), A study on building foundation with deep-soil cement mixing method for thermal-power plat, Part 1: design procedure and its verification by effective stress analysis, 6th US National Conference on Earthquake Engineering No. 488. Kondner R.L. and Zelasko J.S. (1963) A hyperbolic stress-strain formulation for sands, Proc, 2"d Pan American Conference on Soil Mechanics and Foundations Engineering, 289-324. Nagase H. and Ishihara K. (1988) Liquefaction-induced compaction and settlement of sand during earthquake, soil.^ and Foundutions 28 1 Mar 1988, 65-76. Pastor M. and Zienkiewicz O.C. (1986), A Generalised Plasticitv, Hierarchical Model for Sand under Monotonic and Cyclic Lor~ding,(eds. G.N. Pande and W.F. van Impe), 131-150, Jackson and Son, London. Pastor M., Zienkiewicz O.C. and Chan A.H.C. (1988), Simple models for soil behaviour and applications to problems of soil liquefaction, in Numerical Methods in Geon~eclzr~nics, (ed. G. Swaboda) 169-180, A.A. Balkhema. geomechanics -
REFERENCES
303
Pastor M., Zienkiewicz O.C. and Chan A.H.C. (1990), Generalised plasticity and the modelling of soil behaviour, Inr. J. NLIIII. AIINI.Mrtli. Geolllech., 14, 151-1 90. Pastor M., Zienkiewicz O.C. and Leung K.H. (1985). Simple model for transient soil loading in earthquake analysis, I 1 Non-associative models for sands, h t . J. NUIIW.Anrrl. Met11. Geotech., 9, 477-498. Pyke R.M. (1973) Settlement and liquefaction of sands under multi-directional loading, thesis presented to the University of California, Berkeley. California, in partial fulfilment of the requirements for the degree of Ph.D. Seed H.B. (1979) Consideration in the earthquake resistant design of earth and rockfill dams. Geotacl~nique,29, 21 5-263. Seed H.B., Lee K.L., Idriss I.M. and Makdisi F.I. (1975) Analysis of slides of the San Fernando dams during the earthquake of February 9, (1971). J. Geotecl~ Eng. Dir. ASCE, 101. GT7, 651-688. Seed H.B., Pyke R.M. and Martin G.R. (1978). Effect of multidirectional shaking on pore water pressure development on sands, J. Gpotecl~.Big. Div. ASCE, 104(GTI), 27-44. Shiomi T., Shigeno, Y., Sugimoto, M. and Suzuki, Y. (1991) Influence of liquefaction to pilesoil-structure interaction, Second Int. Con$ Geomec/~at~ics/Mi.~.~ouri Rollu, pp. 465-72. Shiomi T. and Shigeno Y. (1993) Consideration of initial shear stress on ground liquefaction. Second Asian-Pasific conference on Computational MechanicslSydney, 1077-1082. Shiomi T., Shigeno Y. and Zienkiewicz O.C. (1993) Numerical prediction for model No. 1. Proceedings q f ' Ve~rijicr~tion of' Nu~?~erical Procwlures for the Antrlysis of Soil Liquefirction ProblemslDavis, California11 7-20 October, 2 13-220. Shiomi T., Muromoto T. and Tani S. (1996) Dynamic behaviours of fill dam during earthquake on liquefaction analysis, Proceedings of the Thirty-First Japan National Conference on Geotechnical Engineering, 1263-4. Shiomi T . (1995) U . w M(IIIN(I/ O ~ M I I D I A Takenaka N. Corp.. Shiomi T., Shigeno Y. and Zienkiewicz O.C. (1993) Numerical prediction for Model No. 1. Verificution c?f'Nutnc,ric~rl Proccdurt,.~,fort11c.Ancr/issisof'Soil Liquclfirction Problems, (eds. K . Arulanandan and R.F. Scott), 1. pp. 213-219. Suzuki K., Saito S., Kimuira T., Kibayashi M. and Hosonii H. (1995), Investigation report on the building foundation made of the lattice wall soil improvement for liquefaction prevention, Kim - Kon, 10, 5 4 5 8 (in Japanese) Tokiniatsu K. and Yoshimi Y. (1983). Empirical correlation of soil liquefaction based on SPT N-value and tines content, Soil unci Fou~~tlutior~, 23, No. 4, p.p. 56-74. Van Genuchten M.Th., Pinder G . F . and Saukin W.P. (1977) Modeling of leachate and soil interactions in an aquifer, Proceedings, 3'-d Annual Municipal Solid Waste Res. Symp. EPA- 60019-77-026, 95-1 03. Vaid Y.P. and Finn W.D.L. (1979), Effect of static shear on liquefaction potential. J. Geotech. Eng. Div., ASCE, 105, (GTIO), 1233-1246. Yoshimi. Y. ( 199 1 ), hzfluer~ceq f Confnit~gPre.s.sure. Liquc~firctiotiqf' Sund (2nd Ed.), Gihodo publisher, Section 3.3., 32-34, (in Japanese) Zienkiewicz O.C., Chang C.T., and Hinton E. (1978). Nonlinear seismic response and liqueFaction. Int. J. Nurw. rind A n d . Method in Geonwcl~.,2(4), 381404. Zienkiewicz O.C. and Xie Y.M. (1991) Analysis of Lower San Fernando dam failure under earthquake, DNIUEtzginrering. 2, Issue 4, 307-322. Zienkiewicz O.C. and Shiomi T . (1984), Dynamic behaviour of saturated porous media. The generalized Biot formula and its numerical solution. It~t.J. Nuw. ( r t d AI?N/.M ~ t h .in Geoniech., 8. pp. 71-96.
Some Special Aspects of Analysis and Formulation: Radiation Boundaries, Adaptive Finite Element Refinement and Incompressible Behaviour
8.1 INTRODUCTION In the presentation of the essential theory and the finite element discretization procedures we have deliberately omitted some 'finer points' which on occasion might be essential to obtain more accurate or more generally applicable solutions to realistic engineering problems. We shall introduce these 'finer points' in the present chapter in sufficient detail to allow the reader to follow the current literature and to devise his or her own programme modifications. The chapter will be divided into three sections corresponding to the topics discussed and each section can be studied independently. These sections are: $8.2 Input for earthquake analysis and the radiation boundary. 48.3 Adaptive refinement for improved accuracy and the capture of localized phenomena. 48.4 Stabilization of computation for nearly incompressible behaviour with equal interpolation.
306
S O M E SPECIAL ASPECTS OF A N A L Y S I S AND FORMULATION
8.2 INPUT FOR EARTHQUAKE ANAL YSIS AND RADIA TION BOUNDARY 8.2.1
Specified earthquake motion: absolute and relative displacements
The input for earthquake analysis is based on measured recorded data of actual earthquakes and is generally presented as the values of the displacement u and or of the acceleration ii at the time interval of 0.02 seconds given for the duration of the earthquake.* If the time history of the input can be specified, we can proceed as outlined in this book to obtain the solution by time integration of the discretized form of the equations of motion such as those given by equations (2.1 1 ) and (2.13) of Chapter 2. The simplest case for the specification of input is illustrated in Figure 8.1 which attempts to model a structure resting on a stratified soil foundation of unlimited extent, by specifying the input motion at some arbitrary internal boundary shown. Such a model corresponds well with such physical models as those of the shaking table or centrifuge where the specified boundary represents a 'box' into which the model is fitted and which moves in a specified manner. In Chapters 6 and 7 we have shown several calculations which correspond to such physical experiments and which model the real phenomena of practice reasonably well. With all displacements or tractions at the boundaries specified, we can use the discretization of Chapter 3 and proceed with the solution of any transient problem. It should, however, be remarked that, if only uniform motion is specified on the boundaries, it is sometimes convenient to recast the equations of motion in terms of the relative displacement u~ which we define as
-
Known motion specified on all boundaries
Figure 8.1 Specified motion on the boundaries of a 'shaking table box' modelling of an infinite foundation
* In the USA such records can be obtained from The Earthquake Research Institute at the University of California at Berkeley and similar sources are available in other countries.
INPUT FOR EARTHQUAKE ANALYSIS
307
where UE = uE(t) is the prescribed earthquake motion which does not depend on the position. The governing equations (2.20) and (2.21) of Chapter 2 become now (neglecting the source terms and putting cy = 1).
with the boundary condition on the input boundary being
which is replaced by
If the relative velocity is used in the finite element discretization of the problem, the numerical computations are identical to those of the absolute displacement if the same initial conditions (e.g. u = 0) are assumed. However, the input is now the acceleration iiE giving a prescribed body force and this is often more accurately known. In a more realistic treatment of the foundation problem, we shall impose somewhat different boundary conditions recognizing the fact that, in the input, only the incoming wave motion is specified and that outgoing waves must leave the problem domain unimpeded. Figure 8.2 again shows the problem initially suggested in Figure 8.1 indicating the position of the same limiting boundary but on which the motion will not now be directly specified. We shall discuss this problem in the next section in more detail and suggest how such problems can be dealt with. First, however, a few words about the way knowledge of the seismic input wave is obtained.
Incoming plane eathquake wave
Figure 8.2 A more realistic model of an 'infinitie' foundation with a specified incoming
wave
308
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
The seismic signal is usually measured at or near the free surface, and it represents the modification of the original seismic wave which is initiated at the earthquake source, caused by passing through different material zones and involving a number of internal reflections and refractions at the interfaces between layers of different material. The geological conditions at the site will very often be such that the so called 'bedrock' exists as a zone of a significantly more rigid material underneath softer soil layers. Any incoming seismic wave passing from the bedrock to the softer soil layers will amplify depending on the material properties of both bedrock and the soil layers. The significant consequence of the presence of the bedrock lies in the fact that all of the reflected waves are practically trapped inside the soft soil layers, as only a small fraction of these can be transmitted back to the bedrock through the interface with the softer soil. If the bedrock is significantly more rigid, the transmitted wave is smaller than the reflected one back towards the soil surface. In such a case, the simple fixed-base approach is valid and no transmitting boundary conditions need be imposed on the bedrock level as practically no waves get transmitted into the bedrock. The need for an arbitrary model truncation emerges in the cases where no distinct base rock exists, or when the extent of the softer soil layers is so great that it would be prohibitive to include the whole zone in a mathematical model. Such a situation may also arise when the non-linear material behaviour can be expected only near the surface and deeper layers (with material properties still far from bedrock-like characteristics) are expected to remain elastic. To model such a case correctly it is necessary to reconstruct the incoming seismic wave at the model truncation boundary. In the simplest case of a one-dimensional elastic, homogeneous, isotropic wave propagation problem involving the free surface, it is very well known that the free surface displacement wave equals the double of the incoming displacement wave. Here the incoming signal can be easily extracted from the recorded total signal on the undisturbed surface. Even in the case of the elastic non-homogeneous domain the incoming signal can again be extracted from the total signal recorded on the surface. Therefore, in the following, it will be assumed that the incoming wave (displacement, velocity or acceleration) is known at a position corresponding to the model truncation boundary, and that outside of this homogeneous elastic conditions pertain.
8.2.2
The radiation boundary condition: formulation of a one-dimensional problem
We return here once again to the problem of a stratified, horizontal, foundation such as we have considered in the previous section but now without a superposed structure. Clearly if we consider a vertical slice shown in Figure 8.3, isolated by cut sections AA and BB we note immediately that the problem is one-dimensional, i.e. that the displacements, stresses, etc., do not vary with the horizontal coordinate x. The equations governing the problem are still (8.2a and b) with the unknown variables remaining as u~ which is now, however, a function of the coordinates y and time t . Thus
309
I N P U T FOR EA R T H O U A K E ANALYSIS
"An
tttttt
Vertical wave PI-opagation
Figure 8.3 A horizontally stratified foundation subject to vertically propagating compression o r shear waves: (a) the corresponding 1-D problem with (b) arbitrary cut-off
Thus, all the derivatives with respect to the x axis are made identically zero. T o demonstrate the wave nature of the problem we shall assume that in the vicinity of the arbitrary, 'input' surface CD (and below this surface) the conditions are such that: (i) only isotropic elastic behaviour exists; (ii) the body forces have been taken into account independently; and finally that (iii) the dynamic phenomena are sufficiently rapid so that the flow in the porous medium can be neglected and k = 0 is assumed. Writing the total relative velocity in terms of its components
the system (8.2) reduces, in the absence of internal flow, to
310
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
where only total stresses are considered. The elastic constitutive relation under isotropic undrained conditions gives
and
where
is the shear modulus and
is the restrained axial modulus. E and u are Young's modulus and Poisson's ratio. Equation (8.6a) becomes on insertion of the above
and (8.6b) becomes
d2uY dy2
--
p..
TU
K
p=
0
(8.9b)
Each of the above equations corresponds to the well known scalar wave equation
a2Ch
1d2d
which has the solution
in which c is the wave velocity and 4,and 4, represent two waves travelling in the positive and negative directions of y respectively (incoming and outgoing waves). Thus, 4 = u, represents shear waves travelling with velocity
INPUT FOR EARTHQUAKE ANALYSIS
311
and q5 = u,, represents compressive waves travelling with velocity
We observe that c, tends to infinity for fully incompressible solid and fluid situations. To obtain the radiation condition we observe the solution sought at the 'cut-off' line C D should represent only an outgoing wave, i.e.
We observe immediately that
where
or that on the boundary
to ensure the existence of outgoing waves alone. Using the relationships (8.7) and noting the definitions of (8.12) and (8.13) we will observe that on the boundary C D the tangential traction becomes
and the normal traction becomes
This is equivalent to the requirement that on the boundary 'dashpots' of suitable strength are imposed in tangential and normal directions. Representation of such radiation (or quiet) boundary conditions in the manner presented above was
312
S O M E SPECIAL ASPECTS OF A N A L Y S I S AND FORMULATION
suggested almost simultaneously by Zienkiewicz and Newton (1969) and Lysmer and Kuhlmeyer (1969). In the one-dimensional case presented here the radiation condition is exact. However, on many occasion it has been used effectively on two or three-dimensional boundary shapes where the conditions of equations 8.17(a) and (b) imply dashpots in the tangential and normal direction at any position of the boundary. The numerical tests of the effectiveness of such a radiation boundary condition are presented by Zienkiewicz et a/. (1987) where it was shown that, for a given wave input form, identical results are obtained independently of the arbitrary cut-off position. Many alternative forms of radiation boundary conditions have been developed. Here the early work of Smith (1973) which was recently generalised by Zienkiewicz et a1 (1987a) is one possibility. Alternative methods are discussed by Wolf and Song in their recent text (1996), White et a/. (1977), Kunar and Marti (1981) and also by Zienkiewicz and Taylor (199 1). It is usual to conduct an analysis in terms of the relative displacement UR defined by (8.1) and to apply the radiation condition to this relative displacement only (see Zienkiewicz et al., 1987a, Clough and Penzien, 1993).
8.2.3
The radiation boundary condition: treatment of two-dimensional problems
A more general situation of engineering interest is the one illustrated in Figure 8.4 where a structure 'perturbs' the simple one dimensional solution of the layered foundation. Once again the horizontal boundary on which the vertically propagating waves enter the problem domain is treated identically to that of the one dimensional
8
8
~ e r t i k awave ~ prbpakation
Figure 8.4 Foundation of Figure 8.3 perturbed by imposition of a structure (a) and the ID problem (b)
INPUT FOR EARTHQUAKE A N A L Y S I S
313
t t t t t t t t t t t t t t Figure 8.5 Repeatable boundary conditions. Displacement at A=displacement at B.
case. Indeed, identical 'dampers' are placed on this boundary to ensure transmission of the exiting waves (but now these are approximate only as the transmission conditions d o not apply exactly to waves exiting obliquely to the boundary). More serious difficulties are, however, posed on the two vertical boundaries AA and BB and the boundary condition which needs to be imposed on these. Clearly, at points which are far away from the superposed structure the solution must be asymptotic to the previously discussed one-dimensional one. A possible way of dealing with the boundary conditions on these sections is therefore to impose the radiation damper between the interior region and the one-dimensional, free field. solution. Such a treatment is suggested by Zienkiewicz r t 01. (1988) but a simpler alternative is that of repeatable boundary conditions which is also given there. The latter achieves identical results more simply. In the repeatable boundary condition, which is illustrated in Figure 8.5, it is assumed that a sequence of structures is placed on the foundation at regular intervals B. Treatment of such repeatable conditions is simple in the finite element context (see Zienkiewicz and Taylor, 1989) as clearly the values of displacement, stresses etc. are identical on such a section as A or B due to periodicity and the assembly of nodal values at these boundaries is ensured by suitable node numbering. In Figure 8.6 we illustrate a test problem where different depths and widths of the analysed domain are used. A homogeneous elastic material is here assumed throughout the space and Figure 8.7 shows the time histories of displacement, acceleration and stress for a typical point at the base of the structure and with different domains of computation. It is surprising to note how little the results are affected by the extent of the domain assumed.
8.2.4
Earthquake input and the radiation boundary condition - concluding remarks
We have limited our discussion so far to that of the behaviour of the two-dimensional foundation layer problems with a vertically propagating wave input. Extension of
314
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION Possible arbitrary model truncation
\\
Shallow and wide mesh (SW)
Vertically propagating incoming seismic wave (El Centro)
Deep and narrow mesh (DN)
Shallow and narrow mesh (SN)
Figure 8.6 Two-dimensional model problem a n d Three mesles (SN, DN a n d SW)
the problem to three dimensions for the same wave input is trivial but of course three dampers will now be necessary on the radiation boundaries. Greater difficulties are presented by problems in which the earthquake (or shock) waves enter the boundary obliquely or indeed horizontally. Here, of course, the input motion history will be dependent on the position and the determination of this in itself is a major problem. However, once such motion is established it is possible to apply radiation boundary conditions throughout. We shall not discuss this difficult problem further as it is not frequently encountered in practice.
8.3 ADAPTIVE REFINEMENT FOR IMPROVED ACCURACY AND THE CAPTURE OF LOCALIZED PHENOMENA 8.3.1 Introduction to adaptive refinement Accuracy control and adaptive finite element refinement are, of course, of importance in all analysis problems, even if the material behaviour is linearly elastic. However, the need for adaptive refinement is even greater when plastic deformations are pronounced as here often very sharp gradients of displacements can occur,
316
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
Figure 8.8 First adaptive solution of a purely plastic deformation problem. A perforated bar (a) initial mesh (b) final adapted mesh with elongation DOF 1039 (c) initial material
configuration (d) final material deformation
leading in the limit to localized displacement discontinuities. In Figure 8.8 we show a typical plastic deformation pattern occurring in a uniformly stretched tensile specimen with a small perforation (viz. Zienkiewicz and Huang 1990). In this analysis the mesh was adaptively refined with elements not only being reduced in size near the displacement discontinuity, but also stretched in the direction of this discontinuity which is indicated by the material deformation pattern. The 'capture' of this discontinuity as it actually develops can be achieved approximately with finite elements using a continuous interpolation. However, by sufficient refinement, the exact solution can be approached as closely as possible. How far the refinement should proceed is a question which is difficult to answer precisely and here we will need to revert to the notion of error tolerance. In adaptive refinement of many engineering problems in which the errors are distributed throughout the domain, it is convenient to introduce error norms (such as the frequently used energy error norm) and to require that the error of that norm be kept below a certain value, usually as a fixed percentage of the total value of energy norm in the domain. Such an approach is not recommended in studies of plastic deformation or localization as our interest cannot in general be described by a single 'number'. We may, for instance, wish to find the maximum values of the loads carried by the structure of a particular size at a particular stage of deformation or indeed the maximum loads occurring throughout the deformation history. Alternatively, the interest may lie in determining precisely the position of the region where large strains or discontinuities occur.
ADAPTIVE REFINEMENT
317
For such problems we can separate the process of error determination and of refinement of the mesh. Thus the latter can be guided, for instance, by requiring that such a quantity (or indicator) as
where C is constant between all the elements, hnli, is their minimum size and s the direction of the maximum gradient of 4, the function of interest. This quantity can be interpreted as the maximum value of the first term of the Taylor expansion defining the local error of the scalar quantity 4. By ensuring that the mesh is generated so that the quantity C is constant throughout all elements, we achieve a solution which captures well all local discontinuities and which is efficient in achieving the progression which gives overall accuracy. At any stage of refinement, estimates of error are possible by using various recovery procedures, (see Zienkiewicz and Zhu, 1992), but alternatively convergence to exact solution can be studied by simply reducing the constant C in refinement. The use of the indicator defined by (8.18) allows element elongation to be included in the refinement, as on many occasions the feature occurring at high gradients is almost one-dimensional. Indeed, in a truly one-dimensional feature the maximum sizes of an element along its direction would be arbitrary and any reasonable value would be fixed on the maximum element size h,,,,. However, if the contours of the function 4 diverge or are curved, the upper limit h,,,, can be specified more closely. Thus, for instance, if the contours separated by the value of h,,,,, diverge by an angle 0 then the limit on /I,~, could be replaced as
as a bound based on the variation of the smallest dimension of the element. With curved contours
it is often specified where cw is circa 0.1 and R is the radius of curvature. These type of procedures are discussed in detail by Zienkiewicz and Wu (1994) in the context of fluid mechanics. An alternative refinement indicator has been used for the longer time in fluid mechanics. This is a requirement that
specifies the minimum size of elements. This specification was first formulated by Peraire et ul. (1987) and is very effective in the capture of shocks. Here, the elongation of elements can be computed directly in terms of principal curvatures.
318
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULA TION
It appears that the first indicator (i.e. that of (8.18)) is most efficient in the capture of narrow discontinuities, but both provide a remeshing which gives a rapid convergence and reduction of both local and global errors. Figure 8.9 shows how an adaptive analysis based on the first indicator can model discontinuity developed during the failure of the foundation under an eccentric load. Here a von Mises type of yield surface is used with ideal plasticity assumptions (Zienkiewicz et a/., 1995a and b). In Chapter 3 we have already mentioned that special conditions have to be satisfied by mixed finite element forms for incompressible, or nearly incompressible, behaviour such as is encountered under undrained conditions. Indeed, such behaviour will
Rigid and rough footing
Figure 8.9 Adaptive solution of the problem of foundation collapse with an ideally plasticelastic material (a) ecentrically loaded footing (b) final adapted mesh and deformed configuration showing displacement discontinuity.
ADAPTIVE REFINEMENT
319
occur in many applications of plasticity using von Mises or Tresca yield surfaces. For adequate solution it is always necessary to use here special mixed forms of elements which are outlined in Chapter 3. In the two examples quoted already we used the T6CI3C triangle where six nodes define a quadratic variation of continuous displacement and three nodes interpolate pressures in a continuous manner. In Figure 8.10 we show again an analysis of an ideal elasto- plastic problem in which a strong localization occurs. Here two forms of regular mesh are comparedone named 'lucky' mesh in which the triangle sub-division lines follow approximately the slip surface, and the other, the 'bad' mesh in which these lines are orthogonal to the slip surface. It is clearly noted that for the same sub-division the 'bad' mesh gives answers which are always inferior to those of the 'lucky' mesh. However, the adaptive solution starting from either refinement shows nearly exact values of the collapse load. A serious problem with adaptive analysis of non-linear problems of plasticity in which the results are path dependent is that of data transfer between the various stages of analysis. In principle, the control of the error should be achieved at each load increment separately and this, of course, necessitates the transfer of history dependent data such as stresses, strains, etc. from the mesh of the previous step to that used in the next increment. To avoid difficulties we have re-analysed the problems in each of the previous cases from the start of loading for every new mesh developed. Indeed such a procedure has also been used quite effectively in transient analysis of the San Fernando dam by Zienkiewicz and Xie (1991) and Zienkiewicz et ul. (1995a, and b) with results shown in Figure 8.11. However, currently new procedures of transferring data have been developed and it is now possible to change the mesh at each load increment, thus ensuring a constant degree of accuracy (Zienkiewicz et al., 1998).
8.3.2 Localization and strain softening: possible non-uniqueness of numerical solutions Strain-softening behaviour is a phenomenon frequently encountered in soils and invariably it leads to a very localized, sliding surface type of deformation. This is well exhibited in the so called 'slickensides', frequently observed in clays. The analysis of plasticity problems with a negative hardening (or softening) modulus, H, is in itself a complex task, but the basic difficulties have been overcome many years ago and are described in Zienkiewicz and Taylor (1991). However, the reason for localization only becomes clear if some specific cases are examined. Consider, for example, the behaviour of a one-dimensional, bar type, problem illustrated in Figure 8.12. In this example we assume plastic, softening, behaviour and consider analysis of the bar divided into a number of equal elements of length 'lz'. Further, statistically, we perturb the yield stress in an arbitrary element so that only that element yields when the load is applied and thus gives the peak yield stress. If the extension imposed on the specimen continues beyond that peak then only that one element shows the plastic deformation, all others unloading elastically. Depending on the ratio of the
320
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
element size h to that of the total length .t a progressively steeper unloading branch of the load deformation plot will occur. This can reach negative slope values and allows only elastic unloading which, obviously, is not correct. But certainly the steepening of this slope will increase to infinity for a finite value of / I and will imply a displacement discontinuity or full localization.
ting
T
Figure 8.10 Failure of a rigid footing on a vertical cut. Ideal, von Mises. plasticity and triangular T6Cl3C element (quadratic, continuous displacements, linear continuous pressure) ( a ) geometry data; (b) Mesh 2 (fine 'lucky'); (c) Mesh 3 (coarse. 'bad'); (d) Mesh 6 (adaptive solution obtained from Mesh 3); (e) displacen~entvectors; (f) Effective strain contour for Mesh 1 (fine. bad) (g) Load, displacement results for various meshes.
ADAPTIVE REFINEMENT
Figure 8.11 Earthquake analysis of lower San Fernando Dam (a) initial mesh refinement at t-75 seconds (c) adaptive refinement at t = 30 seconds.
321
(b) adaptive
In the example quoted, the localization was caused by a small weakness due to the statistical nature of the material strength behaviour. However, in other geometrically more complex problems the stress concentration, etc., will act in precisely the same manner, always causing a localization with softening material behaviour. However, the example discussed shows up another feature of the problem, i.e. that of numerical non-uniqueness as the slope of the unloading portion of the displacement load curve depends largely on the size of the element used. This non-uniqueness of the problem becomes most serious in multi-dimensional behaviour in many structural problems. In Figure 8.13(a) we illustrate the fairly large discrepancies which occur in the estimate of the maximum load for the problem illustrated in Figure 8.10 for which now a softening modulus has been assumed and different mesh subdivisions used in the solution. Even discounting the results obtained by the use of the coarse, bad, mesh as being very unreliable, we note a difference of about 20% in the estimate of the maximum load capacity when the simulation is achieved by meshes which for ideal plasticity give almost identical answers. While the reason for this has been hinted at it in the simple example of Figure (8.12), the manner in which the problem can be overcome has supplied many researchers with material for exercising their ingenuity. De Borst er al. (1993), Ortiz et al. (1987), Bazant and Lin (1988), Belytschko er al. (1988) and Belytschko and Tabarrok (1993) describe some of the possible procedures which range from the consideration of material as a Cosserat medium, through so called gradient plasticity, to a simple failure energy consideration introduced in the last of
322
S O M E SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
Figure 8.12 Non-uniqueness-mesh size dependence in extension of a homogeneous bar with a strain softening material (peak value of yield stress a, perturbed in a single element). (a) stress u versus strain e for material (b) stress ii versus average strain c = u / L assuming yielding in a single element of length h.
Figure 8.13 Strain softening (H = -5000): comparison of reaction vs prescribed displacement for various meshes using T6C13C element. (a) constant plastic modulus (b) mesh dependent plastic modulus
323
ADAPTIVE REFINEMENT
/
Work dissipated in failure per unit volume
Figure 8.14 Work dissipation in failure of the material
these references. We shall only refer here to that last procedure which, in the opinion of the authors, deals adequately and in a simple manner with the difficulties encountered. The procedure considers, in the manner common to that of early theories of fracture, namely Griffiths (1921), the constancy of work required for failing the material and requires the energy to be independent of the discretization used and therefore to be a pure material property. In Figure 8.14 we show a typical stress-strain relation with strain softening in which failure is reached. The area under the full triangle is the work required to cause this failure and, in a unit volume of material, becomes
where H i s the softening modulus. If an element of size h in the direction of maximum straining is to model failure correctly, the work requirement to fail a unit width of the element which must be kept constant i.e., is 1 a;h = constant 2 H
--
This would be invariant only if
where C is a constant. It appears therefore necessary to reduce the softening modulus in the manner of (8.24) as the size of the elements modelling the localization decreases. This indeed was done in the case of a problem illustrated in Figure 8.10(a) and the results are shown in Figure 8.13(a,b) which gives an almost identical failure load obtained by two very different meshes. It will be observed that the above discussion leads to two conclusions:
324
S O M E SPECIAL ASPECTS OF A N A L Y S I S AND FORMULATION
(i) that with strain softening, localization will always occur in the failure zone and this will show a continuously decrease in size with the element size h, and (ii) that the softening modulus cannot remain a material constant but must tend to zero (i.e. giving no softening) as the size of the element tends also to zero to present a consistent work estimation. This idea can be incorporated in a material model with concentrated localization singularity and has been introduced by Simo et ul. (1993) and Oliver (1995). It is clear that such a model will, in the limit, tend to give identical results to the adaptive refinement if equation (8.24) is used. The adaptive refinements of the type here discussed have been introduced by Zienkiewicz et ul. (1995a and b) from which the examples and previous figures have been quoted. This and other papers in the field, indicate that adaptive refinement is a feature which can improve the results of analysis significantly although with experience reasonable engineering estimates can be obtained without this feature.
8.4 STABZLZZA TZON OF COMPUTA TZON FOR NEARL Y INCOMPRESSIBLE BEHA VZOUR WITH MIXED ZNTERPOLA TZON 8.4.1
The problem of incompressible behaviour under undrained conditions
In Chapters 2 and 3 we have already mentioned the difficulties which can be encountered when the standard finite element approximation is used to model incompressible, or nearly incompressible, behaviour. Such behaviour will be obtained when the permeability is very small and when the compressibility (1/Q) decreases. In other words, this happens when the elastic bulk moduls of the pore water is very high. However, in the u-p approximation we have mentioned already that satisfactory behaviour of the solution can be obtained under all circumstances when the Babuska-Brezzi condition or the equivalent of the patch test is satisfied. Some permissible interpolations are shown in both Chapters 2 and 3 and in the two examples illustrating this chapter we shall show how the unstable behaviour of the 'illegal' Q4lP4 element can be eliminated by the use of the acceptable Q9/P4 interpolation. In the first element of quadrilateral form, a bilinear interpolation is used for both the u and p variables, while in the second a quadratic approximation is used for the displacement. While the use of such correct interpolations is desirable and we have based our code on this assumption, in recent years much research effort has been devoted to the introduction of stabilizing procedures which would allow arbitrary interpolation (say equal interpolation) of both variables to be used effectively. Such stabilization can, without doubt, lead to more efficient and simple formulations and a recent paper by Pastor et ul. (1997) shows the various approaches suggested in the literature. Here the work of Schneider et a1 (1978), Brezzi and Pitkaranta (1984), Hughes et al. (1986),
325
S T A B I L I Z A T I O N OF COMPUTATION
Hafez and Soliman (1991) and Sampaio (1991) suggests many alternatives. Some of these were shown by Zienkiewicz and Wu (1991) to derive very simply from the same roots of time-stepping analysis. The motivation for most of this work lies in problems of fluid mechanics and their numerical solution and it was shown recently by Zienkiewicz and Codina (1995) that an algorithm using the operator split procedure suggested by Chorin in 1965 automatically provides the desired stabilization. The use of such stabilization in the context of geomechanics was first made by Zienkiewicz and Wu (1994) and extended by Pastor et al. (1999). In this chapter we shall discuss only this last process of stabilization as it appears to be the most simple and efficient.
8.4.2
The velocity covvection, stabilization process
In this section we shall outline the semi-explicit time-stepping, operator split procedure which is effective in dealing with the incompressibility problems arising in geomechanics and which follows the methodology originally suggested by Chorin (1967 and 1968) and extended by Zienkiewicz and Codina (1995). It is convenient to introduce the velocity, v, as the basic variable and to compute the displacement increment by subsequent integration. Thus we have the definition
and in each time step it is simple to establish
once values of v"+l and p"+l have been computed. The starting points for the development of the algorithm are equations (2.20) and (2.21) of Chapter 2 (or (8.2a) and (8.2b) of this chapter) rewritten in terms of effective stresses with CY = 1, again neglecting the source term and introducing the new variable v. We can write these governing equations as pv =
sT/
-
Vp + pb and
noting that a
=a
da'
-
m p and
= Dds
(8.27~)
326
S O M E SPECIAL ASPECTS OF ANALYSIS AND FORMULATION
is the constitutive relation and that (3.21) needs to be used for strain calculation. The operator split algorithm solves (8.27a) in two steps. In the first the quantity v* is calculated explicitly from
where the RHS is computed at t = t,,. In the second part the velocity v* is corrected implicitly in terms of known pressures using
The above can only be evaluated after Ap is established if H2 # 0. In what follows we shall use O2 = 112 for good accuracy but any values of it in the range
can be chosen, provided that A t satisfies certain stability limits. Equations (8.30) and (8.31) must be discretized in space before proceeding with numerical calculations. Following the standard procedures of Chapter 3 with
we obtain the following after application of the Galerkin process.
and
All these matrices are defined in Chapter 3 in (3.23-3.26) and need not be repeated here. We must, however, mention that the evaluation of both V* and Vn+' is fully explicit if the mass matrix M is diagonalized. This can be done in a variety of ways by well known procedures discussed in finite element texts (see, for example, Zienkiewicz and Taylor, 1989). The determination of the pressure increment A p and hence of p"+l requires the solution of (8.27b). We now write the implicit time approximation as
Here various values of 01 can be used but
S T A B I L I Z A T I O N OF COMPUTATION
327
is particularly convenient and accurate. We must remark that with Qr 5 $ no stable solution is possible. Using the computed values of u* and (8.31) we can rewrite (8.36) as
from which A p can be established after discretization. This again proceeds in the manner previously described and we now have
In the above the matrices S, H and Q are defined in Chapter 3 by (3.29-3.32). The only new matrix occurring now is H* which is the approximation to the Laplacian operator.
By the usual procedures we find
We shall delay the explanation of the reasons why the split operator procedure permits the use of arbitrary interpolations for u and p (N and Np respectively) and shall first illustrate its effectiveness in examples.
8.4.3
Examples illustvating the effectiveness of the opevatov split pvocedure
Two examples are here quoted. The first of these is the soil layer subject to a periodic surface load. Indeed this problem is identical to the one used in Section 2.2.3 where the limits of applicability of various formulations are tested and for which exact solutions are readily available. Here we shall only use the u-p formulation and shall demonstrate how the very oscillatory results obtained by an equal interpolation can be improved by the use of the stabilization just described. In Figure 8.15 we show the details of the problem and in Figure 8.16 we show solutions obtained by the use of 2D elements. The first uses 20 Q4P4 elements and shows oscillations which are very pronounced. The second one shows the very close approximation and suppression of oscillation obtained using the Q9P4 element as well as the new stabilizing algorithm. In Figures 8.17 and 8.18 a fully two-dimensional problem of a foundation load is solved again showing similar results.
328
S O M E SPECIAL ASPECTS OF ANALYSIS A N D FORMULATION
--H Im
-1 q= 100 exp (-lwt) t- = 0 11, =
0
apw --
a.~
k, n E
p,
10-'mn/s 0.333 X 7.492 10 (pa) 0.2 3 2.0 x I0 (N/m3)
PW
1.0 x 10'(N/m3)
v
The height of the column has been taken as L = 30m and the excitation frequency chosen is W = 3.379 radls
Figure 8.15 Example I a saturated soil layer under a periodic load.
8.4.4
The reason for the success of the stabilizing algorithm
In Chapter 3 we have indicated the main reasons for the difficulties encountered in solving the problem where incompressibility is approached. We first made a comment on these difficulties when discussing the Jacobian matrix used in the solution of an iterative step by the Newton-Raphson procedure where the matrices of (3.31) and (3.30) tend to zero, i.e. when
and H
- I,,
( ~ ~ 1 ' ) ~ k ~-+~0 ~ ' d f l
which occurs when the compressibility and the permeability of both tend to zero. This zero limit leads to a zero diagonal which occurs also in steady state equations of Section 3.2.5 giving a linear form
As we mentioned there, satisfactory solutions can still be obtained but these require that the number of parameters describing the variable u must be greater than these describing the variable p i.e.
329
S T A B I L I Z A T I O N OF COMPUTATION
Medium compressibility
Small compressibility
4
Q * = 10 MPa
"."
9
Q*= I0 MPa
0.0
1.5 ~ 1 4 (a) Solution with standard column with 20 Q4P4 elements
(b) Solution with standard column with 20 Q8P4 elements
(c) Stabilized procedure with 20 Q4P4 elements
Figure 8.16 Example 1, vertical pressure amplitude distribution Note: Exact solution is very close to the stabilized solution
330
SOME SPECIAL ASPECTS OF ANALYSIS AND FORMULATION E
(a)
30 MPa
(b)
Example 2: a saturated soil foundation under transient load; (a) the problem domain; (b) transient load applied data
Figure 8.17
Figure 8.18 Example 2: two dimensional foundation pressure contours computed for small permeability and compressibility Q* = lo9 MPa, k = lo-' mis; (a) direct use of implicit algorithm with Q4P4 elements; (b) direct use of implicit algorithm with Q8P4 elements; (c) Q4iP4 elements with stabilized procedure
This is a necessary condition for avoiding singularities and can be readily achieved with certain interpolations. However, if the problem is recast in the manner given in Section 8.4.2. we shall find that even in the limiting case (i.e. with zero compressibility and permeability) a non-zero diagonal will be obtained a n d stability can always be achieved. As we have recast the problem in terms of velocities we shall linearize using these variables and write
where K includes a time integration operator
REFERENCES
331
In the steady state
and we can write the sum of (8.34) and (8.35) as
Eliminating v* from (8.39) by using (8.35) we have
The two equations (8.47) and (8.48) can be written as
and a non-zero diagonal is found to exist in its finite time steps. This seems to achieve complete stabilization and any interpolation of the V / u and p variables can be used with equal interpolation, of course, being the obvious choice. The procedure outlined unfortunately results only in conditional stability, although the time-step length is now given by the speed of the shear wave and hence is not too restrictive.
We find that the integration of the new stabilization procedure into the computer code is reasonably economic and can well be made use of in many programmes, especially those in which nearly explicit solution is going to be used.
REFERENCES Bazant Z. P. and Lin F. B. (1988) Non-local yield limit degradation. Int. J. Nunz. Metl~.Eng.. 26. 1805-1 823. Belytschko T. and Tabarrok M. ( 1 993) H-adaptive finite element methods for dynamic problems with emphasis on localization, Int. J. Num. Metll. Eng., 36, 42454265. Belytschko T.. Fish J. and Englemann B. E. (1988) A finite element with embedded localization zones, Comjl. Metl~.Appl. Mec11. Eng., 79, 59-89. Brezzi F. and Pitkaranta J. (1984) On the stabilization of finite element approximations of the Stokes problem in Efi'cient solutions uf elliptic prohlenn. Notes on Nzrmerical Fluid Mechanics, Vieweg, Wiesbaden.
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S O M E SPECIAL ASPECTS OF A N A L Y S l S AND FORMULATION
Chorin A. J. (1967) A numerical method for solving incompressible viscous problems, J. Cotnputationul Pl~ysics,2. 12-26. Chorin A. J. (1968) Numerical solution of incompressible flow problems, Studies it1 Nunwr. And., 2, 64-7 1. Clough R . W. and Penzien J. (1993) Dyntmiics qf Structures (2nd edn), McGraw-Hill, Inc., New York. De Borst R.. Sluys L. J., Hiihlhaus H. B. and Pamin J. (1993) Fundamental issues in finite element analysis of localization of deformation, Eng. Cornput., 10, 99-121. De Sampaio P. A. B. (1991) A Petrov-Galerkin Formulation for the Incompressible NavierStokes Equations using Equal Order Interpolation for Velocity and Pressure, h ~ t J. . Nutw Metll. Engrg., 31 No. 6, 1135-1 149. Griffiths A. A. (1921), Brittle fracture, Proc. Roy. Soc. (A), 221, 163. Hafez M. and Soliman M. (1991) Numerical solution of the incompressible Navier-Stokes equations in primitive variables on unstaggered grids, Proc. AIAA Conf., 91-1561-CP, 368-379. Hughes T. J. R., Franca L. P. and Balestra M. (1986) A new finite element formulation for fluid dynamics. V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal order interpolation, Comp. Meth. Appl. Mech. Eng., 59, 85-99. Kunar R. R. and Marti J. N. (1981) A non-reflecting boundary for explicit calculation in computational models for infinite domain media-structure interaction, ASME Engineering Mechanics Division, 46, 183-204. Lysmer J. and Kuhlemeyer R . L. (1969) Finite dynamic model for infinite media. ASCE E M , 95, No. EM4, 859-877. Oliver J. (1995) Continuum modelling of strong discontinuities in solid mechanics in Proc. COMPLAS IV, CINME, Barrelonu, p. 455479. (see also Int. J. Num. Metlz. Eng. 1996) Ortiz M . and Quigley J. J. (1991) Adaptive mesh refinement in strain localization problems, Comp. Meth. Appl. Mech. Eng., 90, 78 1-804. Ortiz M., Leroy Y. and Needleman A. (1987) A finite element method for localized failure analysis, Comp. Meth. Appl. Mech. Eng., 61, 189-214. Pastor M., Peraire J. and Zienkiewicz 0 . C. (1991) Adaptive remeshing for shear band localization problem, Archive of Applied Meclzcinics-hzgenieur Arclziv, 61, 30-39. Pastor M.. Rubio C.. Mira P., Peraire J., Vilotte J. P. and Zienkiewicz 0. C. (1992) Numerical analysis of localization in Nutnericcrl Models in Geotnechunics, A. A. Balkema, Rotterdam. Pastor M., Quecedo M. and Zienkiewicz 0 . C. (1997) A mixed displacement pressure formulation for numerical analysis of plastic failure, Comp. Struct, 62, 13-23. Pastor M., Zienkiewicz 0 . C., Li T., Li X. and Huang M. (1999) Stabilized finite elements with equal order interpolation for soil dynamics problems, A r c l z i ~qj' ~ ~Computer ~ Methods in Engineering. Peraire J., Vahadati M., Morgan K. and Zienkiewicz 0. C. (1987) Adaptive remeshing for compressible flow computations, J. Cornput. Phy., 72, 449466. Schneider G . E., Raithby G. D. and Yovanovich M. M. (1978) Finite element analysis of incompressible flow incorporating equal order pressure and velocity interpolation in Nu~nericulMethodsfor Luminur uncl Turbulent Flow, Pentech Press, Plymouth. Simo J. C., Oliver J. and Armero F. (1993) An analysis of strong discontinuities induced by strain softening in rate independent inelastic solids, Conlp. Mech., 12, 277-296. Smith W. D. (1973) A non-reflecting boundary for wave propagation problems, J. Comput. P h j x . 15. 492-503. White W., Valliappan S. and Lee I. K. (1977) Unified boundary for finite dynamic models, ASCE E M , 103, NO. EM5, 949-965. -
REFERENCES
333
Wolf J. P. and Song C. (1996) Finite Element Modelling o f Unbounded Media, John Wiley & Sons, Chichester. Zienkiewicz 0.C. and Codina R. (1995) A general algorithm for compressible and incompressible flow. Part I: The split characteristic based scheme, Int. J. Nurn. Metk. in Fluids, 20. 869-885. Zienkiewicz 0. C. and Huang G . C. (1990) A note on localization phenomena and adaptive finite element analysis in forming processes, Comm. Appl. Num. Meth., 6, 71-76. Zienkiewicz 0.C. and Newton R. E. (1969) Coupled Vibrations of a Structure Submerged in a Compressible Fluid, Proc. Int. Symp. on Finite Element Techniques, Stuttgart, pl-15, May. Zienkiewicz 0. C. and Taylor R. L. (1989) The Finite Element Method - Volunze I : Basic Formulation and Linear Problems (4th edn), McGraw-Hill Book Company, London. Zienkiewicz 0. C. and Taylor R. L. (1991) The Finite Element Method - Volume 2: Solid and Fluid Mcd7cmics, Dynamics und Non-linearity (4th edn), McGraw-Hill Book Company, London. Zienkiewicz 0.C. and Wu J. (1991) Incompressibility without tears! How to avoid restrictions of mixed formulations, Int. J. Nunz. Meth. Eng., 32, 1184-1203. Zienkiewicz 0. C. and Zhu J. Z. (1992) The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comp. Meth. Appl. Mech. Eng., 101, 207-224. Zienkiewicz 0. C., Bicanic N. and Shen F. Q. (1987a) Single step averaging generalized Smith boundary - (a) transmitting boundary for computational dynamics in Proc. Int. Conf Nurn. Mrth. Eng.: Theor,y crnd Applications ( N U M E T A 87) (eds. Pande G. N. and Middleton J.) Vol. 11, paper T4911, Swansea, 6-10 July 1987, Martinus Nijhoff Publishers, Dordrecht. Zienkiewicz 0. C., Clough R. W. and Seed H. B. (1987b) Earthquake analysis procedures for dams, CIGB ICOLD Bulletin 52. Zienkiewicz 0. C.. Bicanic N. and Shen F. Q. (1988) Earthquake input definition and the transmitting boundary condition in Advtrnces in Conl~~urutioncrl Nonlinear. Mechanics, I . St. Doltsinis (ed.), Springer-Verlag, 109-1 38. Zienkiewicz 0.C., Huang M., Wu J. and Wu S. (1992) A new algorithm for coupled soil-pore fluid problem, Shock and Vibration, 1, 215-233. Zienkiewicz 0. C., Huang M., Wu J. and Wu S. (1993) A new algorithm for the coupled soilpore fluid problem, Shock und Vibration, 1, 3-14. Zienkiewicz 0. C., Huang M. and Pastor M. (1995a) Localisation problems in Plasticity using finite elements with adaptive remeshing, Int. J. Num. Anal. Geomech., 19, 127-148. Zienkiewicz 0. C., Pastor M . and Huang M . (1995b) Softening, localisation and adaptive remeshing. Capture of discontinuous solutions, Con~p.Mech., 17, 98-109. Zienkiewicz 0. C., Boroomand B. and Zhu J. Z. (1998) Error Estimate and adaptivity using recovery procedure. Comm. Aj~pl.Nurn. Meth. Zienkiewicz 0.C. and Xie Y. M. (1991) Analysis of the lower San Fernando dam failure under earthquake, Dum Engineering, 2, 307-322. Zienkiewicz 0. C. and Wu J. (1994) Automatic directional refinement in adaptive analysis of compressible flows, Int. J. Nurn. Meth. Eng., 37, No. 13, 2189.
Computer Procedures for Static and Dynamic Saturated Porous Media Finite Element Analysis
9.1 INTRODUCTION This volume on computational geomechanics would not be complete without a description of the finite element implementation. However, due to space limitation, only a brief introduction to the finite element program DIANA-SWANDYNE 11, or simply, SWANDYNE I1 could be described. A limited (not more than 50 elements) executable version of the program together with 1. pre-processing program: DYNMESH 2.
post-processing program: DYNPLT
3.
soil model tester: SM2D
4.
post-processing program for soil model tester: SM2DGRH
have been made available on the World Wide Web at URL: http://www.bham.ac.uk/ CivEngIswandyne as well as the associated example data and result files. Also available is a limited version of the executable of GLADYS-2E which is an explicit implementation of the u-w formulation as described in Chapter 3.
9.2
OUTLINE DESCRIPTION OF DIANA-S WAND YNE I1
DIANA-SWANDYNE I1 is the acronym of Dynamic Interaction And Nonlinear Analysis -SWANsea DYNamic version 11. The program is an improved version of
336
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
the DIANA-SWANDYNE I (Chan 1988). It is intended for static, consolidation and dynamic analysis for problems in geomechanics. It is a two-dimensional program which incorporates plane strain and axisymmetric analysis. The governing equation that is being solved is the fully coupled, finite deformation and finite rotation Biot equation with the u-p simplification with the fluid acceleration neglected as described in Chapter 3. The program was written in standard Fortran 77 with a number of minor Fortran 95 enhancements such as the use of IMPLICIT NONE and END DO. The system dependent functions such as time and date functions are restricted to a limited number of subroutines and they are described in Section 9.6. The programming generally followed the one recommended by Irons, except for codes which were obtained from external source, with features such as five characters for variables and six characters for subroutines and functions used. DIANA-SWANDYNE I1 is capable of performing analysis for static (drained and undrained). consolidation and dynamic (drained, draining and undrained). For drained analysis, the fluid phase is either neglected or its pressure fixed at constant values. As for draining analysis, the pressure is not known for all locations and it will evolve with time. Lastly for undrained analysis, the undrained condition can be imposed at element level or simply by using a global no-drainage condition with a drained or draining analysis. The program used the finite element method with triangular and quadrilateral isoparametric elements in spatial domain. The time integration is done with the Generalised Newmark method (Whitman 1953, Newrnark 1959, Katona 1985, Katona and Ziekiewicz 1985). Both tangential stiffness and BFGS method are available for nonlinear iteration. The solution process used is the profile solver. Within a given time step, the incremental strain can be further subdivided to increase the accuracy of the incremental stress integration. As fully nonlinear behaviour is expected in geomechanics applications, most of the relevant properties can be made variable in the program. This includes the solid density, the fluid density, the void ratio and the Biot alpha. The permeability can be a function of pore pressure and void ratio. Finite deformation is accounted for using the Updated Lagrangian Formulation and finite rotation using Jaumann stress rate is included also. As the soil models used are primarily small strain models, it is not often necessary to use a more rigorous formulation. Thermal phase has also been included. External Loadings can be given in the form of boundary displacement or pressure, boundary traction or influx and pressure loading on solid phase. All the abovementioned can be a function of time. Body force can be applied to each element separately and can be a function of time. The variation is useful for initial stress analysis or centrifuge swing-up operation. The earthquake is prescribed as a boundary acceleration so that no further assumption is needed for its application. Both horizontal andlor vertical earthquakes can be applied. There are data available for earthquake trace of the following: the El Centro N-S earthquake, the San Fernando N-S earthquake, the bumpy road of Cambridge Geotechnical Centrifuge, earthquake trace from various centrifuge sites via the VELACS project and Bristol shaking table type earthquake.
DESCRIPTION OF MAJOR ROUTINES
337
Plotting can be done with all the nodal values of the mesh. This includes displacement, pressure, velocity, acceleration and rate of change of pressure. Effective stress state, incremental strain and pore pressure can be given at the gauss points. Acceleration, absolute and relative displacement, total and excess pore pressure can also be plotted from anywhere in the spatial domain. Deformed shape, pore pressure contour, stress distribution and plastic development can also be traced. Basic mesh plotting facilities are available and the program is accompanied with a post-processing program to handle result processing and plotting. There are quite a number of constitutive models currently available in the program including: Elastic family: Linear Elastic model, Anisotropic Elastic model (Graham and Houlsby 1983), Elastic model with Moduli varying with mean effective confining pressure and with a Coulomb friction envelope. Classical Elasto-plastic family: von Mises, Tresca, Non- associative elastoperfectly plastic Mohr Coulomb and Drucker Prager model (Owen and Hinton 1980), CAP model (Chen and Mizuno 1990; Sander and Rubin 1979) Cam-Clay family: Original Cam-Clay and modified Cam-Clay Kinematic hardening family: Al-Tabbaa and Wood model (Al-Tabbaa and Wood 1989, Al-Tabbaa 1987), Two surface kinematic hardening plane strain model for sand (Hamilton 1997, Hamilton et a1 1998). Generalised plasticity model: Pastor-Zienkiewicz Mark-I11 model (Pastor and Zienkiewicz 1986)
DESCRIPTION OF MAJOR ROUTINES USED IN DIANAS WAND YNE ZI In this section, the major routines used in DIANA-SWANDYNE I1 are described. Due to the size of the program, only major subroutines are described. A number of the significant service subroutines are described in Section 9.4. Simple description of significant constitutive models are given in Section 9.5. The subroutines are listed in the order that they are first called.
9.3.1
The top level routines
The top two level of the program has been given in Figure 9.1. DYNE11 is the dummy program unit. It gives the version number and copyright message before calling the main subroutine DYNMAIN.
338
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC DYNE11 - DYNMAIN INDATA INMESH TMSTEP
- Figure 9.2
- Figure 9.3 - Figure 9.4
Figure 9.1 Top two levels of the program DIANA-SWANDYNE I 1
DYNMAIN is the main calling subroutine in the program. As shown in Figure 9.1, it calls four major subroutines to execute the various functions of the program in the following order. INTIAL checks the hardware and software platform the system is running on. It finds out the machine constants for the software platform. It also checks if the system stores arrays in columnwise format as assumed in the program. The names of files for input and output purposes are also determined. It reads from the DATA file the title and user information about the run up to the line starting with EXEC. INDATA inputs the main control data and material information for the program from the DATA file such as the type of earthquake input and whether or not the global solution matrix is symmetric. Also keywords such as STATIC, CONSOLIDATION or DYNAMIC are used to specify the analysis intended. The major subroutines called by INDATA are given in Figure 9.2 and are described in Section 9.3.2. INMESH inputs the finite element mesh information from DATA and MESH file. The major subroutines called by INMESH given in Figure 9.3 and are described in Section 9.3.3. TMSTEP is the main analysis routine which controls the whole time stepping and iteration process. The major subroutines called by TMSTEP are given in Figure 9.4 and described in Section 9.3.4.
9.3.2
Subroutines for control and material data input
The major subroutines called by subroutine I N D A T A for control a n d material data input are given in Figure 9.2. They are described briefly one by one in this section: GETMAT reads in material data and element type information for each material region. MODNAM reads in the name of the constitutive model to be used and finds out its corresponding internal model number. It then calls subroutine CONSTI with option 1 to read in the constitutive model data. MODNMl compares the input name of the constitutive model with the list of internal names to find out the material model number. CONSTI see Section 9.5
DESCRIPTION O F MAJOR ROUTINES INDATA - GETMAT
MODNAM DAMPNM
339 MODNMI ICONSTI
- DAMPMD
- Figure 9.8 DAMPDF
IDAMPEL
PERMFN
Figure 9.2 Subroutines for Section 9.3.2
DAMPNM reads in the name of the damping model to be used and finds out its corresponding internal model number. It then calls subroutine DAMPMD with option 1 to read in the damping model data. DAMPMD is the standard interface for damping models. It is called and calls to various damping models with the same set of argument. Four functions are expected from the damping models:
1. Reads in the material data for the damping model 2. Given the current velocity, calculates the damping force 3. Forms the symmetric damping matrix 4. Forms the actual damping matrix which could be non-symmetric DAMPDF is the default damping model with no stiffness component for Rayleigh damping. The mass component for the Rayleigh damping is dealt with separately within GETMAT. DAMPEL is the damping model for the stiffness component for Rayleigh damping. PERMFN is to determine the type of permeability variation to be used. At the time of writing, only two formulae have currently been included: 1. k = constant
2 . k cc e3/l
+ e as suggested in Taylor (1948)
The Kozeny-Carmen equation where k x y , , n 3 / p k o ~ ' s ~ ( 1n ) 2 is being considered for future development.
9.3.3 Subroutines for mesh data input T h e major subroutines called by subroutine I N M E S H for mesh d a t a input are given in Figure 9.3. They are described briefly one by one in this section: GETELM reads in element connectivity information and returns the maximum element number used. GETRIG reads in nodes which are connected to the rigid block (if used). GETTIE reads in nodes which are tied in pairs or tied to the rigid block.
340
COMPUTER PROCEDURES FOR S T A T I C AND D Y N A M I C GETELM GETRIG GETTIE CHECK 1 CHKTIE GNSIDE GETNOD GETRGP CHKNOD CHKRIG CHECK2 GETBOU CHECK3 CHECH8
- GNSIDl
7: GENNOD GENFND
- NFIND1
CHECK9
,
CHECK4 GETPRE CHECK6 GETPLT WRITEP FDIMEN CHECK5
Figure 9.3
EQNORD CKCONN MINEQN CKCON2 SLOAN l
-CKCONl
CHECK7
- CHKPLT
Subroutines for Section 9.3.3
CHECK1 deduces the phase of each node using the element connectivity and the number of nodes in each phase for each type of elements. There are five phases used in the program: 1 for solid with 2 dofs, 2 for fluid with 1 dof, 3 for temperature with 1 dof, 4 for rigid block with 3 dofs and 5 for nodes not referenced therefore with 0 dof. CHKTIE checks if the tied nodes are of the same phase. GNSIDE generates mid-side node. It can also generate higher order elements from a lower order mesh e.g. 8-noded quadrilaterals from 4-noded ones. GNSIDl keeps track on the mid-side node number generated for each element side. GETNOD gets nodal coordinates from the MESH file. GETRGP gets material properties of the rigid block and generates the area of the rigid block if necessary. CHKNOD generates nodal coordinate of mid-side nodes and other phases of the same element. For example, if only the coordinates of the solid corner nodes of a 8-4 element are specified, the coordinate of the mid-side nodes of the solid phase and of
DESCRIPTION OF MAJOR ROUTINES
341
all the nodes of the fluid phase will be generated. The order of precedence is solid, then fluid and lastly temperature. GENNOD generates coordinates for mid-side nodes which are not specified. GENFND generates coordinates for nodes of other phases if they are not specified. CHKRIG checks if the nodes connected to the rigid block are of solid phase. NFINDl returns the other tied node of a pair given one of them, zero if the node is not tied. CHECK2 checks which of the phases are present in the analysis. GETBOU gets boundary condition code from data file. CHECK3 determines equation number for each dof in the global equation system. CHECKS performs profile minimisation CHECK9 finds the minimum profile length using either the various Cuthill-McKee methods or the Sloan method (Sloan 1989, Sloan and Ng 1989). EQNORD rearranges the equation numbers so that the solid nodes are eliminated before the fluid nodes of the same element. This is done to avoid excessive numerical error. CKCONN prepares the connectivity list for various Cuthill- McKee methods in MINEQN. CKCONl helps subroutine CKCONN to create the connectivity list. MINEQN minimises the profile length using one of the following methods:
1. Cuthill-McKee method 2. Reversed Cuthill-McKee method 3. Modified reversed Cuthill-McKee method CKCON2 prepares the connectivity list for the Sloan's method in subroutine SLOANl. SLOANl (originally called GRAPH) minimises profile length by minimising the diameter of the linked list as given in Sloan (1989) and Sloan and Ng (1989). The Fortran 77 coding has been obtained from the author.' CHECK7 forms the profile index array for each active equations. CHECK4 forms the element index array from the global profile. GETPRE gets time-independent and time-dependent prescribed values from the DATA file. If the dof involved is fixed, the value is used as prescribed and if the dof is free, the value is used as (generalised) force.
'
The subroutines are free for academic usage although a nominal charge will be made for commercial usage. Please contact Dr. Scott Sloan, Department of Civil Engineering, University of Newcastle, NSW 2308, Australia for more information. His email at the time of writing is cesws @clod.newcastle.edu.au.
342
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
CHECK6 checks the boundary code, time-independent and time- dependent prescribed values for the tied nodes to see if they are consistent. GETPLT gets information for the plotting output file DISP which records the time history at various user-specified locations. CHKPLT checks the plotting informations, calculates and stores the relevant information such as the local coordinate of time history points that do not coincide with a node. WRTEP writes the mesh and other relevant information into the plotting output file PLOT. FDIMEN generates the location of major arrays on the global integer array and double precision array. CHECKS checks the boundary code, time-independent and time- dependent prescribed values for the all nodes to see if they are consistent.
9.3.4 Subroutines called by the main control routine for analysis READEQ reads in earthquake acceleration records or coefficients for numerically generated motion such as sinusodial function from ERQK or DATA input file respectively. PRCONV prints the convergence criteria used in the analysis. FMCART forms Cartesian derivatives of the shape function for each gauss point. GETINT reads from INIT file the initial condition of the analysis. This file could also be an output file from a previous analysis which is to be restarted. CALINT calculates simple initial stress state such as linear variation of stress or constant & It also uses subroutine CLINT1 to analyse the initial stress state given. CLINT1 performs linear regression for CALINT. CONSTI see Section 9.5 CMPROP computes the value of quantities such as average density. WRTMES writes displacement, pressure, temperature, stress, incremental strain and internal parameters for the constitutive model at the current time station to output plotting file DISP. WRTMEl finds the residual force for a dof. WRTPLT writes the current quantities requested in GETPLT to plotting output file PLOT. This is done for every time step. TMCOEF forms the multiplying coefficient for each matrix in accordance to the appropriate order of the Generalised Newmark time stepping scheme. FORMGL forms the global tangential matrix as given in eqn (3.47) FMELOl see Section 9.3.5.1 FORMEL see Section 9.3.5.2 ASSEMB assembles element matrices into the global profile storage.
343
DESCRIPTION OF MAJOR ROUTINES
PRCONV FMCART GETINT
CALINT 7: CONSTI
- CMPROP - WRTMES - WRTPLT - TMCOEF
- FORMGL
FMELOl FORMEL ASSEMB DATRIA
- CALINTl - Figure 9.8
- Figure 9.5 - Figure 9.6
- INFLIG - TFBOUN - TMVALU
- PREACC - UPDISP - CMPINC - RESIDU - TRNFRC - CHKREl - DASOLN - INCRDP - CONVER
- EARTHQ - FDERIV - Figure 9.7 - RFNORM - CHKRE2
- QNBFGS - OUTCRK - RESIDl - UPPROP - UPCORD - CHKNOD - ADVALI - ADVALC -
NADVAL ADVALP ADVARE ADVLN2
SAVFIN
- RESID2
Figure 9.4 Subroutines for Section 9.3.4
DATRIA performs triangular decomposition for a matrix stored in a profile form. This is adopted from Zienkiewicz and Taylor (1989). INFLIG opens, reads then closes an in-flight command data file. For some multitasking operating system, the user would be able to supply a command in the data file leading such as to a soft termination for the analyses. TFBOUN calculates the multplicative coefficient for time- dependent prescribed values. TMVALU calculates piece-wise linear multiplicate coefficient for subroutine TFBOUN. PREACC obtains the prescribed solid acceleration for dynamic analysis. EARTHQ calculates the current acceleration for each direction UPDISP updates the current displacement.
344
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
CMPINC calculates currentlincremental quantities such as gradient of current displacement and incremental temperature. FDERIV calculates Cartesian derivatives for the shape functions from the derivatives in local coordinates. RESIDU see Section 9.3.5.3 TRNFRC transforms residual force vector from containing all dofs to just the active, nonrestraint, dofs. For tied nodes, only the master dofs are used. Round-off errors are removed if the residual force vector is close to zero. RFNORM calculates and outputs the error norm for each phase. CHKREl finds the maximum residual error at each dof for each phase CHKRE2 performs insert sort for subroutine CHKREl. DASOLN finds solution to a system of simultaneous equations stored in profile form and the coefficient matrix already decomposed into triangular form using DATRIA. This is adopted from Zienkiewicz and Taylor (1989). INCRDP calculates the incremental displacement (for STATIC), velocity (for CONSOLIDATION) or acceleration (for DYNAMIC) for the current time step depending on the type of analysis performed. CONVER checks if the convergence criteria is met within the time step. QNBFGS calculates the forward and backward transformation using the quasi-Newton BFGS iterative method. The code has been adapted from Matthies and Strang (1979). OUTCRK outputs information of cracked Gauss points. OUTCRZ outputs cracked Gauss point information on the screen. RESIDl outputs error norm, incremental norm and residual force norm for each phase. UPPROP calculates the updated value of quantities such as average density due to change in void ratio during the finite deformation analysis. UPCORD updates the coordinate for Updated Lagrangian analysis. CHKNOD see Section 9.3.3 ADVALl interface routine for subroutine ADVALC ADVALC calculates the minimum and maximum alpha values (a measure of mesh distortion suggested by Cheung et al., 1996) for the triangular elements in a mesh. For six-noded elements, the maximum angle of deviation from centre line is also calculated.
NADVAL finds the nodes of the required phase within the connectivity of one element ADVALP returns the value of alpha measure of quality of triangle ADVARE returns the area of the triangle. ADVLN2 returns the square of the distance between two points. SAVFIN saves the current state in FINL output file. The format is compatible with the input file INIT. It can be used to obtain the initial state for dynamic analysis from a static analysis or restarting an analysis.
345
DESCRIPTION OF MAJOR ROUTINES
RESID2 outputs a summary of the current analysis such as the average number of iterations per time step for easy reference.
9.3.5 Subroutines for the formation of element matrices and residual calculation
9.3.5.1 Subroutines for the analytical formation of element matrix for 3-noded elements (Figure 9.5) FMELOl forms element matrices for 3-noded element CONSTI see Section 9.5 DAMPMD see Section 9.3.2 FMASOl forms element mass matrix M for 3-noded element PERMFN see Section 9.3.2 FMPMOl forms element permeability matrix H for 3-noded element FMCMOl forms mass part of the element damping matrix C for 3-noded element FMQMOl forms element coupling matrix Q for 3-noded element FMGMOl forms element mass coupling matrix G for 3-noded element
9.3.5.2 Subroutines for the formation of element matrix (Figure 9.6) FORMEL forms element matrices CONSTI see Section 9.5 FMSTIF forms element stiffness matrix K DAMPMD see Section 9.3.2 FMMASS forms element mass matrix M CRACKK modifies material property after a Gauss point is cracked. PERMFN see Section 9.3.2 FMPERM forms element permeability matrix H FMELO1
I
CONSTI FMSTO1 DAMPMD FMASO 1 PERMFN FMPMO 1 FMCMO1 FMQMO 1 FMGMO 1
-
Figure 9.8
Figure 9.5 Subroutines for Section 9.3.5.1
346
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC FORMEL
- CONSTI - Figure 9.8 - FMSTIF - DAMPMD - FMMASS - CRACKK - PERMFN - FMPERM - FMCOMP - GXCOND - GXCOMP - GXJMAT - FMQMAT - FMGMAT - GXLMAT
Figure 9.6 Subroutines for Section 9.3.5.2 FMCOMP forms element compressibility matrix S GXCOND forms element thermal conductivity matrix GXCOMP forms element thermal storage mass matrix GXJMAT forms element fluid and thermal coupling matrix FMQMAT forms element solid and fluid coupling matrix Q FMGMAT forms element solid and fluid mass coupling matrix G GXLMAT forms element solid and thermal coupling matrix
9.3.5.3 Subroutines for the formation of residual forces (Figure 9.7) RESIDU calculates the residual force vector. INTSTR subdivides the strain increment for strain integration CONSTI see Section 9.5 DAMPMD see Section 9.3.2 ASSEMl assembles into the element force vector the force due to stiffness contribution ASSEM2 assembles into the element force vector the force due to mass and mass damping contribution ASSEM4 assembles into the element force vector the force due to fluid compressibility contribution PERMFN see Section 9.3.2 CRACKK see Section 9.3.5.2 ASSEM3 assembles into the element force vector the force due to permeability contribution ASSEMS assembles the element force vector into the global residual force vector
347
MAJOR SERVICE SUBROUTINES RESIDU
- INTSTR - Figure 9.8 - CONSTI - DAMPMD - ASSEMl - ASSEM2 - ASSEM4 - PERMFN - CRACKK - ASSEM3 - PROPFZ - ASSEMS
Figure 9.7 Subroutines for Section 9.3.5.3
9.4 MAJOR SER VICE SUBROUTINES ADDVEC adds two arrays and puts the results into the first one. AGAUSS returns the 2-D Gauss point location and weight for quadrilaterals and triangles. BPLOT interfaces the system dependent plotting routine XPLOT with the higher level plotting subroutines and makes them system independent. CFFINP opens a temporary file for free format read for information stored in a character string. CFORM7 formats the output of a double precision number within a given number of character space while maximising the number of significant figures produced. CFORM8 outputs a double precision array with maximum number of significant figures within the given space using the subroutine CFORM7. CHKDFl checks the difference of two double precision numbers and prints if it exceeds a specified norm. CHKRWl checks if the computer platform uses a columnwise storage scheme as specified in the Fortran 77 standard. CHREAL writes the value of a double precision number into a character string. CHRINT writes the value of an integer number into a character string. CLOFIL closes a unit number and outputs a message to that effect. CPRINT prints out a matrix stored in profile form. CTITLl checks if a specific keyword is present at the beginning of a character string with echo of the input. CTITLZ checks if a specific keyword is present at the beginning of a character string with or without echo of the input. DOEQ checks if two double precision numbers are the same within the machine precision. DOEQO checks if a double precision number is close to zero within the machine precision. DAXPY is the BLAS 3.0 routine for double precision vector operation of y
=y
+ ax.
348
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
DCOPY is the BLAS 3.0 routine for double precision vector operation of y = x. DDOT is the BLAS 3.0 routine for double precision vector operation of d = x.y. DMACH obtains the machine constants such as the smallest difference between a double precision number and unity. DOTPRD is the former function name for vector dot product. It is now an interface for DDOT. FRICTM calculates the friction angle required for the current stress state and the given cohesion. GIVALU gives a double precision array a particular value. GIVINT gives an integer array a particular value. HlSHAP returns one-dimensional finite element shape function. H2CORD returns the local coordinates of the nodes for a two-dimensional finite element. HZSHAP returns two-dimensional finite element shape functions for triangular and quadrilateral elements. HANDLE handles an error condition in the input data and sets a flag so that the execution would stops after the data has been read in. IALLOC allocates a section of the global integer or double precision array to be used. INVARl calculates stress invariants for plane strain condition. IOFMTl ensures the same format statement is used in read and write operation for plotting output file PLOT. PRINT prints an integer matrix. IVECTC prints only one digit, i.e. in compact form, for each element of an integer array. IVECTP prints an integer array. LOADVC copies one double precision vector into another. This is now only an interface to DCOPY. LOWERC converts a character string to lower case. LSAMEl checks if two character strings are the same under case insensitive condition. MBTMUL calls BLAS 3.0 routine DGEMM for double precision matrix multiplication. MPRINT prints a double precision matrix. MROWPR prints a row of a matrix stored in profile form. MSTORE stores message summary for one unified output at the end of the execution. MVECTP prints a double precision vector. NEOLIN finds the end of text within a character string. NFACTL returns the factoral function in integer form. OPENFL is the subroutine to unify the open statements on different computer systems POLYOZ calculates the various properties of a n-sided polygon including the centroid, area and the second moment of inertia
CONSTITUTIVE MODEL SUBROUTINES
349
PSTREZ calculates the principal stress and direction. PSTRE3 calculates second order tensor rotation for, e.g., the stress tensor. RATEPF calculates, if supported by the system, the number of page faults for the underlying virtual memory system. RGAUSS gives one-dimensional Gauss-Legendre integration points. RMCOMM discards the input line if it begins with a letter 'C'. RMISPC reduces intermediate spaces within a character string to single space. RMLSPC removes leading spaces from a character string. SOLVC3 finds the local coordinates for a given point within a given element. SOLVC6 finds the local coordinates for a given point within a given element by looping over all elements within the finite element mesh. STACKP generates a stack dump for debugging purpose when an error occurs. STOPCU stops the execution if the maximum allowable CPU time is exceeded. TERMIN termins the analysis after issuing an error message. TIMEMG outputs the given message if a specified period of time has elapsed since the last time message has been output from this subroutine to the screen. TRIAOl returns the shape function and element matrix components for 3-noded triangular element. UPPERC converts a character string to uppercase. USAGEC checks how much of the global double precision array has been used. USAGE1 checks how much of the global integer array has been used. VALMAX returns the absolute element value within a double precision array. VECTAS performs addition of two double precision vectors by calling DAXPY. VECTSB performs subraction of two double precision vectors by calling DAXPY. WARNED writes out a warning to the screen and stores the message for message summary at the end of the execution.
9.5
CONSTITUTIVE MODEL SUBROUTZNES
Quite a large number of constitutive models have been made available to the computer program DIANA-SWANDYNE 11. They are all linked to the computer program via a standard constitutive model interface CONSTI which is described in Section 9.5.1. Constitutive models available for general dissemination are described in Section 9.5.2. Other models available are described in Section 9.5.3. Their dissemination is restricted both by licensing conditions and difficulties of implementation. Some of them, although connected, have not been thoroughly tested for general application. Due to limited space, the subroutines called by the main constitutive model subroutines listed in Sections 9.5.2 and 9.5.3 will not be shown.
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
350
9.5.1 Standard Constitutive model interface subroutine CONSTI For a fully coupled soil and pore-fluid computer program like DIANA-SWANDYNE 11, facilities must be made available so that further material models can be installed. Despite the wide acceptance of the Biot dynamic formulation and numerical implementation as described in this volume, there is still vigorous on-going research on constitutive models. The program must provide room for expansion especially in this aspect. The single material interface is performed through the subroutine CONSTI. The specification has been given in the user manual of the program SM2D (Chan 1995) and it is reproduced in Appendix 9A. A few modifications have been made to cater for three-dimensional analysis, the provision of Gauss point location and models for partially saturation application. The material interface, besides linking directly to SM2D, DIANA-SWANDYNE I1 and GLADYS-2E, is also available to ABAQUS via their material model interface (MMI) DMAT, CRISP (Britto and Gunn 1987) via CRSM2D (see Al-Tabbaa 1996) and LUSAS's MMI. From CRISP v4.0 onwards, CONSTI has been adopted as the standard material model interface for the computer program. Most of the material models available are listed in Figure 9.8. The standard interface is called and calls to various constitutive models with the same set of arguments. Five standard functions are expected from the constitutive models: 1. Read in the material data for the model 2. Given the current stress state, internal parameters and incremental strain, calculate the incremental stress and changes in internal parameters 3. Form the symmetric D matrix 4.
Form the actual D matrix which could be non-symmetric
5 . Initialise the internal parameter array
Four other functions are prepared but at the time of writing, not, fully implemented: 6. 7. 8. 9.
Return values for optimisation Output internal values to output unit ICOUT Special D-matrix e.g. consistent tangent operator Constant D-matrix e.g. linear elastic
Further suggestions for further standard functions are always welcome.
9.5.2
Constitutive models available for general dissemination (Figure 9.8)
CAP3D1-Interface to subroutine CAPMDL as listed on pages 412-423 of Chen and Mizuno (1990). This is available also for three-dimensional applications. CAPMOD-Interface routine with subroutine CAP as given in Sandler and Rubin (1979).
351
CONSTITUTIVE MODEL SUBROUTINES CONSTI
-
-
ADJCN4 ADJJIM ADJMH4 ALTER0 BRICK1 CAPMOD CJHMOD CSMOOl DEPOIN ELAS3D ELASGM ELASTA EXPERI HASHIS MCOULS MODCAM NCRIS2 SARAH SLIP03 STATE2 TABBA2 TSMODl UNSATS VONBAC VONMIS
Figure 9.8 Subroutines for Section 9.5
CJHMOD-Two-surface plane strain kinematic hardening model for sand. (Hamilton 1997; Hamilton et a1 1998) CSM001-Original
Cam-Clay model adapted from Britto and Gunn (1987)
DEP08N-Pastor-Zienkiewicz mark I11 model (1986) which is a generalized plasticity model for sand as described in Chapter 4 of this volume. ELAS3D-Linear ELASGM-general
Elastic model which can be used in three-dimensional analysis. elastic model, in which:
1. The bulk modulus and the shear modulus can vary with mean effective confining stress. The variation can be linear, square root or generally nonlinear 2. Cohesion and the Mohr Coulumb friction envelope is also available
ELASTA-Anisotropic elastic model for over-consolidated clay (Graham and Houlsby 1983). EXPERI-Experimental new model. A dummy subroutine to provide an easy connection for testing newly implemented constitutive model. MCOULSClassical Elastoplastic model adopted and modified from Owen and Hinton (1980): von Mises, Tresca, non-associative Mohr Coulomb with associative or nonassociative deviatoric response and Drucker-Prager model. MODCAM-Modified
Cam-Clay model
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
352
SLIPOSSimple plane slip model STATE2-State
parameter based one-dimensional model (Muir Wood et a1 1994)
TABBA2-Two 1989).
surface kinematic hardening model for clay (Al-Tabbaa and Wood
VONMIS--von Mises model VONBAC-von
Mises model with backward Euler integration
9.5.3 Other constitutive models implemented (Figure 9.8) The second author would like to take this opportunity to thank all the colleagues who have given kind permission for us to use their constitutive model and supply us with the source code.
ADJCNkInterface to a concrete model (GIBB 1994) ADJJIM-Interface
to a concrete interface model (GIBB 1995a)
ADJMHkInterface to non-associative Mohr Coulomb model with varying friction and dilatancy angle. Backward Euler integration scheme is used in this model together with comer handling strategies. (GIBB 1995b and 1995c) f i T E R L K i n e m a t i c hardening model proposed by Molenkamp (1982, 1987, 1990 and 1992) BRICK1-Simpson HASHI-Hashiguchi
brick mode1 (1992a and l992b) (1989) model
NCRIS1-Cristescu saturated sand model (Cristescu 1989 and 1991, Roatesi and Chan 1994, Roatesi 1995, Chan and Roatesi 1998, Roatesi and Chan 1998) SARAH-Three-surface TSMOD1-Pietruszczak TSMAIN.
kinematic yield surface model (Stallebrass 1990) Two-surface model-adapted
from author supplied program
UNSATSInterface to MODBUSY and JOSAO1, the Barcelona Unsaturated Soil model (Josa 1988; Alonso et a1 1990) VONOOl to VON008-vectorised
von Mises models
VONADl to VONADSAdaptive strategy subroutines to use VONOOl to VON008
9.6 SYSTEM-DEPENDENT SUBROUTINES ANSI01 performs screen control using escape sequence in ANSLSYS for MS-DOS. GETFIL gets the filenames to be used for the current analysis. OPNFLL opens a file and connects it to a unit number. DOSCOM performs a MS-DOS line command.
REFERENCES
353
SYSFUN returns values for various system functions such as system time, CPU time elapsed, page faults and date. SETSUF sets the name of file extension. COMLIN returns the text given on the command line. ZPSYMB draws a symbol on the screen. GETKEY gets one key from the keyboard. DSORTX sorts a double precision array. TIMEXX returns time and date for the current time. XPLOT maintains a set of basic plotting primitives such as pen up, pen down and opening a graphic device.
9.7 REFERENCES Alonso E. E., Gens A. and Josa A. (1990) A Constitutive Model for Partially Saturated Soils, GPotechnique, 40, No. 3, 405430. Al-Tabbaa A. (1987) Permeability and Stress Strain Response of Speswhite Kaolin, Ph.D. Dissertation, Cambridge University Engineering Department. Al-Tabbaa A. (1995) Excess Pore Pressure During Consolidation and Swelling with Radial Drainage, Giotechnique, 45, No. 4, 701-707. Al-Tabbaa A. and Muir Wood D. (1989) An experimentally based 'bubble' model for clay, NUMOG 111, Niagara Falls, 91-99. Britto A. M. and Gunn M. J. (1987) Critical State Soil Mechanics via Finite Elements, Ellis Horwood Ltd, Chichester. Chan A. H. C. (1988) A Unified Finite Element Solution to Static and Dynamic Geomechanics Problems, Ph.D. Thesis, University College of Swansea, Wales. Chan A. H. C. (1995) User Manual for SM2D - Soil Model Tester for 2-Dimensional Application, School of Civil Engineering, University of Birmingham, December, Birmingham. Chan A. H. C. and Roatesi S. (1998) Finite Element Approach in Viscoplasticity for Cristescu Saturated Sand Model, Rev. Roum. Sci. Techn. Mec. Appl., No. 1-2. Chen W. F. and Mizuno E. (1990) Nonlinear Analysis in Soil Mechanics - Theory and Implementation, Elsevier, Amsterdam. Cheung Y. K., Lo S. H. and Leung A. Y. T. (1996) Finite Element Implementation, Blackwell Science, Cambridge, MA 02142. Cristescu N. (1989) Rock Rheology, Kluwer, Dordrecht. Cristescu N. (1991) Nonassociated elastic/viscoplastic constitutive equations for sand, Int. J. Plasticity, 7 , 41-64. Gibb (1994) Nonlinear Concrete Model-Theory and Validation Report, Gibb Ltd, Reading, UK. Gibb (1995a) Concrete Construction Joint Interface Model and Validation Report, Gibb Ltd, Reading, UK. Gibb (1995b) Nonlinear Soil Model-Theory Report, Gibb Ltd, Reading, UK. Gibb (1995~)Nonlinear Soil Model-Validation Report, Gibb Ltd, Reading, UK. Graham J. and Houlsby G. T. (1983) Anisotropic Elasticity of a Natural Clay, GCotechnique, 33, NO. 2, 165-180. Hamilton C. J. (1997) A Plane Strain Constitutive Model for Sands under Non-monotonic Loading, Ph.D. Thesis, University of Birmingham, UK.
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COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
Hamilton C. J., Chan A. H. C. and Muir Wood D. (1998) Dynamic Finite Element Analyses of Sand: Structure Interaction Using a New Kinematic Hardening Model in Seismic Design Practice into the Next Century, Booth E. (ed.), A. A. Balkema, Rotterdam, 179-186. Hashiguchi K. (1988) A Mathematical Modification of Two Surface Model Formulation in Plasticity, Int. J. Solids & Structures, 24, No. 10, 987-1 001. Josa A. (1988) An Elastoplastic Model for Partially Saturated Soils, Ph.D. Thesis, ETSICCP, Barcelona (in Spanish). Katona M. G. (1985) A General Family of Single-step Methods for Numerical Time Integration of Structural Dynamic Equations, NUMETA 85, l , 213-225. Katona M. G. and Zienkiewicz 0 . C. (1985) A Unified Set of Single-step Algorithms Part 3: The Beta-m Method, a Generalisation of the Newmark Scheme, Znt. J. Num. Meth. Engrg., 21, 1345-1359. Matthies H. and Strang G. (1979) The Solution of Nonlinear Finite Element Equations, Int. J. Num. Meth. Engrg., 14, 1613-1626. Molenkamp F. (1982) Kinematic Model for Alternating Loading ALTERNAT, LGM Report Co-218598, Delft, NL. Molenkamp F. (1987) Elasto Plastic Model for Analysis of Liquefaction Under Alternating Loading in Workshop on Constitutive Laws for the Analysis of Fill Retention Structures, Ottawa. Molenkamp F. (1990) Reformulation of ALTERNAT to Minimise Numerical Drift Due to Cyclic Loading, University of Manchester Internal Report, Manchester, UK. Molenkamp F. (1992) Application for Non-linear Elastic Model, Int. J. Num. Anal. Geomech., 16, 131-150. Muir Wood D., Belkheir K. and Liu D. F. (1994) Technical Note - Strain Softening and state parameter for sand modelling, Gotechnique, 44, No. 2, 335-339. Newrnark N. M. (1959) A Method of Computation for Structural Dynamics, Proc. ASCE, 8, 67-94. Owen D. R. J. and Hinton E. (1980) Finite Elements in Plasticity-Theory and Practice, Pineridge Press, Swansea, Wales. Pastor M. and Zienkiewicz 0. C. (1986) A Generalised Plasticity Hierarchical model for Sand Under Monotonic and Cyclic Loading, NUMOG 11, Ghent, April, 131-150. Roatesi S. (1995) Finite Element Approach in Viscoplasticity, Technical Report, No. 887B, Technology and Research Ministry, Bucharest, Romania. Roatesi S. and Chan A. H. C. (1994) Numerical integration of a viscoplastic constitutive equation for geomaterials. Comparison with the analytical solution for step creep in The XVIII National Conference of Solid Mechanics, Brasov, Romania, 31-38. Roatesi S. and Chan A. H. C. (1998) Comparison of Finite Element Analysis and Analytical Solution for Underground Openings Problems in Viscoplastic Rock Mass, Rev. Roum. Sci. Techn. Mec. Appl., No. 1-2. Sandler I. S. and Rubin D. (1979) An Algorithm and a Modular Subroutine for the Cap Model, Znt. J. Num. Anal. Geomech., 3, 173-186. Simpson B. (1992a) Development and Application of a New Soil Model for Prediction of Ground Movements, The Wroth Memorial Symposium-Predictive Soil Mechanics', St. Catherine's College, Oxford, 27-29 July, 628-643. Simpson B. (1992b) Retaining Structures: Displacement and Design, GPotechnique, 42, No. 4, 541-576. Sloan S. W. (1989) A FORTRAN Program for Profile and Wavefront Reduction, Int. J. Num. Meth. Engrg., 28, 2651-2679. Sloan S. W. and Ng W. S. (1989) A Direct Comparison of Three Algorithms for Reducing Profile and Wavefront, Comp. Struct., 33,411-419.
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INTERFACING WITH THE MAIN PROGRAM
Stallebrass S. E. (1990) Modelling the Effect of Recent Stress History on the Deformation of Overconsolidated Soils, Ph.D. Dissertation, City University, London, England. Taylor D. W. (1948) Fundamentals of Soil Mechanics, John Wiley and Sons, Inc., New York. Whitman R. V. (1953) After Marcuson (1995): An Example of Professional Modesty, The Earth. Engineers and Education, MIT, 200-202. Zienkiewicz 0 . C. and Taylor R. L. (1989) The Finite Element Method-Volume I: Basic Formulation and Linear Problems (4th edn), McGraw-Hill Book Company, London.
APPENDIX 9A IMPLEMENTING NEW MODELS INTO SM2D In this chapter, the way to implement new models onto DIANA SWANDYNE-I1 is described. Since it is totally compatible with SM2D, no separate description is required. To test the soil model routine using program SM2D, please refer to SM2D manual chapter (Chan 1995). In this appendix, the implementation of an elastic and an elasto-plastic model von Mises are used as examples. To implement a new model, a slight modification to subroutine MODNMl and CONSTI is required. A model name and a model number has to be assigned to the new model, so that when reading a particular name, subroutine MODNMl will assign the model number to the material array. Subroutine CONSTI will use the model number to call the specific new model.
9A.1 INTERFACING WITH THE MAIN PROGRAM The interface of the program is done with a single subroutine call. A subroutine name which is sufficiently individual should be given. The routine can be written in FORTRAN 77 or any other compatible language. The interface call is as follow: SUBROUTINEELASTI (PROPD,LPRPD,ISWDP,
ICDAT,ICOUT,DSTRE
l,DSTAN,DMATX,ESTRE,PARAM,LPARA, IELEM, I G A U S , V O I D R , I C P R T 2,KTEST,NTEST) C.... C
ELASTICCONSTITUTIVEMODELFORVERIFICATION
L....
INTEGER NSTRE PARAMETER ( N S T R E = 4 )
The subroutine name EXPERD should be used for material model development. Using such would not require the change in the subroutines CONSTI and MODNMl until the model is ready for production.
9A.1.1
Variable NSTRE
NSTRE is the number of stress components in the analysis. For the two-dimensional analysis, d, and T,,. In the case the number of stress components is four. They are ordered as: d,,, dyy, of three-dimensions, two more stress components are required: T,, and T,,. The introduction of parameter NSTRE is to ease the changeover from two-dimensional to three-dimensional stress state.
356
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
9A.2 INPUT ONLY VARIABLES The following variables are for input only and their value should not be changed by the subroutine. All of integer variables should be positive. C .. . .
9A.2.1
INPUT VARIABLES INTEGER IELEM INTEGER I C P R T INTEGER IGAUS INTEGER ISWDP INTEGER LPRPD INTEGER ICDAT INTEGER ICOUT INTEGER NTEST I N T E G E R LPARA d o u b l e p r e c i s i o n D S T A N (NSTRE) DOUBLE P R E C I S I O N V O I D R
Variable IELEM
IELEM is the current element number. This is just for information. If it is desirable to trace the progress of a particular Gauss point, this could be useful.
9A.2.2
Variable ICPR T
ICPRT is the unit number for debugging or information output that can be massive. Unit ICOUT should not be used for this purpose.
9A.2.3
Variable IGA US
IGAUS is the current Gauss point number. This is just for information. If it is desirable to trace the progress of a particular Gauss point, it could be useful. The Gauss point is counted for each element i.e. it is the local Gauss point number.
9A.2.4
Variable ZSWDP
ISWDP is the controlling switch and its function will be fully described later. However, here is a brief overview of its capabilities 1.
Read in material data, check length of array, check input properties
2. Form incremental stress from the given incremental strain 3. Form symmetric D-matrix 4. Form non-symmetric D-matrix (if applicable)
INPUT O N L Y VARIABLES
357
5.
Initialise internal variables, check current stress state. For initial stress state calculated according to the Gauss point coordinate GPCOD, it can be implemented with a flag in the propd array, see e.g. subroutine TABBA2. 6. Return values for optimisation 7. Output internal values to unit icout
8. Special D-matrix e.g. tangent operator 9. Constant D-matrix e.g. linear elastic
9A.2.5
Variable LPRPD
The dimensions of the material property array. All of the elements from PROPD(1) to PROPD(LPRPD) are available in each material.
9A.2.6
Variable ICDA T
The unit number for input data.
9A.2.7
Variable ICOUT
The unit number for echoing the material data input. Minor output (less than two lines) per entry is acceptable. Monitoring of a Gauss point can be done with PRINT statement.
9A.2.8
Variable NTEST
The dimension of the soil model condition array KTEST(NTEST).
9A.2.9
Variable LPARA
The dimension of the soil model internal parameter array. All elements from PARAM(1) to PARAM(LPARA) are available at each Gauss point.
9A.2.10 Array DSTAN(NSTRE) The array for incremental strain input. This is active only for ISWDP = 2. The shear strain is the engineering shear strain which is two times the component of the incremental strain tensor. The incremental strain is tensile positive.
9A.2.11
Variable VOIDR
The current void ratio based on mechanical deformations.
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
358
9A.3
INPUT AND OUTPUT VARIABLES
Variables and arrays described in this section will carry information into the subroutine or out of the subroutine. C.
...
INPUT/OUTPUTVARIABLES: INTEGERKTEST (NTEST) d o u b l e p r e c i s i o n PROPD ( L P R P D ) d o u b l e p r e c i s i o n E S T R E (NSTRE) d o u b l e p r e c i s i o n P A R A M (LPARA)
9A.3.1 Away KTEST (NTEST)
This is the soil model information array. They are defined as follow: 1. KTEST(1) Valid only for ISWDP = 2. 0 indicates this is a trial incremental, maybe using it to determine the number of sub-increments. 1 indicates this increment may be taken as the accepted increment 2. KTEST(2) The type of subdivision that is active (variable NSUBV) 3. KTEST(3) The number of times MDIVT is exceeded 4.
KTEST(4) Number of warnings from soil models. If there is a warning state in the current model, this can be incremented by one. For a serious error, STOP statement can be used.
5. KTEST(5) Not used. 6. KTEST(6) Soil model error counter A: Usually used to denote tension state is encountered and remedial action is needed. If such a case is encountered, increment KTEST(6) by one. 7.
KTEST(7) Soil model error counter B: Usually used to denote allowable stress ratio has been exceeded and remedial action is needed. If such a case occurs, increment KTEST(7) by one.
8. KTEST(8) Soil model error counter C: Usually used to denote CONSl = nTD, . ng is less than zero and remedial action is needed. If such a case occurs, increment KTEST(8) by one. 9.
+
KTEST(9) Soil model error counter D: Usually used to denote the plastic modulus CONSl is less than zero and remedial action is needed. If such a case occurs, increment KTEST(9) by one.
10. KTEST(l0) Soil model error counter E: No specific definition yet. 11. KTEST(11) 0: Elastic D-matrix; 1: Elasto-plastic D-matrix
The programmer of new soil model is free to redefine any or all of these soil model error counters. Please use PRINT statement to indicate your definitions.
OUTPUT ONLY VARIABLES
359
9A.3.2 Away PROPD(LPRPD) The array for material data input. Different material type will have different PROPD even if they share the same model. This array should be written to only for ISWDP = 1 and for all the other ISWDP. This is for inputting information only. The subroutine could check if there is sufficient number of elements in the array for the use of the model. For a given Gauss point, the values within the propd(*) array should remain constant throughout the analysis. The array should be read by via iswdp = 1 and internal processing of the input information is allowed. So if the material information is read in elsewhere, a temporary file could be used so that a proper read-in with iswdp = 1 can be done. This array should be defined before any other option for iswdp other than one is used. A check could be used to keep track of whether iswdp = I has been called for the material group the Gauss point belongs to.
For options ISWDP = 2, 3,4, 5. this contains the current effective stress state (tension positive) which enters. For option ISWDP = 2, it should contain the incremented stress state which exits. The value of ESTRE should not be altered for iswdp = 3 and 4. For iswdp = 2, 3 and 4. the ESTRE should either be the output value of iswdp = 5 or inputloutput value of iswdp = 2.
The array for internal parameters. Different PARAM(LPARA) will be available for each Gauss point. The subroutine should initialise the values within this array for iswdp = 5 and check if the array is long enough for its purpose. The value of PARAM should not be altered for iswdp = 3 and 4. For iswdp = 2, 3 and 4, the value in PARAM should either be the output value of iswdp = 5 or inputloutput value of iswdp = 2. A check should be used within the routine so that iswdp = 5 will be called automatically if the values in PARAM have not be initialised.
9A.4 OUTPUT ONLY VARIABLES This section describes the arrays for output only. C....
OUTPUTVARIABLES: d o u b l e p r e c i s i o n D S T R E (NSTRE) d o u b l e p r e c i s i o n D M A T X (NSTRE,NSTRE)
When ISWDP = 2, DSTRE(NSTRE) should contain the incremental stress i.e. ESTRE(final)-ESTRE(initia1) when exits.
360
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
9A.4.2
Matrix DMA TX(NSTRE, NSTRE)
When ISWDP = 3, DMATX(NSTRE,NSTRE) should contain the symmetric version of the D-matrix on exit. If the soil model is non- associative then a symmetrical version should be given. This can be done either by using the elastic matrix, or the associative version of the matrix, or averaging so that DOISTRE=l,NSTRE DOJSTRE =l,NSTRE DMATX (ISTRE,JSTRE) =0.5DO*(DMATX (ISTRE,JSTRE) +DMATX (JSTRE,ISTRE)) &
END DO END DO
When ISWDP = 4, DMATX(NSTRE, NSTRE) should contain the non-symmetric version of the D-matrix on exit. If the soil model is associative, then ISWDP = 4 can lead to the same code as ISWDP = 3.
9A.4.3
Local Variables
Any number of local variables can be defined in the subroutine, although large arrays are not recommended. Except for very good reasons, COMMON should not be used. If it were used, it must not retain values that are specific to a Gauss point. Note, value of all the local variables is not necessarily retained for the next entry of the soil model nor retained for different Gauss points in the same material region. All real constants should be given in double precision also to preserve accuracy. C....
C..
..
9A.5
LOCAL VARIABLES LOCALDEBUG doubleprecisionYOUNG doubleprecisionPOISS doubleprecisionRLMDA double precision SHEAR INTEGER ICOUN double precision REAL1, REALO, SVOLV, DVOLV PARAMETER (REAL1=1.OdO,REALO=O.OdO) doubleprecisionREAL2 PARAMETER (REAL2=2.OdO) double precision REAP5 PARAMETER (REAP5=0.5DO) CHECK IF (DEBUG)WRITE (ICOUT,") 'VOIDRATIO:',VOIDR
ISWDP BRANCHING
The simpliest way of branching for ISWDP is to use a computed G O TO. It is not advisable, however, to jump from one option to another within the code unless it is a very
IS W D P BRANCHING
361
straight forward jump. The computed GO TO is used in the same way as the CASE statement in Fortran 95.
An error should be given if ISWDP is outside the range and CALL TERMIN will terminate the program after issuing a warning C. ..
.
ERROR CONDITION WRITE (ICOUT,*) 'ISWDP:', ISWDP CALLTERMIN ('ELASTI-1:UNKNOWNOPTION:') GOT0 2999
Label 2999 is the common exit point
9A.5.1 ISWDP=l For Material Data Input C.. . . 1100
2001
READ INMATERIALDATA CONTINUE READ(ICDAT,*,ERR=3100,END=3200) YOUNG,POISS RLMDA=YOUNG*POISS/( G REAL^-REAL~*POISS)*(REAL~+POISS)) SHEAR=YOUNG*REAP5/(REALl+POISS) WRITE (ICOUT,2001) YOUNG,POISS,RLMDA, SHEAR FORMAT ( ' YOUNG:', E11.3, 'POISS:', E11.3 ,/, 'RLMDA:', E11.3, 'SHEAR:', E11.3) 1 PROPD(1) =RLMDA PROPD (2) = SHEAR GOT0 2999
Note that the input data is echoed. Calculated values are stored in the array
9A.5.2 IS WDP = 2 Forming Incremental Stress With the input incremental strain DSTAN, the routine is required to output the incremental stress DSTRE and the new stress state ESTRE. C.... FORMTHE INCREMENTALSTRESS 1200 CONTINUE RLMDA = PROPD ( 1) SHEAR=PROPD ( 2 ) DVOLV=DSTAN(l) fDSTAN(2) +DSTAN(3) SVOLV = RLMDA*DVOLV DSTRE (1) = SVOLV+REAL2*SHEAR*DSTAN (1) DSTRE (2)=SVOLV+REAL2*SHEAR*DSTAN (2) DSTRE (3)=SVOLV+REAL2*SHEAR*DSTAN (3) ESTRE (1)=ESTRE (1)+DSTRE (1) ESTRE (2)=ESTRE (2)+DSTRE (2) ESTRE (3)=ESTRE (3) +DSTRE ( 3 )
362 C....
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC SINCEENGINEERINGSHEARSTRAINISUSED DO ICOUN = 4, NSTRE DSTRE (ICOUN)=SHEAR*DSTAN (ICOUN) ESTRE (ICOUN) =ESTRE(ICOUN)+DSTRE(ICOUN) END DO GOT0 2999
The do-loop is used to facilitate changeover to three-dimensional analysis.
9A.5.3 ZSWDP = 3 Symmetric version of D-matrix As the D-matrix is symmetric in this case, it is merged with option ISWDP C.... 1300
SYMMETRICVERSIONOFDMATX CONTINUE
9A.5.4 ZSWDP = 4 Not Necessary Symmetric D- matrix C.... NOTNECESSARYSYMMETRICVERSIONOFDMATX 1400 CONTINUE RLMDA=PROPD (1) SHEAR=PROPD (2) DMATX (1,l)=RLMDA+REAL2*SHEAR DMATX (1,2)=RLMDA DMATX (1,3)=RLMDA DMATX (1,4)=REAL0 DMATX (2,l)=RLMDA DMATX (2,2)=RLMDA+REAL2*SHEAR DMATX (2,3)=RLMDA DMATX (2,4)=REALO DMATX (3,l)=RLMDA DMATX (3,2)=RLMDA DMATX (3,3)=RLMDA+REAL2*SHEAR DMATX (3,4)=REAL0 DMATX ( 4,l) = REAL0 DMATX (4,2)=REAL0 DMATX (4,3) =REAL0 DO ICOUN=4,NSTRE DMATX ( ICOUN, ICOUN) = SHEAR END DO GOT0 2999
The do-loop is used to facilitate change over to three- dimensional analysis
9A.5.5 IS WDP = 5 Znitialise Internal Variables STORE SOMETHINGINTOPARAM 1500 CONTINUE DO ICOUN=l,LPARA
C....
= 4.
COMMON EXIT
363
PARAM(1COUN) = R E A L 0 END DO GOT0 2 9 9 9
9A.6
COMMON EXIT
It is advisable to use a common exit for all options. This is especially for debugging purpose. C . . .. 2999
COMMONEXIT CONTINUE RETURN
9A.7 ERROR EXIT Subroutine HANDLE will output the character message without terminating the program. In this subroutine, it is used to eliminate error in input data. C.. .. 3100
3200
ERRORCONDITION CONTINUE WRITE ( I C O U T , * ) ' I C D A T : ' , I C D A T C A L L H A N D L E ('ELASTI-2:ERRORINCHANNEL1) RETURN CONTINUE WRITE ( I C O U T , * ) ' I C D A T : ' , ICDAT CALLHANDLE ( ' E L A S T I - 3 : E N D O F F I L E I N C H A N N E L : ' ) RETURN END
9A.8 SOME CURRENT DEVELOPMENTS 9A.8.1
Interface for partially saturated model
A new interface for partially saturated analysis is in the process of development SUBROUTINEMODBUS ( p r o p d , l p r p d , i s w d p , i c d a t , i c o u t , dstre, dstan, d m a t x , e s t r e , p a r a m , lpara, i e l e m , i g a u s , v o i d r , i c p r t , k t e s t ,ntest, porep, dpore, gpcod)
The new arguments are:
1. porep
-
pore pressure corresponding to estre
2. dpore - change in pore pressure, incremental pore pressure applied with dstan 3. gpcod Gauss point coordinate -
364
9A.8.2
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC
Output to screen
A common variable ICHPC is used for the output to the screenlfile determined by the calling program. INTEGER ICHPC COMMON /PCOMOl/ ICHPC
The value ICHPC should not be changed. Direct output to the screen using PRINT statement or WRITE(6,fmt) should be avoided except for debugging purpose.
9A.8.3 Further error code from the material model for a. ktest(l1) can be used as mcode to output a single digit code to indicate the state of the Gauss point. Usually 0 implies elastic response. If NPRIN> 1 then the elasto-plastic code will be output during matrix formation. Future development could include outputing the code for iswdp = 2 to indicate the current state for the material model b. ktest(1) indicates the state for strain subdivision 0: total increment from final state 1: total increment from initial state 2: sub-increment c. ktest(2)=nsubv type of subdivision d. ktest(3) number of cases where mdivt is exceeded
9A.8.4
Time stepping informationfor the material model
A common block was introduced to relay time stepping, iteration and linear elastic step information to the soil model routine: integeritime, i i t e r , k e l a s common / t m s t p l / i t i m e , i i t e r , k e l a s
When kelas is 1, linear elastic response is expected
9A.9 ANOTHER EXAMPLEIMPLEMENTATION An example is von Mises
C
SUBROUTINEEXPERI (PROPD,LPRPD,ISWDP, ICDAT, ICOUT,DSTRE SUBROUTINEVONMIS (PROPD,LPRPD,ISWDP, ICDAT, ICOUT,DSTRE l,DSTAN,DMATX,ESTRE,PARAM,LPARA, IELEM, IGAUS,VOIDR,ICPRT 2, KTEST, NTEST)
C.. . . C
C.. . .
VONMISESELASTOPLASTICMODELWITHLINEARHARDENING
ANOTHER EXAMPLE IMPLEMENTATION C C..
365
updatedon27/3/1991for amistakeinepstn
..
updatedon15/10/1992toconformwith~alford~ortrancompilercheckingstandard
C.. . .
C.... C.. .
.
C....
C.... C.... C.... C....
*
C....
IMPLICIT NONE INTEGER NSTRE PARAMETER (NSTRE = 4) INPUT VARIABLES INTEGER IELEM, ICPRT INTEGER IGAUS INTEGER ISWDP INTEGER LPRPD INTEGER ICDAT INTEGER ICOUT, NTEST INTEGERLPARA, KTEST(NTEST) DOUBLEPRECISIONDSTAN(NSTRE),VOIDR INPUT/OUTPUTVARIABLES: DOUBLEPRECISIONPROPD(LPRPD),ESTRE(NSTRE),PARAM(LPARA) OUTPUTVARIABLES: DOUBLEPRECISIONDSTRE(NSTRE),DMATX(NSTRE,NSTRE) LOCALVARIABLES LOGICAL DEBUG PARAMETER (DEBUG= .FALSE.) ! YOUNG'S MODULUS DOUBLE PRECISIONYOUNG ! POISSON'S RATIO DOUBLEPRECISIONPOISS ! LAME ' S CONSTANT DOUBLE PRECISIONRLMDA ! SHEAR MODULUS DOUBLE PRECISION SHEAR DOUBLEPRECISIONFSTRE (NSTRE) DOUBLEPRECISIONDEVIA (NSTRE) DOUBLEPRECISIONYIELD DOUBLEPRECISIONCURJ2 DOUBLE PRECISIONREQJ2 INTEGER ISTRE, JSTRE , IPARA INTEGER ISTEP, MSTEP DOUBLE PRECISIONRATIO DOUBLE PRECISIONHMODU DOUBLEPRECISIONCONSl, CONS2 DOUBLE PRECISIONPMEAN DOUBLE PRECISIONSMALL PARAMETER ( SMALL = 1. OD 10) DOUBLE PRECISIONvsmal PARAMETER (vsmal= 1. Od-30) INTEGER ICOUN DOUBLEPRECISIONACOEF,BCOEF, CCOEF DOUBLEPRECISIONEPSTN (NSTRE) DOUBLE PRECISIONREALl,REALO, SVOLV, DVOLV PARAMETER (REAL1= 1.ODO, REAL0 = 0. ODO) DOUBLE PRECISIONREAL2 PARAMETER (REAL2= 2. OD0 ) DOUBLE PRECISIONREAP5 PARAMETER (REAP5= 0. 5DO) CHECK IF (DEBUG)WRITE (ICOUT,* ) 'VOIDRATIO:',VOIDR
366
COMPUTER PROCEDURES FOR STATIC AND DYNAMIC G O T 0 (1100,1200,1300,1400,1500), ISWDP
C
. . . . ERROR C O N D I T I O N
WRITE ( I C O U T , * ) ' I S W D P : ' , I S W D P C A L L T E R M I N ('ELASTI-1:UNKNOWNOPTION:') GOT0 2 9 9 9 C . . . . READ I N M A T E R I A L DATA 1100 C O N T I N U E READ ( I C D A T , * , E R R = 3100, E N D = 3 2 0 0 ) YOUNG, P O I S S , HMODU, Y I E L D P R I N T *, 'VON M I S E S MODEL: J 2 ' ' = Y + H * W' ' ' P R I N T * , 'J2" : SECONDDEVIATORIC INVARIANT' P R I N T *, 'W" : P L A S T I C D E V I A T O R I C WORK DONE ' P R I N T * , ' H : HARDENING C O N S T A N T '
p r i n t * , ' e x a m p l e e x p e r i .f o r ' i f ( p o i s s . l t . O . O d 0 ) then poiss = -poiss p r i n t *, ' E l a s t i c D - m a t r i x i s u s e d f o r i t e r a t i o n ' w r i t e ( 6 , * ) ' E l a s t i c D - m a t r i x is used f o r i t e r a t i o n ' propd ( 1 1 ) = l . O d O else propd ( 1 1 ) = O . O d O end i f RLMDA = YOUNG * P O I S S / ( ( R E A L 1 REAL^ * P O I S S ) * REAL^ + POISS) -
S H E A R = YOUNG * R E A P 5 / ( R E A L 1 + P O I S S ) C U R J 2 = Y I E L D * Y I E L D / 3.ODO WRITE ( I C O U T , 2 0 0 1 ) YOUNG,POISS,HMODU,YIELD,RLMDA, SHEAR 2 0 0 1 FORMAT ( ' Y O U N G : ' , E 1 1 . 3 , 'POISS:',E11.3 1 ,/, 'HMODU: ' , E 1 1 . 3 , ' Y I E L D : ' , E 1 1 . 3 2 ,/, ' R L M D A : ' , E 1 1 . 3 , ' S H E A R : ' , E 1 1 . 3 ) \ P R O P D ( 1 ) = RLMDA PROPD ( 2 ) = SHEAR P R O P D ( 3 ) =HMODU PROPD ( 4 ) = Y I E L D PROPD ( 5 ) = CURJ2 GOT0 2 9 9 9 C . . . . FORM T H E I N C R E M E N T A L S T R E S S 1200 CONTINUE C P R I N T * , I E L E M , I G A U S , DSTAN C . . . . E L A S T I C INCREMENT RLMDA=PROPD ( 1 ) SHEAR=PROPD ( 2 ) HMODU=PROPD ( 3 ) C U R J 2 = PARAM ( 1) C . . . . VOLUMETRIC P A R T O F THE I N C R E M E N T , I N VON M I S E S , T H I S I S ALWAYS C ELASTIC DVOLV=DSTAN ( 1 ) +DSTAN ( 2 ) +DSTAN ( 3 ) S V O L V = R L M D A * DVOLV C.... T H I S ISTHEDEVIATORICCOMPONENTOFTHE I N C R E M E N T A L S T R E S S DSTRE ( 1 ) = R E A L 2 * SHEAR*DSTAN ( 1 ) D S T R E ( 2 ) = R E A L 2 * SHEAR * DSTAN ( 2 ) DSTRE ( 3 ) = REAL2 * SHEAR * DSTAN ( 3 ) C . . . . S I N C E ENGINEERING SHEAR S T R A I N I S USED DSTRE ( 4 ) =SHEAR*DSTAN ( 4 ) C . . . . CHECK CURRENT Y I E L D S T R E S S PMEAN=-(ESTRE ( 1 ) +ESTRE ( 2 ) +ESTRE ( 3 ) ) /3.ODO DEVIA ( 1 ) = E S T R E ( 1 ) +PMEAN
)
ANOTHER EXAMPLE IMPLEMENTATION
367
D E V I A ( 3 ) = E S T R E ( 3 ) +PMEAN D E V I A ( 4 ) =ESTRE ( 4 ) PMEAN=PMEAN-SVOLV C . . . . FORM THE C O E F F I C I E N T S ACOEF=DSTRE ( 1 ) *DSTRE ( 1 ) +DSTRE ( 2 ) *DSTRE ( 2 ) 1+2.ODO*DSTRE ( 4 ) "DSTRE ( 4 ) + D S T R E ( 3 ) *DSTRE ( 3 ) B C O E F = D E V I A ( 1 ) * DSTRE ( 1 ) + D E V I A ( 2 ) * D S T R E ( 2 ) 1+2.ODO*DEVIA ( 4 ) "DSTRE ( 4 ) +DEVIA ( 3 ) *DSTRE ( 3 ) BCOEF = 2.ODO * BCOEF C C O E F = D E V I A ( l ) * D E V I A ( l ) +DEVIA ( 2 ) * D E V I A ( 2 ) 1 + 2 . O D O * D E V I A ( 4 ) * D E V I A ( 4 ) +DEVIA ( 3 ) * D E V I A ( 3 ) CCOEF = C C O E F - 2 . O D 0 * C U R J 2 I F ( a b s ( A C O E F ) . l e . v s m a l ) THEN RATIO = 1.ODO ELSE RATIO= (-BCOEF+SQRT (BCOEF*BCOEF-~.ODO*ACOEF*CCOEF)) 1 / (2.ODO"ACOEF) RATIO=MIN (RATIO, 1.ODO) END I F C . . . . HOW FAR OUT I S THE S T R E S S P O I N T ? R E Q J 2 = 0 . 5 D O * ( ( D E V I A (1)+ D S T R E ( 1 ) ) * * 2 + (DEVIA ( 2 ) +DSTRE ( 2 ) ) * * 2 1 + (DEVIA ( 3 ) +DSTRE ( 3 ) ) * * 2 2 3 + 2 . 0 D O k (DEVIA ( 4 ) + D S T R E ( 4 ) ) * * 2 ) C . . . . UPDATE THE S T R E S S S T A T E DEVIA ( 1 ) = D E V I A ( 1 ) + R A T I O k D S T R E ( 1 ) DEVIA ( 2 ) = D E V I A ( 2 ) + R A T I O k D S T R E ( 2 ) DEVIA ( 3 ) = D E V I A ( 3 ) + R A T I O k D S T R E ( 3 ) DEVIA ( 4 ) = D E V I A ( 4 ) + R A T I O k D S T R E ( 4 ) EPSTN ( 1 ) = (1.ODO-RATIO)*DSTAN ( 1 ) EPSTN ( 2 ) = (1.ODO-RAT1OIXDSTAN ( 2 ) EPSTN ( 3 ) = (1.ODO-RATIO)*DSTAN ( 3 ) E P S T N ( 4 ) = (1.ODO-RATIO)*DSTAN ( 4 ) C . . . . THE NUMBER O F I N T E G R A T I O N S T E P S RATIO=SQRT (REQJ2 /CURJ2) MSTEP=MAX ( I N T (lO.ODO* ( R A T I O - O . 9 0 0 0 0 0 0 0 0 0 0 1 ~ O ) ) , 0 ) c .. c o r r e c t i o n on 2 7 / 3 / 1 9 9 1 1 2 4 1 i s t r e = 1, n s t r e do epstn ( i s t r e ) =epstn ( i s t r e ) / d b l e (max ( 1 , m s t e p ) ) 1241 continue c . . . . e n d of c o r r e c t i o n C P R I N T *, 'MSTEP: ' ,MSTEP, R E Q J 2 , CURJ2 C . . . . L O O P OVER THE NUMBER O F S T E P S
..
DO
1
1
+
1250ISTEP=l,MSTEP DSTRE ( 1 ) = 2 . O D O * SHEAR* EPSTN ( 1 ) DSTRE ( 2 ) = 2 . O D O * S H E A R * E P S T N ( 2 ) DSTRE ( 3 ) = 2 . O D O * S H E A R * E P S T N ( 3 ) SHEAR * E P S T N ( 4 ) DSTRE ( 4 ) = C O N S 2 = D E V I A ( 1 ) *DSTRE ( 1 ) + D E V I A ( 2 ) "DSTRE ( 2 ) 2 . 0 D O k D E V I A ( 4 ) " D S T R E ( 4 ) + D E V I A ( 3 ) "DSTRE ( 3 ) CONSl=CONS2*2.0DO*SHEAR/ ((HMODU+~.ODO*SHEAR) *2.ODO*CURJZ) I F ( D E B U G ) PRINT * , 2 . ODO * SHEAR * HMODU "(DEVIA ( 1 ) *EPSTN ( 1 ) +DEVIA ( 2 ) *EPSTN ( 2 )
COMPUTER PROCEDURES FOR S T A T I C AND D Y N A M I C
368 2 3
+DEVIA ( 3 ) *EPSTN ( 3 ) + D E V I A ( 4 ) *EPSTN ( 4 ) ) * CURJ2
/ (HMODU+2.ODO * SHEAR) -CONS2-CONS1*2.ODO CURJ~=CURJ~+~.ODO*SHEAR*HMODU
1 2 3
"(DEVIA ( 1 ) "EPSTN ( 1 ) +DEVIA ( 2 ) *EPSTN ( 2 ) +DEVIA ( 3 ) * E P S T N ( 3 ) + D E V I A ( 4 ) " E P S T N ( 4 ) ) / (HMODU+2.ODO * SHEAR) DEVIA ( 1 ) =DEVIA ( 1 ) * (1.ODO -CONSl) +DSTRE ( 1 ) DEVIA ( 2 ) =DEVIA ( 2 ) * (1.ODO-CONS1) +DSTRE ( 2 ) DEVIA ( 3 ) = D E V I A ( 3 ) * ( 1 . O D O C O N S l ) +DSTRE ( 3 ) DEVIA ( 4 ) =DEVIA ( 4 ) * ( 1 . O D O C O N S l ) +DSTRE ( 4 ) REQJ2=0.5DO* (DEVIA ( 1 ) **2+DEVIA ( 2 ) ** 2 1 +2.ODO*DEVIA ( 4 ) * * 2 + D E V I A ( 3 ) * * 2 ) RATIO=SQRT (CURJ2/REQJ2) I F (DEBUG) P R I N T * , RATIO, C U R J 2 , R E Q J 2 DEVIA ( 1 ) =DEVIA ( 1 ) *RATIO DEVIA ( 2 ) =DEVIA ( 2 ) *RATIO DEVIA ( 3 ) =DEVIA ( 3 ) "RATIO DEVIA ( 4 ) =DEVIA ( 4 ) *RATIO 1 2 5 0 CONTINUE PARAM 1) = C U R J 2 F S T R E 1) = D E V I A 1 ) -PMEAN F S T R E 2 ) = D E V I A 2 ) -PMEAN F S T R E 3 ) = D E V I A 3 ) -PMEAN FSTRE 4 ) = DEVIA 4 ) DSTRE 1 ) = FSTRE 1 ) -ESTRE ( 1 ) DSTRE 2 ) = FSTRE 2 ) E S T R E ( 2 ) DSTRE 3 ) = FSTRE 3 ) -ESTRE ( 3 ) DSTRE ( 4 ) = F S T R E ( 4 ) E S T R E ( 4 ) ESTRE ( 1 ) = F S T R E ( 1 ) ESTRE ( 2 ) = F S T R E ( 2 ) ESTRE ( 3 ) = F S T R E ( 3 ) ESTRE ( 4 ) = F S T R E ( 4 ) GOT0 2 9 9 9 C . . . . SYMMETRIC V E R S I O N O F DMATX 1 3 0 0 CONTINUE C . . . . NOTNECESSARYSYMMETRICVERSIONOFDMATX 1 4 0 0 CONTINUE RLMDA=PROPD ( 1 ) SHEAR=PROPD ( 2 ) HMODU = P R O P D ( 3 ) C U R J 2 =PARAM ( 1 ) DMATX ( 1 , l ) = RLMDA + R E A L 2 * S H E A R DMATX ( 1 , 2 ) = R L M D A DMATX ( 1 , 3 ) = R L M D A DMATX ( 1 , 4 ) = R E A L 0 DMATX ( 2 , l ) = R L M D A DMATX ( 2 , 2 ) = RLMDA + R E A L 2 * S H E A R DMATX ( 2 , 3 ) = R L M D A DMATX ( 2 , 4 ) = R E A L 0 DMATX ( 3 , l ) = R L M D A DMATX ( 3 , 2 ) = R L M D A DMATX ( 3 , 3 ) = R L M D A + R E A L 2 * S H E A R DMATX ( 3 , 4 ) = R E A L 0 DMATX ( 4 , l ) = R E A L 0 DMATX ( 4 , 2 ) = R E A L 0
369
ANOTHER E X A M P L E IMPLEMENTATION
C..
.
DMATX ( 4 , 3 ) = R E A L 0 DMATX ( 4 , 4 ) = S H E A R CHECK C U R R E N T Y I E L D S T R E S S PMEAN=-(ESTRE ( 1 ) + E S T R E ( 2 ) + E S T R E ( 3 ) ) /3.ODO DEVIA ( 1 ) = E S T R E ( 1 ) +PMEAN DEVIA ( 2 ) = E S T R E ( 2 ) +PMEAN DEVIA ( 3 ) = E S T R E ( 3 ) +PMEAN DEVIA ( 4 ) =ESTRE ( 4 ) REQJ2=0.5DO* (DEVIA ( 1 ) * * 2 + D E V I A ( 2 ) * * 2 1 +2.ODO*DEVIA(4) **2+DEVIA ( 3 ) ""2) I F (REQJ2/CURJ2.GT.1.ODO+SMALL) T H E N P R I N T * , 'REQJ2 > CURJ2 I N V O N M I S : ' , R E Q J 2 , C U R J 2 STOP 'STOPPEDINVONMIS' END I F I F (REQJ2/CURJ2.GT.O.99DO.and.abs ( p r o p d ( l l ) ) . l e . v s r n a l ) T H E N
CONS1=2.ODO*SHEAR*SHEAR/
(
(HMODU+~.ODO*SHEAR)*REQJ~)
DO1420JSTRE=l,NSTRE 1 4 1 0 I S T R E = 1, N S T R E DO DMATX ( I S T R E , J S T R E ) =DMATX ( I S T R E , J S T R E ) 1 -(CONSl*DEVIA (JSTRE)) *DEVIA (ISTRE) CONTINUE CONTINUE END I F GOT0 2 9 9 9 C . . . . STORE SOMETHINGINTOPARAM 1500 CONTINUE D O 1 5 1 0 I P A R A = 1, L P A R A PARAM ( I P A R A ) = O . O D O CONTINUE PARAM ( 1 ) = P R O P D ( 5 ) PARAM ( L P A R A ) = - I 9 GOT0 2 9 9 9 C . . . . COMMON E X I T 2 9 9 9 CONTINUE RETURN C . . . . ERROR C O N D I T I O N 3100 C O N T I N U E WRITE ( I C O U T , * ) ' I C D A T : ' , I C D A T CALLHANDLE ( ' E L A S T I - 2 : E R R O R I N C H A N N E L ' ) RETURN CONTINUE WRITE ( I C O U T , * ) ' I C D A T : ' , ICDAT CALLHANDLE ( ' E L A S T I - 3 : E N D O F F I L E I N C H A N N E L : ' ) RETURN END
Author Index
Adkin J. E. 18, 50 Advani S. H. 202, 203. 204, 206, 207, 215 Al-tabbaa A. 337, 350, 352,353 Alonso E. E. 14, 15, 198, 216, 352,353 Anandarajah A. 158, 171 Armero F. 324, 332 Arulanandan K. 5, 15, 218, 230, 231, 247 Arulmoli K. 237, 247 Atkinson J. H. 113, 135, 171 Aubry D. 131 , 159,171 Babuska I. 64, 70, 81 Bacchus D. R . 140, 175 Baggio P. 188, 189, 193, 215 Bahda F. 151, 157, 171 Bailey 184 Baker R. 158, 161, 171 Balakrishna J. 14, 15 Balasubramanian A. S. 134, 135, 138, 139, 171 Balestra M. 324, 332 Banerjee P. K. 158, 172 Bardet J. P. 131, 172 Bathe K. J. 202, 215 Batlle F. 198, 216 Bazant Z. P. 129, 172, 302, 321, 331 Bear J. 14, 15 Belkheir K. 352, 354 Bell 184 Bellman R. 81 Belytschko T. 321, 331 Bergan P. G. 66, 81 Bettess P. 27, 51 Bianchini G. 144, 147, 148, 150, 151, 175 Bicanic N. 18, 36, 52, 84, 188, 209, 212, 216, 312, 313,333
Biot M. A. 8, 12, 15, 18, 49, 50 Bishop A. W. 8, 14, 15, 40, 50 Booker J. R. 202, 207,215 Boroomand B. 3 19,333 Boussinesq J. 6, 15 Bowen R. M. 18, 39,50 Bransby P. L. 113, 171 Brezzi F. 64, 70, 81, 324, 331 Britto A. M. 350, 351, 353 Brooks R. N. 189, 194, 195, 198, 215 Burland J. B. I l l , 123, 172, 175 Cambou B. 172 Carter J. P. 202, 215 Castro G. 144, 145, 153, 154, 172 Chambon R. 133, 172 Chan A. H. C. 18, 25, 26, 50,51, 55, 57, 60, 64, 70, 78, 81, 84, 90, 132, 137, 139, 148, 149, 151, 152, 155, 159, 163, 172, 174, 221, 226, 230, 233, 235, 237, 247, 248, 250, 276, 302, 303, 336, 350, 351,353,354 Chang C. T. 27,51, 62,84, 129, 130, 165, 176, 282, 303 Chaudhry A. R. 134, 135, 138, 139, 171 Chen W.F. 21 1, 337, 350,353 Cheung Y. K. 344, 353 Chorin A. J. 325, 332 Clough R. W. 71, 81,82, 3 12, 332,333 Codina R. 325,333 Cooke B. 221. 248 Corapcioglu M. Y. 14, 15 Corey A. T. 189, 194, 195, 198,215 Coulomb C.A. 85, 100, 175 Coussy 0. 39,50 Cowin S. C. 160, 172 Craig R. F. 25, 50
372 Creager W.P. 284, 295. 302 Crisfield M. A. 63, 82 Cristescu N. 352, 353 Crouse C. B. 236, 249 Cuellar V. 129, 172 Culligan-hensley P. J. 221, 248 Dafalias Y. F. 90, 131, 132, 133, 158, 171, 172, 173, 175 Darve F. 90, 133, 159, 172 De Boer R. 8, 15, 18, 39,50 De Borst R. 321. 332 de Saint-Venant 85, 175 Derski W. 18, 50 Desai C. S. 36, 50, 158, 161, 171 Desrues J. 133, 172 Dewoolkar M.M. 62,221, 227, 245-246,249 Di Prisco C. 158, 172 Dikmen S. U. 255, 261,302 Dobry R. 237, 245,249, 250 Drucker D. C. 86, 111, 120, 144, 172, 188, 215 Ehlers W. 39, 50 Englemann B. E. 32 1, 331 Esrig M. I. 131, 175 Famiyesin 0.0. R. 25,26,50, 55,78,81,226, 233,235,237,248 Felippa C. A. 64. 83 Fillunger P. 6, 7, 10, 15 Finn W. D. L. 129, 267, 303 Fish J. 321, 331 Flavigny E. 133, 172 Franca L. P. 324,332 Frossard E. 135, 172 Fruth L. S. 236, 247 Fuji 267, 302 Fujikawa S. 256, 302 Fukutake K. 256,302 Gantmacher F. R. 82 Gawin D. 40, 51, 188, 189, 193,215 Gens A. 14, 15, 198, 216, 352,353 van Genuchten M. T. 14, 16, 277,303 Ghaboussi J. 25, 50, 72, 82, 131, 158, 172, 255, 261,302 Gibson R. E. 8, 15, 111, 120, 172 Graham J. 337, 351, 353 Gray W.G. 39, 40,42, 50 Green A. E. 18, 39, 50
A UTHOR INDEX
Griffiths A.A. 323, 332 Gudehus G. 188, 215 Gunn M. J. 350, 351, 353 Habib P. 125. 142. 173 Hafez M. 325, 332 Hamilton C. J. 351, 353-354 Hashiguchi K. 131, 158, 173, 352, 354 Hassanizadeh M. 39, 40, 42,50 Hatanaka M. 262, 302 Hellawell E. E. 22 1 , 249 Henkel D. J. 11 1, 120, 131, 140,172,173,175 Herrmann L. R. 131, 132,172 Heyliger P. R. 202. 215 Hight D. W. 14, 15 Hill R. 158, 160, 173 Hinds J. 284, 295,302 Hine N. W. 60. 66.84 Hinton E. 62, 64, 83, 84, 129, 130, 165, 176, 282. 303. 337. 351. 354 Hirai H. 131, 158, 173 Holubec I. 128, 174 Hosomi H. 268, 303 Houlsby C . T. 337, 351, 353 Huang G. C. 333 Huang M. 231, 250-251, 3 16, 318, 319, 324, 332, 333 Huber M.T. 86, 175 Hughes T. J. R. 324, 332 Hulhaus H. B. 321, 332 Hujeux J. C. 131, 159, 171 Humpheson C. 128, 175, 179, 216 Hushmand B. 236, 237, 249 Hyodo M. 267, 302 Iai S. 23 1 , 250 Idriss I. M. 1. 16, 274, 303 Imamura T. 131, 173 IshiharaK. 125, 142, 151, 152, 153, 163, 164, 173, 175, 237, 249, 255, 256, 302 Ito K. 231. 248 Jauman G. 20, 50 Josa A. 352, 353, 354 Justin J.D. 284, 295, 302 Kaliakin V. N. 131, 173 Kansai-kyogikai 257, 302 Karadi G. M. 195, 197,216 Katona M. G. 60.82, 234, 249, 336,354 Kibayashi M. 268, 303
AUTHOR INDEX
Kim C. S. 202. 203. 204. 206, 207, 215 Kimuira T. 268, 303 Kishino Y. 293, 302 KO Hon-Y. 237,250 Kolymbas D. 90, 133, 173, 175 Kondner R. L. 256, 302 Kowalski S. J. 18, 39. 50 Krieg R . D. 131, 173 Krizek R . J. 129, 172, 202, 216,302 Krucinski S. 159, 164, 174 Kuhlemeyer R. L. 312, 332 Kunar R. R . 312, 332 Labanieh S. 90, 133, 159, 172 Lambe P. C. 230,250 Lambe T. W. 25, 50 Lanier J. 158, 172 Lassoudiere F. 131, 159, 171 Law H. K. 231, 250 Ledesma A. 18,36,52,84, 188,209,212,216 Lee F. H. 219, 249 Lee I. K. 312,332 Lee J. H. W. 202,203, 204, 206, 207,215 Lee K. L. 1, 16, 274. 303 Lee T. S. 202, 203, 204, 206, 207, 215 Leliavsky S. 8, 15, 19, 51 Leroy Y. 32 1,332 Leung A. Y. T. 280. 344. 353 Leung K. H. 57, 62, 64,82,83,84, 90, 128, 129, 132, 135, 147. 163, 174. 175. 176 Leung K.H. 303 Levy M. M. 6, 15 Levy M. 86, 175 Lewis R. W. 40, 41, 51, 67, 82, 128, 175, 179, 202,215,216 Li G. C. 36, 50 Li T. 325, 332 Li X. K. 51. 82, 325, 332 Liakopoulos A. C. 14, 15, 36, 51, 188, 189, 215 Liang R. L. 158, 173 Lin F. B. 321, 331 Liu D.F. 352, 354 Lloret A. 14, 15, 198, 216 Lo S. H. 344,353 Luong M. P. 125, 142, 173 Lyell C. 5, 15 Lysmer J. 31 2,332 Madabhushi S. P. G. 240, 249 Makdisi F. I. 1, 16
373 Makdisi F. I. 274, 303 Marti J. N. 312, 332 Martin G. R. 129, 255, 303 Masuda A. 293,302 Matsuoka H. 159, 173 Matthies H. 63, 82, 344, 354 Mchenry D. 8, 15 Meijer K. L. 202, 216 Meimon Y. 131, 159, 171 Meiri D. 195, 197. 216 Melan E. 86, 173 Meroi E. A. 193, 194, 195, 201, 201, 202,216 Mira P. 332 Mitchell R. J. 221, 248 von Mises R. 86, 175 Miura S. 163, 173 Mizuno E. 21 1, 337, 350, 353 Molenkamp F. 352, 354 Mollener 66, 81 Momen H. 131, 158. 172 Monte J. L. 202, 216 Morgan K. 317, 332 Morland L. W. 18, 39, 51 Mroz Z. 90, 92, 131, 158, 173, 174, 175 Muir Wood D. 25, 26, 50, 55, 78, 81, 128, 174, 226, 233, 235, 237, 248, 337, 351. 352. 353,354 Muller L. 1, 2, 15 Muraleetharan M. M. H. 236, 247 Murata H. 267, 302 Muromoto T. 28 1 , 303 Nagase H. 255,302 Nakamura N. 293,302 Nakazawa S. 64. 70. 84 Narasimhan T. N. 14, 15, 188,216 Narikawa M. 293,302 Needleman A. 321,332 Neuman S. P. 14, 15 Newmark N. M. 60, 82, 336, 354 Newton R. E. 312,333 Ng W. S. 341,354 Norris V. A. I3 1, 158, 173, 174 NovaR. 127, 128,143,158, 160, 161,172, 174 Ohtsuki A. 256, 302 Okada S. 151, 152, 153, 173 Oliver J. 324. 332 Ortiz M. 321, 332 Owen D. R. J. 202, 216, 337, 351,354
372 Creager W.P. 284, 295, 302 Crisfield M. A. 63, 82 Cristescu N. 352, 353 Crouse C. B. 236, 249 Cuellar V. 129, 172 Culligan-hensley P. J. 221, 248 Dafalias Y. F. 90, 131, 132, 133, 158, 171, 172, 173, 175 Darve F. 90, 133, 159, 172 De Boer R. 8, 15, 18, 39, 50 De Borst R. 321,332 de Saint-Venant 85, 175 Derski W. 18, 50 Desai C. S. 36, 50, 158, 161, 171 Desrues J. 133, 172 Dewoolkar M.M. 62, 221,227, 245-246, 249 Di Prisco C. 158, 172 Dikmen S. U. 255, 261,302 Dobry R. 237, 245, 249, 250 Drucker D. C. 86, 11 1, 120, 144, 172, 188, 215 Ehlers W. 39, 50 Englemann B. E. 321, 331 Esrig M. I. 131, 175 Famiyesin 0.0.R. 25,26,50,55, 78,81,226, 233,235,237,248 Felippa C. A. 64, 83 Fillunger P. 6, 7, 10, 15 Finn W. D. L. 129, 267,303 Fish J. 321, 331 Flavigny E. 133, 172 Franca L. P. 324, 332 Frossard E. 135, 172 Fruth L. S. 236, 247 Fuji 267, 302 Fujikawa S. 256, 302 Fukutake K. 256,302 Gantmacher F. R. 82 Gawin D. 40, 51, 188, 189, 193,215 Gens A. 14, 15, 198, 216, 352,353 van Genuchten M. T. 14, 16, 277,303 Ghaboussi J. 25, 50, 72, 82, 131, 158, 172, 255, 261,302 Gibson R. E. 8, 15, 111, 120, 172 Graham J. 337, 351,353 Gray W.G. 39,40,42,50 Green A. E. 18, 39, 50
AUTHOR INDEX
Griffiths A.A. 323, 332 Gudehus G. 188, 215 Gunn M. J. 350. 351, 353 Habib P. 125, 142. 173 Hafez M. 325, 332 Hamilton C. J. 351, 353-354 Hashiguchi K. 131, 158, 173, 352,354 Hassanizadeh M. 39, 40, 42, 50 Hatanaka M. 262, 302 Hellawell E. E. 221, 249 Henkel D. J. 1 1 1 , 120, 131, l40,I72,l73,175 Herrmann L. R. 131, 132, 172 Heyliger P. R. 202, 215 Hight D. W. 14, 15 Hill R . 158, 160, 173 Hinds J. 284, 295, 302 Hine N . W. 60, 66,84 Hinton E. 62, 64, 83, 84, 129, 130, 165, 176, 282,303, 337, 351,354 Hirai H. 131, 158, 173 Holubec I . 128, 174 Hosomi H. 268, 303 Houlsby G. T. 337, 351, 353 Huang G . C. 333 Huang M. 231, 250-251, 316, 318, 319, 324, 332, 333 Huber M.T. 86, 175 Hughes T. J. R . 324, 332 Hulhaus H. B. 321, 332 Hujeux J. C. 131, 159, 171 Humpheson C. 128, 175, 179, 216 Hushmand B. 236, 237, 249 Hyodo M. 267,302 Iai S. 231, 250 Idriss I. M. 1, 16, 274. 303 Imamura T. 131, 173 Ishihara K. 125, 142, 151, 152, 153, 163, 164, 173, 175, 237, 249, 255, 256, 302 Ito K. 231, 248 Jauman G. 20,50 Josa A. 352,353, 354 Justin J.D. 284. 295. 302 Kaliakin V. N. 131, 173 Kansai-kyogikai 257, 302 Karadi G. M. 195, 197, 216 Katona M. G . 60,82, 234, 249, 336, 354 Kibayashi M. 268, 303
A UTHOR INDEX
Sugimoto M. 264,303 Suzuki Y. 264, 268, 293,302, 303 Tabarrok M. 321, 331 Taboada V. M. 237,250 Tanaka Y. 143, 175 Tani S. 281, 303 Tanimoto K. 143, 175 Tatsuoka F. 125, 142, 153, 1 302 Taylor D. W. 124. 144, 146, 147, 175, 339, 355 Taylor P. W. 140, 175 Taylor R. L. 21,51, 53, 58, 60, 63, 64, 66, 70, 83,84, 186,312,313,319,326,333,343, 344, 355 Terzaghi K. Von 16, 25,51 Terzaghi K. 181, 216 Thompson P. Y. 230, 250 Thurairajah A. 11 1. 120, 121, 175 Toki S. 163, 173 Tokimatsu K. 257, 303 Tresca H. 85, 99, 175 Uchida A. 262,302 Ueno M. 131. 173 Vahadati M. 317,332 Vaid Y. P. 267, 303 Valliappan S. 312. 332 Venter K. V. 21 8, 221, 250 Vilotte J. P. 332 Wang Z. L. 131, 133, 175 Whitaker S. 39, 51 White W. 312, 332 Whitman R . V. 25, 50, 184, 336, 355 Wilde P. 127, 128, 139, 175 Willis P. G. 8, 15, 18, 50 Wilson E. L. 25, 50, 51, 72, 82, 202, 215
375
Wineman A. S. 175 Witherspoon P. A. 14, 15, 188, 216 Wolf J. P. 312, 333 Wood D. M. see Muir Wood D. Wood W. L. 60, 66, 83,84 Wroth C. P. 1 1 I, 175 Wu J. S. S. 78, 83 Wu J. 317, 325, 333 Wu S. 333 Wu W. 133, 173, 175
Yamada Y. 163, 164, 175 Yamazaki F. 255,302 Yasuda S. 125, 142, 173, 256,267.302 Yasufuku N. 302 Yoshimi Y. 257, 263, 303 Youssif N. B. 158, 172 Yovanovich M. M. 324, 332 Zaremba S. 20, 51 Zelasko J. S. 256, 302 Zeng X. 219, 237,247, 249,250 Z h a n X . Y . 51.83, 188, 189, 190, 195, 196. 198, 199.216 Zhu J. Z. 317, 319,333 Zienkiewicz 0. C. 8, 16, 18, 21,24.25, 26, 27, 36, 51,52, 53, 55, 58, 60, 62, 63, 64, 66. 67. 70, 72, 77, 78, 82, 83, 84, 90, 92, 104, 128, 129, 130, 131, 132, 135, 137, 139, 143, 147. 148, 149, 151, 152, 155, 158, 159, 162, 163. 165, 172, 173, 174, 175, 176, 179, 186, 188. 201, 202, 209, 212, 216, 230, 233, 234, 235, 249, 250, 251, 256, 274, 276, 280, 281, 282. 302, 303, 312, 313, 316, 317, 318, 319, 324, 325, 326, 332, 333, 336, 337, 343, 344, 351. 354, 355
Subject Index
ABAQUS 350 Accoustic emission 143 Accumulated shear strain 255 Accuracy control 3 14 Adaptive refinement 305, 3 14-324 Analysing procedure 233-234 anisotropic material tensor 158 Anisotropy 151, 157-1 63 fabric tensor 159 initial or fabric 158, 159 loadistress induced 151, 158, 159 modified stress invariants - see stress invariants structure tensor 159 transverse isotropic material 160-1 6 1 Autogenous volumetric strain 129, 166 Babuska-Brezzi condition 64, 70, 324 Backward Euler integration scheme 352 Balance equation of fluid and air mass for partially saturated soil 38 of fluid mass for fully saturated soil 22-23 of fluid mass for partially saturated soil 34-35 see ulso general field equation Balance equation for mixture theory macroscopic balance equations 4 2 4 3 microscopic balance equations 42 of energy 42-43 of entropy 4 2 4 3 of mass 4 2 4 3 of momentum 4 2 4 3 Banding sand 145, 154 Bangkok clay 138-1 39 Biot alpha - see effective stress
Biot Theory 18, 256 macroscopic thermodynamical approach 39 Bishop parameter see effective stress Boundary condition 23-24 Bulk modulus of fluid 78 of soil matrix 78 -
capillary pressure 13 Capilliary pressure 40 Centrifuge 6 dynamic compatibility 2 19-22 1 numerical modelling of loose and dense sand bed test 237-245 numerical modelling of retaining dyke test 221-230 numerical modelling of retaining wall test 245-247 numerical modelling of submerged quay wall test 240, 242, 243 numerical modelling of uniform loose sand bed test 237-245 Scaling laws 21 9-22 1 Silicon oil as substitute fluid 221 use of substitute fluid 220-221 validation of numerical solutions 21 8 VELACS study 218, 230-245 Characteristic state line 125, 142, 144, 146. 148, 151 Co-continuous 64 Cohesion due to suction 276 Consistency condition 10&105 Consolidation equation 28, 66, 67 fully and partially saturated soil column 203-206
378 Consolidation equation (contd.) fully and partially saturated two-dimensional soil layer 206-209 small strain and finite deformation 202-203 Terzaghi theory 204 Constitutive equation for gas (Dalton's law) 4 4 4 5 for soil 19, 44 Constitutive models (see ah0 plasticity framework) Al-Tabbaa and Wood 337. 352 alternate model 352 anisotropic Elastic 337, 351 Barcelona unsaturated soil model 352 benchmark tests 1 10 brick model 352 calibration 1 11 Cam-Clay 120-123, 337, 351 CAP model 337. 350 computational aspects 105-1 10 concrete interface model 352 concrete model 352 constitutive model subroutines 349-352 Cristescu visco-elastoplastic model for saturated sand 352 critical state models 120-124 extending to sands 12&129 for normally consolidation clays 111-115 densification models 130 liquefaction parameters 270, 282, 28&285, 296 modified for cyclic mobility 165-1 71, 254 determination of soil parameters 253 Drucker Prager 102-104,113,187-188,337 elastic model with varying moduli 337, 35 1 endochronic theory 129 generalized plasticity model for normally consolidated clays 134-1 39 for overconsolidated clays 139-141 for sands 141-157 Pastor Zienkiewicz mark-111 model 132, 143, 287, 337, 351 parameters for Lower San Fernando dam analysis 276 parameters for N-dam analysis 291; parameters for VELACS study 236 parametric idenfication 235-236 true triaxial path 149-1 51, 164
SUBJECT INDEX
unloading and cyclic loading 151-1 57 for the anisotropic bahaviour of sand 163-165 Hashiguchi subloading surface model 131, 352 hyperbolic stress streain relation for shear 256 implementing new model subroutine into SWANDYNE - see SM2D linear elastic 269, 337, 351 modified Cam-Clay 123-124, 337, 351 Mohr Coulomb model 100-102 applied for LSSGW 269 applied in limit examples 179-1 88 implementation in SWANDYNE 337, 351, 352 lack of plastic strain before yield surface is reached 1 1 3, 11 9-1 20 rounded 104, 143 used in densification model 130, 269 used in densification model for cyclic mobility I65 Pietruszczak two-surface model 352 state parameter based one-dimensional model 352 three-surface kinematic yield surface model 352 Tresca 99-100, 184, 186, 337 two surface kinematic hardening plane strain model for sand 337, 351 used with or without initial shear stress 267 von Mises-Huber 97-99, 184, 186, 337, 352 Constitutive tensor 19-20, 132-1 33, 149-150 inversion 92-93 isotropic linear elastic 25, 91 control volume 43 Cosserat medium 321 Coulomb 85 coupled analysis 4 coupling matrix 59 CRISP 350 critical state line 11 1, 120-122, 136, 144 for sands 124, 128, 148 critical state model Cyclic mobility 129, 153, 165, 166 Damping algorithmic (numerical) damping 62 damping matrices 71, 75-76 ~ a ~ l e damping i ~ h 71, 78
379
SUBJECT INDEX viscous damping 71 Darcy's law, generalized Dashpots 3 1 1 Degree of saturation of air 37 of water 12, 33, 37 Del Monte sand 188 Density fluid 21 solid 21 volume averaged 21 deviatoric stress tensor 95 DIANA-SWANDYNE 11-seeSWANDYNE Dilatancy 122 dilatancy rule for clay 134-135 dilatancy rule for sand 128, 141-142, 155, 163-164 Effective stress path approach 256 negative dilatancy 256 Discontinuity in displacement 3 16 Discrete memory factor 153 Drained Analysis 67-69, 70 Drained behaviour 4 Drucker stability criteria 144 Earth pressure coefficient at rest (KO) 262-263, 265-266 in situ measurement 262 Earthquake El-Centro (I 940) 2 10, 256, 27 1, 298 Hyogoken-Nanbu (1995) 255-256, 257, 258, 269, 273 Kobe (1995) see Hyogoken-Nanbu (1995) Nihonkai Nanseibu (1993) 283 Niigata (1964) 253-254, 287-293 Akita-Kencho record 288-290 Rinkai (1992) artificially generated for Tokyo Bay area 295. 297-298 Taft (1952) 271, 298 USA record 306 Earthquake damage Counter measures deep soil mixing 253, 293-302 lattice-shaped stiffened ground walls (LSSGW) 267-273 Earthquake loading examples earth dam failure in Hokkaido 253-254, 28 1-287 effect of deep soil mixing counter measure 253-254, 293-302 elasto-plastic large strain behaviour 21 1-215
fully saturated soil column 209-21 1, 256, 263, 267, 328 Lower San Fernando dam failure 253-254, 274281, 319 quay wall failure in Kobe City 253-254. 267-273 rock fill dam failure - see Lower San Fernando dam failure soil layer liquefaction at Kobe Port Island 253-254, 254262 soil layer liquefaction during Niigata 1964 earthquake 253-254,262-267 soil structure interaction at Kawagishi-cho during Niigata 1964 earthquake 253-254,287-293 Earthquake motion input 305, 306-315 Earthquake record 66 acceleration records 254 input motion recorded during earthquake 254 Edogawa layer 298 Effective stress 4, 6 Biot alpha 1 1-12, 19, 39 Bishop parameter 40 from mixture theory under the assumption of incompressible grains 3 9 4 0 in fully saturated porous media 6 1 2 , 19 in partially saturated media 12-14, 33 under the assumption of local thermodynamic equilibrium 39 Elastic modulus for generalized plasticity model 137-1 38 one dimensional constrained modulus (or restrained axial modulus) 27, 310 Poisson ratio 27 Young's modulus 27 Equilibrium equation of fluid 22 of mixture 19-21 See ulso general field equations Equilibrium line 131 Error indicator 3 17 Failure mechanism investigation using numerical method 254. 28 1-287 Failures 177-1 78 due to loss of suction 276 dynamic failure I failure behaviour 96-97, 153 local failure 105
SUBJECT INDEX
Failures (contd.) Lower San Fernando dam 1-3 see also earthquake loading examples see ulso earthquake loading examples static failures 1, 4 Vajont 1, 2 Finite Element Quadrilateral Q41P4 324, 327, 330 Quadrilateral Q81P4 330 Quadrilateral Q9/P4 324 Triangular T6CI3C 3 19-320, 322 Finite element discretization 53-55 Finite rotation 20 Fluid compressibility 26 Flushing water 256 Free surface see phreatic surface Fuji river sand 153, 164, 165 -
Galerkin process 54, 326 General field equation Linear momentum balance for fluid phases 4 5 4 6 Linear momentum balance for solid phase 45-46 Mass balance equation 4 6 4 7 Generalized plasticity Geometric non-linearity 209 GLADYS-2E 55. 335 Governing equation fully saturated behaviour with a single pore fluid 19-27 partially saturated behaviour with air flow considered 36-39 partially saturated behaviour with air pressure neglected 3 1-36 Hardening behaviour 9 6 9 7 , 144 anisotropic hardening 158 deviatoric hardening 126-128,139,147,164 isotropic hardening 158 Strain hardening 1 0 4 105 History dependent behaviour 89 Hollow cylinder 150 Hostun Sand 148 Huber 86 Image point 132 Incompressible behaviour 64, 70 Stabilization 305. 32&33 1 operator splitting algorithm 326-328 reason for success 328-331
velocity correction 325-328 undrained condition 3 2 4 3 2 5 Infinite foundation see radiation boundary condition Inhomogeneity 125 Initial matrix method 69 Initial shear stress 253, 262-267. 285 due to Self-weight 262 Type 1 267 Type I1 267 Iteration 63-64 -
Jacobian matrix 63. 66 Jaumann stress rate 336 Jaumann stress rate - see Zaremba-Jaumann stress changes Kinematic equations 41 Kobe City hotel with earthquake damage counter measure 267-268 Ocean Weather Station 271 Port Island 256-257 Laminar box numerical n~odellingof 236 Levy 86 Liquefaction 4, 265, 271 criteria for full liquefaction 267 induced by anisotropy 157 liquefaction strength 255, 270, 282, 284, 295, 297 modelled in densification model 130 modelled in kinematic hardening model 131 modelled in modified densification model 166 modelled under cyclic loading in generalized plasticity model for sand 153 modelled under monotonic loading in generalized plasticity model for sand 144 numerical modelling examples 237-245 see also earthquake loading examples reason for the development of advanced models 129 Loading criteria 90-92, 94 Localized phenomena 3 14-324 non-uniqueness of numerical solution 3 19-324
SUBJECT INDEX
Lode angle 96, 137, 143, 163 modified for anisotropy 162 LSSGW - see earthquake damage counter measures LUSAS 350 Mass conservation - see balance equation Mass lumping 72, 78 Mass matrix 59, 67, 75, 77 diagonal form 64, 326 Mesh bad 319, 322 lucky 319, 322 Mixed interpolation 324-33 1 Mixture Theory 18, 3 9 4 9 MuDIAN 281 Multi-direction loading 253, 254-262 multilinear laws 90 Multi-step methods 60 Neutral criteria 90-92 Newmark method see time stepping scheme Newton Raphson iterative procedure 63, 69, 328 Niigata sand 154 Non-associative flow rule 128 Normal Consolidation Line 113, 120-1 21 relation with Plasticity Index 113 -
Overconsolidation ratio 1 15 partial saturation Partially saturated examples air storage modelling in an aquifer 195-197 consolidation of soil column 203-206 consolidation of two-dimensional soil layer 206-209 elasto-plastic large strain behaviour 21 1-215 flexible footing resting on a partially saturated soil 198-201 one-dimensional column 36, 188-192 subsidence due to pumping from a phreatic aquifer 193-195 Patch test 70, 324 Permeability 65 anisotropic permeability 78 Creager's approach 284, 295 Darcy's Law 22, 37-38
381
effect of permeability on validity of various assumptions 29-3 1 for air flow 37-38,44-45, 189, 195 in partially saturated media 14, 33, 4 4 4 5 , 188-189, 195 Kozeny-Carmen equation 339 See ulso general field equations Phase transformation line 125, 142 Phreatic surface 35, 275 Plastic modulus for anisotropic behaviour of sand 163 for bounding surface model 131-1 32 for critical state model extended to sand 127 for generalized plasticiy framework 91-92 for kinematic hardening models 131 for normally consolidated clay 136 for overconsolidated clay 139 for sand under monotonic loading 143-144 for sand under unloading and cyclic loading 152-1 53, 155-1 56 Plastic potential surface 122 for loose and dense sand 128 Plasticity framework 85-86 (see rrlso constitutive models) bounding surface models 131-1 32 classical plasticity 91, 93-1 10 pressure dependent criteria 100-1 04 pressure independent criteria 97-100 critical state framework 110-129 generalized plasticity 87-92 gradient plasticity 32 1 hypoplasticity and incrementally non-linear models 90, 132-133 kinematic hardening models 130-13 1 multi-surface kinematic hardening model 130-131, 158 multi-laminate 159 phenomenological aspects 86-87 Poncelet 85 Pore water migration 253, 274-281 Power plant 294 Predictions Class A 21 7-218 Class B 2 17-2 18 Class C 217 Profile length minimizer Cuthill-McKee method 341 Sloan method 341 Profile solver 343, 344
382 Quasi-Newton method 69 BFGS 344 Quay wall 267-273 Radiation boundary condition 305, 306-3 15 one-dimensional problem 308-3 12 two-dimensional problem 3 12-3 15 Rankine 85 Reid Sand 150 Relative fluid displacement 26 Repeated boundary condition 3 13 Representative elementary volume 43 Residual condition 143, 147 Roscoe surface 120 Rotation of principal stress axes 150-1 51 Sand boiling 256, 281 saturated, fully Saturation-capillary pressure relationship 188-189, 205 Secant update 63 Shape of yield surface in x-plane 98, 100, 137. 186-188 Shear wave 310 Shock wave 314, 317 Simple shear test apparatus 255 Skempton B soil parameter for pore water pressure 24-25 SM2D 350 example new model subroutine 364-369 implementing new model subroutines 355-364 Softening behaviour 96-97 localization 3 19-324 of sand 147 Soil improvement - see earthquake damage counter measures St. Venant 85 Stability criteria 60, 62 conditional stability 33 1 unconditional stability 66 Stabilization of staggered scheme 64 Staggered procedure - see time stepping Standard penetration test 257 Static analysis 64, 178-188 embankment 179-1 81 footing 179, 181-183 intermediate constraint on deformation 181, I84 small constraint on deformation 179-1 8 1
SUBJECT INDEX
strong constraint - undrained behaviour 182-1 86 Steady state 68-69 Stiffness matrix 63, 7&76 Stress invariants 95, 1 17, 150, 163 computational aspects 105- 1 10 modified stress invariants 158, 160-162, 164 Stress paths anisotropy 110 consolidated drained test 110, 1 1 6 1 19 consolidated undrained test 110, 119-120 conventional triaxial stress path 110, 115-120 modelled by modified densification model 168-1 70 isotropic Compression 11 1 isotropic Consolidation 110, 136 three-dimensional effects 110 unloading, reloading and cyclic loading 110 Stress ratio 124, 134, 163 SWANDYNE 55, 57, 60, 62, 335-337 constitutive model subroutines 349-352 element matrices and residual calculation subroutines 345-347 Internet URL xi major service subroutines 347-349 subroutines for analysis 342-345 subroutines for Control and material data input 338-339 subroutines for mesh data input 339-342 system dependent subroutines 352-353 top level routines 337-338 Tangent stiffness method 69 Three dimensional modelling 257-258, 287-293 Tilted building 253, 287-293 Time step length 66-67 critical time step length for explicit scheme 78 Time stepping scheme central difference scheme 62 error control 67 explicit scheme 55, 62, 64, 72, 77-78, 33 1 Generalized Newmark (GNpj) method 60-62, 77-78, 336 G N 11 scheme 62 GN22 scheme 61
383
SUBJECT INDEX GN32 scheme 61 implicit scheme 55, 66 Newmark method 60, 77. 336 SSpj 60 staggered procedure 64 trapezoidal scheme 62 Tresca 85 Two-phase flow 188 Uncoupled equation 68 Undrained Analysis 64, 67-69, 70, 182 Undrained behaviour 4, 24-25, 27-3 1, 182 .see crlso incompressible behaviour Uniaxial behaviour 86-87 Unloading criteria 90-92, 94 Unloading plasticity 132, 152 u-p formulation 25-27, 55-57 explicit u and implicit p scheme 62 fully Implicit scheme spatial discretization 58-59 structure of numerical equations 69-70 temporal discretization 60-65 tensorial form of the equations 78-81 Updated Lagrangian Formulation 46, 336
u-U formulation 26-27, 71-73, 256 block diagonal structure 78 fully Explicit scheme 72 spatial discretization 73-74 structure of numerical equations 7 4 7 7 u-w formulation u-w-p formulation 19-21. 27 van Genuchten's formula 14, 27C277 Vertical Input motion 259-261, 265 Volume fraction 18 von Mises 86 Wave equation 3 10 Weald clay 140 Westergaard's formula for dynamic water pressure 299 Yield and failure surfaces 9&96 for loose and dense sand 128-129 frequently used criteria 97-1 04 open and closed yield surfaces 1 14 Zaremba-Jaumann stress changes 20
