NOTE / NOTE

Influence of tension cut-off on the stability ofanchored concrete soldier-pile walls in clay

Armando N. Antao, Nuno M. da Costa Guerra, Manuel Matos Fernandes, andAntonio S. Cardoso

Abstract: A previous paper studied the stability of soldier-pile walls in clay under vertical loading using upper boundanalyses. A classical Tresca yield criterion was assumed in that analysis. This paper extends that study by considering atension truncated Tresca yield criterion in an upper bound numerical analysis of the problem. It shows that assuming zerotension soil strength has a significant influence on the values of the minimum soldier-pile resistance required to ensurestability.

Key words: anchored retaining wall, concrete soldier-pile walls, vertical equilibrium, finite elements, limit analysis, soil-to-wall interface shear forces, tension cut-off.

Resume : Dans un article anterieur, on a etudie la stabilite de murs de pieux verticaux dans l’argile soumis a un charge-ment vertical au moyen d’analyses a la limite superieure. On a alors suppose le critere de rupture classique de Tresca. Cetarticle poursuit cette etude en considerant le critere de rupture de Tresca tronque en traction dans une analyse numeriquede limite superieure du probleme. Cette etude montre qu’en supposant qu’il n’y a pas de traction dans la resistance du sol,cette supposition a une influence significative sur les valeurs de la resistance minimale des pieux verticaux requise pourgarantir la stabilite.

Mots-cles : mur de soutenement ancre, murs de pieux verticaux en beton, equilibre vertical, elements finis, analyse limite,forces de cisaillement d’interface sol–mur, coupure en traction.

[Traduit par la Redaction]

Introduction

Cardoso et al. (2006) presented a study on the stability ofconcrete soldier-pile walls in clay under vertical loading ap-plied by the anchors. Figure 1a illustrates the type of struc-ture studied; Fig. 1b shows the forces involved in theequilibrium of the wall (left) and in the soil mass (right);Fig. 1c shows the correspondent stresses applied on the soilmass.

In the aforementioned study, the authors presented the fol-lowing two types of results based on analytical upper boundlimit analysis, numerical upper bound limit analysis, andconventional elastoplastic finite element calculations:

(1) An evaluation of the horizontal force, Fh, applied by theanchors to the ground in a limit state; this force, whichcan be considered as an active thrust, is obtained byanalysing the equilibrium of the soil mass (right side ofFig. 1b);

(2) Using results mentioned in the previous paragraph, anevaluation of the force Npile that is to be supported bythe soldier piles in a limit state (left side of Fig. 1b);this force was proven to be highly dependent on theshear stresses at the soil-to-wall interface, on angle b ofinclination of the anchors, and on the stability number ofthe excavation

½1� Ns ¼�H

cu

All of these results were obtained assuming the clay as apurely cohesive material with shear strength modelled by theclassical Tresca (CT) yield criterion, allowing for thedevelopment of any values of tension stresses (Fig. 2a). Thepresent study is basically an extension of the cited paper andaims at evaluating the influence of considering the tensionusing the truncated Tresca (TT) criterion (Fig. 2b). Thiscriterion makes it impossible to develop tension stress val-ues in the soil mass that are less than a given truncationvalue stt.

Received 20 June 2007. Accepted 25 March 2008. Published onthe NRC Research Press Web site at cgj.nrc.ca on 15 July 2008.

A.N. Antao. New University of Lisbon, FCT – CivilEngineering Department, Monte da Caparica, 2829-516Caparica, Portugal.N.M. da C. Guerra.1 Technical University of Lisbon – IST,Civil Engineering and Architecture Department, AvenidaRovisco Pais 1, 1049-001 Lisbon, Portugal.M. Matos Fernandes and A.S. Cardoso. University of Oporto– FEUP, Civil Engineering Department, Rua Dr. Roberto Frias,4200-465 Oporto, Portugal.

1Corresponding author (e-mail: nguerra@civil.ist.utl.pt).

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The numerical upper bound limit analysis tool used in thispaper is based on the one applied in the previously men-tioned work. This tool makes it possible to automatically de-termine strict upper bound approximations of limit loads ofstructural or geotechnical problems. It considers a continu-ous approximation for the velocity field, using constantstrain finite elements. Details of this tool can be found inAntao (1997) and in Vicente da Silva and Antao (2007).The tool was adapted to make it possible to carry out calcu-lations with the tension TT criterion. The numerical imple-mentation of the truncated criterion was validated in Antaoet al. (2007).

The results presented herein demonstrate that the absenceof tensile strength has, in some cases, a very significant un-favorable influence on the results that were formerly pre-sented.

Determining earth pressures of purelycohesive soils

IntroductionFigure 1c presents the geometry of the problem and the

loads considered for determining the earth pressures usingthe upper bound (UB) numerical method. A vertical cut ofdepth H, in a purely cohesive material with undrained shearstrength cu and unit weight g, is considered. The problemconsists of determining the limit values of the horizontalpressures when a given unit weight is applied to the soiland known uniform tangential stresses are applied to the

cut. The tangential stresses are uniform because they corre-spond to a percentage of the undrained resistance, which isassumed to be constant; the horizontal stress is consideredto be linear, following results previously obtained using fi-nite element elastoplastic analyses (Cardoso et al. 2006).Cardoso et al. (2006) presented the solution for that problemfor the CT yield criterion. Results were shown in a dimen-sionless way — FUB

h =� — considering the resultant of thehorizontal stresses, FUB

h , divided by the factor

½2� � ¼ 1

2�H2

as a function of the dimensionless tangential stress cam

½3� �am ¼ pcaHw

cuH¼ Fl

cuH

where p is the ratio between the adhesion mobilized on thevertical face of the cut and the available adhesion ca (con-sidered positive if the force Fl is applied upwards to thewall), H is the depth of the cut, and Hw is the height of thewall (Fig. 1b). In this study Hw is considered to be equal toH.

Results and main remarksFigure 3 shows the previously obtained FUB

h =� results forthe CT yield criterion, as well as the new results obtainedfrom the calculations performed for this study using differ-ent j�ttj=cu ratios: 0.75, 0.5, 0.25 and 0.001. The latter ratiocorresponds, for practical purposes, to the tension cut-off

Fig. 1. Geometry of the problem (a) and forces (b) and stresses (c) involved in the equilibrium of the soldier-pile wall and of the soil mass.

Fig. 2. Tresca criteria: classical and tension truncated. The dashed circle represents an example of an admissible stress state.

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solutions. A value of this ratio equal to 0.001 was adoptedbecause preliminary calculations using a value of 1 � 10�7

showed identical results with much less computational effi-ciency. The graphs shown in this figure are represented fordifferent values of the stability number, NS.

As expected, the results show both the unfavorable effectof a shear force applied downwards to the soil mass and theunfavorable influence of considering a tension truncated cri-terion on the horizontal force needed to ensure equilibrium.It was interesting to notice that the differences between re-sults for CT and TT criteria are more significant for largervalues of cam and for smaller values of NS.

This can be understood by the analysis of the failuremechanisms obtained, as Figs. 4–6 show. These figuresshow the deformed meshes issued from the finite elementcalculations, for the limit state.

The left side of Fig. 4 presents the results of the CT cri-terion; the right side of the same figure shows the results forthe TT, for j�ttj=cu ¼ 0:001. The deformed meshes are shown

for NS = 3.8 and for four different values of cam: –0.025, 0,0.2, and 0.3. It is possible to observe that: (i) the influence ofthe tension cut-off Tresca criterion is very clear in the de-formed meshes; (ii) the soil volume involved in the develop-ment of failure is smaller for larger values of cam; (iii) thedifferences between the deformed meshes for TT and CTare clearer for larger values of cam. This explains why thedifferences between the results of Fh/m for CT and TT cri-teria are greater for larger values of cam.

Figure 5 presents the results of the deformed meshes ob-tained for constant cam (= 0.3) and j�ttj=cu ¼ 0:001 and forvarying NS. It is possible to observe that the deformedmeshes for the larger values of NS are closer to the one ofthe CT case (Fig. 4g) than they are for the smaller valuesof NS. They are, therefore, more different for the more re-sistant soils. This was expected, and has affected the previ-ously analysed Fh/m values presented in Fig. 3: thedifferences between the results of Fh/m for CT and TT aregreater for the smaller NS values.

Fig. 3. Limit horizontal force versus tangential stresses on the soil mass for different NS values.

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The depth of tension cracks (under Rankine’s conditions)is z0 ¼ 2cu=� and, therefore

½4� z0

H¼ 2cu

�H¼ 2

NS

For NS equal to 3.8, 4.5, 5.2, and 5.9, z0/H is 0.53, 0.44,0.38, and 0.34, respectively. It is interesting to notice that

these depths can be recognized in the deformed meshes pre-sented in Fig. 5.

Figure 6 shows the deformed meshes for constant valuesof cam (= 0) and NS (= 3.8) and for a varying j�ttj=cu ratio.As expected, the deformed meshes for the TT case are pro-gressively closer to the CT for increasing j�ttj=cu, in accord-ance with the results of Fh/m presented in Fig. 3.

The mechanism obtained for CT does not depend on the

Fig. 4. Comparison between the deformed finite element meshes at the limit state for the CT case (left) and for the TT case with j�ttj=cu ¼0:001 (right); constant NS (= 3.8), and varying cam.

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value of NS, although the result of Fh/m is, obviously, differ-ent. In the case TT, on the contrary, the mechanism dependson NS, and, as such, it contributes to changing the Fh/m value,which is added to the change due to the different NS value.

General stability set of forcesIt should also be pointed out that whereas a wide range of

cam could be used to obtain results for CT, this was not pos-sible when the TT was considered. In fact, for cam valuessmaller or larger than those represented by the points inFig. 3, a sudden increase in the results of the horizontalforce was obtained. This also explains the limited range ofvalues of cam chosen to represent the deformed meshes inFig. 4.

To understand this behaviour, further calculations wereperformed for a given NS value (equal to 4.5) and for thetwo extreme situations under analysis (CT and TT withj�ttj=cu ¼ 0:001); these calculations are intended to find thecurve (interaction diagram) that encloses an approximationof the allowable values of the pairs (cam, Fh/m). In fact, thecurves presented in Fig. 3 show estimates of active earthforces, which means that Fh/m values larger than those rep-resented will correspond to stable conditions until the devel-opment of passive earth forces. Furthermore, obviously,there are also limits on the left and right parts of the figures.

Figure 7 represents the interaction diagram for the twocases analysed. The lower part of the curves is the one al-ready shown in Fig. 3b; the upper stretch of the curves rep-resents, therefore, the passive earth forces and the ones onthe left and right show the limit values due to excessiveshear stresses. For practical reasons, the upper points of thesurfaces were obtained by keeping the shear stresses con-stant (constant cam) and by changing Fh/m until collapse oc-curred, as performed for the lower points, whereas the left

and right points were obtained by doing the opposite, thatis, keeping Fh/m constant and changing cam.

The points obtained by these two procedures are identi-fied in Fig. 7 using different symbols. As can be observed,symbols appear mixed in the contact zones of the differentparts of the curves, meaning that the final curve is independ-ent of the technique used.

In this figure a point inside each interaction diagramcorresponds to stable conditions, whereas an outside pointcorresponds to unstable conditions. This demonstrates thatconsidering the TT criterion leads to a significant loss inthe set of the stable pairs of the dimensionless forces (Fh/m,cam). Of note is also the fact that, in the passive zone of thecurves, there is virtually no difference between CT and TT(in a passive case, pure tension is not expected); however, itis interesting to notice that the range of cam in which thepassive state could be obtained for TT is much narrowerthan the range obtained with CT.

The graphs presented in Figs. 4–6 are typical examples ofthe deformed meshes obtained for the points on the lowerpart (active state) of the curves in Fig. 7. Figure 8a showsthe deformed mesh for a typical point on the upper part ofthe curves (passive state) for the TT case, cam = 0.2; resultsfor CT are very similar. Figure 8b shows the deformed meshfor the TT case and Fh/m = 1.25, which is a local mechanismdue to excessive shear stresses on the vertical cut face, typi-cal of the left and right parts of the curves, for both CT andTT.

Application to the stability of flexibleretaining structures

The problem addressed in the previous section wasstudied when analysing the stability of flexible retaining

Fig. 5. Deformed finite element meshes at the limit state for constant cam (= 0.3) and j�ttj=cu (= 0.001), and varying NS.

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walls subject to large vertical loads (see Fig. 1a). Verticalequilibrium requires the following equation to be verified:

½5� Ww þ Fhtan � ¼ Npile þ Fl

From eq. [3]

½6� Fl ¼ �amcuH

From eqs. [6] and [1]

½7� Fl ¼ �am

�H2

Ns

Replacing F1 in eq. [5] with the expression from eq. [7] oneobtains

½8� Npile �Ww ¼ Fhtan� � �am

�H2

Ns

Dividing this equation by m (see eq. [2])

½9� Npile �Ww

�¼ Fh

�tan� � 2�am

NS

Fig. 6. Deformed finite element meshes at the limit state for constant cam (= 0), NS (= 3.8), and varying j�ttj=cu.

Fig. 7. Interaction diagram for Fh/m values for the CT and TT casesðj�ttj=cu ¼ 0:001Þ with NS = 4.5.

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The results shown in Figs. 9 and 10 are obtained using ineq. [9] the FUB

h =� values that were presented in the previoussection. Figure 9 presents the values of ðNUB

pile �WwÞ=� for

constant b and varying NS, and Fig. 10 presents the samevalues for constant NS and varying b.

From the analysis of both figures it is possible to draw the

Fig. 8. Deformed finite element meshes for typical passive state (a) and local mechanism (b).

Fig. 9. Values of (Npile – Ww)/m obtained from the Fh/m values presented in Fig. 3 for different Ns values and constant b = 458.

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expected conclusion that considering tension truncationmeans that larger values of the minimum pile resistance arerequired to ensure stability.

It is also possible to observe that the relative increase inthe vertical pile resistance due to considering tension trunca-tion is greater for smaller values of NS and for larger valuesof b. This means that tension truncation is less relevant forpoorer soil conditions and smaller vertical forces.

Furthermore, the cam value for which the minimum pileresistance is obtained is lower for smaller values of j�ttj=cu.

Actions for designing soldier pilesFor cases where the base stability problem is not an issue

(NS < 6), Terzaghi and Peck’s trapezoidal diagrams with amaximum horizontal stress of 0.2–0.4gH are commonlyused to establish anchor prestress. The resultant force ofthis prestress (ps) diagram is Fh;ps ¼ 1:5�ps�, where rps =0.2–0.4, which gives

Fig. 10. Values of (Npile – Ww)/m obtained from the Fh/m values presented in Fig. 3 for different b values and constant Ns = 4.5.

Fig. 11. Values of rUB obtained from the upper bound FEM calcu-lations: comparison with commonly adopted prestress levels.

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½10� �ps ¼Fh;ps=�

1:5

From this equation, it is also possible to conclude that�UB ¼ ðFUB

h =�Þ=1:5.Comparing (as performed by Cardoso et al. 2006) the ini-

tial horizontal component of anchor prestress with the totalhorizontal force at the limit state using parameter r (rps andrUB), it is possible to obtain Fig. 11.

Conventional elastoplastic FEM results published inCardoso et al. (2006) show that the interface resistance at thelimit state is, in most cases: �0:3 � �am � þ0:3; therefore,the cam values represented in Fig. 11 are �0:3, 0, and þ0:3.

From this figure it is possible to conclude that if no ten-sion cut-off is considered, a value of rps = 0.3 used for an-chor prestress will provide, in most cases, the horizontalforce required for the stability of the soil mass. However, ifsome truncation of tensile stresses is considered, a largervalue may be required. For example, for the tension cut-offsituation, a value of about rps = 0.35 should be appropriate.

Results from this figure, combined with eqs. [9] and [10],can be used for the design of vertical piles in anchoredsoldier-pile walls.

ConclusionsThis work complements results presented by Cardoso et

al. (2006).A finite element implementation of the upper bound theo-

rem was used to calculate active earth pressures of purelycohesive materials, considering the tension truncated Trescacriterion.

Limitations on the obtained results were explained in lightof the interaction diagram of the set of admissible forces.

The results were used to determine the minimum pile re-sistance of anchored soldier-pile walls that ensure stability,and a simple design method was proposed.

Considering tension truncation leads to larger values ofthe minimum pile resistance required for stability. This factis clearer for smaller values of the stability number.

ReferencesAntao, A.N. 1997. Analyse de la stabilite des ouvrages souterrains

par une methode cinematique regularisee. Ph.D. thesis, EcoleNationale des Ponts et Chaussees, Paris.

Antao, A.N., Guerra, N.M.C., Cardoso, A.S., and Matos Fernandes,M. 2007. Earth pressures of soils in undrained conditions. Appli-cation to the stability of flexible retaining walls. In Proceedingsof the 5th International Workshop on Applications of Computa-tional Mechanics in Geotechnical Engineering. Edited by L.R.Sousa, M.M. Fernandes, E.A. Vargas, and R.F. Azevedo.Guimaraes, Taylor & Francis. pp. 247–256.

Cardoso, A.S., Guerra, N.M.C., Antao, A.N., and Matos Fernandes,M. 2006. Limit analysis of anchored concrete soldier-pile wallsin city under vertical loading. Canadian Geotechnical Journal,43: 516–530. doi:10.1139/T06-019.

Vicente da Silva, M., and Antao, A.N. 2007. A non-linear program-ming method approach for upper bound limit analysis. InternationalJournal for Numerical Methods in Engineering, 72: 1192–1218.doi:10.1002/nme.2061.

List of symbols

A anchor force per unit length of the wallca soil-to-wall interface resistance (adhesion)cu soil undrained shear strengthFh horizontal force per unit length applied to the wallFl shear force mobilized per unit length at the back

side of the wallH excavation depth

Hw wall heightNpile vertical reaction mobilized on the soldier-piles per

unit length of the wallNS stability number of the excavation

p fraction of mobilization of the interface resistanceWs weight of the unstable soil mass per unit length of

the wallWw wall weight per unit length

z0 depth of tension cracksb anchor inclination angle with the horizontal plang unit weight of soilm auxiliary symbol, equal to 0.5gH2

r parameter defining horizontal stress of trapezoidaldiagram

rps parameter defining horizontal prestress of trapezoidaldiagram

s total normal stressstt value of tension truncationt shear stress

cam normalized mobilized resistance

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