Limit analysis of anchored concrete soldier-pile walls in clay under vertical loading

Limit analysis of anchored concrete soldier-pilewalls in clay under vertical loading

António S. Cardoso, Nuno M. da Costa Guerra, Armando N. Antão, andManuel Matos Fernandes

Abstract: The vertical stability of anchored concrete soldier-pile walls is highly influenced by the complexity of theinteraction between the different parts of the structure, i.e., wall, anchors, and supported soil mass. The problem is ana-lyzed using upper bound limit analysis through published solutions and proposed closed-form equations. A comparisonis made between these equations and numerical limit analyses. An estimate of the theoretical minimum pile resistancerequired to avoid excavation collapse is presented. Published finite element elastoplastic results are used for comparison.

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

Résumé : La stabilité verticale de murs de pieux verticaux ancrés est fortement influencée par la complexité del’interaction entre les différentes parties de la structure – mur, ancrages et massif de sol retenu. Le problème est étudiéau moyen d’une analyse à la limite supérieure selon des solutions publiées et des équations exactes proposées. On faitune comparaison entre ces équations et les analyses numériques limites. On présente une estimation de la résistancethéorique minimale des pieux pour éviter l’effondrement de l’excavation. Des résultats élastoplastiques d’éléments finispubliés sont utilisés pour fin de comparaison.

Mots clés : mur de soutènement, murs de pieux verticaux en béton, équilibre vertical, éléments finis, analyse limite,forces de cisaillement à l’interface sol-mur.

[Traduit par la Rédaction] Cardoso et al. 530

1. Introduction

The literature reports a number of incidents and accidentsof tied-back flexible retaining walls related with inadequatebearing resistance to vertical loads applied by the anchors(Broms and Stille 1976; Ulrich 1989; Clough and O’Rourke1990; Winter 1990; Stocker 1991; Gould et al. 1992; Barley1997; Cacoilo et al. 1998; Guerra 1999). The walls involvedcan be classified into two types: those for which construc-tion is (mostly) completed before excavation (sheet-pilewalls, diaphragm walls, etc.), known as continuous embed-ded walls; and the ones partially constructed during excava-tion, which are designated as soldier-pile walls.

The vertical equilibrium of continuous embedded retain-ing walls was studied by Matos Fernandes (1985) and byMatos Fernandes et al. (1993, 1994) using finite element

(f.e.) simulations. A parametric study and the analysis of adocumented case history were presented by these authors.

These numerical studies, the description of the above-mentioned incidents and accidents, as well as some resultsof small-scale tests (Hanna and Matallana 1970; Plant 1972)allow for the depiction of the pattern of behavior of an-chored continuous embedded walls under both good andmarginal stability conditions with respect to vertical loading.This is shown in Fig. 1 and is summarized in Table 1.

An interesting point revealed by the f.e. analyses, and cor-roborated by the small-scale tests, is the fact that verticalfailure may occur without full mobilization of upward shearresistance at the back face of the wall (see Fig. 1 and Ta-ble 1).

Some of the incidents and accidents mentioned abovewere also observed in soldier-pile walls. In a few other casesinvolving concrete soldier-pile walls, of which the authorshave direct knowledge, the deficient support of vertical loadswas related to buckling of the soldier piles, which indicatesthat their design loads had been significantly underestimated.

Figure 2 illustrates the typical construction sequence of aconcrete soldier-pile wall. Figure 3 shows the forces in-volved in vertical equilibrium of both the wall and the sup-ported soil mass. It can be understood that the degree ofmobilization of the shear stresses at the back wall face (notethat in this type of wall there is contact with the soil throughthat face only) will considerably affect the magnitude of theload to be supported by the soldier-piles.

Studies on the vertical equilibrium of soldier-pile wallshave been carried out by Guerra (1999) and Guerra et al.

Can. Geotech. J. 43: 516–530 (2006) doi:10.1139/T06-019 © 2006 NRC Canada

516

Received 6 July 2005. Accepted 20 February 2006. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on19 April 2006.

A.S. Cardoso and M. Matos Fernandes. University ofOporto – FEUP, Civil Engineering Department, R. Dr.Roberto Frias, 4200–465 Oporto, Portugal.N.M.C. Guerra.1 Technical University of Lisbon – IST, CivilEngineering and Archit. Department, Av. Rovisco Pais 1,1049–001 Lisbon, Portugal.A.N. Antão. New University of Lisbon, FCT, CivilEngineering Department, Monte da Caparica, 2829–516Caparica, Portugal.

1Corresponding author (e-mail: [email protected]).

(2001, 2004) involving field monitoring, f.e. analyses, andanalytical limit analyses. The results of the f.e. calculations,in which buckling of soldier piles was simulated, confirmedthat in this type of structure the pattern of failure by loss ofvertical equilibrium is similar to the one previously pre-sented for continuous embedded walls. Note that a failure byinsufficient bearing capacity of the foundation of the soldier-

piles, as far as the global failure mechanism is concerned,would lead to similar conclusions.

The results presented by Guerra et al. (2004) demonstratethat the role of the shear stresses mobilized at the soil-to-wall interface needed further clarification. In fact, two iden-tical f.e. analyses, one with null interface shear resistanceand the other with a resistant interface, lead to similar per-

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Cardoso et al. 517

Good stability conditions Marginal stability conditions

Retaining wall movements anddeflection

Negligible settlement. Small to moderate lateraldisplacements. Front face clearly convex.

Large settlements and large lateral displace-ments. Front face may be concave.

Ground surface movements anddeflection

Small to moderate displacement. Surface clearlyconcave.

Large horizontal and vertical displacements.Surface may be convex.

Ground anchor forces Small variations of anchor loads after pre-stressing.

Anchor loads progressively decrease.

Shear stresses in the soil-to-wallinterface—back face

Downward shear stresses applied to the wall. Upward shear stresses applied to the wall.

Shear stresses in the soil-to-wallinterface—front face

Significant mobilization of upward shearstresses applied to the wall.

Full mobilization of upward shear stressesapplied to the wall.

Table 1. Pattern of behavior of anchored continuous embedded walls with good and marginal stability conditions due to vertical loading.

Fig. 1. Pattern of behavior of anchored continuous embedded walls with good and marginal stability conditions due to vertical loading.(a) Good stability conditions. (b) Marginal stability conditions.

formances of the supported soil. The same paper tries to ex-plain these results through a semi-empirical limit analysisapproach.

In brief, the problem can be described as follows:(1) considering the equilibrium of the wall only, it would

appear that higher mobilization of upward tangentialstresses applied to the wall would lead to lower requiredpile resistance;

(2) however, higher mobilized adhesion at the interface in-creases the total vertical downward force on the soilmass;

(3) then, the equilibrium of this mass will demand a largerhorizontal force applied by the anchors;

(4) since anchors are inclined downwards, this will lead to agreater vertical force on the wall;

(5) so, in some circumstances, the mobilization of upwardtangential stresses applied to the back of the wall by thesoil may give rise to higher vertical loads on the soldierpiles.

In this paper, the vertical stability of soldier-pile walls isreassessed by presenting a closed form equation and resultsof f.e. numerical analyses, both based on the upper boundtheorem (UBT).

Conclusions are drawn concerning the proposed theoreti-cal solutions and some practical implications related to thedesign of anchored soldier-pile walls.

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518 Can. Geotech. J. Vol. 43, 2006

Fig. 2. Construction stages of concrete soldier-pile walls.

Fig. 3. Forces involved in the equilibrium of a soldier-pile wall (left) and in the soil mass (right).

2. Equilibrium of the wall and of the soil mass

2.1. IntroductionTo ensure stability of a non-self-standing soil mass, the

retaining wall must apply a sufficiently high horizontalforce. If shear stresses are mobilized at the soil-to-wall inter-face they will influence the horizontal force required to en-sure stability. In soldier-pile walls this horizontal force isalmost exclusively provided by anchor forces because of thevery small contribution from earth pressures in front of thesoldier piles, which are not considered in the paper.

An analysis of this problem will now be presentedwherein the wall and the supported soil are considered sepa-rately.

2.2. Vertical equilibrium of the wallVertical equilibrium of the wall requires that the following

equation be verified (see the left side of Fig. 3):

[1] Ww + Σ A sin β = Npile + Fal

where Ww is the weight of the wall per unit length; Σ A sinβis the vertical force applied by the anchors per unit length ofthe wall; Npile is the load per unit length applied on the sol-dier piles; and Fal is the shear force mobilized at the backsoil-to-wall interface per unit length of the wall and is posi-tive if applied upwards to the wall interface.

The shear force, Fal, can be determined by

[2] Fal = pcaHw

where ca is the soil-to-wall interface adhesion; Hw is the wallheight; p is the fraction of mobilization of the soil-to-wallinterface adhesion ca, considered positive if Fal is appliedupwards to the wall.

For a mobilized fraction p of the interface adhesion ca,eq. [1] can be written as

[3] Σ A N W pc Hsin )β = ( pile w a w− +

= − +( )N WN

pile wS

am2µ χ

where χam is given by

[4] χama w

u

= pc H

c H

NS is

[5] NH

cS

u

= γ

and µ is defined as

[6] µ γ= 12

2H

2.3. Equilibrium of the soil mass

2.3.1. GeneralThe right side of Fig. 3 shows the external forces applied

to the soil mass. The problem to be solved consists of deter-mining the horizontal load Σ Acosβ necessary to ensure sta-bility of the soil mass of total height H, in undrained

conditions, submitted to gravity (total weight of the unstablezone, WS), and to a shear force applied at the cut face, Fal.The soil resistance is modeled by a Tresca yield criterioncharacterized by an undrained shear strength, cu.

The problem described is, in fact, an earth pressure prob-lem in undrained conditions and with a shear force due toadhesion at the soil-to-wall interface.

2.3.2. Published solutionsThe problem described herein has been addressed by sev-

eral authors. Two of the most recent solutions are the onesfrom Chen and Liu (1990) and Soubra and Macuh (2002).Although these solutions were developed mainly for materi-als with both cohesion and friction, their application tofrictionless materials is possible.

Both solutions are based on the UBT. Chen and Liu(1990) assume a mechanism considering a composite slipsurface with a log-spiral surface between two linear ones.Soubra and Macuh (2002) assume a mechanism that consid-ers a log-spiral slip surface.

In both cases, the total (active) force, Pa, can be given by

[7] P K H K qH K cHa a aq ac= + −12

2γγ

where c is the soil cohesion; q is the surcharge applied onthe supported soil surface; and Kaγ, Kaq, and Kac are earthpressure coefficients.

Considering an undrained soil analyzed in total stresses,with an undrained shear strength, cu, interacting with a verti-cal wall, Pa is horizontal and Kaγ = 1; therefore, taking q = 0,the horizontal force needed to ensure stability can be esti-mated by

[8] P A H K c Ha ac ucos= = −Σ β γ12

2

where Kac depends on the geometry assumed for the slip sur-face and on pca /cu. The best solution will correspond to themechanism maximizing Pa. A numerical determination ofKac was performed, and the results obtained are shown in Ta-ble 2. These results, as well as the corresponding slip sur-faces, do not depend on NS. In these solutions, Hw isconsidered to be equal to H. The determination of Kac for thesolution of Soubra and Macuh (2002) considered the correc-

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Cardoso et al. 519

χam

Chen andLiu (1990)

Soubra andMacuh (2002)

–0.4 2.337 2.328–0.2 2.183 2.181

0 2.000 2.0000.2 1.789 1.7890.4 1.550 1.5430.5 1.415 1.4140.6 1.266 1.2650.7 1.097 1.0960.8 0.896 0.894

Table 2. Values of Kac obtained from Chenand Liu (1990) and Soubra and Macuh(2002).

tions to the printed equations provided by the first author ofthis work.

From the analysis of Table 2 it can be seen that the resultsobtained from the two solutions are very similar.

2.3.3. Proposed solutionA simplified method based on the UBT with a semi-

empirical correction was used by Guerra et al. (2004) allow-ing for the determination of Σ Acosβ assuming planarsurfaces.

In the present section, a theoretical analytical solutionbased on an upper bound solution is presented considering amechanism based on a circular slip surface.

Figure 4 shows the conditions of the assumed mechanism:two rigid regions are considered, separated by a circular slipsurface of radius R. No restrictions are made regarding thelocation of the center, O, of the slip surface. This surface isdefined by angles λ and α, as shown in the figure. In allcases, the surface intercepts the excavation toe.

It is assumed that the displacement of the region limitedby the soil surface, the cut face, and the circular slip surfaceis compatible with a rotation δ around point O, the other re-gion remaining fixed.

An upper bound solution is obtained by equalizing thework of external forces in an assumed mechanism to thework of plastic deformation. Appendix A shows that the hor-izontal force Σ Acosβ can be written as a function of anglesα and λ. The best solution corresponds to the maximumvalue of all estimates of the horizontal force. Maximization

of Σ Acosβ relative to α and λ makes it possible to obtainthe following (see details in Appendix A):

[9]( cos )Σ A

TS

N

βµ

UB

S

= −

where S and T have the meaning presented in Appendix Aand are functions of b (see Fig. 4), χam, α*, and λ*. Theselast two variables, α* and λ*, are the values of angles α andλ that optimize the solution and are functions of b, χam, andNS (for b = 2/3 the results do not depend on NS).

The value of (Σ Acosβ)UB from eq. [9] is, therefore, thebest (upper bound) estimate of the horizontal force in a limitcondition using the mechanism previously referred to.

For an unsupported soil mass (ca and Σ Acosβ equal tozero), the stability factor, NS, obtained is equal to 3.831,which is the known upper bound approximation, NS cr

UB− , of

the critical stability factor, NS–cr, using a circular slip surface(Taylor 1948), NS–cr being the exact solution. More accuratebounds for this stability factor were obtained by Pastor et al.(2000): NS cr

LB− = 3.760 (lower bound) and NS cr

UB− = 3.786 (up-

per bound). The value of NS crUB− = 3.831 corresponds to small

errors on the unsafe side relative to the best known solutions(between 1.2% and 1.9%). The obtained solution seems,therefore, to be very close to the exact solution.

Figure 5 shows the results of (Σ A cos β)/µ issued fromChen and Liu (1990), Soubra and Macuh (2002), Guerra etal. (2004), and from the proposed solution (for b = 2/3). Ascould be expected, all results show an unfavorable effect ofdownwards shear force applied to the soil mass: to a larger

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Fig. 4. Conditions of the mechanism assumed in the proposed upper bound solution.

value of χam, corresponds a larger limit horizontal force. Itcan also be seen that the results become less dependent onNS for increasing χam.

The two solutions obtained from Guerra et al. (2004) foreach value of NS correspond to the solutions presented con-sidering NS–cr = 3.75 and 3.85 (semi-empirical estimates ofthe critical stability number of the excavation, as defined inthat work).

The results from Chen and Liu (1990) and Soubra andMacuh (2002) are practically the same (as already seen inTable 2). Considerable differences can be observed betweenthese two solutions and the ones considering planar surfaces.These solutions would be practically the same if the theoreti-cal value of NS–cr = 4 was used in the planar surfaces, in-stead of the semi-empirical one, referred to above. Thismeans that the slip surfaces obtained from these methods arealmost planar, as will be confirmed later.

Differences also occur between Chen and Liu (1990) andSoubra and Macuh (2002) solutions and the one proposed inthis paper. These differences are minimal for χam > 0.3 andincrease for the lower values of χam. They can be explainedby the different shape of the slip surfaces, as shown inFig. 6. This figure also shows that: (i) slip surfaces from theproposed solution and from Chen and Liu (1990) and Soubraand Macuh (2002) are very similar for χam = 0.4 and 0.8;and (ii) differences between the surfaces increase for lesservalues of χam.

It should be noted that results from Soubra and Macuh(2002) correspond to a degeneration of log-spiral slip sur-faces into circular ones; results from Chen and Liu (1990)correspond to composite log-spiral—also degenerated intocircular—between two linear surfaces. In both cases, forhigh values of χam, the geometrical configuration of the opti-mized slip surface is close to planar.

In the proposed solution, for χam = �0.4, 0, and 0.4, thecircular slip surface has upwards concavity, whereas forχam = 0.8 this concavity is directed downwards (this is notclear in the figure because the surface is, in this case, almostplanar).

2.3.4. Solution using numerical finite element limitanalyses

The previous section has shown that the proposed solutiongives results that are different from the ones obtained byother authors. To check the validity of the proposed solution,an alternative limit analysis will now be considered.

This alternative consists of a UBT application to the studyof stability of soil masses using a numerical implementationwith the f.e. method technique.

The application of the UBT to a certain system involvesthe calculation of the work of the external forces in a certaindisplacement field, We, as well as the dissipated plastic workin the same displacement field, Wi. The displacement fieldmust be carefully chosen because, for the same set of forces,some displacement fields will lead to We > Wi and others tothe opposite relation. In the first case, the external forceswill lead to plastic collapse of the structure, and in the othercase nothing could be concluded about these forces.

Kinematical admissible displacement fields accept discon-tinuous components. In fact, the conventional application ofUBT assumes rigid block mechanisms and, therefore, the ex-istence of discontinuities in the displacement field (as per-formed in the previous section). These displacement fieldsare usually made dependent on a small number of variables.A minimization process of Wi – We as a function of thesevariables makes it possible to obtain the closest solution tothe unknown exact one.

Although it is possible, in a reasonably simple way, toconsider either continuous or both continuous and discontin-uous displacement fields (see, for example, Leca andDormieux 1992), the possibility of minimization will alwaysbe dependent on the restrictions imposed to the assumed dis-placement field and on the previous knowledge of the prob-lem under analysis. Therefore, it is convenient to utilize atool capable of finding a good, if not exact, solution for theminimization of We – Wi, without any a priori assumptionsabout the displacement field. A numerical implementation ofthis method, considering a continuous displacement field,was done by Antão (1997) and applied to the stability of soilmasses by Antão (2003).

Using this numerical f.e. model, it is possible to search forcombinations of the two forces (normal and tangential to thesoil vertical cut) that cause collapse of the soil mass repre-sented in Fig. 7. In the situation under study, for each case(H, γ, and cu), the shear stress was fixed and the maximumnormal stress (with triangular distribution) causing collapsewas obtained. Figure 8 presents, for the situation of null ad-hesion (null shear stress), the development of the mechanismautomatically obtained by the numerical limit analysis(NLA).

Successive calculations of the same type lead to the re-sults summarized in Fig. 9, which are superposed with thecurves obtained from eq. [9]. A very good agreement be-tween the equation and the numerical methodology can beconcluded from the figure. For χam < 0.5, the two solutionsare almost perfectly matched; differences can be seen forhigher values of χam with (Σ Acosβ)/µ always assuming lesservalues for the results of eq. [9].

The differences found for χam > 0.5 can be explained bysmall differences in the mechanisms associated with each

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Cardoso et al. 521

Fig. 5. Estimate of the horizontal limit force, Σ A cosβ, given bypublished solutions and the proposed one for b = 2/3.

method—proposed analytical solution and results from nu-merical calculations. In Fig. 10, the slip surfaces of the pro-posed solution, previously shown in Fig. 6, are plotted,being superposed on a representation of the relative magni-tude of the dissipated internal work calculated by the numer-ical method. In fact, as previously mentioned, discontinuitiesare not considered in the numerical method. Therefore, theexistence of concentration of the dissipated work on certainzones makes it possible to predict the location of a slip band.This band can also be seen in the figure.

Analysis of Fig. 10 makes it possible to draw the follow-ing conclusions:

(i) there is a generally good agreement between the twotypes of results;

(ii) for χam = –0.4 and χam = 0, the shape of the slip bandobtained by the numerical method is very similar to thecircular slip surface from the proposed method, withpractically the same radius;

(iii) for χam = 0.4, although there is a spatial coincidence be-tween both plastic dissipation zones, there are differ-ences in these zones: in the case of the proposed

© 2006 NRC Canada


Fig. 6. Shape of the slip surfaces of the published solutions and of the proposed one for b = 2/3 (slip surfaces not dependent on NS).

Fig. 7. Schematic representation of the problem analyzed by NLA. Fig. 8. Representation of the mechanism automatically obtainedby the NLA implementation of the UBT, for the case χam = 0(mechanism not dependent on NS).

method, the slip surface is almost linear, whereas in thecase of the numerical slip band it seems to consist oftwo distinct curved zones, one with an upward concav-ity and the other with a downward concavity;

(iv) for χam = 0.8, this effect is clearer and the dissipationzone of the numerical solution is larger; overall, the dif-ferences are more visible for this situation and this canprobably explain the less accurate agreement seen inFig. 9.

2.4. Equilibrium of both wall and soil massA value of Σ A cos β necessary for the equilibrium of the

soil mass can be obtained from any of the methods presentedin section 2.3. Using such a value in eq. [3], it is possible todetermine Npile – Ww necessary to ensure soil and wall sta-bility

[10] N W N WpileUB

w pile wUB− = −( )

= −( ) tanΣ AN

cos UB

Samβ β µ χ2

The value of NpileUB from eq. [10] gives an upper bound so-

lution of the pile resistance conducing to imminent collapseof the excavation. It is, therefore, an estimate of a minimumrequired value of the pile resistance. This estimate dependson the percentage of shear force mobilization, p, at the soil-to-wall interface, which is unknown.

Figure 11 presents the values of (N WpileUB

w− )/µ corre-sponding to the different solutions seen in section 2.3. Thesevalues were obtained from eq. [10] and are represented as afunction of χam for varying NS (Fig. 11a) and β (Fig. 11b).For the proposed solution, two results are shown: one for b =2/3 and the other for b = 1/2. The solutions from Chen andLiu (1990) and Soubra and Macuh (2002) are almost identi-cal; therefore, they are represented by the same line pattern.The symbols shown in the figure correspond to results of the

FEM calculations, which will be commented on in the fol-lowing section.

From this figure it can be seen that NS and β strongly in-fluence the results of (N Wpile

UBw− )/µ and that the estimate of

the minimum pile resistance to avoid collapse does not nec-essarily diminish with the increase of upward mobilized ad-hesion at the soil-to-wall interface. This means that highermobilization of adhesion may be unfavorable when χam ishigher than a certain value χam–cr. This can be observed forall solutions and was also a conclusion of Guerra et al.(2004).

Considering the vertical equilibrium of the wall only, itwould appear that a higher upward mobilization of adhesionwould lead to a lower required pile resistance (see eq. [3]).However, higher mobilized adhesion increases the totaldownward vertical force on the soil mass. Then, beyond acritical value of χam > χam–cr, the equilibrium requires ahigher horizontal force Σ Acosβ and, since anchors are in-clined downwards, a greater value of Npile.

The following comments are deduced from Fig. 11:(i) the value of χam–cr is independent of NS and varies with

β and, to a lesser extent, with b (for the proposed solu-tion);

(ii) consideration of different values of b in the proposedsolution has significant consequences on the results of(N Wpile

UBw− )/µ; it has more significant influence for

greater values of NS, β, and χam;(iii) solutions from Chen and Liu (1990) and Soubra and

Macuh (2002) become closer to the proposed solutionfor b = 2/3 with an increase in χam; for low values of χamthey clearly diverge, the first solutions leading to lowerresults of Npile

UB; and(iv) NS influences Npile

UB in the expected way: less resistantsoil masses (greater NS) demand more resistant verticalpiles.

3. Comparison with results of conventionalfinite element calculations

The FEM calculation procedure used to obtain the resultsplotted by the symbols in Fig. 11 was described by Guerra etal. (2004) considering the numerical case study representedin Fig. 12. The results presented in Fig. 11 were directly ob-tained from that paper or from calculations using the sametechnique: a series of successive calculations varying pile re-sistance and keeping constant all other variables makes itpossible to estimate the pile resistance that will lead to im-minent soil collapse, with a small margin of error. For thedefinition of such pile resistance, both the convergence crite-rion (tolerance of 0.1% after 20 000 iterations) and the dis-placements were considered. These f.e. analyses alsoenabled the determination of the percentage, p, of adhesionmobilization at the soil-to-wall interface and, consequently,of the value of χam.

The values of (N WpileUB

w− )/µ and χam determined as men-tioned previously are reasonably well adjusted to the pro-posed solution for b = 2/3. Furthermore, it can be observedthat in all cases with ca ≠ 0, the interface resistance for thelimit situation was far from full mobilization. The only situ-ation in which “full mobilization” occurred was, therefore,

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Cardoso et al. 523

Fig. 9. Comparison between the estimate of the horizontal forceobtained by the NLA implementation of UBT and the proposedsolution.

when ca = 0 (which implies that χam = 0). For this situation itcan be seen that the results obtained from f.e. analysis arealmost coincident with the results from the proposed solu-tion for b = 2/3. This is because the resultant forces of thestress distribution obtained from the f.e. analyses have val-ues of b quite close to 2/3.

For ca ≠ 0, it can be concluded that: (i) a very good agree-ment was obtained between FEM results and the proposedsolution (with b = 2/3) for β = 45° (Fig. 11a); and (ii) agree-ment is not so good for other values of β: in fact, for NS =4.5 (Fig. 11b) it can be seen that more significant differencesbetween the FEM results and the proposed solution seem tooccur for higher values of |β – 45°|.

After commenting on the approximation of the numericalresults to the theoretical curves, it is necessary to discuss thespecific positions of the points corresponding to these resultson the graphs in Fig. 11.

The values of χam provided by the f.e. analyses are in allcases to the left of the respective χam–cr, and the discrepancyincreases with NS (see Fig. 11a). This result is reasonable. Infact, the f.e. analyses (see Fig. 12) simulated the essential ofthe whole construction process. In the early constructionstages, the excavation depth is still small or moderate, as isthe vertical force applied by the anchors and the wallweight; so, the structural strength of the soldier piles is ade-quate and the vertical equilibrium is comfortably granted. Insuch a situation, as Fig. 1 and Table 1 illustrate, there is a

tendency for the supported soil to apply downward tangen-tial stresses to the back of the wall. Since these conditionsprevail immediately before the occurrence of soldier-pilebuckling, it is comprehensible that the situation of imminentcollapse is achieved with a distribution of tangential stressesthat corresponds to a value of χam to the left of the criticalvalue.

Two further observations on the results of Fig. 11 seempertinent:(i) in spite of the fact that values of χam are significantly

different from those of χam–cr in some f.e. analyses be-cause of the flattened shape of the theoretical curves,the values of the limit load on the soldier piles providedby the f.e. calculations are quite close to the respectiveminimum analytical values; and

(ii) it is reasonable that the difference between χam andχam–cr increases with NS (see Fig. 11). For less resistantsoils, in the numerical analyses, there is a trend forgreater and earlier downward tangential stresses to beapplied to the wall through the interface, before theplastification of the soldier piles.

In conclusion, the exact position of the point representa-tive, in each case, of the results of the f.e. simulations on thegraphs of Fig. 11 depends on the sequence of the construc-tion, on the soil strength and, presumably, on a number ofother parameters, such as the stiffness of the soil, thesoldier-piles, and their foundation.

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Fig. 10. Comparison of the slip surfaces of the proposed solution with the plastic dissipation zones of NLA (slip surfaces and NLAmechanism not dependent on NS).

4. Actions for the design of vertical soldierpiles

When NS < 6 and the base stability of the excavation isensured, it is reasonable to adopt, when establishing anchorpre-stress, a diagram such as the one presented in Fig. 12with a maximum pressure of 0.2�0.4γH. The resultant forceof this diagram is Σ Aps cos β = 1.5 ρps µ, where ρps =0.2�0.4.

It is interesting to compare the initial horizontal compo-nent of anchor pre-stress to the total horizontal force at thelimit state, although the application point is clearly not thesame in the two cases (b = 2/3 for the proposed analyticalsolution and b = 1/2 for the initial horizontal pre-stress dia-gram). To carry out this comparison, the initial pre-stresshorizontal force must be adimensionalized, which leads to(Σ Apscosβ)/µ = 1.5ρps. Adimensional results are representedin Fig. 13 for ρps = 0.2, 0.3, and 0.4. This figure also pres-ents the results previously shown in Fig. 9, both for the pro-posed analytical solution and the FEM results; the fittingbetween the two solutions being fairly good.

It should be pointed out that in the f.e. calculations, whoseresults are plotted in Fig. 13, ρps is equal to 0.3 (see Fig. 12).It can be observed from Fig. 13 that, in these calculations,the horizontal force applied by the anchors at the limit stateis below the correspondent level of the pre-stress. In fact, asit is shown in Fig. 1 and Table 1, the degradation of the con-ditions of vertical equilibrium normally induces a progres-sive reduction of the anchor forces. From the same figure itcan also be seen that the horizontal force applied by the an-chors can decrease more—from a given initial value to thelimit load—for a more resistant soil (lower NS), which is ex-pected.

The solution proposed in this paper may be used as thebasis for the design of vertical soldier piles but must alsoconsider the usual minimum demands on anchor pre-stress.Equation [9] can be re-written as

[11] ρ βµ

UBUB

S1.5 1.5= = −

⎛

⎝⎜⎜

⎞

⎠⎟⎟

( cos )Σ AT

SN

1

The results obtained using FEM (see Fig. 11) show thatthe interface resistance mobilized at the limit state is, inmost cases, such that –0.3 ≤ χam ≤ 0.3, negative values occur-ring for high values of β or NS and positive ones when β andNS are less than 45° and 4.5, respectively.

Figure 14 shows the results from eq. [11] for the above-mentioned interval and for b = 2/3 and 1/2. It can be seenthat the parameter b does not have a very significant influ-ence. Furthermore, for NS < 6 the average value ρps = 0.3used for anchor pre-stress will ensure in most cases the hori-zontal force needed to guarantee the stability of the soilmass. A minimum value of 0.2 is considered in the figurebecause it corresponds to the usual minimum value appliedto prevent excessive displacements of the wall. Hence, basedon eq. [10] and on the last consideration, the following equa-tion is proposed for the design of soldier piles:

[12] Npile – Ww = 1.5 max (ρps; ρUB) µ tan β –2µNS

χam

© 2006 NRC Canada

Cardoso et al. 525

Fig. 11. Values of the estimate of the limit force on the soldierpiles for (a) β = 45° and varying NS and for (b) NS = 4.5 andvarying β.

© 2006 NRC Canada


where ρUB should be adopted from Fig. 14.It should be noted that eq. [10] takes exclusively into ac-

count the limit situation of the stability of the supported soiland the wall. The alternative equation proposed now—eq. [12]—is justified by some concerns related to the dis-placements which, in most cases, should also be considered.

5. Conclusions

The pattern of behavior of anchored retaining walls inmarginal stability conditions due to deficient vertical supportis characterized by high vertical and horizontal displace-ments of the wall and the supported ground, inducing a pro-gressive loss of anchor forces and, finally, leading to thecollapse of the system.

In the particular case of soldier-pile walls, the loss of ver-

tical equilibrium is triggered by pile buckling (or by abearing capacity failure of their foundation).

To clarify the complex interaction between the wall andthe supported soil, it is crucial to link the equilibrium of thewall with the equilibrium of the soil through the normal andtangential forces at the interface.

The equilibrium of the soil mass was studied using upperbound limit analysis. An analytical solution for this problem,using a rigid block mechanism with a circular slip surface,was developed and compared with other analytical solutionsfrom the literature, as well as with results of numerical limitanalyses using the finite element method. The results of thecomparisons are considered quite encouraging.

On the basis of the analytical solution mentioned above,the vertical equilibrium of the soldier-pile wall was formu-lated, and an equation was obtained for the limit value of theresistance of the soldier-piles required to ensure overall sta-bility. This resistance depends on the geometrical and me-chanical characteristics of the excavation and of the soil andis highly dependent on the anchor inclination.

The analytical values for that resistance have been com-

Fig. 12. Assumptions for the numerical case study of Guerra et al. (2004).

Fig. 13. Horizontal component of the anchor forces at the limitstate for FEM results and for the proposed solution.

Fig. 14. Minimum values of ρUB resulting from eq. [12].

pared to the results of finite element simulations of excava-tions supported by soldier-pile walls in a previous paper ofthe authors. The agreement between the two approaches isconsidered satisfactory.

Based on the results obtained, and taking into account thedesign demands on anchor pre-stressing in common practice,a simple equation for the design of soldier-piles is proposed.

References

Antão, A.N. 1997. Analyse de la stabilité des ouvrages souterrainspar une méthode cinématique régularisée. Ph.D. thesis, ÉcoleNationale des Ponts et Chaussées, Paris, France. [In French.]

Antão, A.N. 2003. Numerical implementation of the upper boundtheorem of limit analysis. Application to the determination ofactive and passive earth pressure coefficients in a cohesionlessmaterial. In Proceedings of the 7th National Conference of The-oretical, Applied and Computational Mechanics, Évora, Portu-gal, 14–16 April 2003. Associação Portuguesa de MecãnicaTeórica, Aplicada e Computational — Portuguese Society ofTheoretical, Applied and Computational Mechanics, Lisbon,Portugal. Vol. IV, pp. 1989–1998. [In Portuguese.]

Barley, A.D. 1997. Contribuição para a discussão na sessão n. 5. InGround anchorages and anchored structures. Edited by G.Littlejohn, American Society of Civil Engineers. Thomas Tel-ford, London, UK. pp. 592–596. [In Portuguese.]

Broms, B.B., and Stille, H. 1976. Failure of anchored sheet pilewalls. ASCE Journal of Geotechnical Engineering Division,102(3): 235–251.

Cacoilo, D., Tamaro, G., and Edinger, P. 1998. Design and perfor-mance of a tied-back sheet pile wall in soft clay. In Design andconstruction of earth retaining systems. ASCE GeotechnicalSpecial Publication No. 83, American Society of Civil Engi-neers, Reston, Va. pp. 14–25.

Chen, W.F., and Liu, X.L. 1990. Limit analysis in soil mechanics.Elsevier, New York.

Clough, G.W., and O’Rourke, T.D. 1990. Construction inducedmovements of in situ walls. In Design and performance of earthretaining structures, ASCE Geotechnical Special PublicationNo. 25, American Society of Civil Engineers, Reston, Va.pp. 439–470.

Gould, J.P., Tamaro, G.J., and Powers, J.P. 1992. Excavation andsupport systems in urban settings. In Excavation and support forthe urban infrastructure, ASCE Geotechnical Special PublicationNo. 33, American Society of Civil Engineers, Reston, Va.pp. 144–171.

Guerra, N.M.C. 1999. Collapse mechanism of Berlin-type retainingwalls by loss of vertical equilibrium. Ph.D. thesis, Instituto Su-perior Técnico, Technical University of Lisbon. [In Portuguese.]

Guerra, N.M.C., Gomes Correia, A., Matos Fernandes, M., andCardoso, A.S. 2001. Modeling the collapse of Berlin-type wallsby loss of vertical equilibrium: a few preliminary results. In Pro-ceedings of the 3rd International Workshop on Applications ofComputational Mechanics in Geotechnical Engineering. Editedby M. Matos Fernandes, L. Ribeiro e Sousa, R.F. Azevedo, andE.A. Vargas. Oporto, Portugal. A.A. Balkema, Rotterdam, TheNetherlands. pp. 231–238.

Guerra, N.M.d.C., Cardoso, A.S., Matos Fernandes, M., andGomes Correia, A. 2004. Vertical stability of anchored concretesoldier-pile walls in clay. ASCE Journal of Geotechnical andGeoenvironmental Engineering Division, 130(12): 1259–1270.

Hanna, T.H., and Matallana, G.A. 1970. The behavior of tied-backretaining walls. Canadian Geotechnical Journal, 7(4): 372–396.

Leca, E., and Dormieux, L. 1992. Contribution à l’étude de lastabilité du font de taille d’un tunnel en milieu cohérent. RevueFrançaise de Géotechnique, 61: 5–19. [In French.]

Matos Fernandes, M.A. 1985. Tied-back retaining walls: bearingcapacity of the soil to vertical forces applied by the anchors.Geotecnia, Journal of the Portuguese Society of Geotechnique,43: 43–64. [In Portuguese.]

Matos Fernandes, M.A., Cardoso, A.J.S., Trigo, J.F.C., and Mar-ques, J.M.M.C. 1993. Bearing capacity failure of tied-backwalls: a complex case of soil–wall interaction. Computers andGeotechnics, 15: 87–103.

Matos Fernandes, M.A., Cardoso, A.J.S., Trigo, J.F.C., and Mar-ques, J.M.M.C. 1994. Finite element modeling of supported ex-cavations – Chapter 9. In Soil-structure interaction: numericalanalysis and modeling. Edited by J.W. Bull. E&FN Spon, Lon-don, UK.

Pastor, J., Thai, T.H., and Francescato, P. 2000. New bounds forthe height limit of a vertical slope. International Journal for Nu-merical and Analytical Methods in Geomechanics, 24: 165–182.

Plant, G.W. 1972. Anchors inclination – its effects on the perfor-mance of a laboratory scale tied-back retaining wall. Proceed-ings of the Institution of Civil Engineers, Part 2, 53: 257–274.

Soubra, A.H., and Macuh, B. 2002. Active and passive earth pres-sure coefficients by a kinematical approach. Geotechnical Engi-neering, 155(2): 119–131.

Stocker, M.F. 1991. Contribution to discussion in session n. 4b. InProceedings of the 10th European Conference on Soil Me-chanics and Foundation Engineering, Firenze, Italy, 26–30 May1991. A.A. Balkema, Rotterdam, The Netherlands. Vol. 4,p. 1368.

Taylor, D.W. 1948. Fundamentals of soil mechanics. John Wiley &Sons, New York.

Ulrich, E.J.J. 1989. Tieback supported cuts in over consolidatedsoils. ASCE Journal of Geotechnical Engineering Division,115(4): 521–545.

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List of symbols

A anchor force per unit length of the wallAps pre-stress anchor force per unit length of the wall

b quotient between the distance from the top of thewall and the application point of the anchor forceresult and the excavation depth H

c cohesionca soil-to-wall interface resistance (adhesion)cu soil undrained shear strength

Eanc Young’s modulus of anchor cablesEpile Young’s modulus of soldier pilesEwall Young’s modulus of the wall

Eu soil undrained Young’s modulusFal shear force mobilized per unit length at the back

side of the wallH excavation depth

Hw wall heightK0 initial horizontal stress coefficient

Kaγ, Kac, Kaq active earth pressure coefficientsNpile vertical reaction mobilized on the soldier piles per

unit length of the wall

© 2006 NRC Canada

Cardoso et al. 527

NS stability number of the excavationNS–cr critical stability number of the excavation

Pa total active forcep fraction of mobilization of the interface resistanceq surcharge applied on the supported soil surfaceR radius of circular slip surfaceS parameter of the proposed solution

Sanc cross-sectional area of anchor cable per unit lengthof the wall

Spile cross-sectional area of soldier piles per unit lengthof the wall

T parameter of the proposed solutionW1, W2 auxiliary soil weights

We work of external forcesWi dissipated plastic workWs weight of the unstable soil mass per unit length of

the wallWw wall weight per unit lengthx, z horizontal and vertical coordinates, respectively, of

the points of the slip surfaceα auxiliary angle for the definition of the slip surface

considered in the proposed solutionα* value of α that optimizes the proposed solutionβ anchor inclination angle with the horizontal plan

χam normalized mobilized resistanceχam–cr critical value of the normalized mobilized resistance

δ rotation of the unstable soil mass around the centerof the slip surface for the proposed solution

γ unit weight of soilλ auxiliary angle for the definition of the slip surface

considered in the proposed solutionλ* value of λ that optimizes the proposed solutionµ auxiliary symbol, equal to 0.5γH2

θ auxiliary angle for the definition of the slip surfaceconsidered in the proposed solution

νu Poisson coefficient of soil in undrained conditions

ρ parameter defining horizontal stress of trapezoidaldiagram

ρps parameter defining horizontal pre-stress of trape-zoidal diagram.

Appendix A.

Figure A1 shows the assumed displacement field and thegeometrical conditions. The following geometrical relation-ships apply:

[A1] RH=

22

sin sinα λ

[A2] θ π α λ= − +⎛⎝⎜

⎞⎠⎟

2 2

[A3]RH

sintan tan

θα λ

= −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

12

1

2

1

[A4]RH

cosθλ α

= +

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

12

1

2

1

tan tan

Table A1 presents the results of the work of the externalforces and the work of plastic deformations.

Use of the upper bound theorem (UBT) implies the equal-ization of the work of the external forces to the work ofplastic deformation. This equality and the geometrical rela-tions shown in eqs. [A1]�[A4] results in the following rela-tionship:

[A5]1

21

2

13

16 2tan

tantan sinα λ α α⎛

⎝⎜

⎞⎠⎟−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟+ +

⎛⎝⎜

⎞⎠⎟−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

χλ α

am

SN1

2

1

tantan

−⎛⎝⎜

⎞⎠⎟+ −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

( cos )

tan tan

Σ Ab

βµ α λ

UB

21

2

2 1

=⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

λ

α λ21

2

2NSsin sin

This equation leads to:

[A6]( cos )

tan tan tan tanΣA β

µ

α λ χ α λUB

am

=

+ ⎛⎝⎜

⎞⎠⎟ + − ⎛

⎝⎜1

13 2

12 2

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ −

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+ −

λα λ

α

sin( ) sin

( ) tan tan

24

1 2 1

N

b

S

λ2

⎛⎝⎜

⎞⎠⎟

© 2006 NRC Canada


Maximization of the right side of eq. [A6] with respect to αand λ leads to

[A7]( cos )Σ A

TS

N

βµ

UB

S

= −

where

[A8] T

b

=+ ⎛

⎝⎜

⎞⎠⎟

− ⎛⎝⎜

⎞⎠⎟

113 2

2

tan * tan*

* tan*

α λ

α λ1 + (2 1) tan

and

© 2006 NRC Canada

Cardoso et al. 529

Forces, Fi Displacements, di

Work, F d

Hi i

µ δ

Work of external forces

W H1

2= =γ

αµα

2

tan tanR

HHcos

tanθ

αδ−⎛

⎝⎜ ⎞

⎠⎟2

3R

Hcos

tan tanθ

α α−⎛

⎝⎜ ⎞

⎠⎟2

31

W R R

H2

2 2

2= − = ⎛

⎝⎜ ⎞

⎠⎟ −( sin ) ( sin )λ λ γ λ λ µ

43

23

R

HH

sin

sinsin

λ

λ λα δ

⎛⎝⎜ ⎞

⎠⎟

−

43 2

3R

Hsin sinλ α⎛

⎝⎜ ⎞

⎠⎟

⎡

⎣⎢

⎤

⎦⎥

pc H pc HH

H

c N Na w a w

u S

am

S

= =2 22

µγ

γ χ µ R

HHcos

tanθ

αδ−⎛

⎝⎜ ⎞

⎠⎟1 R

H Ncos

tanθ

αχ−⎛

⎝⎜ ⎞

⎠⎟1 2 am

S

Σ ΣA Acos cosβ = βµ

µ R

Hb Hsinθ δ+⎛

⎝⎜ ⎞

⎠⎟ R

Hb Asinθ β

µ+⎛

⎝⎜ ⎞

⎠⎟Σ cos

Work of plastic deformation

λ λ µγ

γ λ µRc RcH

H

c N N

R

Hu u

u S S

= =2 22

R

HHδ λ 2

2

N

R

HS

⎛⎝⎜ ⎞

⎠⎟

Table A1. Determination of the work of external forces and of the work of plastic deformation for theproposed solutions.

Fig. A1. Conditions of the mechanism assumed in the proposed upper bound solution.

[A9] S = −− ⎛

⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ −

4

12 2 2

1

χ α λ λα λam

sintan * tan

* *( *) sin *

+ − ⎛⎝⎜

⎞⎠⎟( ) tan * tan

*2 1

2b α λ

wherein α* and λ* are the optimal values of α and λ.For b = 2/3, it turns out that T = 1 and that α*, λ*, and S

do not depend on NS. Table A2 shows the values of α*, λ*,and S as a function of χam for this case. In this table, nega-tive values of λ* correspond to a mechanism with the con-cavity of the surface directed downwards (Fig. A1 presentsthe concavity directed upwards). Table A3 shows the sameresults for b = 1/2 as a function of χam and NS.

List of symbols

A anchor force per unit length of the wallb quotient between the distance from the top of the

wall and the application point of the anchor forceresultant and the excavation depth H

ca soil-to-wall interface resistance (adhesion)

cu soil undrained shear strengthFal shear force mobilized per unit length at the back

side of the wallH excavation depth

Hw wall heightNS stability number of the excavation

p fraction of mobilization of the interface resistanceR radius of circular slip surfaceS parameter of the proposed solutionT parameter of the proposed solution

W1, W2 auxiliary soil weightsα auxiliary angle for the definition of the slip surface

considered in the proposed solutionα* value of α that optimizes the proposed solutionβ anchor inclination angle with the horizontal plan

χam normalized mobilized resistanceδ rotation of the unstable soil mass around the center

of the slip surface for the proposed solutionγ unit weight of soil

λ* value of λ that optimizes the proposed solutionµ auxiliary symbol, equal to 0.5γH2

θ auxiliary angle for the definition of the slip surfaceconsidered in the proposed solution

© 2006 NRC Canada


χam –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

NS = 3.8α* 43.17 44.07 45.09 46.23 47.51 48.95 50.61 52.52 54.80 57.58 61.19 66.42 76.28λ* 43.05 39.54 36.14 32.86 29.73 26.80 24.15 21.88 20.19 19.38 20.06 23.73 35.76S 4.847 4.704 4.549 4.382 4.201 4.007 3.800 3.578 3.344 3.101 2.865 2.701 3.233T 1.123 1.116 1.109 1.103 1.097 1.091 1.087 1.084 1.084 1.090 1.107 1.160 1.440

NS = 4.5α* 44.31 45.23 46.26 47.44 48.77 50.31 52.09 54.21 56.79 60.10 64.75 72.56 —λ* 49.26 45.87 42.64 39.57 36.72 34.15 31.96 30.32 29.50 30.01 33.02 41.85 —S 4.954 4.813 4.661 4.498 4.325 4.141 3.949 3.750 3.552 3.376 3.293 3.737 —T 1.149 1.142 1.136 1.131 1.126 1.123 1.123 1.125 1.134 1.155 1.209 1.406 —

NS = 5.2α* 45.81 46.76 47.84 49.09 50.54 52.24 54.26 56.75 59.94 64.39 71.57 — —λ* 56.42 53.18 50.14 47.35 44.87 42.79 41.29 40.61 41.26 44.26 52.33 — —S 5.124 4.984 4.837 4.684 4.524 4.362 4.202 4.056 3.956 4.000 4.683 — —T 1.184 1.177 1.172 1.169 1.167 1.169 1.174 1.188 1.217 1.283 1.492 — —

NS = 5.9α* 47.86 48.87 50.05 51.44 53.10 55.11 57.61 60.87 65.47 72.86 — — —λ* 65.03 61.98 59.22 56.83 54.91 53.64 53.28 54.36 57.99 66.65 — — —S 5.407 5.273 5.138 5.005 4.879 4.770 4.700 4.722 5.011 6.414 — — —T 1.235 1.229 1.226 1.226 1.231 1.242 1.264 1.307 1.405 1.711 — —

Table A3. Values of α*, λ*, S and T for b = 1/2.

χam –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

α* 43.97 44.71 45.55 46.49 47.55 48.75 50.11 51.66 53.45 55.53 58.03 61.11 65.15λ* 47.50 43.17 38.82 34.44 30.02 25.56 21.05 16.51 11.95 7.39 2.90 –1.40 –5.20S 4.311 4.212 4.100 3.973 3.831 3.672 3.494 3.295 3.071 2.818 2.528 2.190 1.783

Table A2. Values of α*, λ*, and S for b = 2/3.

Limit analysis of anchored concrete soldier-pile walls in clay under vertical loading

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