(David Potts)_Numerical Analysis a Virtual Dream or Practical Reality

Potts, D. M. (2003). Geotechnique 53, No. 6, 535–573

535

Numerical analysis: a virtual dream or practical reality?

D. M. POTTS

The development of numerical analysis and its applica-tion to geotechnical problems over the past 20 years haveprovided geotechnical engineers with an extremely power-ful analysis tool. However, the use of such analysis is stillnot widespread, and when it is used there is all too oftenevidence of bad practice. Part of the reason for this is alack of education and of guidance, especially from codesof practice, as to the appropriate use of such methods ofanalysis. Clearly, some form of initiative is required topromote good practice and allow the full potential of thisanalysis tool to be realised, both from a safety and aneconomy perspective. This lecture begins by reviewingthe key advantages of numerical analysis over conven-tional analysis tools, and then debates whether or not itcan replace the conventional analysis tools in the designprocess. Examples from engineering practice are usedextensively to illustrate the arguments both for andagainst the use of numerical analysis.

KEYWORDS: Rankine Lecture

Le developpement de l’analyse numerique et ses applica-tions aux problemes geotechniques au cours des 20 der-nieres annees ont donne aux ingenieurs geotechniciens unoutil d’analyse extremement puissant. Cependant, l’utili-sation de cette analyse est encore restreinte et memelorsqu’elle est utilisee, elle est trop souvent mal pratiquee.La raison en est, en partie, le manque d’information et deformation, notamment dans les codes de pratique, sur lebon emploi de ces methodes d’analyse. Clairement, unecertaine forme d’initiative est necessaire pour encouragerune bonne pratique et permettre la realisation de tout lepotentiel de cet outil d’analyse, tant sur le plan de lasecurite que d’un point de vue economique. Cette con-ference commence par passer en revue les principauxavantages de l’analyse numerique par rapport aux outilsd’analyse classiques, puis on debat pour savoir si elle peutou non remplacer ces outils classiques dans le processusde conception. On utilise de nombreux exemples tires dela pratique afin d’illustrer des arguments pour et contrel’utilisation de l’analyse numerique.

INTRODUCTIONThe title of this Rankine Lecture, ‘Numerical analysis: avirtual dream or practical reality?’, essentially poses thequestion: ‘Is numerical analysis just an advanced toy foracademics and the privileged few, or is it in a position toprovide a genuine tool for routine geotechnical analysis?’

This is a pertinent question to ask at this time as newcodes of practice, for example Eurocode 7, are not asprescriptive as the older codes and allow the designer tochoose an appropriate method of analysis. Recent compari-sons (Gaba et al., 2002; Ravaska, 2002) indicate that the useof numerical analysis, as opposed to conventional methods,can lead to more economical design, and consequently theuse of this type of analysis is likely to increase in the future.

Although most geotechnical engineers have had contactwith numerical analysis, this has often been at arm’s length,and many do not fully appreciate the complexities andsubtleties involved in its use. For such analysis to be used ina constructive manner, and to avoid future disasters, it isimportant that geotechnical engineers understand both theenormous potential of this type of analysis and its pitfalls.

This lecture begins by discussing the role of analysis indesign. The various different methods of analysis that arecurrently available to geotechnical engineers are then cate-gorised and compared. This leads the author to propose amotion for debate: ‘Numerical methods of analysis havereached the stage where they are superior to conventionalapproaches and can replace them in the geotechnical designprocess.’ Cases both for and against the motion are thenpresented, using a series of practical examples. The lectureconcludes by presenting some views about the future.

THE ROLE OF ANALYSIS IN DESIGNThe role of analysis in the design process becomes clear

once the design objectives have been considered.When designing any geotechnical structure, the engineer

must ensure that it is stable. Stability takes several forms.First, the structure and support system must be stable as awhole. There must be no danger of rotational, vertical ortranslational failure (see Fig. 1). Second, overall stabilitymust be established. For example, if a retaining structuresupports sloping ground, the possibility of the constructionpromoting an overall slope failure should be investigated.

The loads on, and forces in, any structural members suchas walls, slabs, props, anchors or membranes must beestimated under the most adverse conditions, so that theymay be designed to carry them safely. For example, whendesigning an embedded retaining wall to support an excava-tion, it is necessary to estimate the bending moments andaxial and shear forces in the wall elements and any props oranchors.

Movements must be estimated, both of the structure andof the ground. This is particularly important if there areadjacent buildings or sensitive services. For example, if anexcavation is to be made in an urban area close to existingservices and buildings (Fig. 2), one of the key designconstraints is the effect that the excavation has on the

Manuscript received 19 March 2003; revised manuscript accepted25 March 2003Discussion on this paper closes 1 February 2004, for further detailssee p. ii.� Department of Civil and Environmental Engineering, ImperialCollege of Science, Technology and Medicine, London. Fig. 1. Stability

adjacent structures and services. This could involve estimat-ing both movements and induced structural forces.

To satisfy these design objectives, it is necessary for anengineer to perform calculations—for example by introdu-cing a factor of safety into stability calculations, or byperforming calculations to estimate movements and structur-al forces. Such calculations use mathematics and are basedon analysis. A good analysis, which simulates real behav-iour, allows the engineer to understand problems better.Analysis is not the only part of the design process, whichinvolves many facets such as parameter identification andempiricism, but it plays an important role. The remainder ofthis lecture will concentrate on analysis.

METHODS OF ANALYSISIntroduction

The objective here is to provide a framework in which tocompare the different methods of analysis available togeotechnical engineers. This will enable the relative meritsof numerical analysis, compared with conventional methods,to be established. However, before considering the differentforms of analysis, it is instructive to briefly consider thetheoretical requirements that must be satisfied. Essentiallythere are four of these.

Equilibrium. This can be divided into two parts: overallequilibrium and internal equilibrium. Overall equilibrium isassociated with resolving forces and taking moments,whereas internal equilibrium is associated with establishingstress fields that satisfy the following well-known partialdifferential equations:

@�x

@xþ @� yx

@ yþ @�zx

@zþ ª ¼ 0

@�xy

@xþ @� y

@ yþ @�zy

@z¼ 0

@�xz

@xþ @� yz

@ yþ @�z

@z¼ 0 (1)

Note that the above equations are in terms of total stresses(�x, �y, �z, �xy, �xz, � yz) in accordance with a Cartesian

coordinate system, and that self-weight (ª) is assumed to actin the negative x direction. Equilibrium is therefore con-cerned with forces and stresses.

Compatibility. Compatibility, on the other hand, is con-cerned with displacements and strains. It essentially requiresthat, if a body moves under loading, holes do not appearwithin it and overlapping of material does not occur. This isclearly a statement of common sense, but it must beremembered that within an analysis such restrictions mustbe formulated mathematically. In situations in which strainsare small, this is achieved by imposing the condition that thevariation of displacements throughout the body underconsideration satisfies the following strain equations:

�x ¼ � @u

@x; � y ¼ � @v

@ y; �z ¼ � @w

@z

ªxy ¼ � @v

@x� @u

@ y; ªyz ¼ � @w

@ y� @v

@z;

ªxz ¼ � @w

@x� @u

@z(2)

where u, v and w are the displacements in the x, y and zdirections respectively. It can be shown mathematically that,for a compatible displacement field to exist, all the abovecomponents of strain and their derivatives must exist and becontinuous to at least the second order.

Constitutive behaviour. This is a description of materialbehaviour. In simple terms it is the stress–strain behaviour ofthe soil. It usually takes the form of a relationship betweenstresses and strains, and therefore provides a link betweenequilibrium and compatibility. For calculation purposes theconstitutive behaviour has to be expressed mathematically:

f˜�g ¼ [D]f˜�g (3)

These equations can be expressed in terms of either total oreffective stresses. However, if they are expressed in terms ofeffective stress, the behaviour of the pore fluid must beconsidered to enable total stresses to be derived, which canthen be combined with the equilibrium equations. If the soilis linear-elastic, then accumulated stresses and strains can berelated to each other via the elastic properties. However,because soil usually behaves in a non-linear manner, it ismore realistic for the constitutive equations to relate incre-ments of stress (˜�) and strain (˜�), as indicated in equa-tion (3), and for the [D] matrix to depend on the current andpast stress and strain history.

Boundary conditions. These are specific to the situationunder consideration and define the nature of the boundaryvalue problem to be investigated. For example, they coulddefine a displacement constraint, a sequence of loadapplication, excavation, construction, or a pore water pressurechange.

The above discussion is clearly simplified. For example, ifthe flow of pore fluid is involved, then the continuity equa-tion and boundary conditions involving prescribed flows orpore fluid pressures would also have to be accounted for.Likewise, if dynamic loading was involved, the equilibriumequations would have to be extended to account for inertiaand damping effects.

Current methods of analysis can be conveniently groupedinto the following categories: closed form, simple andnumerical analysis. Each of these categories is now consid-ered in turn. The ability of each method to satisfy thefundamental theoretical requirements and provide designinformation is summarised in Tables 1 and 2 respectively.

Services

Tunnel

Fig. 2. Interaction of structures

536 POTTS

Closed form solutionsA closed form solution is the ultimate method of analysis.

For a particular geotechnical structure, if it is possible toestablish a realistic constitutive model for material behav-iour, identify the boundary conditions, combine these withthe equations of equilibrium and compatibility, and thenperform the resulting integrations, an exact theoretical solu-tion can be obtained. The solution is exact in a theoreticalsense, but is still an approximation to the real problem, asassumptions about geometry, the applied boundary condi-tions and the constitutive behaviour have been made inidealising the real physical problem into an equivalentmathematical form. In principle, it is possible to derive acomplete analytical solution for movements, forces, stressesand strains for every point within the problem domain,for all stages of the construction history. A single analysiswill therefore provide information on both movements andstability.

However, as soil is a highly complex multi-phase materialthat behaves non-linearly when loaded, complete analyticalsolutions to realistic geotechnical problems are not usuallypossible. Solutions can be obtained only for two limitedclasses of problem.

First, there are solutions in which the soil is assumed tobehave in an isotropic linear-elastic manner and the bound-ary conditions are simple. Although these can be useful forproviding a first estimate of movements and structuralforces, they are of little use for investigating stability. Com-parison with observed behaviour indicates that such solutionsare often inaccurate.

Second, there are some solutions for problems that containsufficient geometric symmetries that the problem reduces tobeing essentially one-dimensional. Expansion of sphericaland infinitely long cylindrical cavities in an infinite elasto-plastic continuum are examples. Although they are useful,such solutions have restricted practical application.

Simple methodsFaced with the requirement for more appropriate analysis

tools, the pioneers of geotechnical engineering made simpli-fications. These involved relaxing one or more of thesolution requirements, usually the compatibility requirement.The resulting analysis methods can be broadly categorised aslimit equilibrium, stress field and limit analysis methods (seeTable 1).

Examples of limit equilibrium analysis are Coulomb’swedge analysis (Coulomb, 1776), the analysis performed byCaquot and Kerisel to derive the well-known active andpassive earth pressure coefficients (Caquot & Kerisel, 1948),and the methods of slices used in slope stability (Bromhead,1992).

Rankine earth pressure coefficients (Rankine, 1857), thesolutions of Sokolovskii (1960, 1965) and the bearing capa-city coefficients are all examples of stress field solutions.

Limit analysis involves upper bound (that is, unsafe) andlower bound (that is, safe) solutions: see Chen (1975). Theformer neglects equilibrium and the latter compatibility. Inprinciple, if upper bound and lower bound solutions can befound that furnish the same result, then this is the exactsolution of the problem being analysed. However, there arefew cases where this can be achieved, and even in thesecases a very idealised constitutive model is employed.

It is important to note that all of these approachesessentially require an assumption as to the mechanism offailure. They are also restricted to assuming that the soilbehaves either undrained or drained. In the former case aTresca failure condition in terms of undrained strength, su, isassumed. In the latter case a Mohr–Coulomb failure criter-ion in terms of cohesion, c9, and angle of shearing resis-tance, �9, is assumed.

In respect to satisfying the design requirements, thesemethods provide information on stability only. As they failto satisfy all of the four theoretical requirements, they are

Table 1. Basic solution requirements satisfied by various methods of analysis

Method of analysis Solution requirements

Equilibrium Compatibility Constitutive behaviour Boundary conditions

Force Displacement

Closed form S S Linear elastic S SLimit equilibrium S NS Rigid with failure criterion S NSStress field S NS Rigid with failure criterion S NS

Limit analysis Lower bound S NS Ideal plasticity with associated flow rule S NSUpper bound NS S NS S

Full numerical analysis S S Any S S

S, satisfied; NS, not satisfied.

Table 2. Design requirements satisfied by the various methods of analysis

Method of analysis Design requirements

Stability Movements Adjacent structures

Closed form (linear-elastic) No Yes YesLimit equilibrium Yes No NoStress field Yes No No

Limit analysis Lower bound Yes No NoUpper bound Yes Crude estimate No

Full numerical analysis Yes Yes Yes

NUMERICAL ANALYSIS: DREAM OR REALITY? 537

only approximate solutions. For the remainder of this lec-ture, the closed form and simple methods of analysis will bereferred to collectively as conventional methods.

Numerical methodsIn this class of analysis all requirements of a theoretical

solution are considered, but may only be satisfied in anapproximate manner. These methods have been developed toa standard where they are useful to the design process onlyin the last 10–15 years. They were therefore not available tothe pioneers of geotechnical engineering.

Approaches based on finite difference and finite elementmethods are those most widely used. These methods essen-tially involve a computer simulation of the history of theboundary value problem from greenfield conditions, throughconstruction and into the long term. A single analysis canprovide information on all design requirements: see Table 2.

Their ability to accurately reflect field conditions essen-tially depends on the ability of the constitutive models torepresent real soil behaviour and the ability of the geotechni-cal engineer to assign appropriate boundary conditions tothe various stages of construction. Additional advantagesover the conventional methods are that the effects of time onthe development of pore water pressures can be simulatedby including coupled consolidation/swelling, dynamic behav-iour can be accounted for, and—perhaps most importantly—no postulated failure mechanism or mode of behaviour ofthe problem is required, as these are predicted by the analy-sis itself.

In many respects numerical methods are superior to con-ventional methods. However, they do contain approxima-tions. The two main assumptions will now be discussed,using the finite element method as an example.

Discretisation approximation. When modelling a boundaryvalue problem the geometry under investigation is repre-sented by an assemblage of small regions, termed finiteelements (see Fig. 3). These elements have nodes, defined onthe element boundaries or within the element. To proceedwith the method, a primary variable must be selected (such asdisplacement or stress), and rules as to how it should varyover a finite element must be established. This variation isexpressed in terms of nodal values.

In geotechnical engineering it is usual to adopt displace-ment as the primary variable and assume that it varies in a

polynomial manner over the element. For example, a 4-noded plane-strain element assumes a linear variation,whereas an 8-noded element assumes a quadratic variationof displacements u and v: see Fig. 3. An analysis willtherefore be accurate only if such an assumption accuratelyreflects the way the soil wants to deform in the boundaryvalue problem being investigated. However, accuracy can beimproved by using smaller elements and therefore moreelements to represent a fixed volume of soil. This results ina trade-off between accuracy and calculation time. Forfurther details see Potts & Zdravkovic (1999).

Non-linear solution strategy. The basic theory behindnumerical methods (that is, finite elements, finite differencesand boundary elements) assumes that material behaviour islinear and that displacements are small. This in turn impliesthat material behaviour is essentially linear-elastic, with aconstant stiffness and an infinite strength. Unfortunately, soildoes not behave in this manner, and consequently, owing tomaterial and/or geometric non-linearity, soil stiffness changesthroughout an analysis. This is illustrated by the one-dimensional load–displacement curve shown in Fig. 4.

A solution strategy is required that can account for thischanging material behaviour. As a first approximation thischange in stiffness may be accounted for by applying theloading in a series of increments (or steps). For eachincrement an appropriate stiffness is selected, and the incre-mental displacements, stresses and strains are calculated. Asthe load increments are of a finite size, this results in an

u or vLineardisplacementvariation

Undeformedelement

x

u or vQuadraticdisplacementvariation

Undeformedelement

x

∆v

∆u

y

x

y

y

Fig. 3. Discretisation and displacement approximations

Load

Displacement

KG1

KG2

[KG]i{∆d}nGi � {∆RG}i

Fig. 4. Implications of non-linearity

538 POTTS

approximation as the stiffness changes over the increments.Some of the more sophisticated software packages attemptto account for this, but there is no mathematically rigorousway of doing so: see Potts & Zdravkovic (1999). Conse-quently, approximations, albeit of a much smaller value, stillexist.

As the solution strategy is a key component of any non-linear analysis, it can strongly influence the accuracy of theresults and the computer resources required to obtain them.We shall return to this problem later in this lecture.

THE MOTIONBased on the above discussion, numerical analysis would

appear to be able to do everything that the conventionalmethods can do, plus a lot more. This raises the question asto why we still use conventional methods. In the past theremay have been some restrictions imposed by codes of prac-tice, but this is not so with the newer codes (such asEurocode 7).

To explore this question further the remainder of thislecture will be presented in a debate-style format. Themotion to be considered is:

Numerical methods of analysis have reached the stagewhere they are superior to conventional approaches andcan replace them in the geotechnical design process.

Note that the motion is not about whether or not numericalanalysis should be used, but whether numerical analysis canreplace conventional methods of analysis.

Cases for and against the motion will be presented. Eachcase will involve numerical examples: except where specifi-cally stated, these have been performed using the ImperialCollege Finite Element Program (ICFEP) written by theauthor, with help from staff and research students at ImperialCollege.

THE CASE FOR THE MOTIONIntroduction

The case for the motion is based on the premise thatnumerical analysis can:

(a) do everything conventional analysis can do(b) accommodate realistic soil behaviour(c) account for consolidation(d ) provide mechanisms of behaviour(e) accommodate complex soil stratigraphy( f ) account for interaction between structures(g) accommodate three-dimensional geometries.

Each of these postulates will now be considered in turn,using examples from geotechnical practice.

(a) Do everything conventional analysis can doTo illustrate this postulate the behaviour of surface foun-

dations will be considered. This is perhaps the simplestproblem faced by geotechnical engineers. Ideally, for design,the complete load–displacement curve of the foundation isrequired: see Fig. 5. Conventional methods of analysis onlyprovide estimates of the initial gradient of this curve (fromclosed form elastic solutions) and the ultimate load that canbe supported by the soil (from one of the simple methods).For the purposes of this lecture, we shall concentrate on theprediction of ultimate load. Conventionally, this is usuallyexpressed using the following well-known drained bearingcapacity equation:

qmax ¼ Qmax

A¼ icdcsc N strip

c c9þ iqdqsq N stripq q9

þ iªdªsªN stripª Bª9 (4)

where Qmax is the maximum load that can be applied to thefoundation; A is the area of the foundation; B is the halfwidth of the foundation for a strip or rectangular footing, orthe radius for a circular footing; c9 is the drained cohesionof the soil; q9 is the magnitude of any surcharge pressureexisting on the ground surface adjacent to the footing (seeinsert in Fig. 5); ª9 is the effective bulk unit weight of thesoil; N strip

c , N stripq and N strip

ª are bearing capacity coefficientsfor a strip footing and are dependent on the angle ofshearing resistance; �9, sc, sq and sª are shape factors toaccount for the shape of the footing; dc, dq and dª are depthfactors to account for the depth of the footing below groundsurface level; and ic, iq and iª are inclination factors toaccount for the inclination of the load on the footing.

Several restrictions apply to this equation. First, it islimited to either fully saturated or dry soil; second, it isbased on the Mohr–Coulomb failure criterion; and, third, itis not applicable to layered soils. It is also worth noting thatnot all the terms in the equation have a theoretical basis. Inparticular, the shape, depth and inclination factors are semi-empirical.

There is no exact analytical expression for the coefficientN strip

ª , and consequently there are many different expressionscurrently in use (Sieffert & Bay-Gress, 2000). These arebased on limit equilibrium, stress field or limit analysiscalculations. The variation of N strip

ª with �9 from some ofthe more popular expressions in current use is shown in Fig.6. This figure indicates that, for example, for a soil with a�9 ¼ 258 the value of N strip

ª varies from 6·76 to 10·89 (thatis, a difference of 61%), depending on the solution adopted.

The coefficients N stripc and N strip

q are theoretically exact fora strip footing (Prandtl, 1920), being based on a combinationof stress and velocity field calculations. They are given bythe following equations:

Nq ¼ 1 þ sin�9

1 � sin�9e� tan �9 (5)

Nc ¼ (Nq � 1) cot�9 (6)

Note, however, that these expressions are based on theassumption of (i) elasto-plastic soil behaviour with theMohr–Coulomb failure criterion as a yield surface and (ii)associated plasticity. This latter assumption implies that theangle of dilation, �, which controls the plastic volume

Qq′q′

LoadInitial

stiffnessUltimate

load

Displacement

Fig. 5. Typical load–displacement curve for a footing


change, is equal to the angle of shearing resistance, �9. Itseffect on soil behaviour is best illustrated using a drainedtriaxial test on a Mohr–Coulomb soil as an example. In thisrespect Fig. 7 shows two predictions for an ideal (that is, noend effects) triaxial compression test, one based on � ¼ �9and the other on � ¼ 08. All other properties are the same inboth predictions. As can be seen from Fig. 7(a), the magni-tude of the angle of dilation has no effect on the plot ofdeviatoric stress (�91 � �93) against axial strain (�a).However, the angle of dilation has a significant effect on theplot of volumetric strain (�vol) versus axial strain: seeFig. 7(b). In this figure both analyses initially predictidentical elastic volumetric compression (that is, up to pointA). After yielding occurs at point A the plastic volumetricstrain is controlled by the magnitude of �. For any value of� other than zero, plastic dilation occurs indefinitely. Com-pared with real soil behaviour, both the magnitude of thedilation when � ¼ �9 and the indefinite increase in volumewhen � . 08 are unrealistic. If � ¼ 08 no plastic volumetricstrain occurs, and as there is no change in stress whenyielding occurs, there is no further elastic volume straineither. Consequently, there is no further volumetric totalstrain predicted once yielding occurs: see Fig. 7(b). Thereare clearly severe limitations and shortcomings with theconventional bearing capacity equation.

Application of the finite element method to the problemof a rigid surface footing loaded centrally by a vertical loadis now considered. The finite element mesh used for theseanalyses is shown in Fig. 8, and the soil is assumed to obey

the Mohr–Coulomb model, having a Young’s modulus, E9,of 10 MN/m2, a Poisson’s ratio, �, of 0·3, a cohesion, c9, of0·0 and an angle of shearing resistance, �9, of 258. Toinvestigate the effect of the angle of dilation on bearingcapacity, limiting values of � ¼ 08 and � ¼ �9 are used inthe analysis. For this situation equation (4) reduces to

qmax ¼ Qmax

A¼ sq N strip

q q9þ sªN stripª Bª9 (7)

Results from plane-strain analyses of both rough and smoothstrip footings resting on a weightless (ª9 ¼ 0) soilare presented in Fig. 9, in the form of normalised load–displacement curves. For this situation only the first term inequation (7) is non-zero, and sq ¼ 1. The load has thereforebeen expressed in terms of the mobilised value ofNmob

q ¼ Q=(Aq9), where Q is the load applied to thefooting. Initially, before the footing is loaded in the numer-ical analysis, it is assumed to support a surcharge q9 equalto that acting on the ground surface either side of thefooting. Consequently, at the beginning of the analysis,before any displacement occurs, Nmob

q ¼ 1 and it then in-creases as the footing displaces, until failure is reached whenNmob

q ¼ Nq. The settlement of the footing, � has beennormalised by B, the half width of the strip footing.

Results from four analyses with a surcharge loadq9 ¼ 100 kPa and one with q9 ¼ 10 kPa are presented in Fig.9. For the analyses with q9 ¼ 100 kPa, it can be seen that forboth � ¼ 08 and � ¼ �9 the roughness of the footing hasonly a minor influence on both the load–displacement curveand the ultimate load (that is, the Nq values for smooth andrough footings differ by less than 2%). However, the value

Meyerhof (1963)Hansen (1970)Vesic (1973)EC7 (1996)

50

40

30

20

10

015 20 25 30 35

Angle of shearing resistance, φ′: deg

Nγ

Fig. 6. Range of Nª values from different formulae

300

200

100

00 0.4 0.6 0.8 1.0

Aφ′ � 25°, c � 0

ν � 0° and 25°

εa: %

σ 1′ �

σ3′:

kPa

(a)

0.2 0 0.2 0.4 0.6 0.8 1.00.2

0.0

�0.2

�0.4

�0.6

�0.8

�1.0

ν � 0°

ν � 25°

ε vol

: %

A

εa: %

(b)

Fig. 7. Drained triaxial test with Mohr–Coulomb model (a) deviatoric stress against axial strain; (b) volumetric strainagainst axial strain

1 m

20 m

10 m

Fig. 8. Finite element mesh for surface footing problem

540 POTTS

of the angle of dilation has a much greater effect. Theload–displacement curves for the analyses with � ¼ 08 aremuch softer, and indicate an Nq value some 7% lower thanthat for the analyses with � ¼ �9, which themselves give Nq

values within 2% of the theoretically exact value given byequation (5). In this respect note that the elastic propertiesare the same for all analyses, and that the differences inbehaviour are due to the magnitude of � alone.

The effect of the magnitude of the surcharge load, q9, canbe assessed by comparing, for example, the results from theanalyses of a smooth footing with � ¼ 08 and with values ofq9 ¼ 10 kPa and q9 ¼ 100 kPa respectively. Both analysesgive similar values of Nq, but the load–displacement curvefor the analysis with q9 ¼ 10 kPa is much stiffer than thatfor the analysis with q9 ¼ 100 kPa.

The effect of footing shape was investigated by repeatingthe four analyses with q ¼ 100 kPa assuming axisymmetricconditions, which implies that the footing is circular in plan.The results are presented in Fig. 10. These results indicatethat, in contrast to the results for the strip footing, thefooting roughness does affect the ultimate load, there beinga 22% difference in the Nq value predicted for the smoothand rough footings with � ¼ �9. In addition, and again incontrast to the strip footing, the analyses indicate that theangle of dilation has only a small effect on the ultimatefooting load. However, the magnitude of the angle ofdilation does affect the load–displacement curves, the effect

being similar to that observed for the strip footing. Compari-son of the results given in Figs 9 and 10 indicate that theshape factor, sq, is approximately 1·6. Design codes recom-mend shape factors 1·2 , sq , 1·5 (Sieffert & Bay-Gress,2000), and consequently the finite element results predict avalue slightly higher than that used in current practice.

Further analyses were then performed with a surchargeq9 ¼ 0, but with a soil with weight ª9 ¼ 18 kN/m3. For thissituation only the second term in equation (7) is non-zero,and for a strip footing sª ¼ 1. As before, the footing loadhas been expressed in terms of the mobilised value ofN mob

ª ¼ Q=(ABª9), which at the beginning of an analysis isequal to zero and then increases to a maximum value ofN mob

ª ¼ Nª at failure.The results for the strip footing are presented in Fig. 11.

In contrast to the Nq analysis (see Fig. 9), the Nª value isdependent on footing roughness, but is less sensitive to themagnitude of the angle of dilation. In addition, the magni-tude of the angle of dilation does not have a significanteffect on the shape of the load–displacement curve. Theresults for the circular footing are presented in Fig. 12 andindicate similar trends to those of the strip footing, althoughthe magnitudes of Nª are different.

The predicted Nª values are summarised in Table 3, whereit can be seen that the values for rough footings areapproximately twice those for smooth footings. Values of theshape factor sª range from 0·87 to 0·98 and are thereforehigher than typical values of 0·6 , sª , 0·9 recommendedin design codes (Sieffert & Bay-Gress, 2000). Also shown in

Rough, ν � 25°Smooth, ν � 25°Rough, ν � 0°Smooth, ν � 0°

q′ = 100 kPa

q′ = 10 kPaSmooth, ν � 0°,12

11

10

9

8

7

6

5

4

3

2

10 0.05 0.1 0.15 0.2 0.25

Normalised settlement, δ/B

Mob

ilise

d be

arin

g ca

paci

ty fa

ctor

, Nqm

ob

Fig. 9. Load–displacement curves for strip footings on weight-less and cohesionless soil (effect of roughness, dilation andsurcharge)

21

19

17

15

13

11

9

7

5

3

10 0.1 0.2 0.3 0.4 0.5 0.6


q′ = 100 kPa

Smooth, ν � 0°

Rough, ν � 0°Rough, ν � 25°

Smooth, ν � 25°

Mob

ilise

d be

arin

g ca

paci

ty fa

ctor

, Nqm

ob

Fig. 10. Load–displacement curves for a circular footing onweightless and cohesionless soil (effect of roughness anddilation)

Rough, ν � 0°

Rough, ν � 25°

Smooth, ν � 0°Smooth, ν � 25°

7

6

5

4

3

2

1

00 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02


Mob

ilise

d be

arin

g ca

paci

ty fa

ctor

, Nγm

ob

Fig. 11. Load–displacement curves for a strip footing on acohesionless soil (effect of roughness and dilation)

Rough, ν � 0°

Rough, ν � 25°

Smooth, ν � 0°Smooth, ν � 25°

7

6

5

4

3

2

1

00 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02


Mob

ilise

d be

arin

g ca

paci

ty fa

ctor

, Nγm

ob

Fig. 12. Load–displacement curves for a circular footing on acohesionless soil (effect of roughness and dilation)


Table 3 are N stripª values from four widely used expressions.

It can be seen that the finite element results are in goodagreement with the N strip

ª values given by Hansen (1970)and Meyerhof (1963), which have been widely used indesign codes in the past. However, it is interesting to notethat the N strip

ª value given by Eurocode 7 is considerablyhigher, and therefore less conservative, than the finite ele-ment results.

Before leaving this drained bearing capacity problem it isof interest to examine the predicted failure mechanisms.These are shown in Fig. 13 for both the Nq and Nª cases forthe smooth strip footing with � ¼ �9. The arrows shown inthis figure represent the vectors of incremental displacementfor the last increment of each analysis when a limit load hadbeen reached. The length of the arrows and their orientationrepresent the magnitude and direction of the movementrespectively. To identify the failure mechanism it is not theabsolute magnitudes of the movements, but their relativemagnitudes, that are important.

As both the plots shown in Fig. 13 are drawn to the samegeometric scale, it is clear that the Nq failure mechanism ismuch deeper and wider than the Nª mechanism. Thisdifference in the size and extent of the failure mechanismshas important implications for the situation when bothq9 6¼ 0 and ª9 6¼ 0. It also has implications for the conven-tional bearing capacity equation, equation (4), which as-sumes superposition of the Nc, Nq and Nª terms that havebeen determined independently from analyses withq9 ¼ ª9 ¼ 0, c9 ¼ ª9 ¼ 0 and c9 ¼ q9 ¼ 0 respectively.

The superposition assumption has been examined in great-er detail by performing an analysis of a rough strip footingon a soil with weight and with surcharge loading. Values ofc9 ¼ 0, � ¼ �9 ¼ 258, q9 ¼ 10 kPa, ª9 ¼ 18 kN/m3 and

B ¼ 1 m were assumed. The analysis predicted an ultimatefooting load Qmax ¼ 556 kN/m. From the analyses discussedabove, which considered the Nq and Nª cases separately,values of Nq ¼ 11·03 and Nª ¼ 6·72 were obtained. Substi-tuting these values into equation (7) gives an ultimate loadQmax ¼ 463 kN/m. Comparison of these two estimates ofQmax indicates that the superposition assumption used in theconventional bearing capacity equation results in an estimateof Qmax that is conservative by 17%.

The results presented above clearly show that numericalanalysis can do all that conventional methods can do for thesurface footing problem. It can also do more: for example, itcan provide the complete load–displacement curve, canaccount for different values of the angle of dilation, canpredict failure mechanisms, and do all this in a matter ofminutes with modern computer hardware and software.

(b) Accommodate realistic soil behaviour and (c) accountfor consolidation

In the foregoing discussion we have seen that numericalanalysis can deal with non-associated plasticity and enablesolutions in which � 6¼ �9 can be obtained. Although suchsolutions are an improvement on conventional analysis, theMohr–Coulomb model is restricted in its ability to simulatereal soil behaviour. This is clearly evident from Fig. 7,which shows the behaviour of a drained triaxial test on aMohr–Coulomb material. Although the deviator stress–straincurve represents a passable representation of soil behaviour,the unlimited dilation indicated in the volumetric–axialstrain curve is unrealistic.

The Mohr–Coulomb model is adopted by most conven-tional methods of analyses because of its relative simplicity,but it is possible to accommodate more realistic constitutivemodels in numerical analysis. In fact, the only way that themore complex constitutive models that have been developedover recent years can be used to analyse real geotechnicalproblems is by incorporating them into a numerical analysis.Even the relatively simple critical-state-type models, such asCam clay (Roscoe & Schofield, 1963) and modified Camclay (Roscoe & Burland, 1968), must be incorporated into anumerical analysis if they are to be of use in geotechnicalpractice. Such models are an improvement over the Mohr–Coulomb model because they link compression and shearbehaviour in a realistic manner. This can be seen in Fig. 14,which shows the results of ideal drained triaxial tests onboth normally and overconsolidated modified Cam clay. Forthe normally consolidated sample volumetric compression is

Table 3. Bearing capacity factor Nª

Footing Finite elementanalysis

Shape factors

Strip Circle

Smooth, � ¼ 08 3·59 3·14 0·87Smooth, � ¼ 258 3·74 3·31 0·88Rough, � ¼ 08 6·74 6·62 0·98Rough, � ¼ 258 6·72 6·64 0·98Hansen, (1970) &Meyerhof, (1963)

6·76

Vesic, (1973) 10·87Eurocode 7 9·01

Smooth strip footing, φ′ � ν � 25°, c′ � 0

q′ � 0

q′ � 100 kPa

γ � 0

γ � 18 kN/m3

(a) (b)

Fig. 13. Comparison of failure mechanisms: (a) Nq failure mechanism; (b) Nª failuremechanism

542 POTTS

predicted, whereas for the overconsolidated clay sample ini-tial compression followed by dilation is predicted. The rateof volumetric compression (normally consolidated sample)and dilation (overconsolidated sample) reduces as the axialstrain increases, and eventually, when the critical state isreached, the change in volume becomes zero. The stress–strain curve for the overconsolidated sample also indicatesstrain-softening behaviour. Although such behaviour is amore realistic representation of real soil behaviour, and avast improvement on the Mohr–Coulomb model, it is stillapproximate. For example, for the overconsolidated samplethe linear stress–strain curve pre-peak and the suddenchange in gradient of the stress–strain curve at peak strengthare only a crude approximation of real soil behaviour. Toovercome the shortcomings of the simple critical-state mod-els many complex constitutive models have been developed:see Potts & Zdravkovic (1999).

As an example of the use of an advanced constitutivemodel, the problem of a rigid strip footing resting on thesurface of a typical soft clay will be considered. The footingis 2 m wide and the clay stratum is normally consolidatedbelow a depth of 2 m, with a stronger crust at the surface.The initial profile of undrained strength with depth is shownin Fig. 15.

At the design stage the following information is required:

(i) the initial short-term bearing capacity(ii) the settlements, both initially (that is, short term) and in

the long term, of the footing when loaded with a loadfactor of 2·5

(iii) the undrained bearing capacity of the footing in thelong term after all excess pore water pressures due tothe initial loading have dissipated. This is relevant ifadditional load is to be added in the future, or if thefooting is to be reused. This is particularly relevant inthe current climate of sustainable development.

A variant of the modified Cam clay model was used torepresent the clay layer, and the model parameters and initialstress conditions were based on a site investigation from asite in Grimsby (Mair et al., 1992). The soil properties were:�9 ¼ 328; ªsat ¼ 17 kN/m3; slope of the virgin consolidationline, º ¼ 0·22; slope of the swelling lines, k ¼ 0·02; specificvolume at a mean effective stress of 1 kPa on the virginconsolidation line, v1 ¼ 3·0; elastic shear modulus, G ¼1700 kPa; and permeability, k ¼ 5 3 10�10 m/s. The initialundrained strength (su), overconsolidation ratio (OCR) andcoefficient of earth pressures at rest (K0) profiles are givenin Fig. 15. As the information required for the design

Drained triaxial test - OCDrained triaxial test - NC500

250

00

0.08

0.140 0.2 0.4 0 0.1 0.2

0

�0.03

�0.060

600

1200

(σ1

� σ

3): k

Pa

ε vol

εa εa

0.02

Fig. 14. Drained triaxial compression tests with modified Cam clay

0

2

4

6

8

100 10 20 30 40 0 2 4 6

10

8

6

4

2

0

Dep

th: m

Undrained strength, su: kPa OCR, K0

OCR

K0OC

Fig. 15. Undrained strength, OCR, and K0 profiles for soft clay


involved both short-term and long-term predictions, acoupled finite element analysis was undertaken linking themechanical and pore fluid flow behaviour. For further infor-mation see Potts & Zdravkovic (2001).

The results of the analyses are shown in Fig. 16. Initially,an analysis was performed in which the footing was loadedrapidly until failure occurred. This enabled the short-termbearing capacity to be obtained (point A in Fig. 16). Asecond analysis was then undertaken in which the footingwas loaded until 40% of the initial undrained bearingcapacity was mobilised (that is, a load factor of 2·5: O to Bin Fig. 16). This stage of the analysis was again performedin a short time period to maintain undrained conditions.From the results of this analysis it was possible to obtain anestimate of the short-term footing settlement.

The load on the footing was then maintained constant (Bto C in Fig. 16) and a series of time steps were applied untilall excess pore water pressures, arising from the initial rapidloading, dissipated (that is, long-term consolidation wassimulated). During this stage of the analysis the footingsettled (distance BC in Fig. 16).

In the third stage of this analysis further load was rapidlyapplied to the footing to estimate the increase in bearingcapacity as a result of the pre-loading and consolidation(point D in Fig. 16). The analysis indicates that the bearingcapacity has increased by 11% (that is, the difference be-tween footing loads at points A and D).

This example demonstrates the ability of numerical analy-sis to account for complex constitutive models and coupledbehaviour. It has also indicated that the numerical analysiscan furnish estimates to important design questions.

It is interesting to consider how the design requirementscould have been satisfied using conventional analysis. In thisrespect it is noted that some approximation to the initialundrained strength profile would have to be made to estimatethe initial short-term bearing capacity, as conventional meth-ods cannot deal with the complex distribution of strengthshown in Fig. 15. It would also be possible to estimatesettlements, but with difficulty. However, it would not bepossible to estimate the long-term undrained bearing capa-city, as it is not possible to estimate the long-term distribu-tion of undrained shear strength with conventional methods.

(d) Provide mechanisms of behaviour and (e) accommodatecomplex soil stratigraphy

To illustrate the ability of numerical analysis to predictmechanisms of behaviour and account for complex soilstratigraphy, the stabilisation of the Pisa Tower will be

considered. This has been probably the most challengingfoundation problem of modern times.

The impetus to the activities undertaken to stabilise theTower, over the past 10 years, was the collapse, withoutwarning, of the nearby civic tower of Pavia in 1989, whichkilled four people. The Italian government appointed acommission to advise on the stability of the Pisa Tower andto develop and implement measures for stabilising it. One ofthe early decisions of the Commission was to develop anumerical model of the Tower and the underlying groundthat could be used to assess the effectiveness of variouspossible remedial measures. Professor John Burland (a mem-ber of the Commission) and the author were asked to under-take this task.

Figure 17 shows a picture of the Tower and a cross-section through it. It is essentially a hollow cylinder 60 mhigh and the foundations are 19·6 m in diameter. The massof the Tower is 14 500 t, and in 1990 the foundations wereinclined due south at 5·58 to the horizontal.

The Tower is a campanile for the cathedral, constructionof which began on 9 August 1173. By 1178 constructionhad progressed to about one quarter of the way up the fourthstorey, when work stopped. The reason for the stoppage isnot known, but had it continued much further the founda-tions would have experienced an undrained bearing capacityfailure. The work recommenced in 1272, after a pause ofnearly 100 years, by which time the strength of the groundhad increased owing to consolidation under the weight ofthe Tower. By 1278 construction had reached the seventhcornice, when work again stopped owing to military action.Once again there can be no doubt that, had work continued,the Tower would have fallen over. In 1360 work on the bellchamber commenced, and it was completed in about 1370,nearly 200 years after commencement of the work. In 1838a walkway was excavated around the foundations. This isknown as the catino, and its purpose was to expose thecolumn plinths and foundation steps for all to see, as wasoriginally intended.

Figure 18 shows the ground profile underlying the Tower.It consists of three distinct horizons, the properties of whichare described in AGI (1991) and Potts & Burland (2000).Horizon A is about 10 m thick and consists primarily ofestuarine deposits laid down under tidal conditions. As aconsequence, the soil types consist of rather variable sandyand clayey silts. At the bottom of horizon A is a 2 m thickmedium dense fine sand layer (the upper sand). Based onsample descriptions and piezocone tests, the material to thesouth of the Tower appears to be more silty and clayey thanto the north, and the sand layer is locally thinner.

Horizon B consists of marine clay, which extends to adepth of about 40 m. It is subdivided into four distinctlayers. The upper layer is a soft sensitive clay known asthe Pancone. It is underlain by a layer of stiffer clay (theintermediate clay), which in turn overlies a sand layer (theintermediate sand). The bottom of horizon B is a normallyconsolidated clay known as the lower clay. Horizon B islaterally very uniform in the vicinity of the Tower.

Horizon C is a dense sand that extends to a considerabledepth (the lower sand). The water table in horizon A isbetween 1 m and 2 m below ground surface. The manyborings beneath and around the Tower show that the surfaceof the Pancone clay is dished beneath the Tower, from whichit can be deduced that the average settlement, at this level,is approximately 3 m.

Before remedial measures could be designed, it wasimportant to identify the cause of the inclination of theTower. As a first step it was necessary to establish thehistory of inclination. This was accomplished by a fine pieceof detective work by Professor John Burland. Based on the

DA

Initial bearingcapacity

40% pre-loadB C

O

100

80

60

40

20

00 0.2 0.4 0.6 0.8 1.0 1.2

Displacement: m

Load

: kN

Fig. 16. Load–displacement curve for pre-loading on soft clay

544 POTTS

measured thickness of each masonry layer and a hypothesison the manner in which the masons corrected for theprogressive lean of the Tower, Burland deduced the historyof inclination of the foundations of the Tower shown in Fig.19 (Burland & Potts, 1994). In this figure the weight of theTower is plotted against the deduced inclination of itsfoundation.

During the first phase of construction to the fourth storey(1173 to 1178) the Tower inclined slightly to the north. Thisnorthward inclination increased slightly during the rest peri-od of nearly 100 years. When construction recommenced in1272, the Tower began to move towards the south andaccelerated shortly before construction reached the seventhcornice in 1278. The movement continued during the follow-ing 90 years. After the completion of the bell chamber in1370 the inclination of the Tower increased significantly, as

it did when the excavation for the catino was undertaken in1834.

The nature of the movements shown in Fig. 19, and morerecent measurements of the ongoing movements, are notconsistent with a bearing capacity type failure discussedpreviously for the surface footing problem (Potts & Burland,2000). It is therefore of interest to consider the possiblemechanisms associated with the stability of the foundationsof tall towers.

There are two possible mechanisms that could account forfailure of such a tower: (i) bearing capacity failure due to

Seventh cornice

V7

South

V1

Catino

Plane ofplinth

FS

FN

Floor ofinstrument room

First cornice

(a) (b)

5° 1

1′ 2

0″

Fig. 17. (a) Pisa Tower; (b) schematic cross-section through the Tower

�15

0

�15

�30

�45

Ele

vatio

n: m

Laye

r C

Laye

r B

Laye

r A

Sandy and clayey siltsUpper sand

Upper clay (Pancone)

Intermediate clayIntermediate sand

Lower clay

Lower sand

WT

Sealevel

Fig. 18. Soil profile beneath Pisa Tower

15

10

5

0

Load

: ton

nes

� 1

03

0 1 2 3 4 5 6

Loading

Consolidation

Plane strain

3D analysis

Historical

1178 to 1272

1278

1360

1838

Pre

sent

Inclination of foundations: deg

Fig. 19. Historical inclination of Pisa Tower


insufficient soil strength, and (ii) leaning instability due toinsufficient soil stiffness. Bearing capacity failure is themore common type of foundation instability and the onecovered in most textbooks and codes of practice. It isusually investigated using the formulae given in equation(4). Leaning instability is not so common and is onlyrelevant to tall structures. It occurs at a critical inclinationwhen the overturning moment generated by a small increasein inclination is equal to, or greater than, the resistingmoment provided by the foundations and generated by thesame small rotation. In all but very simple cases, it isdifficult, probably impossible, to analyse without using nu-merical analysis.

These two alternative failure mechanisms are best demon-strated by a simple example. Fig. 20 shows a simple towerresting on a uniform deposit of undrained clay. The dimen-sions of the tower are similar to those of the Pisa Tower. Totrigger a rotational failure some initial defect (imperfection)must be present. In this example the tower was given aninitial tilt of 0·58 (that is, the initial geometry of the towerhad a tilt). The self-weight of the tower was then increasedgradually in a plane-strain, large displacement finite elementanalysis, until failure occurred.

The clay was modelled as a linear-elastic Tresca material.Three analyses were performed, each with a different valueof shear stiffness, G, of the soil. All other parameters werethe same. In particular, the undrained strength, su, was80 kPa in all analyses. Therefore, according to conventionalmethods, the bearing capacity of the tower was the same inall three analyses. Consequently, if instability was governedby bearing capacity failure, all three analyses should fail atthe same weight of the tower. However, as can be seen fromFig. 21, this was not the case. This figure shows the increasein rotation of the tower, above the initial 0·58 imperfection,plotted against the weight of the tower, for analyses withG/su values of 10, 100 and 1000. Real soils are likely tohave properties that are between the two extreme values.The results show that failure occurs very abruptly, with littlewarning, and that the weight of the tower at failure isdependent on the shear stiffness of the soil. The weight at

failure for the analysis with the softer soil, G/su ¼ 10, isabout half of that for the analysis with the stiffest soil,G/su ¼ 1000.

It is of interest to examine the analyses with the twoextreme values of G/su in more detail. In particular it isinstructive to consider what is happening in the soil atfailure. Fig. 22 shows vectors of incremental displacementfor the soft soil (G/su ¼ 10) from the last increment of theanalysis. They show that movements are located in a zonebelow the foundation, and indicate a rotational type offailure. At first sight this looks like a plastic-type collapsemechanism. However, examination of the zone in which thesoil has gone plastic (also shown in Fig. 22) indicates that itis very small and not consistent with a plastic failuremechanism. Consequently, this figure indicates a mechanismof failure consistent with a leaning instability.

Considering the results from the analysis performed withthe stiffer soil (G/su ¼ 1000), vectors of incremental displa-cement just before collapse are shown in Fig. 23. Themechanism of failure indicated by these vectors is verydifferent from the one shown in Fig. 22 for the softer soil.Instead of the soil rotating as a block with the foundation,the vectors indicate a more traditional bearing capacity typemechanism, with the soil being pushed outwards on bothsides. The plastic zone, also indicated in Fig. 23, is verylarge, and therefore the results clearly indicate a plasticbearing capacity type mechanism of failure. The mechanismis not symmetrical because the tower is leaning to one side.

In view of the temporary stabilisation scheme, whichinvolved adding lead weights to the north side of the PisaTower and which will be discussed in more detail subse-quently, it is of interest to examine the response of the

Initial tiltof tower � 0.5° 60

m

20 m

Undrained clay(Elasto-plastic)

Tresca model � su � 80 kPa

Fig. 20. Geometry for investigation of leaning instability of atower

4

3

2

1

0

Rot

atio

n: d

eg

G/su � 10 G/su � 100

G/su � 1000

0 50 100 150Weight of tower: MN

Fig. 21. Tower rotation with increase in weight

Plasticzone

G/su � 10

Fig. 22. Vectors of incremental displacements and plastic zoneat failure (low soil stiffness)

546 POTTS

simple tower in the above example if, at the point ofcollapse, weight is added to the higher side of the founda-tion. The effect of a 1·5 MN/m load is shown in Figs 24 and25 for the soft (G/su ¼ 10) and stiff (G/su ¼ 1000) soilsrespectively. Again, vectors of incremental displacement areshown. These indicate the nature of the movements due tothis additional load.

It is noted that for the soft soil (Fig. 23) the sense ofmovement is reversed: the tower rotates back and collapse isarrested. So if the tower is undergoing leaning instability,

adding weight to reduce the overturning moment stabilisesthe tower. In contrast, for the stiffer soil (Fig. 24), the towercontinues to increase its inclination when the load is applied.In fact, it was not possible to obtain a converged solutionwhen the weight was added. The addition of the loadinitiates collapse, even though the load acts to reduce theoverturning moment. This is, of course, consistent with con-ventional bearing capacity theory.

As well as demonstrating the difference between the twotypes of instability and the ability of numerical analysis topredict which is likely to occur, these analyses also indicatethat a counterweight-type scheme will be beneficial to thePisa Tower only if it is suffering a predominantly leaninginstability. To complicate matters further, real soils are likelyto have stiffness values between the two extremes consideredabove, and therefore both mechanisms of behaviour arelikely to be active to some degree.

Returning to the Pisa Tower, before stabilisation schemescould be developed it was necessary to establish whichmechanism was dominant. To achieve this, numerical analy-sis to simulate the past history of the Tower was undertaken.Comparison of the results from such an analysis with theobserved behaviour given in Fig. 19 also allowed the compu-ter model to be validated.

It must be emphasised that a prime objective of theanalysis was to develop an understanding of the mechanismscontrolling the behaviour of the Tower. It was felt that, untilthese had been clarified, it would be unhelpful to attempthighly sophisticated and time-consuming three-dimensionalanalyses. It should also be remembered that the initialanalyses were performed in 1990, when computing resourceswere much more restricted than they are now. Accordingly, aplane-strain approach was used initially, recognising that theinterpretation of the results would require some care. From1994 three-dimensional analyses were carried out using theFourier series aided finite element method (Potts & Zdravko-vic, 1999). Both small and large displacement analyses wereperformed, and all analyses involved coupled consolidation.It is important to note that, in all analyses, the overturningmoment arising from the lean of the Tower was self-gener-ated in the analysis; it was not imposed as an externalboundary condition. As the results of the plane-strain andthree-dimensional analyses were similar, they will simply beidentified as ‘predictions’ in the figures that follow. Forfurther details the reader is referred to Potts & Burland(2000).

The soil stratigraphy shown in Fig. 18 was modelled as13 different soil types. A simple Mohr–Coulomb model wasused to represent the sand layers, whereas a form of themodified Cam clay model was used to represent the claylayers. The soil properties were obtained from laboratorytests, and are described in detail by Potts & Burland (2000).A detail of the plane-strain finite element mesh close to theTower foundation is shown in Fig. 26 (note that the com-plete mesh extended 100 m either side of the axis of theTower and down to the top of horizon C). The properties oflayer A1 varied from north to south, becoming more com-pressible and less permeable. This is believed to haveprovided the defect or trigger that initialised the tilting ofthe Tower.

The results of the analysis simulating the history of theTower from 1173 to 1990 are shown in Figs 19 and 27. Inthis latter figure rotation and settlement of the foundationare plotted against time. Also shown in Fig. 27 are key tiltobservations taken from Fig. 19. The agreement between thepredictions and observations is quite remarkable. The resultsshown were based on initial estimates of soil parameters andgeometric idealisations.

At the end of the analysis (which corresponds to the

Plasticzone

G/su � 1000

Fig. 23. Vectors of incremental displacements and plastic zoneat failure (high soil stiffness)

1.5 MN/m

G/su � 10

Fig. 24. Effect of north weighting (low soil stiffness)

1.5 MN/m

G/su � 1000

Fig. 25. Effect of north weighting (high soil stiffness)


conditions in year 1990), the Tower was only just stable, andany small perturbation would cause collapse. To identify thedominant mechanism of instability it is instructive to exam-ine the zones of soil in which its strength is fully mobilised.These are shown in Fig. 28. There are no zones of containedfailure within the Pancone clay, but there are extensive zoneswithin horizon A. The lowest zones are in the upper sandlayer and result from the lateral extension of this layer.There is a zone beneath and outside the southern edge ofthe foundation, and a smaller zone underneath the northernside. It is evident from this figure that the impendinginstability of the Tower foundations is not due to the onsetof a bearing capacity failure, but can be attributed to the

high compressibility of the Pancone clay, which results inleaning instability.

Having established the likely mechanism controlling thebehaviour of the Tower, and having validated the numericalanalysis, the analysis was then extended to investigatevarious stabilisation schemes. Fully aware that the selection,design and implementation of permanent stabilisation meas-ures would take time, the Commission took an early resolu-tion to implement short-term temporary and fully reversiblemeasures to increase slightly the stability of the foundation.This would then provide time to develop a permanentsolution.

In developing both the short- and long-term stabilisationmethods the numerical analysis proved invaluable. To illus-trate this the temporary counterweight scheme, which wasactually adopted, will be considered. The scheme consistedof constructing a temporary prestressed concrete ring beamaround the base of the Tower and loading it with leadweights: see Fig. 29.

Clearly, such a solution would not have been considered ifit had not been recognised that leaning instability, ratherthan bearing capacity failure, was controlling the behaviourof the Tower. Before implementing such a solution it wasessential that a detailed analysis should be carried out. Thepurpose of such an analysis was twofold: first, to ensure thatthe proposal was safe and did not lead to any undesirableeffects; and, second, to provide a class A prediction thatcould be used to assess the observed response of the Toweras the load was being applied.

The numerical analysis performed to simulate the historyof the Tower was used as the starting point to investigate the

WM

North South

A1

Fig. 26. Detail of finite element mesh close to Tower foundation

5.44°1838

Predictions

Observations

Date197017701570137011700

2°

4°

6° 1272 1360

Rot

atio

n(n

orth

to s

outh

)

0

2

4

Set

tlem

ent:

m

Predictions

Fig. 27. Rotation and settlement of Tower with time

Fully mobilisedstrength

Upper sand

Upper clay

Fig. 28. Zones of fully mobilised strength: Pisa Tower, 1990 Fig. 29. Lead weights on the north side of the Tower

548 POTTS

counterweight scheme. Fig. 30 shows the predicted responseof the Tower due to the application of a counterweight tothe foundation masonry at eccentricities of 6·4 m, 7·8 m and9·4 m to the north. The results show that as long as the loadis less than 1200 t the weighting has the desired effect ofrotating the Tower to the north, with only small settlements.In this load range the behaviour is not sensitive to theeccentricity. In the event, practical considerations requiredthe eccentricity to be 6·4 m, and it was decided that a loadof 690 t would be applied. For this situation the numericalanalysis predicted a rotation of the Tower of 27·5 arcseconds towards the north and a settlement of 2·4 mm.

A comparison of the predictions with the observations thatwere made during application of the lead weights is shownin Fig. 31. Note that in this figure the load does not includethe weight of the concrete ring beam. As can be seen, thepredicted settlements agree very well with those observed,whereas the predicted rotations are 80% of those observed.

The accuracy of the predictions is perhaps surprising as therotations are two to three orders of magnitude smaller thanthe accumulated ones arising from the history of the Towerand used to validate the numerical model.

The measured data were then used to further refine thecomputer model (that is, a class C prediction; Lambe, 1973).To obtain an excellent match with the observed data it wasnecessary only to make a reduction to the shear stiffness ofthe upper soil layers in horizon A, which was done withinthe scatter of experimental data. The results are shown inFig. 32, where the load now includes the weight of theconcrete ring beam.

The refined model was then used to analyse the permanentstabilisation schemes and, in particular, the soil extractionscheme actually used. For further information the reader isreferred to Potts & Burland (2000).

The Pisa Tower example clearly shows that numericalanalysis can deal with complex soil stratigraphy and, perhaps

P

e

North

∆θ: deg

0.08

0

�0.08 South

1000 2000 Load, P: tonnes

e � 6.4 m

e � 7.8 m

e � 9.4 m

0

50

100Settlement: mm

Fig. 30. Predicted response of the Tower due to north weighting

800

400

01/6/93

Date1/2/94A

pplie

d lo

ad: t

onne

s

Observations

Prediction

600400200

Applied load: tonnes

30

20

10

0

1

2

3Av.

set

tlem

ent:

mm

Cha

nge

of in

clin

atio

n: s

Fig. 31. Predicted and observed response due to application of counterweight (class Aprediction)


more importantly, can predict the correct mechanism ofinstability. In the author’s opinion it is the ability to predictmechanisms of behaviour that sets numerical analysis in aleague of its own. The fact that the analysis also gavequantitatively correct predictions is of course also impressiveand extremely useful. At this point it is worth askingthe question: ‘How else could the Pisa Tower have beenmodelled?’

( f ) Account for interaction between structuresAs noted earlier in this lecture, one of the design objec-

tives is to estimate the effect that construction may have onadjacent structures and services. This is difficult, oftenimpossible, to achieve using conventional methods of analy-sis. However, it is relatively easy to accommodate bothadjacent structures and services in a numerical analysis. Toillustrate this point the recent deep excavation performed atWestminster for the new underground station will be used asan example.

The original Westminster station served the District andCircle underground lines. However, construction of the Jubi-lee line extension required that the station be expanded,which in turn involved a deep excavation some 75 m by28 m in plan and 37 m deep—one of the deepest in London.Above the station a new building, Portcullis House, was tobe constructed. An aerial photo of the site is shown in Fig.33. The Big Ben clock tower and the Houses of Parliamentcan be seen on the right of the picture, with the RiverThames behind and Bridge Street to the left. The excavationsite is clearly visible to the left of Bridge Street. The Districtand Circle underground lines run approximately parallel tothe River Thames on the river side of the excavation, andthe tunnels for the Jubilee line extension run below BridgeStreet. Clearly, there are many important buildings andservices adjacent to the excavation that are likely to be

affected by its construction. In particular there was concernabout the potential movement of the Big Ben clock tower.To complicate matters, construction of both the deep excava-tion and the tunnels for the Jubilee line extension were totake place simultaneously.

A cross-section through the excavation, approximatelyparallel to the River Thames, is shown in Fig. 34, and it canbe seen that the tunnels for the Jubilee line extension arevery close to the excavation. The sequence of constructionwas as follows: the westbound (WB) running tunnel, thediaphragm wall for the excavation, the eastbound (EB)running tunnel, enlargement of the WB running tunnel toform the WB station tunnel, excavation to a level of 83·6 m,enlargement of the EB running tunnel to form the EB stationtunnel, and finally full excavation to a level of 69·5 m.

A considerable number of numerical analyses were per-formed during the design process, most of which wereplane-strain finite element analyses, as at the time computerresources were not available to perform three-dimensionalanalyses. An important aspect of these analyses was thatthey incorporated constitutive models for the soils thatsimulated their small-strain behaviour. This was essential ifrealistic predictions of movements of adjacent structureswere to be accurately predicted. These analyses indicated:

(i) A bottom prop was required to support the walls of thestation box before any excavation took place. Withoutthis prop movements adjacent to the excavation werepredicted to be too large and detrimental. In the eventthese props were constructed by digging shafts and thenhorizontal adits, which were filled with reinforcedconcrete.

(ii) Even with the provision of the bottom props, themovements of the Big Ben clock tower were likely tobe too large. This resulted in the provision ofcompensation grouting facilities, which in the eventwere used on several occasions during the constructionperiod to correct for movements of the clock towerfoundations.

(iii) Possible movements and structural forces in the wallsand props of the excavation, in the Jubilee line tunnellinings and other adjacent structures.

As an example of some of the predictions that were made,Fig. 35 shows a comparison between predicted and observed

30

20

10

0

Observations

3D prediction

Plane strainprediction

200 400 600

Applied load: tonnes

1

2

3

Av.

set

tlem

ent:

mm

Cha

nge

of in

clin

atio

n: s

Fig. 32. Predicted and observed response due to application ofcounterweight (class C prediction)

Fig. 33. Jubilee line extension site at Westminster

550 POTTS

lateral movements of the wall of the excavation shown inthe cross-section in Fig. 34. Results are given for threeexcavation depths. Note that the predictions were madebefore construction, and therefore they are class A predic-tions in Lambe (1973) terminology. For further details of theconstitutive models and material parameters the reader isreferred to Higgins et al. (1996).

Inspection of Fig. 35 indicates that there is good agree-ment between the predictions and observations, with thepredictions being slightly larger than the observations. Thisis probably due to the fact that the analysis assumed plane-strain conditions, whereas in reality the problem is morethree-dimensional. This will be discussed further in the nextsection.

Clearly, it would have been extremely difficult, if notimpossible, to have modelled this construction with conven-tional methods of analysis.

(g) Accommodate three-dimensional geometriesIn most conventional analyses it is common to assume

one- or two-dimensional geometries. This often involvesassuming plane-strain or axisymmetric conditions. Until veryrecently this was also true for numerical analysis, if realisticconstitutive models were to be included. However, in manyreal situations such assumptions are questionable. Over thelast four years developments in computer hardware haveenabled limited three-dimensional analyses to be performed,and with further hardware developments such analyses arelikely to become more widespread in the future.

To illustrate the use of three-dimensional finite elementanalysis, another deep excavation problem will be consid-ered. This is associated with the Crossrail undergroundscheme. Although this scheme is still in the planning stages,the route of the proposed tunnels between Paddington andLiverpool Street is fixed. At one location on this route it isthe intention to construct an excavation 35 m by 35 m inplan and 40 m deep. Its purpose is to act as a launchingchamber for tunnel-boring machines and then subsequentlyto form part of a station complex. Clearly, the excavationwill affect adjacent structures and services, and analysis willbe required to quantify these effects.

At present, the Moor House site, which is directly adja-cent to the proposed Crossrail excavation (see Fig. 36), isunder redevelopment, and as part of the design process theeffect of the excavation on the foundations must be assessed.Current practice would be to perform two-dimensionalplane-strain and/or axisymmetric numerical analysis to

14.0 m 33.0 m 14.0 m

Big Ben

69.5m

CL

8.1 m 1.0

85.0 mEB

SR

SR WB

75.7 m

64.0 m1.0

Thames gravels

London clayS

R

Station tunnel

Running tunnel

60.5 m

52.0 m

41.0 m

Lambeth Group clay

Lambeth Group sand

Thanet sand

106.0 m

Made ground/alluvium

100.0 m

92.0 m

Fig. 34. Cross-section through excavation at Westminster

100

90

80

70

Ele

vatio

n: m

60

50�10 0 10 20 30 40 50

Displacement: mm

FE prediction

Observed

69.5 m

83.6 m

Elevation level � 93.1 m

Fig. 35. Comparison of measured and predicted displacementsof Westminster station wall


predict ground movements and then use these movements toassess the effect on adjacent structures. Clearly, engineeringjudgement is involved in this process.

The finite element mesh shown in Fig. 37, consisting of800 eight-noded quadrilateral elements, has been used toperform such analyses. The concrete wall is assumed to be1·2 m thick and have a total depth of 46·7 m (6 m below

the bottom of the final excavation level). It is propped atseven levels. A realistic excavation procedure, in whichexcavation is performed to a depth below the intended propposition before the prop is inserted and the process re-peated, was assumed. The soil layers were modelled with aconstitutive model that accounts for small-strain stiffnesschanges. Excavation was performed relatively quickly so

EB Crossrail running tunnel

WB Crossrail running tunnel

Moor Housedevelopment

35 m

40 m deepexcavation 35

m

Fig. 36. Proposed Crossrail excavation near the Moor House development

Made ground

Terrace gravel

London clay

Lambeth Group clay

Thanet sand

y

x

101.0 m0.0�53.0

�33.0

�27.0

�22.5

�17.5

�12.5

�7.5

�3.0

�2.5

�6.5

�13.7

Fig. 37. Finite element mesh used in plane-strain and axisymmetric analyses

552 POTTS

that there was no time for excess pore water dissipation inthe clay soils.

Results from a plane-strain analysis and two axisymmetricanalyses are presented in Fig. 38 in the form of profiles oflateral wall deflection and surface settlement of the retainedground. In one of the axisymmetric analyses (labelled ‘stiffout of plane’ in Fig. 38) isotropic properties were assumedfor the wall, which implies the same stiffness in the circum-ferential and vertical directions. This analysis predicts verylittle wall movement and surface settlement, and is thereforeunrealistic. This is not surprising, as it is difficult toconstruct a wall that has substantial circumferential stiffness.The second axisymmetric analysis (labelled ‘soft out ofplane’ in Fig. 38) assumes anisotropic wall properties with alow stiffness (Young’s modulus) in the circumferential direc-tion. This analysis gives larger and more realistic move-ments. Of course, the out of plane stiffness of the wall doesnot affect the results of a plane-strain analysis and, as canbe seen from Fig. 38, such an analysis produces larger wallmovements and surface settlements than the ‘soft out ofplane’ axisymmetric analysis. This is particularly true forthe settlements of the retained ground, where the magnitudeof movements from the plane-strain analysis are more thandouble those of the axisymmetric analysis.

Design engineers would have to use engineering judge-ment as to which of the two predictions, plane strain oraxisymmetric (‘soft out of plane’), is appropriate to theirdesign. In this respect they might use the plane-strain resultsas being appropriate to the mid side of the excavation, andthe axisymmetric results for the corner. Some sort of inter-polation would then be required for wall positions inbetween.

This problem can be resolved by performing a three-dimensional analysis. The finite element mesh used for suchan analysis is shown in Fig. 39. It is similar to the two-

dimensional mesh in the vertical plane (Fig. 37), andconsists of 4500 20-noded hexahedral elements. Because ofthe eightfold symmetry of the excavation, only one eighth ofthe problem has been modelled in the analysis.

The same constitutive models for the soils and structuralmembers and the same construction procedure as adopted inthe two-dimensional analysis presented above were used.However, three-dimensional analysis presents additionalcomplications. For example, the modelling of the wallrequires particular attention. Fig. 40 shows plan views oftypical diaphragm and contiguous pile walls. The diaphragmwall has been assumed to consist of a series of reinforcedconcrete panels separated by unreinforced joints. Ideally, theindividual structural forms (panels or piles) and the jointsbetween them should be modelled in any analysis. However,this leads to an excessive number of elements and complica-tions, especially with modelling the joints. Consequently, itis common practice to use either solid or shell elements tomodel the wall, even though these are continuous in thehorizontal direction. Care must therefore be exercised whenchoosing appropriate wall properties. If isotropic propertiesare used, as is usual for plane-strain analysis, then the wallswill have the same axial and bending stiffness in both thevertical and horizontal directions. This is clearly unrealisticfor the same reasons discussed above for the axisymmetricanalysis.

A typical diaphragm wall will have a very low bendingstiffness in the horizontal direction, but a finite horizontalaxial stiffness, albeit considerably smaller than the verticalaxial stiffness. It is not possible to model such behaviourwith solid elements, and consequently shell elements mustbe used, as the bending and axial stiffness can be inde-pendently varied in such elements. Note that, at present,most commercial software packages do not have suchfacilities.

Plane strainAxisymmetric (soft out of plane)Axisymmetruc (stiff out of plane)

0.01

0

�0.01

�0.02

�0.03

�0.04

0 10 20 30 40 50 60 70 80 90

Distance from the wall: m(b)

Sur

face

dis

plac

emen

t: m

0.10.080.060.040.020�35

�30

�25

�20

�15

�10

�5

0

10

5

15Wall

Ele

vatio

n:m

Wall deflection (m)(towards excavation)

(a)

Fig. 38. Comparison of results for different geometries: (a) wall deflection; (b) surface displacements of retained ground


As both the horizontal bending and axial stiffness are verysmall for a contiguous pile wall, it can be modelled witheither solid or shell elements, combined with an anisotropicelastic constitutive model giving a low Young’s modulus inthe horizontal direction.

Two three-dimensional analyses, one for a diaphragm wallwhere the horizontal axial stiffness is one fifth of the verticalstiffness and one for a contiguous pile wall, were performed.The diaphragm wall was represented using shell elements,whereas the contiguous pile wall was modelled using solidelements. Results showing the profiles of lateral wall move-ment and settlement of the retained ground surface at mid

excavation and in the corner of the excavation are presentedin Figs 41 and 42. Also shown in these figures for compari-son are results from plane-strain and axisymmetric (‘soft outof plane’) analyses.

Considering Fig. 41, it is evident that at mid excavationboth three-dimensional analyses give similar predictions.The lateral wall movements and surface settlements arebetween those predicted from the plane-strain and axisym-metric analyses, with the wall movements being nearer tothose predicted by the plane-strain analysis, whereas thesurface settlement trough is nearer to the axisymmetricresults. The plane-strain analysis therefore overpredicts wall

Plan view

�13.7 m

�53.0 m0.0 101.0 m

0.0

y

z

x

101.0 m

Fig. 39. Finite element mesh used in 3D analysis

No out-of-planebending stiffness

Limited horizontalaxial stiffness

JointPanel

Corner

(a)

No out-of-planebending stiffness

No horizontalaxial stiffness

Pile

(b)

Fig. 40. 3D modelling of different wall types: (a) diaphragm wall; (b) contiguous pilewall

554 POTTS

Axisymmetric (soft out of plane)

Plane strain

Diaphragm wall

Contiguous wall

9080706050403020100

Distance from the wall: m(b)

�0.04

�0.03

�0.02

�0.01

0

Sur

face

dis

plac

emen

t: m

0.10.080.060.040.020�35

�30

�25

�20

�15

�10

�5

0

5

10

15Wall

Ele

vatio

n: m

Wall deflection: m(towards excavation)

(a)

Fig. 41. Comparison of results from different geometries in the centre of the excavation: (a) wall deflections; (b) surfacedisplacements of retained ground

Axisymmetric (soft out of plane)

Plane strain

Diaphragm wall

Contiguous wall

9080706050403020100Distance from the wall: m

(b)

�0.04

�0.03

�0.02

�0.01

0

Sur

face

dis

plac

emen

t: m

0.10.080.060.040.020�35

�30

�25

�20

�15

�10

�5

0

5

10

15Wall

Ele

vatio

n: m

Wall deflection: m(towards excavation)

(a)

Fig. 42. Comparison of results from different geometries in the corner of the excavation: (a) wall deflections; (b) surfacedisplacements of retained ground


movements by a small margin compared with the three-dimensional analysis, and this is consistent with the com-parison of predictions and observations for the Westminsterbox presented in Fig. 35.

At the corner of the excavation (Fig. 42) the lateral wallmovements for the diaphragm wall are very small and verydifferent from those for the contiguous pile wall. However,the surface settlement troughs from the two analyses arevery similar and are also similar to that obtained from theaxisymmetric analysis.

These results clearly show that the type of wall can havea significant effect on wall movements. It can also affect thestructural forces induced in it.

In summary, detailed three-dimensional analyses of realgeotechnical problems are now possible, although they re-quire considerable computing resources. Their use will in-crease in the future, as they can provide more detailedresults, which should lead to optimised designs and costsavings.

From the above discussion it is clear that numerical analy-sis has many advantages over conventional methods ofanalysis.

Here ends the case for the motion.

THE CASE AGAINST THE MOTIONIntroduction

The case against the motion is based on the followingfour points:

(a) There is no standard non-linear strategy.(b) Some constitutive models cannot give sensible pre-

dictions.(c) It is difficult to analyse some simple problems.(d ) Results can be user dependent.

This compares with the seven points used for the case forthe motion. At this point some readers may be thinking thatthis imbalance might reflect the author’s personal views andallegiances. This is not unreasonable, as the author has spent25 years promoting numerical analysis. However, as willbecome evident, these four points are significant and morethan adequate for the case against.

(a) No standard non-linear strategyAs noted at the beginning of this lecture, one of the main

approximations involved in numerical analysis is related tothe solution strategy adopted to solve the governing equa-tions. The difficulty arises because the basic theory behindthe numerical methods assumes material behaviour to beessentially linear-elastic. Application to problems in whichmaterial behaviour is non-linear then becomes problematic,because the material stiffness changes during the analysis, asillustrated in Fig. 4. As a first step to overcoming thisproblem the loading can be applied in a series of increments(or steps). However, because the increments are finite, thematerial stiffness will still change over the increment andshould be accounted for. Unfortunately, there is no exactmathematically rigorous way of dealing with this, and conse-quently approximations must be introduced. These tend tobe camouflaged in technical jargon.

The various finite element and finite difference programsdeal with this problem in different ways, which results ininconsistencies. To illustrate this point three classes of non-linear solution strategy used in finite element analysis willbe considered. All of these can be found in softwarecurrently used to analyse geotechnical problems. It shouldbe stressed that it is not necessary for the reader to under-stand the finer details of the strategies; however, it is

important that the reader gets an overall grasp of the mainassumptions made.

In all three cases the governing equations are reduced tothe following incremental form:

[KG]if˜d nGgi ¼ f˜RGgi (8)

where [KG]i is the incremental global stiffness matrix,{˜dnG}i is the vector of incremental nodal displacements,{˜RG}i is the vector of incremental nodal forces, and i isthe increment number. To obtain a solution to a boundaryvalue problem, the change in boundary conditions is appliedin a series of increments, and for each increment equation(8) must be solved. The final solution is obtained bysumming the results from each increment. Because of thenon-linear constitutive behaviour, the incremental globalstiffness, [KG]i, is dependent on the current stress and strainlevels and therefore is not constant but varies over anincrement. Hence the solution of equation (8) is not straight-forward.

Tangent stiffness algorithm. The tangent stiffness method,sometimes called the variable stiffness method, is thesimplest solution strategy. In this approach, the incrementalstiffness matrix [KG]i in equation (8) is assumed to beconstant over each increment and is calculated using thecurrent stress state at the beginning of each increment. Thisis equivalent to making a piece-wise linear approximation tothe non-linear constitutive behaviour. To illustrate theapplication of this approach, the simple problem of auniaxially loaded bar of non-linear material is considered.If this bar is loaded, the true load–displacement response isas shown in Fig. 43. This might represent the behaviour of astrain-hardening plastic material that has a very small initialelastic domain.

In the tangent stiffness approach the applied load is splitinto a sequence of increments. In Fig. 43 three incrementsof load are shown, ˜R1, ˜R2 and ˜R3. The analysis startswith the application of ˜R1. The incremental global stiffnessmatrix [KG]1 for this increment is evaluated on the basis ofthe unstressed state of the bar corresponding to point a. Foran elasto-plastic material [KG]1 might be constructed usingthe elastic material properties. Equation (8) is then solved todetermine the nodal displacements {˜dnG}1. As the materialstiffness is assumed to remain constant, the load–displace-ment curve follows the straight line ab9 in Fig. 43. In reality,the stiffness of the material does not remain constant duringthis loading increment, and the true solution is representedby the curved path ab. There is therefore an error in the

Load

Tangent stiffness solution

d

c

b

a

True solution

Displacement

d′

c′

b′

∆R3

∆R2

∆R1

∆dnG1 ∆dnG

2 ∆dnG3

KG3

KG2

KG1

Fig. 43. Finite element application of the tangent stiffnessalgorithm

556 POTTS

predicted displacement equal to the distance b9b; however,in the tangent stiffness approach this error is neglected. Thesecond increment of load, ˜R2, is then applied, with theincremental global stiffness matrix [KG]2 evaluated usingthe stresses and strains appropriate to the end of increment1—that is, point b9 in Fig. 43. Solution of equation (8) thengives the nodal displacements {˜dnG}2. The load–displace-ment curve follows the straight path b9c9 in Fig. 43. Thisdeviates further from the true solution, the error in thedisplacements now being equal to the distance c9c. A similarprocedure now occurs when ˜R3 is applied. The stiffnessmatrix [KG]3 is evaluated using the stresses and strainsappropriate to the end of increment 2—that is, point c9 inFig. 43. The load–displacement curve moves to point d9 andagain drifts further from the true solution. Clearly, theaccuracy of the solution depends on the size of the loadincrements. For example, if the increment size was reducedso that more increments were needed to reach the sameaccumulated load, the tangent stiffness solution would benearer to the true solution.

From the above simple example it may be concluded that,in order to obtain accurate solutions to strongly non-linearproblems, many small solution increments are required. Theresults obtained using this method can drift from the truesolution, and the stresses can fail to satisfy the constitutivemodel. Thus the basic solution requirements may not befulfilled. Potts & Zdravkovic (1999) have shown that themagnitude of the error is problem dependent and is affectedby the degree of material non-linearity, the geometry of theproblem, and the size of the solution increments used.Unfortunately, in general, it is impossible to predeterminethe size of the solution increment required to achieve anacceptable error.

Visco-plastic algorithm. This method uses the equations ofvisco-plastic behaviour and time as an artifice to calculate thebehaviour of non-linear, elasto-plastic, time-independentmaterials (Owen & Hinton, 1980; Zienkiewicz & Cormeau,1974).

The method was originally developed for linear-elasticvisco-plastic (that is, time-dependent) material behaviour.Such material can be represented by a network of the simplerheological units shown in Fig. 44. Each unit consists of anelastic and a visco-plastic component connected in series.The elastic component is represented by a spring, and thevisco-plastic component by a slider and dashpot connectedin parallel. If a load is applied to the network, then one oftwo situations occurs in each individual unit. If the load issuch that the induced stress in the unit does not causeyielding, the slider remains rigid and all the deformationoccurs in the spring. This represents elastic behaviour. Alter-natively, if the induced stress causes yielding, the sliderbecomes free and the dashpot is activated. As the dashpottakes time to react, initially all deformation occurs in thespring. However, with time the dashpot moves. The rate ofmovement of the dashpot depends on the stress it supportsand its fluidity. With time progressing, the dashpot moves ata decreasing rate, because some of the stress the unit iscarrying is dissipated to adjacent units in the network, whichas a result suffer further movements themselves. This repre-sents visco-plastic behaviour. Eventually, a stationary condi-tion is reached where all the dashpots in the network stopmoving and are no longer sustaining stresses. This occurswhen the stress in each unit drops below the yield surfaceand the slider becomes rigid. The external load is nowsupported purely by the springs within the network, but,importantly, straining of the system has occurred, not onlydue to compression or extension of the springs, but also dueto movement of the dashpots. If the load was now removed,

only the displacements (strains) occurring in the springswould be recoverable, the dashpot displacements (strains)being permanent.

Application to finite element analysis of elasto-plasticmaterials can be summarised as follows (see also Fig. 45).On application of a solution increment the system is as-sumed to instantaneously behave linear-elastically. If theresulting stress state lies within the yield surface, the incre-mental behaviour is elastic and the calculated displacementsare correct. If the resulting stress state violates yield, thestress state can be sustained only momentarily, and visco-plastic straining occurs. The magnitude of the visco-plastic

σ

Spring

Dashpot

Slider(rigid for F � 0free for F � 0)

σ

ενp

εe

Time

Strain

Fig. 44. Rheological model for visco-plastic algorithm

True solutionLoad

Initialelasticsolution

Displacement

∆Ri

t0 t1 t2

∆di

∆d ie ∆d i

vp

∆d2vp

∆d1vp

Fig. 45. Finite element application of the visco-plastic algorithm


strain rate is determined by the value of the yield function,which is a measure of the degree by which the current stressstate exceeds the yield condition. The visco-plastic strainsincrease with time, causing the material to relax, with areduction in the yield function and hence in the visco-plasticstrain rate. A marching technique is used to step forward intime until the visco-plastic strain rate is insignificant. At thispoint, the accumulated visco-plastic strain and the associatedstress change are equal to the incremental plastic strain andstress change respectively. This process is illustrated for thesimple problem of a uniaxially loaded bar of non-linearmaterial in Fig. 45. For further details see Smith & Griffiths(1988) or Potts & Zdravkovic (1999).

Owing to its simplicity, the visco-plastic algorithm hasbeen widely used. However, the method has its limitationsfor geotechnical analysis. First, the algorithm relies on thefact that for each increment the elastic parameters remainconstant. The simple algorithm cannot accommodate elasticparameters that vary during the increment because, for suchcases, it cannot determine the true elastic stress changeassociated with the incremental elastic strains. The bestprocedure is to use the elastic parameters associated withthe accumulated stresses and strains at the beginning of theincrement to calculate the elastic constitutive behaviour, andto assume that this remains constant for the increment. Sucha procedure yields accurate results only if the increments aresmall and/or the elastic non-linearity is not great.

A more severe limitation of the method arises when thealgorithm is used as an artifice to solve problems involvingnon-viscous material (that is, elasto-plastic materials). Ascan be seen from Fig. 45, the initial elastic solution predictsa stress state above the true solution. These stresses are thenused in the algorithm to calculate the plastic strains. For anelasto-plastic material this implies that the differentials ofthe plastic potential that are used to evaluate the plasticstrains are evaluated at an illegal stress state, which liesoutside the yield surface. The process of using illegalstresses to determine the plastic behaviour continues forsubsequent time steps. As previously noted, for the tangentstiffness algorithm, this is theoretically incorrect and resultsin failure to satisfy the constitutive equations. The magni-tude of the error depends on the constitutive model and inparticular on how sensitive the partial derivatives of theplastic potential are to the stress state.

Potts & Zdravkovic (1999) show that, although the visco-plastic algorithm works well for simple elasto-plastic consti-tutive models such as Tresca and Mohr–Coulomb, it hassevere limitations when used with critical-state type models.The accuracy of the algorithm is therefore dependent on theconstitutive model used.

Modified Newton–Raphson algorithm. The previous dis-cussion of both the tangent stiffness and visco-plasticalgorithms has highlighted the fact that errors can arisewhen the constitutive behaviour is based on illegal stressstates. The modified Newton–Raphson (MNR) algorithmattempts to rectify this problem by evaluating the constitutivebehaviour only in, or very near to, legal stress space.

The MNR method uses an iterative technique to solveequation (8). The first iteration is essentially the same asthat of the tangent stiffness method. However, it is recog-nised that the solution is likely to be in error, and thepredicted incremental displacements are used to calculatethe residual load, a measure of the error in the analysis.Equation (8) is then solved again with this residual load,{ł}, forming the right-hand-side vector. Equation (8) can berewritten as

[KG]i(f˜d nGgi) j ¼ fłg j�1 (9)

The superscript j refers to the iteration number, and{ł}o ¼ {˜RG}i. This process is repeated until the residualload is small. The incremental displacements are equal tothe sum of the iterative displacements ({˜dnG}i) j. Thisapproach is illustrated in Fig. 46 for the simple problem of auniaxially loaded bar of non-linear material. In principle, theiterative scheme ensures that for each solution increment theanalysis satisfies all solution requirements.

A key step in this calculation process is to determine theresidual load vector. At the end of each iteration the currentestimate of the incremental displacements is calculated andused to evaluate the incremental strains at each integrationpoint. The constitutive model is then integrated along theincremental strain paths to obtain an estimate of the stresschanges. These stress changes are added to the stresses atthe beginning of the increment and used to evaluate consis-tent equivalent nodal forces. The difference between theseforces and the externally applied loads (from the boundaryconditions) gives the residual load vector. A difference existsinitially because a constant incremental global stiffnessmatrix [KG]i is assumed over the increment. Because of thenon-linear material behaviour [KG]i is not constant but varieswith the incremental stress and strain changes.

As the constitutive behaviour changes over the increment,care must be taken when integrating the constitutive equa-tions to obtain the stress change. Methods of performing thisintegration are termed stress point algorithms, and bothexplicit and implicit approaches have been proposed in theliterature. There are many of these algorithms in use, and, asthey control the accuracy of the final solution, users mustverify the approach used in their software. In this respectnote that all stress point algorithms contain additionalassumptions: see Potts & Zdravkovic (1999). Consequently,some algorithms are more accurate than others, and, inaddition, no two computer programs will do exactly thesame thing.

This process is called a Newton–Raphson scheme if theincremental global stiffness matrix [KG]i is recalculated andinverted for each iteration on the basis of the latest estimateof the stresses and strains obtained from the previous itera-tion. To reduce the amount of computation, the modifiedNewton–Raphson (MNR) scheme calculates and inverts thestiffness matrix only at the beginning of the increment anduses it for all iterations within the increment. Sometimes theincremental global stiffness matrix is calculated using theelastic constitutive matrix, rather than the elasto-plasticmatrix. Clearly, there are several options here, and manysoftware packages allow the user to specify how the MNR

True solutionLoad

Displacement

∆RiK0

∆di

∆d1 ∆d2

Ψ1Ψ2

Fig. 46. Finite element application of the modified Newton–Raphson algorithm

558 POTTS

algorithm should work. In addition, some form of accelera-tion technique is often applied during the iteration process(Thomas, 1984; Crisfield, 1986). The various ways in whichthe MNR algorithm works can have an effect on the resultsof the analysis, but this is generally small compared withthat of the stress point algorithm.

Idealised triaxial test. A comparison of the three solutionstrategies presented above suggests the following. Thetangent stiffness method is the simplest, but its accuracy isinfluenced by increment size. The accuracy of the visco-plastic approach is also influenced by increment size, ifcomplex constitutive models are used. The MNR method ispotentially the most accurate, and is likely to be the leastsensitive to increment size.

Potts & Zdravkovic (1999) provide an extensive compari-son of the three methods by considering a range of boundaryvalue problems. Their results for an ideal drained triaxialcompression test are presented here as an example. The testwas deemed ideal, as the end effects at the top and bottomof the sample were considered negligible, and the stress andstrain conditions were uniform throughout. Consequently, thesample can be represented by a single quadrilateral element.There are therefore no discretisation errors, and the analysesare only testing the ability of the different solution strategiesto accurately integrate the constitutive equations.

The sample was assumed to be initially isotropicallynormally consolidated to a mean effective stress, p9, of200 kPa, with zero pore water pressure. The modified Camclay model was used to represent the soil behaviour, and thesoil parameters used for the analyses are shown in Table 4.Increments of compressive axial strain were applied to thesample until the axial strain reached 20%, while maintaininga constant radial stress and zero pore water pressure. Theresults are presented as plots of volumetric strain anddeviatoric stress (�91 – �93) against axial strain.

Results from three MNR analyses are presented in Fig.47. The analyses differed in the size of the strain incrementused, and the label associated with each line in the figureindicates the magnitude of the axial strain applied at eachincrement of that analysis. Also shown in Fig. 47 are resultsfrom the closed form analytical solution to this problemgiven by Potts & Zdravkovic (1999).

It can be seen that there is some dependence of the resultson the increment size, but that this is small. The resultsagree well with the analytical solution. Note that the stresspoint algorithm used to obtain these results is one of themost accurate that there is, and therefore the results pre-sented give a good indication of the best that can becurrently achieved.

The tangent stiffness results are presented in Fig. 48,where it can be seen that they are sensitive to incrementsize. Large errors occur for the larger increment sizes, butthe accuracy improves as the increment size reduces. For thesame increment size the results are much less accurate thanthose from the MNR analyses.

The results of the visco-plastic analyses are shown in Fig.49, where it can be seen that they are also very sensitive to

the increment size. Even the results from the analysis withthe smallest increment size of 0·1% are in considerableerror. It is interesting to note that the visco-plastic algorithmis much more accurate when the soil behaves undrained and/or when it is used with simpler constitutive models of theTresca and Mohr–Coulomb types.

Table 4. Soil parameters for idealised triaxial test

Description Value

Overconsolidation ratio 1·0Specific volume at unit pressure on virgin consolidation line, v1 1·788Slope of virgin consolidation line in v–ln p9 space, º 0·066Slope of swelling line in v–ln p9 space, k 0·0077Slope of critical-state line in J–p9 plane, MJ 0·693Elastic shear modulus G /Preconsolidation pressure, p90 100

0.5%1.0%2.0%

Analytical solution

MNR

2010Axial strain: %

2.0%1.0%0.5%

4

2

0

200

400

Dev

iato

ric s

tres

s: k

Pa

Vol

umet

ric s

trai

n: %

Fig. 47. Drained triaxial compression test prediction using theMNR algorithm

2.0%

1.0%

0.5%

Analytical solution

Tangent stiffness

10 20Axial strain: %

2.0%

1.0%

0.5%4

2

0

200

400D

evia

toric

str

ess:

kP

aV

olum

etric

str

ain:

%

Fig. 48. Drained triaxial compression test prediction using thetangent stiffness algorithm


It can be concluded from the above results, and fromthose presented by Potts & Zdravkovic (1999), that for boththe tangent stiffness and visco-plastic algorithms the incre-ment size required for accurate predictions depends on thenature of the boundary value problem under investigation,the complexity of the constitutive model and whether thesoil behaves drained or undrained. This is also true for someMNR approaches that use less accurate stress point algo-rithms than that used above.

Summary. There is no accepted non-linear solution strat-egy. Most of the finite element and finite difference computerprograms currently available use different approaches. Con-sequently, they are likely to produce results of differingaccuracy. It is a bit like having two calculators that, whenasked to add 2 and 2 together, give different answers, forexample 3 and 5. As engineers we like our calculations to beaccurate (or at least within a known small accuracy tolerance)and our assumptions and approximations to be confined toparameter selection and problem idealisation.

Although the solution schemes discussed above are applic-able to the finite element method, similar alternatives arisewith the other numerical methods.

(b) Some constitutive models cannot give sensiblepredictions

The fact that some of the most popular constitutivemodels can give non-sensible predictions is a serious pro-blem and a pitfall that many users encounter, but are oftenunaware of. To illustrate the sorts of problem that can arise,we shall begin by considering the popular Mohr–Coulombmodel. As noted previously in this lecture, this model canbe used with an angle of dilation ranging from � ¼ 08 to� ¼ �9. This parameter controls the magnitude of the plasticdilation (plastic volume expansion), and remains constantonce the state of stress in the soil is on the yield surface:see Fig. 7. This implies that the soil will continue to dilateindefinitely if shearing continues. Clearly, such behaviour isnot realistic, as most soils will eventually reach a critical-state condition, after which they will deform at constantvolume if sheared any further. Such unrealistic behaviour

does not have a great influence on boundary value problemsthat are kinematically unconfined. For example, it has al-ready been shown that the dilation angle has only a rela-tively small effect on the bearing capacity of surfacefootings. However, it can have a major effect on problemsthat are kinematically confined (such as drained cavityexpansion or drained axial pile loading), owing to therestrictions on volume change imposed by the boundaryconditions. In particular, unrealistic results can be obtainedin undrained analysis in which there is a severe constraintimposed by the zero total volume change restriction asso-ciated with undrained soil behaviour. This could arise in acoupled consolidation analysis when loading occurs rela-tively rapidly, or in an undrained stage of an analysis inwhich the soil is modelled in terms of an effective stressconstitutive model, and a high bulk compressibility is as-signed to the pore fluid. To illustrate these problems threeexamples are presented below.

The first example considers the problem of predicting thebehaviour of a 1 m diameter, 20 m deep pile subjected toaxial loading. The finite element mesh indicating the geome-try and boundary conditions is shown in Fig. 50. Note thefine mesh around the pile, which meant that there was noneed to complicate matters by adding interface elementsbetween the pile and the soil. The pile was assumed to bewished in place (that is, installed without changing the in-situ stress conditions in the soil) and then loaded verticallyby displacing its top downwards. The soil was assumed tobe dry (that is, no water present) and to behave in a drainedmanner using the Mohr–Coulomb constitutive model, with aYoung’s modulus, Es ¼ 105 kN/m2, a Poisson’s ratio, �s

¼ 0·3, a cohesion, c9 ¼ 0 and angle of shearing resistance,�9 ¼ 258. Initial stresses in the soil were based onªdry ¼ 18 kN/m3 and K0 ¼ 1. The pile was assumed tobehave linear-elastically, with a Young’s modulus, Ec ¼ 20 3106 kN/m2 and a Poisson’s ratio, �c ¼ 0·15.

The results of two analyses are presented in Fig. 51. Theonly difference between these analyses was that in oneanalysis the angle of dilation, � ¼ �9 ¼ 258, whereas in theother analysis � ¼ 08. For both analyses the load–displace-ment behaviour is presented in terms of the total load on thepile against pile head displacement and in terms of the shaftand base components of the total load.

Considering the analysis with � ¼ �9 ¼ 258 it can be seenthat no ultimate total load is indicated in Fig. 51(a). Boththe shaft and base components of load continue to increasewith pile head displacement. Although results are shownonly for a pile head displacement of up to 0·02 m in Fig.51(a), the analysis was continued to much greater displace-ments (2·0 m). Even at these larger displacements both theshaft and base loads continued to increase, and no ultimatecapacity was predicted.

In contrast, the analysis with � ¼ 08 does predict anultimate load at a pile head displacement of 2 m. Resultsfrom this analysis are shown in Fig. 51(b) for displacementsup to 0·09 m (9% of pile diameter). It can be seen that theshaft component of the pile capacity is fully mobilised at apile head displacement of only 0·005 m, whereas the basecomponent is still increasing, and only reaches a maximumat a displacement of 2 m.

Comparison of the results in Figs 51(a) and 51(b) indi-cates that not only does dilation affect the limit loads, but italso dominates the load–displacement behaviour. Furtheranalyses with angles of dilation 08 , � , 258 also indicatethat no ultimate limit load is predicted (at least for a pilehead displacement of 2 m), and that the higher the dilationangle the stiffer the load–displacement curve. Consequently,the only analysis that predicts an ultimate pile load is that inwhich � ¼ 08. However, as most sands in the field exhibit

0.1%

0.5%

1.0%

10 20 1.0%

0.5%

Analytical solution

Visco-plastic solution

0.1%

400

200

0

2

4

Vol

umet

ric s

trai

n: %

Dev

iato

ric s

tres

s: k

Pa

Axial strain (%)

Fig. 49. Drained triaxial compression test prediction using thevisco-plastic algorithm

560 POTTS

some fixed amount of dilation, predictions based on � ¼ 08are likely to be too conservative.

The second example considers ideal (no end effects)undrained triaxial compression tests on a linear-elasticMohr–Coulomb plastic soil, with parameters E9 ¼ 10 MPa,� ¼ 0·3, c9 ¼ 0 and �9 ¼ 248. As there are no end effects, asingle quadrilateral finite element is used to model thetriaxial sample with appropriate boundary conditions. The

samples were assumed to be initially isotropically consoli-dated with p9 ¼ 200 kPa and zero pore water pressure. Aseries of finite element analyses were then performed, eachwith a different angle of dilation, �, in which the sampleswere sheared undrained. Undrained conditions were enforcedby setting the bulk modulus of the pore water to be 1000times larger than the effective elastic bulk modulus of thesoil skeleton, K9skel.

50 m

0.5 m

A20

m

55 m

Pile

Detail A

0.5 0.5 0.5

Fig. 50. Finite element mesh for single pile analysis

Soil: φ′ � ν � 25° Soil: φ′ � 25°; ν � 0°

10000

8000

6000

4000

2000

00 0.005 0.01 0.015 0.02

Base

Shaft

Total

Ver

tical

forc

e: k

N

Displacement: m(a)

0 0.03 0.06 0.09

Total

Shaft

Base

Displacement: m(b)

Fig. 51. Behaviour of a pile in drained soil: (a) soil with v �9; (b) soil with v 08


The results are shown in Figs 52(a) and 52(b) in the formof deviatoric stress (�91 – �93) against mean effective stress,p9, and deviatoric stress against axial strain, �a, respectively.It can be seen from Fig. 52(a) that all analyses follow thesame stress path. However, the rate at which the stress statemoves up the Mohr–Coulomb failure line differs for eachanalysis. This can be seen from Fig. 52(b). The analysis withzero plastic dilation, � ¼ 08, remains at a constant deviatoricand mean effective stress when it reaches the failure line.However, all other analyses move up the failure line, thosewith the larger dilation moving up more rapidly. Theycontinue to move up the failure line indefinitely with con-tinued shearing. Consequently, the only analysis that indi-cates failure (that is a limiting value of deviator stress) isthe analysis performed with zero plastic dilation.

The third example considers the undrained loading of asmooth, rigid strip surface foundation. The soil was assumedto have the same parameters as those used for the triaxialtests above. The initial stresses in the soil were calculated onthe basis of a saturated bulk unit weight of 20 kN/m3, a

groundwater table at the soil surface, and a K0 ¼ 1 � sin�9.The footing was loaded by applying increments of verticaldisplacement, and undrained conditions were again enforcedby setting the bulk modulus of the pore water to be 1000times K9skel. The results of two analyses, one with � ¼ 08 andthe other with � ¼ �9, are shown in Fig. 53 in the form ofload–displacement curves. The difference in the results isquite staggering: while the analysis with � ¼ 08 reaches alimit load, the analysis with � ¼ �9 shows a continuingincrease in load with displacement. As with the undrainedtriaxial test, a limit load is obtained only if � ¼ 08.

It can be concluded from these examples that a limit loadwill be obtained only if � ¼ 08. Consequently, great caremust be exercised when using the Mohr–Coulomb model inboth confined and undrained analysis. It could be argued thatthe model should not be used with � . 08 for such analysis.However, reality is not so simple, and often an analysisinvolves both an undrained and a drained phase (for exam-ple, undrained excavation followed by drained dissipation ofexcess pore water pressures).

600

450

300

150

00 100 200

(σ1

� σ

3): k

Pa

(σ1

� σ

3): k

Pa

600

450

300

150

00 2 4 6 8 10

εa: %

(b)

300 400 500 600

p′: kPa(a)

ν � 0°

ν � 31 φ′ � 8°

ν � 32 φ′ � 16°

ν � φ′ � 24°

Fig. 52. Undrained triaxial compression with Mohr–Coulomb model: (a) stresspaths; (b) stress–strain curves

562 POTTS

Although the examples described above have used theMohr–Coulomb model, similar problems will occur withany constitutive model that predicts finite plastic volumechanges at large strains. Consequently, some of the moreadvanced constitutive models that are currently available willsuffer similar problems.

One way to avoid these problems is to use a model thatreproduces critical-state conditions where, after a certainamount of shear strain, the model predicts no further changein volume. However, there can be other subtle problems thatcan arise with the use of these more sophisticated constitu-tive models. As an example, the use of the modified Camclay model will be considered.

The original model combines compression and shear be-haviour. However, the shear behaviour is formulated only in

triaxial space, where two of the principal stresses have equalmagnitude. For use in numerical analysis the model musttherefore be extended to general stress space. This involvesfurther assumptions. In this respect it is usual to assume soilbehaviour to be isotropic. This enables the state of stress tobe represented by three stress invariants, which in turnimplies that the yield and plastic potential surfaces must alsobe expressed in terms of the three stress invariants. A logicalchoice for these invariants is the mean effective stress, p9,the deviatoric stress, J, and the Lode’s angle, Ł. These aredefined in equations (9)–(11):

p9 ¼ 1

3(� 91 þ � 92 þ � 93) (9)

J ¼ 1ffiffiffi6

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(� 91 � � 93)2 þ (� 91 � � 92)2 þ (� 92 � � 93)2

p(10)

Ł ¼ tan �1 1ffiffiffi3

p 2(� 92 � � 93)

(� 91 � � 93)� 1

� �� (11)

The choice of these invariants is not arbitrary, because theabove quantities have geometric significance in principaleffective stress space. For example, consider the stress staterepresented by point S (�91S, �92S, �93S), which has invariantsp9S, JS and ŁS in Fig. 54. The distance of the deviatoricplane in which S lies, from the origin, is

ffiffiffi3

pp9S: see Fig.

54(a). The distance of S from the space diagonal in thedeviatoric plane is given by

ffiffiffi2

pJS, and the orientation of S

within this plane by the value of ŁS: see Fig. 54(b). In thisfigure, (�91)pr, (�92)pr and (�93)pr refer to the projections of theprincipal stress axes onto the deviatoric plane. If �91S

> �92S > �93S, S is constrained to lie between the linesmarked as Ł ¼ �308 and Ł ¼ +308. These limiting values ofŁ correspond to triaxial compression (�91S > �92S ¼ �93S) andtriaxial extension (�91S ¼ �92S > �93S) respectively.

The original modified Cam clay model uses two stress

70

60

50

40

30

Load

2 m

Load

: kN

/m

20

10

00 5 10

Vertical displacement, ν: mm

15 20 25

ν � 0°

ν � φ′ � 24°

Fig. 53. Load–displacement curves for undrained loading of astrip footing, using the Mohr–Coulomb model with differentangles of dilation

Deviatoric plane

Space diagonal

Deviatoric plane

σ2′

σ1′

σ3′

3p′S

S

σ 1′ � σ 2′ �

σ 3′(σ2′)

pr

(σ3′)pr (σ1′)

pr θ � �30°

θ � 0°

θ � 30°

Sθ S

(b)

(a)

2J S

Fig. 54. Invariants in principal stress space


values, the mean effective stress, p9, and the shear stress,q ¼ �91 � �93. To generalise the model, p9 is retained and q isreplaced by J. Although this is sufficient, it implies that themodel is independent of the Lode’s angle, Ł. In the devia-toric plane this means that the yield and failure (criticalstate) surfaces are circular.

An alternative approach is to allow the parameter MJ , theslope of the critical-state line in J–p9 space, to vary with Ł.This will then control the shapes of the yield and failuresurfaces in the deviatoric plane. Although there are variousassumptions that could be made here, it will be assumed thatthe angle of shearing resistance, �9, is a constant, anassumption consistent with conventional practice. This leadsto the following definition of MJ (Potts & Zdravkovic,1999):

M J ¼ g(Ł) ¼ sin�9

cosŁþ sin Ł sin�9ffiffiffi3

p(12)

When plotted in the deviatoric plane the yield and failuresurfaces plot as an irregular hexagon: see Fig. 55. Alsoplotted in this figure is the circle corresponding to theconstant MJ assumption. Both assumptions have been cho-sen to agree in triaxial compression. However, as the Lode’sangle, Ł, changes, the constant MJ assumption (circle in Fig.55) implies higher J values than if MJ is based on equation(12) (Mohr–Coulomb hexagon in Fig. 55). This results inthe angle of shearing resistance, �9, varying with Lode’sangle and having a magnitude higher than that for triaxialcompression. This variation in �9, for the constant MJ

assumption, can be expressed by

�9 ¼ sin �1 M J cosŁ

1 � M J sin Łffiffiffi3

p

0B@

1CA (13)

From this equation it is possible to express MJ in terms ofthe angle of shearing resistance, �9TC, in triaxial compres-sion, that is, Ł ¼ �308:

M J (�9TC) ¼ 2ffiffiffi3

psin�9TC

3 � sin�9TC

(14)

Figure 56 shows the variation of �9 with Ł, given by equa-tion (13), for three constant values of MJ . The values of MJ

have been determined from equation (14) using �9TC ¼ 208,258 and 308. If the plastic potential is circular in the

deviatoric plane, Potts & Zdravkovic (1999) show thatplane-strain failure occurs when the Lode’s angle, Ł ¼ 08.Fig. 56 indicates that for all values of MJ there is a largechange in �9 with Ł. For example, if MJ is set to give�9TC ¼ 258, then under plane-strain conditions �9PS ¼ 34·68.This difference is considerable, and much larger than in-dicated by laboratory testing. The difference between �9TC

and �9PS becomes greater the larger the value of MJ .To investigate the effect of adopting the two options for

MJ in a boundary value problem, two analyses of a roughrigid strip footing, 2 m wide, have been performed. Themodified Cam clay model was used to represent the soil,which had the following parameters: OCR ¼ 6, v1 ¼ 2·848,º ¼ 0·161, k ¼ 0·0322, � ¼ 0·2. In one analysis the yieldand plastic potential surfaces were assumed to be circular inthe deviatoric plane. A constant value of MJ ¼ 0·5187 wasused for this analysis, which is equivalent to �9TC ¼ 238. Inthe second analysis a constant value of �9 ¼ 238 was used,giving a Mohr–Coulomb hexagon for the yield surface inthe deviatoric plane. However, the plastic potential still gavea circle in the deviatoric plane, and therefore plane-strainfailure occurred at Ł ¼ 08, as for the first analysis.

In both analyses the initial stress conditions in the soilwere based on a saturated bulk unit weight of 18 kN/m3, agroundwater table at a depth of 2·5 m, and a K0 ¼ 1·227.Above the groundwater table the soil was assumed to besaturated and able to sustain pore water suctions. Coupledconsolidation analyses were performed, but the permeabilityand time steps were chosen such that undrained conditionsexisted. Loading of the footing was simulated by imposingincrements of vertical displacement.

In summary, the input to both analyses is identical, exceptthat in the first the strength parameter, MJ , is specified,whereas in the second �9 is input. In both analyses�9TC ¼ 238, and therefore any analysis in triaxial compressionwould give identical results. However, the strip footingproblem is plane strain and therefore differences are ex-pected. The resulting load–displacement curves are given inFig. 57. The analysis with a constant MJ gave a collapseload some 58% larger than the analysis with a constant �9.The implications for practice are clear: if a user is not awareof this problem, or is not fully conversant with the constitu-tive model implemented in the software being used, theycould easily base the input on �9TC ¼ 238. If the model usesa constant MJ formulation, this would then imply a�9PS ¼ 31·28, which in turn leads to a large error in theprediction of any collapse load.

There are limited experimental data on soil behaviour inthe deviatoric plane, and consequently several alternativesfor the shapes of the yield and plastic potential surfaces are

σ2′Mohr–Coulomb

Circle

σ3′ σ1′

Fig. 55. Failure/yield surfaces in deviatoric plane

50

40

30

20

45

35

�30 �20 �10 0 10 20 30

Triaxialextension

Plane strainTriaxialcompression

φ′TC � 30°

25

φ′:

deg

Lode’s angle, θ: deg

φ′TC � 20°

φ′TC � 25°

Fig. 56. Variation of �9 with Ł for constant MJ

564 POTTS

available in the literature. Two of these have been discussedabove, and they are shown along with those suggested byMatsuoka & Nakai (1974) and Lade & Duncan (1975) inFig. 58. Clearly, more research is needed here, and in thisrespect laboratory tests using true triaxial and hollow cylin-der apparatus are necessary.

The shapes of the yield, plastic potential and failuresurfaces in the deviatoric plane can have a significant effecton predictions. This applies to all constitutive models, notjust the modified Cam clay model considered in detailabove. It is therefore essential that a software user under-stands these effects. At this point readers who use numericalanalysis should ask themselves whether they actually knowthe assumptions made in the software they are using. In thisrespect the author’s experience is that most software manualsdo not supply such detailed information.

(c) Difficult to analyse some simple problemsThis point for the case against appears to contradict the

case for, which stated that numerical analysis could doeverything that conventional analysis could do. However, asin all debates, there is room to be economical with the truthon both sides. Not everything in life is black and white, anddifferences in opinion lead to large grey areas.

The problem to be considered here is concerned withexcavation in front of an embedded cantilever wall retainingundrained clay: see Fig. 59. The wall is 1 m wide and 20 mdeep, with E ¼ 20 3 106 kPa and � ¼ 0·15. The clay hasthe following properties: Eu ¼ 60 MPa, �u ¼ 0·499, su

¼ 60 kPa, ªsat ¼ 20 kN/m3 and K0 ¼ 1·0. The adhesion, cw,between the soil and the wall is assumed to be 0·5su. It isalso assumed that, if tension cracks form, they will remaindry. This assumption is not essential, but it simplifies thepresentation. The same conclusions apply if the cracks arefilled with water. The problem is to determine the depth ofexcavation in front of the wall to initiate collapse.

A conventional limit equilibrium calculation indicates thatthe limiting depth of excavation is 11·6 m, and at this levelthe depth of the tension crack at the back of the wall is7·35 m.

As the soil behaves undrained, with a specified value ofundrained strength su, it can be represented by the Trescaconstitutive model in any numerical analysis. In addition,interface elements are needed between the wall and the soilbecause the wall adhesion, cw ¼ 0·5su. Feeding this informa-tion, the soil properties and the initial stress conditions intoa finite element analysis (Run 1) results in a predictedlimiting depth of excavation of 13·8 m, considerably greaterthan that predicted by the simple limit equilibrium calcula-tion (11·6 m).

The displaced shape of the wall at an excavation depth of12 m and vectors of incremental displacement at the incre-ment of the analysis just prior to collapse are shown in Fig.60. Both plots indicate larger horizontal movements at andbelow excavation level than at the top of the wall. Clearly,this is unrealistic, as the wall is expected to behave as anembedded cantilever with greater movements at its top. Thereason for this unrealistic prediction becomes clear when thestresses in the soil are investigated. The horizontal stressesacting on the retained side of the wall and the zone of soilin which tensile stresses occur are shown in Fig. 61, whereit can be seen that tension extends to a depth of approxi-mately 6·5 m. These tensile stresses restrain the movementof the top of the wall. Clearly, this is unrealistic and occursbecause the Tresca model allows the soil to sustain tensilestresses.

One possible way of overcoming this problem is to changethe properties of the interface elements around the wall sothat they cannot sustain tensile normal stresses. Once thenormal stress is reduced to zero from its initial compressivevalue, the interface element simply opens without sustainingany further reduction in normal stresses. The interfaceelement therefore simulates a crack forming between the soil

250

200

150

100

50

00 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Vertical displacement, ν: m

Ver

tical

load

: kN

/m

MJ constant

φ′ constant

0.05

Fig. 57. Load–displacement curves for strip footing

σ3′

σ2′

σ1′

Lade

Matsuoka–Nakai

Mohr–Coulomb

Circle

Fig. 58. Failure/yield surfaces in deviatoric plane

H � ?20 m

Undrained clay:

su � 60 kPa

Eu � 60 MPa

µ � 0.499

cw � 0.5 su

Fig. 59. Geometry of embedded cantilever wall problem and soilconditions


LH2604

Highlight

and the wall. In all other respects the analysis (Run 2) is thesame as the previous one.

Lateral wall displacements at an excavation depth of 12 mand vectors of incremental displacement just prior to failurefrom this analysis are presented in Fig. 62. These are nowmore realistic, indicating that the largest movements occur atthe top of the wall, and that failure involves active andpassive shear zones behind and in front of the wall respec-tively. The analysis indicates a limiting depth of excavationof 12·9 m, which, although less than the first analysis, is still1·3 m greater than the limit equilibrium value.

The zone of soil sustaining tensile stresses and thehorizontal stress distribution acting on the back of the wallare shown in Fig. 63. It can be seen that there are nohorizontal stresses acting on the back of the wall down to adepth of approximately 10 m, but there is still a large zoneof tension in the retained clay. At first this might seemcontradictory, but it arises because the interface elementslimit only the horizontal stress between the soil and the wall.There is no restriction imposed on the stresses within thesoil mass, and consequently tensile stresses, not necessarilyin the horizontal direction, have occurred. These stresses

will affect the results. Using interface elements that cannotsustain tension has therefore not solved the problem.

An alternative way of overcoming this problem is not toallow tension to occur in the soil. This requires modificationof the Tresca constitutive model. There are several ways ofdealing with this, some being better than others (seeNyaoro, 1989). The most accurate and theoretically correctway is to introduce a tension cut-off, in the form of asecond yield surface, and its associated plastic potential.This means that the software has to deal with both this andthe Tresca model simultaneously. Clearly, this is computa-tionally more difficult than dealing with a single yieldsurface. Note also that such modelling is still an approxima-tion to real soil behaviour, in which tension first occurs indirections not consistent with the orientations of the finaltension cracks. Clearly, rotation of the direction of tensilestress must occur, and although this is accounted for by adouble yield surface model of the type described above, thevariation in stiffness and plastic strains is probably a grossapproximation.

Results from an analysis (Run 3) using a double yieldsurface model for the soil indicate a limiting depth of

0

5

10

15

20�0.12 �0.06 0

Wall deflection: m

Dep

th: m

Hlim

� 1

3.8

m

H �

12

m

Fig. 60. FE analysis with Tresca model in both interface and soil elements (Run 1)

Tension in soil

0

5

10

15

20

Dep

th: m

�100 0 100 300

σh: kPa

Fig. 61. Tension zone in soil for the conditions in Run 1

566 POTTS

excavation of 12·1 m. In this analysis no restriction wasimposed on the normal stresses in the interface elements.However, no tensile stresses were predicted in the soil norbetween the soil and the back of the wall. The lateral walldisplacements at an excavation depth of 12 m and vectors ofincremental displacement just prior to failure from such ananalysis are presented in Fig. 64.

The limiting depth of excavation and maximum wallbending moment for all three analyses, along with thosefrom the limit equilibrium calculation, are compared inTable 5. It can be seen that, although the numerical pre-dictions become closer to those from the limit equili-brium calculation as the numerical analysis improves on itsability to simulate tension cracks, there are still considerabledifferences.

It can be concluded that it is not straightforward to modelthis simple problem using numerical analysis. The resultsobtained depend on how the no-tension requirement ismodelled. Numerical analysis produces results that are lessconservative (that is, deeper excavation depths and smallerbending moments) than conventional methods, which them-selves are already viewed cautiously.

(d) Results can be user dependentThis is the most powerful argument for the case against

the motion. It is also the most worrying and, from acomputational analyst’s point of view, extremely depressing.To illustrate this point, results from two benchmarking exer-cises will be considered. Benchmarking is involved in theprocess of testing, validating, verifying or checking the per-formance or operation of computer software. One approachinvolves devising a problem, asking a group of people toproduce independent analysis, and then correlating the re-sults. The two examples to be considered here involve atunnel and an excavation.

Undrained analysis of a shield tunnel. This benchmarkingexercise was organised by the Co-Operation in Science andTechnology (COST) Action C7 for soil–structure interactionin urban civil engineering. The problem consists of excavat-ing a circular hole, under plane-strain conditions, in either anelastic or an elasto-plastic soil. For the latter situation, thesoil is assumed to have an undrained strength constant withdepth.

Undrained conditions were stipulated, and a set of

0

5

10

15

Dep

th: m

20�0.12 �0.06 0

Hlim

� 1

2.9

m

Wall deflection: m

H �

12

m

Fig. 62. FE analysis with no tension permitted in interface elements (Run 2)

Tension in soil

0

5

10

15

20

Dep

th: m

�100 0 100 300

σh: kPa

Fig. 63. Tension zone in soil for the conditions in Run 2


analyses was required from each participant, all performedin terms of total stresses under plane-strain conditions. Thedetailed results from this benchmarking exercise can befound in Potts et al. (2002). Only some of the results fromtwo of the analyses will be presented here.

The tunnel diameter is 10 m and the depth of cover abovethe tunnel crown is 15 m. At a depth of 45 m below groundsurface bedrock occurs: see Fig. 65. Consequently, the onlygeometric dimension not prescribed is related to the positionof the lateral boundaries.

In the first analysis to be considered here the participantswere asked to excavate and construct the tunnel with aprescribed volume loss in an elasto-plastic soil with aconstant value of su ¼ 60 kPa. They were told that the soilhad a saturated bulk unit weight of 20 kN/m3 and a K0 (interms of total stress) of 0·75. They were also given theelastic properties of the soil and the lining, but here we shallconcentrate only on the initial stress conditions.

The distributions of the initial vertical and horizontal totalstresses with depth in the ground, prior to tunnel construc-tion, are shown in Fig. 66. As K0 ¼ 0·75, the differencebetween the vertical and horizontal total stresses increases inmagnitude with depth. As the undrained strength remainsconstant (that is, su ¼ 60 kPa), then below a certain depththe deviator stress [¼ 0·5(�v – �h)] will exceed su. As canbe seen from Fig. 66, this occurs at a depth of 26 m, justbelow the position of the tunnel invert. Clearly this is notpossible, and there is an inconsistency between the initialstresses and the undrained strength. However, most of thesoftware programs used by the participants accepted theseinitial conditions and produced results without flagging thisinconsistency as an error or issuing a warning message.Some of these programs were commercial software regularlyused in geotechnical practice. In the author’s opinion thisstate of affairs is totally unacceptable.

In the second analysis to be reported here, the participantswere asked to repeat the above analysis but with K0 ¼ 1·0,which of course avoids the inconsistency between initialstresses and undrained strength. They were provided with allthe soil and lining properties: Gs ¼ 12 MPa, �s ¼ 0·495,su ¼ 60 kPa, ªs ¼ 20 kN/m3, K0 ¼ 1, El ¼ 21 GPa, �l ¼0·18, ªl ¼ 24 kN/m3. They were also instructed to perform afull tunnel excavation, inserting the lining at an appropriatestage of the analysis so that after construction the volumeloss was 2%. Consequently, the participants had only todecide on the lateral extent of their finite element mesh orfinite difference grid, generate the mesh or grid, input thematerial properties and boundary conditions and run theanalysis.

The results from the 11 participants are shown in Fig. 67in the form of ground surface settlements at the end of theanalysis. Note that the participants were from a wide rangeof European countries, and were experienced analysts.Nevertheless, the scatter in the results is very large. Perhapseven more worrying are the unrealistic shapes of many ofthe settlement troughs and the fact that many of these donot correlate with a volume loss of 2%. In this respect onlytwo of the participants stated that they had checked theachieved volume loss, and therefore it can be concluded thatmost of the other participants did not.

A review of the results revealed that the large settlementsobtained in one of the analyses could be explained by amisinterpretation of volume loss (4% had been applied), andthat in another analysis a vertical restraint had been imposed

0

5

10

15D

epth

: m

20�0.12 �0.06 0

Hlim

� 1

2.1

m

Wall deflection: mH

� 1

2 m

Fig. 64. FE analysis with no tension permitted in soil (Run 3)

Table 5. Summary of maximum excavation depths and bendingmoments

Method of analysis Maximum excavationdepth: m

Maximum bendingmoment: kNm/m

Limit equilibrium 11·6 498FE, Run 1 13·8 1491FE, Run 2 12·9 194FE, Run 3 12·1 244

0.0 m (ground surface)

15 m

Tunnel diameter � 10 m

Lining thickness � 0.3 m

�45.0 m (bedrock)

Fig. 65. Geometry for the analysis of a shield tunnel

568 POTTS

on the far field vertical boundary, which influenced theresult. However, in the other solutions no obvious cause forthe differences could be found, except that the far fieldvertical boundary had been placed at different distancesfrom the symmetry axis of the tunnel. Consequently, aftersome discussion the participants were asked to repeat theiranalyses, but with the distance from the axis of symmetry tothe far field vertical boundary set at 100 m. It was alsospecified that only a lateral restraint was to be applied onthis far field vertical boundary. The results from theseanalyses are presented in Fig. 68, again in the form of thepredicted surface settlement troughs. The scatter in results isless than for the initial analyses, but it is still significant,especially above the tunnel crown. Again, many of theanalyses did not achieve the specified volume loss.

Deep excavation. This second benchmarking example wasorganised by Working Group 1·6 ‘Numerical methods ingeotechnics’ of the German Society for Geotechnics. In fact,it is one of a series of three that the Society has organised todate. The other two involve a shotcrete tunnel and a tied-backexcavation.

The problem geometry is shown in Fig. 69, and Table 6lists the relevant material properties and construction stages.

Note that the soil is assumed to be dry. The participantswere also instructed to assume plane-strain conditions, use alinear-elastic perfectly plastic Mohr–Coulomb model to re-present the soil, assume perfect bonding between the dia-phragm wall and adjacent soil, model the struts as rigid,neglect the influence of wall construction (that is, assume itwas ‘wished in place’), and to model the wall with eitherbeam or continuum elements. Consequently, the participantshad only to generate a mesh, input the material propertiesand boundary conditions and run the analysis.

10 m

15 m

su � 60 kPa

40020000

4

8

12

16

20

24

28

26 m 2σvo � σho � 60 kPa

σvoσho

σho � 0.75 σvo

σ: kPa

Dep

th: m

Fig. 66. Initial stress conditions for the original analysis of a shield tunnel

100755025

Distance from tunnel axis: m

060

50

40

30

20

10

0

Sur

face

set

tlem

ent:

mm

Fig. 67. Surface settlement troughs obtained from differentparticipants (first attempt; after Potts et al., 2002)

�5

0

5

10

15

200 25 50 75 100

Distance from tunnel axis: m

Sur

face

set

tlem

ent:

mm

Fig. 68. Surface settlement troughs obtained from differentparticipants (second attempt; after Potts et al., 2002)

�4.0 Excavation step 1

�8.0 Excavation step 2

�12.0 Final excavation

Diaphragmwall

�20.0

�26.0

15 m0.8 m0.0Ground surface

Layer 1

Layer 2

Layer 3

60 m

40 m

Strut 1

Strut 2

�3.0

�7.0

∆

∆

∆

∆

∆

∆

∆

∆

CL

Fig. 69. Geometry for the analysis of a deep excavation


There were 12 participants, and five of these used thesame commercially available computer program. Predictedhorizontal displacements of the top of the wall after con-struction stage 1, excavation to –4 m, are presented in Fig.70. Each of the vertical bars in this figure represents theprediction from a single participant. Not only are themagnitudes of the movements all different, but there is alsono consensus as to the direction of movement, with 50% ofthe participants predicting movements towards the excavationand 50% predicting movements into the retained ground. Asthe wall is acting as an embedded cantilever during thisstage of the analysis, the latter predictions are somewhatunrealistic.

Vertical displacements of the retained ground surface afterthe final excavation stage are shown in Fig. 71. The resultsare almost evenly distributed between the limiting values,showing a very large scatter. Further information on thisexample and the other benchmark problems organised by the

German Society for Geotechnics can be found in Schweiger(1997, 1998, 2000) and Potts et al. (2002)

Comments. It should be stressed that the results of the twoexamples presented above are typical of other benchmarkingexercises. In addition, both examples are relatively simple.They did not involve the participants in the selection ofconstitutive models, nor in assigning parameter values. Therehave been benchmarking exercises where this has been thecase, and these have resulted in considerably greater scatterin the results: see Schweiger (2000) or Potts et al. (2002).

In summary, benchmarking exercises have shown that it isnot possible for different users to obtain consistent results,even for problems that are essentially completely defined.As some of the participants used the same software package,this cannot be the fault of the software alone. A significantamount of human error is involved, and this implies that theusers have insufficient training and/or experience in the useof advanced numerical analysis.

Here ends the case against the motion.

DISCUSSIONThe motion under debate is ‘Numerical methods of analy-

sis have reached the stage where they are superior to con-ventional approaches and can replace them in thegeotechnical design process’. The cases for and against havebeen presented and can be briefly summarised as follows:

Case for

(a) Numerical analyses can do everything conventionalanalysis can do, plus much more.

(b) Their ability to predict mechanisms of behaviour is amajor advantage.

(c) Numerical analysis can deal with both simple andcomplex problems.

Case against

(a) There are uncertainties in the numerical algorithms.(b) There are limitations with current constitutive models.(c) Results from numerical analysis are user dependent.

As in all debates, it is up to you, the reader, to decidewhether or not the motion is carried.

Irrespective of your decision, the author hopes that he hasdemonstrated the enormous power and potential of numericalanalysis. It is by far the best analysis tool that geotechnicalengineers have at their disposal, and its use in the designprocess will increase in the future. However, like all tools, itneeds a skilled operator. In this respect the author believesthat to perform useful numerical analyses (and the emphasisis on ‘useful’) requires:

(a) an in-depth understanding of soil mechanics and thetheory behind numerical analysis

(b) an appreciation of the limitations of constitutive models

Table 6. Material properties for deep excavation benchmark problem

Material E: kPa � �9: degrees c9: kPa K0 ª: kN/m3

Layer 1 20 000 0·3 35 2 0·5 21Layer 2 12 000 0·4 26 10 0·65 19Layer 3 80 000 0·4 26 10 0·65 19Wall (t ¼ 0·8 m) 21 3 106 0·15 – – – 22

Construction sequence:Step 1: Excavation to level �4·0 m.Step 2: Excavation to level �8·0 m with strut 1 (at �3·0 m) active.Step 3: Full excavation to level �12·0 m with struts 1 and 2 (at �3·0 m and �7·0 mrespectively) active.

Towards excavation

Towards retained soil

Participants

0.01

0.005

0.00

�0.05

�0.01

Hor

izon

tal d

ispl

acem

ent

of to

p of

wal

l: m

Fig. 70. Horizontal displacement of the top of the wall, stage 1(after Schweiger, 1998)

Construction stage 3: final excavation,struts 1 and 2 active

2520151050�0.03

0.00

0.03

0.06

Ver

tical

sur

face

dis

plac

emen

t beh

ind

wal

l: m

Distance from wall: m

Fig. 71. Surface displacements of the retained ground behindthe wall, stage 3 (after Schweiger, 1998)

570 POTTS

(c) familiarisation with the software used to perform theanalysis.

At this point those readers who undertake numerical analysismight wish to reflect on how well their own experiencesatisfies these requirements. The benchmarking exercisesimply that many users are not proficient enough. This isperhaps not surprising, as our undergraduate and postgradu-ate courses do not cover all of these requirements insufficient depth. This position must clearly change if numer-ical analysis is to fulfil its enormous potential. Thishas implications for education, the profession and softwaresuppliers.

Ideally, the education system should include more numer-ical analysis and constitutive modelling in the teachingsyllabus. However, this implies a potential increase in theteaching load on the students. As both undergraduate andpostgraduate courses are currently full, this then raises thequestion as to what material can be omitted. Perhaps there isroom to teach less of peripheral subjects, such as manage-ment, or teach less on conventional analysis. Clearly, somedifficult decisions will have to be made, and in this respectacademia will need help from both the profession andindustry.

The profession and industry should provide guidance onbest practice for the use of numerical analysis and provideappropriate training. This must be based on fair and impar-tial advice. It is worth noting that although several advicedocuments exist relating to laboratory and field testing(AGS, 1998, 2000; BSI, 1999; Head, 1992, 1994, 1998;Mair & Wood, 1987; Meigh, 1987), few exist covering bestpractice for numerical analysis. Earlier, while presenting thebenchmark examples, the author referred to the work ofCOST Action C7. Working group A of this Action hasproduced a document entitled Guidelines for the use ofadvanced numerical analysis, which has recently been pub-lished by Thomas Telford Ltd (Potts et al., 2002). Thisdocument does not address all of the problems associatedwith numerical analysis, but it is a start. It was the intentionthat Technical Committee 12 (TC12) of the InternationalSociety of Soil Mechanics and Geotechnical Engineering(ISSMGE) should take this document and improve on it.However, regretably a decision has been made by ISSMGEto disband TC12.

The only other advice documents, apart from textbooks,related to the use of numerical analysis in geotechnicalengineering that the author is aware of are those producedby the German Society for Geotechnics. There are three ofthese, all of which are based on the results of benchmarkingexercises. They cover general recommendations (Meissner,1991), recommendations for numerical simulations in tunnel-ling (Meissner, 1996), and recommendations for deep exca-vations (Meissner, 2002).

It is also imperative that software suppliers provide a fulldescription of how their software works, especially theirnon-linear solution strategy. They should also disclose allapproximations and limitations. It would also be useful ifthey could implement some standard constitutive models.This would, at least in theory, allow users of two differentsoftware programs to analyse the same problem. At presentthe only constitutive models that appear in most geotechnicalsoftware are Tresca and Mohr–Coulomb, although, evenhere, the formulations are not always standard. As alreadynoted, these models are relatively simple and do not fullytest the accuracy of a non-linear solution strategy. Thesemodels also have restrictions and shortcomings, as demon-strated in this lecture. The specification of standard modelsshould ideally be made by some independent body, and as aminimum should include Tresca, Mohr–Coulomb and a

simple critical-state model such as modified Cam clay. Thesemodels should be fully prescribed so that there are noambiguities. The author is not recommending that only thesestandard models be used to analyse geotechnical problems.The reason for including them is so that users of differentsoftware can, in principle, analyse the same problem andconsequently determine where differences, not associatedwith the constitutive model, arise. As more complex modelsbecome accepted, then these too could be included asstandard.

CONCLUSIONSAs noted at the beginning of this lecture, analysis is only

part of the design process. However, it is an important part.The author is in no doubt that, in the future, numericalanalysis will play a pivotal role in this process. With therapid advances in both computer hardware and software thisis likely to be sooner rather than later. The pitfalls discussedin this lecture are not insurmountable. However, more workis needed, and each of us has a role to play. Benchmarkingmust continue and the lessons learnt used to produce guide-lines for best practice. Collection and publication of casehistories are essential to aid the validation process. Acade-mia must adapt its teaching. Site investigation practice andthe form of investigations will have to change so thatrelevant soil parameters can be collected. All too oftenroutine investigations are undertaken with no regard to howthe results may be used in analysis. Even those engineersnot directly involved in performing or checking numericalanalysis have a role in recognising where numerical analysismay bring benefits and cost savings. The end result will besafer, more appropriate and more economical geotechnicalstructures.

ACKNOWLEDGEMENTSThe author wishes to acknowledge the staff and research

students of the Soil Mechanics Section at Imperial Collegefor their support during the preparation of this lecture. Inparticular he would like to thank Dr Lidija Zdravkovic, whoperformed some of the analysis, produced the figures andprovided academic support. He is also grateful to Dr H.Schweiger for permission to use some of the benchmarkingresults, and to Crossrail and the Moor House Partnership forpermission to use the results from the three-dimensionalanalyses of the Moor House site.

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AGS (1998). The selection of geotechnical soil laboratory testing.Association of Geotechnical and Geoenvironmental Specialists,London.

AGS (2000). Guidelines for combined geoenvironmental and geo-technical investigations. Association of Geotechnical and Geoen-vironmental Specialists, London.

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sign, DD ENV 1997-1. Milton Keynes: BSI.British Standards Institution (1999). Code of practice for site

investigation, BS 5930. Milton Keynes: BSI.Burland, J. B. & Potts, D. M. (1994). Development and application

of a numerical model for the leaning tower of Pisa. In Pre-failure deformation of geomaterials (eds S. Shibuya, T. Mitachiand S. Miure), pp. 715–737. Rotterdam: Balkema.

Caquot, A. & Kerisel, J. (1948). Tables for the calculationof passive pressure, active pressure and bearing capacity of


foundations (translated by Ministry of Works, Chief ScientificAdvisors Division, London). Paris: Gauthier-Villars.

Chen, W. F. (1975). Developments in geotechnical engineering, 7,Limit analysis and soil plasticity. Amsterdam, Elsevier.

Coulomb, C. A. (1776). Essai sur une application des regles deminimis a quelques problemes de statique, relatifs a l’architec-ture. Mem. Math. Phys. Acad. R. Sci. 7, 343–382.

Crisfield, M. A. (1986). Finite elements and solution procedures forstructural analysis. Swansea: Pineridge Press.

Gaba, A. R., Simpson, B., Powrie, W. & Beadman, D. R. (2002).Embedded retaining walls: guidance for economic design, RP629. London: Construction Industry Information and ResearchAssociation.

Hansen, J. B. (1970). A revised and extended formula for bearingcapacity, Bulletin No. 28. Danish Geotechnical Institute.

Head, K. H. (1992). Manual of soil laboratory testing, 2nd edn,vol. 1, Soil classification and compaction tests. Copenhagen,Pentech Press.

Head, K. H. (1994). Manual of soil laboratory testing, 2nd edn,vol. 2, Permeability, shear strength and compressibility tests.London, Pentech Press.

Head, K. H. (1998). Manual of soil laboratory testing, 2nd edn,vol. 3, Effective stress tests. London, Pentech Press.

Higgins, K. G., Mair, R. J. & Potts, D. M. (1996). Numericalmodelling of the influence of the Westminster Station excavationand tunnelling on the Big Ben clock tower. In Geotechnicalaspects of underground construction in soft clay (eds R. J. Mairand R. N. Taylor), pp. 525–530. Rotterdam: Balkema.

Lade, P. V. & Duncan, J. M. (1975). Elasto-plastic stress–straintheory for cohesionless soil. J. Geotech. Engng Div., ASCE 101,1037–1053.

Lambe, T. W. (1973). Predictions in soil engineering. Geotechnique23, No. 2, 149–202.

Mair, R. J., Hight, D. W. & Potts, D. M. (1992). Finite elementanalyses of settlements above a tunnel in soft ground, ContractorReport 265. Crowthorne: Transport and Road Research Lab-oratory.

Mair, R. J. & Wood, D. M. (1987). Pressuremeter testing; London:CIRIA-Butterworths.

Matsuoka, H. & Nakai, T. (1974). Stress-deformation and strengthcharacteristics of soil under three different principal stresses.Proc. Jpn Soc. Civ. Engrs 232, 59–70.

Meigh, A. C. (1987). Cone penetration testing. London: CIRIA-Butterworths.

Meissner, H. (1991). Empfehlungen des Arbeitskreises 1·6, ‘Numer-ik in der Geotechnik’, Abschnitt 1, ‘Allgemeine Empfehlungen’.Geotechnik 14, 1–10.

Meissner, H. (1996). Tunnelbau unter Tage-Empfehlungen desArbeitskreises 1·6, ‘Numerik in der Geotechnik’, Abschnitt 2.Geotechnik 19, 99–108.

Meissner, H. (2002). Baugruben-Empfehlungen des Arbeitskreises1·6, ‘Numerik in der Geotechnik’, Abschnitt 3. Geotechnik 25,45–56.

Meyerhof, G. G. (1963). Some recent research on the bearingcapacity of foundations. Can. Geotech. J. 1, No. 1, 16–26.

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Potts, D. M. & Burland, J. B. (2000). Development and applicationof a numerical model for simulating the stabilisation of theLeaning Tower of Pisa. In Developments in theoretical geome-chanics (eds D. W. Smith and J. P. Carter), pp. 737–758.Rotterdam: Balkema.

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VOTE OF THANKSPROFESSOR P. R. VAUGHAN, Imperial College ofScience, Technology and Medicine

It gives me great personal pleasure to propose this vote ofthanks.

Not very long ago I asked Skem what he thought was themost significant development in our subject over his career.Without significant pause for thought he said, ‘numericalanalysis’. When you think about it, this is self-evident. Ourability to synthesise our knowledge of material propertiesand to compare the results with observations of reality haschanged beyond measure.

David Potts came to Imperial College in 1979 and livedjust down the corridor. I watched with fascination his attach-ment to a succession of computers that he purchased atknock-down prices from his own consulting income. One atleast was reputed to have fallen off the back of a lorry.

David brings rare attributes to this new world: a highlyfocused competence in numerical methods themselves, awide knowledge of the mathematical models on which thesemethods feed, and a great interest in finding out how toapply them to real engineering problems. He has shown usthese interests tonight. And I do not forget that he did partof his PhD at Cambridge in the laboratory (or at least in thecentrifuge pit).

This is the 42nd Rankine Lecture. The use of computersin geotechnics started at about the same time as this lectureseries. Finite element analyses started not long after, yet thisis only the second time that we have had a Rankine Lecturelargely or wholly about numerical analysis. Does this tell usthat we are not confident or proficient in our use of thesenew techniques? David has asked us to consider whether we

572 POTTS

get the best out of them. They can turn our calculations intoanalyses, which are so much more powerful and informative.They can tell us what we do not expect. Used intelligently,they will even save us money. However, it has not madeanalysis easier or cheaper or accessible to staff of lowerskill.

I am reliably informed that, when David was interviewedfor his chair, the Rector asked, ‘If your program is soimpressive, why don’t you market it?’ David replied, ‘Comewith me to Heathrow.’ ‘What?’ said the Rector. ‘Come withme to Heathrow,’ said David, ‘I want you to walk across thetarmac, climb into a Jumbo, and sit in the pilot’s seat. Theplane’s yours. Now fly it.’ Maybe one day planes will flyautomatically, but what price someone entering ‘Paris’ in thecomputer, instead of ‘New York’?

The new methods are at least as easy to abuse as the oldones. What do we sometimes do? Go down to the DIY shopand buy a program? Give it to the newest recruit at the backof the drawing office, genuflect towards it once a month,and make our reports look much more expensive by stickinga hundred pages of computer printout in the back, offeringup a prayer the while that no-one will ask us what it means?

Leave the engineering science and the decision-making tothe people at the software house who wrote the program,because we have more important things to do?

The questions David has addressed will not go away.Neither could we make them do so if we wanted. Burningbooks had little effect; neither would burning programs.Numerical analysis is here to stay. Currently we do not doas well as we should. We have an obligation to use the beststate-of-the-art tools to solve our problems, and an obliga-tion to use them effectively. If we do not, the courts maydraw our omission to our attention.

David has shown us that, to use this technique effectively,we need to understand the approximations involved in themethod itself, and how to simulate the behaviour of ourmultiplicity of engineering materials. How do we effectivelyintegrate this technique into our procedures? How do westaff it? How do we educate people to be able to do it? Howdo we pay for it?

So, what are we going to do? David has challenged usto think about it. There is much to think about. Thankyou David, for a most stimulating and thought-provokinglecture.


(David Potts)_Numerical Analysis a Virtual Dream or Practical Reality

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