Clarke, SD

Gilbert, M

Smith, CC

English

White Rose Research Online

Analysis of the stability of sheet pile walls using discontinuity layout optimization

promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in the Proceedings of the 7th European Conference in Numerical Methods in Geotechnical Engineering.   White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/75932     Published paper  Clarke, SD, Smith, CC and Gilbert, M (2010) Analysis of the stability of sheet pile walls using discontinuity layout optimization. In: Proceedings of the 7th European Conference in Numerical Methods in Geotechnical Engineering. 7th European Conference in Numerical Methods in Geotechnical Engineering, June, 2010, Trondheim, Norway. CRC Press , 163 - 168. ISBN 9780415592390  http://dx.doi.org/10.1201/b10551-30    Analysis of the stability of sheet pile walls using Discontinuity LayoutOptimizationS. D. Clarke, C. C. Smith & M. GilbertThe University of Sheffield, UKABSTRACT: In this paper it is demonstrated that one-dimensional rigid-plastic elements can be used in con-junction with the recently developed Discontinuity Layout Optimization (DLO) procedure Smith & Gilbert(2007a) to permit the modelling of sheet pile walls. The resulting procedure allows identification of a widevariety of failure modes, including those involving wall translation and / or rigid body rotation, and also rigid-plastic bending of the wall due to the formation of one or more plastic hinges. Results from the procedure arecompared with those obtained (i) using classical retaining wall theory, and (ii) from other numerical limit anal-ysis procedures described in the literature, demonstrating its efficacy. A series of increasingly complex exampleproblems are then studied, showing the ability of the procedure to treat problems involving water and a varietyof wall support arrangements.1 INTRODUCTIONSheet pile wall design requires knowledge of how theground and structure can be made to work togetherin order to produce a safe design. Sheet pile wallsare most often designed to resist the effects of ac-tive earth pressures. The Rankine or Coulomb meth-ods of analysis are normally used to estimate thesepressures once the mechanical characteristics of thesoil to be retained have been suitably approximated.These pressures allow a factor of safety against col-lapse to be determined for any given wall embedmentdepth.This paper aims to demonstrate that the Disconti-nuity Layout Optimization (DLO) procedure can beused to model sheet pile walls. In the paper computedcollapse loads are compared with those calculated us-ing established methods, and subsequently with re-sults obtained using a lower bound finite element limitanalysis method (Krabbenhoft et al. 2005). In the lat-ter case, more complex problems, which include a wa-ter table and ground anchors, are considered.2 DISCONTINUITY LAYOUT OPTIMIZATIONAt the present time, there are two main numericallimit analysis methods available for geotechnical ap-plications, Finite Element Limit Analysis (FELA)(Lysmer 1970, Sloan 1988, Makrodimopoulos &Martin 2006) and Discontinuity Layout Optimiza-tion (DLO) (Smith & Gilbert 2007a, 2007b, 2008).Limit analysis can also be carried out using conven-tional elasto-plastic finite element analysis, by iter-ating towards a collapse state; however this paper isconcerned only with direct limit analysis approaches.With both FELA and DLO the collapse state can beidentified directly using optimization techniques.A numerical limit analysis problem can be formu-lated as an upper or lower bound problem. Using anupper bound ‘kinematic’ formulation, the DLO pro-cedure defines a discontinuous velocity field cover-ing the entire problem domain, utilising a set of prob-lem variables which represent deformations along po-tential discontinuities (of which there may be manymillions). A rigorous mathematical optimization ap-proach (e.g. linear programming) is then used to se-lect the set of variables that minimises the energy dis-sipated in order to find the critical collapse mecha-nism.The DLO method has been implemented intothe geotechnical stability software package, Limit-State:GEO (LimitState 2009), which has been used toanalyse all example wall problems considered in thispaper.3 CANTILEVER WALL ANALYSIS3.1 Sliding failureSliding failure may be analysed using a classicalRankine approach assuming a rigid retaining wall. Inthis example a smooth interface is assumed between1Ground level, 8.00mBase level, 0.00mWallTip level, -4.00mFigure 1: Rankine analysis geometryTable 1: Soil parametersUnit weight, γ 18 kN/m3Cohesion intercept, c′ 0 kPaAngle of friction, φ′ 30◦the soil and the retaining wall. The geometry for thebasic analysis is shown in Figure 1 and the soil pa-rameters given in Table 1 have been used. A slidingonly failure can be modelled in LimitState:GEO bysetting Model rotations to be false.Using the Rankine earth pressure coefficients(Eqs. 1, 2), the pressures acting on the active (Ka)and passive (Kp) sides of the wall can be calculated.Ka =1− sinφ′1 + sinφ′(1)Kp =1+ sinφ′1− sinφ′ (2)The active force is given by Pa = 0.5H2γKa whereH is the total height of the retaining wall from crestto tip. The passive force is similarly given by Pp =0.5H2γKp. By examining horizontal equilibrium, therequired depth below the base of the wall to the tip(dw) can be calculated, which for the geometry inFigure 1 is 4.0m. To show that the DLO method cancalculate the same mechanism and critical depth, theproblem was set up in LimitState:GEO with the ge-ometry as specified in Figure 1. In LimitState:GEOthe sheet pile wall is defined as an engineered ele-ment (LimitState 2009) with an infinite moment re-sistance to match the rigid sheet pile modelled in theRankine analysis. The wall must also have an infinitelateral capacity, N , so that the DLO method treats thewall as an impenetrable barrier as far as the soil isconcerned. The interface between the retaining walland the soil can be defined within LimitState:GEO byadding a second material to the wall. In this case, asa smooth interface is assumed, the second material isset with φ′ = 0 to remove frictional effects.Figure 2 shows the failure mechanism obtained us-ing LimitState:GEO. To find a collapse mechanismFigure 2: DLO failure mechanismTable 2: Variation of adequacy factor with nodal densityNodal density No. of nodes Adequacy factorCourse 250 1.012072156Medium 500 1.003030377Fine 1000 1.001162663Custom 3000* 1.000544063*User definedan ‘adequacy factor’ must be applied to a force or selfweight in the system to precipitate failure. For thisscenario, adequacy was applied to the retained soilweight, and was found to be 1.001162663 for a ‘Fine’nodal density. The adequacy factor is dependent onthe nodal density of the analysis, and for an increasingnumber of nodes (which provide the end-points of po-tential lines of discontinuity) the adequacy factor willconverge towards the analytical solution of 1.0. Thisis shown in Table 2.By modelling the sheet pile wall as a rigid mate-rial the stress distribution around the wall can be plot-Figure 3: Stress distribution for Rankine DLO analysis(simple rotational mechanism)2ted. In a conventional sliding analysis active and pas-sive earth pressures are assumed to vary linearly withdepth. However this is not a strict requirement in plas-tic limit analysis. Only the resultant of the stress dis-tributions must be the same. The DLO method with-out model rotations only examines horizontal and ver-tical equilibrium and generates a mechanism involv-ing a single wedge either side of the wall. This doesnot necessarily require a linear variation of pressurewith depth. However if the wall is allowed to rotate,and fail in the expected mode for a cantilever wall,then the soil must also yield throughout the adjacent‘wedges’ and the pressure distribution becomes well-defined. A linear variation with depth above the pointof rotation is then predicted, as shown in Figure 3.3.2 Rotational and bending failureThe previous section has shown that the DLO methodis capable of accurately analysing a sheet pile retain-ing wall for failure against sliding. In this sectionmore complex modelling is undertaken with the in-clusion of both rotational failure mechanisms, a watertable and yielding of the sheet pile wall. The DLOsolutions have then been compared with results ob-tained using a lower bound finite element limit anal-ysis (FELA) method (Krabbenhoft et al. 2005). Theproblem geometry is defined in Figure 4, with thedepth to the wall tip varying depending upon the sup-port conditions and friction on the soil-wall interface.Figure 4: Problem geometry, Krabbenhoft et al. (2005)Table 3: Cantilever pile wall parametersPlastic moment resistance, Mp 982 kNm/mTip level, dw -9.6 mAngle of wall friction, φ′ 0◦Figure 5 shows soil displacement vectors generatedby the FELA method (Krabbenhoft et al. 2005) forthe cantilevered sheet pile wall with no anchor. Thesoil and wall parameters are given in Tables 1 & 3.Specifying this problem in LimitState:GEO was doneas for the sliding analysis but with the addition ofFigure 5: FELA vector plot for a smooth cantilever wall,Krabbenhoft et al. (2005)Figure 6: DLO failure mechanism for a smooth cantileverwallvertices being placed on the sheet pile wall aroundthe expected point of yield (from Figure 5). Limit-State:GEO 2.0 is able to model rotations at vertices.By manually adding vertices the sheet pile wall is al-lowed to yield and bend at these locations. Figure 6shows the failure mechanism predicted by the DLOmethod, indicating that the sheet pile wall yields atthe same depth as predicted by the FELA analysis(Krabbenhoft et al. 2005). As the DLO method com-putes upper bound solutions, it would be expected togive a higher predicted factor of safety than a lowerbound limit analysis method. Factor of safety is nor-mally defined on soil strength rather than self weightfor retaining walls, thus partial factors are applied totanφ′ until the adequacy factor on the retained soilself weight becomes 1. For the cantilever wall anal-ysis the required factor was 1.035, meaning that theDLO method gave a result 3.5% higher than the lower3bound FELA analysis result given in Krabbenhoftet al. (2005). The true solution will lie somewhere be-tween the two results.4 ANCHORED WALL ANALYSISThe previous examples have looked at embedded can-tilever sheet pile walls where the failure mechanisminvolves a combined failure of the sheet pile and sur-rounding soil. To reduce the required embedded depthand required moment resistance of the pile, groundanchors can be used to help stabilize the retainingwall. The location of the anchor is given in Figure 4,being 1m down from the crest of the wall. Generallyground anchors can be modelled in two distinct ways:• by an equivalent prop force acting on the face ofthe wall or,• as a discrete soil reinforcement element.Both approaches can be specified within Limit-State:GEO but to correctly assess the interaction be-tween the ground anchor and the failure mechanismthe anchor should be modelled as a discrete soil rein-forcement element.4.1 Anchor capacityIf the anchor is located too close to the wall, then thefailure mechanism directly affects the pull-out resis-tance of the soil reinforcement element as the slip sur-face may cut through the position of the anchor. In thissituation the capacity of the anchor is reduced to thatequal to only the embedded end, thus giving a real-istic failure mechanism. The pull-out capacity of theanchor T can be assumed to vary proportionally withthe vertical effective stress:T = α(c′ + σ′v tanφ′mob)a (3)where α = interaction coefficient; c′ = drained cohe-sion intercept; σ′v = vertical effective stress; and φ′mob= mobilized angle of friction between the anchor andsoil mass. For essentially 1D reinforcement such assoil nails, a = nπD where D is the diameter of thesoil nail, and n is the number of soil nails per unitwidth.In certain situations the capacity of the anchor permetre length is unknown, but an ultimate pull-outresistance for a given anchor is known. This hap-pens most frequently when benchmarking the DLOmethod against work done by other authors, wherethe exact location of the generated slip surface is un-known (Krabbenhoft et al. 2005). Modelling a sin-gle, ultimate, pull-out resistance in LimitState:GEOwhich is independent of the failure mechanism can beachieved in several ways. Here it will be implementedby using an engineered element to tie the sheet pile to1mRigidelementRetained soilSheet pile wallLinking engineered elementAnchor elementFigure 7: Specifying an ultimate anchor capacity in Limit-State:GEOan anchor at the boundary of the model, as shown inFigure 7.There are three key components to this implemen-tation in LimitState:GEO, which are:• the requirements of the linking engineered ele-ment,• the interface between the sheet pile wall andlinking engineered element and• the specification of the anchor element.The linking engineered element (LEE) is requiredto attach the sheet pile wall to the anchor that pro-vides the pull-out resistance. To have no impact on thefailure mechanism the LEE must have zero bendingmoment, lateral and pull-out resistances, thus solelyacting as a inextensible tie (or debonded tendon) be-tween the sheet pile wall and the anchor.Figure 7 shows that the point at which the linkingengineered element attaches to the sheet pile wall isnot located directly on the wall. If the LEE (with abending moment resistance of zero) is attached di-rectly to the sheet pile wall then this results in Limit-State:GEO modelling a zero-bending moment also atthis point in the sheet pile wall. By attaching the LEEto an extended section of sheet pile wall this problemis eliminated.Finally the capacity of the tie-back system is de-fined by the pull-out resistance of the anchor elementwithin the rigid zone on the boundary, as seen at theright of Figure 7. This anchor element is modelled asa distinct engineered element with a specified pull-outcapacity T . The anchor capacity is thus 1× T .4.2 Failure mechanism independent analysisThe above method of specifying the ground anchorhas been tested by comparing the results obtainedwith those from Krabbenhoft et al. (2005), where aconstant anchor yield force, T , of 112 kN was used tostabilize the retaining wall in Figure 4, reducing the4Figure 8: FELA vector plot for a rough anchored wall,Krabbenhoft et al. (2005)Table 4: Anchored wall parametersPlastic moment resistance, Mp 115 kNm/mTip level, dw -2.0 mAngle of wall friction, φ′ 30◦Anchor force, T 112 kNembedded depth to 2 m and the required moment re-sistance to 115 kNm/m. This analysis was done to mo-bilize full soil friction on the pile. The problem wasset up in LimitState:GEO using the geometry givenin Figure 4 and the soil and wall parameters given inTable 4. As was the case for the cantilevered wall, thefactor of safety is defined in terms of a partial factoron soil strength. For an adequacy factor of 1, a par-tial factor of 1.032 was required, showing that the dif-ference between the lower and upper bound methodsis 3.2%. From a comparsion between the vector plotand failure mechanism, it can be seen that the modeof failure is the very similar in both models.Figure 9: DLO failure mechanism for a rough anchoredwall4.3 Failure mechanism dependent analysisOne of the main strengths of the DLO method is itsability to evaluate the stability of geotechnical prob-lems of any geometry, always finding the lowest en-ergy solution. The anchored retaining wall analysis inthe last section showed that DLO method is capableof closely matching the failure mechanism and factorof safety obtained using a lower bound FELA anal-ysis of a problem involving an anchor whose capac-ity is independent of the failure mechanism. As dis-cussed earlier, the DLO method can also model dis-crete reinforcement, modelling the interaction of ananchor with the surrounding soil. An example of thisis shown in Figure 10 where the LEE was replaced byan engineered element with a capacity T of 20 kN/malong its full length, and the single anchor adjacent tothe boundary was removed. Now the factor of safetybecomes a function of the length of the anchor and itsinfluence on the failure mechanism. The parametersfor the sheet pile wall remain unchanged from thosegiven in Table 4. The analysis indicated that an anchorlength of 11.45m was required to ensure wall stability.By comparing Figures 9 & 10 the effect of thediscrete anchor on the failure mechanism can clearlybe seen, with the failure mechanism in Figure 10 at-tempting to circumnavigate the anchor.Figure 10: DLO failure mechanism for a rough anchoredwall with a discrete anchor5 CONCLUSIONSThis paper has demonstrated that the discontinuitylayout optimization (DLO) procedure can be used tomodel sheet pile walls, and also, using a variety ofmethods, connected ground anchors.A classical Rankine analysis was first used todemonstrate that the DLO procedure is capable ofmaking accurate predictions when only sliding fail-ures are involved. More complex sheet pile wall sys-tems were then modelled, and the results were bench-marked against those presented by Krabbenhoft et al.(2005). It was found that the upper bound DLO resultswere within 3.2-3.5% of the results obtained using alower bound FELA method. It was also found that theDLO method is capable of correctly identifying thefailure mechanisms for a variety of sheet pile retain-ing wall problems. The problems are easily specified5in the LimitState:GEO software, allowing rapid anal-ysis of retaining wall problems (run times were de-pendent on the nodal density specified, ranging fromseconds to minutes on a modern desktop PC).REFERENCESKrabbenhoft, K., Damkilde, L. & Krabbenhoft, S. (2005).Ultimate limit state design of sheet pile walls by fi-nite elements and nonlinear programming, Comput-ers and Structures 83 pp. 383–393.LimitState (2009). LimitState:GEO Manual VERSION 2.0,sept 3 edn, LimitState Ltd.Lysmer, J. (1970). Limit analysis of plane problems in soilmechanics, Journal of the Soil Mechanics and Foun-dations Division ASCE 96 , 4: 1311–1334.Makrodimopoulos, A. & Martin, C. (2006). Lower boundlimit analysis of cohesive-frictional materials usingsecond-order cone programming, Int. J. Num. Meth.in Eng. 6 , 4: 604–634.Sloan, S. (1988). Lower bound limit analysis using finiteelements and linear programming, Int. J. Num. Anal.Meth. in Geomech. 12 , 4: 61–77.Smith, C. & Gilbert, M. (2008). Limit analysis of the sta-bility of foundations on inclined ground, 2nd BGAInternational Conference on Foundations, Dundee,pp. 1683–1692.Smith, C. C. & Gilbert, M. (2007a). Application of discon-tinuity layout optimization to plane plasticity prob-lems, Proceedings of the Royal Society A: Mathemat-ical, Physical and Engineering Sciences 463, 2086pp. 2461–2484.Smith, C. C. & Gilbert, M. (2007b). New upper boundsolutions for layered soil bearing capacity problemsusing discontinuity layout optimization, 10th Aus-tralia New Zealand Conference on Geomechanics,Brisbane, pp. 250–255.6

Analysis of the stability of sheet pile walls using discontinuity layout optimization

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