COMPUTING AND COMPUTER MODELLING INGEOTECHNICAL ENGINEERING

J.P. Carter1, C.S. Desai2, D.M. Potts3, H.F. Schweiger4 and S.W. Sloan5

1.0 ABSTRACTA broad review is presented of the role of computing in geotechnical engineering. Included in the

discussions are the conventional deterministic techniques for numerical modelling, stochastic techniques fordealing with uncertainty, ‘soft-computing’ tools, as well as modern database software for geotechnicalapplications. Considerable emphasis is given to the methods commonly used for the solution of boundaryand initial value problems. Constitutive modelling of soil and rock mass behaviour and material interfaces isan essential component of this type of computing, and so a review of recent developments and capabilities ofconstitutive models is also included. The importance of validating computer simulations and geotechnicalsoftware is emphasised, and some methodologies for achieving this are suggested. A description of severalpreviously conducted validation studies is included. The paper also includes discussion of the limitations ofvarious numerical modelling techniques and some of the more notable pitfalls. The concepts described inthe paper are illustrated with examples taken from research and practice. In presenting these concepts andexamples, emphasis has been placed on the behaviour of soil, but it is noted that many of the models andtechniques described also have application in rock engineering.

2.0 INTRODUCTION

The desire to understand the physical world and to be able to describe it using mathematical concepts andnumbers has long been a goal of scientists and engineers. This desire has been evident since at least the timeof Pythagoras. The discipline of geotechnical engineering is no exception, as first researchers and nowpractitioners routinely make use of mathematical models and computer technology in their day-to-day work,trying to understand and predict their world of geomaterials. For example, simple numerical procedureshave been used for many years in geotechnical practice in the assessment of strength, analysis of soilconsolidation, and estimation of slope stability. With the introduction of electronic computers after WorldWar II came the opportunity for engineers to make much more use of numerical procedures to solve theequations governing their practical problems. This new computing power has made possible the solution ofquite complicated non-linear, time-dependent problems, boundary and initial value problems that were oncetoo tedious and intractable using hand methods of calculation.

The ready availability of desktop and portable computers has meant that these numerical tools for thesolution of boundary and initial value problems are no longer the preserve of academic and researchengineers. With the spectacular improvements in hardware have come major developments in software, andvery sophisticated packages for solving geotechnical problems are now available commercially. Thisincludes a wide range of software for solving problems from the more routine type, such as limitingequilibrium calculations, to the most powerful non-linear finite element analyses.

The availability of this powerful hardware and sophisticated geotechnical software has allowedgeotechnical engineers to examine many problems in much greater depth than was previously possible. In

1 Challis Professor, Department of Civil Engineering, The University of Sydney, Sydney, NSW, AUSTRALIA2 Regents Professor, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson,

Arizona, USA3 Professor of Analytical Soil Mechanics, Department of Civil and Environmental Engineering, Imperial College of

Science, Technology and Medicine, London, UK4 Associate Professor, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology,

Graz, AUSTRIA5 Professor of Civil Engineering, Department of Civil, Surveying and Environmental Engineering, University of

Newcastle, Newcastle, NSW, AUSTRALIA

particular, they have allowed the possibility of using numerical methods to examine the importantmechanisms that control the overall behaviour in many problems. Further, they can be used to identify thekey parameters in any problem, thus indicating areas that require more detailed and thorough investigation.These developments have also meant that generally better quality field and laboratory data are required asinputs to the various models. However, often precise values for some of the input data for these numericalmodels will not be known, but knowledge of which parameters are most important allows judgements to bemade about what additional information should be collected, and where additional resources are bestdirected in any particular problem. Furthermore, knowledge of the key model parameters together with theresults of parametric investigation of a problem can often allow engineering judgements to be made withsome confidence about the consequences of a particular decision, rather than in complete or partialignorance of them. Increasingly, the tools for deterministic modelling of geotechnical problems are beingcoupled with statistical techniques, to provide a means of dealing with uncertainties in the key problemparameters, and of associating probabilities with the predicted response.

The application of computer technology in geotechnical engineering has not been confined to the area ofanalysis and mathematical modelling. The availability of sophisticated information technology tools hasalso had noticeable impact on the way geotechnical engineers record, store, retrieve, process, visualise anddisplay important geotechnical data. These tools range from the ubiquitous spreadsheet packages todatabase software and high-end graphics visualisation tools.

This paper provides a broad review of the role of computing and computer technology in geotechnicalengineering. A major part of the review includes discussion of some of the more common analytical andnumerical modelling techniques available to geotechnical engineers, particularly those used today to obtainsolutions on personal computers. The use of the numerical methods and the associated software is illustratedby applications taken from geotechnical research and practice. Throughout this discussion, emphasis isgiven to the role of these deterministic techniques in identifying mechanisms and providing the user withexplanations of observed behaviour, as well as the consequences of planned actions and proposedconstruction activities. The examples considered involve analysis of strength and deformation of soil, andinclude pre-failure behaviour as well as the development of failure mechanisms within soil bodies. The needto validate and calibrate numerical models, particularly those now being used commonly in practice isemphasised, and some examples of validation studies are described. Some of the potential pitfalls ofnumerical analysis are also described.

Included in the paper is a brief review of the stochastic and ‘soft computing’ tools that are increasinglybeing applied in practice to model geotechnical problems. In particular, an application of the artificial neuralnetwork technique is described. The role of database packages in geotechnical practice is also illustrated bya recent application on a large-scale construction project.

3.0 GEOTECHNICAL DESIGN

Traditionally, geotechnical design has been carried out using simple analysis or empirical approaches.The ready availability of inexpensive, but sophisticated, computer hardware and software has lead toconsiderable advances in analysis and design of geotechnical structures, with much progress made recentlyin the application of the modelling techniques to geotechnical structures.

In common with other branches of engineering, the design objectives in geotechnics may be identified asfollows:• Local stability of the structure and its support system as well as overall stability should be ensured.• The induced movements must be tolerable, not only for the structure being designed but also for any

neighbouring structures and services.The design process usually consists of some form of assessment of these important aspects of ground

behaviour, and often this will include calculations to provide estimates of stability and the deformationsresulting from the proposed works. Analysis therefore provides the mathematical framework for thesecalculations, and almost invariably the analytical tools are embodied in computer software allowingnumerical analysis to be efficiently and conveniently executed.

However, it is important to recognise that geotechnical design involves much more than just analysis, andit often includes data gathering prior to analysis, as well as observation and monitoring during and followingconstruction. Construction issues may also be significant and should be included in the decision makingprocess that constitutes geotechnical design.

The analytical tools used most often in the geotechnical design process are still those based ondeterministic analysis. It is therefore appropriate to begin a review of computing in geotechnicalengineering by discussing the various deterministic analysis methods. However, as there is growingdevelopment and use of computing tools that can deal with imprecise information, e.g., stochastic methods,neural networks and fuzzy logic, these will also be discussed later in this paper. As already mentioned, datagathering and processing is also important in geotechnical design, and so computer tools to assist with thesetasks are also described.

4.0 DETERMINISTIC GEOTECHNICAL ANALYSIS

Usually, geotechnical analysis involves the solution of a boundary value or initial value problem andmost often this is achieved by some form of numerical solution procedure. In all geotechnical applicationsinvolving numerical analysis it is essential, for economic computer processing and to obtain a reliablenumerical solution, to provide a good model of the physical problem. In finite difference and finite elementanalysis, for example, this normally involves at least two distinct phases, i.e.,• idealisation, and• discretisation.

Idealisation is achieved by breaking down the physical problem into its component parts, e.g., continuumcomponents such as elastic regions, and discrete structural components such as beams, columns and plates.At this stage of the modellingprocess, reliable knowledge ofthe site geology is of paramountimportance. In addition, variousconstitutive models that will beemployed in the analysis must bedetermined at this stage. Thefinal subdivision of the problemdomain should be only asdetailed as is necessary for thepurpose at hand. Too muchdetail only clutters the analysisand may cloud important aspectsof the behaviour. The choice ofan adequate level of detail is amatter for experience andjudgement.

In a finite element procedure,for example, the idealised modelis then further subdivided ordiscretised using an appropriatesubdivision of elements. Theaim here is usually to satisfy thegoverning equations of theproblem separately within eachelement. Details, such as node and element numbers, will be assigned as part of this subdivision process.Each of these phases is illustrated schematically in Figure 1.

As indicated previously, most geotechnical analysis involves an assessment of stability and deformation.Once the problem has been idealised, there are four fundamental conditions that should then be satisfied bythe solution of the boundary or initial value problem. These are:• equilibrium,• compatibility,• constitutive behaviour, and• boundary and initial conditions.Unless all four conditions are satisfied (either exactly or approximately), the solution of the ideal problem isnot rigorous in the mathematical sense.

Material 1

Material 2

Fixed base

Smooth, rigidboundary

Fill

Sand

Clay

Rock

Fill

Sand

Clay

Rock

(a) Physical problem

(b) Idealisation

Nodal points

Elements

Lc

(c) Discretisation

Figure 1 : Steps in the modelling process

5.0 METHODS OF ANALYSIS

Some of the most common methods of analysis used in geotechnical engineering to solve boundary valueproblems are listed in Table 1. Included are numerical methods as well as some more traditional techniquesthat may be amenable to hand calculation. The numerical methods may be classified as follows:• the finite difference method (FDM),• the finite element method (FEM),• the boundary element method (BEM), and• the discrete element method (DEM).

Table 1 also provides an indication of whether the four basic requirements of a mathematically rigoroussolution are satisfied by each of these techniques. It is clear that only the elastoplastic analyses are capableof providing a complete solution while also satisfying (sometimes approximately) all four solutionrequirements. The difficulty of obtaining closed-form elastoplastic solutions for practical problems meansthat numerical methods are the only generally applicable techniques.

Detailed descriptions of each of the numerical methods listed in Table 1 may be found in a large numberof textbooks (e.g., Zienkiewicz, 1967; Desai and Abel, 1972; Britto and Gunn, 1987; Smith and Griffiths,1988; Beer and Watson, 1992; Potts and Zdravkovic, 1999, 2000), so there is no need to duplicate such

detail here. However, it is probably worth noting that the FDM, FEM and DEM methods consider the entireregion under investigation, breaking it up, or discretising it, into a finite number of sub-regions or elements.The governing equations of the problem are applied separately and approximately within each of theseelements, translating the governing differential equations into matrix equations for each element.

Table 1. Summary of common analysis methods

Method of Analysis

Limit Bound Theorems Elastic Elastoplastic Analysis

Equilibrium Lower Upper Analysis Closed-form Numerical

EquilibriumOverall 4Locally 6

4 6 4 4 4

Compatibility 6 6 4 4 4 4 (1)

BoundaryConditions

Force only Force only Displacementonly

4 4 4

ConstitutiveModel

Failure criterion Perfectly rigid plasticity Elastic Elastoplastic Any (2)

CollapseInformation

4 4 4 6 4 4

Informationbefore Collapse

6 6 6 4 4 4

Comment SimpleSafe or unsafe?

Safe estimateof collapse

Unsafe estimateof collapse

Closed formsolutionsavailable

Complicated Powerfulcomputertechniques

ExamplesSlip circle,

WedgeMethods

- - Many Limited FDM, FEM,BEM and

DEM

(1) Inherent and induced material discontinuities can be simulated.(2) Includes perfect plasticity and models that can allow for complicated behaviour such as

discontinuous deformations, degradation (softening), and non-local effects.

Compatibility, equilibrium and the boundary conditions are enforced at the interfaces between elements andat the boundaries of the problem.

On the other hand, in the BEM only the boundary of the body under consideration is discretised, thusproviding a computational efficiency by reducing the dimensions of the problem by one. The BEM isparticularly suited to linear problems. For this reason, and because it is well suited to modelling infinite orsemi-infinite domains, the BEM is sometimes combined with the finite element technique. In this case, aproblem involving non-linear behaviour in part of the infinite domain can be efficiently modelled by usingfinite elements to represent that part of the domain in which non-linear behaviour is likely, while alsomodelling accurately the infinite region by using boundary elements to represent the far field.

For all methods, an approximate set of matrix equations may be assembled for all elements in the regionconsidered. This usually requires storage of large systems of matrix equations, and the technique is knownas the “implicit” solution method. An alternative method, known as the “explicit” solution scheme, is alsoemployed in some software packages. This method usually involves solution of the full dynamic equationsof motion, even for problems that are essentially static or quasi-static. The explicit methods do not requirethe storage of large systems of equations, but they are known to present difficulties in determining reliablesolutions to some problems in statics.

In the following, brief details of each of the main methods of numerical analysis are presented togetherwith a summary of their main advantages and disadvantages, and their suitability for various geotechnicalproblems. Much of this discussion is based on suggestions proposed by Schweiger and Beer (1996).

5.1 Finite Element Method

The finite element method is still the most widely used and probably the most versatile method foranalysing boundary value problems in geotechnical engineering. The main advantages and disadvantagesfor geotechnical analysis may be summarized as follows.

5.1.1 Advantages

• nonlinear material behaviour can be considered for the entire domain analysed.• modelling of excavation sequences including the installation of reinforcement and structural support

systems is possible.• structural features in the soil or rock mass, such as closely spaced parallel sets of joints or fissures, can

be efficiently modelled, e.g., by applying a suitable homogenisation technique.• time-dependent material behaviour may be introduced.• the equation system is symmetric (except for non-associated flow rules in elasto-plastic problems using

tangent stiffness methods).• the conventional displacement formulation may be used for most load-path analyses.• special formulations are now available for other types of geotechnical problem, e.g., seepage analysis,

and the bound theorem solutions in plasticity theory.• the method has been extensively applied to solve practical problems and thus a lot of experience is

already available.

5.1.2 Disadvantages

The following disadvantages are particularly pronounced for 3-D analyses and are less relevant for 2-Dmodels.• the entire volume of the domain analysed has to be discretised, i.e., large pre- and post-processing

efforts are required.• due to large equation systems, run times and disk storage requirements may be excessive (depending on

the general structure and the implemented algorithms of the finite element code).• sophisticated algorithms are needed for strain hardening and softening constitutive models.• the method is generally not suitable for highly jointed rocks or highly fissured soils when these defects

are randomly distributed and dominate the mechanical behaviour.

5.2 Boundary Element Method

Significant advances have been made in the development of the boundary element method and as aconsequence this technique provides an alternative to the finite element method under certain circumstances,particularly for some problems in rock engineering (Beer and Watson, 1992). The main advantages anddisadvantages may be summarized as follows.

5.2.1 Advantages

• pre- and post-processing efforts are reduced by an order of magnitude (as a result of surfacediscretisation rather than volume discretisation).

• the surface discretisation leads to smaller equation systems and less disk storage requirements, thuscomputation time is generally decreased.

• distinct structural features such as faults and interfaces located in arbitrary positions can be modelledvery efficiently, and the nonlinear behaviour of the fault can be readily included in the analysis (e.g.,Beer, 1995).

5.2.2 Disadvantages

• except for interfaces and discontinuities, only elastic material behaviour can be considered with surfacediscretisation.

• in general, non-symmetric and often fully-populated equation systems are obtained.• a detailed modelling of excavation sequences and support measures is practically impossible.• the standard formulation is not suitable for highly jointed rocks when the joints are randomly distributed.• the method has only been used for solving a limited class of problems, e.g., tunnelling problems, and

thus less experience is available than with finite element models.

5.3 Coupled Finite Element - Boundary Element Method

It follows from the arguments given above that it should be possible to minimise the respectivedisadvantages of both methods by combining them. This is in fact true and very efficient numerical modelscan be obtained by discretising the soil or rock around the region of particular interest, e.g., representing theregion around a tunnel by finite elements and the far field by boundary elements (e.g., Beer and Watson,1992; Carter and Xiao, 1993). Two disadvantages however remain, namely the cumbersome modelling ofmajor discontinuities intercepting the region of interest in an arbitrary direction, e.g., a tunnel axis, and thenon-symmetric equation system that is generated by the combined model. The latter problem may beresolved by applying the principle of minimum potential energy for establishing the stiffness matrix of theboundary element region (Beer and Watson, 1992). If this is done, then after assembling with the finiteelement stiffness matrix, the resulting equation system remains symmetric.

5.4 Explicit Finite Difference Method

The finite difference method does not have a long-standing tradition in geotechnical engineering, perhapswith the exception of analysing flow problems including those involving consolidation and contaminanttransport. However, with the development of the finite difference code FLAC (Cundall and Board, 1988),which is based on an explicit time marching scheme using the full dynamic equations of motion, even forstatic problems, an attractive alternative to the finite element method was introduced. Any disturbance ofequilibrium is propagated at a material dependent rate. This scheme is conditionally stable and small timesteps must be used to prevent propagation of information beyond neighbouring calculation points within onetime step. Artificial nodal damping is introduced for solving static problems in FLAC. The method iscomparable to the finite element method (using constant strain triangles) and therefore some of thearguments listed above basically hold for the finite difference method as well. However, due to the explicitalgorithm employed some additional advantages and disadvantages may be identified.

5.4.1 Advantages

• the explicit solution method avoids the solution of large sets of equations.

• large strain plasticity, strain hardening and softening models and soil-structure interaction are generallyeasier to introduce than in finite elements.

• the model preparation for simple problems is very easy.

5.4.2 Disadvantages

• the method is less efficient for linear or moderately nonlinear problems.• until recently, model preparation for complex 3-D structures has not been particularly efficient because

sophisticated pre-processing tools have not been as readily available, compared to finite element pre-processors.

• because the method is based on Newton’s law of motion no “converged solution” for static problemsexists, as is the case in static finite element analysis. The decision whether or not sufficient time stepshave been performed to obtain a solution (below but close to failure) has to be taken by the user, and thisjudgement may not always be easy under certain circumstances, although several checks are possible(e.g., unbalanced forces, velocity field).

5.5 Discrete Element Method

The methods described so far are based on continuum mechanics principles and are therefore restricted toproblems where the mechanical behaviour is not governed to a large extent by the effects of joints andcracks. If this is the case discrete element methods are much better suited for numerical solution. Thesemethods may be characterised as follows:• finite deformations and rotations of discrete blocks (deformable or rigid) are calculated.• blocks that are originally connected may separate during the analysis.• new contacts which develop between blocks due to displacements and rotations are detected

automatically.Several different approaches to achieve these criteria have been developed, with probably the most

commonly used methods being the discrete element codes UDEC and 3-DEC (Lemos et al., 1985), whichboth employ an explicit finite difference scheme, as in the program FLAC.

Due to the different nature of a discontinuum analysis, as compared to continuum techniques, a directcomparison seems to be not appropriate. The major strength of the distinct element method is certainly thefact that a large number of irregular joints can be taken into account in a physically rational way. Thedrawbacks associated with the technique are that establishing the model, taking into account all relevantconstruction stages, is still very time consuming, at least for 3-D analyses. In addition, a lot of experience isnecessary in determining the most appropriate values of input parameters such as joint stiffnesses. Thesevalues are not always available from experiments and specification of inappropriate values for thesep ar amet e rs ma y le ad to c o mp ut at i on al pr ob le ms. I n ad di t io n, ru n ti mes f o r 3- D a na ly s es a re us ua l ly q ui t e hi g h.

5.6 Which Method For Which Problem?

Having discussed the main advantages and disadvantages of the most common numerical methods areasonable question is which method should be used for any particular problem. Of course, the answer tothis question will be very problem dependent. In many cases several methods may be appropriate and thedecision on which to use will be made simply on the basis of the experience and familiarity of the analystwith these techniques.

However, it is possible in certain classes of problem to provide broad guidelines. One example concernsthe analysis of tunnels. This problem area has been addressed by Schweiger and Beer (1996), whosuggested guidelines for tunnelling by considering the separate problems of shallow and deep tunnels. Inmaking suggestions they discussed the importance of the input parameters and what can be expected fromnumerical analyses. For example, they concluded that the finite element and finite difference method aremost suitable for shallow tunnels in soil, whereas the boundary element and discrete element method aremost suitable for deep tunnels in jointed and faulted rocks. They also concluded that the most promisingcontinuum approach is a combination of finite elements and boundary elements, because the merits of eachmethod can be fully exploited as appropriate.

In the following sections of this paper some of the methods of numerical analysis described above andsome of their essential components are described further. Their use is illustrated by applications fromgeotechnical practice. As indicated in Table 1, not all techniques are designed to provide a solution for thecomplete load-deflection behaviour of a soil or rock structure up to the point of collapse. Some address onlythe pre-failure behaviour and some the ultimate condition.

The first method to be described in detail involves the use of the bound theorems of classical plasticitytheory together with special finite element formulations to assess stability. These well-known methods forbracketing the true collapse load do not require a sophisticated constitutive model for the soil or rock mass.They are based on the assumption of a rigid plastic model for the soil or rock and require only the definitionof a failure criterion and a plastic flow rule. With recent developments in computer technology, linear andnon-linear programming and the finite element method, these classical theorems have a renewed andimportant role to play in geotechnical analysis, particularly as they may now be used routinely for three-dimensional problems. After all, determining when collapse will occur, and avoiding that condition inpractice, is one of the fundamental requirements of geotechnical design.

6.0 LIMIT ANALYSIS USING FINITE ELEMENTS

Stability analysis in geotechnical engineering has traditionally been carried out using either slip–line fieldor limit equilibrium techniques. Whilst slip–line methods have the advantage of being mathematicallyrigorous, they are notoriously difficult to apply to problems with complex geometries or complicatedloading. A further shortcoming of these techniques is that the boundary conditions need to be treatedspecifically for each problem, thus making it difficult to develop general purpose computer programs whichcan analyse a broad range of cases. Despite these limitations, slip–line analysis has provided manyfundamental solutions that are used routinely in geotechnical engineering practice (Sokolovskii, 1965).Limit equilibrium methods, although less rigorous than slip–line methods, can be generalised to deal with avariety of complicated boundary conditions, soil properties and loading conditions. The accuracy of limitequilibrium solutions is often questioned because of the assumptions that are needed to make the methodwork. Nonetheless, this approach is often favoured by practising engineers because of its simplicity andgenerality.

Another approach for analysing the stability of geotechnical structures is to use the upper and lowerbound limit theorems developed by Drucker et al. (1952). These theorems can be used to bracket the exactultimate load from above and below and are based, respectively, on the notions of a kinematically admissiblevelocity field and a statically admissible stress field. A kinematically admissible velocity field is simply afailure mechanism in which the velocities (displacement increments) satisfy both the flow rule and thevelocity boundary conditions, whilst a statically admissible stress field is one where the stresses satisfyequilibrium, the stress boundary conditions, and the yield criterion.

The bound theorems assume the material is perfectly plastic and obeys an associated flow rule. The latterassumption, which implies the strain increments are normal to the yield surface, is often perceived to be ashortcoming for frictional soils as it predicts excessive dilation upon shear failure. For geotechnicalproblems which are not strongly constrained in a kinematic sense (e.g., those with a freely deforming surfaceand a semi–infinite domain), the use of an associated flow rule for frictional soils may in fact give goodestimates of the collapse load. This important result is discussed at length by Davis (1969) and has beenconfirmed in a number of finite element studies (e.g., Sloan, 1981).

Put simply, the upper bound theorem states that the power expended by the external forces may beequated to the power dissipated in a kinematically admissible failure mechanism to compute anunconservative estimate of the true collapse load. In geotechnical engineering, the simplest form of upperbound calculation is based on a mechanism comprised of rigid blocks where power is dissipated solely at theinterfaces between adjacent blocks. Once a kinematically admissible mechanism has been formulated, thebest upper bound is found by optimising the geometry of the blocks to yield the minimum dissipated powerand, hence, the corresponding collapse load. This type of calculation is very useful for undrained stabilityanalysis of clays where the soil can be modelled using a Tresca yield condition and deformation occurs atconstant volume. For drained loading, however, where the soil is often assumed to obey a Mohr–Coulombfailure criterion, this type of computation is more difficult because of the dilation that accompanies plasticshearing along the discontinuities.

The lower bound theorem states that the collapse load obtained from any statically admissible stress fieldwill under–estimate the true collapse load. Generally speaking, the upper bound theorem is applied morefrequently than the lower bound theorem to predict soil behaviour, since it is usually easier to construct agood kinematically admissible failure mechanism than it is to construct a good statically admissible stressfield. Although an upper bound solution often gives a useful estimate of the ultimate load, a lower boundsolution is more desirable in engineering practice as it results in a safe design.

The bound theorems are especially powerful when both types of solution can be computed so that theactual collapse load can be bracketed from above and below. This feature is invaluable when an exactsolution cannot be determined, since it provides a built–in error check on the accuracy of the approximatecollapse load.

6.1 Lower Bound Limit Analysis Formulations

The use of finite elements and linear programming to compute rigorous lower bounds fortwo–dimensional stability problems appears to have been first proposed by Lysmer (1970). Lysmer’sformulation is based on a simple three–noded triangular element with the nodal normal and shear stressesbeing taken as the problem variables. Following previous studies, the linearised yield condition is obtainedby adopting an internal polyhedral approximation to the parent yield surface, so that each nonlinearinequality is replaced by a series of linear inequalities. By assuming a linear approximation for the stressfield inside each element, it can be guaranteed that this yield condition is satisfied throughout the discretisedregion. The presence of statically admissible discontinuities, which are permitted between adjacentelements, greatly improves the accuracy of the final results. Application of the stress–boundary conditions,equilibrium equations, and linearised yield criterion generates the linear constraints on the stress field, whilethe objective function, which is maximised, corresponds to the collapse load. In Lysmer’s originalformulation, the optimal solution to the linear programming problem, and hence the statically admissiblestress field, was isolated by using the simplex algorithm.

Though Lysmer’s method represented a significant advance, it had a number of shortcomings. The firstof these resulted from the choice of problem variables which, although ingenious, led to a poorlyconditioned constraint matrix. The second limitation was one of computational efficiency. With thesimplex solution technique used by Lysmer, the analyses had to be restricted to meshes with only a fewelements in order to avoid excessive computer times. This is a serious limitation and probably explains whythe technique did not achieve the prominence that it deserved. The third and final shortcoming of themethod was its inability to generate a complete stress field for a semi–infinite continuum. Semi–infinite soilmasses arise frequently in geotechnical engineering, and it is necessary to extend the stress field throughoutthe entire domain for the solution to be classed as a rigorous lower bound. In Lysmer’s method, the solutionobtained is an incomplete one, as the resultant stress field is statically admissible only in the region limitedby the boundaries of the finite element mesh.

Following Lysmer, other investigators, including Anderheggen and Knopfel (1972), Pastor (1978), andBottero et al. (1980), proposed alternative two–dimensional lower bound techniques which are based on thelinear programming method. These studies led to a number of key improvements, such as the developmentof extension elements for analysing semi–infinite media and the use of cartesian stresses as problemvariables to simplify the formulation. In 1982, Pastor and Turgeman generalised their lower boundtechnique to deal with the important case of axisymmetric loading.

Although potentially powerful, these early lower bound formulations were restricted by the performanceof their underlying linear programming (LP) solvers. Because the techniques often lead to very largeoptimisation problems, their evolution has been linked closely with the development of efficient LPalgorithms. Indeed, special features of the lower bound formulation, such as the extreme sparsity of theoverall constraint matrix, must be exploited fully to avoid excessive computation times. In the late eighties,Sloan (1988a) introduced a formulation based on an active set algorithm that permits large two–dimensionalproblems to be solved efficiently on a PC or workstation. Practical applications of this scheme to dateinclude tunnels (Sloan et al., 1990; Assadi and Sloan, 1991; Sloan and Assadi, 1991, 1992), foundations(Utrichton et al., 1998; Merifield et al., 1999) and slopes (Yu et al., 1998).

Despite the success of LP based lower bound formulations for the solution of two–dimensional andaxisymmetric problems, these approaches are unsuitable for performing general three–dimensional limit

analysis. Although possible, the linearisation process for 3D yield functions inevitably generates hugenumbers of linear inequalities. This in turn, results in unacceptably long solution times for any LP solverwhich conducts a vertex–to–vertex search (such as the traditional simplex method or any active set method).One alternative for finite element formulations of the lower bound theorem is to use nonlinear programming(NLP) solution methods. Irrespective of the approximation chosen to represent the stress field in thecontinuum, this approach assumes that the yield criterion is employed in its original nonlinear form.Three–dimensional stress fields thus present no special difficulty, apart from the extra variables involved.One such formulation, which uses linear stress finite elements, incorporates nonlinear yield conditionsexplicitly, and exploits the underlying convexity of the corresponding optimisation problem, has beendeveloped recently by Lyamin and Sloan (1997, 2000a) and Lyamin (1999). In this method, the lowerbound solution is found very efficiently by solving the system of nonlinear equations that define theKuhn–Tucker optimality conditions. The solver used for this purpose, a variant of the one originallydeveloped by Zouain et al. (1993) in their study of mixed limit analysis formulations, is a two–stagequasi–Newton scheme. After accounting for the nature of the lower bound optimisation problem anddeveloping a new deflection strategy in the solution phase, the resulting optimisation procedure is manytimes faster than an equivalent LP formulation. Indeed, comparisons to date suggest that the new techniquetypically offers at least a 50–fold reduction in CPU time for large scale two–dimensional applications. Thescheme can deal with any (convex) type of yield criterion and permits optimisation with respect to surface aswell as body forces, which can be of unknown distribution. Because of its speed, the NLP formulation ofLyamin and Sloan (1997, 2000a) is ideally suited to three–dimensional applications, where the number ofunknowns is usually very large.

6.2 Upper Bound Limit Analysis Formulations

General formulations of the upper bound theorem, based on finite elements and linear programming,have been investigated by Anderheggen and Knopfel (1972), Maier et al. (1972) and Bottero et al. (1980).These methods permit plastic deformation to occur throughout the continuum and inherit all of theadvantages of the finite element technique, but have tended to be computationally cumbersome due to thelarge linear programming problems they generate. In the plate studies performed by Anderheggen andKnopfel (1972), an attempt was made to address this shortcoming by suggesting various solution strategiesbased on the revised simplex optimisation algorithm. This type of algorithm was also used by Bottero et al.(1980), who generalised the method of Anderheggen and Knopfel (1972) to include velocity discontinuitiesin plane strain limit analysis. In their formulation, a three–noded constant strain triangular element is usedto model the velocity field, each node has two unknown velocities, and each triangular element is associatedwith a specified number of unknown plastic multiplier rates (one for each side of the linearised yieldsurface). To be kinematically admissible, the velocities and plastic multiplier rates are subject to a set oflinear constraints arising from the flow rule and the velocities must match the appropriate boundaryconditions. For a given set of prescribed velocities, the finite element formulation works by choosing the setof velocities and plastic multiplier rates which minimise the dissipated power. This power is then equated tothe power dissipated by the external loads to yield a strict upper bound on the true limit load. To ensure thatthe finite element formulation leads to a LP problem, the actual yield surface is linearised by using anexternal polyhedral approximation.

Turgeman and Pastor (1982), in an important generalisation, extended their linear programming schemeto handle axisymmetric problems for the case of a Von Mises or Tresca material. Although significant, theformulations of Bottero et al. (1980) and Turgeman and Pastor (1982) both suffer from the disadvantage thatthe direction of shearing must be specified for each discontinuity a priori. This precludes the use of a largenumber of discontinuities in an arbitrary arrangement, since it is generally not possible to determine thesedirections so that the mode of failure is kinematically acceptable. The revised simplex optimisationprocedure used by Bottero et al. (1980) also appears to be rather slow and, indeed, they suggested that amore efficient solution strategy needed to be found.

One effective means of solving large, sparse LP problems is the steepest edge active set scheme (Sloan,1988b). Although it was originally proposed for the solution of problems arising from the finite elementlower bound method (Sloan, 1988a), this algorithm has also proved efficient for upper bound analysis(Sloan, 1989). This is because the form of the dual upper bound LP problem is very similar to the form of

the lower bound LP problem, with more rows than columns in the constraint matrix and all of the variablesunrestricted in sign.

All LP formulations mentioned so far are based on the three–noded triangular element. With this simpleelement it is necessary to use a special grid arrangement, in which four triangles are coalesced to form aquadrilateral with the central node lying at the intersection of the diagonals. If this pattern is not used, thenthe elements cannot provide a sufficient number of degrees of freedom to satisfy the incompressibilitycondition, as discussed in detail by Nagtegaal et al. (1974). In response to this shortcoming, Yu et al. (1994)developed a linear strain element for upper bound limit analysis. A major advantage of using this element,rather than a constant strain one, is that the velocity field can be modelled accurately with fewer elementsand the incompressibility condition can be satisfied without resorting to a restrictive grid arrangement.

More recently, the upper bound formulation of Sloan (1989) was generalised by Sloan and Kleeman(1995) to allow a large number of discontinuities in the velocity field. In the latter formulation, a velocitydiscontinuity may occur at any edge that is shared by two adjacent triangles, and the sign of shearing ischosen automatically during the optimisation process to give the least amount of dissipated power. Eachdiscontinuity is defined by four nodes and requires four unknowns to describe the tangential velocity jumpsalong its length. Although it is still based on a LP solver, the Sloan and Kleeman (1995) formulation iscomputationally efficient for two–dimensional applications and gives good estimates of the true limit loadwith a relatively coarse mesh. Moreover, it does not require the elements to be arranged in a special patternin order to model the incompressibility condition satisfactorily.

As the finite element formulation of the upper bound theorem is inherently nonlinear, a number of upperbound formulations have been based on nonlinear programming (NLP) methods. Early studies focusedmainly on plates and shells and include the work of Hodge and Belytschko (1968), Biron and Chasleux(1972), and Nguyen et al. (1977). The most commonly used solution algorithm in these formulations is thesequential unconstrained minimisation technique (McCormick and Fiacco, 1963) with a Carroll penaltyfunction. Unfortunately, this scheme is not computationally attractive for large scale problems and fewpractical applications have been considered in the literature. Another drawback, which also persists in morerecent upper bound formulations, such as those of Huh and Yang (1991) and Capsoni and Corradi (1997), isthat these formulations can only be used with a limited variety of yield functions.

In a completely different vein, Jiang (1994) proposed an upper bound formulation based onvisco–plasticity theory. Following the practice of others, Jiang (1994) employed an augmented Lagrangianmethod to solve the resulting nonlinear optimisation problem, but implemented it in conjunction with thealgorithm of Uzawa. One year later, Jiang (1995) demonstrated that the same nonlinear programmingscheme can be applied to direct upper bound analysis. Despite its good performance for two–dimensionalexamples, Jiang’s formulation has not been extended to deal with three–dimensional problems. Moreimportantly, because the method does not permit discontinuities in the velocity field, it is unlikely to giveaccurate results for cohesive–frictional materials.

A straightforward method for evaluating the ultimate loads of structures has been suggested recently byde Buhan and Maghous (1995). This scheme is based upon the kinematic approach of yield design theoryand leads to the problem of minimising a function of a finite number of variables without constraints. Intheir implementation, the optimisation procedure was carried out by means of the simplex method of Nelderand Mead (1965). Though this formulation was claimed to be computationally efficient, it needed severalminutes of CPU time just to solve a plane strain problem with a grid of around one hundred triangles. Thesetimings suggest the technique would have difficulty in coping with large scale three–dimensionalgeometries, where the number of unknowns is typically much greater than two–dimensional cases.

In the same year as the work of de Buhan and Maghous, Liu et al. (1995) proposed a method forperforming three–dimensional upper bound limit analysis which uses a direct iterative algorithm. The basiccharacteristic of this algorithm is that, at each iteration, the rigid zones are distinguished from the plasticzones and are constrained accordingly. This involves modifying the goal function and the constraintconditions and neatly avoids the numerical difficulties that are caused by undetermined rigid zones and anundifferentiable objective function. In essence, the problem is reduced to one of solving a series of relevantelastic problems. According to their paper the process is efficient, numerically stable, and can beimplemented easily in an existing displacement finite element code.

A new two– and three–dimensional upper bound formulation, which is based on nonlinear programming,has very recently been proposed by Lyamin and Sloan (2000b). This scheme uses a similar solver to theirlower bound method (Lyamin and Sloan 1997, 2000a) and employs nodal velocities, element stresses, and a

set of discontinuity variables as the unknowns. Over each element, the velocity field is assumed to varylinearly while the stresses are assumed constant. As in the formulation of Sloan and Kleeman (1995),velocity discontinuities are permitted at shared edges between two elements, and the sign of shearing ischosen automatically to minimise the dissipated power. The upper bound solution is found by using atwo–stage quasi–Newton scheme to solve the system of nonlinear equations that define the Kuhn–Tuckeroptimality conditions. This strategy is very efficient, with preliminary comparisons suggesting a 100–foldspeedup over an equivalent linear programming formulation for large scale two–dimensional applications.The scheme can deal with any (convex) type of yield criterion and permits optimisation with respect to bothsurface and body forces. Because of its speed, the NLP formulation of Lyamin and Sloan (2000b) is wellsuited to three–dimensional applications.

6.3 Applications

Some typical soil stability problems are now studied to demonstrate the efficiency of the upper and lowerbound formulations of Lyamin and Sloan (1997, 2000a, 2000b). Three different meshes have beengenerated for each of the examples and these can be treated as coarse, medium and fine models of theoriginal problem. All runs use linear elements and the timings are for a Dell Precision 220 PC with aPentium III 800MHz processor.

6.3.1 Rigid Strip Footing On Cohesive–Frictional Soil

The exact collapse pressure for a rigid strip footing on a weightless cohesive–frictional soil with nosurcharge is given by the Prandtl (1920) solution:

( ) φφπφπ ′

−

′

+′=′

cot124

tantanexp 2

c

q (1)

where c′ and φ′ are, respectively, the effective cohesion and the effective friction angle. For a soil with afriction angle of φ′ = 35° this equation gives q/c′ = 46.14. The boundary conditions and material propertiesused in the lower and upper bound analyses, together with some typical meshes, are shown in Figure 2 andFigure 3.

Figure 3 : Upper bound mesh for strip footingFigure 2 : Lower bound mesh for strip footing

The lower bound results presented in Table 2 compare the performance of a recent nonlinearprogramming formulation (Lyamin and Sloan 1997, 2000a) with that of an equivalent linear programmingformulation (Sloan 1988a, 1988b). The newer procedure demonstrates fast convergence to the optimumsolution and, most importantly, the number of iterations required is essentially independent of the problemsize. For the coarsest mesh, the nonlinear formulation is nearly four times faster than the linearprogramming formulation and gives a lower bound limit load that is 2% better. For the finest mesh thespeed advantage of the nonlinear procedure is more dramatic, with a 54–fold reduction in the CPU time. Inthis case, the lower bound limit load is 1.3% below the exact limit load and the analysis uses just 3.6 secondsof CPU time. Because the number of iterations with the new algorithm is essentially constant for all cases,its CPU time grows only linearly with the problem size. This is in contrast to the linear programmingformulation, where the iterations and CPU time grow at a much faster rate, and the CPU time savings fromthe nonlinear method become larger as the problem size increases.

The upper bound results presented in Table 3 compare the performance of a new nonlinear programmingformulation (Lyamin and Sloan, 2000b) with that of an equivalent linear programming formulation (Sloan1989, 1988b; Sloan and Kleeman, 1995). As in the lower bound analyses, the new procedure demonstratesfast convergence to the optimum solution with an iteration count that is essentially independent of theproblem size. For the coarsest mesh, the nonlinear formulation is nearly four times faster than the linearprogramming formulation and gives an upper bound limit load which is 1% better. For the finest mesh thespeed advantage of the nonlinear procedure is more dramatic, with a 155 fold reduction in the CPU time. Inthis case, the upper bound limit load is 2.5% above the exact limit load and the analysis uses just 13.6seconds of CPU time.

Comparing the results in Table 2 and Table 3 it may be seen that, for the fine meshes, the nonlinearprogramming formulations bracket the true collapse load to within 3.8%. The solution times required toachieve this level of accuracy are modest, and the methods can be run easily on a desktop PC. For a similar

Table 2. Lower bounds for smooth strip footing on weightless cohesive–frictional soil.

Mesh Linear programmingNSID=24

Nonlinear programming Ratio of LP/NLPvalues

q/c′ Error*

(%)

CPU(Sec)

Iter. q/c′ Error*

(%)

CPU(Sec)

Iter. Collapsepressure

CPU Iter.

coarse 36.70 -20.0 1.19 288 37.64 -18.4 0.31 21 0.98 3.8 13.7

medium 42.52 -7.8 25.9 1845 43.79 -5.1 1.72 33 0.97 15 56

fine 43.99 -4.6 197 5552 45.53 -1.3 3.63 29 0.97 54 191

* With respect to exact Prandtl (1920) solution qexact = 46.12c′NSID = number of sides in linearised yield surface.

Table 3. Upper bounds for smooth strip footing on weightless cohesive–frictional soil

Mesh Linear programming NSID=24

Nonlinear programming Ratio of LP/NLP values

q/c′ Error*

(%)

CPU(Sec)

Iter. q/c′ Error*

(%)

CPU(Sec)

Iter. Collapsepressure

CPU Iter.

coarse 50.48 +9.4 6.76 1396 49.82 +8.0 1.82 28 1.01 3.7 49.9

medium 48.77 +5.7 133.7 6875 48.00 +4.0 5.5 29 1.02 24.3 237

fine 48.03 +4.1 2105 16771 47.30 +2.5 13.6 30 1.02 155 559

* With respect to exact Prandtl (1920) solution qexact = 46.12c′

discretisation, the lower bound scheme is generally more accurate than the upper bound scheme, especiallyfor analyses involving frictional soils with high friction angles.

6.3.2 Square And Rectangular Footings On Weightless Cohesive–Frictional Soil

Rigorous solutions for the bearing capacity of rectangular footings are difficult to derive andapproximations are usually adopted in practice. The most common approach is to apply empirical shapefactors to the classical Terzaghi formula for a strip footing (see, for example, Terzaghi, 1943 and Vesic,1973, 1975).

It is of much interest to compare the lower bound bearing capacity values for square and rectangularfootings with approximations that are commonly used in practice. Many empirical and semi–empiricalengineering solutions for the bearing capacity q have been proposed, but focus here is placed on two that usethe modified Terzaghi (1943) equation:

γγγγ γλλλλλλλλλ BNNqNcq idsqsqiqdqsccicdcs

++′=

2

1(2)

where B is the footing width, qs is the ground surcharge, γ is the soil unit weight, Nc, Nq, Nγ are bearing

capacity factors, λ cs, λqs, λγs are shape factors, λcd, λqd, λγd are embedment factors and λci, λqi, λγi areinclination factors. In the case of a surface footing on a weightless soil with vertical loading, Equation (2)reduces to the simple form:

ccs Ncq ′= λ (3)

In his original study, Terzaghi (1943) presented a table of values for Nc and suggested the use of λcs =1.3

for square and circular footings. Thirty years later Vesic (1973), on the basis of experimental evidence,advocated the shape factors λcs = 1+(B/L)(Nq/Nc) for rectangular footings and λcs = 1+(Nq/Nc) for squarefootings. In these approximations, N c and N q are the Prandtl values N c = (Nq–1)cotφ′ andNq = exp(π tanφ′ )tan2(π/4+φ′ /2).

Table 4 summarises various bearing capacity values for a rough square footing and two types of roughrectangular footing. These results were obtained using the boundary conditions and mesh layout shown inFigure 4. All of the cases assume a cohesive–frictional soil with zero self–weight and results are presentedfor the lower bound method, the original Terzaghi (1943) method, and the Vesic (1973) method. TheTerzaghi estimates exceed the square footing lower bounds by an average of around 19%, and would appear

Table 4. Predicted bearing capacities q/c′ = λcsNc for rough square footing and rough rectangular footingson weightless cohesive–frictional soil.

φ′(°) Square Rectangular (L/B = 2) Rectangular (L/B = 5)

Lower

bound

Terzaghi

(1943)

Vesic

(1973)

Lowerbound

Vesic

(1973)

Lower

bound

Vesic

(1973)

0 5.54 7.41 (25%)

λcs=1.3

Nc=5.7

6.14 (10%)

λcs=1.19

Nc=5.14

5.35 5.64 (5%)

λcs=1.10

Nc=5.14

5.17 5.34 (3%)

λcs=1.04

Nc=5.14

10 9.83 12.48 (21%)

λcs=1.3

Nc=9.6

10.82 (9%)

λcs=1.30

Nc=8.35

9.16 9.59 (4%)

λcs=1.15

Nc=8.35

8.55 8.84 (3%)

λcs=1.06

Nc=8.35

20 20.33 23.14 (12%)

λcs=1.3

Nc=17.8

21.23 (4%)

λcs=1.43

Nc=14.83

17.56 18.03 (3%)

λcs=1.22

Nc=14.83

15.58 16.11 (3%)

λcs=1.09

Nc=14.83

to be unsafe. The Vesic (1973) approximations, on the other hand, are within 5% of the rigorous lower bound results for rectangular footings of all shapes, and their accuracy increases with increasing friction angle.

The average number of nodes for the finite element models considered here was around 12000, while the largest mesh analysed had 13392 nodes, 3348 elements and 6444 discontinuities. Bearing in mind that each of these nodes has six unknown stresses, these grids generate very large optimisation problems and it comes as no surprise that the method is computationally demanding. On average, the CPU time for the analyses shown in Table 4 was 4500 set for a Dell Precision 220 PC with a Pentium III 8OOMHz processor. Although large, these timings are certainly competitive with those that would be needed for a three- dimensional incremental analysis with the displacement finite element method. Moreover, the solutions have the advantage that the limit load, which is obtained explicitly and does not have to be inferred from a load-deformation plot, is a rigorous lower bound on the true collapse load.

regular mesh

Mesh

nodes 1 13392 c’ z

#f z ;t200 elements 3348

y=o I discontinuities 6444

Figure 4 : Smooth rigid rectangular footing on cohesive-frictional soil: (a) general view with applied boundary conditions and (b) typical generated mesh

CONSTITUTIVE MODELS FOR GEOMATERIALS AND INTERFACES

As identified previously, one of the primary aims of geotechnical design is to ensure an adequate margin of safety, and this normally involves some form of stability analysis. In the previous section, it was demonstrated how modern computing methods could be combined with the classical limit theorems of plasticity to conduct very accurate and very sophisticated stability analyses of geotechnical problems. Of course, these analyses provide no information about the behaviour of the soil or rock mass prior to collapse. They are not designed to do so. In order to predict the response prior to collapse a complete constitutive model is required for the soil or rock material. Such models must be capable of predicting not only the onset of failure, but also the complete stress-strain response leading up to failure. In many cases the ability to predict the post-failme (post-peak) behaviour is also desirable. For this type of calculation a complete constitutive model is required.

It is often said that the developments in computer solution methods are far ahead of those related to the characterisation of the mechanical behaviour of geomaterials and interfaces or joints. Fortunately, the recent research emphasis placed on the development and application of constitutive models, so as to account for important factors that were not included in previous empirical and simplified models, has been well directed. Unless the materials and interfaces in geotechnical systems are characterised in such a way that they realistically account for the important factors that influence the material behaviour, results from sophisticated computer methods will have only limited validity, if at all.

The behaviour of geomaterials and interfaces is affected significantly by factors such as the state of stress or strain, the initial or in situ conditions, the applied stress path, the volume change response, and environmental factors such as temperature and chemicals, as well as time dependent effects such as creep

15

and the type of loading (static, repetitive and dynamic). In addition, natural geomaterials are oftenmultiphase, and include fluids (water) and gas as well as solid components. The pursuit of the developmentof constitutive models for these complex materials has involved progressive consideration of the foregoingfactors. Most often, the models that have been developed are designed to allow for a limited number offactors only. Recently, efforts have been made to develop more unified models that can allow inclusion of alarger number of factors in an integrated mathematical framework. A review of some of the recentdevelopments is presented below, together with comments on their capabilities and limitations. In view ofspace limitations and the large number of available publications, such a review can only be brief andsomewhat selective.

7.1 Elasticity Models

The theory of linear elasticity has a long history of application to geotechnical problems. Initially,emphasis was placed on the use of linear elasticity for a homogeneous isotropic medium. With thedevelopment of numerical solution techniques, the behaviour of non-homogeneous and anisotropic mediawas addressed.

Nonlinear elastic or piecewise linear elastic models were subsequently developed to account mainly forthe influence of the state of stress or strain on the material behaviour. These include the models based on thefunctional representation of one or more observed stress-strain and volumetric response curves. Thehyperbolic representation (Kondner, 1963; Duncan and Chang, 1970; Desai and Siriwardane, 1984; Faheyand Carter, 1993) has been often used for static and quasi-static behaviour, and can provide satisfactoryprediction of the load-displacement behaviour under monotonic loading. However, such functionalsimulation of curves is not capable of accounting for other factors such as stress path, volume change, andrepetitive and dynamic loading. Furthermore, they are usually not appropriate when evaluation of stressesand deformations (strains) are required in local zones in geotechnical systems.

Over the past few decades, great emphasis has been placed on research into the small strain behaviour ofgeomaterials. This has been seen as a fundamental issue for many problems in geotechnical engineering(e.g., Burland, 1989; Atkinson et al., 1990; Atkinson, 1993; Ng et al., 1995; Puzrin and Burland, 1998) Inparallel with detailed experimental work to investigate the small strain response (e.g., Shibuya et al., 1995;Jamiolkowski et al., 1999), several constitutive models that include the influence of strain level on thestress-strain response have been developed. There are also examples of their application to boundary valueproblems documented in the literature (e.g., Jardine et al., 1986; Powrie et al., 1998; Addenbrooke et al.,1997; Schweiger et al., 1999).

7.2 Hyperelasticity And Hypoelasticity

Hyperelastic or higher order elastic models have been considered for geological materials. However,they do not allow for factors such as stress path and irreversible deformations. Hence, their applications togeomaterials have been quite limited. However, it has been demonstrated that hypoelasticity and itscombination with plasticity can provide useful models for some geomaterials (e.g., Desai and Siriwardane,1984; Kolymbas, 1988; Desai, 1999).

7.3 Plasticity Models

The available elastoplastic models can be divided into two categories: (a) classical plasticity models, and(b) continuous yielding or hardening plasticity models. In the classical models, such as those based on thevon Mises, Mohr-Coulomb and Drucker-Prager criteria, plastic yielding occurs after the elastic response,when a specific yield stress or criterion is exceeded. These models are capable of predicting ultimate orfailure stresses and loads. However, they cannot allow properly for factors such as continuous yielding thatin many cases occurs from the very beginning of loading, volume change response and the effect of stresspath on the failure strength.

Models based on the critical state concept (Roscoe et al., 1958), cap models (e.g., DeMaggio andSandler, 1971) and their various modifications were proposed largely to allow for the continuous yielding

r es po ns e e xh i bi te d b y ma n y ge ol o gi ca l mat er i al s. As a r es ul t , th ey ha ve be en o f te n u se d in co mp u te r methods.At the same time, it should be recognised that they do suffer from some limitations, including the following:• they cannot predict dilative volume change before the peak stress,• they do not allow for different strengths mobilised under different stress paths,• they do not include the effect of deviatoric plastic strains on the yielding behaviour, and• they are usually based on associative plasticity, and hence they do not always adequately include the

effects of friction.In order to overcome these limitations, a number of advanced hardening models have been proposed with

continuous yield surfaces. These include the hierarchical single surface (HISS) model (Desai et al., 1986;Desai, 1995; 1999), as well as those proposed by other investigators (e.g., Lade and Duncan, 1975; Kim andLade, 1988; Nova, 1988; Lagioia and Nova, 1995; Whittle, et al., 1994).

Kinematic and anisotropic hardening models have been proposed to account for cyclic behaviour underdynamic loading (Mroz, 1967; Dafalias, 1979; Mroz et al., 1978; Prevost, 1978; Somasundaram and Desai,1988; Wathugala and Desai, 1993). Most of these models involve moving yield surfaces and may involveassociated computational difficulties. Models based on the unified disturbed state concept (DSC), do notinvolve moving surfaces, and generally they are computationally more efficient than the more advancedhardening and softening models (Katti and Desai 1995; Shao and Desai, 2000).

7.4 Hypoplasticity

Of course, other possibilities of describing the mechanical behaviour of soils have been proposed in theliterature such as hypoplastic formulations (e.g., Kolymbas, 1991; Niemunis and Herle, 1997). Contrary toelastoplasticity, in hypoplastic models no distinction is made between elastic and plastic deformation and noyield and plastic potential surfaces or hardening rules are needed. The aim in developing hypoplasticmodels for soils is to discover a single tensorial equation that can adequately describe many importantfeatures of the mechanical behaviour of the material. It is also desirable that the parameters of the model fora granular material depend directly on the properties of the grains.

There are several useful treatments in the literature of hypoplastic models applied to soils (e.g.,Kolymbas, 1991; Gudehus, 1996; von Wolffersdorff, 1996; Wu et al., 1996; Bauer, 1996; Herle andGudehus, 1999). Despite their apparent simplicity some indeed are able to reproduce features of real soils,such as pressure and density coupling, dilatancy, contractancy, variable friction angle and stiffness.Although some of the features of these models seem to be very attractive, so far they have been applied onlyto sands. More experience in the application of these models is still required.

7.5 Degradation And Softening

Plasticity based models that allow for softening or degradation have also been proposed. In these modelsthe yield surfaces expand during loading up to the peak stress, and then during softening they are allowed tocontract. This approach may suffer from non-uniqueness of computer solutions because the discontinuousnature of the material is not adequately taken into account.

The classical continuum damage models (Kachanov, 1986) allow for degradation in the material stiffnessand strength due to microcracking and resulting damage. However, since the coupling between theundamaged and damaged parts is not accounted for, these models are essentially local and can entailspurious mesh dependence when used in numerical solutions. In order to render damage models to be non-local, various enrichments have been proposed. These include consideration of microcrack interaction onthe observed response (Bazant, 1994), and gradient and Cosserat theories (de Borst et al., 1993; Mühlhaus,1995). Such enhanced models can provide computations that are independent of the mesh layouts.However, they can be complex and may entail other computational difficulties. The DSC also allows for thecoupling within its framework, and hence, leads to non-local models that are free from spurious meshdependence (Desai et al., 1997; Desai, 1999).

The problem of strain localisation that accompanies softening and degradation has received a great dealof attention in the literature, ever since the developments in numerical methods made possible the solution ofboundary value problems with such material features. It is in problems such as these that the inherentmaterial response, i.e., softening, and the numerical solution strategy have a strong interaction. Numerical

procedures that deal inappropriately with post-peak material response are likely to provide spurious answers.It is therefore worth providing a brief summary of the main approaches that have been developed to dealwith the problem of strain localisation. It should be noted here that to date the methods discussed belowhave been applied almost exclusively to two-dimensional (plane strain) problems.

7.5.1 Enhanced Finite Element Models

In this approach the domain undergoing localisation is divided into two regions: the localised zone inwhich large displacements take place and the non-localised zone in which the strains remain small. Theactual boundary of these two zones is not very well defined, but provided that the dimension of therepresentative domain under consideration is large compared to the thickness of the localised zone, the shearband may be treated, at least conceptually, as a plane of discontinuity in displacement.

Two conceptually different approaches have been proposed. One way is to use special finite elementswith appropriate discontinuous or modified shape functions (Belytschko et al., 1988; Ortiz et al., 1987). Inthis approach, firstly, a bifurcation analysis is carried out at the element level. When the onset oflocalisation is detected, the element interpolation is extended by adding to it suitably defined shape functionsthat reproduce the localised deformation modes.

The other approach is to introduce the effect of the shear band after the onset of localisation through aconstitutive framework (Pietruszczak and Mróz, 1981), which is generally known as the homogenisationtechnique.

7.5.2 Non-Local Models

A fully non-local model is obtained by introducing a relationship between the average stresses and theaverage strains, which complies with the relationship between macroscopic and microscopic stress forgranular bodies. The macroscopic stress can be thought of as some average of the more rapidly varyingmicroscopic stress (Brinkgreve and Vermeer, 1995).

In most applications, the non-local formulation is restricted to a specific part of the constitutive equation.Bazant and Lin (1988) averaged the invariant of plastic shear strains, but alternatively an average of theplastic strain rate tensor could be taken and then the invariant formulated. Either approach leads to a set ofrate equations, which remain elliptic after the onset of localisation.

A modified non-local model overcoming some of the numerical problems generally associated with non-local models has been presented by Vermeer and Brinkgreve (1994) and Brinkgreve and Vermeer (1995)with the considerable advantage of a relatively easy implementation into existing finite element codes. Thenumerical results for a purely cohesive material presented by Brinkgreve and Vermeer (1995) showed thatthe results are mesh independent.

7.5.3 Gradient Plasticity Models

Gradient dependent models can be viewed as an approximation of fully non-local models. The spatialderivatives of the inelastic state variables enter the continuum description of the material in addition to theinelastic state variables. The gradient terms can be introduced either into the flow rule (Vardoulakis andAifantis, 1991) or the dilatancy constraint (Vardoulakis and Aifantis, 1989). Due to the gradient term thetangent stiffness matrix becomes non-symmetric even for associated plasticity. Although gradient plasticitytheories are highly versatile for describing localisation of deformation in a continuum, a disadvantage of theapproach is the introduction of an additional variable at global level in addition to the conventionaldisplacement degrees of freedom.

7.5.4 Micropolar Continua

In this approach rotational degrees of freedom are added to the conventional translational degrees offreedom. Additional strain quantities (relative rotation and micro-curvatures) and additional stress quantities(couple stresses) enter into the continuum description (Mühlhaus and Vardoulakis, 1987; de Borst, 1991).Since these static and kinematic components are related through a ‘bending’ modulus, which has thedimension of length incorporated, an internal length scale is introduced, even in the elastic regime.

The determination of the material properties from test data has been given relatively little attention so far.A further disadvantage is that the rotations emerge as additional degrees-of-freedom at global level, whichincreases the computational effort.

More recently Cosserat continua have been successfully applied to model strain localisation within theframework of hypoplasticity (Tejchman and Bauer, 1996) using the mean grain diameter as the characteristiclength. Although results look promising the required fineness of the finite element mesh in order to avoidmesh dependence seems to prohibit practical applications without significant further enhancements.Cosserat-type models have also been adapted to the solution of boundary value problems in rock mechanics,for cases where the rock mass has a distinct structure, e.g., foliation (Adhikary et al., 1995, 1996)

7.5.5 Viscoplastic Models

In viscoplasticity the strain rate included in the constitutive equation prevents the set of equationsdescribing the dynamic motion of the softening solid from becoming elliptic. Failure modes are usuallyaccompanied by high strain rates and therefore the inclusion of the strain rate into the constitutive equationseems natural (Perzyna, 1992).

This kind of approach is purely mathematical in the sense that the additional parameters needed are notdirectly derivable from experiments and may have little physical meaning. Therefore a semi-inverseapproach is needed (Sluys and de Borst, 1992). A basic drawback of the use of viscosity to restore well-posedness of the governing equations is the fact that the regularising effect gradually vanishes for the rate-independent limit or a very slow process, i.e., the method is generally not suitable for static loading. Somerecently published results by Oka et al. (1994) however suggest that by introducing additional materialfunctions the viscoplastic approach still offers some potential for analysing quasi-static strain localisationproblems.

7.5.6 Mesh Adaptivity Techniques

Mesh adaptation is a numerical approach to capture the localisation pattern by refining the mesh in theregions where the strain gradients are highest and vice versa for regions with low strain gradients. The totalnumber of elements may also be increased so that the global error associated with the mesh meets certainaccuracy requirements (Hicks, 1995).

In the adaptive procedure the first step is to define the remeshing criterion. This is closely linked to errorestimation (Zienkiewicz and Zhu, 1987) and the mesh is redefined when the specified error has beenexceeded. It is also possible to redefine the mesh at fixed intervals, but an error analysis is still needed informing the new mesh.

The second step is to generate the new mesh by increasing the mesh in the areas where the error ishighest. The third step is to map the current state variables, i.e., stresses, strains and strain hardening orsoftening parameters, from the old mesh to the new one, in a process which is called smoothing. For boththese steps several possibilities exist and the final accuracy of the analysis is dependent on the choices made.

With standard finite elements, the mesh geometry may induce the formation of unrealistic and wrongfailure mechanisms - especially when low order triangles are used (Pastor et al., 1992); this is overcome byusing adaptive meshes. However one problem still remains: as the mesh is refined, the shear band thicknesswill decrease to zero leading to zero energy dissipation. Therefore, other of limitations need to be imposed,such as the minimum element size should be limited to a finite proportion (e.g., 1/3) of the actual shear bandwidth (Zienkiewicz et al., 1995).

7.6 Saturated And Unsaturated Materials

The development of models for saturated geomaterials has been the subject of research for a long time.Terzaghi’s effective stress concept (Terzaghi, 1943) is considered to be the earliest model for saturated soilsand forms the basis of most of the stress-strain models developed since then.

Constitutive models capable of describing at least some aspects of the complex behaviour of partiallysaturated soils were first explored seriously in relation to foundation problems associated with shrinking andswelling clay soils (e.g., Aitchison, 1956, 1961; Richards, 1992). The recent emphasis on geoenvironmentalengineering has spurred considerable renewed interest in the characterisation of constitutive models forpartially or unsaturated materials. The literature on the subject is now wide in scope and a review isavailable in the text by Desai (1999). Some of the recent models based on the critical state concept,plasticity theory and the DSC are given in publications by Alonso, et al. (1990), Bolzon et al. (1996),Pietruszczak and Pande (1996), and Geiser et al. (1997).

7.7 Liquefaction

Liquefaction represents the phenomenon of instability of the saturated geomaterial’s microstructure atcertain critical values of stress, pore water pressure, (plastic) strains and plastic work or dissipated energy, asaffected by the initial conditions, e.g., initial effective mean pressure and physical state (density).

A number of empirical approaches, based on experimentally observed states of pore water pressure andstress at which liquefaction can occur, and index properties have often been used to predict the onset ofliquefaction (Casagrande, 1976; Castro and Poulos, 1977; Seed, 1979; National Research Council, 1985;Ishihara, 1993). Although such conventional approaches have provided useful results, they are not based onmechanistic considerations that can account for the microstructural modifications in the deforming material.Dissipated energy has been proposed as a criterion for the identification of liquefaction potential (Nemat-Nasser and Shokooh, 1979; Davis and Berrill 1982, 1988; Figueroa et al., 1994; Desai, 2000). Recently, theDSC has been proposed as a mechanistic procedure that is fundamental yet simple enough for practicalapplication to liquefaction problems (Desai et al., 1998; Park and Desai 1999; Desai, 2000).

7.8 Comments

Most of the constitutive models in the past have addressed a limited number of factors at any time. As aresult, separate models are often warranted for different behavioural features of the same material. This canlead to complexities such as a greater number of parameters, which is the sum of the parameters for eachmodel, for given characteristic(s). Hence, it is desirable and useful to develop unified models that canpermit consideration of the behavioural features as special cases. A unified approach can lead to compactmodels that are significantly simplified, involve fewer parameters, and are easier to implement in computerprocedures. Such a unified concept, called the disturbed state concept (DSC), has been developed and foundto be successful for numerousgeological materials and interfaces.The DSC allows implicitly for thenonlocal effects and thecharacteristic dimension and as aresult does not suffer from spuriousmesh dependence (Desai, 1999).Because of its potentially unifyingqualities, a brief summary of theDSC is provided in the followingsection.

7.9 Disturbed State Concept(DSC)

The recently developed DSCrepresents a unified approach forconst i tu t ive model l ing ofgeomaterials and interfaces andjoints. It allows for various factorssuch as elastic, plastic and creepstrains, microcracking leading todegradation or damage, andstiffening or healing underthermomechanical loading. Itshierarchical framework permits theuser to adopt specialised version(s),e.g., including elasticity, plasticity,viscoplasticity, with disturbance(degradation and stiffening),

Figure 5 : Schematic illustration of stress-strain behaviour andmaterial disturbance assumed in DSC models

depending upon the material and application need. Details of the DSC are given in various publications(Desai, 1995, 2000; Desai et al., 1986, 1991, 1995, 1997, 1998; Desai and Salami, 1987; Desai andVaradarajan, 1987; Desai and Fishman, 1991; Desai and Ma, 1992; Desai and Rigby, 1997; Desai and Toth,1996; Liu et al., 2000; Park and Desai, 1999; Shao and Desai, 2000; Wathugala and Desai, 1993) and atextbook (Desai, 1999).

The essential idea of the DSC is that the response of a material to variations in both internal and externalconditions can be described in terms of the responses of the constituent material parts in two reference states,the “relatively intact” state (RI) and the “fully adjusted” state (FA). The overall response of the material,i.e., the observed response, is then related to the responses at the reference states through a disturbancefunction, which provides a coupling and interpolation mechanism between the responses of the material inthe RI and FA states. A summary of the formulation of DSC models is provided below.

7.9.1 Incremental Equations

The incremental constitutive equations for the DSC are given by:

( ) ( ) dd 1 iccia dDDDd σσσσσ −++−= (4)

or

( ) ( ) dCd C1 c icciia dDDDd σσεεσ −++−= (5)

where σ and ε are the stress and strain vectors, respectively, and superscripts a, i, and c denote observed,relatively intact (RI), and fully adjusted (FA) states, respectively. D is the disturbance assumed to be scalarfunction (it can also be treated as a tensiorial quantity), dD denotes increment or rate of disturbance, and Cdenotes the constitutive matrix.

In the DSC, a material element during deformation is considered to be composed of material parts indifferent reference states, as indicated in Figure 5. For example, in the case of a dry material, the initialcontinuum state is considered to be a reference state and is called the relative intact (RI) state. Duringdeformation, the material transforms to a fully adjusted (FA) state at disturbed locations, as a consequence ofthe changes in its microstructure due to relative particle motions, microcracking or stiffening. The observedor average response of the material (a) is then expressed in terms of the responses of the material parts in theRI (i) and FA (c) states. The disturbance, D, acts as the coupling and interpolation mechanism between theRI and FA states so as to yield the observed response.

The RI response can be characterised by using continuum models such as those based on elasticity,plasticity or elastoviscoplasticity. For instance, the RI response can be characterised by using thehierarchical single surface (HISS) plasticity models. For the associative case, the yield surface, F, in theHISS-δ0 model is given by:

( )( ) 01 5.02112 =−+−−= −

rn

D SJJJF βγα (6)

where J1 is the first invariant of the stress tensor, σij,J2D is the second invariant of the deviatoric stress tensor, Sij,

Sr is the stress ratio defined as 23232

27 /DD JJ −⋅ , and

J3D is the third invariant of Sij.The superior bar denotes a quantity non-dimensionalised with respect to atmospheric pressure (pa). J1 = J1

+ 3R where 3R is the bonding (tensile or cohesive strength), γ and β are ultimate (failure) parameters, n isthe phase change parameter associated with the state at which volume change from contraction to dilationoccurs, and α is the hardening or growth function, which, in a simple form, is expressed as:

1

1ηξ

α a= (7)

where ( ) ( )∫

=

1/2

pTp dd εεξ is the trajectory of total plastic strains, and a1 and η1 are the hardening

parameters.The FA response can be characterised in different ways. If the FA (microcracked or damaged) material

is assumed to carry no stress at all, i.e., it acts as a “void” as in the classical damage model, Equation (4) willspecialise to that for the classical damage model (Kachanov, 1986). This characterisation is considered to beinappropriate, as it does not include the coupling between the RI and FA states. If the FA material isassumed to carry hydrostatic stress and no shear stress, its behaviour (Cc) can be characterised on the basisof its bulk response. In a general method, the FA response can be characterised by using the critical stateconcept (Roscoe et al., 1958), in which the FA material continues to carry the shear stress and deform inshear without change in its volume, under a given mean pressure. The critical state equations to characterisethe FA response are given by (Roscoe et al., 1958; Desai, 1999):

ccD JmJ 12 = (8)

( ) 3ln 11 acc

oc p/Jee λ−= (9)

where e is the voids ratio and λ is a material parameter.

7.9.2 Disturbance

Disturbance is expressed as the ratio of the volume of the material in the FA state to the total volume. Ina phenomenological context, D is expressed on the basis of observed response (e.g., stress-strain, volumetricor void ratio, effective stress or pore water pressure and nondestructive properties such as P- or S-wavevelocities), and in terms of the deviatoric plastic strain trajectory or plastic work (Desai, 1999). Forexample:

ci

ai

Dσσσσ

−−= (10)

and

1

−= − ZDA

u eDD ξ (11)

where σ is an appropriate stress measure, i.e., σ1, σ1 - σ 3 or DJ 2 . ξD is the deviatoric plastic strain

trajectory, and A, Z and Du are disturbance parameters.

7.9.3 Specialisations

Equation (4) includes various continuum and damage models as special cases. For example, if D = 0, thefollowing results:

iii dCd εσ = (12)

where Ci can be based on elasticity, plasticity or elastoviscoplasticity theories. If the FA part cannot carryany stress at all, Equation (4) gives the classical damage model:

( ) iiia dDdCDd σεσ −−= 1 (13)

which does not include the effect of the microcracked or damage parts on the observed behaviour.

7.9.4 Interfaces And Joints

An interface or joint in a geological system constitutes a junction between two (similar or dissimilar)materials and represents a discontinuity where relative motions (slip, debonding or separation, rebondingand interpenetration) can occur. In order to allow for the relative motions, it becomes necessary to developconstitutive models based on appropriate and specialised laboratory tests. The force and kinematicconditions at the interface are introduced by developing various models: springs to represent normal andshear response, special joint elements to allow for relative displacements (Goodman et al., 1968; Ghaboussiet al., 1973) and force constraints (Katona, 1983). A number of models have been proposed asmodifications or variations of the foregoing approaches. An alternative, called the thin-layer elementapproach, has been proposed (Desai et al., 1984). In this approach the “smeared” interface zone is treated asa “solid” element, and its stiffness properties are evaluated in the same manner as the other solid elements inthe neighbouring regions. However, the material parameters are obtained from special shear tests, such assimple shear or direct shear tests (Desai and Rigby, 1997).

The DSC can also be applied for the characterisation of interfaces and joints. The same mathematicalframework, Equation (4), is applicable. As a result, the models for geological materials and interfaces areessentially the same. This eliminates the inconsistency in previous approaches in which different modelswere used for soils and interfaces. For example, if the HISS-δ0 model is used to define the RI behaviour, theyield function is given by (Desai and Fishman, 1991; Desai and Ma, 1992):

02 =−+= qn

nnF γσαστ (14)

where τ and σn are the shear andnormal stresses in a two-dimensional interface, respectively,γ is the ultimate or failureparameter, n is the phase changeparameter, q allows a curvedultimate (failure) envelope, and α isthe hardening function expressed interms of the plastic relative shearand normal displacements. Thedisturbance is defined on the basisof results from tests involving theapplication of shear and normalloads.

7.9.5 Parameters

The material parameters in theDSC have physical meanings in thatthey are related to physical statesduring deformation. Their numberis usually smaller than that in otheravailable models of comparablecapabilities. For instance, with twoelastic (E , ν), and five plasticity (γ,β, a1, η1, R) parameters, the HISS-δ0 model can include the effects ofthe factors previously mentioned,that are not present in the criticalstate models (with 6 parameters)and Cap models (with 10parameters).

The parameters in the DSCmodel can be found from laboratory Figure 6 : Finite element mesh used for field pile test

uniaxial, shear, triaxial or multiaxial tests. The standard triaxial compression (and extension) test is suitableto quantify the parameters defining the elasticity, plasticity and disturbance. For viscous behaviour, creep orrelaxation tests are required. Details of procedures for finding the parameters are available in variouspublications (e.g., Desai and Ma 1992; Desai et al., 1995; Desai, 1999).

7.9.6 Validations and Applications

The DSC model and its specialised versions have been calibrated with respect to test data for a widerange of geological materials and interfaces, and other materials such as concrete ceramics, metals andalloys (Desai, 1995, 1999; Desai et al., 1984, 1991, 1995, 1997, 1998; Desai and Salami, 1987; Desai andFishman, 1991; Desai and Ma, 1992; Desai and Toth, 1996; Desai and Rigby, 1997). They have beenvalidated with respect to tests used to find the parameters and independent tests not used in finding them.

The DSC has been implemented in nonlinear finite element codes for static and dynamic analysisincluding dry and saturated materials. The codes are used to predict observed behaviour of a number ofpractical problems in geotechnical engineering. These include two- and three-dimensional analysis offootings and pile foundations, reinforced retaining walls, dams and excavations, underground works(tunnels), anchor-soil systems, landslides, consolidation problems, and seepage and liquefaction in shaketable tests. Two typical examples are given below to illustrate some of the capabilities of these sophisticatedmodels.

7.9.7 Example - Cyclic Analysis Of Piles In Marine Clay

Figure 6 shows the finite element mesh for a displacement-controlled field load test performed by EarthTechnology Corp. (1986) on a 76.2 mm diameter pile segment. The numerical calculations involvedsimulation of the in situ stress conditions, driving effects, consolidation, tension tests and finally, axial cyclicloading (Shao and Desai 2000).

Figure 7 : Comparison between field measurements and predictions from DSC and HISS models: one-waycyclic load tests

The material properties for the DSC with the HISS-δ0 model for the RI behaviour of the marine clay weredetermined from static and cyclic triaxial tests on cylindrical specimens and multiaxial tests on cubicalspecimens (Katti and Desai, 1995; Shao and Desai 2000; Wathugala and Desai, 1993).

The parameters for the interface between the steel pile and the clay were determined from laboratory testsusing the cyclic multi degree-of-freedom (CYMDOF) shear device, including measurements of pore waterpressures. Details of the parameters are given elsewhere (Katti and Desai, 1995; Shao and Desai 2000;Wathugala and Desai, 1993).

The finite element analyses were performed by using the DSC model and the kinematic hardening HISS-δ0* model (Wathugala and Desai, 1993) implemented in the DSC-DYN2D computer code. The latter allowsfor cyclic degradation and interface effects.

Figures 7 and 8 show comparisons between the computed results using the DSC and δ0* model and theobserved data for one-way and two-way cyclic load tests, respectively. These figures also include thepredicted growth of disturbance during the loading. It can be seen that the DSC model provides very goodcorrelation with the observed results. Furthermore, it shows improved predictions compared to those fromthe HISS-δ0* plasticity model because the DSC model allows for cyclic degradation and interface effects.

7.9.8 Example - Dynamic Analysis Of Shake Table Test And Liquefaction

Figure 9(a) shows the shake table test configuration (Akiyoshi et al., 1996). The applied loadinginvolved specifying horizontal displacement, X, at the bottom nodes, given by the following function:

( )tfuX x π2sin= (15)

Figure 8 : Comparison between field measurements and predictions from DSC and HISS models: two-waycyclic load tests

where ux is the amplitude (= 0.0013 m), f is the frequency (= 5 Hz) and t is time.Fuji river sand was used in the shake table test. Because various physical properties of the Fuji river sand

and the Ottawa sand are similar, e.g., the grain-size distribution, Figure 9(b), the multiaxial cyclic test datafor the Ottawa sand were used in the finite element analysis (Park and Desai 1999).

Shaking direction Shaking machine

100

300

700

800 10

00

Sand

1500

1700

AccelerometerPore water pressure meterUnit: mm

0.01 0.1 1 10

(b) Grain Size Distribution Curves of Ottawa Sand and Fuji River Sand

0

20

40

60

80

100

Figure 5 Shake Table Setup and Grain Size Distributions for Sands

Gradation Curve

Fuji SandOttawa Sand

(a) Shake Table Test Set-Up (Akiyoshi, et al., 1996)

%Finer

Particle size (mm)

Figure 9 : Shake table test setup and grain size distribution for sands

The finite element mesh isshown in Figure 10. The steelin the test box was assumed tobe linear elastic, while theDSC model was used torepresent the behaviour of thesand. The idea of therepeating side boundaries wasemployed, in which thedisplacements of the sideboundary nodes on the samehorizontal planes wereassumed to be the same. Theanalysis used the computercode DSC-DYN2D.

F i g u r e 1 1 s h o w scomparisons between themeasured and computedexcess pore water pressureswith time, at the point (depth= 300 mm) shown as a soliddot in Figure 12. The test dataindicate liquefaction afterabout 2.0 secs when the porewater pressure equals theinitial effective stress. It canbe seen from Figure 11 that thefinite element predictionscompare well with the testresults.

Figures 12(a) to (d) showgrowth of disturbance in themesh at typical times = 0.5,1.0, 2.0 and 10.0 secs,respectively; the plot ofcomputed disturbance at depth= 300 mm is shown in Figure12(e). Laboratory tests on theOttawa sand showed that theliquefaction initiates at anaverage value of the criticaldisturbance, Dc = 0.84 (Desaiet al., 1998). At time t = 0.5sec, the computed disturbancein the sand is well below thecritical value in all elements.At t = 1.0 sec, the disturbancehas grown, and its value isbetween 0.50 and 0.70 in theelements in the middle zone,which is below Dc = 0.84. Att = 2.0 secs, the disturbance has reached values higher than 0.80 at and below the depth of 300 mm; thisindicates that liquefaction initiates at about 2.0 secs. At t=10secs, the disturbance in about 80% of the testbox has grown to a value equal to or greater than the critical value, indicating that the soil has liquefied andfailed.

Figure 11 : Excess pore pressures at a depth of 300 mm

Figure 10 : Finite element mesh used for simulation of shake table test

These and other results show that the DSC model can provide a fundamental and simplified approach forthe evaluation of liquefaction potential (Desai, 2000; Desai et al., 1998).

0.30.40.50.60.70.80.91.01.1

(a) Time = 0.5 sec (b) Time = 1.0 sec

(d) Time = 10.0 secs(c) Time = 2.0 secs

0 1 2 3 4 5 6 7 8 9 10Time (sec)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Disturbance(D)

(e) Disturbance vs. time at depth = 300mm

Figure 8 Growth of Disturbance in Sand and at Depth = 300mmFigure 12 : Growth of the disturbed zone in sand

7.10 Structured Soils

Much of the research on constitutive models conducted to date has focused on the behaviour of soilsamples reconstituted in the laboratory. The emphasis on reconstituted materials arose because of bothconvenience and the need to maintain control over test and sample conditions, if models truly reflecting theresponse to changing applied stresses were to be developed. However, natural soils (and rock masses) differsignificantly from laboratory prepared specimens, and these differences need to be accounted for if thedeveloped models are to be applicable to materials found in nature. For example, there still exists a need todevelop constitutive models that can take into account the structure and fabric of most natural materials.

The important influence of structure on the mechanical properties of soil has long been recognised (e.g.,Casagrande, 1932; Mitchell, 1976). The term “soil structure” is used here to mean the arrangement andbonding of the soil constituents, and for simplicity it encompasses all features of a soil that cause itsmechanical behaviour to be different from that of the corresponding reconstituted soil. The removal of soilstructure is referred to as destructuring, and destructuring is usually a progressive process.

In recent years, there have been several studies in which a theoretical framework for describing thebehaviour of structured soils has been formulated (e.g., Burland, 1990; Leroueil and Vaughan, 1990; Gensand Nova, 1993; Cotecchia and Chandler, 1997; Liu and Carter, 1999). It is rational to study the behaviourof natural soils by using knowledge of the corresponding reconstituted soils as a frame of reference(Burland, 1990, Liu and Carter, 1999). The disturbed state concept theory (DSC), described previously,provides a convenient framework to predict the difference in structured soil behaviour from that of thereconstituted soil, and it may also have the potential to describe the destructuring of a natural soil withloading. One such DSC model, suitable for isotropic and one-dimensional compression, has beenformulated by Liu et al. (2000), based on a particular disturbance function. Simulations of the proposedDSC model for both structured soil and reconstituted soil have been made and compared to experimentaldata. An illustration of the accuracy of this method for incorporating the effects of soil structure is includedhere.

The disturbed state concept allows flexibility in choosing ways to define the two reference statesaccording to the practical problem of interest and the knowledge available, including its response inlaboratory and field tests. A special selection of the reference states for quantifying the influence of soilstructure on soil behaviour has been suggested by Liu et al. (2000). The fully adjusted state (FA) waschosen to be the corresponding reconstituted state. The fully adjusted state for the virgin compression of astructured soil is therefore based on the following two assumptions: (1) the material is reconstituted and hasthe same mineralogy as the structured soil, and (2) the soil is in a state of virgin compression and the stressstate is the same as that applied to the structured soil. The relative intact state (RI) was chosen to be the“zero state”, i.e., the state with no response to stress (a perfectly rigid material).

As indicated previously, a key step in building a DSC model lies in finding a disturbance function.Based on a study of a large body of experimental data on the compression behaviour of structured clays andother soils (e.g., Burland, 1990; Leroueil and Vaughan, 1990; Smith et al., 1992), the following disturbancefunction, Dεv, for the compression behaviour of naturally structured soils was proposed by Liu et al. (2000).

′

′+=ε p

pbD i,y

v 1 (16)

where b is the disturbance index for compression, p′ is the mean effective, and p′y,i represent the meaneffective stress at which virgin yielding first occurs. Further details of the mathematical formulation of thisapproach can be found in Liu et al. (2000).

An example of the application of the DSC to the destructuring of clay during one-dimensionalcompression is shown in Figure 13 for Winnipeg clay (after Graham and Li, 1985). Simulations using theDSC for Winnipeg clay and six other soils over a range of mean effective stresses from 10 kPa to 2,000 kPahave been made by Liu et al. (2000). For one soil (Mexico City clay) the magnitude of volumetric strainreached was as high as 120%. It was found that the proposed DSC model can describe successfully thecompression behaviour of both naturally structured and artificially structured soils, no matter if thebehaviour of the corresponding reconstituted soil is linear or non-linear in the e-lnp′ space. A complete

three-dimensional model ,capable of quantifying theshearing response of structuredsoil using the DSC is nowrequired before this approachcan be adopted in situationsother than one-dimensionalcompression.

7.11 Which ConstitutiveModel Should BeChosen?

Choos ing a su i tab leconstitutive model for any givensoil for use in the solution of aparticular boundary valuep r o b l e m c a n p r o v eproblematical. Many theoretical models have been published in the literature and each has its own particularstrengths and all have their limitations. Determining the key model parameters is often a major difficulty,either because some model parameters may have no real physical meaning, or inadequate test data may beavailable. Considerable experience and judgement are therefore required in order to make a sensible,meaningful selection of the most appropriate model and the most suitable values of its parameters.

It is unlikely that a universal constitutive model, capable of providing accurate predictions of soilbehaviour for all soil types, all loading and all drainage conditions, will be proposed, at least in theforeseeable future. Although the DSC seems to offer a framework for unifying many of these models, itseems likely, at least in the short to medium term, that progress will continue to be made through thedevelopment and use of models of limited applicability, but nevertheless capable of accurate predictions.Knowing the limitations of such models is as important as knowing their strengths.

7.11.1 Example – Foundations On Carbonate Sands

The issue of selecting the most suitable stress-strain model and its parameter values comes into sharpfocus for the case of shallow foundations resting on structured carbonate soils of the seabed. This is becausenatural samples of these materials are often lightly to moderately cemented and the degree of cementationcan by highly variable. There are difficulties associated with obtaining high quality samples of these naturalsoils. In order to avoid these difficulties, and as a means of initiating meaningful studies of cemented soils,laboratory test samples have been prepared by artificially cementing reconstituted samples of carbonatesands recovered from the sea floor (e.g., Huang, 1994; Huang and Airey, 1998; Carter and Airey, 1994).The results of a sequence of related studies on artificially cemented calcareous soils (Huang, 1994; Yeoh,1996; Pan, 1999) and attempts to model them numerically, are described briefly here.

7.11.2 Typical Behaviour

The volumetric behaviour of the artificially cemented sands is illustrated in Figure 14. Typical behaviourduring drained triaxial compression is indicated in Figures 15 and 16.

Figure 14 includes results for samples of cemented and uncemented soils, prepared with different initialdensities. All specimens exhibit consolidation behaviour resembling overconsolidated clay. Initially theresponse is stiff, but eventually at a sufficiently high stress level the rate of volume change with stress levelexhibits a marked change and the samples become much more compressible. The transition is a result ofbreakdown in the cement bonding accompanied by some particle crushing and perhaps particlerearrangement. High volume compressibility is eventually exhibited by all specimens, and the presence ofan initial cementation only slightly affects the position of the consolidation curve.

Figures 15 and 16 indicate that the cemented material exhibits relatively brittle behaviour at lowconfining pressures, accompanied by a tendency to dilate. However, even at relatively moderate confining

0

5

10

15

20

25

0 100 200 300 400 500

Vertical effective stress s ' v (kPa)

Vo

lum

etri

c st

rain

ev (

%)

Winnipeg clays'vy,I = 200 kPa

bv = 0.28

Structured Reconstituted

Structured Reconstituted

simul.

test

Figure 13 : Comparison of DSC predictions and test results for astructured clay

stresses shearing inducessignificant volume reduction in thespecimens and the behaviourappears to be much more ductile,although there is still somebrittleness due to cementation.

7.11.3 Stress-Strain Models

A selection of constitutivemodels was investigated todetermine their suitability torepresent the stress-strainbehaviour of the artificiallycemented carbonate soil in both“single element” (i.e., triaxial)tests and boundary value problems(Islam, 1999). The modelsconsidered are listed in Table 5,together with the number ofparameters required to describecompletely the constitutivebehaviour. None employs theDSC.

All models are elastoplasticand involve either strain hardeningor softening. All have only oneyield surface, except theMolenkamp (1981) model (alsoreferred to as “Monot”), that hastwo. Some adopt associatedplastic flow, while others do not.None includes the effects ofcementation directly by includinga finite cohesion and tensilestrength. Rather, they incorporateits effects indirectly by regarding ac e m e n t e d s o i l a s a noverconsolidated material.

The model developed byLagioia and Nova (1995) wasdesigned specifically for acemented carbonate soil, thoughnot the soil specifically consideredhere. The Molenkamp model hasbeen used previously in acomprehensive study of thebehaviour of foundations oncalcarenite on the North-WestShelf of Australia (Smith et al.,1988), as well as in studies ofother granular media (e.g., Hicks,1992). Although formally it requires specification of a relatively large number of input parameters (23),many of these parameters may be assigned “standard” values.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.1 1 10 100

Mean effective stress (MPa)S

pec

ific

vo

lum

e (v

=1+

e)

Uncemented

20% Cement

Figure 14 : Isotropic compression of carbonate sands

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 5 10 15 20 25 30 35

Axial strain (%)

Dev

iato

r st

ress

(M

Pa)

1.2

0.6

0.3

0.1

Effective confining stress (MPa)

Figure 15 : Typical stress-strain behaviour of carbonate sand

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35

Axial strain (%)

Vo

lum

etri

c st

rain

(%

)

0.1

0.3

0.6

1.2 Effective confining stress (MPa)

Figure 16 : Typical volumetric behaviour of carbonate sand

The model labelled “SU2” is based veryclosely on the well-known Modified CamClay (MCC) model, but with one importantdistinction (Islam, 1999). It does not use theMCC ellipse as a yield surface, however itdoes use the original ellipse as a plasticpotential. In SU2 the yield surface in p´-qspace is a flatter ellipse, since this shape bettermatches the experimental data for theuncemented carbonate sand. However, it does underpredict the peak strength of the cemented soil. The newflattened ellipse corresponds to an increased separation of the isotropic consolidation and critical state linesin voids ratio – effective stress space. Whereas in Modified Cam clay this separation corresponds to a ratioof mean effective stresses of 2, in the model SU2 a ratio of 5 has been found to fit the data for carbonatesoils more accurately. These details are illustrated graphically in Figure 17. It is evident from this figurethat the flow law resulting from the adoption of the MCC ellipse as a plastic potential produces goodpredictions of the volumetric response of both the cemented and uncemented soil.

Values for the parameters of each model have been selected to provide a good fit to the triaxial test data.The fitting process was conducted for a comprehensive series of test results, using data from both drainedand undrained tests, in order to obtain the best possible overall agreement between model predictions andmeasured behaviour. A typical comparison between the model predictions and the experimental data isgiven in Figures 18 and 19 for the case of a drained triaxial test conducted at a confining pressure of300 kPa.

Figure 18 indicates that most of the models considered predict the strength of the cemented soil at largestrains and under-predict the shear strength at smaller values of axial strain. The models can reasonablypredict the final “critical state” shear strength, but not the larger peak strengths due to the contribution ofcementation. In this case the Modified Cam Clay model provides the best prediction of the stress-strain

Table 5. Stress-strain models

Model ParametersModified Cam Clay 5Lagioia & Nova 6SU2 6Molenkamp 23

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Normalized mean stress

No

rmal

ised

dev

iato

ric

stre

ss Modified Cam Clay yield locus SU2 yield locus Critical State Line Experimental data

Cemented soil

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Normalized mean stress

No

rmal

ised

dev

iato

ric

stre

ss Modified Cam Clay yield locus SU2 yield locus Critical State Line Experimental data

Uncemented soil

-1.0

0.0

1.0

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Stress ratio q/p'

Dila

tan

cy

Cemented soil-1.0

0.0

1.0

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Stress ratio q/p'

Dila

tan

cy

Uncemented soil

Figure 17 : Details of the model SU2 for uncemented (left) and artificially cemented (right) carbonate soils

curve, consistent with the fact thatit was also able to predictreasonably accurately the peakstrengths (Figure 17). However,Figure 19 reveals that the MCCmodel is a poor predictor of thevolumetric behaviour duringdrained shearing. It predictsdilation, while the test dataindicate that the sample contractedduring shearing. All other modelsprovide reasonable predictions ofthe volume reduction. Clearly theadoption of an associated flowrule in the MCC model isinappropriate for this material.

7.11.4 Model Footing Tests –Vertical Loading

Model footing tests have beencarried out on 300 mm diametersamples of the artificiallycemented carbonate sanddescribed previously (Yeoh,1996). An isotropic effectiveconfining pressure of 300 kPa wasapplied to the sample and a50 mm diameter rigid footing waspushed into its surface at aconstant rate, slow enough forfully drained conditions to beachieved. A typical set of resultsfrom this series of tests ispresented in Figure 20, which alsoindicates the model predictionsobtained using the same set ofmodel parameters obtained byfitting each model to the triaxialtest data. It is evident that mostmodels provide reasonablepredictions of the performance ofthe footing throughout the entirerange of footing displacementsconsidered, i.e., up to a settlementequal to 30% of the footingdiameter. However, the MCCmodel generally overpredicts thestiffness of the footing. This isnot unexpected, since for this caseMCC predicts dilation duringshearing rather than the observedcompression.

7.11.5 Model Footing Tests – Inclined Loading

0

2

4

6

8

10

12

0 5 10 15 20 25 30

Normalised displacement d/B (%)

Bea

rin

g p

ress

ure

(M

Pa)

Modified Cam Clay

Molenkamp Nova Sydney Uni. 2 Experimental data

Finite element simulation of axisymmetric model footing on artificially cemented carbonate soil

Figure 20 : Predictions of model footing tests

0

500

1000

1500

0 10 20 30 40 50Axial strain (%)

Dev

iato

r st

ress

(kP

a)

Modified Cam Clay

Molenkamp

Nova

Sydney Uni. 2

Experimental data

Carbonate sand, North West shelf, AustraliaTriaxial test, Cell pressure : 300 kPaDensity : 13 kN/m3, Cement content : 20 %

Figure 18 : Comparison of stress-strain predictions

-5.0

0.0

5.0

0 10 20 30 40 50

Axial strain (%)

Vo

lum

e st

rain

(%

)

Modified Cam Clay

Molenkamp Nova

Sydney Uni. 2Experimental data

Carbonate sand, North West shelf, Australia Triaxial test, Cell pressure : 300 kPa Density : 13 kN/m3, Cement content : 20 %

Figure 19 : Comparison of volumetric-strain predictions

Dilation

Predictions of the SU2 model forcases of circular footings subjected toinclined loading (Islam, 1999) havealso been compared with experimentalmeasurements (Pan, 1999) made at 1gin the laboratory. These comparisonsare shown in Figure 21, wheresatisfactory agreement can beobserved over a large range of valuesof the normal ised foot ingdisplacement.

7.11.6 Centrifuge Tests

Further validation of the SU2model in predicting the behaviour offootings on carbonate sand isdemonstrated in Figure 22 (Islam,1999), where model predictions arecompared with centrifuge test resultsobtained by Finnie (1993). Thesetests involved the vertical loading of arigid circular footing resting on afinite layer of cemented carbonatesand, overlying carbonate silt. Thefailure mechanism in this caseinvolved a punching mechanism withthe region of cemented soilimmediately under the footingpenetrating the underling silt.

7.11.7 Discussion

It may be concluded from this study of stress-strain behaviour that the Modified Cam Clay modelmatches well the yield surface of the artificially cemented soil but is a poor predictor of the volume changethat accompanies shearing. This has important consequences when the model is used in the prediction of aboundary value problem, as demonstrated. The other models considered in this study provide reasonablematches to the single element behaviour, although they are generally incapable of predicting accurately thepeak strengths. However, these same models provide quite reasonable predictions of the behaviour of modelfootings, especially at very large footing displacements. This indicates that the peak strengths of thematerials have only a small influence on the overall response of the footings. Excluding MCC, it wouldseem that there is little to choose between the models for use in analysing the behaviour of vertically loadedfootings on this type of material. However, the model SU2 has the attraction of being closely based on thewell-known MCC model, with only one simple but significant change, and it has relatively few parametersto be quantified, all of which have a clear physical meaning. Again, it is emphasised that none of the modelsconsidered here incorporates explicitly the effects of cementation or the break down of that cementation dueto increasing isotropic and deviatoric stress. For cases where these factors may be important, differentconstitutive models will be required, and the disturbed state concept described previously, is one method thatshows promise for capturing these effects.

8.0 SOME LIMITATIONS AND PITFALLS OF NUMERICAL ANALYSIS

8.1 Introduction

When using numerical analysis it is necessary to discretise the geometry of the boundary value probleminto a finite number of sub-regions. For example, when using the finite element method the geometry is

0

2500

5000

7500

10000

0 5 10 15 20 25 30

Normalised displacement d/B (%)

Ave

rag

e tr

acti

on

(kP

a)

Load inclination, degrees

0

3020

10

0

30

20

10

Predictions

Data

Figure 21 : Circular footing on cemented sand

0

600

1200

1800

0 5 10 15 20 25 30

Normalised vertical displacement d/B (%)

Ave

rag

e b

eari

ng

pre

ssu

re q B : 4m

t/B : 0.5Strong crust

Weak crust

Medium crust

Prediction

Data

Figure 22 : Circular footing on cemented layer over silt

divided into an assemblage of finite elements, whereas in the finite difference approach the geometry isrepresented by a grid of points. Approximations are then made as to how the primary variables, usuallydisplacements, vary within these sub-regions. Clearly this introduces approximations and to obtain accurateresults it is necessary to have sufficiently small sub-regions in areas where there are high gradients of stressand strain. This problem is fundamental to all numerical analysis and occurs no matter what constitutivemodel is being used, although the use of a nonlinear constitutive model can complicate the issue. The othermain source of error involved in nonlinear numerical analysis is associated with the integration of theconstitutive equations. Approximations must be made for this to be achieved and many solution strategiesexist for performing this task.

Other approximations that are involved in numerical analysis are those arising from the idealisationsmade when reducing the real problem to a form which can be analysed. This usually involves geometricapproximations and idealisations as to material behaviour. A potential source of error is associated with alack of ‘in depth’ understanding of the constitutive models employed to represent soil behaviour. This is acommon source of error, due to the complexities of many of the constitutive models currently available.

To illustrate the potential pitfalls associated with the above approximations a short discussion onnonlinear solution strategies and two common problems that occur with constitutive models are presentedbelow. An in depth discussion of many of the restrictions and pitfalls involved in numerical analysis ofgeotechnical problems is given by Potts and Zdravkovic (1999, 2000).

8.2 Nonlinear Numerical Analysis

When analysing any boundary value problem, four basic solution requirements need to be satisfied:equilibrium, compatibility, constitutive behaviour and the boundary conditions. Nonlinearity introduced bythe constitutive behaviour causes the governing equations to be reduced to an incremental form. Forexample for the finite element method the equations take the form:

[ ] { } { } iG

inG

iG RdK ∆=∆ (17)

where [KG]i is the incremental global system stiffness matrix, {∆d}inG is the vector of incremental nodal

displacements, {∆RG}i is the vector of incremental nodal forces and i is the increment number. To obtain asolution to a boundary value problem, the change in boundary conditions is applied in a series of incrementsand for each increment Equation (17) must be solved. The final solution is obtained by summing the resultsof each increment. Due to the nonlinear constitutive behaviour, the incremental global stiffness matrix [KG]i

is dependent on the current stress and strain levels and therefore is not constant, but varies over anincrement. Unless a very large number of small increments are used this variation should be accounted for.Hence, the solution of Equation (17) is not straightforward and different solution strategies exist. Theobjective of all such strategies is the solution of Equation (17), ensuring satisfaction of the four basicsolution requirements listed above.

This strategy is a key component of a nonlinear analysis, as it can strongly influence the accuracy of theresults and the computer resources required to obtain them. Many different solution strategies exist but fewcomparative studies have been performed to establish their merits for geotechnical analysis. To illustrate thelimitations and pitfalls that can arise three different categories (namely, tangent stiffness, visco-plastic andmodified Newton-Raphson) of solution algorithm are considered and compared.

8.3 Tangent Stiffness Method

8.3.1 Introduction

The tangent stiffness method, sometimes called the variable stiffness method, is the simplest solutionstrategy. This is the method implemented in a variety of computer codes including CRISP (Britto and Gunn,1987), which is widely used in engineering practice.

In this approach, the incremental stiffness matrix [KG]i in Equation (17) is assumed to be constant overeach increment and is calculated using the current stress state at the beginning of each increment. This isequivalent to making a piece-wise linear approximation to the nonlinear constitutive behaviour. To illustrate

the application of this approach, the simple problem of a uniaxiallyloaded bar of nonlinear material is considered, see Figure 23. Ifthis bar is loaded, the true load displacement response is shown inFigure 24. This might represent the behaviour of a strainhardening plastic material which has a very small initial elasticdomain.

8.3.2 Numerical Implementation

In the tangent stiffness approach the applied load is split into asequence of increments. In Figure 24 three increments of load areshown as ∆R1, ∆R2 and ∆R3. The analysis starts with theapplication of ∆R1. The incremental global stiffness matrix [KG]1

for this increment is evaluated based on the unstressed state of thebar corresponding to point ‘a’. For an elasto-plastic material thismight be constructed using the elastic constitutive matrix [D].Equation (17) is then solved to determine the nodal displacements{∆d}1

nG . As the material stiffness is assumed to remain constant,the load displacement curve follows the straight line‘ab′’ on Figure 24. In reality, the stiffness of thematerial does not remain constant during thisloading increment and the true solution isrepresented by the curved path ‘ab’. There istherefore an error in the predicted displacementequal to the distance ‘b′b’, however in the tangentstiffness approach this error is neglected. Thesecond increment of load, ∆R2, is then applied, withthe incremental global stiffness matrix [KG]2

evaluated using the stresses and strains appropriateto the end of increment 1, i.e., point ‘b′’ on Figure24. Solution of Equation (17) then gives the nodaldisplacements {∆d}2

nG. The load displacementcurve follows the straight path ‘b′c′’ on Figure 24.This deviates further from the true solution, theerror in the displacements now being equal to thedistance ‘c′c’. A similar procedure now occurswhen ∆R3 is applied. The stiffness matrix [KG]3 isevaluated using the stresses and strains appropriateto the end of increment 2, i.e., point ‘c′’ on Figure 24. The load displacement curve moves to point ‘d′’ andagain drifts further from the true solution. Clearly, the accuracy of the solution depends on the size of theload increments. For example, if the increment size were reduced so that more increments were needed toreach the same accumulated load, the tangent stiffness solution would be nearer to the true solution.

From the above simple example it may be concluded that in order to obtain accurate solutions to stronglynonlinear problems many small solution increments are required. The results obtained using this method candrift from the true solution and the stresses can fail to satisfy the constitutive relations. Thus the basicsolution requirements may not be fulfilled. As shown later in this paper, the magnitude of the error isproblem dependent and is affected by the degree of material nonlinearity, the geometry of the problem andthe size of the solution increments used. Unfortunately, in general, it is impossible to predetermine the sizeof solution increment required to achieve an acceptable error.

The tangent stiffness method can give particularly inaccurate results when soil behaviour changes fromelastic to plastic or vice versa. For instance, if an element is in an elastic state at the beginning of anincrement, it is assumed to behave elastically over the whole increment. This is incorrect if during theincrement the behaviour becomes plastic and results in an illegal stress state that violates the constitutivemodel. Such illegal stress states can also occur for plastic elements if the increment size used is too large,for example a tensile stress state could be predicted for a constitutive model that cannot sustain tension.

Figure 24 : Application of the tangent stiffnessalgorithm to the uniaxial loading of a bar of

nonlinear material

Figure 23 : Uniaxial loading of a bar

This can be a major problem with critical state type models, such as modified Cam clay, which employ a v-ln p ′ relationship (v = specific volume, p′ = mean effective stress), since a tensile value of p′ cannot beaccommodated. In that case, either the analysis has to be aborted or the stress state has to be modified insome arbitrary way, which would cause the solution to violate the equilibrium condition and the constitutivemodel.

8.3.3 Uniform Compression Of A Mohr-Coulomb Soil

To illustrate some of the above deficiencies, drainedone-dimensional loading of a soil element (i.e., an idealoedometer test) is considered. This is shown graphically inFigure 25. Lateral movements are restrained and the soilsample is loaded vertically by specifying verticalmovements along its top surface. No side friction isassumed and therefore the soil experiences uniform stressesand strains. Consequently, to model this using finiteelements, only a single element is needed. There is nodiscretisation error, the finite element program isessentially used only to integrate the constitutive model over the loading path.

Firstly, it is assumed that the soil behaves according to a linear elastic perfectly plastic Mohr-Coulombmodel with Young’s modulus, E′ = 10000kPa, Poisson’s ratio, ν = 0.2, cohesion, c′ = 0, angle of shearingresistance φ′ = 30o and angle of dilation, ψ = 30o. As the angles of dilation, ψ, and shearing resistance, φ′,are the same, the model is associated. For this analysis the yield function and plastic potential are given by:

{ }{ }( )( )

01

tan

=−

′+′

′=′

θφ

σgp

c

Jk,F (18)

where

( )

3

sinsincos

sinφθθ

φθ ′+

′=g , (19)

mean effective stress

( ) 3/321 σσσ ′+′+′=′p , (20)

deviatoric stress

( ) ( ) ( )213

232

221

6

1 σσσσσσ ′−′+′−′+′−′=J , (21)

Lode’s angle

( )( )

−

′−′′−′

= − 123

1tan

31

321

σσσσ

θ , (22)

and {k} is a vector of hardening or softening parameters, which in turn may depend on the state parameters.For this material the values of the vector {k} are all zero.

It is also assumed that the soil sample has an initial isotropic stress σ v′ = σ h′ = 50kPa and that loading isalways sufficiently slow to ensure drained conditions.

Figure 26 shows the stress path in J-p′ space predicted by a tangent stiffness analysis in which equalincrements of displacement were applied to the top of the sample. Each increment gave an incremental axialstrain ∆εa = 3%. Also shown on the figure is the true solution. This was obtained by noting that initially thesoil is elastic and that it only becomes elasto-plastic when it reaches the Mohr-Coulomb yield curve. In J-p′space it can be shown that the elastic stress path is given by:

Figure 25 : Uniform one-dimensionalcompression

( )( )

( ) ( )50866.013

213−′=′−′

+−

= pppJ iνν (23)

The Mohr-Coulomb yield curve is given by Equation (18), which, with the parameter values listed above,gives:

pJ ′= 693.0 (24)

Combining Equations (23) and (24)gives the stress state at which the stresspath reaches the yield surface. Thisoccurs when J = 173kPa andp′ = 250kPa. It can be shown that thisoccurs when the applied axial strainεa = 3.6%. Consequently, the truesolution follows the path ‘abc’, where‘ab’ is given by Equation (23) and ‘bc’is given by Equation (24).

Inspection of Figure 26 indicates adiscrepancy between the tangentstiffness and the true solution, with theformer lying above the latter andindicating a higher angle of shearingresistance, φ′ . The reason for thisdiscrepancy can be explained asfollows.

For the first increment of loading thematerial constitutive matrix is assumedto be elastic and the predicted stresspath follows the path ‘ab′’. Because the applied incremental axial strain is only ∆εa=3%, this is less thanεa=3.6% which is required to bring the soil to yield at point ‘b’. As the soil is assumed to be linear elastic,the solution for this increment is therefore correct. For the second increment of loading the incrementalglobal stiffness matrix [KG]2 is based on the stress state at the end of increment 1 (i.e. point ‘b′’). Since thesoil is elastic here, the elastic constitutive matrix [D] is used again. The stress path now moves to point ‘c′’.As the applied strain εa = (∆εa

1+∆εa2) = 6% is greater than εa = 3.6%, which is required to bring the soil to

yield at point ‘b’, the stress state now lies above the Mohr-Coulomb yield surface. The tangent stiffnessalgorithm has overshot the yield surface. For increment three the algorithm realises the soil is plastic atpoint ‘c′’ and forms the incremental global stiffness matrix [KG]3 based on the elasto-plastic matrix, [Dep],consistent with the stress state at ‘c′’. The stress path then moves to point ‘d′’. For subsequent incrementsthe algorithm uses the elasto-plastic constitutive matrix, [Dep], and traces the stress path ‘d′e′’.

The reason why this part of the curve is straight, with an inclination greater than the correct solution, path‘bc’, can be found by inspecting the elasto-plastic constitutive matrix, [Dep], defined by Equation (25). Inthis expression F is the yield surface, P is the plastic potential, {m} is a vector of state parameters, the valuesof which are immaterial, and the parameter A reflects the influence of hardening or softening on theincremental stress-strain response. For perfect plasticity A = 0.

[ ] [ ][ ] { }{ }( ) { }{ }( ) [ ]

{ }{ }( ) [ ] { }{ }( )A

mPD

kF

DkFmP

D

DDT

T

ep

+

∂∂

∂∂

∂∂

∂∂

−=

σσ

σσ

σσ

σσ

,,

,,

(25)

For the current model the elastic [D] matrix is constant, and as the yield and plastic potential functionsare both assumed to be given by Equation (18), the variation of [Dep] depends on the values of the partialdifferentials of the yield function with respect to the stress components. In this respect, it can be shown thatthe gradient of the stress path in J-p′ space is given by:

Figure 26 : Oedometer stress path predicted by the tangentstiffness algorithm

{ }{ }( )

{ }{ }( ) p

J

J

kF

p

kF

′=

∂′∂

′∂′∂

,

,

σ

σ

(26)

As this ratio is first evaluated at point‘c′’, which is above the Mohr-Coulombyield curve, the stress path sets off at thewrong gradient for increment three. Thiserror remains for all subsequentincrements.

The error in the tangent stiffnessapproach can therefore be associated withthe overshoot at increment 2. If theincrement sizes had been selected such thatat the end of an increment the stress pathjust reached the yield surface (i.e., at point‘b’), the tangent stiffness algorithm wouldthen give the correct solution. This isshown in Figure 27 where, in analysislabelled A, the first increment was selectedsuch that ∆εa = 3.6%. After this incrementthe stress state was at point b, which iscorrect, and for increment 2 andsubsequent increments the correct elasto-plastic constitutive matrix [Dep] was used to obtain the incremental stiffness matrix [KG]i. As the matrix[Dep] remains constant along the stress path ‘bc’, the solution is independent of the size of the incrementsfrom point ‘b’ onwards. In analysis labelled B in Figure 27, a much larger first increment, ∆εa = 10%, wasapplied. This causes a large overshoot on the first increment and results in a significant divergence from thetrue solution even if the subsequent increments are reduced to 1%. As noted above, once the stress state hasovershot, making subsequent increments smaller does not improve the solution.

It can be concluded that, for this particular problem, the tangent stiffness algorithm is always in error,unless the increment size is such that at the end of an increment the stress state happens to be at point ‘b’.Because the solution to this simple one dimensional problem is known, it can be arranged for this to occur,as for analysis A in Figure 27. However, in general multi-axis boundary value problems, the answers towhich are not known, it is impossible to choose the correct increment sizes so that overshoot never occurs.The only solution is to use a very large number of small load increments and hope for the best.

Another source of error arising from the way the tangent stiffness method works is that the answersdepend on the way the yield function is implemented. While it is perfectly acceptable, from a mathematicalpoint of view, to write the yield surface in either of the forms shown in Equations (27) or (18), thepredictions from the tangent stiffness algorithm will differ if, as is usually the case, overshoot occurs.

{ }{ }( ) ( ) 0tan

, =

′+′

′−=′ θ

φσ gp

cJkF (27)

For the simple oedometer situation, this can be seen by calculating the partial differentials in Equation(26), for the yield function given by Equation (27). This gives:

{ }{ }( )

{ }{ }( ) ( )θσ

σ

g

J

kF

p

kF

=

∂′∂

′∂′∂

,

,

(28)

Figure 27 : Effect of the first increment size on a tangentstiffness prediction of an oedometer stress path

As noted above, this equation gives thegradient of the resulting stress path in J-p′ space.Whereas Equation (26) indicates that theinclination depends on the amount of overshoot,Equation (28) indicates that the inclination isconstant and equal to the gradient of the Mohr-Coulomb yield curve. The two results arecompared in Figure 28. The stress path based onthe yield function written in the form of Equation(18) appears to pass through the origin of stressspace, but to have an incorrect slope, indicating avalue of φ′ that is too high, but the correct value ofc′. In contrast, the stress path based on the yieldfunction written in the form of Equation (27) isparallel to the true solution, but does not passthrough the origin of stress space, indicating thatthe material has a fictitious c′, but the correct φ′.Clearly, if there is no overshoot, both formulations give the same result, which for this problem agrees withthe true solution. The reason for this inconsistency is that, in theory, the differentials of the yield functionare only valid if the stress state is on the yield surface, i.e., F({σ′},{k}) = 0. If it is not, it is thentheoretically incorrect to use the differentials and inconsistencies will arise. The implications for practiceare self-evident. Two different pieces of software which purport to use the same Mohr-Coulomb conditioncan give very different results, depending on the finer details of their implementation. This is clearly yetanother draw back with the tangent stiffness algorithm for nonlinear analysis.

The analysis labelled A in Figure 27 was performed with a first increment of axial strain of ∆εa = 3.6%and subsequent increments of ∆εa = 1%. As the first increment just brought the stress path to the yieldsurface, the results from this analysis are inagreement with the true solution. The situation isnow considered where, after being loaded to point‘c’, see Figure 27, the soil sample is unloaded withtwo increments of ∆εa = -1%. The results of thisanalysis are shown in Figure 29. The predictedstress path on unloading is given by path ‘cde’,which indicates that the soil remains plastic and thestress path stays on the yield surface. This isclearly incorrect as such behaviour violates thebasic postulates of elasto-plastic theory. Whenunloaded, the soil sample should become purelyelastic, and the correct stress path is marked as path‘cf’ on Figure 29. Because the soil has constantelastic parameters, this path is parallel to the initialelastic loading path ‘ab’. The reason for the errorin the tangent stiffness analysis arises from the factthat when the first increment of unloading occurs,the stress state is plastic, i.e., point ‘c’. The algorithm does not “know” that unloading is going to occur, sowhen it forms the incremental global stiffness matrix, it uses the elasto-plastic constitutive matrix [Dep]. Theresult is that the stress path remains on the yield surface after application of the unloading increment. Sincethe soil is still on the yield surface, the same procedure occurs for the second increment of unloading.

8.3.4 Uniform Compression Of A Modified Cam Clay Soil

The above one-dimensional loading problem is now repeated with the soil represented by a simplifiedform of the modified Cam clay model. The soil parameters are listed in Table 6.

Because a constant value of the critical state stress ratio in J-p′ space, MJ, has been used, the yield (andplastic potential) surface plots as a circle in the deviatoric plane. A further simplification has been made for

Figure 29 : Example of an unloading stress pathusing the tangent stiffness algorithm

Figure 28 : Effect of yield function implementationon errors associated with the tangent stiffness

algorithm

(18)

(27)

Table 6. Properties for modified Cam clay model

Specific volume at unit pressure on virginconsolidation line, v1

1.788

Slope of virgin consolidation line in v-ln p′ space, λ 0.066

Slope of swelling line in v-ln p′ space, κ 0.0077

Slope of critical state line in J-p′ space, MJ 0.693

the present analysis. Instead ofusing the slope of the swellingline to calculate the elastic bulkmodulus, constant elasticparameters, E′ = 50000kPa andν = 0.26, have been used. Thissimplification has been made tobe consistent with resultspresented in the next section ofthis paper. For the presentinvestigation, it does notsignificantly affect soilbehaviour and therefore any conclusions reached are valid for the full model. Again the initial stresses areσ v′ = σ h′ = 50kPa, and the soil is assumed to be normally consolidated. This later assumption implies thatthe initial isotropic stress state is on the yield surface.

Three tangent stiffness analyses, with displacement controlled loading increments equivalent to∆εa = 0.1%, ∆εa = 0.4% and ∆εa = 1% respectively, have been performed. The predicted stress paths areshown in Figure 30. Also shown in this figure is the true solution. Consider the analysis with the smallestincrement size, ∆εa = 0.1%. Apart from the very first increment the results of this analysis agree with thetrue solution. This is not so for the other two analyses. For the analysis with ∆εa = 0.4% the stress path is inconsiderable error for the first threeincrements. Subsequently the stresspath is parallel to the true solution,however, there is still a substantialerror. Matters are even worse for theanalysis with the largest incrementsize, ∆εa = 1%. This has very largeerrors initially.

The reason for the errors in theseanalyses is the same as that explainedabove for the Mohr-Coulomb analysis.That is the yield (and plastic potential)derivatives are evaluated in illegalstress space, i.e., with stress valueswhich do not satisfy the yield (orplastic potential) function. This ismathematically wrong and leads toincorrect elasto-plastic constitutivematrices. The reason why the errorsare much greater than for the Mohr-Coulomb analyses is that the yield(and plastic potential) derivatives are not constant on the yield (or plastic potential) surface, as they are withthe Mohr-Coulomb model, but vary. Matters are also not helped by the fact that the model is strainhardening/softening and, once the analysis goes wrong, incorrect plastic strains and hardening/softeningparameters are subsequently calculated.

The comments made above for the Mohr-Coulomb model on implementation of the yield function and onunloading also apply here. In fact, they apply to any constitutive model because they are caused by flaws inthe tangent stiffness algorithm itself.

Figure 30 : Effect of increment size on the tangent stiffnessprediction of an oedometer stress path

8.4 Visco-plastic Method

8.4.1 Introduction

This method uses the equations of visco-plastic behaviour andtime as an artifice to calculate the behaviour of nonlinear, elasto-plastic, time independent materials (Owen and Hinton, 1980;Zienkiewicz and Cormeau, 1974). The method was originallydeveloped for linear elastic visco-plastic (i.e., time dependent)material behaviour. Such a material can be represented by anetwork of the simple rheological units shown in Figure 31. Eachunit consists of an elastic and a visco-plastic component connectedin series. The elastic component is represented by a spring and thevisco-plastic component by a slider and dashpot connected inparallel. If a load is applied to the network, then one of twosituations occurs in each individual unit. If the load is such that theinduced stress in the unit does not cause yielding, the slider remainsrigid and all the deformation occurs in the spring. This representselastic behaviour. Alternatively, if the induced stress causesyielding, the slider becomes free and the dashpot is activated. Asthe dashpot takes time to react, initially all deformation occurs inthe spring. However, with time the dashpot moves. The rate ofmovement of the dashpot depends on the stress it supports and itsfluidity. With time progressing, the dashpot moves at a decreasing rate, because some of the stress the unitis carrying is dissipated to adjacent units in the network, which as a result suffer further movementsthemselves. This represents visco-plastic behaviour. Eventually, a stationary condition is reached where allthe dashpots in the network stop moving and are no longer sustaining stresses. This occurs when the stressin each unit drops below the yield surface and the slider becomes rigid. The external load is now supportedpurely by the springs within the network, but, importantly, straining of the system has occurred not only dueto compression or extension of the springs, but also due to movement of the dashpots. If the load was nowremoved, only the displacements (strains) occurring in the springs would be recoverable, the dashpotdisplacements (strains) being permanent.

8.4.2 Numerical Implementation

Application to finite element analysis of elasto-plastic materials can be summarised as follows. Onapplication of a solution increment the system is assumed to instantaneously behave linear elastically. If theresulting stress state lies within the yield surface, the incremental behaviour is elastic and the calculateddisplacements are correct. If the resulting stress state violates yield, the stress state can only be sustainedmomentarily and visco-plastic straining occurs. The magnitude of the visco-plastic strain rate is determinedby the value of the yield function, which is a measure of the degree by which the current stress state exceedsthe yield condition. The visco-plastic strains increase with time, causing the material to relax with areduction in the yield function and hence the visco-plastic strain rate. A marching technique is used to stepforward in time until the visco-plastic strain rate is insignificant. At this point, the accumulated visco-plasticstrain and the associated stress change are equal to the incremental plastic strain and stress changerespectively. This process is illustrated for the simple problem of a uniaxially loaded bar of nonlinearmaterial in Figure 32.

For genuine visco-plastic materials the visco-plastic strain rate is given by:

{ } { }{ }( ) { }{ }( ){ }σ

σσγε∂

=

∂∂ mP

F

kFf

t o

vp ,,

(29)

where γ is the dashpot fluidity parameter and Fo is a stress scalar to non-dimensionalise F({σ},{k})(Zienkiewicz and Cormeau, 1974). When the method is applied to time independent elasto-plastic materials,both γ and Fo can be assumed to be unity (Griffiths, 1980).

Figure 31 : Rheological model forvisco-plastic material

In order to use the procedure described above,a suitable time step, ∆t, must be selected. If ∆t issmall many iterations are required to obtain anaccurate solution. However, if ∆t is too largenumerical instability can occur. The mosteconomical choice for ∆t is the largest value thatcan be tolerated without causing such instability.An estimate for this critical time step is suggestedby Stolle and Higgins (1989).

Due to its simplicity, the visco-plasticalgorithm has been widely used. However, themethod has severe limitations for geotechnicalanalysis. Firstly, the algorithm relies on the factthat for each increment the elastic parametersremain constant. The simple algorithm cannotaccommodate elastic parameters that vary duringthe increment because, for such cases, it cannotdetermine the true elastic stress changesassociated with the incremental elastic strains.The best that can be done is to use the elasticparameters associated with the accumulatedstresses and strains at the beginning of the increment to calculate the elastic constitutive matrix, [D], andassume that this remains constant for the increment. Such a procedure only yields accurate results if theincrements are small or the elastic nonlinearity is not great.

A more severe limitation of the method arises when the algorithm is used as an artifice to solve problemsinvolving non-viscous material (i.e., elasto-plastic materials). As noted above, the visco-plastic strains arecalculated using Equations (29). In Equation (29) the partial differentials of the plastic potential areevaluated at an illegal stress state {σ}t, which lies outside the yield surface, i.e., F({σ′},{k})>0. As noted forthe tangent stiffness method,this is theoretically incorrectand results in failure to satisfythe constitutive equations.The magnitude of the errordepends on the constitutivemodel and in particular onhow sensitive the partialderivatives are to the stressstate. This is now illustratedby applying the visco-plasticalgorithm to the one-dimensional loading problem(i.e., ideal oedometer test)considered above for thetangent stiffness method.

8.4.3 Uniform CompressionOf A Mohr-CoulombSoil

As with the tangentstiffness method, the problem shown graphically in Figure 25, with the soil properties given in the previousMohr-Coulomb example is considered. Figure 33 shows the stress path in J-p′ space predicted by a visco-plastic analysis in which equal increments of vertical displacement were applied to the top of the sample.Each increment gave an axial strain ∆εa = 3% and therefore the predictions in Figure 33 are directlycomparable to those for the tangent stiffness method given in Figure 26. The results were obtained using thecritical time step. It can be seen that the visco-plastic predictions are in remarkably good agreement with the

Figure 32 : Application of the visco-plastic algorithmto the uniaxial loading of a bar of a nonlinear material

Figure 33 : Oedometer stress path predicted by the visco-plastic algorithm

true solution. Even when the increment size was doubled (i.e., ∆εa = 6%), the predictions did not changesignificantly. Due to the problem highlighted above, concerning evaluation of the plastic potentialdifferentials in illegal stress space, there were some small differences, but these only caused changes in thefourth significant figure for both stress and plastic strains. Predictions were also insensitive to values of thetime step as long as it was not greater than the critical value.

The results were therefore not significantly dependent on either the solution increment size or the timestep. The algorithm was also able to deal accurately with the change from purely elastic to elasto-plasticbehaviour and vice versa. In these respects the algorithm behaved much better than the tangent stiffnessmethod.

It can therefore be concluded that the visco-plastic algorithm works well for this one-dimensional loadingproblem with the Mohr-Coulomb model. It has also been found that it works well for other boundary valueproblems, involving either the Tresca or the Mohr-Coulomb model.

8.4.4 Uniform Compression Of A Modified Cam Clay Soil

The one-dimensional loading problem was repeated with the soil represented by the simplified modifiedCam clay model described previously. This model has linear elastic behaviour and therefore the problem ofdealing with nonlinear elasticity is not relevant. In fact, it was because of this deficiency in the visco-plasticalgorithm that the model was simplified. The soil properties are given in Table 6 and the initial conditionsare discussed above.

Results from four visco-plastic analyses, with displacement controlled loading increments equivalent to∆εa = 0.01%, ∆εa = 0.1%, ∆εa = 0.4% and ∆εa = 1%, are compared with the true solution in Figure 34. Theresults have been obtained using the critical time step and the convergence criteria was set such that theiteration process stopped when there was no change in the fourth significant figure of the incrementalstresses and incremental plastic strains.

Only the solution with the smallest increment size (i.e., ∆εa = 0.01%) agrees with the true solution. It isinstructive to compare these results with those given in Figure 30 for the tangent stiffness method. In viewof the accuracy of the analysis with the Mohr-Coulomb model, it is, perhaps, surprising that the visco-plasticalgorithm requires smaller increments than the tangent stiffness method to obtain an accurate solution. It isalso of interest to note that when the increment size is too large, the tangent stiffness predictions lie abovethe true solution, whereas for the visco-plastic analyses the opposite occurs, with the predictions lying belowthe true solution. The visco-plastic solutions are particularly in error during the early stages of loading, seeFigure 34b.

To explain why the visco-plastic solutions are in error, consider the results shown in Figure 35. The truesolution is marked as a dashed line on this plot. A visco-plastic analysis consisting of a single increment,equivalent to ∆εa = 1%, is performed starting from point ‘a’, which is on the true stress path. To do this inthe analysis, the initial stresses are set appropriate to point ‘a’: σ v′ = 535.7kPa and σ h′ = 343.8kPa. This

loading increment should move the stress path from point ‘a’ to point ‘e’. The line ‘ae’ therefore represents

Figure 34 : Effect of increment size on the visco-plastic prediction of an oedometer stress path

the true solution to which the visco-plasticanalysis can be compared. However, the visco-plastic analysis actually moves the stress pathfrom point ‘a’ to point ‘d’, thus incurring asubstantial error.

To see how such an error arises, theintermediate steps involved in the visco-plasticalgorithm are plotted in Figure 35. These canbe explained as follows. Initially, on the firstiteration, the visco-plastic strains are zero andthe stress change is assumed to be entirelyelastic. This is represented by the stress state atpoint ‘b’. This stress state is used to evaluatethe first contribution to the incremental visco-plastic strains. These strains are thereforebased on the normal to the plastic potentialfunction at ‘b’. This normal is shown on Figure 35 and should be compared to that shown for point ‘a’,which provides the correct solution. As the directions of the normals differ significantly, the resultingcontribution to the visco-plastic strains is in error (note: along path ‘ae’ of the true solution, the normal to theplastic potential does not change significantly, being very similar to that at point ‘a’). This contribution tothe visco-plastic strains is used to calculate a correction vector. They are also used to update thehardening/softening parameter for the constitutive model. A second iteration is performed which, due to thecorrection vector, gives different incremental displacements and incremental total strains. The incrementalstresses are also now different as they depend on these new incremental total strains and the visco-plasticstrains calculated for iteration 1. The stress state is now represented by point ‘c’. A second contribution tothe visco-plastic strains is calculated based on the plastic potential at point ‘c’. Again, this is in error becausethis is an illegal stress state. The error is related to the difference in direction of the normals to the plasticpotential surfaces at points ‘a’ and ‘c’. This second contribution to the incremental plastic strains is used toobtain an additional correction vector. A third iteration is then performed which brings the stress state topoint ‘d’ on Figure 35. Subsequent iterations cause only very small changes to the visco-plastic strains andthe incremental stresses and therefore the stress state remains at point ‘d’. At the end of the iterativeprocess, the incremental plastic strains are equated to the visco-plastic strains. As the visco-plastic strainsare the sum of the contributions obtained from each iteration, and as each of these contributions has beencalculated using the incorrect plastic potential differentials (i.e., wrong direction of the normal), theincremental plastic strains are in error. This is evident from Figure 36, which compares the predicted andtrue incremental plastic strains. Since thehardening parameter for the model is calculatedfrom the plastic strains, this is also incorrect. Itis therefore not surprising that the algorithmends up giving the wrong stress staterepresented by point ‘d’ in Figure 35.

If the soil sample is unloaded at any stage,the analysis indicates elastic behaviour andtherefore behaves correctly.

It is concluded that for complex critical stateconstitutive models the visco-plastic algorithmcan involve severe errors. The magnitude ofthese errors depends on the finer details of themodel and, in particular, on how rapidly theplastic potential differentials vary with changesin stress state. The problems associated withthe implementation of a particular constitutivemodel, as discussed for the tangent stiffnessmethod, also apply here. As the plastic strainsare calculated from plastic potential differentials evaluated in illegal stress space, the answers depend on the

Figure 36 : Comparison of incremental plastic strainsfrom a single increment of a visco-plastic analysis and

the true solution

Figure 35 : A single increment of a visco-plasticanalysis

finer details of how the model is implemented in the software. Again, two pieces of software which purportto use the same equations could give different results.

The above conclusion is perhaps surprising as the visco-plastic algorithm appears to work well for simpleconstitutive models of the Tresca and Mohr-Coulomb types. However, as noted previously, in these simplermodels the differentials of the plastic potential do not vary by a great amount when the stress state movesinto illegal stress space.

8.5 Modified Newton-Raphson Method

8.5.1 Introduction

The previous discussion of both the tangent stiffness and visco-plastic algorithms has demonstrated thaterrors can arise when the constitutive behaviour is based on illegal stress states. The modified Newton-Raphson (MNR) algorithm described in this section attempts to rectify this problem by only evaluating theconstitutive behaviour in, or very near to, legal stress space.

The MNR method uses an iterative technique to solve Equation (17). The first iteration is essentially thesame as the tangent stiffness method. However, it is recognised that the solution is likely to be in error andthe predicted incremental displacements are used to calculate the residual load, a measure of the error in theanalysis. Equation (17) is then solved again with this residual load, {Ψ }, forming the incremental righthand side vector. Equation (17) can be rewritten as:

[ ] { }( ) { } 1−= jjinG

iG dK Ψ∆ (30)

The superscript ‘j’ refers to the iteration number and{Ψ }0 = {∆RG}i. This process is repeated until theresidual load is small. The incremental displacementsare equal to the sum of the iterative displacements.This approach is illustrated in Figure 37 for the simpleproblem of a uniaxially loaded bar of nonlinearmaterial. In principle, the iterative scheme ensures thatfor each solution increment the analysis satisfies allsolution requirements.

A key step in this calculation process is determiningthe residual load vector. At the end of each iteration thecurrent estimate of the incremental displacements iscalculated and used to evaluate the incremental strainsat each integration point. The constitutive model isthen integrated along the incremental strain paths toobtain an estimate of the stress changes. These stresschanges are added to the stresses at the beginning of theincrement and used to evaluate consistent equivalentnodal forces. The difference between these forces andthe externally applied loads (from the boundaryconditions) gives the residual load vector. A differencearises because a constant incremental global stiffnessmatrix [KG]i is assumed over the increment. Due to thenonlinear material behaviour, [KG]i is not constant but varies with the incremental stress and strain changes.

Since the constitutive behaviour changes over the increment, care must be taken when integrating theconstitutive equations to obtain the stress change. Methods of performing this integration are termed stresspoint algorithms and both explicit and implicit approaches have been proposed in the literature. There aremany of these algorithms in use and, as they control the accuracy of the final solution, users must verify theapproach used in their software. Two of the most accurate stress point algorithms are describedsubsequently.

The process described above is called a Newton-Raphson scheme if the incremental global stiffnessmatrix [KG]i is recalculated and inverted for each iteration, based on the latest estimate of the stresses and

Figure 37 : Application of the modified Newton-Raphson algorithm to the uniaxial loading of a

bar of linear material

strains obtained from the previous iteration. To reduce the amount of computation, the modified Newton-Raphson method only calculates and inverts the stiffness matrix at the beginning of the increment and uses itfor all iterations within the increment. Sometimes the incremental global stiffness matrix is calculated usingthe elastic constitutive matrix, [D], rather than the elasto-plastic matrix, [Dep]. Clearly, there are severaloptions here and many software packages allow the user to specify how the MNR algorithm should work. Inaddition, an acceleration technique is often applied during the iteration process (Thomas, 1984).

8.5.2 Stress Point Algorithms

Two classes of stress point algorithms are considered. The substepping algorithm is essentially explicit,whereas the return algorithm is implicit. In both the substepping and return algorithms, the objective is tointegrate the constitutive equations along an incremental strain path. While the magnitudes of the strainincrement are known, the manner in which they vary during the increment is not. It is therefore not possibleto integrate the constitutive equations without making an additional assumption. Each stress point algorithmmakes a different assumption and this influences the accuracy of the solution obtained.

The schemes presented by Wissman and Hauck (1983) and Sloan (1987) are examples of substeppingstress point algorithms. In this approach, the incremental strains are divided into a number of substeps. It isassumed that in each substep the strains {∆εss} are a proportion, ∆T, of the incremental strains {∆εinc}. Thiscan be expressed as:

{ } { }incss T εε ∆∆=∆ (31)

It should be noted that in each substep, the ratio between the strain components is the same as that for theincremental strains and hence the strains are said to vary proportionally over the increment. The constitutiveequations are then integrated numerically over each substep using either an Euler, modified Euler or Runge-Kutta scheme. The size of each substep (i.e., ∆T) can vary and, in the more sophisticated schemes, isdetermined by setting an error tolerance on the numerical integration. This allows control of errors resultingfrom the numerical integration procedure and ensures that they are negligible.

The basic assumption in these substepping approaches is therefore that the strains vary in a proportionalmanner over the increment. In some boundary value problems, this assumption is correct and consequentlythe solutions are extremely accurate. However, in general, this may not be true and an error can beintroduced. The magnitude of the error is dependent on the size of the solution increment.

The schemes presented by Borja and Lee (1990) and Borja (1991) are examples of one-step implicit typereturn algorithms. In this approach, the plastic strains over the increment are calculated from the stressconditions corresponding to the end of the increment. The problem, of course, is that these stress conditionsare not known, hence the implicit nature of the scheme. Most formulations involve some form of elasticpredictor to give a first estimate of the stress changes, coupled with a sophisticated iterative sub-algorithm totransfer from this stress state back to the yield surface. The objective of the iterative sub-algorithm is toensure that, on convergence, the constitutive behaviour is satisfied, albeit with the assumption that the plasticstrains over the increment are based on the plastic potential at the end of the increment. Many differentiterative sub-algorithms have been proposed in the literature. In view of the findings presented previously, itis important that the final converged solution does not depend on quantities evaluated in illegal stress space.In this respect some of the earlier return algorithms broke this rule and are therefore inaccurate.

The basic assumption in these approaches is therefore that the plastic strains over the increment can becalculated from the stress state at the end of the increment. This is theoretically incorrect as the plasticresponse, and in particular the plastic flow direction, is a function of the current stress state. The plastic flowdirection should be consistent with the stress state at the beginning of the solution increment and shouldevolve as a function of the changing stress state, such that at the end of the increment it is consistent with thefinal stress state. This type of behaviour is exemplified by the substepping approach. If the plastic flowdirection does not change over an increment, the return algorithm solutions are accurate. Invariably,however, this is not the case and an error is introduced. The magnitude of any error is dependent on the sizeof the solution increment.

Potts and Ganendra (1994) performed a fundamental comparison of these two types of stress pointalgorithm. They conclude that both algorithms give accurate results, but, of the two, the substeppingalgorithm is better.

Another advantage of the substepping approach is that it is extremely robust and can easily deal withconstitutive models in which two or more yield surfaces are active simultaneously and for which the elasticportion of the model is highly nonlinear. In fact, most of the software required to program the algorithm iscommon to any constitutive model. This is not so for the return algorithm, which, although in theory canaccommodate such complex constitutive models, involves some extremely complicated mathematics. Thesoftware to deal with the algorithm is also constitutive model dependent. This means considerable effort isrequired to include a new or modified model.

As the MNR method involves iterations for each solution increment, convergence criteria must be set.This usually involves setting limits to the size of both the iterative displacements, ({∆d}i

nG)j, and the residualloads, {Ψ }

j. As both these quantities are vectors, it is normal to express their size in terms of the scalarnorms.

8.5.3 Uniform Compression Of Mohr-Coulomb And Modified Cam Clay Soils

The MNR method using a substepping stress point algorithm has been used to analyse the simple onedimensional oedometer problem, considered previously for both the tangent stiffness and visco-plasticapproaches. Results are presented in Figures 38a and 38b for the Mohr-Coulomb and modified Cam claysoils, respectively. To be consistent with the analyses performed with the tangent and visco-plasticalgorithm, the analysis for the Mohr-Coulomb soil involved displacement increments which gaveincremental strains ∆εa = 3%, whereas for the modified Cam clay analysis the increment size was equivalentto ∆εa = 1%.

The predictions are in excellent agreement with the true solution. An unload-reload loop is shown ineach figure, indicating that the MNR approach can accurately deal with changes in stress path direction. Forthe modified Cam clay analysis it should be noted that at the beginning of the test the soil sample wasnormally consolidated, with an isotropic stress p′ = 50kPa. The initial stress path is therefore elasto-plasticand not elastic. Consequently, it is not parallel to the unload/reload path. Additional analysis, performedwith different sizes of solution increment, indicate that the predictions, for all practical purposes, areindependent of increment size.

These results clearly show that, for this simple problem, the MNR approach does not suffer from theinaccuracies inherent in both the tangent stiffness and visco-plastic approaches. To investigate how thedifferent methods perform for more complex boundary value problems, a small parametric study has beenperformed, see Potts and Zdravkovic (1999). Some of the main findings of this study are presented next.

Figure 38 : Oedometer stress paths predicted by the MNR algorithm: a) Mohr-Coulomb and b) modifiedCam Clay models

Table 7. Modified Cam clay properties for drained triaxial tests

Overconsolidation ratio 1.0

Specific volume at unit pressure on virgin consolidationline, v1

1.788

Slope of virgin consolidation line in v-lnp′ space, λ 0.066

Slope of swelling line in v-lnp′ space, κ 0.0077

Slope of critical state line in J-p′ space, MJ 0.693

Elastic shear modulus, G/Preconsolidation pressure, p0′ 100

8.6 Comparison Of Solution Strategies

8.6.1 Introduction

A comparison of the three solution strategies presented above suggests the following. The tangentstiffness method is the simplest, but its accuracy is influenced by increment size. The accuracy of the visco-plastic approach is also influenced by increment size, if complex constitutive models are used. The MNRmethod is potentially the most accurate and is likely to be the least sensitive to increment size. However,considering the computer resources required for each solution increment, the MNR method is likely to be themost expensive, the tangent stiffness method the cheapest and the visco-plastic method is probablysomewhere in between. It may be possible though, to use larger and therefore fewer increments with theMNR method to obtain a similar accuracy. Thus, it is not obvious which solution strategy is the mosteconomic for a particular solution accuracy.

All three solution algorithms have been incorporated into the single computer program, ICFEP.Consequently, much of the computer code is common to all analyses and any difference in the results can beattributed to the different solution strategies. The code has been extensively tested against availableanalytical solutions and with other computer codes, where applicable. The program was used to compare therelative performance of each of the three schemes in the analysis of two simple idealised laboratory tests andthree more complex boundary value problems. Analyses of the laboratory tests were carried out using asingle four-noded isoparametric element with a single integration point, whereas eight-noded isoparametricelements, with reduced integration, were employed for the analyses of the boundary value problems.

As already shown, the errors in the solution algorithms are more pronounced for critical state type modelsthan for the simpler linear elastic perfectly plastic models (i.e., Mohr-Coulomb and Tresca). Hence, in thecomparative study the soil has been modelled with the modified Cam clay model. To account for thenonlinear elasticity that is present in this model, the visco-plastic algorithm was modified to incorporate anadditional stress correction based on an explicit stress point algorithm, similar to that used in the MNRmethod, at each time step, see Potts and Zdravkovic (1999).

8.6.2 Ideal Drained TriaxialTest

Idealised drained triaxialcompress ion tes ts wereconsidered. A cylindricalsample was assumed to bei s o t r o p i c a l l y n o r m a l l yconsolidated to a mean effectivestress, p ′ = 200kPa, with zeropore water pressure. The soilparameters used for the analysesare shown in Table 7.

Increments of compressiveaxial strain were applied to thesample until the axial strain reached 20%, while maintaining a constant radial stress and zero pore waterpressure. The results are presented as plots of volumetric strain and deviatoric stress, q, versus axial strain.Deviatoric stress q is defined as:

( ) ( ) ( )[ ]conditionstriaxialfor

2

13

31

213

232

221

σσ

σσσσσσ

′−′=

′−′+′−′+′−′== Jq(32)

Results of these analyses are presented in Figures 39, 40 and 41. The label associated with each line inthese plots indicates the magnitude of axial strain applied at each increment of that analysis. The tests weredeemed ideal as the end effects at the top and bottom of the sample were considered negligible and the stressand strain conditions were uniform throughout. Analytical solutions are also provided on the plots forcomparison purposes.

Results from the MNR analyses are compared withthe analytical solution in Figure 39. The results arenot sensitive to increment size and agree well with theanalytical solution. The tangent stiffness results arepresented in Figure 40. The results are sensitive toincrement size, giving very large errors for the largerincrement sizes. The deviatoric stress, q at failure(20% axial strain) is over predicted. The results of thevisco-plastic analyses are shown in Figure 41.Inspection of this figure indicates that the solution isalso sensitive to increment size. Even the results fromthe analyses with the smallest increment size of 0.1%are in considerable error.

It is of interest to note that results from the tangentstiffness analyses over predict the deviatoric stress atany particular value of axial strain. The opposite istrue for the visco-plastic analysis, where q is underpredicted in all cases. This is similar to theobservations made earlier for the simple oedometertest.

8.6.3 Footing Problem

A smooth rigid strip footing subjected to vertical loading, as depicted in Figure 42, has been analysed.The same soil constitutive model and parameters as used for the idealised triaxial test analyses, see Table 7,have been employed to model the soil which, in this case, was assumed to behave undrained. The finiteelement mesh is shown in Figure 43. Note that due to symmetry about the vertical line through the centre ofthe footing, only half of the problem needs to be considered in the finite element analysis. Plane strainconditions are assumed. Before loading the footing, the coefficient of earth pressure at rest, Ko, wasassumed to be unity and the vertical effective stress and pore water pressure were calculated using asaturated bulk unit weight of the soil of 20kN/m3 and a static water table at the ground surface. The footing

Figure 41 : Visco-plastic: Drained triaxial test

Figure 40 : Tangent stiffness: Drained triaxial testFigure 39 : Modified Newton-Raphson: Drainedtriaxial test

was loaded by applying a series of equally sized increments of vertical displacement until the totaldisplacement was 25mm.

The load-displacement curves for the tangent stiffness, visco-plastic and MNR analyses are presented inFigure 44. For the MNR method, analyses were performed using 1, 2, 5, 10, 25, 50 and 500 increments toreach a footing settlement of 25mm. With the exception of the analysis performed with only a singleincrement, all analyses gave very similar results and plot as a single curve, marked MNR on this figure. TheMNR results are therefore insensitive to increment size and show a well-defined collapse load of 2.8kN/m.

For the tangent stiffness approach, analyses using 25, 50, 100, 200, 500 and 1000 increments have beencarried out. Analyses with a smaller number of increments were also attempted, but illegal stresses(negative mean effective stresses, p′) were predicted. As the constitutive model is not defined for suchstresses, the analyses had to be aborted. Some finite element packages overcome this problem by arbitrarilyresetting the offending negative p′ values. There is no theoretical basis for this and it leads to violation ofboth the equilibrium and the constitutive conditions. Although such adjustments enable an analysis to becompleted, the final solution is in error.

Results from the tangent stiffness analyses are shown in Figure 44. When plotted, the curve from theanalysis with 1000 increments is indistinguishable from those of the MNR analyses. The tangent stiffnessresults are strongly influenced by increment size, with the ultimate footing load decreasing from 7.5kN/m to

2.8kN/m with reduction in thesize of the applied displacementincrement. There is also atendency for the load-displacement curve to continueto rise and not reach a well-defined ultimate failure load forthe analysis with large applieddisplacement increments. Theresults are unconservative, overpredicting the ultimate footingload. There is also no indicationfrom the shape of the tangentstiffness load-displacementcurves as to whether the solutionis accurate, since all the curveshave similar shapes.Figure 44 : Footing load-displacement curves

Figure 43 : Finite element mesh for footing analysisFigure 42 : Geometry of

footing problem

Visco-plastic analyses with 10, 25, 50, 100 and500 increments were performed. The 10 incrementanalysis had convergence problems in the iterationprocess, which would initially converge, but thendiverge. Similar behaviour was encountered foranalyses using still fewer increments. Results fromthe analyses with 25 and 500 increments are shownin Figure 44. The solutions are sensitive toincrement size, but to a lesser degree than thetangent stiffness approach.

The load on the footing at a settlement of 25mmis plotted against number of increments, for alltangent stiffness, visco-plastic and MNR analyses,in Figure 45. The insensitivity of the MNR analysesto increment size is clearly shown. In theseanalyses the ultimate footing load only changedfrom 2.83kN/m to 2.79kN/m as the number ofincrements increased from 2 to 500. Even for theMNR analyses performed with a single increment,the resulting ultimate footing load of 3.13kN/m isstill reasonable and is more accurate than the value of 3.67kN/m obtained from the tangent stiffness analysiswith 200 increments. Both the tangent stiffness and visco-plastic analyses produce ultimate failure loadswhich approach 2.79kN/m as the number of increments increase. However, tangent stiffness analysesapproach this value from above and therefore over predict, while visco-plastic analyses approach this valuefrom below and therefore under predict. This trend is consistent with the results from the triaxial test, wherethe ultimate value of q was over predicted by the tangent stiffness and under predicted by the visco-plasticapproach.

It can be shown that, for the material properties and initial stress conditions adopted, the undrainedstrength of the soil, su, varies linearly with depth below the ground surface. Davis and Booker (1973)provide approximate solutions for the bearing capacity of footings on soils with such an undrained strengthprofile. For the present situation their charts give a collapse load of 1.91kN/m and it can be seen (Figure 45)that all the finite element predictions exceed this value. This occurs because the analytical failure zone isvery localised near the soil surface. Further analyses have been carried out using a refined mesh in whichthe thickness of the elements immediately below the footing has been reduced from 0.1m to 0.03m. TheMNR analyses with this mesh predict an ultimate load of 2.1kN/m. If the mesh is further refined the Davisand Booker solution will be recovered. For this refined mesh even smaller applied displacement incrementsizes were required for the tangent stiffness analyses to obtain an accurate solution. Analyses with anincrement size of 0.125mm displacement (equivalent to the analysis with 200 increments described above)or greater, yielded negative values of p′ in the elements below the corner of the footing and therefore theseanalyses could not be completed.

8.6.4 Comments

Results from the tangent stiffness analyses of both the idealised triaxial tests and the more complexboundary value problems are strongly dependent on increment size. The error associated with the tangentstiffness analyses usually results in unconservative predictions of failure loads and displacements in mostgeotechnical problems. For the footing problem large over predictions of failure loads are obtained, unless avery large number of increments (≥1000) is employed. Inaccurate analyses based on too large an incrementsize produced ostensible plausible load displacement curves. Analytical solutions are not available for mostproblems requiring a finite element analysis. Therefore it is difficult to judge whether a tangent stiffnessanalysis is accurate on the basis of its results. Several analyses must be carried out using different incrementsizes to establish the likely accuracy of any predictions. This could be a very costly exercise, especially ifthere was little experience in the problem being analysed and no indication of the optimum increment size.

Results from the visco-plastic analyses are also dependent on increment size. For boundary valueproblems involving undrained soil behaviour these analyses were more accurate than tangent stiffnessanalyses with the same increment size. However, if soil behaviour was drained, visco-plastic analyses were

Figure 45 : Ultimate footing load against number ofincrements

only accurate if many small solution increments were used. In general, the visco-plastic analyses used morecomputer resources than both the tangent stiffness and MNR approaches. For the triaxial tests, footing andpile problems (not presented here), the visco-plastic analyses under-predicted failure loads if insufficientincrements were used and were therefore conservative in this context.

For both the tangent stiffness and visco-plastic analyses the number of increments to obtain an accuratesolution is problem dependent. For example, for the footing problem the tangent stiffness approach requiredover 1000 increments and the visco-plastic method over 500 increments, whereas for an excavation problem(not presented here) the former required only 100 and the later 10 increments. Close inspection of theresults from the visco-plastic and tangent stiffness analyses indicated that a major reason for their poorperformance was their failure to satisfy the constitutive laws. This problem is largely eliminated in theMNR approach, where a much tighter constraint on the constitutive conditions is enforced.

The results from the MNR analyses are accurate and essentially independent of increment size. For theboundary value problems considered, the tangent stiffness method required considerably more CPU timethan the MNR method to obtain results of similar accuracy, e.g., over seven times more for the foundationproblem and over three and a half times more for the excavation problem. Similar comparisons can befound between the MNR and visco-plastic solutions. Thus not withstanding the potentially very largecomputer resources required to find the optimum tangent stiffness or visco-plastic increment size, thetangent stiffness or visco-plastic method with an optimum increment size is still likely to require morecomputer resources than an MNR analysis of the same accuracy. Though it may be possible to obtaintangent stiffness or visco-plastic results using less computer resources than with the MNR approach, this isusually at the expense of the accuracy of the results. Alternatively, for a given amount of computingresources, a MNR analysis produces a more accurate solution than either the tangent stiffness or visco-plastic approaches.

The study has shown that the MNR method appears to be the most efficient solution strategy forobtaining an accurate solution to problems using critical state type constitutive models for soil behaviour.The large errors in the results from the tangent stiffness and visco-plastic algorithms in the present studyemphasise the importance of checking the sensitivity of the results of any finite element analysis toincrement size.

8.7 Using The Mohr-Coulomb Model In Constrained Problems

The Mohr Coulomb model can be used with a dilation angle ranging from ψ = 0 to ψ = φ'. Thisparameter controls the magnitude of the plastic dilation (plastic volume expansion) and remains constantonce the stress state of the soil is on the yield surface. This implies that the soil will continue to dilateindefinitely if shearing continues. Clearly such behaviour is not realistic, as most soils will eventually reacha critical state condition, after which they will deform at constant volume if sheared any further. While suchunrealistic behaviour does not have a great influence on boundary value problems that are unrestrained (e.g.,the drained surface footing problem), it can have a major effect on problems which are constrained (e.g.,drained cavity expansion, end bearing of a deeply embedded pile), due to the restrictions on volume changeimposed by the boundary conditions. In particular, unexpected results can be obtained in undrained analysisin which there is a severe constraint imposed by the zero total volume change restriction associated withundrained soil behaviour. To illustrate this problem two examples will now be presented.

The first example considers ideal (no end effects) undrained triaxial compression (∆σv > 0, ∆σh = 0) testson a linear elastic Mohr Coulomb plastic soil with parameters E' = 10000kPa, ν = 0.3, c' = 0, φ' = 24o. Asthere are no end effects, a single finite element is used to model the triaxial test with the appropriateboundary conditions. The samples were assumed to be initially isotopically consolidated with p' = 200kPaand zero pore water pressure. A series of finite element runs were then made, each with a different angle ofdilation, ψ, in which the samples were sheared undrained. Undrained conditions were enforced by settingthe bulk modulus of water to be 1000 times larger than the effective elastic bulk modulus of the soilskeleton, K'. The results are shown in Figure 46a and 46b in the form of J-p' and J-εz plots. It can be seenthat in terms of J-p' all analyses follow the same stress path. However, the rate at which the stress statemoves up the Mohr Coulomb failure line differs for each analysis. This can be seen from Figure 46b. Theanalysis with zero plastic dilation, ψ = 0, remains at a constant J and p' when it reaches the failure line.However, all other analyses move up the failure line, with those with the larger dilation moving up more

rapidly. They continue to move up this failure line indefinitely with continued shearing. Consequently, theonly analysis that indicates failure (i.e., a limiting value of J) is the analysis performed with zero plasticdilation.

The second example considers theundrained loading of a smooth rigid stripfooting. The soil was assumed to have thesame parameters as those used for the triaxialtests above. The initial stresses in the soilwere calculated on the basis of a saturatedbulk unit weight of 20kN/m3, a ground watertable at the soil surface and a Ko = 1-sinφ'.The footing was loaded by applyingincrements of vertical displacement andundrained conditions were again enforced bysetting the bulk modulus of the pore water tobe 1000 times K'. The results of two analyses,one with ψ = 0o and the other with ψ = φ', areshown in Figure 47. The difference is quitestaggering: while the analysis with ψ = 0o

reaches a limit load, the analysis with ψ = φ'shows a continuing increase in load withdisplacement. As with the triaxial tests, alimit load is only obtained if ψ = 0o.

It can be concluded from these two examples that a limit load will only be obtained if ψ = 0o.Consequently great care must be exercised when using the Mohr Coulomb model in undrained analysis. Itcould be argued that the model should not be used with ψ > 0 for such analysis. However, reality is not thatsimple and often a finite element analysis involves both an undrained and a drained phase (i.e., undrainedexcavation followed by drained dissipation). Consequently it may be necessary to adjust the value of ψbetween the two phases of the analysis. Alternatively, a more complex constitutive model which betterrepresents soil behaviour may have to be employed.

Figure 46 : Prediction of dilation a) stress paths and b) stress-strain curves in triaxial compression, usingMohr-Coulomb model with different angles (Note: ν ≡ ψ in this figure)

Figure 47 : Load-displacement curves for strip footing,Mohr Coulomb model with different angles of dilation

ψ

ψ

8.8 Influence Of The Shape Of The Yield And Plastic Potential Surfaces

As noted by Potts and Gens (1984) and Potts and Zdravkovic (1999) the shape of the plastic potential inthe deviatoric plane can affect the Lode’s angle θ at failure in plane strain analyses. This implies that it willaffect the value of the soil strength that can be mobilised. In many commercial software packages, the userhas little control over the shape of the plasticpotential and it is therefore important that itsimplications are understood. This phenomenon isinvestigated by considering the modified Cam-clay constitutive model.

Many software packages assume that both theyield and plastic potential surfaces plot as circlesin the deviatoric plane. This is defined byspecifying a constant value of the parameter MJ

(i.e., the slope of the critical state line in J-p′stress space). Such an assumption implies that theangle of shearing resistance, φ', varies with theLode’s angle, θ. By equating MJ to theexpression for g(θ ) given by Equation (19) andre-arranging, gives the following expression for φ'in terms of MJ and θ:

−′ −

3

sin 1

cos sin = 1

θθφ

M

MJ

J

(33)

From this equation it is possible to express MJ in terms of the angle of shearing resistance, φ'TC, in triaxialcompression (θ = -30o):

TC

TCTCJM φ

φφ

′−′

′sin 3

sin 3 2 = )(

(34)

In Figure 48 the variation of φ' with θ, given by Equation (33), are plotted for three values of MJ. Thevalues of MJ have been determined from Equation (34) using φ'TC = 20o, 25o and 30o. If the plastic potentialis circular in the deviatoric plane, it can be shown that plane strain failure occurs when the Lode's angleθ = 0o. Inspection of Figure 48indicates that for all values of MJ thereis a large change in φ' with θ. Forexample if φ' is set to give φ'TC = 25o,then under plane strain conditions themobilised φ' value is φ'PS = 34.6o. Thisdifference is considerable and muchlarger than indicated by carefullaboratory testing. The differencesbetween φ'TC and φ'PS becomes greaterthe larger the value of MJ.

The effect of θ on undrainedstrength, su, for the constant MJ

formulation is shown in Figure 49. Thevariation has been calculated forOCR = 1, g(θ ) = MJ, Ko = 1-sinφ'TC andκ/λ = 0.1. The equivalent variationbased on the formulation which

Figure 49 : Effect of θ on su, for the constant MJ formulation

Figure 48 : Variation of φ′ with θ for constant MJ

assumes a constant φ', instead of a constant MJ, is given in Figure 50. This is also based on the aboveparameters, except that g(θ ) is now given by Equation (18). The variation shown in this figure is in muchbetter agreement with the available experimental data than the trends shown in Figure 49.

To investigate the effect of theplastic potential in a boundaryvalue problem two analyses of arough rigid strip footing have beenperformed. The finite elementmesh was similar to the one shownin Figure 43. The modified Camclay model was used to representthe soil which had the followingmaterial parameters, OCR = 6,v1 = 2.848, λ = 0.161, κ = 0.0322and ν = 0.2. In one analysis theyield and plastic potential surfaceswere assumed to be circular in thedeviatoric plane. A value ofMJ = 0.5187 was used for thisanalysis, this is equivalent toφ'TC = 23o. In the second analysis aconstant value of φ' = 23o was usedgiving a Mohr Coulomb hexagonfor the yield surface in thedeviatoric plane. However, theplastic potential still gave a circlein the deviatoric plane andtherefore plane strain failureoccurred at θ = 0o, as for the firstanalysis. In both analysis theinitial stress conditions in the soilwere based on a saturated bulk unitweight of 18kN/m3, a ground watertable at a depth of 2.5m and aKo = 1.227. Above the groundwater table the soil was assumed tobe saturated and able to sustainpore water pressure suctions.Coupled consolidation analyseswere performed but thepermeability and time steps werechosen such that undrainedconditions occurred. Loading ofthe footing was simulated byimposing increments of vertical displacement.

In summary, the input to both analysis is identical, except that in the first, the strength parameter MJ isspecified, whereas in the second, φ' is input. In both analyses φ'TC = 23o and therefore any analyses intriaxial compression would give identical results. However, the strip footing problem is plane strain andtherefore differences are expected. The resulting load displacement curves are given in Figure 51. Theanalysis with a constant MJ gave a collapse load some 58% larger than the analysis with a constant φ'. Theimplications for practice are clear, if a user is not aware of the plastic potential problem and is not fullyconversant with the constitutive model implemented in the software being used, they could easily base theinput on φ'TC = 23o. If the model uses a constant MJ formulation, this would then imply a φ'PS = 31.2o, whichin turn leads to a large error in the prediction of any collapse load.

Figure 51 : Load-displacement curves for two different approaches

Figure 50 : Effect of θ on su, for the constant φ′ formulation

9.0 VALIDATION AND CALIBRATION OF COMPUTER SIMULATIONS

To date, less attention has been paid in the literature to the important issue of validation and reliability ofnumerical models in general, and specific software in particular, than has been paid to the development ofthe methods themselves. The work by Schweiger (1991) is one of the limited studies on the subject ofmodel validation. There is now a strong need to define procedures and guidelines to arrive at reliablenumerical methods and, more importantly, input parameters which represent accurately the strength andstiffness properties of the ground in situ.

As should be evident from previous discussion, benchmarking is of great importance in geotechnicalengineering, probably more so than in other engineering disciplines, such as structural engineering. Reasonsfor the importance of benchmarking may be summarized as follows:• the domain to be analysed is often not clearly defined by the structure,• it is not always clear whether continuum or discontinuum models are more appropriate for the problem

at hand,• a wide variety of constitutive models exists in the literature, but there is no “approved” model for each

type of soil,• in most cases construction details cannot be modelled very accurately in time and space (e.g., the

excavation sequence, pre-stressing of anchors, etc.), at least not from a practical point of view,• soil-structure interaction is often important and may lead to numerical problems (e.g., with certain types

of interface elements),• the implementation details and solution procedures may have a significant influence on the results of

certain problems, but may not be important for others, and• there are no approved implementation and solution procedures for commercial codes (such as implicit

versus explicit solution strategies, return algorithms, etc.).Obviously, there is currently considerable scope for developers and users of numerical models to exercise

their personal preferences when tackling geotechnical problems. From a practical point of view, it istherefore very difficult to prove the validity of many calculated results because of the numerous modellingassumptions required. So far, no clear guidelines exist, and thus results for a particular problem may varysignificantly if analysed by different users, even for reasonably well-defined working load conditions.

Difficult issues such as these have been addressed by various groups, including a working group of theGerman Society for Geotechnics, viz., “1.6 Numerical Methods in Geotechnics”. It is the aim of this groupto provide recommendations for numerical analyses in geotechnical engineering. So far the group haspublished general recommendations (Meissner, 1991), recommendations for numerical simulations intunnelling (Meissner, 1996) and it is expected that recommendations for deep excavations will also bepublished shortly. In addition, benchmark examples have been specified and the results obtained by varioususers employing different software have been compared. Some of the work presented by this group issummarised here. The efforts of the working group may be seen as a first step towards greater objectivity innumerical analyses of geotechnical problems in practice.

To date, three example problems have been specified by the working group, and these have beendiscussed in two separate workshops. The first two examples, involving a tunnel excavation and a deepopen excavation problem, have been rather idealised problems, with a very tight specification so that littleroom for interpretation was left to the analysts. Despite the simplicity of the examples and the rather strictspecifications, significant differences in the results were obtained, even in cases where the same softwarehas been utilised by different users. Further details of these examples are given below.

The third example, which represents an actual application (a tied back diaphragm wall in Berlin sand),which was only slightly modified in order to reduce the computational effort, will also be presented.Limited field measurements are available for this example, providing information on the order of magnitudeof the deformations to be expected. Whereas in the first two examples, the constitutive model and theparameters were pre-specified, in the latter example the choice of constitutive model has been left to the userand the parameter values had to be selected either from the literature, on the basis of personal experience, ordetermined from laboratory tests which were made available to the analysts.

9.1 Specifications For Benchmark Examples

Keeping in mind the purpose of benchmarking, from a practical point of view the following requirementsfor benchmark examples are suggested:• no analytical solution is available but commercial codes are capable of solving the problem,• the actual practical problem should be addressed, and simplified in such a way that the solution can be

obtained with reasonable computational effort,• no calibration of laboratory tests has been performed (this is done extensively in research and is of minor

interest to engineers in practice),• preferably the examples should be set in such a way that, in addition to global results, specific aspects

can also be checked (e.g., handling of initial stresses, dilation behaviour, excavation procedures, etc.),• the influence of different constitutive models on the predicted results should become apparent, and• the problem of parameter identification for various constitutive models should be addressed.

Once a series of examples has been designed and solutions are available they could serve as:• a check of commercial codes,• learning aids for young geotechnical engineers to help them to become familiar with numerical analysis,

and• verification examples for proving competence

in numerical analysis of geotechnical problems.In addition, these examples will identify

limitations of the present state of the art innumerical modelling in practice, provide thepossibility to show alternative modellingassumptions, and highlight the importance ofappropriate constitutive models.

9.2 Tunnel Excavation Example

Figure 52 depicts the geometry of the firstvalidation example, a tunnel excavation problem,and Table 8 contains a list of the materialparameters given to all participants in this exercise.Additional specifications are as follows.

9.2.1 General Assumptions

• plane strain conditions apply,• a linear elastic - perfectly plastic analysis with

the Mohr-Coulomb failure criterion wasrequired,

• perfect bonding existed between the shotcreteand the ground,

• the shotcrete lining should be represented bybeam or continuum elements, with 2 rows of elements over the cross-section if continuum elements withquadratic shape function are used, and

60 m

layer 2

50 m

30 m

5 m

II

ground surface12

.05

m

layer 1

II

I

I

Figure 52 : Geometry of tunnel excavation example

Table 8. Material parameters for the “tunnel excavation” example

E (kN/m2) ν φ (o) c (kN/m2) Ko γ (kN/m3)

Layer 1 50000 0.3 28 20 0.5 21

Layer 2 200000 0.25 40 50 0.6 23

Shotcrete (d = 250 mm): linear elastic - E1 = 5 000 MPa, E2 = 15 000 MPa, ν= 0.15

• to account for deformations occurring ahead of the face (pre-relaxation), the load reduction method or asimilar approach should be adopted.

9.2.2 Computational Steps

The following computational steps had to be performed by the analysts:• the initial stress state was set to σv = γH, σh = KoγH, and subsequently the deformations were set to zero,• the pre-relaxation factors given are valid for the load reduction method,• construction stage 1: 40% pre-relaxation of the full cross section, and• construction stage 2: excavation of full cross section, installation of shotcrete with E = E2.

In addition to a full face excavation, excavation of a top heading and bench was also considered, butthese results will not be discussed here (see Schweiger, 1997; 1998).

9.2.3 Selected Results

In the following, some of the mostinteresting results are presented. InFigure 53, surface settlements obtainedfrom 10 different analyses are compared.50% of the calculations predict 52 or53 mm as the maximum settlement andmost of the others were withinapproximately 20% of these values.However, the calculations identified asTL1A and TL10 show significantly lowersettlements. In both cases the reason wasthat it was not possible to apply the loadreduction method correctly and thereforeother methods have been used. TL10used the stiffness reduction method and itis known that it is difficult to match thesetwo methods (Schweiger et al., 1997).TL9 also employed the stiffnessreduction method but obtained largersettlements, which is rather unusual.

distance from tunnel axis [m]

0 5 10 15 20

surface settlement [cm

]

0

1

2

3

4

5

6

TL 1ATL 2TL 4TL 5TL 6TL 7TL 8TL 9TL 10TL11

Figure 53 : Surface settlements for tunnel excavation in one step

TL 1: -99TL 4: -176TL 9: -317TL 10: -236

TL 1: 74

TL 2: -178TL 8: -165

TL 2: 133TL 4: 144TL 5: 185TL 8: 108TL 10: 200

TL 5: -234

TL 9: 113

[kNm/m]calculation

TL1 TL2 TL4 TL5 TL8 TL9 TL10

max. norm

al force [kN]

0

1000

2000

3000

4000

5000

6000

Figure 54 : Comparison of maximum normal forces and bending moments in tunnellining

Figure 54 shows the calculated normal forces and bending moments in the shotcrete lining. Reasonableagreement is observed for normal forces, with the exception of TL1 and TL10, but a wide scatter is obtainedfor bending moments. Figure 54 indicates also the location of maximum bending moments and thesignificant differences in magnitude (approximately 300%) and location are obvious. Even if TL1, TL9 andTL10 are excluded because they did not refer exactly to the problem specification, the variation is still 70%.Unfortunately, it was not possible from the information available to identify clearly the reasons for thesediscrepancies but most probably they are due to differences in modelling the lining and in evaluation of theinternal forces.

9.3 Deep ExcavationExample

Figure 55 illustrates theg e o m e t r y a n d t h eexcavation stages analysedin this problem, and Table 9lists the relevant materialparameters. Additionalspecifications are asfollows.

9.3.1 GeneralAssumptions

• plane strain conditionsapply,

• a linear elastic -p e r f e c t l y p l a s t i canalysis with the Mohr-C o u l o m b f a i l u r ecriterion was required,

• perfect bonding was tobe assumed between the diaphragm wall and the ground,

• struts used in the excavation could be modelled as rigid members (i.e., the horizontal degree of freedomwas fixed),

• any influence of the diaphragm wall construction could be neglected, i.e., the initial stresses wereestablished without the wall, and then the wall was “wished-in-place”, and

• the diaphragm wall was modelled using either beam or continuum elements, with 2 rows of elementsover the cross section if continuum elements with quadratic shape function were adopted.

9.3.2 Computational Steps

layer 1

layer 2

diaphragmwall

strut No. 1

excavation step 1

strut No. 2

excavation step 2

-7.0

-8.0

-4.0 -3.0

-12.0final excavation

-20.0

40 m

60 m

15.000.80

ground surface = 0.0

-26.0

layer 3

x

y

x'

Figure 55 : Geometry of the deep excavation example

Table 9. Material parameters for “deep excavation” example

E (kN/m2) ν φ (o) c (kN/m2) Ko γ (kN/m3)

Layer 1 20000 0.3 35 2.0 0.5 21

Layer 2 12000 0.4 26 10.0 0.65 19

Layer 3 80000 0.4 26 10.0 0.65 19

Diaphragm wall (d = 800 mm): linear elastic E = 21 000 MPa, ν = 0.15, γ = 22 kN/m3

The following computational steps had to be performed by the various analysts: the initial stress state was set to G = J&T, 4 = K&Y?, all deformations were set to zero and then the wall was “wished-in-place”, construction stage 1: excavation step 1 to a level of -4.0 m, construction stage 2: excavation step 2 to a level of -8.0 m, and strut 1 installed at -3.0 m, and construction stage 3: final excavation to a level of -12.0 m, and strut 2 installed at -7.0 m.

Comparison of Results

It is worth mentioning that 5 out of the 12 calculations submitted for comparison were made by different analysts using the same computer program. Figure 56 compares surface displacements for construction stage 1 and shows 2 groups of results. The lower values for the heave fi-om calculations BGl and BG2 may be explained because of the use of interface elements, which were used despite the fact that the specification did not require them. The results of BG3 and BG12 could not be explained in any detail. There were indications though that for the particular program used a significant difference in vertical displacements was observed depending whether beam or continuum elements were used for modelling the diapbragm wall. This emphasises the significant influence of different modelling assumptions and the need for evaluating the validity of these models under defmed conditions. It may be worth mentioning that this effect was not observed to the same extent in the other programs used. Figure 57 shows the same displacements for the final excavation stage, and the results are now almost evenly distributed between the limiting values.

3.0

2.5

E 2.0

is

j 1.5

St 5 1.0

Y 5 IO 15 20 25

distance from wall [m]

Figure 56 : Vertical displacement of surface behind wall: construction stage 1

6

distance from wall [m]

Figure 57 : Vertical displacement of surface behind wall: final construction stage

61

It is apparent from Figures 56 and 57 that elasto-perfectly plastic constitutive models are not well suited for analysing the displacement pattern around deep excavations, especially for the surface behind the wall because the predicted heave is certainly not realistic. However, it was not the aim of this exercise to compare results with actual field observation, but merely to see what differences are obtained when using slightly different modelling assumptions within a rather tight problem specification.

It is interesting to compare predictions of the horizontal

_ 0.50

6 z 0.25

6 E

$

0.00

a -0.25 g

3 -0.50

c 0 .w -0.75

z c -1.00

-1.25

BGI BG2 BG3 BG4 BG5 BG6 BG7 BG8 BG9 BGlOBGll BG12

calculation

Figure 58 : Horizontal displacement of head of wall

displacement of the head of the wall for excavation step 1. Only 50% of the analyses predict displacements towards the excavation (+ve displacement in Figure 58) whereas the other 50% predict movements towards the soil, which seems to be not very realistic for a cantilever situation. Significant differences were not found in the predictions for the horizontal displacements of the bottom of the wall, the heave inside the excavation and the earth pressure distributions. Calculated bending moments varied within 30% and strut forces for excavation step 2 varied from 155 to 232 kN/m. A more detailed examination of this example can be found in Schweiger (1997, 1998).

Tied-Back Deep Excavation Example

The final example presented here is closely related to an actual project in Berlin. Slight modifications have been introduced in modelling of the construction sequence, in particular the groundwater lowering, which was performed in various steps in the field, but was modelled in one step prior to excavation.

In this example the constitutive model to be used was not prescribed but the choice was left to the analysts. Some basic material parameters have been taken from the literature and additional results from one-dimensional compression tests on loose and dense samples were given to the participants, together with the results of niaxial tests on dense samples. Thus the exercise represents closely the situation that is ofkn faced in practice. Inclimometer measurements made during construction provided information of the actual behaviour in situ, although due to the simplifications mentioned above a one-to-one comparison is not possible. Of course, the measurements were not disclosed to the the participants prior to them submitting their predictions.

It is the aim of this section to demonstrate the necessity of performing exercises of this kind. However, due to space limitations only the most relevant aspects of the problem specification will be given here (Figure 59). For the same reason only a limited number of results will be presented. They indicate however the wide scatter of results that can be produced due to different interpretations of the available data. A much more comprehensive discussion of this validation problem may be found in Schweiger (2000).

Some reference values for stifmess and strength parameters, obtained from the literature and frequently used in the design of excavations in Berlin sand, are given below.

62

- 3 2 . O O m = b a s e o f d i a p h r a g m w a l l

S p e c i f i c a t i o n f o r a n c h o r s : p r e s t r e s s e d a n c h o r f o r c e : 1 . r o w : 7 6 8 K N

2 . r o w : 9 4 5 K N 3 . r o w : 9 8 0 K N

d i s t a n c e o f a n c h o r s : 1 . r o w : 2 . 3 O m 2 . r o w : 1 . 3 5 m 3 . r o w : I . 3 5 m

c r o s s s e c t i o n a r e a : 1 5 c m 2 Y o u n g ’ s m o d u k ~ s E = 2 . 1 e 8 k N / m 2

F i g u r e 5 9 : G e o m e t r y a n d e x c a v a t i o n s t a g e s f o r t h e t i e d b a c k e x c a v a t i o n

ES = 2 0 0 0 0 & k P a , f o r 0 < z 2 2 0 m ( z = d e p t h b e l o w s u r f a c e ) ,

J % = 6 0 0 0 0 & k P a , f o r z > 2 0 m , 4 = 3 s ” ( m e d i u m d e n s e ) Y = 1 9 k N / m 3 ,

Y / = 1 0 k N / m 3 , a n d K , , = 1 - s i n + !

T h e m o d u l i o b t a i n e d f r o m o e d o m e t e r t e s t s c o u l d a l s o b e u s e d t o e s t i m a t e a p p r o x i m a t e v a l u e s f o r t h e Y o u n g ’ s m o d u l e f o r c a l c u l a t i o n s w i t h l i n e a r e l a s t i c - p e r f e c t l y p l a s t i c m a t e r i a l m o d e l s ( M o h r - C o u l o m b ) a s s u m i n g a n a p p r o p r i a t e P o i s s o n ’ s r a t i o .

I n a d d i t i o n t o t h e s e v a l u e s , r e s u l t s f r o m o e d o m e t e r t e s t s o n l o o s e a n d d e n s e s a m p l e s a n d t r i a x i a l t e s t s ’ w i t h C T ; = 1 0 0 , 2 0 0 a n d 3 0 0 k P a w e r e p r o v i d e d t o p a r t i c i p a n t s . I t w a s n o t p o s s i b l e t o i n c l u d e a s i g n i f i c a n t l y l a r g e n u m b e r o f t e s t r e s u l t s a n d t h u s t h e q u e s t i o n a r o s e l a t e r w h e t h e r t h e s t i f f n e s s v a l u e s o b t a i n e d f r o m t h e o e d o m e t e r t e s t s w e r e a c t u a l l y r e s p r e s e n t a t i v e . I n p a r t i c u l a r , i f t h e c o n s t i t u t i v e m o d e l r e q u i r e d a s a n i n p u t p a r a m e t e r a n o e d o m e t e r s t i f f h e s s a t a r e f e r e n c e p r e s s u r e o f 1 0 0 k P a , a v a l u e o f o n l y E s = 1 2 0 0 0 k P a w a s f o u n d i n t h e d a t a p r o v i d e d , a n d t h i s w a s c o n s i d e r e d t o b e t o o l o w b y m a n y a u t h o r s . H o w e v e r , a g a i n t h i s r e p r e s e n t s a s i t u a t i o n t h a t i s c o m m o n i n p r a c t i c e , a n d l e f t r o o m f o r i n t e r p r e t a t i o n b y t h e i n d i v i d u a l a n a l y s t s .

T h e p r o p e r t i e s o f t h e d i a p h r a g m w a l l w e r e s p e c i f i e d a s f o l l o w s . E = 3 0 0 0 0 M P a V = 0 . 1 5

Y = 2 4 k N / m 3

6 3

The angle of wall friction was specified as φ′/2.

9.4.1 General Assumptions

Additional specifications for this example are as follows:• plane strain conditions could be assumed,• any influence of the diaphragm wall construction could be neglected, i.e., the initial stresses were

established without the wall, and then the wall was “wished-in-place” and its different unit weightincorporated appropriately,

• the diaphragm wall could be modelled using either beam or continuum elements,• interface elements existed between the wall and the soil,• the domain to be analysed was as suggested in Figure 59,• the horizontal hydraulic cut-off that existed at a depth of –30.00 m was not to be considered as structural

support, and• the pre-stressing anchor forces were given as design loads.

9.4.2 Computational Steps

The following computational steps had to be performed by the various analysts:• the initial stress state was given by σv = γH, σh = KoγH,• the wall was “wished-in-place” and the deformations reset to zero,• construction stage 1: groundwater-lowering to –17.90 m,• construction stage 2: excavation step 1 (to level -4.80 m),• construction stage 3: activation of anchor 1 at level –4.30 m and prestressing,• construction stage 4: excavation step 2 (to level -9.30 m),• construction stage 5: activation of anchor 2 at level –8.80 m and prestressing,• construction stage 6: excavation step 3 (to level -14.35 m),• construction stage 7: activation of anchor 3 at level –13.85 m and prestressing, and• construction stage 8: excavation step 4 (to level –16.80 m).

The length of the anchors and their prestressing loads are indicated in Figure 59.

9.4.3 Brief Summary Of Assumptions Of Submitted Analyses

In Tables 10 and 11 the main features of all analyses submitted have been summarised in order tohighlight the different assumptions made, according to the personal preferences and experience of theparticipants.

It follows from Tables 10 and 11 that a wide variety of programs and constitutive models has beenemployed to solve this problem. Only a limited number utilised the laboratory test results provided in thespecification to calibrate their models. Most of the analysts used data from the literature for Berlin sand ortheir own experience to arrive at input parameters for their analysis. Close inspection of Tables 10 and 11reveals that only marginal differences exist in the assumptions made about the strength parameters for thesand (everybody believed the experiments in this respect), and the angle of internal friction φ′ was taken as36° or 37° and a small cohesion was assumed by many authors to increase numerical stability. A significantvariation is observed however in the assumption of the dilatancy angle ψ, with values ranging from 0º to 15º.An even more significant scatter is observed in the assumption of the soil stiffness parameters. Althoughmost analysts assumed an increase with depth, either by introducing some sort of power law, similar to theformulation presented by Ohde (1951), which in turn corresponds to the formulation by Janbu (1963), or bydefining different layers with different Young’s moduli. Additional variation is introduced by differentformulations for the interface elements, element types, domains analysed and modelling of the prestressedanchors. Some computer codes and possibly some analysts may have had problems in modelling theprestressing of the ground anchors, and actually part of the force developed due to deformations occurring inthe ground appears lost. Where this is the case a remark has been included in Table 10. The constitutivemodel “Hardening Soil” corresponds to a shear and volumetric hardening plasticity model provided in thecommercially available finite element code PLAXIS. It features also a stress dependent stiffness, differentfor loading and unloading or reloading paths.

9.4.4 Results

A total of 15 organisations (comprising University Institutes and Consulting Companies from Germany,Austria, Switzerland and Italy), referred to as B1 to B15 in the following, submitted predictions. Figure 60shows the deflection curves of the diaphragm wall for all entries. It is obvious from the figure that theresults scatter over a very wide range, which is unsatisfactory and probably unacceptable to most criticalobservers of this important validation exercise. For example, the predicted horizontal displacement of thetop of the wall varied between –229 mm and +33 mm (-ve means displacement towards the excavation).Looking into more detail in Figure 60, it can be observed that entries B2, B3, B9a and B7 are well out of the“mainstream” of results. These are the ones which derived their input parameters mainly from theoedometer tests provided to all analysts, but it should be remembered that these tests showed very lowstiffnesses as compared to values given in the literature. Some others had small errors in the specific weight,but these discrepancies alone cannot account for the large differences in predictions.

As mentioned previously, field measurements are available for this project and although the example herehas been slightly modified in order to facilitate the calculations, the order of magnitude of displacements isknown. Figure 61 shows the measured wall deflections for the final construction stage together with thecalculated results. Only those calculations which are considered to be “near” the measured values areincluded. The scatter is still significant. It should be mentioned that measurements have been taken byinclinometer, but unfortunately no geodetic survey of the wall head is available. It is very likely that thebase of the wall does not remain fixed, as was assumed in the interpretation of the inclinometermeasurements, and that a parallel shift of the measurement of about 5 to 10 mm would probably reflect thein situ behaviour more accurately. This has been confirmed by other measurements under similar conditionsin Berlin. If this is true a maximum horizontal displacement of about 30 mm can be assumed and all entriesthat are within 100% difference (i.e., up to 60 mm) have been considered in the diagram. The predictedmaximum horizontal wall displacements still varied between 7 and 57 mm, and the shapes of the predictedcurves are also quite different from the measured shape. Some of the differences between prediction andmeasurements can be attributed to the fact that the lowering of the groundwater table inside the excavationhas been modelled in one step whereas in reality a stepwise drawdown was performed (the same has beenassumed by calculation B15). Thus the analyses overpredict horizontal displacements, the amount beingstrongly dependent on the constitutive model employed, as was revealed in further studies. In addition, itcan be assumed that the details of the formulation of the interface element have a significant influence on thelateral deflections of the wall, and arguments similar to those discussed in the previous section forimplementing constitutive laws also hold, i.e., no general guidelines and recommendations are currentlyavailable. A need for them is clearly evident from this exercise.

Figure 62 depicts the calculated surface settlements, again only for the same solutions that are presentedin Figure 61. These key displacement predictions vary from settlements of up to approximately 50 mm tosurface heaves of about 15 mm. Considering the fact that calculation of surface settlements is one of themain goals of such an analysis, this lack of agreement is disappointing. It also highlights the pressing needfor recommendations and guidelines that are capable of minimising the unrealistic modelling assumptionsthat have been adopted and consequently the unrealistic predictions that have been obtained. Theimportance of developing such guidelines should be obvious.

Figure 63 shows predictions of the development of anchor forces for the upper layer of anchors.Maximum anchor forces for the final excavation stage range from 106 to 634 kN/m. As mentionedpreviously, some of the analyses did not correctly model the prestressing of the anchors because they do notshow the specified prestressing force in the appropriate construction step. Predicted bending moments,important from a design perspective, also differ significantly from 500 to 1350 kNm/m.

Taking into account the information presented in Figures 60 to 63 and Table 10, it is interesting to notethat no definitive conclusions are possible with respect to the constitutive model or assumptions concerningelement types and so on. It is worth mentioning that even with the same finite element code (PLAXIS) andthe same constitutive model (Hardening Soil Model) significant differences in the predicted results areobserved. Clearly these differences depend entirely on the personal interpretation of the stiffness parametersfrom the information available. Again, it is noted that a more comprehensive coverage of this exercise isbeyond the scope of this paper but further details may be found in Schweiger (2000).

Table 10. Summary of analyses submitted for the tied back excavation problem

No. CodeConstitutive

Modelν(-)

φ′(o)

ψ(o) Ko

Domainanalysed

width x height(m)

Elementtype -soil

Elementtype - wall

Element type -anchor / grout

body

Elementtype -

InterfaceRemark

B1 Tunnel Mohr-Coulomb 0.3 35 5 100 x 649 noded

continuumbar /

membraneprestressforce ?

B2 Plaxis

Hardening(z < 40 m)Mohr-Coulomb(z > 40 m)

0.2 36 6 0.41 100 x 100 quadratic beambar /

membrane

Rinter = .5Einter > EBoden

B2a Plaxis

Hardening(z < 32 m)Mohr-Coulomb(z > 32 m)

0.2 36 6 0.41 100 x 100 quadratic beambar /

membraneRinter = .5

B3 Abaqus

Hypoplasticwithoutintergranularstrains

0.42 161 x 162 linear4 noded

continuumbar / bar ψ = 20

prestressforce ?

B3a Abaqus

Hypoplasticwithintergranularstrains

0.42 161 x 162 linear4 noded

continuumbar / bar ψ = 20

prestressforce ?

B4 Ansys Drucker Prager 0.3 35 0 0.43 105 x 107 linear4 noded

continuumbar / bar

prestressforce ?

B5 Sofistik Mohr-Coulomb 0.3 35 15 80 x 60 beam springno

interfaceprestressforce ?

B6 Z_Soil Mohr-Coulomb 0.3 35 15 122 x 90prestressforce ?

B7 FeerepgtZienkiewicz/Pande/Schad

0.26 40.5 13.5 0.35 90 x 60 quadratic continuumContinuum /continuum

after GE,ψ =20.25

B8 Plaxis

Hardening2 layers(z < 20 m)(z > 20 m)

0.2 35 10 0.43 90 x 70 quadratic beambar /

membranePlaxis

B9 Plaxis Mohr-Coulomb 0.35 35 5 0.43 150 x 100 quadratic beambar /

membraneRinter =0.5

B9a Plaxis Hardening 0.2 35 10 0.43 150 x 100 quadratic beambar /

membraneRinter =0.5

B10 Plaxis Hardening 0.2 36 6 0.41 100 x 72 quadratic beambar /

membraneRinter=0.61

B11 Plaxis

Hardening2 layers(z < 20 m)(z > 20 m)

0.3 35 0 0.45 150 x 120 quadratic beambar /

membraneRigid ?

prestressforce ?

B12 Plaxis Mohr-Coulomb 0.3 35 4 90 x 92 quadratic beambar /

membraneRinter=0.5

prestressforce ?

B13 Abaqus

Hypoplasticwithintergranularstrains

100 x 100linear +internal

nodebeam bar ψ = 15.5

anchorsfixed atbound’y

B14 Befe

Shear HardeningModel withsmall strainstiffness

0.2 35 5 0.43 120 x 100 quadratic8 noded

continuumbar / bar

Dicke = 0ψ = 17.5

B15 Plaxis

Hardening2 layers(z < 20 m)(z > 20 m)

0.20.2

3541

714

0.430.34

96 x 50 quadraticbeam +

continuumbar /

membraneRinter =0.8

9.5 Conclusions Of Validation Studies

Some results from three benchmark examples suggested by the working group 1.6 of the German Societyfor Geotechnics have been presented. These problems have allowed a comparison of results obtained fromvarious methods of analysis, geotechnical models and different analysts. Although quite tight specificationshave been given for the first two examples, significant differences in predicted results have been observedeven in these rather simple cases.

The third example was by far the most difficult of the three problems described, because the choice ofconstitutive law and parameter identification was left entirely to the analyst. Perhaps, as might have beenexpected, this freedom of choice lead to large differences in results. The reasons for the differences couldnot be fully identified except for the fact that information given on the soil properties leads to quite differentinput parameters, depending on personal interpretation. However, it is believed that this exercise closelyrepresents the situation in practice and reveals once again that numerical modelling in geotechnics is a verydifficult task. A lot of experience is required in order to make sensible assumptions for material and modelparameters, which are not explicitly known beforehand, in order to arrive at a reasonable model of the realsituation in the field.

The exercise presented here very clearly indicates the need for guidelines and training for numericalanalysis in geotechnical engineering in order to achieve reliable solutions for practical problems. However,at the same time these examples demonstrate the power of numerical modelling techniques providedexperienced users apply them. A full analysis of the benchmark example No. 3 will be given in aforthcoming report of the working group of the German Society, and the problem will be further dealt withalso by Technical Committee 12 (TC 12 – Validation of Computer Simulations) of the International Societyfor Soil Mechanics and Geotechnical Engineering (ISSMGE).

Table 11. Summary of stiffness parameters used in tied back excavation analyses

No. Constitutive Model E-dependenceNumber of

layersPower

Eref,loading or Emin

(kN/m2)Eref,unloading or Emax

(kN/m2)

B1 Mohr-Coulomb similar to Ohde 2 0.5z<20m: 14 900z>20m: 44 700

B2Hardening SoilMohr-Coulomb

similar to Ohdelinear increase

2 0.85z<40 m: 15 000

z=40 m: 253 00039 000

B2aHardening SoilMohr-Coulomb

similar to Ohdelinear increase

2 0.5z<32 m: 60 000

z=32 m: 226 300180 000

B4 Drucker Prager Layers 43 - z=1m: 10 500 z = 105m: 457 000B5 Mohr-Coulomb Layers 6 - z=4.8m: 32 640 z = 60m: 303 200

B6 Mohr-Coulomb Layers ? -z<20m: 20 000 z

z>20m: 44 700 z

B7Zienkiewicz/Pande/Schad

- - - 60 000

B8 Hardening Soil similar to Ohde 2 0.5z<20m: 20 000z>20m: 60 000

z<20m: 74 400z>20m: 120 000

B9 Mohr-Coulomb Layers 3 - z<20m: 39 400 z>40m: 310 000B9a Hardening Soil similar to Ohde - 0.65 25 000 100 000B10 Hardening Soil similar to Ohde - 0.5 60 000 180 000

B11 Hardening Soil similar to Ohde 2 0.5z<20m: 20 000z>20m: 60 000

z<20m: 100 000z>20m: 300 000

B12 Mohr-Coulomb Layers 9 - z=0-4.8m: 23 000 z = 42-92m: 365 000

B14Shear Hardening

Model with "smallstrain stiffness"

similar to Ohde - 0.5Gmin = 72 000

Gmax = 576 000 (atvery small strains)

Gentl = 288 000Gentl = 2 304 000 (at very

small strains)

B15Hardening SoilHardening Soil

similar to Ohdesimilar to Ohde

2 0.5z<20 m: 32 000z>20 m: 96 000

192 000384 000

horizontal displacement [mm]

-250 -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25 50

-250 -225 -200 -175 -150 -125 -100 -75 -50 -25 0 25 50

depth below surface [m

]

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

B1B2B2aB3 B3a B4 B5 B6B7 B8B9B9aB10B11B12 B13 B14B15

Figure 60 : Wall deflection all solutions submitted: tied back excavation

horizontal displacement [mm]

-60 -50 -40 -30 -20 -10 0 10

-60 -50 -40 -30 -20 -10 0 10depth below

surface [m]

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

B2a B4B5 B6B8 dB9 B11B12 B13 B14B15measurements

measurement

Figure 61 : Wall deflection for selected solutions and inclinometer measurements: tied backexcavation

Outlook And Further Validation Work

As mentioned earlier, the working group 1.6 of the German Society for Geotechnics will continue its efforts in this area. In addition, the Working Group A of the COST-Action of the European Community is also making efforts to provide guidelines for applications of numerical models in geotechnical engineering. Finally, ERTC7 and in particular TC12 of ISSMGE have started working in this area. The terms of refaences of TC12 are as follows: l to promote cooperation

and exchange of information on the development and

0 ! /; 1 1 2 3 4 5 6 7 8

prestress force specifii computational step in step 3 = 334 kN/m

application of computer simulations in geotechnical engineering.

Figure 63 : Forces in fast layer of anchors for selected solutions: tied back excavation

l to foster the development of recommendations and guidelines for the application of numerical analysis in geotechnical engineering.

l to organise workshops devoted to the themes pertinent to the work of TC12 during general and specialty international conferences, e.g., ICSMGE, IACMAG and NUMOG.

DISCRETE ELEmNT MODELS

In the 1970s Cundall developed a computer program known as BALL. This development heralded the birth of a revolutionary computational technique in geotechnics, which is now termed the Discrete Element Method (DEM). It has also been termed the distinct element method. The aim of the original model was to describe the fundamental behaviour of granular materials en musse by focusing on their basic constituents, i.e., the grains themselves and their interactions. The technique is able to deal with both non-cohesive grains and cohesive grains. The method was validated by Cundall and Strack (1979) by comparing force vector plots obtained ti-om the computer program BALL, with the corresponding plots obtained from photoelastic analysis conducted by de Jossehn de Jong and Verruijt (1969). The correspondence between these plots was sufficiently good to mark the distinct element method as a valid tool for fundamental research into the behaviour of particulate materials.

Since the fast proposal of this numerical method, various researchers have developed and adapted it to \study the behaviour of a variety of materials composed of individual particles or blocks. Geotechnical problems where the mechanical behaviour of a soil or rock mass is influenced strongly by the effects of joints and cracks are well suited to analysis by the discrete element methods. These methods may be characterised by a number of distinguishing capabilities, such as their ability to deal with finite deformations and rotations of discrete blocks (deformable or rigid), blocks that are originally connected may separate during the analysis, and the possibility that new contacts may develop between blocks due to displacements and rotations.

Several different approaches to achieve these criteria have been developed, with probably the most commonly used methods now being the distinct element codes UDEC and 3-DEC (Lemos et d., 1985),

which both employ an explicit finite difference scheme to solve the governing equations. These codes solve

70

which both employ an explicit finite difference scheme to solve the governing equations. These codes solvethe equations of motion explicitly and material damping must be included to allow approximations of quasi-static behaviour of granular assemblages or blocky rock masses. With this method stability cannot beguaranteed for the solutions of problems in statics. An alternative implicit equation solution technique hasalso been proposed for the discrete element method, e.g., van Baars (1996), Hocking (1977). The implicitmethod is able to provide a stable solution to quasi-static problems. A useful summary of DEM theory ispresented in the text by Pande et al. (1990).

These techniques are particularly useful for revealing the dominant mechanisms of deformation ofmaterials that posses a definite structure. Examples of this usage may be seen in Figures 64 to 66.

10.1 Biaxial Compression Of Granular Material

Figure 64 shows the simulation of a biaxial test of sandstone (van Baars, 1996). In this simulation thecohesive-granular sandstone has been modelled by a relatively large number of discrete (approximatelycircular) particles. The grains are initially cemented together so that the contacts between the particles havecohesive and frictional properties. Figure 64 shows representations of the particle assemblage at variousstages of the biaxial loading, and the sequence of diagrams reveals the failure mechanism. If cementedcontacts are broken, then a thick line has been drawn perpendicular to the contact. The horizontal surfacesof the loading platens are also shown as thick lines. The vertical boundaries are subjected to a confiningpressure.

Numerical studies such as those shown in Figure 64 can produce interesting findings. van Baars (1996)suggested the following failuremechanism.

In Phase A, some of thecontact forces become tensile,even in the presence of acompressive confining pressure.As a result, micro-cracks formand particles separate. In phaseB the crack growth continuesand weakens the surroundingregion and a failure surfaceeventually forms. It is worthnoting that although the surfaceformed by adjacent cracks isdiagonal, the microcracks areformed by the separation ofparticles along approximatelyhorizontal contacts. As thevertical load applied at thespecimen boundary is increased,the grains with broken contactsact as rollers (phase C), and theoverall response of the specimeninvolves reduction in the verticalload (softening).

10.2 Bearing Response Of Jointed Rock

In a study by Singh et al. (1996) the bearing capacity of blocky media was investigated using bothnumerical and experimental techniques. In the experiments, a physical model consisting of prismatic blocksof a commercially available engineering plastic DELRIN (polyacetal) was constructed. The model wasconfined within a rigid box. In this plane strain problem a rigid footing was placed on the surface of the

Figure 64 : Failure mechanism of a cohesive-granular material in

biaxial compression (after van Baars, 1996)

blocky medium. A vertical load was applied to the footing, and as the load increased the deformation of theblocky medium was observed.

Numerical modelling of the sameproblem has also been carried out usingthe UDEC discrete element program.During both the physical and numericalmodel tests, the average contact stressgenerated under the footing was loggedagainst the vertical displacement of thefooting. A typical plot of the contactstress against the settlement S, normalisedby the footing width B , is shown inFigure 65. Both numerical and physicaltest results are shown on this figure. Thenumerical results plot as a curve withthree distinct stages. Stage 1 starts at theorigin of the plot and is limited by thefirst peak in the load-displacement curve.This peak can be regarded as the bearingcapacity of the blocky medium, because thenumerical model reveals that for thegeometry considered, it corresponds toopening of some of the joints and upwardbuckling of the most highly stressed inclinedlayer of blocks in the medium, generally alayer that is in direct contact with thefooting. The development of this failuremechanism in the numerical model can beseen in Figure 66, and this was also evidentin the physical experiments. The secondstage of the load-displacement curve starts atthe first peak and extends down to the lowesttrough in the curve. This corresponds to astress-relieving phase. In the physical modelthe drop in the load is very sudden and isassociated with the rapid release of storedenergy as a column of blocks buckles. Asthe footing is pushed further into the blockymedium the load rises again and this istermed stage 3 in the load-displacementcurve. The load continues to increase untilanother layer of blocks undergoes an upwardbuckling failure.

The ability and power of the distinctelement method to give a clear andunequivocal indication of the likely failuremechanism is very evident in this problem.When the blocks themselves are very strong,failure can occur by sliding on the joints orelastic buckling of layers of blocks. In theexample considered here, the numericaltechnique indicated clearly that failure isdominated by buckling, with only a minoramount of energy dissipation in shearingalong joint planes in the wedge-shape region of blocks above the buckled layers - see Figure 66. Buckling

Figure 66 : (a) Initial arrangement of blocks, (b) blocksdeformed by footing penetration

Figure 65 : Load-settlement curves for a blocky mass

of inclined layers of blocks was confirmed as the dominant failure mechanism in the physical experiments.Further research work is required to determine if such methods can also give quantitative predictions offailure loads that are in close agreement with those measured in the laboratory, but there can be no doubtingits ability to capture the important mechanisms.

10.3 STOCHASTIC TECHNIQUES

In the past two decades the importance of reliability analyses and the stochastic nature of soil propertieshas been recognised in practical geotechnical engineering. Although the advantages of probabilistic toolsare now generally accepted, there is still only a limited number of applications of these tools to practicalproblems documented in the literature. The reason for this situation is probably that closed form solutionsare not available for most practical cases and numerical methods involve large computational efforts.However recent achievements in numerical analysis and the fast development of hardware have significantlyreduced computer run times and pre- and post-processing efforts so that a probabilistic approach based ondeterministic finite element analysis is now a feasible technique, worthy of further investigation. Several

researchers have already moved in this direction(e.g., Thurner and Schweiger, 2000; Kaggwa,2000).

In general, three major sources of uncertaintyin soil profile modelling can be distinguished(Vanmarcke, 1977). These are the naturalheterogeneity, the limited availability ofinformation on subsurface conditions andmeasurements errors. In practice, mostgeotechnical problems are still solved using adeterministic approach employing soil parameters

which are judged to be “on the safe side”. In order to assess the behaviour of a geotechnical structure or thechange of the deterministic factor of safety with respect to variable parameters such as shear strength forexample, sensitivity analyses are usually carried out.

The uncertainty in the real value of each soil property at a given point can be defined as a randomvariable described by a mean value, coefficient of variation (COV) and a probability distribution function.Common probability distribution functions are the normal and the lognormal distribution; the latter ispreferred for variables which cannot have negative values. Although more sophisticated distributions likethe Gumbel or Beta distributions may improve results for special cases, applications are rare because theyrequire much more computational effort.

To determine the distribution function for a particular soil property more than 40 data points are needed,but it is usually not feasible to obtained this amount of data in most cases. Consequently, a coefficient ofvariation has to be obtained from other projects in similar soil conditions or from the literature (e.g., Lacasseand Nadim, 1997; Rackwitz and Peintinger, 1981; Cherubini et al., 1993). For example, typical coefficientsof variation for the (drained) friction angle φ′ and the cohesion (undrained shear strength) c are shown inTable 12.

A simple but effective procedure to combine different sources of information is presented in Rackwitzand Peintinger (1981). Another helpful tool to take different sources of information into account is theprogram CombInfo (1999).

In addition to the variability of the shear strength parameters (c and φ′ ) a weak negative correlationshould be taken into account. The assumption of independent parameters may lead to conservative results(Mostyn and Li, 1993). Another important aspect that needs to be considered is autocorrelation (e.g.,Lacasse and Nadim, 1997; Mostyn and Li, 1993; Wickremesinghe and Campanella, 1993; Rackwitz andPeintinger, 1981).

Table 12. Typical coefficients of variation

Soil COV φ′(%)

COV c(%)

Gravel 2 - 15 -Sand 2 - 15 -Silt 5 20 - 30Clay 10 15 - 25

10.4 Safety Factor Or Reliability Analysis Of Geotechnical Structures?

10.4.1 Deterministic Analysis

In geotechnical engineering the traditional factor of safety, Fs, for a slope, for example, is usuallydetermined using deterministic methods and can be defined as (e.g., Brinkgreve and Bakker, 1991):

ccs c

cF

φσφσ

tan

tan′+′+= (35)

where c is the cohesion, φ is the friction angle and σ´ is the effective normal stress. c and φ are the peak orultimate strength parameters and the subscript ‘c’ indicates the mobilised strength parameter, just highenough to ensure equilibrium. For slope stability problems this definition is used in limit-equilibriummethods as well as in non-linear finite element analyses. Some results applying analytical methods havebeen published by Fredlund (1984) and Duncan (1996). It can be shown that for homogeneous conditionsthe results obtained from common methods of limit equilibrium deviate at most by ±6%. These types ofsolutions are comparable with special finite element formulations (Yu et al., 1998; Jiang and Magnan,1997).

The above definition of the factor of safety (Equation 35) has been implemented into the finite elementcode PLAXIS (Brinkgreve and Vermeer, 1998) where an automatic reduction of strength parameters isperformed until equilibrium can be no longer satisfied thus indicating the global factor of safety. Forhomogeneous conditions this procedure works very well, but for more complicated structures reliable resultsmay be as difficult to achieve as by methods of slices.

As shown by Li et al. (1993), Fs is a variant index because it depends on the interpretation of resistingand driving forces. In combination with the uncertainties of the soil properties and the fact that geotechnicalengineers tend to use conservative design values, this may lead to very unrealistic reliability levels. Forthese and other reasons, attempts have been made to employ more rational reliability analyses forgeotechnical problems. These are based soundly on probability theory.

10.4.2 Probabilistic Analysis

The key feature of a probabilistic approach is the definition of a so called performance function G(X)which can be written in general form as:

( ) ( ) ( )XSXRXG −= (36)

where R(X) is the “resistance”, S(X) is the “action”, and X is the collection of random input parameters. ForG(X) < 0 failure is implied, while G(X) > 0 indicates stable behaviour. The boundary defined by G(X) = 0separating the stable and unstable state is called the limit state boundary (Mostyn and Li, 1993).

Determination or definition of the resistance R(X) is generally not straightforward, especially when R(X)is represented by a threshold value for serviceability states or when different failure modes are possible.

Th e pro bab il ity of f ail ure pf i s def ine d as:

( )[ ] ( )( )∫

≤

=≤=0

0XG

f dxXfXGpp

(37)

where f(X) is the common probability density function of the vector formed by the variables X, which isusually unknown.

To solve this integral many different possibilities have been proposed in the literature (e.g., Li, 1992;Zhou and Nowak, 1988). Thurner and Schweiger (2000) suggest the use of numerical integration todetermine the kth moment of G (E[Gk(X)]), due to the fact that simulation methods like FORM (First OrderReliability Method) or SORM (Second Order Reliability Method) require the partial derivatives of theperformance function. In the context of numerical methods these approaches involve significantmodifications to techniques such as standard finite element formulations, and these do not seem to befeasible for practical application in the near future.

10.4.3 Transformation And Numerical Integration

The method proposed by Zhou and Nowak (1988) can be used to integrate Equation (37). It is anumerical procedure for computing the statistical parameters of a function G(X) of multiple randomvariables. The sample of basic variables is obtained by transforming a priori the selected points from thestandard normal space to the basic variable space. Thus this method differs from a standard point estimatemethod in the sense that points and weights are predetermined in the standard normal space.

If the basic variables are standard normal distributed the performance function in Equation (37) can bewritten as:

( ) ( )nZZZGZG ,,, 21 K= (38)

The exact kth moment of G, E[Gk(Z)], may be obtained by evaluating the integral:

( )[ ] ( ) ( )∫+∞

∞−

Φ= dzzGzZGE kk

(39)

where Φ(z) is the cumulative distribution function of standard normal variable Z . The integration inEquation (39) can be approximated using the formula:

( )[ ] ( )∑=

≅m

jj

kj

k zGwZGE1 (40)

where m is the number of points considered, wj are the weights and zj are the typical normal points.

10.5 Finite Element Methods Applied To Reliability Analysis

The combination of the point estimate method described above with the well known finite elementanalysis has been demonstrated by a number of authors, e.g., Thurner and Schweiger (2000), Kaggwa(2000). The advantages of this combined approach compared to say limit equilibrium methods or othersimilar methods are: on one hand that all features of numerical modelling remain, and on the other hand thatwith finite element calculations usually a number of relevant system parameters are readily obtained. Theseparameters are the basis for estimation of the performance function, which is evaluated using the numericalprocedure described previously. Thurner and Schweiger (2000) illustrate this approach by taking atunnelling problem as their example.

Kaggwa (2000) has adopted a similar approach and shown how probabilistic methods may be applied toa variety of problems, to determine the significance of spatial variability on the key aspects of the problem.These examples included an examination, using the finite element methods, of the influence of spatialvariability of soil parameters on prediction of displacements and pore pressures in soft clay under the Muartest embankment in Malaysia (Brand and Premchitt, 1989). Kaggwa showed how the field measurementsshould be viewed in the light of observed spatial variation of the soil parameters. From this study heconcluded that the incorporation of spatial variability in finite element analyses should be seen as a naturalprogression of the discretisation process. In addition to geometric discretisation, the values of materialparameters can be varied randomly within a soil layer to simulate better the natural variability of soils. It isnoted that with sufficient repetitions, information on the range of predictions and their statistics can beevaluated.

11.0 ‘SOFT COMPUTING’ TOOLS

In the recent past, a new field of ‘soft computing’ has emerged for solving decision-making, modelling,and control problems. This approach parallels the remarkable ability of the human mind to reason and learnin an environment of uncertainty and imprecision. Soft computing consists of many complementary tools:fuzzy logic, neuro-computing, probabilistic reasoning, genetic algorithms, and others. A brief description ofone of these tools and its application in geotechnical engineering are described in more detail below in orderto illustrate the power of these methods.

An artificial neural network (ANN) is a “computational mechanism able to acquire, represent, andcompute a mapping from multivariate space of information to another, given a set of data representing thatmapping” (Garret, 1994). The basic architecture of neural networks has been covered widely (Rumelhartand McClelland, 1986; Lippman, 1987; Flood and Kartam, 1994). A typical artificial neural network isshown in Figure 67. The most frequently used neural-network paradigm is the back-propagation learningalgorithm (Rumelhart et al., 1986). Neural networks are trained by the presentation of a set of examples ofassociated input and output (target) values. The hidden and output layer neurons process their inputs bymultiplying each of their inputs by the corresponding weights, summing the product, and then processing thesum using a nonlinear transfer function to produce a result. The S-shaped sigmoid function is commonlyused as the transfer function. The neural-network “learns” by adjusting the weights between the neurons inresponse to the errors between actual output values and target output values. At the end of this trainingphase, the neural network represents a model, which should be able to predict a target value given the inputvalue.

Artificial neural network models are adaptive and capable of generalisation. They can handle imperfector incomplete data, and can capture nonlinear and complex interactions among variables of a system.Because of these strengths, the artificial neural network method is emerging as a powerful tool formodelling. Of course, their major limitation is that they contain no information about the underlyingmechanics of the problem being modelled.

11.1 Artificial Neural Network Architecture

A typical back-propagation artificial neural network is shown in Figure 67. A network can have severallayers. The outputs of each intermediate layer are the inputs to the following layer. Each layer has a weightmatrix W, a bias vector b, and an output vector a. Each element of the input vector p is connected to eachneuron input through the weight matrix W. The i-th neuron has a summer that gathers its weighted inputsand bias to form its own scalar output n(i). The various n(i) taken together form an S-element vector n.Finally, the neuron layer outputs form a column vector a. The layers of a multilayer network play differentroles. The layer that produces the network output is called an output layer. The layer that gets the inputs iscalled the input layer. All other layers are called hidden layers. It is common for the number of inputs to a

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Where… R = number of input variables S1 = number of hidden nodes S2 = number of output nodes

∑

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Figure 67 : Schematic representation of an artificial neural network (ANN)

layer to be different from the number of neurons. The network shown in Figure 67 has R inputs (R neuronsin the input layer), S1 neurons in the hidden layer, and S2 neurons in the output layer. The number of hiddenlayers can be varied based on the application. A constant input value of 1 is fed to the biases for eachneuron.

11.2 Learning Rule

The learning rule refers to the mechanism that is used to adjust the weights and biases of the networks toachieve some desired network behaviour. Several learning rules have been developed by different authors.The back-propagation learning rule is the most popular one that has been used to train nonlinear,multilayered networks to perform function approximation and pattern classification. The back-propagationlearning rule can be used to adjust the weights and biases of networks in order to minimise the sum-squarederror of the network. This is done by continually changing the values of the network weights and biases inthe direction of steepest descent with respect to error. Derivatives of the error vector are calculated for thenetwork’s output layer and then back-propagated through the network until derivatives of error are availablefor each hidden layer.

11.3 Training

Training refers to the process that repeatedly applies input vectors to the network and calculates errorswith respect to the target vectors and then finds new weights and biases with the learning rule. It repeats thiscycle until the sum-squared error falls beneath an error goal, or a maximum number of epochs haveoccurred. Training a feed-forward network with the back-propagation learning rule is most frequently usedin function approximation and pattern recognition. The training parameters specify the number of epochsbetween displaying progress, the maximum number of epochs to train, the sum-squared error goal, and thelearning rate. The size of changes that are made in the weights and biases at each epoch are specified withthe learning rate. Training continues until the specified error goal is met, or the minimum error gradientoccurs, or the maximum number of epochs has finished. The implementation of theses training techniques isnow provided in commercially available software such as MATLAB (e.g., Demuth and Beale, 1996).

11.4 Example - Predicting Uplift Capacity Of Suction Caissons

Several applications of the neural network method to civil and geotechnical engineering problems havebeeen reported already in the literature (e.g., Flood and Kartam, 1994; Garrett, 1994; Goh, 1996; Ghaboussiand Sidarta, 1998; Rahman et al., 2000). Considered here is just one of these applications, in which theartificial neural network method was applied to the prediction of the uplift capacity of suction caissonfoundations in a variety of soils (Rahman et al., 2000). The problem is illustrated schematically in Figure68.

A database consisting of theresults of 60 individual test datasets from 12 independentexperimental studies wasavailable both to train andevaluate the artificial neuralnetwork. These data have beenobtained from both laboratoryand field-scale experiments. Themeasured results have been usedseparately in two stages. 50 datasets were used in the trainingstage and the remaining 10 datasets were used in the testingstage. The data available in theliterature are mostly for clays.Few data are available for sands

d

L

P

θ

D

L = the length of the caisson D = the diameter of the caisson in plan D = the depth of the load application point from the soil surface P = load applied to caisson θ = angle of load inclination

Figure 68 : Definition of suction caisson uplift problem

and therefore sands have not included in this example.The training data were used to develop the model. The testing data were used for testing the

generalization capability of the artificial neural network system. The 10 testing data sets were chosenrandomly from the database. However, the L/d values of the testing data set cover the same range as thoseused in the training stage. Therefore the testing result is expected to reflect accurately the prediction ability

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Figure 70 : Comparison of results of ANN and FEM Modelling for testing data

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Figure 69 : Comparison of results of ANN and FEM modelling for training data

of the model that is developed with the training data set. The testing data have the same format as thetraining data in the artificial neural network models, but values of variables are different from those of thetraining data. It is emphasised that in general, the aim of this example was not to look for an artificial neuralnetwork system that can best fit the training data. Rather, the aim was to look for a system trained on thetraining data that could subsequently respond to the testing data in a satisfactory manner.

11.4.1 Model Evaluation

A comparison of the experimental measurements of uplift capacity results and the outcome of the trainingphase of the ANN is provided in Figure 69. An indication of the ability of the ANN to provide predictionsafter appropriate training is provided in Figure 70, which shows a comparison of the ANN predictions andthe remaining, independent experimental data for caisson uplift capacity.

In order to assess whether the ANN model could provide improved predictions of the uplift capacity ofsuction caissons, the outcome of the ANN model was also compared with the results obtained independentlyfrom finite element modelling (FEM) of the caisson uplift problem (Deng and Carter, 1999), and thiscomparison is also provided in Figure 69. The problems considered here all correspond to cases for whichthere also exists experimental data, including measured uplift capacities. The values of the undrained shearstrengths of the soils considered in this study were taken as those quoted in the original source papers. Asuitably large value of the loading rate was adopted in the finite element predictions to simulate theundrained response.

It can be observed in Figures 69 and 70 that the results from the ANN model are closer to the observedvalues for the high range of uplift capacities, and the FEM model predictions are closer to the observedvalues for the lower uplift capacities. The reason is that the high values of uplift capacity have the largesteffect on the basic measure of performance of the ANN model used in this example, so it will try to fit thehigh uplift capacity cases as much as possible in order to maximise performance. However, for the finiteelement model, the minimum value of the sum-squared error cannot be constrained, and therefore the errorwill be fairly uniformly distributed.

This example demonstrates that the predictions from the ANN are as good or better than those obtainedfrom FEM-based solutions. However, the development of the ANN model is much easier as compared tothe FEM-based models.

As a soft-computation tool, the artificial neural network is quite robust in nonlinear relationshipmodelling. However, the underlying assumption is that the input parameters are inherently reliable. This isnot always the case, particularly where uncertainties exist about the soil strength characteristics. Since fuzzylogic can provide a systematic method to deal with imprecise and incomplete information, some authors arein the process of developing fuzzy neural network models for the prediction in geotechnical engineering,including the prediction of the uplift capacity of suction caissons. Others pursue the combination ofdeterministic and stochastic methods of analysis, as described in a previous section of this paper.

12.0 DATABASE SOFTWARE

Most practical geotechnics involves the accumulation of observations and measurements. Obviousexamples are the results of site investigations and project monitoring, and typically include data such asborehole logs, in situ test data, laboratory test data and the results of engineering calculations. Thisinformation may take the form of text, numerical data or graphical images, and frequently includes all typesfor any given project. Storing this data in a form that makes its retrieval quick and convenient is arequirement of modern geotechnical practice. In most professions, digital data storage is now the norm, anda plethora of commercial and non-commercial software has been developed to meet the growing need forstorage and retrieval of information. Geotechnical engineering is no exception. There exist a number ofcommercial software packages to assist with these tasks, and many geotechnical firms have developed theirown in-house solutions to digital storage of their vital information.

12.1 Case Study - Hong Kong International Airport

A striking example of just what can be done, and the efficiency that is achievable, in the storage andtransmission of geotechnical information via a digital database can be seen in the case study provided by theconstruction of the Hong Kong International Airport. Vast amounts of geotechnical data were generated on

this project and shared electronically between interested parties. It is probably one of the biggest, if not thebiggest, geotechnical data management project of its type to date in the world.

The platform on which the new Hong Kong International Airport has been built was created in just twoand a half years, commencing at the end of 1992, and its construction involved a unique combination ofdredging, mining and seawall operations. Three-quarters of the airport platform was reclaimed from the seaand the remaining quarter is the result of the excavation of two islands – Chek Lap Kok and its smallerneighbour, Lam Chau. The construction project is one of the largest civil engineering endeavours of thistype ever undertaken. Coordination and control of the construction project was vested in the AirportAuthority, Hogn Kong, a body established by an act of the Hong Kong legislature in 1995. A detailedaccount of the design and construction of the airport platform has been provided in the book by Plant, Coviland Hughes (1998).

Extensive site investigation was carried out prior to the reclamation works for the airport, but more wasrequired during the course of the Site Preparation Contract. Investigations to assess the performance of thereclamation were particularly important. In addition, investigations were required to refine constructionprocedures, to assess the density of the as-placed fill materials, and for quality control. Additionalinformation was required for the design of subsequent works, such as tunnel alignments, runways, buildingsand services. Instrumentation was installed to monitor the settlement of the reclamation and to assess theground treatment measures used on site.

The investigations carried out at the various stages were numerous and varied. They included drill holes,trial pits, vibrocores, cone penetration tests (CPTs), in situ field vane tests, seismic surveys, SpectralAnalysis of Surface Waves (SASW), and associated laboratory testing. All factual data collected during siteinvestigations, and some of the interpretative reports produced using this data, were made available by theAirport Authority to relevant contractors. The information comprised factual reports, interpretative reports,raw data, drawings, seismic surveys, digital data and photographic records.

According to Plant et al. (1998), “the [Airport] Authority realised early in the project that vast amounts ofground investigation and instrumentation would be generated during site preparation and this corpus of datacould become unwieldy unless efficiently managed”. The Authority’s strategy for managing this body ofdata was to establish and maintain a comprehensive database for airport ground investigations.

The database was designed to incorporate the following functions:• a repository of digital ground investigation data,• stratigraphic correlation,• geological correlation with instrument installations,• correlations between geology, field tests, laboratory tests and stratigraphy, and• comparison of geotechnical, survey and engineering information.

According to Plant et al. (1998) the contents of over 400 site investigation reports were translated orinput as digital data. The database contains an accessible master copy of all available geotechnicalexploration point data, including boreholes, prebores, CPTs, instrument installations, laboratory test resultsand in situ tests.

As indicated in the list above, one of the important uses of the database appears to have been in thecorrelation of data of various types, thus providing a degree of confidence in the various geological andgeotechnical models developed. In particular, geophysical methods were used to assess the stratigraphy andthese data were correlated with the geological descriptions from boreholes and with the CPT data.

In May 1993, the Authority purchased the software program gINT (Geotechnical ComputerApplications, 1991) to provide electronic database facilities. In common with a number of the availabledatabase packages, the gINT program organises data into a tiered system. In this particular case the threetiers consisted of project level data, hole level data, and depth level data. In common with the architecture ofmost electronic databases, the data are stored in specified fields. Access to the data as output is via a set ofstandard templates, created to produce output of the desired form. The data sets used within the database forthe Hong Kong International Airport are indicated in Table 13, which provides an indication of the scale ofthis data management exercise. An example of the graphical output providing a representation of a smallportion of this data is presented in Figure 71.

The need for a standard format for interchanging geotechnical data should be self-evident. Its desirabilityhas been well documented, e.g., Greenwood and Rathery (1992), Nicholls et al. (1996). A standardinterchange format enables different organisations who have different database management systems or whohave different system configurations to pass data from one to another in a standard way.

Plant et al. (1998) note that the United Kingdom-based Association of Geotechnical Specialists (AGS)was probably the first organisation to attempt standardisation of an interchange format for groundinvestigation data (AGS, 1992, 1994). Contractors working on the airport project were obliged to presentfactual data digitally in the specified AGS format. The AGS system comprises a set of “rules” for setting upelectronic files and a data dictionary. As noted by Plant et al., it is necessary sometimes to establishalternative or additional data dictionaries to suit individual projects.

Significant use was made of the gINT system to produce output in the form of cross-sections andinterpretation of seismic reflectors. The output was enhanced by the use of CADD software for productionof final drawings, e.g., Figure 71.

The Hong Kong International Airport case study provides a very convincing illustration of theimportance of database software in the large-scale construction projects, particularly where vast amounts ofgeotechnical data need to be stored, processed, retrieved, interpreted, visualised and analysed, by a variety ofinterested parties. It is difficult to imagine coping with such quantities of data and such heavy processingdemands without the availability of modern computer technology and database software. This particularcase study also demonstrates the importance of standards, such as the one developed by AGS, for theinterchange of geotechnical information. The need for these standards should be self-evident.

Table 13. Data sets in the Hong Kong International Airport database (after Plant et al., 1998)

Dataset Description ContentsNo. of

explorationpoints

PAA Main database All the more important platform-wide data,including instrumentation, laboratory testing,deep CPTs and investigation penetrating into thenatural geology below the base of the fill ormarine (previous) layers.

4499

CPT Shallow CPTdatabase

Platform-wide dredge level CPTS. 3086

QCPT Quality controldatabase

CPTs through the Type C, sand, fill in themiddle and northern parts of the platform,carried out to demonstrate the density of thesand fill.

522

TB Passengerterminal buildingfoundations

All the prebores carried out for the foundationworks of the passenger terminal building.

792

VIBRO Vibrocompactiondatabase

CPTs carried out in the Sand, Fill Type C, and inSPC Works Areas HI/6 and H8/1 wherevibrocompaction ground treatment was carriedout.

748

302 302 Foundations Prebores for the area of Contract 302. 280303 303 Foundations Prebores for the area of Contract 303. 118

INST Instrumentationdatabase

Duplicate set of exploratory hole data ofInstrumentation holes only, rationalised forspecific plots of instrument data against drillingrecords.

399

While the scale of theHong Kong a i rpor tconstruction project mayhave demanded the use ofm o d e r n c o m p u t e rtechnology to handle all thegeotechnical data, it hasalso pointed the wayforward for projects of moremodest scale. The benefitsof convenient data retrievalwould seem to be wellworth the investment in thetechnology, at least for allbut the smallest projects. Atthe very least, in mostgeotechnical consultancies,being able to convenientlylocate and retrieve datafrom a history of similar orrelated projects would be aboon on many jobs.

13.0 EXPERTSYSTEMS

Expert system technology started to emerge from the Artificial Intelligence research laboratories duringthe 1980s. However, even now, more than a decade later, there are few expert systems that have beendeveloped for geotechnical work. The difficulties with the wider adoption of this technology are probablyrelated to both technical and psychological factors. These events are probably not dissimilar to the inertiathat apparently existed when computers were first introduced into common usage.

Expert systems are computer programs that collect human experience in solving problems by usingheuristics (rules of thumb). Forsyth (1986) proposed that expert systems should:• be able to generate conclusions or advice from uncertain, incomplete, or conflicting information or

knowledge,• must be able to explain their reasoning processes and conclusions,• must be able to separate stored knowledge and inference mechanisms so that they can be modified or

edited without affecting each other,• be able to modify, delete or enrich the stored knowledge incrementally,• deliver advice, usually non-numerical, as its output.

Expert systems are typically rule-based, containing numerous rules-of-thumb that express the judgementprocesses of experts. Generally they do not involve the use of procedural algorithms, as are common indeterministic numerical analysis.

There are several reasons for developing special expert systems software in geotechnical engineering,and these include possibly the need to compensate for a lack of available experts, the need to save the cost ofemploying experts, the need to provide additional assistance to experts, and the need to train future experts.A detailed exposition on these needs, and the techniques adopted recently to fulfill them, has been providedby Wong (1990).

Many of the major problems confronted in geotechnical engineering are caused by or related to thecomplicated engineering behaviour of soils and rock masses. This behaviour depends on the completegeological history of the deposit, as represented by the size, shape, mineral composition and packing of theparticles, the stress history that has been experienced, the pore fluid and other factors (Wroth and Houlsby,1985). In addition, many of these properties also vary with time and the applied stress level. Success in thepractice of geotechnical engineering has usually required the accumulation and appropriate application ofexperience, sometimes hard-won experience. Experience and knowledge of soil or rock mechanics and

Figure 71 : Example of CADD output – Hong Kong International Airport(Plant et al., 1998) [Supplied by courtesy of Airport Authority Hong Kong].

geology may be obtained by formal training, but this does not guarantee that reliable practice of geotechnicalengineering will follow. There are several problems in applying what amounts to “second-hand” experience,including the following:• difficulties in identifying potential problems,• problems with interpretation of existing knowledge,• lack of appreciation of the assumptions on which underlying theories are based,• difficulties in justifying various assumptions for a given site,• difficulties in keeping up-to-date with the latest knowledge,• difficulties in learning how various facets of knowledge are integrated into practice, and• difficulties in focusing on critical information when a problem is encountered for the first time.

Assessing and assimilating allthe necessary factors involved inany given geotechnical problem,and being able to assess theirrelative importance, requiresconsiderable skill and experience.Of course, it is possible thatgeotechnical engineers will learnfrom their own mistakes, butgenerally this is not an efficient oreconomical means of gainingexperience. More efficient waysof passing on experience for thefuture need to be found.

A comparison of the way inwhich humans conventionallysolve problems of this type, andthe way in which an expertsystem is usually designed tosolve such problems has beenindicated by Wong (1990), seeFigure 72. It is apparent from thiscomparison that expert systemtechnology is well-suited to assistin the task of making the sort ofjudgements that are often required in geotechnical practice. It is perhaps curious then that so few examplesexist today in geotechnical engineering practice: most of those that do exist were developed in the 1980s. Itis also curious that there seems to have been little progress in the application of this technology togeotechnical practice in recent years. This is possibly related to the difficulty and costs associated with theirdevelopment. Examples of those that have been developed include:• SOILIDEN (Alim and Munro, 1987) – an expert system to assist with the identification of soil on the

basis of field observations and soil characteristics as described in the British code of practice,CP20004:1972.

• CONE (Mullarkey, 1985; as extracted from Siller, 1987) – a knowledge-based system to classify soilsand assess their shear strength based on cone penetrometer data

• RETWALL (Hutchinson, 1985) – a knowledge-based system to evaluate whether a retaining wall isrequired in a given situation and to evaluate which wall type is most appropriate.

• SOILCON (Wharry and Ashley, 1986; as extracted from Siller, 1987) – an expert system designed toadvise the user about the level of geotechnical investigation necessary in order to reduce the associatedrisk.

• LOGS (Lok, 1987; as extracted from Adams et al., 1989) – a system developed to treat information formseveral boring logs and to provide the user with two-dimensional profiles.

• FOOTER (Shukula, 1988) – a system designed to propose a footing design for each column or shearwall in a structure.

CONVENTIONALAPPROACH

EXPERT SYSTEMAPPROACH

1. Understand the problem 1-Provide information needby the expert system

2. Divide the problem intosmaller sub-problems ifnecessary

2. Solve the problem bydrawing conclusions fromstored knowledge andgiven information

3. Solve the problem or sub-problems by:-Using own knowledge and

experience,-Getting help from

experienced people,-Getting help from published

literature.

3. User justifies the solution

4. Justify the solution

Figure 72 : Comparison of problem solving techniques(after Wong, 1990)

• PILE (Santamarina and Chameau, 1987) - a system to assist users in selecting the appropriate type ofpile foundation according to aspects such as soil characteristics, pile loads, installation conditions, andother factors.

• SUPILE (Wong, 1990) – a system designed to assist foundation engineers with pile design.• SITECLAS (Wong et al., 1989) – an expert system designed to assist with site classification.

14.0 COMPUTER VISUALISATION IN GEOTECHNICSLargely because of the complexities in geometry of many geotechnical problems, and the often vast

quantities of data required to describe such problems, computer visualisation tools are being usedincreasingly in geotechnical research and practice. It is now possible to gain a comprehensive and relativelyrapid appreciation of the geometric complexity of many geotechnical challenges with the aid of moderncomputing technology.

An excellent example of this use has been the use of Geographic Information Systems (GIS) for themapping of geology and landslides and representation and assessment of the risk of landsliding in an area ofNew South Wales, near the city of Wollongong, south of Sydney, Australia. This study is being conductedby the University of Wollongong with the collaboration of the Wollongong City Council, the Rail ServicesAustralia and the Australian Geological Survey Organisation (Flentje, 2000). Illustrations of the power ofcomputer visualisation tools being utilised on this project may been seen in Figures 73, which showisometric views of the geomorphology of the study area, the slopes in the area, as well as a compositerepresentation of the geology superimposed on known landslide areas. The ability of these graphicaltechniques to represent clearly the salient aspects of such a study are obvious.

There can be no doubt that the declining cost of both hardware and graphics software tools, and theincrease in “user friendliness” of these tools will lead to much greater use of them in the future. Theattraction of presenting such vivid and meaningful representations of this type of data should ensure that inthe near future these graphics tools will become as common as the geologist pick in the tool bag of mostgeotechnical specialists.

Figure 73 : Example of computer visualisation (Flentje, 2000)

15.0 THE FUTURE

Like most branches of engineering, geotechnics will continue to benefit in the future from the on-goingdevelopments in computer hardware and software. The ability to model real problems using mathematicaltechniques will need to be enhanced. Computer hardware will become more affordable and developments insoftware, though understandably often lagging behind those in hardware, will also make the process ofmodelling more convenient and cost-effective. Potentially, numerical modelling could become routine inthe very near future for a large range of geotechnical problems. What then are the future requirements, andwhat are the impediments, if any, to this advance? Are there areas where major research effort is required?

In the past, most of the geomechanics applications software was developed in universities and researchinstitutions, or their spin-off organisations. When suitable hardware platforms became ubiquitous, as aresult of the fall in hardware prices, a number of these software packages, developed first as research orteaching tools, were commercialised. This development has occurred almost entirely as a cottage industry,labour intensive and with only a relatively small niche market. It has meant that the unit price of manysoftware packages has remained relatively high. It is likely that this trend will continue in the future.Because of limited demand, it is unlikely that major software houses will take up the development of highlyspecialised geotechnical software. Possible exceptions include the general-purpose finite element packagesnow reaching the market, some of which contain a variety of useful soil models, together with their othermultifarious capabilities.

It is possible to identify some areas in which further development is required, and these are:• constitutive models for many geomaterials, interfaces and joints and their critical evaluation, including:

constitutive models for cyclic behaviour and fundamental approaches for liquefaction (instability),constitutive models for natural soils (with structure), e.g., sands, as opposed to reconstituted soils,constitutive models for multi-phase materials, and constitutive models for composite material systems inreinforced soil construction,

• controlled laboratory and field tests to validate numerical and constitutive models,• micromechanical modelling of assemblages of particles using the distinct element model or other

suitable approaches to infer macroscopic behaviour,• appropriate test devices, particularly for interfaces including measurement of pore water pressures,• robust and reliable numerical procedures to obtain unique and realistic solutions for problems involving

discontinuities, degradation and softening,• robust and reliable numerical procedures based on mesh adaptation,• more comprehensive expert systems for geotechnical practice, and• numerical procedures to model accurately in situ penetration tests, e.g., the CPT test.

The analyses required for penetration tests must involve very large strains, rupture and movingboundaries. The work of Hu and Randolph (1996) shows great promise in this area.

The recent development of two and three-dimensional finite element procedures for limit analysis andload-path analysis, and software encoding sophisticated constitutive models, for instance the heirachicalDSC models, now provide the means to solve a wide range of geotechnical problems involving static,repetitive, dynamic and time-dependent loading and material response.

It is clear from the above list, which is by no means complete, that much work is still required in the fieldof constitutive modelling. Probably the greatest impediment to the use of many models proposed already isthe difficulty in assessing what values should be assigned to the input parameters. Many models in theliterature have a number of parameters that are difficult to evaluate directly from either field or laboratorytests. If model parameters have a physical basis, then there is greater chance of parameter measurement, andhence the eventual adoption of the model in practice. Obviously, developments in this area must go hand-in-hand with parallel developments in experimental techniques. It has not always been possible to arrange thissimultaneous advance.

Calibration of new and existing models is also an essential requirement if genuine advances are to bemade. In this respect, the recent efforts made by various workers (a) to calibrate their models against theresults of field and laboratory tests, especially centrifuge data, and (b) to compare the predictions of variousnumerical tools for the same set of control problems, is to be applauded. Such comparisons have alwaysbeen an essential component of previous advances, and they continue to point the direction forward.

As demonstrated in this paper, there is a pressing need for more and comprehensive validation studies fornumerical simulations. Ultimately these should lead to sound guidelines for geotechnical practitioners tofollow.

The future should also see the advancement of the so-called “soft computing” tools. The need to developexpert systems for geotechnical practice is an obvious example.

16.0 CONCLUSIONSThis paper has provided a description of the use of computers and numerical methods in geotechnical

practice and research. It has been intentionally and necessarily selective, reflecting the experience of theauthors and their colleagues.

Sophisticated numerical models have been used regularly in geotechnical research for the past thirtyyears or more, but they have now moved beyond the boundaries of research institutions. It should be clearfrom the limited set of examples presented in this paper that the use of relatively sophisticated non-linearand time-dependent numerical models is now a common occurrence in geotechnical practice. Theavailability of economical personal computers, the release commercially of a number of robust softwarepackages, the training of a new generation of engineers, familiar and comfortable with the world ofelectronic computing (including many older colleagues willing to embrace the new technology), have allplayed their part in this evolution. Practitioners are now using these tools in their day-to-day work.Numerical models have their place in the toolkit of the modern geotechnical engineer. That place may be toexamine possible behavioural mechanisms, or to throw light on the dominant parameters of a problem, or tohelp understand field performance.

A discussion has also been presented about some limitations of the current techniques and a few of thedevelopments required in the future. In particular, constitutive models that capture the essential features ofsoil or rock behaviour, and are based on physically meaningful and simple to measure parameters, areneeded. Models for cyclic loading and for natural (structured) soils are examples. The incorporation ofthese deterministic constitutive models in a probabilistic assessment of future behaviour also requiresdevelopment work.

Further research and development work is also required in the field of “soft computing” as applied togeotechnical problems. Much greater effort is required to validate and calibrate numerical models and theassociated computer software.

17.0 ACKNOWLEDGEMENTS

Some of the results presented in this paper were the outcome of research supported by various grantsfrom the Australian Research Council and the National Science Foundation, Washington, D.C. The authorswish to acknowledge the following individuals for kindly providing some of the material presented in thispaper and for valuable assistance in its preparation: Dr H. Taiebat, Dr M. Rahmann, Mr K. Islam, Dr M. Liu,Mr W. Deng, Mr C. Covil, Dr P. Flentje. The permission of Dr P. Flentje to reproduce the image taken fromthe Wollongong Landslide Hazard and Risk study conducted jointly by the Australian Geological SurveyOrganisation and the University of Wollongong is gratefully acknowledged. The permission of Mr CraigCovil to publish the CADD image from the Hong Kong International Airport project, supplied by courtesyof the Airport Authority Hong Kong, is also gratefully acknowledged.

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