geotechnical stability analysis

Sloan, S. W. (2013). Geotechnique 63, No. 7, 531–572 [http://dx.doi.org/10.1680/geot.12.RL.001]

531

Geotechnical stability analysis

S. W. SLOAN�

This paper describes recent advances in stability analysis that combine the limit theorems of classicalplasticity with finite elements to give rigorous upper and lower bounds on the failure load. Thesemethods, known as finite-element limit analysis, do not require assumptions to be made about themode of failure, and use only simple strength parameters that are familiar to geotechnical engineers.The bounding properties of the solutions are invaluable in practice, and enable accurate limit loads tobe obtained through the use of an exact error estimate and automatic adaptive meshing procedures.The methods are very general, and can deal with heterogeneous soil profiles, anisotropic strengthcharacteristics, fissured soils, discontinuities, complicated boundary conditions, and complex loadingin both two and three dimensions. A new development, which incorporates pore water pressures infinite-element limit analysis, is also described. Following a brief outline of the new techniques,stability solutions are given for several practical problems, including foundations, anchors, slopes,excavations and tunnels.

KEYWORDS: anchors; bearing capacity; excavation; numerical modelling; plasticity; slopes; tunnels

STABILITY ANALYSISIn geotechnical engineering, stability analysis is used topredict the maximum load that can be supported by ageostructure without inducing failure. This ultimate load,which is also known as the limit or collapse load, can beused to determine the allowable working load by dividing itby a predetermined factor of safety. The precise value of thisfactor depends on the type of problem, with, for example,lower values being appropriate for slopes and higher valuesbeing adopted for foundations. Rather than impose a factorof safety on the ultimate load to obtain the allowable work-ing load, it is also possible to apply a factor of safety to thestrength parameters prior to performing the stability analysis.In some procedures – finite-element strength reductionanalysis, for example – the actual safety factor on thestrength can be found for a given set of applied loads andmaterial parameters (which typically comprise the cohesionand friction angle).

Once the allowable load is known, the working deform-ations are usually determined using some form of settlementanalysis. Historically, these deformations have been predictedusing elasticity theory, but they are now often found from avariety of numerical methods, including non-linear finite-element analysis. In some cases, particularly those involvingdense sands, serviceability constraints on the deformationsmay actually control the allowable load rather than theultimate load-carrying capacity.

Broadly speaking, there are four main methods for per-forming geotechnical stability analysis: limit equilibrium,limit analysis, slip-line methods, and the displacement finite-element method. In the following, all these techniques willbe discussed except the slip-line methods. This family ofprocedures is omitted, not because they are considered to beineffective, but simply because they are not well suited tothe development of general-purpose software which can dealwith a wide variety of practical problems.

Limit equilibriumLimit equilibrium is the oldest method for performing

stability analysis, and was first applied in a geotechnicalsetting by Coulomb (1773). In its most basic form, thisapproach presupposes a failure mechanism, and implicitlyassumes that the stresses on the failure planes are limited bythe traditional strength parameters c and �. The chiefadvantages of the limit-equilibrium method are its simplicityand its long history of use, which have resulted in widelyavailable software and extensive collective experience con-cerning its reliability. Its main disadvantage, on the otherhand, is the need to guess the general form of the failuresurface in advance, with poor choices giving poor estimatesof the failure load. In practice, the correct form of thefailure surface is often not intuitively obvious, especially forproblems with an irregular geometry, complex loading, orcomplicated stratigraphy. There are other shortcomings ofthe technique, as follows.

(a) The resulting stresses do not satisfy equilibrium at everypoint in the domain.

(b) There is no simple means of checking the accuracy of thesolution.

(c) It is hard to incorporate anisotropy and inhomogeneity.(d ) It is difficult to generalise the procedure from two to three

dimensions.

Despite these limitations, a multitude of limit-equilibriummethods have been proposed and implemented, particularlyfor slope stability analysis. Indeed, early examples of widelyused slope stability methods include those of Janbu (1954,1973), Bishop (1955), Morgenstern & Price (1965), Spencer(1967) and Sarma (1973, 1979). A more recent procedure,described by Donald & Giam (1989b), is also noteworthy,since it gives a factor of safety that is a strict upper boundon the true value.

The key principles of the limit-equilibrium approach can beillustrated by considering the classical bearing capacity pro-blem for a smooth strip footing, of width B, resting on a deeplayer of undrained clay of strength su, as shown in Fig. 1.

To begin the analysis, the supposition is made that failureoccurs along a circular surface whose centre lies at somepoint directly above the edge of the footing, as shown inFig. 2. In addition, undrained failure along this surface is

Manuscript received 14 September 2012; revised manuscript accepted22 January 2013.� ARC Centre of Excellence for Geotechnical Science and Engineer-ing, University of Newcastle, NSW, Australia.

assumed to be governed by the Tresca criterion, with themaximum shear strength being fully mobilised at everypoint, so that the shear stress is given by � ¼ su:

Taking moments about the centre of the failure surface,O, the following is obtained.

qBð Þ3B

2¼ 2RŁ 3 suð Þ3 R

or

q ¼ 4suŁ

sin2 Ł(1)

The lowest value of q, and hence the geometry of the criticalsurface, can be found by setting dq/dŁ ¼ 0. This leads to thesimple non-linear equation tan Ł � 2Ł ¼ 0, which can besolved to yield the critical angle Łc ¼ 66.88. Inserting thiscritical angle in equation (1) gives the approximate bearingcapacity as

q ¼ 5:52su

which is approximately 7% above the exact solutionq ¼ (2 + �)su derived by Prandtl (1920).

Limit analysisLimit analysis is based on the plastic bounding theorems

developed by Drucker et al. (1951, 1952), and assumessmall deformations, a perfectly plastic material (Fig. 3(a)),and an associated flow rule (Fig. 3(b)). The last assumption,which is often termed the normality rule, implies that theplastic strain rates _�p

ij are normal to the yield surface, f (�ij),so that _�p

ij ¼ _º@ f =@� ij, where _º is a non-negative plasticmultiplier. For this type of plasticity model it is necessary towork with velocities and strain rates, rather than displace-ments and strains, as the latter become undefined at col-lapse.

The lower-bound theorem is based on the principle of astatically admissible stress field. Such a stress field is definedas one that satisfies equilibrium, the stress boundary condi-

tions and the yield criterion. For a perfectly plastic materialmodel with an associated flow rule, it can be shown that theload supported by a statically admissible stress field is alower bound on the true limit load. Although the limit loadfor such a material is unique, the optimum stress field isnot, and thus it is possible to have a variety of stress fieldsthat furnish the same lower bounds. To illustrate the applica-tion of lower-bound limit analysis, the smooth rigid footingproblem shown in Fig. 1 is considered again. The simplestress field shown in Fig. 4, which consists of three distinctzones separated by two vertical stress discontinuities, isstatically admissible since it satisfies equilibrium, the stressboundary conditions, and the undrained (Tresca) yield criter-ion �1 � �3 ¼ 2su everywhere in the domain. Note that eachstress discontinuity is statically admissible because the nor-mal and shear stresses are the same on both of its opposingsides, and that equilibrium is automatically satisfied every-where in each zone because the unit weight is zero and thestress field is constant. Although the normal and shearstresses must be continuous across an admissible stressdiscontinuity, the normal stress on a plane orthogonal to thediscontinuity is permitted to jump. This feature can beexploited in the construction of stress fields to give usefullower bounds, and is shown in Fig. 4.

Since the stress field supports a vertical principal stress of�1 ¼ 4su in the zone beneath the footing, this defines a lowerbound on the bearing capacity of qlow ¼ 4su:

In contrast to the lower-bound theorem, the upper-boundtheorem requires the computation of a kinematically admis-sible velocity field that satisfies the velocity boundary condi-tions and the plastic flow rule. For such a field, an upperbound on the collapse load is obtained by equating thepower expended by the external loads to the power dissi-pated internally by plastic deformation. Note that althoughthe true limit load from such a calculation is unique, theactual failure mechanism is not. This implies that multiplemechanisms may give the same limit load, and it is neces-sary to seek the mechanism that gives the lowest upperbound. A simple upper-bound mechanism for this stripfooting example, shown in Fig. 5, assumes that failureoccurs by the rigid-body rotation of a circular segment, with

Bearing capacity ?� �q

Saturated clayUndrained shear strength � su

B

Fig. 1. Smooth strip footing on deep layer of undrained clay

0

θ

τ � su

σn

B

R

q

Fig. 2. Limit-equilibrium failure surface for strip footing on clay

σ

ε

σij

σij

f ( ) 0σij �

ε λpij �

∂∂

fσij

. .

(a) (b)

Fig. 3. (a) Perfectly plastic material model and (b) associated flowrule

q slow u4�

σ3 0�

σ1 u2� s

σ1 u4� s

σ3 u2� s σ1 u2� s

σ3 0�

Stress discontinuity

Fig. 4. Lower-bound stress field for strip footing on clay

532 SLOAN

all the internal energy being dissipated along the velocitydiscontinuity. From the geometry of Fig. 5, the rate ofinternal energy (i.e. power) dissipation is

_W int ¼ Pint ¼ð˜ussudL

¼ R _øð Þ3 su 3 2RŁð Þ

where _ø is the angular velocity of the segment about point0, and ˜us is the tangential velocity jump across thediscontinuity. Equating this quantity to the rate of work (i.e.power) expended by the external forces

_W ext ¼ Pext ¼ quppB 3 v

¼ quppB 3 Rc _ø sinŁ

2

� �

¼ quppB 3B

2_ø

and substituting for R gives

qupp ¼4suŁ

sin2 Ł

Setting dqupp/dŁ ¼ 0 furnishes the critical angle Łc ¼ 66.88,which in turn gives the lowest (optimal) upper bound forthis mechanism as

qupp ¼ 5:52su

Combining this result with the previous lower-bound esti-mate, the exact bearing capacity for the footing on the idealmaterial in this analysis must lie within the range

4su < q < 5:52su

Although the limit-equilibrium and upper-bound calculationsgive the same estimate of the bearing capacity for this case,their results are generally different for more complex failuremechanisms where the limit-equilibrium solution may not bekinematically admissible. Notwithstanding the limitationsthat stem from the assumption of a simple perfectly plasticmaterial model, the ability of the limit theorems to providerigorous bounds on the collapse load is one of their greatattractions. Indeed, for complex practical problems where thefailure load is difficult to estimate by other methods, this isa compelling advantage, and one of the few instances innon-linear mechanics where the error in an approximatesolution can be bounded exactly.

Although the limit theorems can be applied in an analy-tical setting to give useful bounds for simple problems,discrete numerical formulations provide a more generalmeans of harnessing their power. In particular, finite-elementlimit analysis formulations have evolved rapidly in recentyears, and are now sufficiently developed for large-scalepractical applications in geotechnical engineering. These

procedures will be the focus of attention in this paper, andinevitably lead to some form of optimisation problem, thesolution of which defines either a statically admissible stressfield or a kinematically admissible velocity field. Finite-element formulations of the limit theorems inherit all theadvantages of the finite-element method, and can modelcomplex geometries, layered soils, anisotropy, soil–structureinteraction, interface effects, discontinuities, complicatedloadings, and a wide variety of boundary conditions. Thesuccess of this approach, however, hinges on the develop-ment of formulations and solution algorithms that are robust,efficient and extendable to three dimensions. Moreover,some means of refining the mesh is needed to ensure thatthe ‘gap’ between the upper- and lower-bound limit loads issufficiently small.

Displacement finite-element analysisAs a result of the rapid evolution of powerful user-friendly

software, displacement finite-element analysis is now widelyused in geotechnical practice – not only for the predictionof deformations, but also for the prediction of stability. Thismethod is very general, and can accommodate advancedconstitutive models that incorporate non-associated flow,heterogeneity, anisotropy, and work/strain-hardening andsoftening. In addition, robust procedures are available formodelling interface behaviour, soil–structure interaction andlarge deformations, as well as fully coupled consolidationand dynamics. When it is used to predict stability understatic loading, displacement finite-element analysis can beused in two different modes.

(a) The loads are applied in increments until the deformationresponse indicates that a state of collapse has beenreached. The approximate ultimate load so obtainedfurnishes a safety factor in terms of force, not strength,and requires a complete simulation of the load–deformation response (e.g. Sloan, 1979, 1981; Toh &Sloan, 1980; Sloan & Randolph, 1982; De Borst &Vermeer, 1984). Unless advanced procedures (such asarc-length methods) are used, this approach leads toinstability in the calculations at collapse if the problem isloaded by prescribed forces rather than by prescribeddisplacements.

(b) Successive analyses with reduced strengths are conducteduntil equilibrium can no longer be maintained (e.g.Zienkiewicz et al., 1975; Dawson et al., 1999; Griffiths &Lane, 1999). This approach, known as strength reductionanalysis, involves monitoring deformations at specifiedcontrol points in the soil, and gives the safety factor interms of strength – much like the safety factor that iscomputed in traditional slope stability calculations usinglimit equilibrium. Since the method relies on non-convergence of the finite-element simulations to indicatefailure, considerable care must be exercised to ensure thatthe non-convergence is caused by genuine collapse, andnot some other numerical effect.

Figure 6 shows the load–deformation response for a smooth,rigid strip footing on clay, computed using the displacementfinite-element program SNAC (Abbo & Sloan, 2000), for asoil with a rigidity index G/su ¼ 100 and undrained Poisson’sratio �u ¼ 0.49. In this example the rigid foundation issimulated by the application of uniform vertical displace-ments to nodes underneath the footing, and 15-noded (quar-tic) triangles are used to ensure that the soil deformationsare modelled accurately under incompressible conditions(Sloan, 1979, 1981; Sloan & Randolph, 1982). The finite-element program SNAC, developed at The University ofNewcastle over the past two decades, employs adaptive

v

ω.θ/2

θR BR B

sin/2sin( /2)

��

θθc

θ/2 θ/2qupp

R

B Rigid

Rigid zone, zero velocity

Rc

Circular failuresurface

0

Fig. 5. Upper-bound failure mechanism for strip footing on clay

GEOTECHNICAL STABILITY ANALYSIS 533

explicit methods to integrate the stress–strain relations andload–deformation response to within a specified accuracy,and is thus well suited to collapse predictions (Sloan, 1987;Abbo & Sloan, 1996; Sheng & Sloan, 2001; Sloan et al.,2001). For the mesh shown, the displacement finite-elementanalysis indicates a clear collapse pressure of 5.19su, whichis within 1% of Prandtl’s exact result of (2 + �)su: Unlikethe methods discussed previously, stability analysis with thedisplacement finite-element method requires not only theconventional strength parameters, but also the deformationparameters (Poisson’s ratio and shear modulus in this case).

Displacement finite-element analysis computes the form ofthe failure mechanism automatically, and can model a varietyof complicated loadings and boundary conditions. Themethod is not for the naıve user, however, and even with theadvent of sophisticated geotechnical software considerablecare and experience are required to use the procedure withconfidence in geotechnical practice (Potts, 2003). Since adisplacement finite-element solution satisfies equilibrium andthe flow rule only in a ‘weak’ sense over the domain, thequality of the resulting collapse load prediction is oftencritically dependent on the mesh adopted. Sensitivity studies,using successively finer meshes, are generally advisable toconfirm the accuracy of the computed limit load, since noreliable error estimate is available for the elasto-plasticmodels commonly used in geotechnical analysis. In addition,the accuracy of the limit load can be affected by the numberof load steps used in the analysis (Sloan, 1981; Abbo &Sloan, 1996; Sheng & Sloan, 2001), the numerical integra-tion scheme used to evaluate the elasto-plastic stresses (Potts& Gens, 1985; Sloan, 1987; Sloan et al., 2001), the toler-ances used to check convergence of the global equilibriumiterations, and the type of element employed (Nagtegaal etal., 1974; Sloan, 1979, 1981; Toh & Sloan, 1980; Sloan &Randolph, 1982). Of these factors, the correct choice ofelement is particularly crucial for stability analysis, since theincompressibility constraint imposed by undrained analysismay lead to ‘locking’ where the load–deformation responserises continuously with increasing deformation, regardless ofthe mesh discretisation adopted. This phenomenon is due toconstraints on the nodal displacements, generated by theincompressibility condition, multiplying at a faster rate thanthe degrees of freedom as the mesh is refined, and it isespecially pronounced for axisymmetric loading with low-order elements (such as the linear three-noded triangle and

four-noded quadrilateral). Locking can also occur for dis-placement finite-element analysis with the Mohr–Coulombmodel, which involves dilatational plastic shearing and iswidely used for drained stability predictions (Sloan, 1981).To ensure that an element is suitable for accurate collapseload predictions, under both undrained and drained condi-tions, three different strategies have been proposed.

(a) The use of ‘reduced’ integration in forming the elementstiffness matrices (e.g. Zienkiewicz et al., 1975; Zienkie-wicz, 1979; Griffiths, 1982). This approach, which hasbeen widely used with the quadratic eight-noded quad-rilateral, reduces the number of constraints on the nodaldegrees of freedom at collapse, and introduces additional‘flexibility’ into the displacement field by approximatenumerical integration of the element stiffness matrices. Ingeneral, the method gives good estimates of the collapseload, but may generate unrealistic deformation patternsfor some problems (Sloan, 1983; Sloan & Randolph,1983). Selective integration methods, which under-integrate the volumetric stiffness terms while fullyintegrating the deviatoric stiffness terms, may also beused in some cases to alleviate the problem of locking forlow-order elements (Malkus & Hughes, 1978).

(b) The use of high-order triangular elements, with fullintegration of the stiffness matrices. This approach, firstadvocated by the author (Sloan, 1979, 1981; Sloan &Randolph, 1982), follows from the observation that, asmeshes of high-order triangles are refined, the newdegrees of freedom are added at a faster rate than thenodal constraints imposed by the incompressibilitycondition, thus avoiding the problem of locking. Sincethese elements use full integration, no difficulties areencountered with spurious deformation patterns.Although a variety of triangular elements can be shownto be suitable for geotechnical stability analysis, the 15-noded triangle, with a quartic displacement expansion,gives good collapse load predictions under both plane-strain and axisymmetric loading. This element is alsohighly effective for drained stability applications invol-ving dilatational plasticity models, and can be imple-mented to give efficient run times (Sloan, 1979, 1981;Sloan & Randolph, 1982). For plane-strain deformation,which generates fewer constraints than axisymmetricdeformation, the six-noded quadratic triangle with full

(2 )� π su

G s/ 1000·490

u

u

u

��

νφ

15141312111098761 2 3 4 50

1

2

3

4

5

6

0

Pre

ssur

e/s u

100 (footing displacement)/B

48 quartic triangles825 degrees of freedom

Smooth

Sm

ooth

Sm

ooth

Prescribed displacement analysis

Smooth footing

Fig. 6. Displacement finite-element analysis of strip footing on clay

534 SLOAN

integration is a viable alternative to the 15-noded triangle,and gives reliable estimates of the collapse load.

(c) The use of mixed pressure–displacement formulations.To avoid numerical oscillations in the solutions, theseelements traditionally use a pressure expansion that is oneorder lower than the displacement expansion (e.g. a six-noded quadratic displacement triangle with a linearpressure variation interpolated at the corner nodes), butthey can also be used in a ‘stabilised’ form where thepressure and displacement expansions are of equal order(Pastor et al., 1997, 1999). Although they appear to givegood results, these formulations are more complicatedthan the previous two options, and have not been widelyadopted for geotechnical stability analysis.

In geotechnical applications, undrained and drained stabilityanalyses can be performed as limiting cases of fully coupledBiot consolidation, with the former case corresponding to avery fast loading rate and the latter case corresponding to avery slow loading rate. Interestingly, when using this ap-proach for stability calculations with a Mohr–Coulomb yieldcriterion, a non-associated flow rule with a zero (or small)dilation angle should be used to obtain realistic estimates ofthe collapse load (Small et al., 1976; Small, 1977; Sloan &Abbo, 1999). If a finite dilation angle is adopted, the load–deformation response will display a hardening characteristicand fail to asymptote towards a clear collapse state.

Comparison of methods for stability analysisTable 1 summarises the key features of the limit equili-

brium, limit analysis and displacement finite-element ap-proaches for assessing geotechnical stability. Clearly, thelimit-equilibrium method has shortcomings, some of whichwill be explored further in a later section of this paper, whilethe displacement finite-element method is the most general.Conventional limit analysis has the intrinsic advantage ofproviding solutions that bound the collapse load from aboveand below, but it is restricted to the use of simple soil modelsand is often difficult to apply in practice. The results in Table1 suggest that finite-element limit analysis, which combinesthe generality of the finite-element approach with the rigourof limit analysis, is an appealing alternative to traditionalstability prediction techniques. The potential of this type ofmethod will be explored fully in this paper, with a particularfocus on its practical utility and scope for future development.

FINITE-ELEMENT LIMIT ANALYSISThe theory of finite-element limit analysis is quite differ-

ent from that of displacement-based finite-element analysis,even though both methods are rooted in the concept of adiscrete formulation. Before discussing the fundamental de-tails of finite-element limit analysis, a brief historical review

of its development will be given. This review serves tohighlight some of the advantages and drawbacks of theapproach, as well as its application to practical examples.

Historical development of finite-element lower-bound analysisLysmer (1970) was an early pioneer in applying finite

elements and optimisation theory to compute rigorous lowerbounds for plane-strain geotechnical problems. Lysmer’sformulation was based on a linear three-noded triangle, withthe unknowns being the normal stresses at the end of eachside, plus another ‘internal’ normal stress, and he employedlinear programming to solve the resulting optimisation pro-blem. To satisfy the Mohr–Coulomb yield function in itsnative form, the Cartesian stresses at every point in anelement must satisfy a non-linear (quadratic) inequalityconstraint. To avoid this type of constraint, and thus generatea linear programming problem, Lysmer (1970) linearised theyield surface using an internal polyhedral approximation thatreplaced each non-linear yield inequality constraint by aseries of linear inequalities. The accuracy of the resultinglinearisation can be controlled by varying the number ofsides in the polyhedral approximation, with the highestaccuracy being obtained at the cost of additional constraintsand increased solution times. Because the stress field insideeach element is assumed to vary linearly, it is sufficient toimpose these inequalities at each node to ensure that thelinearised yield condition is satisfied throughout the domain,thereby satisfying a key condition of the lower-bound theo-rem. In addition to the triangular elements used for model-ling the soil, Lysmer’s formulation also included staticallyadmissible stress discontinuities along the edges betweenadjacent elements. These greatly enhance the accuracy of afinite-element lower-bound formulation, especially when sin-gularities are present in the stress field (such as at the edgeof a rigid footing), and feature prominently in most subse-quent implementations of the method. Application of theelement equilibrium equations, the discontinuity equilibriumequations and the stress boundary conditions leads to a setof equality constraints on the unknown stresses, while, asdescribed above, the linearised yield criterion generates alarge set of linear inequality constraints. The objective func-tion, which corresponds to the collapse load, is a linearfunction of the stresses. After assembling all the elementand nodal contributions for the mesh, the collapse load,denoted by the quantity cT�, is maximised by solving alinear programming problem of the form

Maximise cT� collapse load

subject to A1� ¼ b1 continuum and discontinuity

equilibrium, stress boundary

conditionsA2� < b2 linearised yield conditions

(2)

Table 1. Properties of traditional methods used for geotechnical stability analysis

Property Limitequilibrium

Upper-bound limitanalysis

Lower-bound limitanalysis

Displacement finite-elementanalysis

Assumed failure mechanism? Yes Yes – NoEquilibrium satisfied everywhere? No (globally) – Yes No (nodes only)Flow rule satisfied everywhere? No Yes – No (integration points only)Complex loading and boundary conditionspossible?

No Yes Yes Yes

Complex soil models possible? No No No YesCoupled analysis possible? No No No YesError estimate? No Yes (with lower bound) Yes (with upper bound) No


where c, b1 and b2 are vectors of constants; A1 and A2 arematrices of constants; and � is a global vector of unknownnormal and ‘internal’ stresses acting on the element edges.

Although Lysmer’s finite-element approach for computinglower bounds was a pivotal conceptual advance, it has threesignificant limitations that prevented it from being usedwidely in practice. The first of these stems from the choiceof variables used in the formulation, which leads to a poorlyconditioned system of constraint equations that is highlysensitive to the shape of the elements in the mesh. Thesecond shortcoming of the method is its computationalinefficiency, which follows from the use of the simplexalgorithm to solve the linear programming problem definedby equation (2). Since the iterations required by this algo-rithm grow rapidly with the size of the optimisation problembeing tackled, the number of elements that can be used in amesh is severely restricted. The third limitation of theformulation is that it does not include a strategy for ‘extend-ing’ the stress field over a semi-infinite domain so that theequilibrium, stress boundary and yield conditions are satis-fied everywhere. This process, also known as ‘completing’the stress field, is necessary for the solution to be classed asa rigorous lower bound.

Following Lysmer’s seminal work, Anderheggen & Knop-fel (1972), Pastor (1978) and Bottero et al. (1980) proposedvarious discrete methods for two-dimensional lower-boundlimit analysis that were all based on linear triangles andlinear programming. These procedures introduced a numberof key improvements, including the use of Cartesian stressesas problem variables to simplify the formulation, and thedevelopment of special extension elements for generatingcomplete solutions in semi-infinite media. Soon after, Pastor& Turgeman (1982) proposed a lower-bound technique formodelling the important case of axisymmetric loading.Although potentially powerful, these early methods werelimited by the computational performance of the linearprogramming codes at the time, and could solve only rel-atively small problems. Indeed, the practical utility of dis-crete limit analysis techniques has been strongly linked tothe development of efficient algorithms for solving theassociated optimisation problems. These problems have spe-cial features, including extremely sparse and unsymmetricconstraint equations, which must be exploited fully in orderto solve large cases efficiently. In an effort to address thisissue, Sloan (1988a, 1988b) proposed a fast linear program-ming formulation that can solve small- to medium-scaletwo-dimensional problems on a standard desktop machine.This procedure is based on a novel active set algorithm,which employs a steepest-edge search in the optimisationiterations, and fully exploits the highly sparse nature of thelower-bound constraint matrix. The method has been usedsuccessfully to predict the stability of a wide variety of two-dimensional problems, including tunnels (Assadi & Sloan,1991; Sloan & Assadi, 1991, 1992), slopes (Yu et al., 1998),foundations (Ukritchon et al., 1998; Merifield et al., 1999),anchors (Merifield et al., 2001, 2006a), braced excavations(Ukritchon et al., 2003), and longwall mine workings (Sloan& Assadi, 1994).

Although lower-bound methods based on linear program-ming are capable of providing useful solutions for two-dimensional problems of moderate size, they are poorlysuited to three-dimensional analysis, as huge numbers ofinequalities arise when the yield criterion is linearised.Moreover, it is not always clear how to linearise a three-dimensional yield surface in an optimal manner. Both ofthese issues can be avoided by leaving the yield constraintsin their native form and adopting non-linear programmingalgorithms to solve the resulting optimisation problem. In-deed, with this approach, three-dimensional formulations

present no special difficulties, other than adding geometricalcomplexity and increasing the number of unknowns. Anearly discrete lower-bound formulation based on non-linearprogramming was described in Belytschko & Hodge (1970).This procedure used piecewise-quadratic equilibrium stressfields, and maximised the collapse load, subject to the non-linear yield constraints, by means of a sequential uncon-strained minimisation technique. Although it furnishes rigor-ous lower bounds, the method proved to be slow for large-scale problems. In a subsequent modification of Lysmer’sformulation, Basudhar et al. (1979) incorporated the non-linear yield constraints directly, converted the constrainedoptimisation problem to an unconstrained one using theextended penalty method of Kavlie & Moe (1971), andcomputed the optimal solution (best lower bound) using avariant of the sequential unconstrained minimisation tech-nique (Powell, 1964). Following this work, Arai & Tagyo(1985) used constant-stress elements, and the sequentialunconstrained minimisation technique with the conjugategradient algorithm of Fletcher & Reeves (1964), to obtain astatically admissible stress field for geotechnical problems.Although both these non-linear formulations require only amodest number of inequality constraints to ensure that thestresses satisfy the yield criterion, they still proved unsuita-ble for large-scale geotechnical problems, owing to thecomputational inefficiency of the methods employed to solvethe corresponding optimisation problem.

Lyamin (1999) and Lyamin & Sloan (2002a) dramaticallyimproved the practical utility of the discrete lower-boundmethod by employing linear stress elements, imposing thenon-linear yield conditions in their native form, and solvingthe resulting non-linear optimisation problem using a variantof an algorithm developed for mixed limit analysis formula-tions (Zouain et al., 1993). After assembling all the elementand nodal contributions, the load carried by the unknownstresses and body forces, denoted by cT

1� and cT2 h respec-

tively, is maximised by solving the following non-linearprogramming problem

Maximise

cT1� þ cT

2 h collapse load or body force

subject to

A11� þ A12h ¼ b1 continuum equilibrium

A2� ¼ b2 discontinuity equilibrium,

stress boundary conditions

f (� i) < 0 yield conditions for each node i

(3)

where c1, c2, b1 and b2 are vectors of constants; A11, A12

and A2 are matrices of constants; f is the non-linear yieldcriterion; � i is a local vector of Cartesian stresses at node i;� is a global vector of unknown Cartesian stresses; and h isa global vector of unknown body forces acting on eachelement. Including the body forces in the formulation per-mits stability numbers based on the unit weight to beoptimised, and is especially useful in predicting the loadcapacity of slopes, tunnels and excavations. The solutionmethod used by Lyamin & Sloan (2002a) is an interiorpoint, two-stage, quasi-Newton scheme that exploits theunderlying structure of the lower-bound optimisation pro-blem. Since its iteration count is largely independent of thegrid refinement for a given problem, the method is able tohandle large-scale two-dimensional meshes with severalthousand elements in a few seconds, and is many timesfaster than traditional linear programming formulations. Thedetailed timing comparisons presented by Lyamin & Sloan

536 SLOAN

(2002a) suggest that, compared with the linear programmingapproach of Sloan (1988a), their technique typically gives atleast a 50-fold reduction in CPU time for large two-dimen-sional problems. Further advantages include the ability tomodel three-dimensional problems, where the number ofunknowns can be huge, as well as any type of convex yieldcriterion. Thanks to its efficiency and robustness, the lower-bound method of Lyamin & Sloan (2002a) has been used topredict the stability of a wide range of geotechnical pro-blems, including tunnels (Lyamin & Sloan, 2000), sinkholesand cavities (Augarde et al., 2003a, 2003b), two- and three-dimensional foundations on clay and/or sand (Shiau et al.,2003; Hjiaj et al., 2004, 2005; Salgado et al., 2004), anchorsin clay or sand (Merifield et al., 2003, 2005, 2006a),foundations on rock (Merifield et al., 2006b), and slopes insoil or rock (Li et al., 2008, 2009a, 2009b, 2010). Followingthe work of Lyamin & Sloan (2002a), Krabbenhøft &Damkilde (2003) proposed another efficient lower-boundmethod, aimed primarily at solving structural engineeringproblems, based on non-linear programming.

Owing to the presence of singularities in their yieldsurfaces, where the gradients with respect to the stressesbecome undefined, the Tresca and Mohr–Coulomb criteriapose special difficulties in finite-element limit analysis.Lyamin & Sloan (2002a) overcame this difficulty by localsmoothing of the yield surface vertices, with an accompany-ing modification to the search direction to preserve feasibil-ity during the optimisation iterations. An attractivealternative method for solving lower-bound limit analysisproblems, which does not require differentiability of theyield surface in the optimisation process, is to use second-order cone programming (Ciria, 2004; Makrodimopoulos &Martin, 2006). This solution method can be applied to avariety of yield criteria in two dimensions, including theTresca and Mohr–Coulomb models, and has proved to berobust and efficient for large-scale geotechnical problems(Krabbenhøft et al., 2007). In three-dimensional cases, sec-ond-order cone programming can be used for Von Mises andDrucker–Prager yield criteria, but not for Tresca or Mohr–Coulomb models. For the latter, which are of particularinterest in geotechnical applications, it is possible to use adifferent cone-based solution algorithm that is known assemi-definite programming (Krabbenhøft et al., 2008). Likethe second-order cone programming method, this approachdoes not require smoothing of any yield surface vertices,and it has proved to be both robust and efficient for large-scale applications (Krabbenhøft et al., 2008). In summary,the second-order cone programming and semi-definite pro-gramming methods are, respectively, the solution methods ofchoice for the Tresca/Mohr–Coulomb models under two-and three-dimensional conditions. For yield criteria that arecurved in the meridional plane, however, such as the Hoek–Brown model for rock, these procedures are inapplicable,and the more general interior point solution algorithmproposed by Lyamin & Sloan (2002a) is appropriate.

Historical development of finite-element upper-bound analysisEarly discrete formulations of the upper-bound theorem,

based on finite elements and linear programming, wereproposed by Anderheggen & Knopfel (1972) and Maier etal. (1972). Although quite general, these methods wereconcerned primarily with structural applications. The subse-quent plane-strain procedures of Pastor & Turgeman (1976)and Bottero et al. (1980), which focused on geotechnicalapplications with Tresca and Mohr–Coulomb yield criteria,permit a limited number of velocity discontinuities to occurbetween elements, but require the direction of shearing to bespecified a priori. These formulations assume a piecewise

linear velocity field using three-noded triangles, with eachnode having two unknown velocities, and each elementbeing associated with a fixed number of unknown plasticmultiplier rates. To ensure the solution is kinematicallyadmissible, the velocities and plastic multiplier rates mustsatisfy a set of linear constraints arising from the flow rule,with the former unknowns also being subject to the appro-priate boundary conditions. For a given set of prescribedvelocities, the finite-element formulation optimises the velo-cities and plastic multiplier rates to minimise the powerdissipated internally minus the rate of work done by fixedexternal forces. Once this quantity is known, it can beequated to the power expended by the external loads tofurnish a strict upper bound on the true limit load. Togenerate a linear programming problem with an upper-boundfinite-element formulation, it is again necessary to linearisethe yield criterion. The polyhedral approximation must beexternal to the parent yield surface to ensure a rigorousupper bound, and each face of the linearised surface isassociated with a single plastic multiplier. After assemblingall the element and nodal contributions, the power dissipa-tion in the triangles and the discontinuities, denoted by thequantities cT

1 u and cT2

_º, minus the rate of work done by anyfixed external forces, denoted by cT

3 u, is minimised bysolving a linear programming problem of the form

Minimise

cT1 u þ cT

2_º � cT

3 u power dissipation minus rate of

work done by fixed external forces

subject to

A11u þ A12_º ¼ 0 continuum flow rule

A2u ¼ 0 discontinuity flow rule

A3u ¼ b3 velocity boundary conditions

A4u < 0 discontinuity signs

_º > 0 plastic multiplier

where c1, c2, c3 and b3 are vectors of constants; A11, A12,A2, A3 and A4 are matrices of constants; u is a globalvector of nodal velocities; and _º is a global vector ofelement plastic multipliers.

Following these early procedures that focused on planeproblems, Turgeman & Pastor (1982) extended the upper-bound formulation of Bottero et al. (1980) to handleaxisymmetric geometries, but only for Von Mises and Trescamaterials.

While the above upper-bound methods inherit all the keyadvantages of the finite-element technique, and hence canmodel complex problems in two dimensions, they were notwidely applied in practice because of the CPU time requiredto solve their associated linear programming problems. In aneffort to rectify this handicap, Sloan (1989) proposed anupper-bound method based on the steepest-edge active setsolution scheme (Sloan, 1988b), which had proved successfulfor lower-bound limit analysis. Although it still suffered fromthe shortcoming of having to specify the direction of shearingalong the velocity discontinuities a priori, the resulting meth-od was subsequently used to generate useful upper bounds fora variety of underground structures including trapdoors (Sloanet al., 1990) and tunnels (Assadi & Sloan, 1991; Sloan &Assadi, 1991, 1992, 1994). Owing to the nature of thealgorithm used to solve the associated linear optimisationproblem, however, the procedure proved to be inefficient forlarge-scale examples involving thousands of elements.

Most early discrete formulations of the upper-bound theo-rem employed the three-noded triangle with a linearised yieldfunction, since this leads to an optimisation problem with


linear constraints where the power dissipation can be ex-pressed solely in terms of the element plastic multipliers. Byusing an element with a constant-strain field, it is sufficientto enforce the flow rule over each triangle to define akinematically admissible velocity field. Additional flow ruleconstraints, of course, are needed to define kinematicallyadmissible velocity jumps across each discontinuity. If dis-continuities are not included in a mesh of three-nodedtriangles, the elements should be arranged so that fourtriangles form a quadrilateral, with the central node lying atthe intersection of the diagonals. Failing to observe this rulefor undrained (incompressible) problems may lead to ‘lock-ing’, where the elements cannot provide enough degrees offreedom to satisfy the constant-volume condition (Nagtegaalet al., 1974). In response to this shortcoming, Yu et al.(1994) developed a six-noded linear strain triangle for upper-bound limit analysis. This element can model a velocity fieldaccurately with fewer elements than the constant-strain trian-gle and, in the absence of discontinuities, no special gridarrangement is required for incompressible deformation.

The need to specify both the location and the direction ofshearing for each discontinuity in an upper-bound analysis isa significant drawback, since it requires a good guess of thelikely collapse mechanism in advance. This shortcoming wasaddressed by Sloan & Kleeman (1995), who generalised theupper-bound formulation of Sloan (1989) to include velocitydiscontinuities at all edges shared by adjacent triangles. Intheir formulation, the direction of shearing is found as partof the optimisation process, and discontinuities are eitheractive or inactive, depending on which deformation patterngives the least amount of dissipated power. Each discontinu-ity is defined by four nodes, and requires four additionalplastic multipliers to describe the normal and tangentialvelocity jumps along its length. The upper-bound procedureof Sloan & Kleeman (1995) assumes a linearised yieldcriterion, and gives rise to a linear programming problemthat can be solved using the active set solution algorithm ofSloan (1988b). It has proved to be computationally efficientfor small- to medium-scale problems in two dimensions,and, because of the presence of velocity discontinuities at allshared element edges, gives good estimates of the limit loadwithout the need for special grid arrangements. Exampleswhere the method has provided useful upper bounds includeslopes (Yu et al., 1998), foundations (Ukritchon et al., 1998;Merifield et al., 1999), anchors (Merifield et al., 2001,2006a), and braced excavations (Ukritchon et al., 2003).

It is not straightforward to develop discontinuous upper-bound formulations that can model an arbitrary yield condi-tion. This is because the internal power dissipation dependson the state of stress as well as on the strain rates, so that inaddition to finding the velocity and plastic multiplier fieldsthat satisfy the flow rule, it is also necessary to compute astress field that satisfies the yield criterion. Moreover, kine-matically admissible discontinuities are difficult to incorpo-rate at all inter-element edges in three dimensions.

The plate formulation described by Hodge & Belytschko(1968) was one of the first attempts to develop a finite-element upper-bound method based on non-linear program-ming. Their analysis used classical theory to specify thedeformation field solely by the velocity normal to the originalmiddle surface of the plate. This normal velocity wasapproximated within each element by a second-order poly-nomial, composed of independent nodal parameters, and thetotal internal power included contributions from plastic defor-mation through the elements, across hinge lines betweenelements and along clamped boundaries. The resulting uncon-strained optimisation problem requires the ratio of the inter-nal and external energy dissipation rates to be minimised, andwas solved using the simplex method of Nelder & Mead

(1965). Hodge & Belytschko (1968) reported slow conver-gence of the procedure, owing to the complex nature of theobjective function. Following this initial work, various othernon-linear programming formulations were proposed for com-puting upper bounds on the load capacity of plates, shells andstructures (Biron & Charleux, 1972; Nguyen et al., 1978).

Huh & Yang (1991) developed a general upper-boundprocedure for plane stress problems using triangular elementswith a linear velocity field. Their method focused on a so-called ‘�-norm’ family of yield criteria, which includes thevon Mises model as a special case, and the results suggestthat it is accurate and efficient for relatively large two-dimensional problems. In a further development, Capsoni &Corradi (1997) proposed another discrete upper-bound ap-proach where the straining modes are modelled indepen-dently of rigid-body motions. This allows finite elements thatare not involved in the collapse mechanism to be omittedfrom the dissipated power summation, and avoids problemswith non-differentiability of the upper-bound functional.

In a different non-linear approach, Jiang (1994) proposedan upper-bound formulation, based on a regularised modelof limit analysis (Friaa, 1979), which assumes the materialis visco-plastic, and uses two parameters to characterise itscreep behaviour. By fixing the first of these parameters tounity, and letting the second one tend to infinity, it can beshown that the visco-plastic power dissipation converges tothe plastic power dissipation, and a rigorous upper bound isobtained. Although this is an indirect method, the visco-plastic functional is always convex, even for three-dimen-sional Mohr–Coulomb and von Mises yield criteria, andthere is always a unique solution that minimises it. To solvethe resulting non-linear optimisation problem, Jiang (1994)employed the augmented Lagrangian method in conjunctionwith the algorithm of Uzawa (Fortin & Glowinski, 1983). Ina later paper, Jiang (1995) established that the same non-linear programming scheme can be applied to performupper-bound limit analysis directly. Jiang’s formulations per-form well for a variety of two-dimensional examples, buthave not been extended to deal with discontinuities in thevelocity field or three-dimensional geometries. Parallel tothis development, Liu et al. (1995) proposed a direct itera-tive method for performing three-dimensional upper-boundlimit analysis. This scheme treats the rigid zones separatelyfrom the plastic zones during each iteration, and neatlyavoids the numerical difficulties that stem from a non-differentiable objective function in the former. Their papersuggests that the process is efficient and numerically stable,and can be implemented easily in an existing displacementfinite-element code. It has not, however, been widely used togenerate rigorous upper bounds for geotechnical problems.

Following in the footsteps of their successful lower-boundformulation, Lyamin & Sloan (2002b) developed an upper-bound finite-element method that was also based onnon-linear programming. This procedure assumes that thevelocities vary linearly over each element, and that eachelement is associated with a constant-stress field and a singleplastic multiplier rate. Flow rule constraints are imposed onthe nodal velocities, element plastic multipliers and elementstresses to ensure that the solution is kinematically admis-sible. In addition, to satisfy the consistency condition, theelement stresses are constrained to obey the yield criterion,and the plastic multipliers are constrained to be non-negative.Using the approach developed in Sloan & Kleeman (1995),the formulation of Lyamin & Sloan (2002b) allows velocitydiscontinuities along shared element edges, with the velocityjumps across each discontinuity being defined by additionalnon-negative unknowns (plastic multipliers). Their procedureappears to be the first rigorous upper-bound method thatincorporates both continuum and discontinuity deformation

538 SLOAN

in two and three dimensions. Although the yield behaviour inthe discontinuities is restricted to models with a linear yieldenvelope (e.g. Tresca and Mohr–Coulomb), it is otherwisequite general. The resulting optimisation problem can besolved in terms of the nodal velocities and element stressesalone by applying a two-stage, quasi-Newton algorithmdirectly to the Kuhn–Tucker optimality conditions (Lyamin& Sloan, 2002b). Consequently, the element plastic multi-pliers do not need to be included explicitly as variables. Thisformulation has been used to compute accurate upper boundsfor a wide range of important geotechnical problems, includ-ing sinkholes (Augarde et al., 2003a), tunnels (Lyamin &Sloan, 2000), mines (Augarde et al., 2003b), foundationbearing capacity (Shiau et al., 2003; Hjiaj et al., 2004, 2005;Salgado et al., 2004) and anchors (Merifield et al., 2003).

Krabbenhøft et al. (2005) modified the upper-bound for-mulation of Lyamin & Sloan (2002b) by proposing a newstress-based method that uses patches of continuum elementsto incorporate velocity discontinuities in two and threedimensions. The elements in these patches have zero thick-ness, with opposing nodal pairs having the same coordinates,and the scheme results in a simple and efficient structure forprogramming. Moreover, the procedure can accommodateyield criteria that have curved envelopes, such as the Hoek–Brown model. Interestingly, the same idea can also be usedto incorporate stress discontinuities in discrete formulationsof lower-bound limit analysis (Lyamin et al., 2005a).

To avoid the problems associated with non-smooth yieldsurfaces in the optimisation process, second-order coneprogramming can be used to solve discrete formulations ofthe upper-bound theorem (Ciria, 2004; Makrodimopoulos &Martin, 2007). This class of solution scheme is highly effec-tive for lower-bound formulations, as discussed previously,and has proved to be equally effective for upper-boundformulations. Indeed, second-order cone programming is themethod of choice for solving the optimisation problems thatarise from finite-element upper-bound formulations, providedthe yield function can be expressed in a conic quadraticform (such as the von Mises/Drucker–Prager model in planestrain or three dimensions, and the Tresca/Mohr–Coulombmodel in plane strain). For the Tresca/Mohr–Coulomb mod-el in three dimensions, the resulting upper-bound optimisa-tion problems can be solved efficiently using semi-definiteprogramming, just as in the lower-bound case.

FINITE-ELEMENT LOWER-BOUND FORMULATIONAn efficient formulation of the lower-bound method will

now be briefly described. This section follows the formula-

tion of Lyamin & Sloan (2002a), with some importantmodifications to handle stress discontinuities, and highlightsthe fundamental differences between finite-element limitanalysis and displacement finite-element analysis. It alsoillustrates the power of using a discrete formulation of theclassical limit theorems.

Figure 7 shows a soil mass, with volume V and boundaryarea A, subject to a set of fixed surface stresses (tractions) tacting on the boundary At, as well as an unknown set oftractions q acting on the boundary Aq: In practice t mightcorrespond, for example, to a prescribed surcharge while qmight correspond to an unknown bearing capacity. Alsoshown in Fig. 7 is a system of fixed body forces g andunknown body forces h acting over the volume V. Theformer is typically a prescribed unit weight, while the latter,which corresponds to an unknown body force capacity, willbe shown to be very useful in computing the stability ofslopes, tunnels and excavations.

Recalling the problem solved earlier in the section ‘Limitanalysis’, a lower-bound calculation seeks to find a staticallyadmissible stress field � ¼ � xx, � yy, � zz, �xy, �yz, �xzf gT

that satisfies equilibrium throughout V, balances the pre-scribed tractions t on At, nowhere violates the yield criterionf so that f (�) < 0, and maximises the collapse load

Q ¼ð

Aq

Q1 qð ÞdAþð

V

Q2 hð ÞdV

In the above, the functions Q1 and Q2 depend on the case athand. For example, in a bearing capacity problem, Q2 ¼ 0,and one typically wants to maximise the load carried by thetractions normal to a boundary edge, qn, so that Q1 ¼ qn and

Q ¼ð

Aq

qndA (4)

For slope, tunnel and excavation problems, on the otherhand, one often wants to maximise a dimensionless stabilitynumber that is a function of the soil unit weight ª. In thiscase Q1 ¼ 0 and Q2 ¼ ª, giving

Q ¼ð

V

ª dV

where ª is a variable which can be optimised.

Aw h Aq

Atg

V

uy g hy y�

g hz z�

g hx x�

σyy

σxx

y

τyz

τyx

w

uz

ux

q

τxy

τxz

σzz

τzx

τzy

xz

t

Fig. 7. Surface and body forces acting on soil mass


Lower-bound finite elementsFollowing Lyamin & Sloan (2002a), linear elements are

used to discretise the domain. These elements, shown in Fig.8, enable a statically admissible stress field to be found in arigorous manner, and have proved to be highly effective inlarge-scale applications.

The lower bound is found by formulating and solving anon-linear optimisation problem, where the nodal stressesand/or element body forces are the unknowns, and theobjective function to be maximised corresponds to thecollapse load. The unknowns are subject to equilibriumequality constraints for each continuum element, equilibriumequality constraints for each discontinuity, stress boundaryconditions, and a yield condition inequality constraint foreach node.

Figure 9 shows a very simple lower-bound mesh for thestrip footing problem considered previously in Fig. 1. In thismesh, each node i is associated with a vector of threeunknown stresses, and each element e is associated with avector of two unknown body forces (which are not used inthis example, but are included for the sake of generality).Owing to the presence of stress discontinuities between allinter-element edges, multiple nodes may share the samecoordinates, and each node is unique to an element. Acrosseach stress discontinuity, the normal and shear stresses arecontinuous. To satisfy the stress boundary conditions indi-cated, the nodal stresses along the corresponding edges inthe grid are subject to appropriate equality constraints (usingthe standard stress transformation relations), and the stressesat each node i in the grid, � i, are subject to the yieldcondition f (� i) < 0. The load to be maximised, given by

equation (4), can be expressed in terms of the vertical nodalstresses at the nodes underneath the footing.

The key steps in formulating an efficient lower-boundmethod using finite elements are now outlined for the two-dimensional case. A similar approach holds for three dimen-sions.

Objective functionIn many applications, such as bearing capacity calcula-

tions, the objective function to be maximised corresponds toa force acting along the boundary of the domain (thecollapse load). The common case of optimising the externaltraction q along a boundary segment in two dimensions isshown in Fig. 10.

Since the stresses vary linearly throughout an element, thenormal and shear loads acting on an edge of length L andunit thickness are given by

Qn

Qs

� �¼ L

2

q1n

q1s

� �þ q2

n

q2s

� �� (5)

where the local surface stresses qn and qs can be expressedin terms of the Cartesian stresses at node i using thestandard transformation equations

qin

qis

� �¼ cos2 � sin2 � sin 2��1

2sin 2� 1

2sin 2� cos 2�

" # � ixx

� iyy

�ixy

8><>:

9>=>; (6)

When summed over each loaded boundary edge, the con-tributions Qn and Qs give the total force acting on the soil

, σ τiyy

ixy, }TNode {σ i i

xx� σ

Element { , }he ex

ey� h h T

, σ σ τ τiyy

izz

ixy

iyz, ,Node {σ i i

xx� σ

Element { , , }he ex

ey

ez� h h h T

, }τ ixz

T,

Fig. 8. Linear elements for lower-bound limit analysis

�

Qs �⌠⌡q Asd

Qn �⌠⌡q And

q

q 2n

q 2s

1

2

x

y

ns

Segment length � Lq 1

n

q 1s

Fig. 10. Optimising the load along a boundary

qn

σ τn 0� �

Maximise d collapse loadQ q A� �n⌠⌡

Nodes { , , }σ i ixx

iyy

ixy� σ σ τ T

Triangles {h he ex� , }h e

yT

Stresses in triangles satisfy equilibrium

Stresses in discontinuities satisfy equilibrium

Stresses at nodes satisfy yield condition ( ) 0f σ i �

σ τnn nsand continuous for adjacent elements

∂∂σxx

x�

∂∂τxy

y� � � 0h gx x

∂∂σyy

y�

∂∂τxy

x� � � 0h gy y

τ0

�τ

0�

y

x

n

s

Fig. 9. Illustrative lower-bound mesh for strip footing problem

540 SLOAN

mass per unit thickness. Typically, one of these quantities ismaximised, but it is also possible to maximise a forceresultant in a specified direction. Using equations (5) and(6), and summing over all the loaded edges, the objectivefunction (collapse load) can be expressed as

cT1�

where c1 is a vector of constants, and � is the global vectorof unknown nodal stresses. At first sight, it would appearthat the above approach is restricted to problems with lineargeometry and linear loading. This limitation can be relaxed,however, by using the approach of Lyamin & Sloan (2002a),which uses a local coordinate system.

For body loads he ¼ hex, he

y

� �Tacting on an element of

area Ae and unit thickness, the corresponding resultantforces, Qx ¼ Aehe

x and Qy ¼ Aehey, are shown in Fig. 11.

These forces may be assembled over the grid to give thetotal load produced as

cT2 h

where c2 is a vector of constants (element areas) and h is aglobal vector of element body loads. In practice, the mostcommon case of body force optimisation involves a variableunit weight ª so that he ¼ {0, �ª}T: For pseudo-dynamicstability analysis, however, it is also useful to be able tooptimise the lateral body force component he

x:

Continuum equilibriumIn order to be statically admissible, the stresses in each

element must satisfy the equilibrium equations

@� xx

@xþ @�xy

@yþ hx þ gx ¼ 0

@� yy

@yþ @�xy

@xþ hy þ gy ¼ 0

(7)

Over each triangle, the stresses vary linearly according tothe relations

� ¼X3

i¼1

N i�i (8)

where Ni are linear shape functions that are dependent on xand y and the element nodal coordinates. Inserting equation(8) into equation (7) yields the pair of equilibrium equations

BT1 BT

2 BT3

� �e ¼ �(he þ ge) (9)

where the terms Bi are the standard strain–displacement(compatibility) matrices, defined by

BTi ¼

1

Ae

bi 0 ci

0 ci bi

�(10)

and bi and ci are constants that depend on the nodalcoordinates. Rather than impose the equilibrium constraintsin their native form, it is convenient to multiply both sidesof equation (9) by the element area Ae: This permits anelegant implementation of stress discontinuities, as describedin the next section, and leads to the modified equilibriumrelations

�BBT1

�BBT2

�BBT3

� �e ¼ � he þ geð ÞAe (11)

where

�BBTi ¼ Ae BT

i ¼bi 0 ci

0 ci bi

�(12)

Imposing the constraints in equation (11) ensures that thestresses satisfy the equilibrium conditions at every point inan element, thus satisfying a key requirement of the lower-bound theorem.

Discontinuity equilibriumStress discontinuities can dramatically improve the accu-

racy of the collapse load obtained from lower-bound calcula-tions, and are introduced along all inter-element edges.Following the formulation of Lyamin et al. (2005a), eachdiscontinuity is modelled by a patch of continuum elementsof zero thickness, with opposing nodal pairs having the samecoordinates, as shown in Fig. 12.

To satisfy equilibrium, and thus be statically admissible,the normal and shear stresses must be the same on bothsides of the discontinuity according to the relations

� 1nn

�1ns

� �¼ � 2

nn

�2ns

� �,

� 3nn

�3ns

� �¼ � 4

nn

�4ns

� �(13)

where for node i

� inn

�ins

� �¼ cos2 � sin2 � sin 2��1

2sin 2� 1

2sin 2� cos 2�

" # � ixx

� iyy

�ixy

8><>:

9>=>; (14)

Equations (13) and (14) imply that each pair of nodes on astress discontinuity must obey two equality constraints ontheir associated Cartesian stresses. Summing these con-straints over all nodal pairs on the discontinuities gives theglobal set of conditions that must be satisfied for disconti-nuity equilibrium.

Since equation (11) holds true for any value of theelement area, it is possible to set ! 0 for the triangles D1

and D2 in Fig. 12, so that (x1, y1) ¼ (x2, y2) and(x3, y3) ¼ (x4, y4). Considering triangle D1, it can be shownthat the equilibrium relations in equation (11) then become

Qx

x

y

Qy

Element area � Ae

Fig. 11. Optimising body forces over an element

�

δ 0→

1

2

( , ) ( , )( , ) ( , )x y x yx y x y

1 1 2 2

3 3 4 4

�

�

3

4

x

yn

s

n

s

D2

D1

Fig. 12. Statically admissible stress discontinuity


�BBT1 ��BBT

1 0�

�e ¼ �BBT1 ��BBT

1 0�

�1 �2 �3� �T ¼ 0

which implies

�BBT1�

1 ¼ �BBT1�

2

Hence the left pair of relations in equation (13) are satisfied.A similar argument for triangle D2 yields the right pair ofrelations in equation (13), so that all four of the discontinu-ity equilibrium conditions (equation (13)) are satisfied.Although the normal and shear stresses are continuous alongeach discontinuity, the tangential normal stress �ss mayjump, which means that the stresses can potentially differ atnodes that share the same coordinates. This type of formula-tion permits discontinuities to be modelled using standardcontinuum elements, and is simple to implement in both twoand three dimensions. Other alternatives for implementingstress discontinuities are possible, such as imposing theconstraints in equation (13) on the nodal stresses explicitly,and these have been used by a variety of researchers,including Pastor & Turgeman (1976), Sloan (1988a) andLyamin & Sloan (2002a).

Stress boundary conditionsTo satisfy equilibrium, the stresses for any boundary node

must match the prescribed surface tractions (stresses) t.These boundary conditions may be specified in a Cartesianreference frame, but are more commonly defined in terms ofnormal and tangential components along a boundary edge,as shown in Fig. 13.

Noting that the stresses vary linearly along an edge, thestress boundary conditions take the form

� 1nn

�1ns

� �¼ t1

n

t1s

� �,

� 2nn

�2ns

� �¼ t2

n

t2s

� �

where f� inn , �i

nsgT

for node i are again given by equation(14). These constraints must be applied to all edges wheresurface stresses are specified, and they ensure that theboundary conditions are satisfied exactly for a linear finite-element model.

Yield conditionsProvided the stresses vary linearly over an element and

the yield function f (�) is convex, the yield condition issatisfied at every point in the domain if the inequalityconstraint f (� i) < 0 is imposed at each node i. In the two-dimensional case, this implies that the nodal stresses foreach triangle are subject to three non-linear inequality con-straints, as shown in Fig. 14.

Extension elementsFor problems involving semi-infinite domains, special ‘ex-

tension’ elements are needed to complete the stress field sothat the equilibrium, stress boundary and yield conditionsare satisfied everywhere. These elements are placed aroundthe periphery of a standard mesh, and although their effectis often small for a grid that is sufficiently large to capturethe zone of plastic yielding, they do guarantee that thesolution is a rigorous lower bound. For two-dimensionalapplications where the yield surface has a linear envelope,complete stress fields can be found using the unidirectionaland bidirectional extension elements shown in Fig. 15. Theequilibrium and stress boundary conditions for these exten-sion elements are identical to those for the standard con-tinuum elements, with the only change being the differentyield conditions (Pastor, 1978). In the latter, the functionF(�) is defined by the relation f (�) ¼ F(�) � k, where k is anon-negative constant (Makrodimopoulos & Martin, 2006).

For the unidirectional extension element, node 4 is adummy node, since its stresses are not independent and canbe expressed as linear combinations of the stresses at nodes 1,2 and 3. This node is included for the sole purpose of beingable to accommodate a stress discontinuity along the edgedefined by nodes 3 and 4, and it is not subject to constraintsother than those imposed by discontinuity equilibrium.

For cases where the envelope of the yield surface is notlinear, such as the widely used Hoek–Brown criterion, theabove extension elements are inapplicable, and hence thecomputed lower bounds are based on an ‘incomplete’ stressfield. Although theoretically undesirable, this is not a seriousshortcoming in practice, since the extension conditions seldomhave a significant effect on the collapse load for a well-constructed mesh that includes all the zones of plastic yielding.

Lower-bound non-linear optimisation problemFor a given mesh, summing the various objective function

coefficients and constraints described above leads to the non-

�

t 1s

t 2n

t 2s

1

2

x

y

ns

t

t 1n

Fig. 13. Stress boundary conditions

2

3

f ( ) 0σ 1 �

f ( ) 0σ 3 �

1

x

yf ( ) 0σ 2 �

Fig. 14. Yield conditions

f ( ) 0σ 2 �

f ( ) 0σ 3 �

f ( ) 0σ 2 �

F( ) 0σ σ1 2� �

3

2F( ) 0σ σ1 2� �

F( ) 0σ σ3 2� �

1

x

y

12

3 4Unidirectional extension zone

Bidirectional extension zone

Dummy node

Fig. 15. Extension elements

542 SLOAN

linear optimisation problem in equation (3), where theunknowns are the nodal stresses � and element body loadsh. The solution to this non-linear programming problem,which constitutes a statically admissible stress field, can befound efficiently by solving the system of non-linear equa-tions that define its Kuhn–Tucker optimality conditions.Such a strategy was proposed by Lyamin (1999) and Lyamin& Sloan (2002a), who developed a two-stage quasi-Newtonsolver that typically requires less than about 50 iterations,regardless of the problem size. Because it does not requirethe yield surface to be linearised, this type of approach isapplicable in three dimensions for a wide range of smoothyield criteria, including those with curved envelopes in themeridional plane. As noted previously, however, for theTresca and Mohr–Coulomb yield functions, the vertices mustbe smoothed to remove the singularities in the gradients andobtain good convergence. Alternatively, second-order coneprogramming and semi-definite programming algorithms arefast and efficient solution methods for the Tresca/Mohr–Coulomb models under two- and three-dimensional condi-tions respectively (Ciria, 2004; Makrodimopoulos & Martin,2006; Krabbenhøft et al., 2007). Both these procedures areapplicable to non-smooth yield criteria, and are thus ideallysuited to the Tresca and Mohr–Coulomb models.

FINITE-ELEMENT UPPER-BOUND FORMULATIONAn efficient formulation of the finite-element upper-bound

method will now be briefly outlined. This follows theformulation of Lyamin & Sloan (2002b), with importantmodifications to handle velocity discontinuities as describedby Krabbenhøft et al. (2005). Recalling the problem solvedin the section ‘Limit analysis’ and with reference to Fig. 7,an upper-bound calculation searches for a velocity distribu-tion u ¼ {ux, u y, uz}

T that satisfies compatibility, the flowrule and the velocity boundary conditions w on the surfacearea Aw, and minimises the internal power dissipation (dueto plastic shearing) less the rate of work done by the fixedexternal loads. Mathematically, the latter quantity can bewritten as

_W ¼ Pint �ð

At

tTudA�ð

V

gTudV (15)

where Pint is the plastic dissipation, defined by

Pint ¼ð

V

�T _�pdV (16)

and

_�p ¼ f _�pxx, _�p

yy, �pzz, _ªp

xy, _ªpyz, _ªp

xzgT

are the plastic strain rates. An upper bound on the limit load isthen found by equating the optimised value of _W to the powerexpended by the external loads, which may be written as

Pext ¼ð

Aq

qTudAþð

V

hTudV (17)

As in the lower-bound formulation, it is possible to optimiseeither the total force carried by the external tractions, q, orsome set of body forces h (typically the unit weight).

Upper-bound finite elementsFollowing Lyamin & Sloan (2002b), linear elements are

used to discretise the domain. These elements, shown in Fig.16, enable a kinematically admissible velocity field to be

found in a rigorous manner, and are combined with a patch-based method for modelling velocity discontinuities alongall inter-element edges (Krabbenhøft et al., 2005). Theprocedure of Lyamin & Sloan (2002b) was the first toincorporate velocity discontinuities in three dimensions, andhas proved highly effective for solving large-scale stabilityproblems in geotechnical engineering. The scheme of Krab-benhøft et al. (2005), which models each discontinuity by apatch of continuum elements of zero thickness, is particu-larly advantageous in three dimensions, is applicable togeneral types of yield criterion, and parallels the formulationdescribed above for the lower-bound method. The elementsshown in Fig. 16 adopt a linear variation of the velocities utogether with a constant-stress field �. Although it is possi-ble to develop discrete upper-bound formulations that do notinclude the stresses as unknowns, these are restricted to yieldmodels with a linear envelope. Moreover, as discussed later,inclusion of the stresses provides a very convenient platformfor developing a mesh refinement strategy, with an exacterror estimate, for minimising the gap between the solutionsfrom the upper- and lower-bound formulations.

The upper-bound procedure is formulated as a non-linearoptimisation problem, where nodal velocities, elementstresses and plastic multipliers are the unknowns, and theobjective function to be minimised is the internal powerdissipation less the rate of work done by fixed externalforces. To satisfy the requirements of the upper-boundtheorem, the unknowns are subject to constraints arisingfrom the flow rule, the velocity boundary conditions and theyield condition.

Figure 17 shows a very simple upper-bound mesh for thestrip footing problem considered previously in Fig. 1. Eachnode i is associated with a vector of two unknown velocities,and each element e is associated with a vector of threeunknown stresses and an unknown non-negative plasticmultiplier rate _º: In each triangle the plastic strains (velo-cities) are subject to the constraints imposed by the asso-ciated flow rule, and also satisfy the consistency requirement_º f (�e) ¼ 0: The latter condition ensures that plastic defor-mation takes place only for points on the yield surface.Owing to the presence of velocity discontinuities along allinter-element edges, multiple nodes may share the samecoordinates, and each node is unique to an element. Ingeneral, two plastic multipliers are used to model the normaland tangential velocity jumps in each of these discontinu-ities, and these are governed by the corresponding associatedflow rule. To satisfy the velocity boundary conditions, therelevant nodal velocities on the periphery of the grid aresubject to appropriate equality constraints. The upper boundon the limit load is found by using equations (15) and (17),noting that t ¼ g ¼ h ¼ 0 for this case, to minimise Pint andhence Q.

The key steps in formulating an efficient upper-boundmethod using finite elements are now outlined for the two-

, u iy }TNode {u i i

x� u

Element { , ,σ e exx

eyy

exy� σ σ τ T}

Node { , , }u i ix

iy

iz� u u u T

Element { , , , , , }σ e exx

eyy

ezz

eyz

exz� σ σ σ τ τ τe

xyT

Fig. 16. Linear elements for upper-bound limit analysis


dimensional case. A similar formulation for three dimen-sions can be found in Lyamin & Sloan (2002b).

Objective functionIn the finite-element upper-bound formulation, the objec-

tive function corresponds to the internal rate of energydissipated by plastic shearing less the energy expended bythe fixed external forces, and it is given by equations (15)and (16). Noting that the stresses and plastic strain rates areconstant over each element, and summing over all theelements, the internal power dissipation may be written as

Pint ¼ð

V

�T _�pdV ¼X

e

(�T _�pV )e (18)

This quantity can be evaluated conveniently by observingthat the (constant) plastic strain rates are related to the nodalvelocities by the strain–displacement relations

_�p ¼ Beue (19)

where for the linear triangle

Be ¼ B1 B2 B2

� , ue ¼ u1

x , u1y , u2

x , u2y , u3

x , u3y

n oT

and Bi is given by equation (10). Using the matrix�BB

e ¼ Ae Be defined in equation (12), equations (18) and (19)furnish the total internal dissipated power for the mesh, interms of the unknowns stresses and velocities, as

Pint ¼ �T �BBu (20)

where � is a global vector of element stresses, u is a globalvector of nodal velocities, and �BB ¼

Pe�BB

e:

Using the formulation of Krabbenhøft et al. (2005), whichmodels each discontinuity by a patch of continuum elementsof zero thickness, equation (20) can also be used, withoutmodification, to compute the plastic dissipation in the velo-city discontinuities due to plastic shearing. Thus Pint is foundby summing over both the continuum elements and thediscontinuity elements, as they can be treated identically.

The remaining two integrals in equation (15), involvingthe fixed tractions t and body forces h, can be evaluatedusing the linear expansions for the velocities u, and lead toan expression of the form

cTu ¼ð

At

tTudAþð

V

gTudV (21)

where c is a vector of known constants. After combiningequations (15), (20) and (21), the final objective function forthe upper-bound formulation can be written as

�T �BBu� cTu

Continuum flow ruleTo give an upper bound on the limit load, the velocity

field must be kinematically admissible and satisfy the con-straints imposed by an associated flow rule. For the triangu-lar element shown in Fig. 16, the flow rule conditions maybe written as

_�pxx ¼ _º@ f =@� xx

_�pyy ¼ _º@ f =@� yy, _º > 0, _º f �eð Þ ¼ 0

_ªpxy ¼ _º@ f =@�xy

or

_�p ¼ _º= f �eð Þ, _º > 0, _º f �eð Þ ¼ 0 (22)

where _º is the plastic multiplier. After combining equations(19) and (22), and then multiplying both sides by theelement area, the flow rule constraints for each element maybe expressed as

�BBeue ¼ _Æ= f �eð Þ, _Æ > 0, _Æ f �eð Þ ¼ 0 (23)

where _Æ ¼ Ae _º denotes the conventional plastic multipliertimes the element area. Thus, for the two-dimensional case,the continuum flow rule generates four equality constraintsand one inequality constraint on the element unknowns.Unless the yield criterion is a linear function of the stresses,all the equality constraints are non-linear.

Discontinuity flow ruleThe patch-based formulation of Krabbenhøft et al. (2005)

incorporates velocity discontinuities using a procedure iden-tical to that mentioned previously for the lower-boundmethod. For the two-dimensional case, shown in Fig. 18,

wn 1� �

u x0

�

u ux y 0� �

uu

xy

0�

�

Minimise dP V Q wintT p� � �

V nσ ε⌠⌡

.

Nodes { , }u i ix

iy� u u T

Triangles { , , }σ e exx

eyy

exy� σ σ τ T

Strains in triangles satisfy flow rule

Velocities in discontinuities satisfy flow rule

Stresses in elements satisfy yield condition ( ) 0f σ e �

ε λ σpxx xx/� ∂ ∂f

. .

ε λ σpyy yy/� ∂ ∂f

. .

γ λ τpxy xy/� ∂ ∂f

. ., λ λ0, ( ) 0� �f σ e. .

Δu fn n n( , )/� λ σ τ σ∂ ∂.

Δu fs n( , )/� λ σ τ τ∂ ∂. λ λ σ τ0, ( , ) 0� �f n

. .

y

x

n

s

Fig. 17. Illustrative upper-bound mesh for strip footing problem

544 SLOAN

each discontinuity comprises two triangles of zero thickness,and thus has six unknown stresses. Both triangles are subjectto the flow conditions defined by equations (23).

Across the discontinuity, velocity jumps can occur in thenormal and tangential directions, so that the velocities canpotentially differ at nodes that share the same coordinates.In the following, the implications of enforcing the flow ruleconditions in equations (23) for zero-thickness elements arediscussed, and it is shown that the resulting discontinuityformulation is equivalent to that proposed by Sloan & Klee-man (1995).

Using the strain–displacement relations equations (10)and (19) with the discontinuity width ! 0, it is straightfor-ward to show that the local strains in triangle D1 approachthe values

_�pss ! 0

_�pnn ! ˜u12

n =

_ªpns ! ˜u12

s =

(24)

where

˜u12n , ˜u12

s

� ¼ u1

n � u2n, u1

s � u2s

� are the normal and tangential velocity jumps at the nodalpair (1, 2). From equation (24) it is clear that the strainsbecome infinite as ! 0, but multiplying them by theelement area 0.5L gives finite quantities according to

Ae _�pss ¼ 0

Ae _�pnn ¼ ˜u12

n L=2

Ae _ªpns ¼ ˜u12

s L=2

where a unit out-of-plane element thickness has been as-sumed. Similar relations hold for triangle D2, with thesuperscript pair (1, 2) being replaced by (3, 4). These

relations confirm that discontinuous velocity jumps can bemodelled by using two zero-thickness continuum elementswith (x1, y1) ¼ (x2, y2) and (x3, y3) ¼ (x4, y4), provided theflow rule constraints (equations (23)) are satisfied over eachtriangle. Note that _Æ in equation (23) is well-defined as! 0, even though it is the product of a quantity that iszero (Ae) and a quantity that is infinite ( _º). For a plane-strain discontinuity, the general yield condition f (� e) can,without loss of generality, be replaced by its planar counter-part f (�n, �), where �n denotes �nn and � denotes �ns: Theflow rule conditions (equations (23)) that define (˜un, ˜us)are then given by

˜un ¼ _Æ @ f =@� n

˜us ¼ _Æ @ f =@�, _Æ > 0, _Æ f � n, �ð Þ ¼ 0

The common case of a velocity discontinuity in a Mohr–Coulomb material is shown in Fig. 19. Using Koiter’stheorem for composite yield surfaces, the jumps in thenormal and tangential directions are given by

˜un ¼ ˜uþn þ ˜u�n ¼ _Æþ þ _Æ�ð Þ tan�

˜us ¼ ˜uþs þ ˜u�s ¼ _Æþ � _Æ�

where

_Æþ > 0

_Æ� > 0

and

_Æþ f þ � n, �ð Þ ¼ 0

_Æ� f � � n, �ð Þ ¼ 0

Noting that the normal and tangential jumps at a nodal pair(i, j) can be expressed in terms of the Cartesian velocityjumps through the relations

�

Area of and LD D1 212� δ δ 0→

3

4

Δun

Δus

( , ) ( , )( , ) ( , )x y x yx y x y

1 1 2 2

3 3 4 4

�

�

1

2

x

yn

sD1

D2

L

Fig. 18. Kinematically admissible velocity discontinuity

f c� tan� � �τ σ φn

c

f c� tan� � � �τ σ φn

φ

φ

τ

Δu f� � �/ 0s � � �α τ α∂ ∂. .

Δu f� � �n n tan 0� � �α σ α φ∂ /∂. .

Δu f� � �n n tan 0� � �α σ α φ∂ /∂. .

Δu f� � �s / 0� � � α τ α∂ ∂. .

Multipliers ( , ) 0α α� � �. .

σn

c

Mohr–Coulomb| | tanf c� � �τ σ φn

Fig. 19. Mohr–Coulomb yield criterion for velocity discontinuity


˜uijn

˜uijs

� �¼ cos � sin �� sin � cos �

�˜uij

x

˜uijy

( )

it follows that the complete set of flow rule constraints forthe discontinuity is given by

cos � sin �� sin � cos �

�˜u12

x

˜u12y

( )¼ _Æþ12 þ _Æ�12ð Þ tan�

_Æþ12 � _Æ�12

� �

(25)

cos � sin �� sin � cos �

�˜u34

x

˜u34y

( )¼ _Æþ34 þ _Æ�34ð Þ tan�

_Æþ34 � _Æ�34

� �

(26)

_Æþ12 > 0

_Æ�12 > 0

_Æþ34 > 0

_Æ�34 > 0

(27)

_Æþ12 f þ � 1n, �1

� ¼ 0

_Æ�12 f � � 1n, �1

� ¼ 0

_Æþ34 f þ � 2n, �2

� ¼ 0

_Æ�34 f � � 2n, �2

� ¼ 0

where ( _Æþij, _Æ�ij) denotes the values of the plastic multi-pliers ( _Æþ, _Æ�) at the nodal pair (i, j), (� 1

n, �1) are thestresses in triangle D1, and (� 2

n, �2) are the stresses in D2:Equations (25)–(27) are identical to the formulation pro-posed in Sloan & Kleeman (1995), which uses line elementsto model a velocity discontinuity without including elementstresses. Thus imposing the constraints in equation (23) overthe zero-thickness triangles D1 and D2 is sufficient to modela velocity discontinuity.

Velocity boundary conditionsTo be kinematically admissible, the velocity field must

satisfy the prescribed boundary conditions. These boundaryconditions may be specified in a Cartesian reference framebut, as shown in Fig. 20, are more commonly defined interms of normal and tangential velocity components along aboundary edge.

Noting that the velocities vary linearly along each edge,the general form of the boundary conditions may be ex-pressed as

u1n

u1s

� �¼ w1

n

w1s

� �,

u2n

u2s

� �¼ w2

n

w2s

� �(28)

where, for some node i, the transformed nodal velocities arerelated to the Cartesian velocities by the standard equations

uin

uis

� �¼ cos � sin �� sin � cos �

�ui

x

uiy

( )(29)

These constraints must be applied to all boundary nodes thathave prescribed velocities.

Load constraintsTo perform an upper-bound analysis, various additional

constraints are imposed on the velocity field to match thetype of loading. For the case shown in Fig. 17, the boundaryconditions defined by equations (28) and (29) can be used tomodel the loading associated with the rigid footing bysetting the normal velocities w1

n ¼ w2n ¼ �C along the ap-

propriate element edges, where C is some constant. Theactual magnitude of C does not matter, since it cancels whenthe loads are computed using equations (15)–(17). For a‘smooth’ interface the tangential velocities ws underneaththe footing are unrestrained, whereas for a ‘rough’ interfacews ¼ 0. These types of velocity boundary conditions may beused to define the ‘loading’ caused by any type of stiffstructure, such as a retaining wall or a pile.

For problems where part of the body is loaded by anunknown uniform normal pressure q, such as a flexible stripfooting, it is appropriate to impose constraints on the surfacenormal velocities of the formð

Aq

undA ¼ C (30)

where C is a prescribed rate of flow of material across theboundary, typically set to unity. Noting that the velocitiesvary linearly, substituting equation (29) into equation (30)yields the following equality constraints on the nodal velo-cities

1

2

Xedges

Lij uix þ ui

x

� cos �ij þ ui

y þ uiy

� �sin �ij

h i¼ C

where Lij and �ij denote the length and inclination of anedge with nodes (i, j), and a unit thickness is assumed. Thistype of constraint, when substituted into the rate of workdone by the external forces, given by equation (17), permitsan applied uniform pressure to be minimised directly.

Another common type of loading constraint, which isuseful when a body force such as unit weight is to beoptimised, takes the formð

V

uydA ¼ �C (31)

where C is a constant that is typically unity. This constraintpermits a vertical body force to be minimised directly whenthe power expended by the external loads is equated to theinternal power dissipation, and is particularly useful whenanalysing the behaviour of slopes. Noting again that thevelocities vary linearly over each element, the condition inequation (31) gives rise to the following constraints on thenodal velocitiesð

V

uydA ¼ 1

3

Xelements

uiy þ uj

y þ uky

� �Ae ¼ �C

where (i, j, k) denote the nodes for some element e, and Ae

is the element area.

�

w 1s

w 2n

w 2s

1

2

x

y

n

s

w

w 1n

Fig. 20. Velocity boundary conditions

546 SLOAN

Yield conditionsTo be kinematically admissible, the stresses associated

with each element (including the zero-thickness discontinu-ities) must satisfy the yield condition f (� e) < 0. Since theelement stresses are assumed to be constant, this requirementgenerates one non-linear inequality constraint for each con-tinuum triangle and each discontinuity triangle.

Upper-bound non-linear optimisation problemAfter assembling the objective function coefficients and

constraints for a mesh, the upper-bound non-linear optimisa-tion problem can be expressed as

Minimise

�T �BBu � cTu power dissipation – rate of work done

by fixed external forces

subject to

�BBeue ¼ _Æe= f �eð Þ flow rule conditions for each

element e

_Æe > 0 plastic multiplier times Ae

for each element e

_Æe f �eð Þ ¼ 0 consistency condition for

each element e

Au ¼ b velocity boundary conditions,

load constraints

f �eð Þ < 0 yield condition for each

element e

(32)

where � is a global vector of unknown element stresses, u isa global vector of unknown nodal velocities, �BB

eis the

element compatibility matrix defined by equation (12),�BB ¼

P�BB

eis a global compatibility matrix, �T �BBu is the

power dissipated by plastic shearing in the continuum anddiscontinuities, cTu is the rate of work done by fixed tractionsand body forces, _Æe is the plastic multiplier times the areafor element e, f (� e) is the yield function for element e, A isa matrix of equality constraint coefficients, and b is a knownvector of coefficients. The solution to equation (32) constitu-tes a kinematically admissible velocity field, and can befound efficiently by treating the system of non-linear equa-tions that define the Kuhn–Tucker optimality conditions.Interestingly, these optimality conditions do not involve _Æe,so these quantities do not need to be included as unknowns.The two-stage quasi-Newton solver proposed by Lyamin(1999) and Lyamin & Sloan (2002b) typically requires lessthan about 50 iterations, regardless of the problem size, andresults in very efficient formulations for two- and three-dimensional problems. This type of solver has the advantagethat it can be used for general types of yield surfaces,including those with curved failure envelopes. As with thelower-bound case, however, its rate of convergence is affectedby non-smooth yield criteria, and the vertices in the Trescaand Mohr–Coulomb surfaces must be smoothed to obtaingood performance. Alternatively, second-order cone program-ming and semi-definite programming algorithms can be em-ployed to solve equation (32) for the Tresca/Mohr–Coulombmodels under two- and three-dimensional conditions respec-tively. As mentioned previously, both these procedures areapplicable to non-smooth yield criteria, and are thus ideallysuited to the Tresca and Mohr–Coulomb models. Yet anotheroption, which is adopted in this paper, is to consider the dual

of equation (32), which gives a stress-based upper-boundmethod (Krabbenhøft et al., 2005). The optimisation problemthat results from this approach can be solved using any of thealgorithms discussed above.

ADAPTIVE MESH REFINEMENTLimit analysis is most useful when tight bounds on the

collapse load are obtained. For the finite-element limit analy-sis methods described above, the size of the ‘gap’ betweenthe bounds depends strongly on the discretisation adopted,and it is therefore desirable to investigate the possibility ofdeveloping automatic mesh refinement methods.

A comprehensive discussion of mesh generation for thelower-bound method, including stress discontinuities and‘fans’ that are centred on stress singularities (such as thosethat occur at the edge of a rigid footing), has been given byLyamin & Sloan (2003). Their procedure uses a parametricmapping technique to automatically subdivide a specifiednumber of subdomains in both two and three dimensions,and has proved invaluable for solving large-scale problemsin practice. To optimise the lower bound, however, a trial-and-error procedure is needed, where successively finermeshes are generated until no improvement in the limit loadis found. Owing to their strong similarities to lower-boundgrids, this approach can also be used to generate upper-bound grids, with a similar trial-and-error approach beingrequired.

Since a priori error estimates are not available in discretelimit analysis, a posteriori techniques are needed to predictthe overall discretisation error, and hence extract a mean-ingful mesh refinement indicator. In one of the few studiesof adaptive mesh generation for discrete limit analysis,Borges et al. (2001) presented an adaptive strategy for amixed formulation. Their approach employed a directionalerror estimator, with the plastic multiplier field taken as thecontrol variable, and permitted anisotropic mesh refinement,where elements can stretch or contract by different amountsin different directions. Computational results show that itsuccessfully localises the elements in zones of intense plasticshearing, and significantly improves the predicted collapseloads. Although the lower-bound formulation described inthe section ‘Finite-element lower-bound formulation’involves only stress fields, it is possible to obtain ‘quasivelocities’ and ‘quasi-plastic multipliers’ from the dual solu-tion to the optimisation problem described by equation (3).Exploiting this fact, Lyamin et al. (2005b) adapted theapproach of Borges et al. (2001) to their lower-boundformulation, and used it to study the effects of variouscontrol variables, isotropic and anisotropic element refine-ment, and special ‘fan’ zones centred on stress singularities.They employed a modified form of the advancing-frontalgorithm (Peraire et al., 1987) to generate the grid, andpermitted the elements to grow as well as shrink during therefinement process. The results of Lyamin et al. (2005b)show that the quasi-plastic multipliers, when used with avariety of error indicators based on recovered Hessianmatrices and gradient norms, can lead to lower bounds thatlie within a few per cent of the exact limit load. Forproblems involving strong singularities in the stress field,however, the best performance is obtained by incorporatingfan zones, as these are able to model the strong rotation inthe principal stresses that occurs.

More recently, exact a posteriori techniques for estimatingthe discretisation error in discrete limit analysis formulationshave been proposed by Ciria et al. (2008) and Munoz et al.(2009). These approaches rely on identical meshes beingused for the upper- and lower-bound analyses, and providedirect measures of the contributions from each element to


the overall bounds gap. Since the latter is precisely thequantity that needs to be minimised in practical stabilitycalculations, this type of error estimator is innately attrac-tive, and performs well for a wide variety of cases. In theformulations proposed by Ciria et al. (2008) and Munoz etal. (2009), the contribution of each element to the boundsgap is found through an elaborate series of volumetric andsurface integrations that include the effects of the discon-tinuities. For the discrete limit analysis procedures describedin the sections on finite-element lower-bound and upper-bound formulations these integrations are much simpler,because the discontinuities are modelled as standard con-tinuum elements (with zero thickness), and because theupper-bound method includes stresses as unknowns as wellas velocities.

To derive the element contributions to the bounds gap, theprinciple of virtual power is invoked for the case whereidentical meshes are used for the upper- and lower-boundanalyses. In the upper-bound case the total plastic dissipationfor the whole mesh is defined byð

V

�TUB _�pdV ¼

ðAq

qTUBudAþ

ðV

hTUBudV

þð

At

tTudAþð

V

gTudV

(33)

where the subscript UB denotes upper-bound values for theunknown stresses, surface tractions and body forces. Notingthat the velocities u and plastic strain rates _�p are kinemat-ically admissible throughout the domain, including the velo-city discontinuities, the principle of virtual power for thecomputed lower-bound stresses �LB, tractions qLB and bodyforces hLB givesð

V

�TLB _�pdV ¼

ðAq

qTLBudAþ

ðV

hTLBudV

þð

At

tTudAþð

V

gTudV

(34)

where the prescribed tractions t and body forces h are thesame for each analysis. Subtracting equation (34) from equa-tion (33) furnishes the ‘dissipation gap’ ˜ as

˜ ¼ð

V

�UB � �LBð ÞT _�pdV

¼ð

Aq

qUB � qLBð ÞTudAþð

V

hUB � hLBð ÞTudV

(35)

For the common case of proportional loading, with theupper- and lower-bound multipliers (ºUB, ºLB) defined sothat qUB ¼ ºq

UBq, qLB ¼ ºqLBq, hUB ¼ ºh

UBh and hLB ¼ ºhLBh,

equation (35) becomes

˜ ¼ð

V


¼ ºqUB � ºq

LB

� ðAq

qTudAþ ºhUB � ºh

LB

� � ðV

hTudV

(36)

In the above, if both the tractions and body force loads areoptimised simultaneously, their corresponding multipliersmust be related (e.g. by an equation such as ºq ¼ �ºh, with� being a prescribed constant). Equation (36) shows that thedissipation gap defined by

˜ ¼ð

V


provides a direct measure of the difference between theupper- and lower-bound loads. Noting the usual assemblyrules for a grid, it follows that

˜ ¼X

elements

˜e

with ˜e denoting the bounds gap contribution from eachelement. To allow for the contributions of both continuumelements and zero-thickness discontinuity elements, it isconvenient to compute ˜e using the relations

˜e ¼ �eUB � �e

LB

� T �BBeue (37)

where �BBe

is the standard compatibility matrix times theelement volume (defined for the two-dimensional case byequation (12)). Since the element quantities defined by equa-tion (37) are always positive (Ciria et al., 2008), they can beused to identify elements that make large contributions tothe bounds gap and are thus in need of refinement. More-over, for the upper- and lower-bound formulations describedin ‘Finite-element lower-bound formulation’ and ‘Finite-element upper-bound formulation’, all the quantities neededto compute the error estimator are readily available, regard-less of whether the element is a continuum element or adiscontinuity element. As mentioned previously, the solerestriction on this type of refinement process is that identicalmeshes must be adopted for both the upper- and lower-bound analyses.Using the exact error estimate provided by equation (37),the following procedure is used to adaptively refine the meshto give tight bounds on the limit load.

1. Specify the maximum number of continuum elementsallowed, Emax, and generate an initial mesh.

2. Perform upper- and lower-bound analyses using the samemesh.

3. If the gap between the upper and lower bounds is lessthan a specified tolerance, or if the maximum number ofcontinuum elements Emax is reached, exit with upper- andlower-bound estimates of the limit load.

4. Specify a target number of continuum elements for thecurrent mesh iteration, Ei, with Ei < Emax:

5. For each element, compute its contribution to the boundsgap ˜e using equation (37). In the case of a discontinuityelement, its bounds gap contribution is added to theneighbouring continuum element with which it shares themost nodes.

6. Scale the size of each continuum element to be inverselyproportional to the magnitude of ˜e, subject to theconstraint that the new number of continuum elements inthe grid matches the predefined target number ofcontinuum elements for the current iteration Ei:

7. Go to step 2.

In the above algorithm, the target number and maximumnumber of continuum elements, Ei and Emax, are included togive the user additional control over the adaptive refinementprocess. In step 6, some supplementary constraints may beincluded to limit the rate of decrease or increase in theelement size from iteration to iteration. Typically, the maxi-mum decrease in element size is set to a factor of 4, and themaximum increase in size is set to a factor of 2. Theselimits serve to reduce the oscillations in the size of elementsas the optimum mesh is sought, and do not greatly affect thenumber of iterations that are needed.

548 SLOAN

APPLICATIONS: UNDRAINED STABILITY ANALYSISThe finite-element limit analysis formulations described

above are fast and robust, and can model cases that includeinhomogeneous soils, anisotropy, complex loading, naturaldiscontinuities, complicated boundary conditions, and threedimensions. They not only give the limit load directly, with-out the need for an incremental analysis, but also bracketthe solution from above and below, thereby giving an exactestimate of the mesh discretisation error. These featuresgreatly enhance the practical utility of the bounding theo-rems, especially in three dimensions, where conventionalincremental methods are often expensive and difficult to use.In this section, the finite-element limit analysis methods areused to study a variety of undrained stability problems. Theresults will serve to illustrate the types of case that can betackled, and the quality of the solutions that can be ob-tained.

Bearing capacity of rigid strip footing on clay withheterogeneous strength

First a rigid footing is considered, of width B, resting onclay that has an undrained surface shear strength of su0 anda rate of strength increase with depth equal to r, as shownin Fig. 21. Following Davis & Booker (1973), the bearingcapacity can be expressed in the form

Qu

B¼ F 2þ �ð Þsu0 þ

rB

4

�

where F is a factor that depends on the dimensionlessquantity rB/su0 and the footing roughness.

Figure 22 compares the bearing capacity factors fromfinite-element limit analysis, using adaptive mesh refinement

with a maximum of 4000 continuum elements, with thesolutions of Davis & Booker (1973) obtained using themethod of characteristics. Results are presented for bothsmooth and rough footings, and the split scale on thehorizontal axis accounts for the two limiting cases of uni-form shear strength (rB/su0 ¼ 0) and zero surface strength(su0/rB ¼ 0). Whereas there is close agreement between thelimit analysis and characteristics solutions for rB/su0 < 8,significant differences are evident for low values of su0/rBwhere the surface shear strength is small. To resolve thissurprising inconsistency, the problem was reanalysed usingthe stress characteristics program ABC, developed at Oxfordby Martin (2004). This program provides a partial lower-bound stress field, and adaptively refines the mesh of charac-teristics to ensure the solution is accurate. In a privatecommunication, Martin (personal communication, 2011)confirmed that, for each of the cases considered, the lower-bound stress field from ABC can be extended throughout thesoil mass without violating equilibrium or yield, and that itcan also be associated with a velocity field that gives acoincident upper-bound collapse load. This suggests that thesolutions from ABC, shown in Fig. 23, are exact estimatesof the bearing capacity. Indeed, there is excellent agreementbetween these characteristics solutions and the new finite-element limit analysis solutions, with a maximum differenceof less than 1%.

When the quantity su0/rB is small, the failure mechanisminvolves soil being squeezed out in a thin band underneaththe footing. Unless adaptive meshing is employed for thesecases, such as that used in the program ABC, the method ofcharacteristics will be unable to model the true failuremechanism with high accuracy. This explains the discrepan-cies observed with the solutions of Davis & Booker (1973).The finite-element limit analysis methods have no difficultyin dealing with this extreme case, since the adaptive mesh-ing strategy, with the bounds gap error estimator, automatic-ally concentrates the elements where they are needed.

Figure 24 further highlights the difficulties that can arisewhen numerical methods are used to predict the limit loadassociated with a highly localised failure mechanism. Theplot shows the ratio of the bearing capacity found from thelimit-equilibrium analysis of Raymond (1967) to the bearingcapacity found from finite-element limit analysis (taken asthe average of the upper and lower bounds, which are within1% of the characteristics solutions of Martin (personalcommunication, 2011). Except for the case of uniformstrength (rB/su0 ¼ 0), the limit-equilibrium method, which

su0

ρ

s zu( )

Saturated clayUndrained shear strength ( )

Rate of strength increase d /dTresca material 0°

� � �

� �

�

s z s zs z

u u0

u

u

ρρ

φ

Q Bu/

B

z

1

Fig. 21. Rigid footing on clay whose undrained strength increaseswith depth

F

FRough LB

FSmooth UB

FSmooth LB

FRough UB

2016128400·010·020·030·04

ρB s/ u0 s Bu0/ρ

1·0

1·1

1·2

1·3

1·4

1·5

1·6

1·7

1·8

1·9

2·0

00·05

FRough Davis & Booker

FSmooth Davis & Booker

Fig. 22. Bearing capacities predicted by finite-element limitanalysis and method of characteristics (Davis & Booker, 1973)

F

20161284

ρB s/ u0

00·010·020·030·04s Bu0/ρ

FRough LB

FSmooth UB

FRough UB

FSmooth LB

1·0

1·1

1·2

1·3

1·4

1·5

1·6

1·7

1·8

1·9

2·0

00·05

FRough Martin ABC

FSmooth Martin ABC

Fig. 23. Bearing capacities predicted by finite-element limitanalysis and method of characteristics (Martin, personal com-munication, 2011)


assumes a circular slip surface, furnishes solutions that aretypically two to three times greater than the exact values.For the worst case, where the surface shear strength is zeroand the failure mechanism is highly localised, the limit-equilibrium solution overestimates the exact bearing capacityby a factor of approximately 4.5.

The large error in the bearing capacity predictions indi-cated in Fig. 24 is due to the inability of a circular slipsurface to model the actual mode of failure, especially forcases where su0/rB is small. This is shown clearly in Fig.25, which compares the failure surfaces predicted by limitequilibrium with the failure mechanisms (contours of plasticdissipation) predicted from adaptive upper-bound limit ana-lyses for a rough footing on two soils with rB/su0 ¼ 4 andrB/su0 ¼ 100. For the latter case, the zones of plasticdeformation at collapse are highly localised, and occur inclose proximity to the underside of the footing. The limit-equilibrium method, because it assumes a circular failuresurface, is unable to replicate this mode of deformation, andthus overpredicts the bearing capacity. Contours of plasticdissipation, like those shown in Fig. 25, provide a clearindication of zones of intense plastic shearing, and areuseful tools for visualising collapse mechanisms when usingfinite-element limit analysis to solve practical stability prob-lems in geotechnical engineering.

The meshes generated by the adaptive mesh refinementscheme, using the bounds gap error indicator described inthe section ‘Adaptive mesh refinement’ with a maximumlimit of 2000 elements, are shown in Fig. 26 for the caserB/su0 ¼ 4. These plots show that the adaptive scheme

converges to the optimum mesh after four cycles of refine-ment, and gives bounds that bracket the exact solutions towithin 1%. Subsequent mesh refinement cycles do notimprove the estimate of the bearing capacity, owing to therestriction of using 2000 elements, and more accurate pre-dictions would require this limit to be increased. Note thatextension elements were used to check the completeness ofthe lower-bound stress field for the finest mesh, but thesehave been omitted from the plot for clarity. The bounds gaperror indicator clearly concentrates the elements in the zonesof intense plastic shearing that are shown in Fig. 25.Computationally, the upper- and lower-bound limit analysismethods are very fast, with each solution requiring around2 s of CPU time on a standard desktop machine for a gridwith 2000 elements.

Strip footing under inclined eccentric loadingNow the problem, defined in Fig. 27, of a rigid strip

footing, subject to an inclined eccentric load, resting on asoil with uniform undrained shear strength su is considered.To predict the magnitude of the load P, the influence ofthree different footing interface models is examined.

(a) Tension permitted with a shear capacity equal to theundrained strength.

(b) No tension permitted, but no limit on the shear capacity.(c) No tension permitted, with a shear capacity equal to the

undrained strength.

The first of these models is often assumed in practicebecause of its simplicity, while the third model provides abetter representation of actual interface behaviour.

The optimised meshes and associated failure mechanismsfor the three cases are shown in Fig. 28. Using the averageof the upper and lower bounds to estimate the collapse load,and a maximum number of elements equal to 3000, theexact solutions are bracketed to within �1.6% for thevarious interface conditions. As expected, the flow rule forcase (a) ensures that no interface separation occurs, whilethe flow rule for case (b) dictates that no relative sheardeformation arises, with all motion being normal to theinterface. Owing to the effect of the no-tension constraint,case (c), which is a reasonable approximation to the ‘nosuction’ conditions that might apply in practice, gives anaverage collapse load that is approximately 19% lower thanthat for case (a). The power dissipation plots in Fig. 28highlight the different collapse mechanisms that occur forthe three cases, with case (b) exhibiting an interestingdouble failure surface. These examples underscore the versa-

()

limit

equi

lib

()

limit

anal

ysis

Q Qu u

2016128400·010·020·030·04

s Bu0/ρρB s/ u0

1·0

1·5

2·0

2·5

3·0

3·5

4·0

4·5

5·0

00·05

Rough

Smooth

Fig. 24. Bearing capacities predicted by finite-element limitanalysis and limit equilibrium (Raymond, 1967)

ρB s/ 4u0 � ρB s/ 100u0 �

Fig. 25. Failure mechanisms predicted by upper-bound limit analysis and limit equilibrium (Raymond, 1967)

550 SLOAN

tility of the finite-element limit analysis formulations inmodelling complex interface conditions, as well as thebenefits of adaptive mesh generation.

In closing, it is noted that a detailed study of the behav-iour of strip footings under combined vertical, horizontaland moment (V, H, M) loading can be found in Ukritchonet al. (1998). Using modified versions of the early finite-element limit analysis programs developed by Sloan(1988a) and Sloan & Kleeman (1995), they derive compre-hensive three-dimensional failure envelopes that account forthe effects of underbase suction and heterogeneous un-drained strength profiles. These envelopes suggest that the

traditional empirical bearing capacity factors for inclinedeccentric loading are conservative, often underestimatingthe exact values by more than 25%. Moreover, for problemswhere there is a significant strength gradient, these empiri-cal factors are unreliable, and not recommended for practi-cal use.

Stability of plane-strain tunnel and tunnel headingThe undrained stability of a circular tunnel in clay, whose

shear strength increases linearly with depth, has been studiedby several researchers, including Davis et al. (1980), Sloan

100 elements

0 1

2 3

4 5

2000 elements

543211·2

1·3

1·4

1·5

1·6

1·7

0

UB

LB

Iteration

FRough

Fexact

Fig. 26. Adaptive mesh refinement for finite-element limit analysis

P

P PH V/ 0·2�

e B0·45�

Saturated clayUndrained shear strength

Tresca material 0°�

�su

uφ

τ, Δus τ, Δus τ, Δus

fi 0� σn n, Δu fi 0� σn n, Δu fi 0� σn n, Δu

Case (a)Tension allowed| |τmax u� s

Case (b)No tension allowedNo limit on τmax

Case (c)No tension allowed| |τmax u� s

Fig. 27. Rigid footing subject to an inclined eccentric load


& Assadi (1992) and Wilson et al. (2011). The problem isdefined in Fig. 29, where a tunnel of diameter D and coverC is embedded in a soil with a surface undrained strengthsu0 and a strength gradient with depth r. This idealised casemodels a bored tunnel in soft ground where a rigid lining isinserted as the excavation proceeds, and the unlined heading,of length P, is supported by an internal pressure �t: Collapseof the heading is driven by the action of the surcharge �s

and the soil unit weight ª. The assumption of plane strain isclearly valid only when P� D, but the stability for this caseis more critical than that of a three-dimensional tunnelheading, and thus it yields a conservative estimate of theloads needed to trigger collapse. For the purposes of analy-sis, it is convenient to describe the stability of the tunnel bytwo dimensionless load parameters, (�s � �t)/su0 and ªD/su0:In practice, the unlined heading is typically supported by

either compressed air or clay slurry as the tunnel is exca-vated, and the known quantities are C/D, P/D, ªD/su0 andrD/su0, with the value of (�s � �t)/su0 at incipient collapsebeing unknown.

Before tackling the stability of a three-dimensional tunnelheading, first the plane-strain problem shown as sectionA–A is considered. For this case, P/D may be omitted fromthe analysis, and the relevant stability parameter is

� s � � t

su0

¼ fC

D,ªD

su0

,rD

su0

� �

To analyse this problem, the quantities su0, r, ª, H, D and�s are fixed, and the value of ��t (i.e. the tensile stress onthe face of the tunnel) is optimised. Alternatively, it ispossible to fix the value of �t and optimise the surcharge �s:

Case (a): 1·78 , 1·82P Bs P BsLB u UB u� �

Case (b): 1·53 , 1·58P Bs P BsLB u UB u� �

Case (c): 1·45 , 1·48P Bs P BsLB u UB u� �

Fig. 28. Meshes and failure mechanisms for rigid footing subject to an inclined eccentric load

552 SLOAN

Figure 30 shows upper and lower bounds on the stabilityparameter (�s � �t)/su0, plotted as a function of the dimen-sionless unit weight ªD/su0 and the soil strength factor rD/su0, for two tunnels with cover-to-diameter ratios of C/D ¼ 4and C/D ¼ 10. These bounds bracket the exact stabilityparameter to within a few per cent, and were found fromadaptive finite-element limit analysis using a maximum ofaround 4000 elements.

By definition, a negative value of N ¼ (�s � �t)/su0 indi-cates that a compressive normal stress of at least j� s � Nsu0jmust be applied to the tunnel wall to support the imposedloads, whereas a positive value of N implies that no internaltunnel support is required to maintain stability provided� s < Nsu0. Indeed, in the latter case, the tunnel is theoreti-cally capable of sustaining a uniform tensile pressure up toj� s � Nsu0j without undergoing collapse. Points that lie onthe horizontal axis defined by �s � �t ¼ 0 indicate configura-tions for which the tunnel pressure must precisely balancethe ground surcharge in order to prevent collapse.

Figure 31 shows the optimised limit analysis mesh and

power dissipation plot for a tunnel with C/D ¼ 4 in a soilwith ªD/su0 ¼ 3 and a uniform strength profile. In the lower-bound analyses, extension elements were added around theborder of the grid (not shown) to propagate the staticallyadmissible stress field over the semi-infinite domain. Thisstep has a negligible effect on the computed stability param-eter, but ensures that the lower-bound results are trulyrigorous. For the finest grid, with around 4000 elements,each bound calculation required about 4 s of CPU time ona desktop machine, and the optimum arrangement wasdeduced after four cycles of refinement. The ability of theadaptive mesh refinement scheme to concentrate the ele-ments where they are needed is again apparent. Fig. 32shows the failure mechanism for the same example, but witha finite strength gradient. Compared with the case withuniform strength, shown in Fig. 31, the zone of plasticdeformation is much more localised, and, as expected, doesnot extend below the invert, where the strength is higher.

The predictions from limit analysis theory are comparedwith the centrifuge results of Mair (1979) in Fig. 33. For the

ρ

s z s zu u0( ) � � ρ

A

σt

Surcharge σs

C

D

su0

1

z

σt Tunnel lining

SectionA–A

A

Surcharge σs

Unit weight � γP

Stability number ( )/ ( / , / , / , / )σ σ γs t u0 u0 u0� �s f C D P D D s D sρ

Fig. 29. Stability of circular tunnel in undrained clay

()/

σσ

st

u0�

s

54321γD s/

(a)u0

ρD s/ u0

1·00

0·75

0·50

0·25

0�20

�15

�10

�5

0

5

10

15

20

25

0

UB

LB

C D/ � 4

ρD s/ u0

1·00

0·75

0·50

0·25

0

()/

σσ

st

u0�

s

54321γD s/

(b)u0

0�50

�40

�30

�20

�10

0

10

20

30

40

50

60

70UB

LB

C D/ � 10

Fig. 30. Stability bounds for plane-strain tunnel: (a) C/D 4; (b) C/D 10


case of a plane-strain tunnel in kaolin clay with a uniformstrength profile and zero surcharge, the stability boundspredicted by limit analysis are in excellent agreement withthe experimental observations.

Finite-element limit analysis results for a three-dimen-sional tunnel heading in kaolin clay with a uniform strength

profile are shown in Fig. 34. These curves indicate thetunnel pressure required to maintain stability, �t/su, as afunction of the unsupported heading length P/D for a casewhere C/D ¼ 3 and ªD/su ¼ 3.6. Note that, since the tunnelsupport pressure works against failure in this instance, anupper-bound calculation actually gives a lower bound on thesupport pressure, and vice versa. Fig. 34 also shows thetheoretical collapse pressure (accurate to 1%) for the corres-ponding case of a plane-strain tunnel where P/D!1, as

C D

D s

D s

/ 4

/ 3

/ 0

4000 elements

�

�

�

γ u0

u0ρ

Fig. 31. Optimised mesh and failure mechanism for circular tunnel with C/D 4 and uniformstrength

C D

D s

D s

/ 4

/ 3

/ 1

�

�

�

γ u0

u0ρ

Fig. 32. Failure mechanism for circular tunnel with C/D 4 and finite strength gradient

�σ t

u/s

σ

γs

u

u

0

/ 2·6

Uniform

�

�D s

s

3·02·52·01·5�5

�4

�3

�2

�1

0

1·0

UB

LB

Centrifuge(Mair, 1979)

C D/

Fig. 33. Comparison of limit analysis predictions with centrifugeresults for plane-strain tunnel

0

2

4

6

8

10

0 1 2 3 4 5 6

σ tu

/s

P D/

P D/ → ∞

C DD s/ 3/ 3·6

��γ u

UBLB(UB LB)/2�

Centrifuge (Mair, 1979)

Fig. 34. Comparison of limit analysis predictions with centrifugeresults for three-dimensional tunnel heading

554 SLOAN

well as the centrifuge results taken from Mair (1979). In thisexample there is a bigger gap between the upper and lowerbounds, since a trial-and-error meshing procedure was neces-sary using 20 000–30 000 tetrahedral elements (the three-dimensional adaptive limit analysis methods are still underdevelopment). Overall, however, the lower-bound tunnelpressure predictions are close to the observed results of Mair(1979), and asymptote clearly towards the limiting value forplane-strain conditions. For values of P/D > 1, the averageof the upper and lower bounds underpredicts the measuredtunnel pressures by a maximum of 15%. Not surprisingly,this problem is computationally demanding, and typicallyrequired around 4 h of CPU time for each analysis with agrid of 30 000 tetrahedral elements.

Figure 35 shows the power dissipation plots for two of thecases in Fig. 34, where P/D ¼ 0 and P/D ¼ 4. In the former,the failure mechanism involves mostly soil that is directlyabove or in front of the tunnel face, and comprises severalzones of intense plastic shearing. As expected, failure in thecase of P/D ¼ 4 is associated with a much larger zone ofplastic deformation, including collapse of the tunnel roofand heave of the tunnel floor. Clearly, however, the mode ofplastic deformation is not uniform along the length of thetunnel, which suggests that the condition of plane strain hasnot been reached. For all the lower-bound analyses, three-dimensional extension elements were added to the edge ofthe grids (not shown in Fig. 35) to ensure that the stressfields were statically admissible over the semi-infinitedomain.

CONSEQUENCES OF AN ASSOCIATED FLOW RULEFor total stress analysis of the undrained stability of clays,

where the friction angle is assumed to be zero and alldeformation takes place at constant volume, the assumptionof an associated flow rule has little influence on the failureload. For drained stability analysis involving soils with highfriction angles, however, the use of an associated flow rulepredicts excessive dilation during shear failure, and raisesthe question of whether the bound theorems will providerealistic estimates of the limit load.

Theorems for non-associated flow rulesIn a pioneering investigation of the crucial issue of non-

associated flow, Davis (1968) argued that the flow rule willnot have a major influence on the limit load for frictional

soils unless the problem is strongly constrained in a kine-matic sense. A precise definition of the degree of kinematicconstraint is elusive, but many geotechnical collapse modesare not strongly constrained, since they involve a freelydeforming ground surface and a semi-infinite domain. Forthese cases, Davis (1968) conjectured that it is reasonable toassume that the bound theorems will give acceptable esti-mates of the true limit load. In addition, by examining thefailure behaviour on slip-lines for a non-associated Mohr–Coulomb material, he established that the shear and normalstress are related by

� ¼ � n tan�� þ c�

where c� and �� are ‘reduced’ strength parameters, definedby

c� ¼ �c9

tan�� ¼ � tan�9

�� ¼ cosł9 cos�9

1� sinł9 sin�9(38)

and c9 is the effective cohesion, �9 is the effective frictionangle and ł9 is the dilation angle. The use of these reducedstrengths provides a practical means for dealing with non-associated flow in limit analysis, and will be explored laterin this paper. When considering the behaviour of real soil itshould, of course, be remembered that the dilation angleactually varies during the plastic deformation that precedesfailure, and in fact approaches zero at the critical state.Nonetheless, in the absence of laboratory or field data, aconstant rate of dilation is often assumed in practice, withvalues in the range 0 < ł9 < �9/3 being typical.

Apart from the approach suggested above by Davis(1968), very few useful theoretical results are available formodelling non-associated flow in a cohesive-frictional soil.If a plastic potential g is defined so that the plastic strainrates are now given by _�p

ij ¼ _º@g=@� ij, where g is convexand contained within the yield surface f, the impact of theflow rule can be estimated by using the following results.

(a) A conventional upper-bound calculation gives a rigorousupper bound on the limit load for an equivalent materialwith a non-associated flow rule (Davis, 1968).

(b) A rigorous lower bound on the limit load for a non-associated material can be obtained by substituting theplastic potential for the yield criterion in the staticadmissibility conditions (Palmer, 1966).

Although conceptually valuable, these two theorems fre-quently furnish weak bounds if the dilation angle is consider-

P D/ 0� P D/ 4�

Fig. 35. Failure mechanisms for three-dimensional tunnel headings with C/D 3 and ªD/su 3.6


ably less than the friction angle. Unfortunately, this is oftenthe case for materials with high friction angles, such as densesands. Drescher & Detournay (1993), in a stronger result,proved that the limit load obtained from a rigid block mechan-ism with Davis’ discontinuity strengths c� and ��, as definedin equations (38), gives an upper bound on the true limit loadfor a non-associated material with parameters (c9, �9, ł9).This theorem suggests that limit analysis with Davis’ reducedstrength parameters may provide useful estimates of the limitload, provided collapse is triggered by localised plastic de-formation along a well-defined failure surface.

Volume change behaviour of real soilFor a Mohr–Coulomb material undergoing plastic deform-

ation, the shear strength is governed by the effective cohe-sion c9 and friction angle �9, while the volume change iscontrolled by the dilation angle ł9. With an associated flowrule it is assumed implicitly that ł9 ¼ �9, whereas for a realsoil ł9 , �9, so that plastic deformation obeys a non-associated flow rule. Fig. 36 shows the dilation predicted bythese two assumptions for plastic shearing along a planarfailure surface. For the same shear displacement increment(velocity jump) ˜us, the associated flow rule gives a largernormal displacement increment (velocity jump) ˜un, andhence a larger volume change in the material.

Under a general state of stress, the volumetric plasticstrain rate is related to the maximum principal strain rate by_�p

v ¼ [tan2 (458þ ł9=2)� 1] _�p1, where tensile strains are

taken as positive. Typical plots of _�pv against _�p

1 for a varietyof soils, shown in Fig. 37, indicate clearly that the dilationangle varies throughout the process of failure, and eventuallyapproaches zero at the critical state. Moreover, even in stressranges where the rate of volume change is constant, thedilation angle is often appreciably less than the correspond-ing friction angle. All the above observations suggest thatgreat care should be exercised when using a simple Mohr–Coulomb model with an associated flow rule to predict thelimit load under drained loading conditions, particularly forsoils with high friction angles.

Biaxial test with Mohr–Coulomb materialTo further investigate the influence of the flow rule on the

collapse load for a cohesive-frictional problem, the biaxialcompression of a plane-strain block of Mohr–Coulombmaterial is now considered, as shown in Fig. 38. Two length-to-width ratios of L/B ¼ 1 and L/B ¼ 3 are analysed, eachusing a rigidity index G/c9 ¼ 300 and Mohr–Coulombparameters of �9 ¼ 308 and ł9 ¼ 08, 158, �9. Provided thesample length is such that L > B tan(458 + ł9/2), a failureplane is free to form across the specimen at an angle ofŁ ¼ 458 + ł9/2 to the horizontal, and the exact collapsepressure is given by qu ¼ 2c9tan(458 + �9/2). For shortersamples where L , B tan(458 + ł9/2), the exact collapsepressure is unknown and must be determined numerically.

To begin the investigation, the displacement finite-elementcomputer program SNAC (Abbo & Sloan, 2000) was usedto analyse this problem with both associated and non-associated flow rules. The mesh employed for L/B ¼ 1 isshown in Fig. 39, and comprises 800 quartic triangles. Asimilar mesh is used for the case L/B ¼ 3, except that the

Non-associated flow rule

ψ φ �

Δ Δu un s| |tan� φ

Δus

ψ φ Δ Δu un s| |tan� ψ

Δus

Associated flow rule

Fig. 36. Dilation during shearing on a planar failure surface

σσ

13

�

ε pv

.

Dense sandOC clay

Loose sandNC clay

ψ φ

ψ � 0

ψ � 0

Axial strain

Axial strain

Dense sandOC clay

Loose sandNC clay

Fig. 37. Drained triaxial test behaviour

θ ψ45° /2� �

qu

Rough platen

Rough platen

qu

L

B Mohr–Coulomb/ 1, 3/ 300

30°0°, 15°,

L BG c

� �

� �

φψ φ

Fig. 38. Biaxial compression of Mohr–Coulomb block

556 SLOAN

mesh is replicated three times in the vertical direction,giving a total of 2400 elements. As discussed in ‘Displace-ment finite-element analysis’, the 15-noded element is ableto simulate plastic collapse accurately for both isochoric anddilatational constitutive models, and it is computationallyefficient. To further enhance the reliability of the numericallimit load estimates, the adaptive load stepping and stressintegration features of SNAC were used with very tighttolerances on the unbalanced forces, stress error and yieldsurface drift (Abbo & Sloan, 1996; Sloan et al., 2001).

The results from the displacement finite-element analysesare shown in Fig. 40. These indicate that, for the long samplewith L/B ¼ 3, the ultimate pressure matches the theoreticalvalue of qu/c9 ¼ 2 tan(458 + �9/2), and is unaffected by theflow rule. Indeed, the load–deformation responses for thethree cases with ł9 ¼ 08, ł9 ¼ 158 and ł9 ¼ �9 ¼ 308 areindistinguishable. This is because the failure plane is free toform across the specimen without intersecting the loadingplatens, and confirms the proposition of Davis (1968) that theflow rule will not have a major influence on the collapse loadfor problems where the degree of kinematic constraint is low.The load–deformation results for the short sample, on theother hand, are more sensitive to the value of the dilationangle, and, for the non-associated models, exhibit oscillationsafter the onset of plastic deformation. In the most extremecase of a strongly non-associated flow rule with zero plasticvolume change (ł9 ¼ 08), the numerical solution processbecomes unstable at a total vertical strain of around 0.8%.This difference in sensitivity to the flow rule is a direct result

of the fact that the failure plane cannot form at an angle ofŁ ¼ 458 + ł9/2 without intersecting the end platens, whichcauses the deformation field to be kinematically constrained.The numerical instability shown in Fig. 40 is not unusual fordisplacement finite-element analysis with a non-associatedMohr–Coulomb model, and can be especially severe whenthe friction angle is large and ł9� �9 (e.g. De Borst &Vermeer, 1984).

To investigate the collapse pressures and failure mechan-isms for the cases with an associated flow rule, the biaxialtest problem was reanalysed using discrete limit analysiswith adaptive grid refinement and a maximum of 20 000elements. The results for the short specimen with L/B ¼ 1,shown in Fig. 41, bracket the exact collapse pressure towithin 1%, so that 4.48c9 < qu < 4.53c9. The correspondingfailure mechanism, as indicated by the power dissipationcontours, is clearly constrained to intersect the edges of theloading platens, and is highly localised. In contrast, for thelonger specimen with L/B ¼ 3, the failure mechanism ismore diffuse and shows no tendency to localise, even with amesh of 20 000 elements. The predicted collapse pressurefor this case matches the exact analytical value ofqu/c9 ¼ 2 tan(458 + �9/2).

Finally, to investigate the potential of using the reducedstrength parameters proposed by Davis (1968), equation (38)was employed to evaluate c� and �� for the short non-associated sample with c9 ¼ 1, �9 ¼ 308 and ł9 ¼ 158. Theresulting strength parameters c� ¼ 0.96 and �� ¼ 29.028were then adopted to estimate the collapse pressure usingthe adaptive discrete limit analysis methods with a cap of20 000 elements. This process gave a collapse pressure inthe range 4.12c9 < qu < 4.16c9, which is approximately8% below the best estimate of qu � 4.47c9 from the non-associated displacement finite-element results shown in Fig.40. Bearing in mind the uncertainty that is linked to thenon-associated displacement finite-element results, this sug-gests that the ‘Davis parameters’ defined by equation (38)may provide a viable option for predicting the limit loadsfor frictional materials with non-associated flow rules. In-deed, this option will be explored further in the next section,and is supported independently by the recent work ofKrabbenhøft et al. (2012).

APPLICATIONS: DRAINED STABILITY ANALYSISThe cases considered in the section on ‘Undrained stabil-

ity analysis’ demonstrated the potential and versatility offinite-element limit analysis for predicting the stability of awide range of problems under undrained conditions. In thissection, these methods are applied to problems involvingdrained loading for both purely frictional and cohesive-frictional Mohr–Coulomb materials.

Pullout capacity of rough circular anchor in sandSoil anchors are widely used to provide uplift or lateral

resistance for structures such as transmission towers, sheet-pile walls and buried pipelines. Although most plate anchorsare usually square, circular or rectangular in shape, manyexisting solutions have been developed for plane-strainstrips, since these are significantly easier to analyse. Acomprehensive survey of solutions that are available forpredicting the capacity of various types of anchor in sandcan be found in Merifield et al. (2006a) and Merifield &Sloan (2006). These authors also summarise the results of alarge number of chamber, centrifuge and field tests that canbe used to verify theoretical predictions.

Here the pullout capacity, Qu, of a rough circular anchorin sand with unit weight ª and friction angle �9 is consid-

u ux y0, prescribed�

800 quartic triangles

u ux y 0� �

y

x

Rough platen

Fig. 39. Displacement finite-element mesh for biaxial compressionof Mohr–Coulomb block (L/B 1)

1·21·00·80·60·40·2

q cu/ 2 tan(45° /2) � � φ

SNAC finite element

L B/ 1 30°� � �ψ φ

L B/ 1 15°� �ψ

L B/ 1 0°� �ψ

L B/ 3 0°, 15°,� � ψ φ

Vertical strain: %

0

1

2

3

4

5

0

Pre

ssur

e/c

Fig. 40. Displacement finite-element results for biaxial compres-sion of Mohr–Coulomb block


ered, as shown in Fig. 42. For an anchor of diameter Dburied at depth H, the ultimate load capacity can be ex-pressed in the form Qu ¼ ªHANª, where A ¼ �D2/4 is theanchor area, and Nª is a dimensionless ‘breakout’ factor thatis a function of �9 and H/D. Fig. 42 illustrates the finite-element limit analysis mesh used for an anchor with H/D ¼ 2. Even though the anchor problem is axisymmetric innature, a three-dimensional slice is analysed to obtain fullyrigorous upper- and lower-bound solutions that properlyaccount for the hoop components of velocity and stress. Thenumber of tetrahedra used in the limit analysis calculationsranged from approximately 2000 (for H/D ¼ 2) to 14 000(for H/D ¼ 10), with corresponding CPU times of 5–80 min.In all lower-bound analyses, three-dimensional extensionelements were employed to extend the stress field over thesemi-infinite domain (these are not shown). To estimate Nª

for each geometry, the vertical pullout force Qu was opti-mised directly after specifying the material properties andanchor dimensions.

For the case of an anchor in a medium-dense sand with�9 ¼ 43.18 and ł9 ¼ 13.68, Fig. 43 shows the breakout factorNª predicted from discrete limit analysis, as well as theSNAC displacement finite-element code. In the former set ofanalyses, the upper and lower bounds on Nª were computedby adopting the reduced friction angle of �� ¼ 38.48,defined by Davis’ equation (38), to account for the influenceof non-associated flow. Fig. 43 also shows the laboratory test

results reported by Pearce (2000) for a sand with anidentical friction angle and dilation angle, and by Ilampar-uthi et al. (2002) for a sand with a friction angle of�9 ¼ 438. Overall, discrete limit analysis provides goodpredictions of Nª for all anchor depths, although there issome discrepancy with the observations of Pearce (2000) forhigh values of H/D (where the author reported that theeffects of his chamber dimensions could be significant).Indeed, although they are preliminary, these limit analysisresults provide encouraging support for the option of usingDavis’ reduced strengths for soils with high friction angles,where the influence of non-associated flow is most likely tobe significant. Interestingly, the limit analysis estimates ofNª also compare well with the displacement finite-elementpredictions, which were based on the actual measured fric-tion and dilation angles of �9 ¼ 43.18 and ł9 ¼ 13.68. Itshould be noted, however, that considerable judgement wasneeded to determine the values for Nª from some of thedisplacement finite-element computations, owing to oscilla-tions in the load–deformation response. These oscillationswere similar in magnitude to those in observed in Fig. 40for the non-associated analyses of the biaxial test, and in afew cases led to numerical problems associated with poorconvergence. No such problems occur when the ‘Davisparameters’ are adopted in the limit analysis formulations,since these assume an associated flow rule.

For a cohesionless material such as sand, the quantity to

L Bc

q c

/ 11

30°4·48 / 4·53

� � � �

� �φ ψ

u

L Bc

q c

/ 31

30°/ 3·46 exact

� � � �

� �φ ψ

u

Fig. 41. Finite-element limit analysis meshes and failure mechanisms for biaxial compression of Mohr–Coulomb block(associated flow rule)

558 SLOAN

be minimised in an upper-bound calculation is simply therate of work done by any set of fixed external tractions orbody forces, since the internal dissipation, defined by equa-tion (16), is identically zero. To visualise the failure mechan-ism for this type of material it is convenient to plot contoursof the plastic multipliers, since these indicate the magnitudesof the plastic strain rates and thus can be used to identifyzones of intense plastic deformation. Two such plots for acircular anchor in medium-dense sand with �9 ¼ ł9 ¼ 358are shown in Fig. 44. Both the shallow (H/D ¼ 2) and deep(H/D ¼ 10) cases show clearly defined failure mechanismsand yield bounds on Nª that differ from their averages by�6% and �4% respectively.

Stability of an unsupported circular excavation in cohesive-frictional material

The stability of an unsupported circular excavation, of depthH and radius R, in a cohesive-frictional Mohr–Coulombmaterial is now considered (Fig. 45). Like the previous anchorexample this case is axisymmetric, but was treated using athree-dimensional 158 slice to account properly for the hoopterms in the bound calculations. To simplify the study thesame meshes were used for the upper- and lower-boundanalyses, except that extension elements were not required inthe former. The stability number for the excavation, ªH/c9,was found by optimising the unit weight ª after fixing thecohesion c9, the friction angle �9 and the ratio H/R. Thisexample thus illustrates the benefits of being able to optimise abody force directly in the discrete limit analysis formulations.

Figure 46 shows finite-element limit analysis solutions forthe stability number ªH/c9 where H/R ¼ 1, 2, 3 and �9 ¼ 08,108, 208. For the deepest excavation the upper and lowerbounds differ from their average by a maximum of �2.5%,while for the shallowest excavation the bounds are evencloser, with a difference of less than �0.5%.

Also shown in Fig. 46 are solutions for the purelycohesive case derived by Britto & Kusakabe (1982), Pastor& Turgeman (1982) and Turgeman & Pastor (1982). Britto& Kusakabe’s upper bounds, obtained from an axisymmetricmechanism, compare reasonably well with the finite-elementlimit analysis results for all the geometries considered.Similarly, the upper-bound solution of Turgeman & Pastor(1982), found from an axisymmetric finite-element formula-tion based on linear programming, also gives a good esti-mate of ªH/c9 for the case H/R ¼ 1.

The influence of the friction angle on the shape of thefailure mechanism can be seen from Fig. 47, which showscontour plots of the element plastic multipliers for thedeepest excavation with �9 ¼ 08 and �9 ¼ 208. As expected,the zone of plastic deformation is much more extensive forthe purely cohesive case, being roughly twice as wide asthat for the excavation with �9 ¼ 208.

q H Nu � γ γ N f H Dγ ( , / )� φ

σ τ τnn tn sn 0� � �

u u ux y z 0� � �

τ τtn sn 0� �

uu

ux

yz

0�

��

Qu � q Au

D

15°

Mesh for H D/ 2�

H

Rough anchor

Unit weight � γFriction angleCohesion 0

� � �

φc

un 0�

Fig. 42. Circular anchor in sand: problem definition and limit analysis mesh

108642

LB 43·1°, 13·6°, * 38·4°φ ψ φ � � �

SNAC FEA 43·1°, 13·6°φ ψ � �

Pearce (2000) 43·1°, 13·6°φ ψ � �

Ilamparuthi . (2002) 43°et al φ �

0

20

40

60

80

100

120

140

0

Nγ

H D/

UB 43·1°, 13·6°, * 38·4°φ ψ φ � � �

Fig. 43. Comparison of limit analysis and displacement finite-element predictions with chamber test results for circular anchorin sand


INCORPORATION OF PORE PRESSURES IN LIMITANALYSIS

Pore water pressures have a major effect on the stabilityof many geotechnical structures, and it is important that theyare properly accounted for. In this section, a new approachis described that incorporates the effects of steady-stateseepage in finite-element limit analysis. To find the steady-state pore pressures, the governing seepage equation issolved using optimisation theory and finite elements. Bothconfined and unconfined seepage flow conditions are mod-elled efficiently, and the problem of locating the phreaticsurface in the latter presents no special difficulty. Since theproposed method employs the same mesh as the upper- and

lower-bound analyses, there is no need to import and inter-polate the pore pressures from another grid (or program),which is a significant practical benefit.

During the iterative solution process, a Hessian (curva-ture)-based error estimator is applied to the pore pressurefield to generate a mesh that gives accurate pore pressures.Simultaneously, the ‘bounds gap’ error estimator of thesection ‘Adaptive mesh refinement’ is employed to identify aseparate mesh that gives accurate upper and lower boundson the limit load. By combining these two strategies, ahybrid refinement strategy is developed that minimises boththe bounds gap and the error in the computed pore pres-sures.

H BN

/ 26·38 7·17

�� γ

H BN

/ 1077·15 83·18

�� γ

γ φ ψ20 kN/m , 35°, 0� � � �2 c

Fig. 44. Plastic multiplier (strain) contours for shallow and deep circular anchors in sand

H

2R

τ τtn sn 0� �

uu

ux

yz

0�

��

σnn 0�τ τtn sn 0� �

15°

Extension mesh(LB only)

Unit weight � γFriction angle 0°,10°, 20°Cohesion 1

� ��

φc

Stability number / ( , / )� � γ φH c f H R

un 0�

Fig. 45. Circular excavation in cohesive-frictional soil: problem definition and limit analysis mesh

560 SLOAN

Determination of steady-state pore pressuresWhen seepage flow is present, the pressure head needs to

be found in order to compute the pore pressure and effectivestress at any point. Thus, during each iteration, the relevantseepage problem must first be solved before the stabilityanalysis can be carried out. In the case of confined flow allthe boundary conditions are known a priori, and thehydraulic head can be found by solving the governingseepage equation. For unconfined seepage, however, theconditions on some sections of the boundary are unknown,and must be determined as part of the solution. An exampleof the latter case is the flow of water through an earth dam,where the hydraulic head on either side of the dam is knownbut the precise location of the phreatic surface within thedam is not. Interestingly, both types of flow can be modelledin a single optimisation formulation that is based on avariational inequality (e.g. Crank, 1984).

By combining the fluid balance equations with Darcy’slaw for two-dimensional seepage through an isotropic porousmedium, the governing equation for the total head H isobtained as

k x, yð Þ=2H ¼ k x, yð Þ@2H

@x2þ k x, yð Þ

@2H

@y2¼ 0

where k(x, y) is the soil permeability. Using standard vari-

ational calculus, the solution to this equation can be writtenas the optimisation problem

Minimise 12

ðV

=Hð ÞTk=HdV (39)

subject to appropriate boundary conditions on the total headH.

For the two-dimensional case, this problem can be dis-cretised using the linear triangular element shown in Fig. 48according to

H ¼X3

i¼1

NiHi ¼ NeHe (40)

where Ni are linear shape functions, Ne ¼ [N1, N2, N3] is theelement shape function matrix, and He ¼ {H1, H2, H3}T isthe element vector of unknown nodal heads. Substituting theexpression for H from equation (40) into equation (39)gives, after some manipulation, the discrete optimisationproblem

Minimise 12HTKH

subject to AH ¼ H0(41)

where H is a global vector of unknown nodal heads, A is amatrix describing the constant head boundary conditions,and K is a flow matrix defined by

K ¼XE

e

ðAe

=NeTk x, yð Þ=NedA (42)

in which E is the number of triangular elements, and =Ne

denotes the gradient of the shape function matrix for ele-ment e. Using numerical integration to evaluate K, thesolution to the quadratic optimisation problem in equation(41) is straightforward, and is defined by the linear relations

KH ¼ 0 (43)

subject to the boundary conditions AH ¼ H0: Note that, inthe formulation used here, which employs the same meshfor the pore pressure and the limit analysis calculations, thematrix A also contains terms to enforce continuity of thehead across the discontinuities between adjacent elements.After solving equation (43) for the total head at each node,the corresponding pore pressures, p, are readily obtained asp ¼ (H � z)ªw, where z is a vector of the nodal elevationheads and ªw is the unit weight of water.

For problems involving unconfined seepage flow, thequadratic optimisation problem (equation (41)) is augmentedby the constraint p ¼ (H � z)ªw > 0. This additional condi-tion can be used to compute the pore pressures and phreaticsurface using the following algorithm.

3·02·52·01·5

LB Pastor & Turgeman (1982) φ � 0

φ � 20°

φ � 10°

φ � 0°

0

2

4

6

8

10

12

14

16

18

1·0

γHc/

H R/

UB

LBUB Britto & Kusakabe (1982) 0φ �

UB Turgeman & Pastor (1982) φ � 0

Fig. 46. Stability of unsupported circular excavation in cohesive-frictional soil

φγ

� ��

0°, / 36·65 / 6·74

H RH c �

φγ

� �� 20°, / 3

12·79 / 13·1H R

H c �

Fig. 47. Plastic multiplier (strain) contours for collapse ofcircular excavation

H 1

y

x

H 2

H 3

1

3

2

Fig. 48. Linear finite-element for modelling total head


1. Solve equation (43) to give the nodal heads H.2. Compute the nodal pore pressures using the relation

p ¼ (H � z)ªw:3. If the change in objective function HTKH is less than a

small tolerance, exit with the final pore pressures.4. For all nodes i where the pore pressure pi , 0, adjust the

nodal permeability using the relation �kki ¼ s( p)ki, wheres(p) is a smoothed step function that ranges between 0and 1 (see Fig. 49).

5. Recompute K using equation (42) with the adjusted nodalpermeabilities for each element �kki; then go to step 1.

This process, although relatively crude, typically locates thephreatic surface in five or six iterations, and thus imposesonly a small overhead on the overall limit analysis computa-tion. The smoothed step function in step 4 is introduced tominimise the occurrence of pore pressure oscillations in thevicinity of the phreatic surface. This function can take avariety of forms, although the simple expressions(p) ¼ 1

2[1þ tanh (Æp)], shown in Fig. 49, works well in

practice with Æ ¼ 50. When computing the contributions toK in step 5, the permeability is assumed to vary linearlyover each element. This gives a ‘weighted’ permeability forelements that are bisected by the phreatic surface, and aidsconvergence of the iteration scheme.

Lower-bound formulation with steady-state pore pressureThe inclusion of pore water pressure in the lower-bound

method involves the use of effective stresses when enforcingthe yield constraints, whereas total stresses are employed whenimposing the equilibrium and stress boundary conditions.Since it uses the same mesh, the pore pressure is treated as anauxiliary variable that, like the effective stress, varies linearlyover each element. This is shown for the two-dimensional casein Fig. 50. Note that, during the limit analysis calculations foreach mesh, the pore pressure field is fixed.

Upper-bound formulation with steady-state pore pressureThe inclusion of pore pressure in the upper-bound method

requires the use of effective stresses when enforcing theyield condition and flow rule. There is also an additionalterm in the governing equation (15) that is due to the rate ofwork done by the pore pressure field. The pore pressure fieldis again treated as an auxiliary variable that varies linearlyover each element, as shown for the two-dimensional case inFig. 51.

Following Kim et al. (1999), the additional term in equa-tion (15) due to the rate of work done by the static porepressures means that the quantity to be minimised becomes

_W ¼ð

V

�T _�pdV �ð

At

tTudA�ð

V

gTudV �ð

V

=pTudV

where =p ¼ @p=@x, @p=@yf gTis the gradient of the pore

pressure field. Assuming that the pore pressure varies lin-early, these derivatives are uniform over each element, andare given by the equations

@p

@x¼X3

i¼1

@N i

@xpi ¼

X3

i¼1

bipi

@p

@y¼X3

i¼1

@Ni

@ypi ¼

X3

i¼1

cipi

where pi are nodal pore pressures, and the constants bi andci depend on the element nodal coordinates.

Limit analysis with adaptive mesh refinement in presence ofpore pressures

As mentioned previously, a hybrid mesh refinement strat-egy can be developed that minimises the error in both thepore pressures and the upper and lower bounds. This isbased on predicting good element sizes for the pore pres-sures using a Hessian (curvature)-based error estimator(Almeida et al., 2000), together with element sizes thatdirectly minimise the bounds gap (as described in thesection ‘Adaptive mesh refinement’). Where the elementsizes predicted by these two separate approaches differ, thehybrid scheme simply chooses the smallest one. Details ofthe Hessian-based scheme for selecting element sizes, in thecontext of lower-bound limit analysis, can be found inLyamin et al. (2005b). Exactly the same approach is usedhere, with the ‘isotropic’ form of the method being imple-mented, which omits element ‘stretching’.

The steps involved in performing finite-element limitanalysis with adaptive mesh refinement, allowing for thepresence of steady-state pore pressures, may be summarisedas follows.

1. Specify the maximum number of continuum elementsallowed, Emax, and generate an initial mesh.

2. Compute the nodal pore pressures for the mesh using thealgorithm described in the section ‘Determination ofsteady-state pore pressures’, allowing for unconfined flowif needed.

3. Perform upper-bound and lower-bound analyses using thesame mesh as in step 2.

4. If the gap between the upper and lower bounds is lessthan a specified tolerance, or if the maximum number ofcontinuum elements Emax is reached, exit with upper- andlower-bound estimates of the limit load.

5. Specify a target number of continuum elements for thecurrent mesh iteration, Ei, with Ei < Emax:

6. Using the nodal pore pressure field and the Hessian-basederror estimator of Almeida et al. (2000), compute the

1 s p p( ) [1 tanh( )]� �12 α

k s p k( )�

s p( )

p

Fig. 49. Smoothed step function in permeability for locatingphreatic surface

Node { , , , }σ � σ σ τxx yy xy p Ti i i

Element { , }he ex

ey� h h T

yx

Fig. 50. Lower-bound element with auxiliary pore pressure

Node { , , }ui ix

iy� u u p T

Element { , , }σe T� σ σ τxx yy xye e e

yx

Fig. 51. Upper-bound element with auxiliary pore pressure

562 SLOAN

optimum size of each element, subject to the constraintthat the new number of continuum elements in the gridmatches the predefined target number of continuumelements for the current iteration Ei:

7. For each element, compute its contribution to the boundsgap ˜e using equation (37). In the case of a discontinuityelement, its bounds gap contribution is added to theneighbouring continuum element with which it shares themost nodes. Then scale the size of each continuumelement to be inversely proportional to the magnitude of˜e, subject to the constraint that the new number ofcontinuum elements in the grid matches the predefinedtarget number of continuum elements for the currentiteration Ei:

8. Compare the predicted size for each element from steps 6and 7, and choose the smallest one. Then scale theelement sizes to meet the target number of continuumelements for the current iteration Ei:

9. Go to step 2.

APPLICATIONS: SLOPE STABILITY ANALYSISNow the classical problem of slope stability is considered,

and the solutions from finite-element limit analysis arecompared with those found by conventional methods. Twocases are considered, one with no seepage flow and one withunconfined seepage flow, and both have a weak layer thatcauses a non-circular failure surface to develop. To permitdirect comparisons with conventional methods of stabilityanalysis, an efficient strength reduction scheme is describedthat gives the safety factor in terms of the shear strengthrather than the applied load.

Slope in cohesive-frictional soil with weak layerThe first example, taken from the benchmark prediction

exercise documented in Donald & Giam (1989a), is shownin Fig. 52. The problem is designed to develop a non-circular failure plane that propagates along the weak zone,and is a useful test for conventional slope stability methodsas well as finite-element limit analysis.

Using the algorithm described in the section ‘Adaptivemesh refinement’, adaptive finite-element limit analysis wasperformed with a maximum of 4000 continuum elements.Unlike previous examples, however, a strength reductionprocess was followed to compute the safety factor in termsof the shear strength (rather than the applied load). Thisprocess, shown graphically in Fig. 53, can be summarised bythe following steps.

1. Start by assuming a trial safety factor, F0 ¼ 1.2. Compute the available strengths c9a ¼ c9=F0 and

�9a ¼ tan�1 (tan�9=F0):3. Using the available strengths (c9a, �9a) and the adaptive

finite-element limit analysis algorithm given in the

section ‘Adaptive mesh refinement’, compute upper andlower bounds on the unit weight that can be supported bythe slope (ªLB, ªUB). Then compute the mean of thesebounds according to �ªª ¼ (ªUB þ ªLB)=2 and the gravitymultiplier m0 ¼ �ªª=ª, where ª is the actual unit weight.

4. If m0 , 1, set ˜F ¼ �0.1; else set ˜F ¼ 0.1.5. Compute F1 ¼ F0 + ˜F.6. Compute the available strengths c9a ¼ c9=F1 and

�9a ¼ tan�1 (tan�9=F1):7. Using the available strengths (c9a, �9a), compute upper

and lower bounds on the unit weight (ªLB, ªUB). Thencompute the mean according to �ªª ¼ (ªUB þ ªLB)=2 andthe multiplier m1 ¼ �ªª=ª:

8. If (m1 � 1)(m0 � 1) . 0, then set m0 ¼ m1 and F0 ¼ F1

and go to step 5.9. Linearly interpolate the factor of safety according to

F ¼ F0 + (F1 � F0)(m0 � 1)/(m0 � m1).

This process starts by assuming a trial estimate of the safetyfactor, and continues with a simple marching scheme untilthe factor of safety is found that gives a gravity multiplieron the unit weight, m, of unity. Instead of taking the averageof the upper and lower bounds on the unit weight tocompute this multiplier in steps 3 and 7, it is of coursepossible to use the actual lower or upper bounds, and hencecompute an upper or lower bound on the safety factor F.This is an attractive feature, but it is generally unnecessaryowing to the very tight bounds (better than 1%) that aregenerated by the finite-element limit analysis approach. Forthis particular example, the safety factor F ¼ 1.27 was foundafter four iterations, and required around 30 s of CPU time.

The optimised mesh at the completion of the strengthreduction process, shown in Fig. 54, indicates that thebounds gap error estimator has concentrated the elementsalong the failure surface, precisely where they are needed.The corresponding plots of the velocity vectors and plasticmultipliers (strains), shown in Figs 55 and 56 respectively,

c � � �0, 10°, 18·84 kN/mφ γ 3

c � ��

28·5 kN/m20°

18·84 kN/m

2

3φγ

26·6°

0·5 m

0·75 m

12·25 m

Fig. 52. Slope with weak layer: no seepage flow

m(

)/2

��

γγ

γU

BLB

1·401·301·201·10

Trial 1, m � 2·059

Trial 2, m � 1·477

Trial 3, m � 1·141

Trial 4, m � 0·926

0

0·5

1·0

1·5

2·0

2·5

1·00Factor of safety, F

F � 1·27

Fig. 53. Strength reduction process for slope with weak layer


confirm that the mode of failure is dominated by intenseshear deformation in the weak layer of cohesionless material.Interestingly, the latter plot indicates that a secondary failuremechanism also occurs along a plane at right angles to theslope face.

Figure 57 compares the factors of safety computed fromfinite-element limit analysis and a variety of conventionallimit-equilibrium methods. The latter, reported in Donald &Giam (1989a), indicate significant variations in the safetyfactor, even for analyses with the same procedure. Thesevariations reflect the difficulty in locating the critical limit-equilibrium failure surface, which is actually an uncon-strained optimisation problem that demands sophisticatedstrategies to obtain a reliable solution (especially if thefailure surface is permitted to be non-circular). In contrast,the solutions from the finite-element limit analysis methodare guaranteed to give the best possible upper and lower

bounds for a specified mesh, since the governing optimisa-tion problem is both constrained and convex (provided theyield surface is convex).

A further complication with limit-equilibrium proceduresis that they each make different assumptions in order toobtain a solution, some of which are physically morejustified than others. This has resulted in a multitude oftechniques being proposed in the literature, as well as end-less debates on which one is the best. A detailed discussionof the theory and merits of various limit-equilibrium ap-proaches can be found in Duncan & Wright (2005). Withregard to the results compared in Fig. 57, the methods ofMorgenstern & Price (1965), Spencer (1967) and Sarma(1973, 1979) may be viewed as ‘complete equilibrium’techniques, since they satisfy both force and moment equi-librium for each slice. Compared with the limit analysisprediction of F ¼ 1.27, the various implementations of the

Lower-bound extension elementsnot shown

uu

xy

0�

�

uu

xy

0�

�

u ux y 0� �

σ τnn sn 0� �

Fig. 54. Optimised mesh for slope with weak layer

Fig. 55. Velocity vectors at collapse for slope with weak layer

Fig. 56. Plastic multiplier (strain) contours at collapse for slope with weak layer

564 SLOAN

Morgenstern–Price, Spencer and Sarma methods reported inDonald & Giam (1989a) gave, respectively, F ¼ (1.242, 1.2),F ¼ (1.31, 1.24) and F ¼ (1.27, 1.273, 1.51). Hence, to threesignificant figures, two of the Sarma predictions coincidewith those of the limit analysis method, whereas the averageof the Spencer estimates is F ¼ 1.275. The generalisedwedge method of Donald & Giam (1989b) also gives asafety factor of 1.27. Interestingly, it can be shown (Giam &Donald, 1989a) that this technique gives answers identical tothose of the rigorous upper-bound wedge method of Giam &Donald (1989b) and Donald & Chen (1997), which furthercorroborates the finite-element limit analysis estimate. Theoverestimates of the safety factor provided by the simplifiedBishop method reflect the fact that it is better suited to caseswhere the failure surface can be approximated by a circle.

For completeness, Fig. 57 also shows the factor of safetycomputed by the displacement finite-element code PLAXIS2D (2011) using strength reduction. The estimate from thismethod of F ¼ 1.20 is slightly low, possibly because theprogram assumes non-associated flow for the Mohr–Coulombmodel in the strength reduction iteration process. In using

strength reduction with the displacement finite-elementmethod, there is also the question of which monitoringpoints should be chosen to detect non-convergence of theiterations, as different points can give slightly different safetyfactors.

Slope in cohesive-frictional soil with weak layer andunconfined seepage flow

The final example is identical to the preceding case,except that the slope is now subject to the effects of porepressures that are generated by unconfined seepage flow(Fig. 58). Ignoring, for the moment, the limit analysis phase,Figs 59 and 60 show, respectively, the optimised mesh andthe pore pressure head generated by the methods describedin ‘Determination of steady-state pore pressures’ and ‘Limitanalysis with adaptive mesh refinement in presence of porepressures’. In these results, for a mesh with a maximum of2000 elements, the Hessian-based refinement scheme clearlyidentifies the phreatic surface and concentrates the elementsin its vicinity. Moreover, the contours of the pore pressure

1·0

1·1

1·2

1·3

1·4

1·5

1·6

1·7

1·8

1·9

2·0

2·1

2·2

Fac

tor

ofsa

fety

Method

Bis

hop

sim

plifi

ed

Sar

ma

Janb

u ge

nera

lised

Janb

u si

mpl

ified

Spe

ncer

Gen

eral

ised

wed

ge

Mor

gens

tern

–Pric

e

PLA

XIS

Limitanalysis

Fig. 57. Comparison of factors of safety for slope with a weak layer

c � � �0, 10°, 18·84 kN/mφ γ 3

c � ��

28·520°

18·84 kN/m

kN/m2

φγ 3

26·6°

0·5 m0·75 m

12·25 m1 m

2 m

Fig. 58. Slope with weak layer: unconfined seepage flow


head are smooth, and have values that were verified indepen-dently using the program SEEP/W in GeoStudio (2007).

The strength reduction process for this example is againconducted using the algorithm described in ‘Slope in cohe-sive-frictional soil with weak layer and unconfined seepageflow’, except that the hybrid mesh refinement scheme of‘Limit analysis with adaptive mesh refinement in presence ofpore pressures’, which accounts for pore pressures, is usedin steps 3 and 7. Starting with an initial safety factor ofunity, only two strength reduction trials are needed toidentify the safety factor F ¼ 0.96, as shown in Fig. 61. Thisresult was obtained with a maximum of 4000 elements inthe adaptive limit analysis calculations, and required a totalof around 14 s of CPU time. The optimised mesh at thecompletion of the strength reduction process, shown in Fig.62, indicates that the hybrid adaptivity scheme of ‘Limitanalysis with adaptive mesh refinement in presence of pore

2000 elementsHessian-based refinement of pore pressures

Fig. 59. Optimised pore pressure mesh for unconfined seepage flow in slope with weak layer

p /γw

18·017·016·015·014·013·012·011·010·09·08·07·06·05·04·03·02·01·00·0

Fig. 60. Pore pressure head for unconfined seepage flow in slope with weak layer

1·101·000·90

m(

)/2

��

γγ

γU

BLB

Trial 1, m � 0·721

Trial 2, m � 1·489

0

1

2

3

0·80Factor of safety, F

F � 0·96

Fig. 61. Strength reduction process for slope with weak layer andunconfined seepage flow

4000 elementsLower bound extension elements not shown

Fig. 62. Optimised mesh for slope with weak layer and unconfined seepage flow

566 SLOAN

pressures’ has, simultaneously, concentrated elements alongthe failure surface and the phreatic surface. The correspond-ing velocity vectors at collapse, shown in Fig. 63, confirmthat the mode of failure is again dominated by intense sheardeformation in the weak layer of cohesionless material.Compared with the case with no water table (Fig. 55), thefailure mechanism is much more extensive, and generatesgreater lateral deformation on the face of the slope.

Figure 64 compares the factors of safety computed fromfinite-element limit analysis and a variety of conventionallimit-equilibrium methods as implemented in SLOPE/W inGeoStudio (2007). To obtain the values for the latter, eachmethod was run with a variety of options (where applicable)until the lowest factor was found. Compared with the limitanalysis estimate of F ¼ 0.96, the SLOPE/W implementa-tions of the Morgenstern–Price, Spencer and Sarma methodsgive values of F ¼ 0.94, F ¼ 0.94 and F ¼ 0.95 respectively.Slightly lower values are obtained from the less rigoroussimplified Bishop and generalised Janbu procedures, whichpredict, respectively, F ¼ 0.9 and F ¼ 0.93. For this case,PLAXIS 2D with strength reduction gives the lowest safetyfactor of F ¼ 0.86. These results confirm that the finite-element limit analysis method, incorporating pore pressuresand strength reduction, gives believable slope stability pre-dictions. Moreover, with the development of efficient adap-tive meshing for this technique, tight upper and lowerbounds on the safety factor can be found at low computa-tional cost. Bearing in mind that a low factor of safetyobtained by an approximate limit-equilibrium method is notnecessarily correct, this feature is invaluable in practice.

CONCLUSIONSNew methods for performing geotechnical stability analy-

sis in two and three dimensions have been described. Thetechniques are based on finite-element formulations of thelimit theorems of classical plasticity, and incorporate anadaptive meshing strategy to give tight bounds on thecollapse load. Unlike many limit-equilibrium methods, noassumptions regarding the shape of the failure surface needto be made in advance.

Finite-element limit analysis has several other importantadvantages that make it a very attractive option for geotech-nical stability analysis. In no particular order of importance,these advantages include the following.

(a) The methods require only conventional strength param-eters, such as su, c9 and �9.

(b) The methods are ideally suited to strength reductionanalysis, and hence they can provide a safety factor onstrength as well as on load.

(c) The methods can model the effect of the pore pressuresgenerated by steady-state seepage in a rigorous manner.

(d ) The methods give the limit load directly, without the needto perform a complete incremental analysis. This is amajor advantage in large-scale three-dimensional appli-cations, where stability calculations using conventionaldisplacement finite-element analysis are both difficult andtime-consuming.

(e) The numerical solutions fulfil all the conditions of thelimit theorems, so that the difference between the upperand lower bounds provides a direct estimate of the meshdiscretisation error. This is an invaluable feature inpractice, especially for cases where it is difficult toestimate the collapse load by other approximate techni-ques.

( f ) The bounding property of the methods provides someinsurance against operator error. Owing to the complexityof many geotechnical stability problems, this type oferror can be difficult to detect with conventionalapproaches.

(g) The lower-bound solution can be used as the basis fordesign, with the upper-bound solution providing anaccuracy check as well as an insight into the failuremechanism.

(h) Because they are founded on the finite-element concept,the methods can model heterogeneity, anisotropy, com-plex boundary shapes, complicated loading conditionsand arbitrary geometries.

(i) Because the methods incorporate discontinuities in thestress and velocity fields, they are well suited tomodelling jointed media and soil/structure interfaces.

Fig. 63. Velocity vectors at collapse for slope with weak layer and unconfined seepage flow

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

1·0

1·1

Fac

tor

ofsa

fety

Method

Limitanalysis

Bis

hop

sim

plifi

ed

Sar

ma

Janb

u ge

nera

lised

Spe

ncer

Mor

gens

tern

–Pric

e

PLA

XIS

Fig. 64. Comparison of factors of safety for slope with weak layerand unconfined seepage flow


( j) Owing to recent advances in non-linear optimisation, theprocedures are robust, efficient and straightforward to use.

(k) For materials with high friction angles, the effects of non-associated flow can be modelled using the modifiedstrength parameters proposed by Davis (1968).

ACKNOWLEDGEMENTSThe fundamental research reported in this lecture could not

have taken place without the financial support of the AustralianResearch Council, which is currently funding the author’sAustralian Laureate Fellowship on ‘Failure analysis of geotech-nical infrastructure’, as well as the ARC Centre of Excellencefor Geotechnical Science and Engineering (headquartered atThe University of Newcastle, Australia). Special thanks arealso due to several industry partners, including Coffey Geotech-nics, Douglas Partners and Advanced Geomechanics Pty Ltd,for their generous support of the Centre of Excellence.

During my career I have been especially fortunate to havebenefited from the sage advice of several mentors, includingthe late John Booker, John Carter, Ian Donald, Harry Poulosand Mark Randolph. The geotechnical group at Newcastle hasbeen a special place to work, and heartfelt thanks are due toAndrei Lyamin, Kristian Krabbenhøft, Richard Merifield, Dai-chao Sheng, Peter Kleeman, Andrew Abbo, Jim Hambletonand Majid Nazem. Internationally, Dave Potts and LidijaZdravkovic at Imperial College were a great help in clarifyingmy ideas for the lecture, and Chris Martin from Oxfordcontributed important results. Other international collaboratorson the work reported here include Charles Augarde (Durham),Antonio Gens (UPC Barcelona), Rodrigo Salgado (Purdue),Andrew Whittle (MIT) and Hai-Siu Yu (Nottingham).

Last, but not least, I should like to thank my wife Denise,my daughter Erica, and my sons Rory and Oscar for theirpatience and support during the writing of this lecture.

NOTATIONA matrix of constantsA boundary area of soil mass; area of circular anchor

Ae area of element eAq boundary area of soil mass subjected to unknown

surface tractionsAt boundary area of soil mass subjected to fixed surface

tractionsAw boundary area of soil mass subjected to fixed

velocities�BB global strain–displacement matrix for mesh

multiplied by the element areasBe strain–displacement matrix for element e�BB

estrain–displacement matrix for element e multipliedby its area

Bi strain–displacement matrix for node i of an element�BBi strain–displacement matrix for node i of an element

multiplied by the element areaB width of footing; width of biaxial sampleb vector of constantsC tunnel coverc vector of constantsc cohesion

c� reduced cohesion parameter proposed by Davisc9 drained cohesionc9a drained cohesion divided by factor of safetyD tunnel diameter; diameter of circular anchorEi target number of elements in current iteration of

adaptive meshing processEmax maximum number of elements allowed in adaptive

meshing processe eccentricity of load applied to strip footingF bearing capacity factor for strip footing on clay with

heterogeneous strength; factor of safety for slopebased on shear strength

˜F increment in factor of safety for slopefþ, f� positive and negative branches of planar Mohr–

Coulomb yield criterionf (�ij), f (�) yield surface

=f (�) gradient of yield surface with respect to stressesG elastic shear modulusg vector of fixed body forces at a point

ge vector of fixed body forces for element eg plastic potential

gx, g y fixed body forces in x- and y-directionsge

x, gey fixed body forces in x- and y-directions for element e

H global vector of unknown nodal headsHe vector of unknown nodal heads for element eH0 global vector of fixed nodal headsH depth of circular anchor; depth of circular

excavation; total headHi total head at node ih global vector of unknown body forces; vector of

unknown body forces at a pointhe vector of unknown body forces for element e

hLB vector of lower-bound body forceshUB vector of upper-bound body forces

hx, h y unknown body forces in x- and y-directionshe

x, hey unknown body forces in x- and y-directions for

element eK flow matrixk soil permeability

ki permeability at node i�kki smoothed permeability at node iL length of discontinuity; length of element edge;

height of sample in biaxial testm gravity multiplier for slope equal to �ªª=ª

n, s local Cartesian coordinates in normal and tangentialdirections

Ne shape function matrix for element eNi linear shape function for node iNª breakout factor for circular anchorP eccentric load applied to strip footing; unsupported

length of tunnel headingPH, PV horizontal and vertical components of eccentric load

applied to strip footingPLB, PUB lower and upper bounds on inclined eccentric load

applied to strip footingp global vector of unknown nodal pore pressures

pi pore pressure at node i=p gradient of pore pressure fieldQ collapse load

Qn, Qs normal and tangential (shear) loads per unitthickness acting on element edge of length L

Qx, Qy element body force loads per unit thickness acting inx- and y-directions

Qu load capacity of strip footing; load capacity ofcircular anchor

q vector of unknown tractions acting on area Aq

qLB vector of lower-bound tractions acting on area Aq

qUB vector of upper-bound tractions acting on area Aq

q bearing capacityqu collapse pressure for biaxial test; pullout pressure for

circular anchorqn, qs unknown normal and tangential (shear) stresses

acting on element edgeqi

n, qis unknown normal and tangential (shear) stresses

acting on element edge at node iR radius of circular failure surface about origin; radius

of circular excavationRc radial distance to centre of footing from origin of

circular failure surfacesu undrained shear strength

su0 undrained shear strength at ground surfaces(p) smoothed step function that lies between 0 and 1

t vector of fixed surface tractions acting on area At

tin, ti

s fixed surface tractions in normal and tangential(shear) directions at node i

u global vector of unknown nodal velocities; vector ofvelocities at a point

ue vector of unknown nodal velocities for element e

568 SLOAN

u i vector of unknown velocities at node iun, us unknown velocities in normal and tangential (shear)

directions˜un, ˜us velocity jumps across discontinuity in normal and

tangential directionsui

n, uis unknown velocities in normal and tangential (shear)

directions for node i˜uij

n , ˜uijs velocity jumps across discontinuity in normal and

tangential directions for nodal pair (i, j)ux, u y unknown velocities in x- and y-directionsui

x, uiy unknown velocities in x- and y-directions for node iV volume of soil mass_W rate of internal energy dissipation less rate of work

done by external loads_W ext, Pext rate of work expended by external forces_W int, Pint rate of internal energy dissipation

w fixed velocities on surface Aw

wn, ws fixed velocities in normal and tangential (shear)directions

win, wi

s fixed velocities in normal and tangential (shear)directions for node i

x, y Cartesian coordinatesz global vector of nodal elevation heads_Æ plastic multiplier rate multiplied by element area

_Æþ, _Æ� plastic multiplier rate for positive and negativebranches of planar Mohr–Coulomb yield criterion

_Æþij, _Æ�ij plastic multiplier rate for positive and negativebranches of planar Mohr–Coulomb yield criterion atnodal pair (i, j)

� angle of normal to element edge; constant used inDavis’ strength reduction formulae

˜ dissipation gap for mesh˜e dissipation gap for element eª unit weight�ªª mean of upper and lower bounds on maximum unit

weight for slopeªLB, ªUB lower and upper bounds on unit weight

ªw unit weight of water width of velocity discontinuity_�p vector of plastic strain rates_�p

ij plastic strain rates_�p

v volumetric plastic strain rate_�p

1 maximum principal plastic strain rateŁ angle subtended by circular failure surface_º global vector of unknown plastic multiplier rates_º plastic multiplier rate

ºhLB, ºh

UB lower- and upper-bound load multipliers on bodyforces

ºqLB, ºq

UB lower- and upper-bound load multipliers on surfacetractions

�u undrained Poisson’s ratior rate of undrained strength increase with depth� global vector of unknown nodal stresses; vector of

stresses at a point�e vector of unknown stresses for element e� i vector of unknown stresses at node i

�LB vector of lower-bound stresses�UB vector of upper-bound stresses�e

LB vector of lower-bound stresses for element e�e

UB vector of upper-bound stresses for element e�ij stress tensor

�xx, �yy, �xy Cartesian stresses� i

xx, � iyy, �i

xy Cartesian stresses at node i�n normal stress� i

nn normal stress at node i�s ground surcharge�t internal tunnel pressure�1 major principal stress�3 minor principal stress� shear stress

�ins shear stress at node i� friction angle

�� reduced friction angle proposed by Davis�9 drained friction angle�9a drained friction angle divided by factor of safety�u undrained friction angle

ł9 dilation angle_ø angular velocity of rigid rotating segment

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VOTE OF THANKSPROFESSOR H. G. POULOS, Coffey Geotechnics Pty

Ltd, Australia.When I first met Professor Sloan about 30 years ago, the

late Professor Peter Wroth made a ‘Class A’ prediction thatScott Sloan would make an impact on the geotechnicalworld. Unlike some of our geotechnical predictions, PeterWroth’s was accurate, and over the following three decadesScott Sloan’s career has blossomed. In particular, he hasdeveloped innovative applications of the finite-element meth-od and applied these to a wide range of geotechnical

stability problems. He has transformed a numerical methodthat had focused on load–deformation behaviour to one thatcan be applied to a range of problems that had previouslydefied confident analysis with the traditional finite-element,displacement-based approach. In addition, he has developedhighly valuable parametric solutions, some of which he haspresented this evening. Such solutions can be used bothdirectly for routine geotechnical design and also for check-ing the results of more complex numerical techniques. Thelatter application is particularly important these days, whenmany analysts accept the results of their complex analyseswithout an adequately critical appraisal of their relevanceand applicability to the problem in hand.

In this context, it is appropriate that we recall the follow-ing words of a former Rankine Lecturer, Professor DavidPotts: ‘The potential of the numerical analysis in solvinggeotechnical problems is enormous. The potential for disas-ter is equally great if it is used by operators who do notunderstand soil mechanics principles and the concept ofgeotechnical design.’ The work described by Professor Sloanthis evening will assist in reducing the potential for disasterto which Professor Potts refers.

In recent years, Professor Sloan has built up a world-classresearch group at The University of Newcastle, a group thathe leads with enthusiasm and aplomb, and in which thecooperative spirit that he embraces is strongly evident. Thescope of research within this group has become quite broad,embracing not only traditional geotechnical engineering, butalso materials technology and geoenvironmental and geo-chemical science. While much of the research is numerical,there is also, rightly, an emphasis on the verification oftheoretical analyses via laboratory and field experiments.While his focus has been on research, Professor Sloan hasalso applied his techniques to practical problems involvingconsiderable geological complexity – for example, thestability of retaining structures in stiff fissured clays existingin Botany Bay in Sydney Australia.

This Rankine Lecture has had something for everyoneattending this evening

(a) intricate numerical details for the advanced analysts andsoftware developers

(b) design charts and parametric solutions for the practitioner(c) an increased understanding of failure mechanisms for all

present.

We have had the privilege of listening to a person with aremarkable knowledge of numerical analysis and its applica-tion to geotechnical problems. It has been a stimulating andthought-provoking lecture, and members of our profession,both in academia and in practice, will eagerly await theculmination of his work through the software package towhich he has referred. It is with great pleasure that I inviteyou all to show your appreciation to Professor Sloan for amost memorable Rankine Lecture.

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geotechnical stability analysis

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