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The use of adaptive finite-element limit analysis to reveal slip-linefields

C. M. MARTIN*

The numerical method known as finite-element limit analysis (FELA) is generally employed as a toolfor obtaining lower and upper bounds on the exact collapse load of a perfectly plastic structure orcontinuum. Most applications of FELA in geotechnical engineering have focused on plane strainproblems involving the classical Tresca and MohrCoulomb yield criteria, and considerablecomputational effort has been expended on the calculation of lower- and upper-bound solutionsfor particular problems. This paper discusses and demonstrates an alternative use of FELA as a toolfor ascertaining slip-line fields for plane strain problems. A simple but effective strategy for adaptivemesh refinement is a key feature of the process; it allows the layout of plastic regions, rigid regionsand velocity discontinuities to be determined by inspection of the FELA mesh. The correspondingslip-line field can then be constructed numerically in the usual way. The examples presented arerestricted to purely cohesive soil, but the same approach is applicable in principle to frictional orcohesive-frictional materials.

KEYWORDS: limit state design/analysis; numerical modelling; plasticity

ICE Publishing: all rights reserved

INTRODUCTIONMany problems of limit analysis in plane strain can besolved very efficiently and potentially exactly byconstructing the relevant slip-line field. As noted by Hill(1950):

The equations of plane strain are hyperbolic, and the charac-teristics are the slip-lines. This property compels us, if we wish

to solve special problems, to treat the field of slip-lines as thefundamental unknown element to be determined.

The aim of this paper is to show that a range ofpreviously challenging geotechnical stability problems cannow be solved by using finite-element limit analysis(FELA) primarily as a tool for revealing the slip-line field,rather than as a means of obtaining direct lower and upperbounds on the exact collapse load (though the bounds fromFELA still provide a valuable check on the correctness ofthe inferred slip-line solution).

For a given problem, the approach followed hereproceeds in two stages. The first involves FELA using theauthors program OxLim, which combines the lower- and

upper-bound methods of Makrodimopoulos & Martin(2006, 2007, 2008) with a simple strategy for adaptive meshrefinement. Essentially, this strategy attempts to equalisethe quantity

:

cmax dA (integral of the maximum shearstrain rate) over all elements of the mesh, subject to the pro-viso that no element be de-refined from its initial size. Afterseveral cycles of adaptive refinement, the concentration of ele-ments begins to reflect the intensity of

:

cmax. For a materialthat is both homogeneous and purely cohesive, the plasticwork rate is directly proportional to the maximum shearstrain rate, so in this case the adaptivity strategy may eq-

ually well be viewed as one that seeks to equalise each ele-ments contribution to the total internal work rate

:

Wint (Mar-tin, 2009). In a highly refined mesh, velocity discontinuitiesappear as dark bands with numerous tiny elements, wher-eas rigid blocks remain unrefined (except as required to main-tain element quality in the vicinity of an adjacent plasticregion or velocity discontinuity). Regions of diffuse plasticshearing undergo intermediate levels of refinement. The ge-

neration of unstructured triangular meshes in OxLim is per-formed using the program Triangle (Shewchuk, 2002).

The second stage of the solution process involvesconstruction and verification of the slip-line field. Thesecalculations are performed numerically, using standardfinite-difference procedures derived from the method ofcharacteristics (e.g. Hill, 1950; Houlsby & Wroth, 1982;Sokolovskii, 1965). Within a plastic region, the calculatedstress field automatically satisfies both equilibrium and theyield criterion. If any rigid blocks are present, however,additional constraints must be imposed to ensure that theytoo are in equilibrium. The calculated velocity field within aplastic region is automatically consistent with the stressfield in the sense that the principal strain directions are

certain to be aligned with the principal stress directions. Itis possible, however, for the two sets of directions tobecome mismatched in some areas, such that the majorprincipal stress is aligned with the minor principal strainand vice versa. For a purely cohesive soil (i.e. undrainedanalysis with c 5 su and Q 5 0) there is actually no need tocheck for this occurrence because the associated flow ruleremains satisfied regardless; any inconsistency will manifestitself as a difference between the lower and upper boundsthat are calculated from the slip-line stress and velocityfields, respectively.

A slip-line field is generally set up as a function ofvarious geometrical parameters (angles, lengths, radii, etc.)that need to be adjusted until all boundary conditions and

rigid block equilibrium constraints, if any are satisfied forthe problem at hand. For non-trivial problems this requiresthe numerical solution of n non-linear equations in nunknowns, as discussed by Martin (2005) in the context of

Manuscript received 1 March 2011; first decision 24 March2011; accepted 5 April 2011.

Published online at www.geotechniqueletters.com on 11 May2011.

* Department of Engineering Science, University of Oxford,Oxford, UK

Martin, C. M. (2011) Geotechnique Letters 1, 2329, http://dx.doi.org/10.1680/geolett.11.00018

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bearing capacity. In this study, the equation system issolved using the hybrid algorithm of Powell (1970). It isworth pointing out, however, that several of the slip-linefields in the examples given are not statically determinate,meaning that both the stress field and (part of) the velocityfield need to be constructed at each iteration of the solver.If the adjusted slip-line field:

(a) gives a collapse load that falls between the FELA lower

and upper bounds, and(b) gives a collapse mechanism that matches the one

obtained from FELA

there is a strong indication that the correct slip-line fieldhas been identified. Of course, it cannot be stated withabsolute certainty that the exact solution has been founduntil the slip-line stress field has been extended into allrigid regions in a manner that is both statically andplastically admissible. This is beyond the scope of thepresent paper, but it has been verified that the slip-linestress fields given later are all locally extensible; that is,there are no salient points where elements of rigid soil areoverstressed with respect to adjacent slip-lines or bound-

aries (see Hill (1954) for further details of the checksinvolved).

SHALLOW CIRCULAR TUNNELPlasticity solutions for the collapse of circular tunnels havebeen developed by several researchers, notably Davis et al.(1980) and Sloan & Assadi (1993). For homogeneousundrained soil, the problem is usually posed in terms of theparameters C/D (cover to diameter ratio) and cD/su(dimensionless self-weight to strength ratio), with the resultexpressed in terms of the stability number

N~ss{st

su

(1)

where ss is the surcharge pressure acting on the groundsurface and st is the support pressure inside the tunnel.Two illustrative cases are considered here.

When C/D 5 3 and cD/su 5 0 (i.e. the academic case ofa weightless soil), OxLim gives 4?063 # N # 4?140 usingits default bracketing target of 1%. Some furtheradaptive refinement (to reveal the arrangement of plasticregions, rigid blocks and velocity discontinuities withgreater clarity) gives the mesh shown in Fig. 1(a). The slip-line field can immediately be recognised and constructedas shown in Fig. 1(b). Note that in these and allsubsequent diagrams, the coordinates are normalised bythe relevant dimension of the problem (in this case the

tunnel diameter), crosses are used to mark centres ofrotation and heavy lines denote velocity discontinuities.As the subdivision counts in the slip-line analysis aresystematically increased, the calculated stability numberconverges to N 5 4?132. As expected, this lies between theOxLim bounds, and it is also within the range 3 ?78 # N #4?51 obtained by Sloan & Assadi (1993). The FELA andslip-line collapse mechanisms are compared in Fig. 2. Theclose agreement, although qualitative, gives additionalconfidence that the slip-line field has been identifiedcorrectly. To give some idea of the run times involved, ona 3 GHz computer it took OxLim a total of 10 s tobracket N to 1%, and 154 s to reach the level of meshrefinement shown in Fig. 1(a). The slip-line calculations,

starting with Fig. 1(b) and using successive doublings ofthe subdivision counts, required less than 1 s on a2?4 GHz computer to obtain N converged to four-digitprecision.

Repeating the process for C/D 5 3 and cD/su 5 3,OxLim gives 26?158 # N# 26?079 and the mesh shown inFig. 3(a). It is clear that the collapse mechanism is nowmore deep-seated, with the whole tunnel being surroundedby plastically deforming soil. Close inspection also revealsthat two velocity discontinuities intersect near the tunnelinvert. Nevertheless, it is fairly straightforward to constructthe slip-line field, which is shown in Fig. 3(b) and gives N52

6?085. This is consistent with the bounds from OxLim,and it is also within the range 26?49 # N # 25?63

obtained by Sloan & Assadi (1993). Figure 4 compares theFELA and slip-line collapse mechanisms; the agreement isagain convincing.

It is interesting to note that these slip-line fields havemuch in common with the solution recently identified forthe trapdoor problem (Martin, 2009).

STRIP FOOTING UNDER ECCENTRIC LOADINGVarious solutions for the undrained bearing capacity of arigid strip footing under combined vertical, horizontal andmoment (V, H, M) loading are available (see, for example,

Salencon & Pecker (1995), Bransby & Randolph (1998)and Ukritchon et al. (1998)). The (V, H) interaction can bedetermined analytically (Green, 1954), but much less isknown about the (V, M) interaction and the associatedslip-line fields. Only two levels of vertical load areconsidered here: V/V0 5 0?5 and 0?75, where (V0 5 (2 +p)Bsu and B is the width of the footing). For simplicity, itis assumed that the soilfooting interface can sustainunlimited tension.

When V/V0 5 0?5, OxLim gives 0?6689 # M/B2su #

0?6815 and the adaptively refined mesh in Fig. 5(a) revealsan interesting variation on the standard scoop mechan-ism. The slip-line field, shown in Fig. 5(b), is easy toconstruct and gives M/B2su 5 0?6749. It has been verified

that the soil at the left-hand edge of the footing is notoverstressed with respect to the free surface, so there is noobvious barrier to extension of the stress field. Figure 6confirms the expected agreement between the FELA andslip-line collapse mechanisms.

When V/V0 5 0?75, the bounds from OxLim are 0?5131# M/B2su # 0?5216, and the mesh in Fig. 7(a) shows thatthe collapse mechanism is surprisingly complicated. Inparticular, there are now two rigid blocks meeting at ahinge point; there is also a band of diffuse shear emanatingfrom the fan region and propagating to the left, rather thana single slip-line (with velocity discontinuity) in the shapeof a circular arc. The slip-line field, shown in Fig. 7(b),gives M/B2su 5 0?5187. The FELA and slip-line collapse

mechanisms are compared in Fig. 8. In this instance theagreement is particularly compelling because of thecomplexity of the mechanism.

VERTICAL CUTThe final example is one of the fundamental problems ofgeotechnical stability analysis: the collapse of a vertical cutof height H in a homogeneous soil with unit weight c andundrained strength su. The exact value of the stabilitynumber

N~cH

su(2)

has so far defied solution, but the latest bounds from large-scale FELA (Kammoun et al., 2010; Pastor et al., 2009) are

3:77522 N 3:77756 (3)

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Figure 9(a) shows an adaptively refined mesh for thisproblem obtained using OxLim. The picture reveals a

curious arrangement of plastic regions, along with tworigid blocks that meet at a hinge point. It is also apparentthat the velocity discontinuity along the right-hand

perimeter of the mechanism is the only one occurring.The slip-line field is shown in Fig. 9(b), and there is a

persuasive resemblance to Fig. 9(a). The FELA and slip-line collapse mechanisms, shown in Fig. 10, also agreevery well. Note the slight clockwise rotation of the large

_1

_2

_3

_4

_50 1 2

(a)

3 4

0

_1

_2

_3

_4

_50 1 2

(b)

3 4

0

Fig. 1. Tunnel collapse with C/D 5 3 and cD/su 5 0: (a) adaptively refined mesh from FELA; (b) slip-line field

_1

_2

_3

_4

_50 1 2

(a)

3 4

0

_1

_2

_3

_4

_50 1 2

(b)

3 4

0

Fig. 2. Tunnel collapse with C/D 5 3 and cD/su 5 0: comparison of collapse mechanisms from (a) FELA and (b) slip-line solution

The use of adaptive finite-element limit analysis to reveal slip-line fields 25


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rigid block that is visible in both Fig. 10(a) andFig. 10(b).

Table 1 shows that as the slip-line subdivision counts aresuccessively doubled, the calculated stability numberconverges to

N~3:77649... (4)

The final analysis took less than 2 min on a 2?4 GHzcomputer. As expected, the slip-line stability factor fallswithin the narrow range of equation (3). Although it seems

_1

_2

_3

_4

_50 1 2

(a)

3 4

0

_1

_2

_3

_4

_50 1 2

(b)

3 4

0

Fig. 3. Tunnel collapse with C/D 5 3 and cD/su 5 3: (a) adaptively refined mesh from FELA; (b) slip-line field

_1

_2

_3

_4

_50 1 2

(a)

3 4

0

_1

_2

_3

_4

_50 1 2

(b)

3 4

0

Fig. 4. Tunnel collapse with C/D 5 3 and cD/su 5 3: comparison of collapse mechanisms from (a) FELA and (b) slip-line solution

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_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

(a) (b)

Fig. 5. Strip footing moment capacity when V/V0 5 0?5: (a) adaptively refined mesh from FELA; (b) slip-line field

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

(a) (b)

Fig. 6. Strip footing moment capacity when V/V0 5 0?5: comparison of collapse mechanisms from (a) FELA and (b) slip-line solution

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

(a) (b)

Fig. 7. Strip footing moment capacity when V/V0 5 0?75: (a) adaptively refined mesh from FELA; (b) slip-line field

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

_1.0

_0.5

_0.5

0.0

0.0 1.0 1.5

0.5

0.5_1.0_1.5

(a) (b)

Fig. 8. Strip footing moment capacity when V/V0 5 0?75: comparison of collapse mechanisms from (a) FELA and (b) slip-linesolution



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highly likely that the exact solution has now been found, atleast to six-digit precision, formal confirmation will requirethe stress field to be extended into the two rigid blockswithin the mechanism, as well as the semi-infinite soil massoutside. This could well prove to be a difficult task, butpreliminary checks have confirmed that there are noproblems with extensibility at critical locations. In parti-cular, the soil is not overstressed at the hinge point, nor is itoverstressed at the toe of the cut.

CONCLUSIONSThis paper has shown how FELA, when combined with asimple strategy for adaptive mesh refinement, can be used

to reveal the arrangement of plastic regions, rigid blocksand velocity discontinuities in a range of geotechnicalstability problems in plane strain. Using the FELAprogram OxLim, the resolution that can be achieved is sosharp that the relevant slip-line field can readily be deducedfrom inspection, and then constructed in a separateanalysis using the method of characteristics.

The procedure has been demonstrated for a few selectedproblems of undrained stability, but the same strategy shouldalso be applicable to problems involving frictional or cohesive-

frictional materials. The new approach shows great promisefor obtaining exact solutions to all manner of plane strainplasticity problems where it has, to date, proved impossible toidentify the correct slip-line field by trial and error.

0.0

0.0

0.5

0.5

1.0

1.0

0.0

0.0

0.5

0.5

Note: centres of rotation are at

(1.42,0.52) and (_2.91,3.24),

out of range of axes

1.0

1.0

(a) (b)

Fig. 9. Collapse of vertical cut: (a) adaptively refined mesh from FELA; (b) slip-line field

0.0

0.0

0.5

0.5

1.0

1.0

0.0

0.0

0.5

0.5

Note: centres of rotation are at

(1.42,0.52) and (_2.91,3.24),

out of range of axes

1.0

1.0

(a) (b)

Fig. 10. Collapse of vertical cut: comparison of collapse mechanisms from (a) FELA and (b) slip-line solution

28 Martin


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REFERENCESBransby, M. F. & Randolph, M. F. (1998). Combined loading of

skirted foundations. Geotechnique 48, No. 5, 637655.Davis, E. H., Gunn, M. J., Mair, R. J. & Seneviratne, H. N. (1980).

The stability of shallow tunnels and underground openings incohesive material. Geotechnique 30, No. 4, 397416.

Green, A. P. (1954). The plastic yielding of metal junctions due tocombined shear and pressure. J. Mech. Phys. Solids 2, No. 3,197211.

Hill, R. (1950). The mathematical theory of plasticity. Oxford:Clarendon.Hill, R. (1954). On the limits set by plastic yielding to the intensity

of singularities of stress. J. Mech. Phys. Solids 2, No. 4, 278285.

Houlsby, G. T. & Wroth, C. P. (1982). Direct solution of plasticityproblems in soils by the method of characteristics. Proc. 4thInt. Conf. on Numerical Methods in Geomechanics, Edmonton 3,10591071.

Kammoun, Z., Pastor, F., Smaoui, H. & Pastor, J. (2010). Largestatic problem in numerical limit analysis: a decompositionapproach. Int. J. Numer. Anal. Methods Geomech. 34, No. 18,19601980.

Makrodimopoulos, A. & Martin, C. M. (2006). Lower bound

limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Methods Engng 66,No. 4, 604634.

Makrodimopoulos, A. & Martin, C. M. (2007). Upper boundlimit analysis using simplex strain elements and second-ordercone programming. Int. J. Numer. Anal. Methods Geomech. 31,No. 6, 835865.

Makrodimopoulos, A. & Martin, C. M. (2008). Upper boundlimit analysis using discontinuous quadratic displacement

fields. Commun. Numer. Methods Engng24, No. 11, 911927.Martin, C. M. (2005). Exact bearing capacity calculations using

the method of characteristics. Proc. 11th Int. Conf. onComputer Methods and Advances in Geomechanics, Turin 4,441450.

Martin, C. M. (2009). Undrained collapse of a shallow plane-strain trapdoor. Geotechnique 59, No. 10, 855863.

Pastor, F., Loute, E. & Pastor, J. (2009). Limit analysis andconvex programming: a decomposition approach of thekinematic mixed method. Int. J. Numer. Methods Engng 78,No. 3, 254274.

Powell, M. J. D. (1970). A hybrid method for nonlinear algebraicequations. In Numerical methods for nonlinear algebraicequations (Rabinowitz P. (ed.)), 87114. London: Gordonand Breach.

Salencon, J. & Pecker, A. (1995). Ultimate bearing capacity ofshallow foundations under inclined and eccentric loads. Part I:purely cohesive soil. Eur. J. Mech. A Solids 14, No. 3, 349375.

Shewchuk, J. R. (2002). Delaunay refinement algorithms fortriangular mesh generation. Comput. Geom. 22, No. 13, 2174.

Sloan, S. W. & Assadi, A. (1993). Stability of shallow tunnels insoft ground. In Predictive soil mechanics (Houlsby G. T. &Schofield A. N. (eds)), 644663. London: Thomas Telford.

Sokolovskii, V. V. (1965). Statics of granular media. Oxford:Pergamon.

Ukritchon, B., Whittle, A. J. & Sloan, S. W. (1998). Undrainedlimit analyses for combined loading of strip footings on clay.J. Geotech. Geoenv. Engng124, No. 3, 265276.

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Table 1. Convergence of lower- and upper-bound stabilityfactors calculated from slip-line field for vertical cut

Slip-line subdivisioncounts with referenceto Fig. 9(b)

Lower bound(from stress

field)

Upper bound(from velocity

field)

61 3?7759789 3?776611662 3?7763630 3?7765205

64 3?7764590 3

?7764983

68 3?7764829 3?7764928616 3?7764889 3?7764914632 3?7764904 3?7764911


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