stability analysis of slopes reinforced with piles.pdf

Stability analysis of slopes reinforced with piles

E. Ausilio*, E. Conte, G. Dente

Dipartimento di Difesa del Suolo, Universita della Calabria, 87036 Rende, Cosenza, Italy

Received 18 August 2000; received in revised form 22 March 2001; accepted 29 March 2001

Abstract

In this paper, the kinematic approach of limit analysis is used to analyse the stability of

earth slopes reinforced with piles. First, the case of slope without piles is considered and aprocedure is developed to calculate the safety factor for the slope. Results are compared withthose obtained using both the limit equilibrium method and more complex upper and lower

bound limit analysis solutions. Then, the stability of slopes reinforced with piles is analysed.Expressions are derived allowing the force needed to increase the safety factor to a desiredvalue and the most suitable location of piles within the slope to be evaluated. A study is car-ried out to illustrate the effect of piles on slope stability. # 2001 Elsevier Science Ltd. All

rights reserved.

1. Introduction

Slope stability can be increased in different ways such as: flattening of slopes bymodifying the ground surface geometry, carrying out surface and subsurface drai-nage, using soil improvement techniques, installing continuous or discrete retainingstructures such as walls or piles. The first remedy leads to a reduction of the drivingforces for failure; the other measures in general produce an increase of the resistingforces.Piles have been used successfully in many situations in order to stabilise slopes or

to improve slope stability [1–9], and numerous methods have been developed for theanalysis of piled slopes.The finite element method is certainly the most comprehensive approach to study

pile-slope stability, as this method simultaneously solves pile response and slope

Computers and Geotechnics 28 (2001) 591–611

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* Corresponding author.

E-mail addresses:[email protected] (E. Ausilio)

stability. However, its use generally requires high numerical costs and accuratemeasurements of material properties. This makes the use of the method ratherunattractive for current applications. The finite element method has been recentlyused by Cai and Ugai [10] to analyse the effect of piles on slope stability.In practical applications, the study of a slope reinforced with piles is usually car-

ried out by extending the methods commonly used for the stability analysis of slopesto incorporate the reaction force exerted on the unstable soil mass by the piles. Todate, the limit equilibrium method is the most widely used approach to analyse slopestability due to its simplicity of use. Moreover, this method allows for the effect ofseepage, loading and general soil conditions without requiring additional computa-tional efforts. Major criticisms of the limit equilibrium method are that it is generallybased on simplified assumptions, and the results obtained from this method are, inthe light of limit analysis, neither upper bounds nor lower bounds on the true solu-tion [11].The limit equilibrium method was used by Ito et al. [12] to deal with the problem

of the stability of slopes containing piles. In this approach the safety factor of thepiled slope was defined as the ratio of the resisting moment to the overturning momentacting on the potentially unstable soil mass. The resisting moment consists of twocomponents: the moment due to soil shearing resistance along the sliding surface andthe moment provided by the reaction force from the piles. The driving moment and theresisting moment due to soil shearing resistance were obtained applying the ordinaryslice method. To calculate the resisting moment due to the piles, Ito et al. [12] pro-posed the use of the theoretical equation, derived previously by Ito and Matsui [3],to evaluate the lateral force acting on a row of piles due to soil movement.A similar approach was developed by Lee et al. [13] in which Bishop’s simplified

method [14] was employed to find the critical sliding surface for the slope as well asthe driving moment and resisting moment due to soil shearing resistance. The resist-ing moment generated by the piles was obtained from the shear force and bendingmoment developed in the pile at the depth of the sliding surface by the lateral soilmovement. These forces were calculated using a procedure based on the boundaryelement method which was earlier proposed by Poulos [15] and later developed byLee et al. [16].Recently, Hassiotis et al. [17] have extended the friction circle method to incor-

porate the pile reaction in slope stability analysis. The Ito and Matsui equation [3]has been used to evaluate the lateral force that the failing soil mass exerts on a rowof piles.The limit equilibrium method was also used by Chugh [18] and Poulos [8] to

analyse the stability of piled slopes. In both these approaches, it is assumed that thepiles provide an additional shear resistance along the critical sliding surface whichshould increase the safety factor of the slope to a selected value.In this paper, the stability of slopes reinforced with piles is analysed using the kine-

matic approach of limit analysis. The case of a slope without piles is first considered,and a solution is proposed to determine the slope safety factor, which is here definedas a reduction coefficient for the strength parameters of the soil. Then, the stabilityof a slope containing piles is analysed. To account for the presence of the piles, it is

592 E. Ausilio et al. / Computers and Geotechnics 28 (2001) 591–611

assumed that a lateral force and a moment are applied at the depth of the potentialsliding surface. Theoretical solutions are derived that allow the values of these forcesto be calculated. Moreover, conclusions are drawn regarding the most suitablelocations of the piles within the slope.

2. Method of analysis

Limit analysis takes advantages of the static and kinematic theorems of plasticitytheory to find the range in which the true solution of a stability problem falls. Thisrange can be narrowed finding the highest possible lower-bound solution and thelowest possible upper-bound solution. The unknown quantity may be the bearingcapacity of a foundation, the earth pressure on a retaining wall, the safety factor orcritical height of a slope, etc. In limit analysis, soil is assumed to deform plasticallyaccording to the normality rule associated with the Coulomb yield condition.The static theorem considers stress fields which are in equilibrium with surface

tractions and body forces, and do not violate the yield criterion anywhere in the soilmass (statically admissible stress field). Application of the static theorem leads to aset of differential equations which may be solved numerically using the finite elementmethod [11,19].To solve slope stability problems, use of limit analysis has almost exclusively

concentrated on the kinematic theorem [20–25], because under certain assumptions,this is generally simpler to use than the static approach. For instance, when thefailing soil mass is assumed to move as a rigid body, the kinematic theorem neces-sitates the solving of a simple equation.Application of the kinematic theorem requires to equate the rate of work done by

tractions and body forces to the internal energy dissipation rate, for any assumedstrain rate field which is governed by the normality rule and is compatible with thevelocities at the boundary of the failing soil mass (kinematically admissible failuremechanism). This can be expressed by the following work equation:

ðS

Ti vi dSþðV

Xi vi dV ¼ðV

�ij ":ij dV i; j ¼ 1; 2; 3 ð1Þ

where Xi= body forces; Ti= traction; vi= kinematically admissible velocity field;":ij= strain rate field compatible with vi; �ij= stress field relating to Xi and Ti.Moreover, S and V are, respectively, the loaded boundary and the volume of thesliding soil mass. When the unknown quantity is a force that makes the soil massunstable, application of the kinematic theorem leads to an upper bound for the truesolution. On the contrary, this theorem yields a lower bound solution when a stabi-lising force has to be determined.In this study, the kinematic approach is employed to calculate the stabilising force

which must be provided by a retaining structure to increase the safety factor for aslope of homogeneous soils to a selected value. For simplicity, the effect of pore-

E. Ausilio et al. / Computers and Geotechnics 28 (2001) 591–611 593

water pressure on slope stability is not considered in the present work. However,under drained loading conditions, this effect can be accounted for by expressing thesecond term on the left-hand side of Eq. (1) as the rate of work done by the sub-merged soil weight and that due to the seepage forces. In addition, the term �ij onthe right-hand side of (1) is the effective stress tensor.The kinematic approach is first applied to analyse slope stability without retaining

structures.

3. Stability analysis of slopes without piles

In limit analysis, the solution of a slope stability problem is usually expressedeither in terms of the critical slope height [21] or a limit load applied on some por-tion of the slope boundary [23]. If there is no boundary loading, collapse may becaused by the weight of the soil itself. Thus, the limit condition has been alsoexpressed in terms of the unit weight of soil [11].Slope stability analysis is traditionally formulated in terms of the safety factor

with respect to soil shearing strength parameters [26], which is analytically definedas

FS ¼ c

cm¼ tg’

tg’mð2Þ

where FS indicates the safety factor; c and ’ are the cohesion and the shearingresistance angle of the soil, respectively; cm is the mobilized cohesion, and ’m is themobilized angle of shearing resistance. In other words, FS is defined as the factor bywhich the soil shearing strength parameters should be divided to give the conditionof incipient failure. Karal [22] and Donald and Chen [25] accepted Eq. (2) as thedefinition of the safety factor to analyse slope stability using the kinematic approachof limit analysis. As pointed out by Karal [22], a direct consequence of Eq. (2) isthat, for frictional materials, the sliding surfaces are surfaces of potential yield, andthe displacements and the failure mechanism depend on the safety factor. This defi-nition of FS is also adopted in the present study.The kinematically admissible mechanism considered is shown in Fig. 1, where the

sliding surface is described by the log-spiral equation

r ¼ r0 e ��0ð Þ tg’FS ð3Þ

where r0=radius of the log-spiral with respect to angle �0. The failing soil massrotates as a rigid body about the centre of rotation with angular velocity !

:. This

mechanism, which was earlier considered by Chen [21], is geometrically defined byangles �0, �0, �h (Fig. 1) and mobilized angle of shearing resistance tg’

FS. The slopegeometry is specified by height H, and angles � and � which are also indicated inFig. 1.


The rate of external work is due to soil weight and surcharge boundary loads.These two components of the external work rate are indicated in this study as W

:and

Q:, respectively. The rate of work due to soil weight takes the form [21]

W: ¼ � r30!

:f1 � f2 � f3 � f4½ � ð4Þ

where �=soil unit weight; functions f1–f4 depend on the angles �0, �h, �, � and �0,and the mobilized angle of shearing resistance. Expressions for f1–f4 can be found inChen [21]; for the sake of completeness they are also reported in the Appendix ofthis paper. In deriving Eq. (4), it is assumed that the sliding surface passes below thetoe of the slope (Fig. 1). However, for the case in which the sliding surface passesthrough the toe of the slope, the same expression for W

:can be used provided f4 =0

and �0=�.When the slope is subjected to a surcharge boundary load, as shown in Fig. 1, the

rate of work done by this load is

Q: ¼ q L !

:r0 cos �0 þ �ð Þ � L

2

� �þ s L !

:r0 sin �0 þ �ð Þ ð5Þ

where L=distance between the failure surface at the top of the slope and the edge ofthe slope (Fig. 1); q=applied normal traction; s=applied tangential traction.For the rigid-block mechanism considered, the only energy dissipation takes place

along the sliding surface. The rate of energy dissipation, D:, can be written as [21]

D: ¼ c r20 !

:

2 tg’e2 �h��0ð Þ tg’FS � 1h i

ð6Þ

Fig. 1. Slope failure mechanism.


By equating the rate of external work to the rate of energy dissipation, we have

W: þQ

: ¼ D: ð7Þ

and substituting the expressions for W:, Q:and D

:into Eq. (7) yields

�H

Af1 � f2 � f3 � f4ð Þ þ q B cos �0 þ �ð Þ � B

2

� �þ s B sin �0 þ �ð Þ

¼ c


ð8Þ

where [21]

A ¼ sin�0

sin �0 � �ð Þ sin �h þ �ð Þe �h��0ð Þ tg’FS � sin �0 þ �ð Þn o

ð9Þ

B ¼ sin �h � �0ð Þsin �h þ �ð Þ �

sin �h þ �0ð Þsin �h þ �ð Þ sin �0 � �ð Þ sin �h þ �ð Þe �h��0ð Þ tg’FS � sin �0 þ �ð Þ

n o

ð10Þ

The quantities A and B can be related to H and L, respectively, by the followingexpressions

H ¼ A r0 ð11aÞ

L ¼ B r0 ð11bÞ

where distance L is indicated in Fig. 1.For a given FS value, an upper bound for the slope height is obtained solving Eq.

(8), i.e.

H ¼ A

�

c

2 tg’

e2 �h��0ð Þ tg’FS � 1� � q B cos �0 þ �ð Þ � B

2

� �� s B sin �0 þ �ð Þ

f1 � f2 � f3 � f4ð Þ

2664

ð12Þ

The least upper bound for H can be found minimising the function H ¼f �0; �h; �

0ð Þ with respect to �0, �h and �0 [21]. The angles thus obtained define thepotential sliding surface. In addition, substituting these angles into Eq. (12) yieldsthe critical height of the slope. This is the maximum height at which it is possible forthe slope to be stable with the assumed FS value. The true value of the safety factorcould be then found by an iterative procedure in which the resistance parameters of


the soil are progressively changed according to Eq. (2), until the critical height isequal to the actual height of the slope.Alternatively, the safety factor can be directly found by solving the following set

of equations

@ H

@ �0¼ 0

@ H

@ �h¼ 0

@ H

@ �0 ¼ 0

H ¼ Hactual

8>>>>>>>><>>>>>>>>:

ð13Þ

where Hactual denotes the actual slope height. In Eq. (13), the unknown quantitiesare �0, �h, �

0 and FS. Therefore, the solution of Eq. (13) gives both the values of FSand the position of the potential sliding surface.Comparisons of the FS values derived from Eq. (13) to those obtained by other

authors using different methods are presented below.Table 1 shows a comparison of the safety factor calculated by Eq. (13) and that

obtained by Cao and Zaman [27] using three different methods: Bishop’s method

Table 1

Comparison of slope safety factor calculated using different methods (adapted from Ref. [27])

b(ratio)

c(kPa)

’(degree)

FSanalytical method

FS localFS method

FSBishop’s method

FSEq. (13)

1:1 25 20 1.81 1.87 1.74 1.73

1:1 20 20 1.60 1.68 1.50 1.511:1 15 20 1.39 1.46 1.29 1.281:1 10 20 1.17 1.00 1.05 1.04

1:1 30 15 1.81 1.85 1.75 1.761:1 25 15 1.60 1.65 1.53 1.551:1 20 15 1.40 1.45 1.32 1.34

1:1 15 15 1.19 1.24 1.11 1.121:1 10 15 0.98 1.00 0.89 0.891:1 25 10 1.40 1.42 1.35 1.38

1:1 20 10 1.20 1.23 1.15 1.171:1 15 10 1.00 1.00 0.97 0.962:1 20 20 2.01 2.05 2.09 2.07

2:1 15 20 1.76 1.85 1.82 1.812:1 10 20 1.51 1.60 1.54 1.532:1 5 20 1.24 1.23 1.21 1.21

2:1 25 15 1.98 1.87 2.05 2.052.1 20 15 1.74 1.72 1.78 1.792:1 15 15 1.49 1.54 1.53 1.54

2:1 10 15 1.25 1.29 1.29 1.272:1 5 15 0.99 1.00 0.99 0.982:1 15 10 1.23 1.19 1.27 1.27

2:1 10 10 0.99 1.00 1.03 1.02


[14], the local minimum factor-of-safety method proposed by Huang and Yamasaki[28], and the analytical method developed recently by Cao and Zaman [27]. Asshown in Table 1, safety factors derived from Eq. (13) are very close to the valuescalculated using Bishop’s method, and are in good agreement with all the resultspresented by Cao and Zaman [27].Yu et al. [11] have recently presented rigorous upper and lower bound solutions

for the stability analysis of slopes. These solutions have been achieved using twonewly developed numerical procedures that are based on finite element formulationsof the bound theorems of limit analysis and linear programming techniques. Forcomparison, Yu et al. [11] also applied Bishop’s limit-equilibrium method [14] tocalculate the slope safety factor. Results have been presented by Yu et al. [11] ingraphic form in terms of the stability number NF ¼ � H FS

c against the dimensionlessparameter lc’ ¼ � H tg’

c earlier introduced by Janbu [29]. Figs. 2 and 3 show a com-parison of the values of NF obtained in this study with those provided by Yu et al.[11], for two values of slope angle �. As can be seen, Eq. (13) gives results that aresubstantially in good agreement with those calculated using both the finite elementmethod and Bishop’s method. Moreover, it should be noted that the proposed upperbound solution, based on a simple rigid-block mechanism, provides FS values thatare smaller than those obtained by Yu et al. [11] using a more complex upper boundsolution in which a failure mechanism including both rigid body motion and con-tinuous deformation was considered.Finally, a slope with H=13.7 m and �=30� is analysed as another example. Soil

properties are: c=23.94 KPa, ’=10�, and �=19.63 KN/m3. This case was exam-ined by Hassiotis et al. [17]; they calculated a FS value for the slope equal to 1.08.

Fig. 2. Comparison of stability number NF, for a slope with �=45� (adapted from Ref. [11]).


Hassiotis et al. [17] used the friction circle method. Applying Bishop’s method, Hulland Poulos [30] found that the safety factor for the same slope is 1.12. The FS valueobtained solving Eq. (13) is 1.11, midway among those calculated by the otherauthors. The potential sliding surfaces found applying the three different methodol-ogies are shown in Fig. 4. Their positions are consistent with the corresponding FSvalues.

4. Stability analysis of slopes reinforced with piles

When the safety factor for a slope is considered to be inadequate, slope stabilitymay be improved installing a retaining structure such as a row of piles (Fig. 5). Thepiles should be designed to provide the stabilising force needed to increase the safetyfactor to a selected value.In this section, the kinematic approach is applied to assess the additional force that the

piles must provide to increase slope stability. To account for the presence of the piles, a

Fig. 3. Comparison of stability number NF, for a slope with �=60� (adapted from Ref. [11]).

Fig. 4. The critical sliding surfaces found by Hassiotis et al. [17], Hull and Poulos [30], and using Eq. (13).


lateral force and a moment are assumed to be applied at the depth of the potentiallysliding surface. Under this assumption, the rate of energy dissipation becomes

D: ¼ c r20 !

:


þ F r0 sin�F !:e �F��0ð Þtg’FS �M !

: ð14Þ

in which FS is the target safety factor of the slope; angle �F specifies the position ofthe retaining structure along the sliding surface (Fig. 5); F is the stabilising force,per unit width of soil, which the piles have to provide to improve slope stability;moment M accounts for F distribution with depth in the portion of the piles abovethe sliding surface, it is given by

M ¼ F m h ð15Þ

where h is the height of the portion of the piles above the sliding surface (Fig. 5), andm is a coefficient less than unity. For instance, if F is assumed to be linearly dis-tributed between the ground surface and the sliding surface, m is set equal to 1/3.When m=0, the presence of the piles on slope stability is expressed by an additionalshearing resistance along the potential sliding surface, as assumed also by Poulos [8].Height h can be calculated using one of the following expressions according to theabscissaxF which is measured from the slope toe (Fig. 5):

h ¼ rF sin�F � rh sin�h if �D4xF < 0 ð16aÞ

Fig. 5. Piled slope stability problem.


h ¼ rF sin�F � rh sin�h þ xF tg� if 04xF4H ctg� ð16bÞ

h ¼ rF sin�F � rh sin�h þHþ xF �H ctg�ð Þtg� if xF > H ctg� ð16cÞ

where

xF ¼ rF cos#F � rh cos�h �D ð17aÞ

D ¼ sin �� 0ð Þsin� sin�0 H ð17bÞ

rF ¼ H

Ae �F��0ð Þtg’FS and rh ¼ H

Ae �h��0ð Þ tg’FS ð17cÞ

For a selected FS value, h is a function of angles �0, �h, �0 and �F.

The rate of external work is given again by the sum of W:and Q

:. These latter are

expressed by Eqs. (4) and (5), respectively. Therefore, equating the rate of externalwork to the rate of energy dissipation leads to the following expression for F:

F ¼� H

Af1� f2� f3� f4ð ÞþqB cos �0þ�ð Þ� B

2

� �þ s B sin �0þ�ð Þ� c

2 tg’e2 �h��0ð Þ tg’FS�1h i

A

Hsin�F e

�F��0ð Þ tg’FS�mhA

H

� �

ð18Þ

Eq. (18) gives the force, per unit width of soil, which must be provided by aretaining structure to achieve the desired value of the safety factor of the slope. If theretaining structure consists of a row of piles, the lateral force acting on each pile maybe obtained in an approximate manner multiplying F by the centre to centre spacingbetween the piles. To evaluate more suitably the force acting on the piles, archingbetween adjacent piles should be considered.When a retaining structure is inserted in a slope, the additional resistance pro-

vided by this structure changes both the slope safety factor and potential failuremechanism with respect to the case without piles. As a consequence, other possiblesliding surfaces could be more critical than the one found for the slope without piles.The most critical surface is that for which the highest F value is required to increasethe safety factor to the desired value. From the computational point of view, thissurface can be found maximising function F ¼ F �0; �h; �F; �

0ð Þ with respect to angles�0, �h and �0 under the condition that the position of piles within the slope is given.To this end, the following set of equations has to be solved


@ F

@ �0¼ 0

@ F

@ �h¼ 0

@ F

@ �0 ¼ 0

xF ¼ H

Acos�F e �F��oð Þtg’FS �cos�h e �h��oð Þ tg’FS

h i�H

sin ��0ð Þsin� sin�0

8>>>>>>>>>>><>>>>>>>>>>>:

ð19Þ

where xF specifies the position of the piles with respect to the toe of the slope (Fig. 5).The value of xF should be assumed keeping in mind the critical sliding surface foundfor the slope without piles. This surface indicates, in fact, the range of positionswhere piles have to be placed for increasing effectively slope stability. A retainingstructure outside the region of soil affected by this sliding surface could have noinfluence on slope stability [30].In Eq. (19), the unknown quantities are �0, �h, �

0 and �F. Angles �0, �h, and �0

specify the critical potential sliding surface, and the maximum F value is calculatedsubstituting these angles into Eq. (18). However, it should be noted that, if m isassumed not to be zero, F depends on height h which can be determined from �0, �h,�F and �0 using Eqs. (16) and (17). This implies that Eq. (19) has to be solved con-sidering the expression for h relevant with the assumed value of xF, according toEqs. 17.Once force F is obtained, pile geometry, centre-to-centre distance at which the

piles have to be placed, and structural requirements for the piles can be determinedfrom a pile-soil interaction analysis [1,4,12,16,17,31–36]. Maximum displacement,shear and bending moments acting on the piles should be considered to assurethat the design is adequate. This matter is however outside the scope of the presentwork.The outlined approach is illustrated considering the same slope shown in Fig. 4, as

an example. The safety factor for this slope without pile reinforcement is 1.11. Thecritical sliding surface is also indicated in Fig. 4. Since a safety factor of 1.11 isconsidered inadequate, the slope may be reinforced installing a row of piles toincrease the safety factor to a selected value. For this example, it is assumed that therequired safety factor is 1.50. The piles are assumed to be located at xF ¼13.7 m.The stabilising force, for unit width of soil, which has to be provided by the piles toincrease slope stability, is evaluated using Eqs. (18) and (19) in which m is set equalto 1/3. In the case examined, this force is equal to 515 kN/m, and is assumed to belinearly distributed between the ground surface and the sliding surface. Moreover,from Eq. (16) the height of the portion of the pile above the sliding surface ish=12.7 m. Therefore, the total length of the piles may be preliminarily assumed asLp�2 h=25 m [8]. The potential sliding surface for the slope without piles and thatfor the piled slope are shown in Fig. 6. As can be noted, the sliding surface for theslope reinforced with piles is deeper and passes beneath the toe of the slope. Thewriters have found that this occurs generally for low values of the soil shearing


resistance angle or when the achievement of a high FS value is required, especiallywhen the slope is gentle.

5. Results

As can be noted from Eq. (18), force F depends on the position of the piles withinthe slope which is specified by angle �F or equivalently by abscissa xF. The mostsuitable position for the piles is that where the piles are most effective for improvingslope stability.Many studies have been conducted in order to establish the optimal location of

the piles within a slope. However, the results obtained are rather different, and insome cases even contrasting.Ito et al. [12] showed that the maximum effect of piles on slope stability is when

they are placed in the upper-middle part of the slope. Hassiotis et al. [17] arrived atsimilar conclusions. According to these latter authors, the piles should be locatedclose to the top of the slope to achieve the maximum safety factor, especially whenthe slope is steep. Lee et al. [13] analysed the case of a purely cohesive soil slope.They found that when the piles are installed into a homogeneous soil the mosteffective pile positions are the toe and crest of the slope. By contrast, the piles havelittle effect on stability when they are located close to the middle of the slope. For atwo-layered soil slope where the upper soft layer is underlain by a stiff layer, Lee etal. [13] showed that the piles are more effective when installed between the middleand the crest of the slope. However, if the soil profile is reversed, according to Lee etal. [13] the most effective positions for the piles are again the toe and the crest of theslope. Recently, Cai and Ugai [10], using the finite element method, have pointedout that the piles should be located in the middle of the slope to achieve the max-imum safety factor for the slope. The same authors have also applied a modifiedversion of Bishop’s method in which the reaction force from the piles is expressed byIto–Matsui’s equation [3]. Using this approach, Cai and Ugai [10] have found thatthe piles have to be installed closer to the top of the slope to give the best result.In order to illustrate the effect of the pile position on slope stability, the example

shown in Fig. 7 is considered. Soil strength parameters are assumed to be: c=4.7

Fig. 6. Critical sliding surface for the slope without piles and for the piled slope.


kPa and ’=25�; unit weight is 20 kN/m3. For this slope, the safety factor is equal to1.19 as derived from Eq. (13). The optimal position of the piles within the slope isdetermined when the stabilising force needed to increase the safety factor to thedesired value takes the minimum value. Assuming that the pile position variesbetween the base and the top of the slope, the force provided by the piles is calcu-lated using Eqs. (18) and (19), and is plotted against the dimensionless abscissa xF

Lx,

where Lx ¼ H ctg� (Fig. 7). It should be noted that when xFLx

¼ 0 or xFLx

¼ 1 pileposition is the toe or the crest of the slope, respectively. Moreover, values of xF

Lx

greater than unity indicate that the piles are located at the top of the slope. The

stabilising force is expressed in a dimensionless form as K ¼ F1=2 �H 2. Coefficient m is

Fig. 7. Illustrative example of a slope reinforced with piles.

Fig. 8. Effect of pile location on dimensionless force K, when m=0.


assumed to be 0, 1/3 or 1/2. The results are given in Figs. 8–10 for three differentvalues of the improvement ratio � which is defined as

� ¼ FS

FS0ð20Þ

where FS=safety factor of the piled slope problem, and FS0=safety factor of theslope without piles.As can be expected, K increases with increasing �. The increase in K is greater

when m=1/2, although the results appear not to be greatly affected by the value ofm. In all the cases examined, the optimal location of the piles is near the toe of theslope, where the force provided by the piles to achieve the selected value of theimprovement ratio takes the lowest value. This is due to the shape of the slidingsurface which is a log-spiral curve having a radius that increases as the surfacedevelops from the top to the base of the slope. For a rotational failure mechanism asshown in Fig. 5, the required stabilising moment due to F, with respect to the rota-tion centre, has an arm that increases as the location of the piles approaches to theslope toe, and consequently force F decreases. However, Figs. 8–10 show that thepiles are also very effective when they are located between the middle and the toe ofthe slope, especially when m is assumed to be zero. The figures also show that theregion where the piles are more effective reduces as � increases, and is located closerto the toe of the slope. Therefore, when the achievement of a high improvementratio value is required, the piles should be located with greater care within the slope.

Fig. 9. Effect of pile location on dimensionless force K, when m=1/3.


Fig. 10. Effect of pile location on dimensionless force K, when m=1/2.

Fig. 11. Safety factor for the slope without piles against lc’.


In the following, the results of a parametric study are also presented in order toillustrate the effect of several factors on the stabilising force that the piles have toprovide to increase slope stability. The calculations have been carried out with threevalues of the slope angle, �=30, 45 and 60�. The results are given for different valuesof � which are chosen to adequately improve slope stability. In all the calculations, itis assumed that m is equal to 1/3.The slope safety factor without piles, FS0, can be determined from the results

presented in Fig. 11 which have been obtained using Eq. (13). As can be noted, thevalues of FS0 decrease when increasing the slope angle and increasing the parameterlc’ that has been defined in a previous section. It should be noted that the value oflc’ also indicates the position of the potential sliding surface within the slope. Aspointed out by Duncan and Wright [37], when lc’ is small the sliding surfacebecomes deeper and expands into the soil, especially when the slope is gentle. Bycontrast, as the value of lc’ increases, the critical sliding surface becomes increas-ingly shallow. Duncan and Wright [37] considered sliding surfaces of circular shape.However, this also occurs when log-spiral sliding surfaces are considered, as shownin Fig. 12.

Fig. 12. Critical sliding surfaces of a slope with �=30� for different values of lc’.

Fig. 13. Force K against lc’ for a slope with �=30�.


The piles are assumed to be located at the middle of the slope which should be asuitable location for the piles. Following Poulos [8], a retaining structure which islocated near the toe or the crest of the slope could restrain only a small mass of thesoil, while a lot of the soil mass behind or in front of the structure could be unstable.Moreover, this location is consistent with the assumption that the piles are laterallyloaded. At the crest of the slope, the axial response could be more important thanthe lateral one, because of soil movement is here predominantly down the piles andnot laterally across them [30].




The results are presented in Figs. 13–15 in terms of K against Janbu’s dimension-less parameter lc’. These figures show that the stabilising force provided by the pilesare affected by both � and lc’. As previously observed, K strongly increases as theimprovement ratio increases. On the contrary, the stabilising force decreases withJanbu’s parameter, and the lowest values of K occur when lc’ is high. This impliesthat the use of piles is a very effective measure to increase slope stability especiallywhen the sliding surface for the slope without piles is shallow. In this case, in fact,the piles have to provide a smaller force to achieve the desired value of theimprovement ratio.

6. Conclusions

A kinematic approach has been described for the stability analysis of slopes rein-forced with piles. The first step of the approach consists in finding the critical slidingsurface and the safety factor for the slope without piles. To this purpose, a proce-dure has been developed in which the solution is expressed in terms of the safetyfactor that is defined as a reduction coefficient for the shearing resistance parametersof the soil. The results obtained using the proposed procedure are found to be ingood agreement with those derived from both Bishop’s method and more complexupper and lower bound solutions of limit analysis.For slopes containing piles, analytical expressions have been derived that allow

the force needed to increase the safety factor to a desired value, and the most sui-table location of the piles within the slope to be evaluated. These expressions may befound useful for designing piles to reinforce slopes. The calculations carried outusing the expressions obtained show that installing a row a piles is an effectiveremedy to improve slope stability especially when the sliding surface for the unrein-forced slope is relatively shallow. The results also indicate that the optimal locationof the piles within the slope is near the toe of the slope where the stabilising forceneeded to increase the safety factor to the desired value takes a minimum value. Pilesappear also to be very effective when they are installed in the region from the middleto the toe of the slope. However, this region reduces when the achievement of highsafety factor values is required.

Appendix

f1 ¼3tg’�cos�h þ sin�hð Þ exp 3 �h � �0ð Þ tg’�½ � � 3 tg’� cos�0 � sin�0

� �3 1þ 9 tg2’�ð Þ

f2 ¼ 1

6

L

ro2cos�0 � L

rocos�

� �sin �0 þ �ð Þ


f3 ¼ 1

6exp �h � �0ð Þtg’�½ � sin �h � �0ð Þ � L

r0sin �h þ�ð Þ

� �

� cos�0 � L

r0cos�þ cos�hexp �h � �0ð Þtg’�½ �

�

f4 ¼ H2

r20

sin �� 0ð Þ2sin�sin�0 cos�0 � L

r0cos�� 1

3

H

r0cotg�0 þ cotg�ð Þ

� �

In these expressions, the quantities Hr0and L

r0are given by Eqs. (11a) and (11b),

respectively, and tg’� ¼ tg’FS .

References

[1] De Beer E, Wallays M. Stabilization of a slope in schist by means of bored piles reinforced with steel

beams. In: Proc. 2th International Congress on Rock Mechanics, Beograd, 1970. p. 361–9.

[2] Esu F, D’Elia B. Interazione terreno-struttura in un palo sollecitato da una frana tipo colata. Rivista

Italiana di Geotecnica 1974;8(1):27–38.

[3] Ito T, Matsui T. Methods to estimate lateral force acting on stabilizing piles. Soils and Foundations

1975;15(4):43–59.

[4] Fukuoka M. The effects of horizontal loads on piles due to landslides. In: Proc. 9th International

Conference on Soil Mechanics and Foundation Engineering, Tokyo, 1977. p. 27–42.

[5] Sommer H. Creeping slope in a stiff clay. In: Proc. 9th International Conference on Soil Mechanics

and Foundation Engineering, Tokyo, 1977. p. 113–8.

[6] Gudehus G, Schwarz W. Stabilisation of creeping slopes by dowels. In: Proc. 11th International

Conference on Soil Mechanics and Foundation Engineering, San Francisco, 1985. p. 1697–700.

[7] Carruba P, Maugeri M, Motta E. Esperienze in vera grandezza sul comportamento di pali per la

stabilizzazione di un pendio. In: XVII Convegno Nazionale di Geotecnica, Taormina, 1989. p. 81–90.

[8] Poulos HG. Design of reinforcing piles to increase slope stability. Canadian Geotechnical Journal

1995;32(5):808–18.

[9] Hong WP, Han JG. The behavior of stabilizing piles installed in slopes. In: Proc. 7th International

Symposium on Landslides, Rotterdam, 1996. p. 1709–14.

[10] Cai F, Ugai K. Numerical analysis of the stability of a slope reinforced with piles. Soils and Foun-

dations 2000;40(1):73–84.

[11] Yu HS, Salgado R, Sloan SW, Kim JM. Limit analysis versus limit equilibrium for slope stability.

Journal of Geotechnical and Geoenvironmental Engineering, ASCE 1998;124(1):1–11.

[12] Ito T, Matsui T, Hong WP. Design method for the stability analysis of the slope with landing pier.

Soils and Foundations 1979;19(4):43–57.

[13] Lee CY, Hull TS, Poulos HG. Simplified pile-slope stability analysis. Computers and Geotechnics

1995;17:1–16.

[14] Bishop AW. The use of slip circle in the stability analysis of earth slopes. Geotechnique 1955;5(1):7–17.

[15] Poulos HG. Analysis of piles in soil undergoing lateral movement. Journal of Soil Mechanics and

Foundation Division, ASCE 1973;99(SM5):391–406.

[16] Lee CY, Poulos HG, Hull TS. Effect of seafloor instability on offshore pile foundations. Canadian

Geotechnical Journal 1991;28(5):729–37.

[17] Hassiotis S, Chameau JL, Gunaratne M. Design method for stabilization of slopes with piles. Jour-

nal of Geotechnical and Geoenvironmental Engineering, ASCE 1997;123(4):314–23.

[18] Chugh AK. Procedure for design of restraining structures for slope stabilization problems. Geo-

technical Engineering 1982;13:223–34.


[19] Lysmer J. Limit analysis of plane problems in soil mechanics. Journal of Soil Mechanics and Foun-

dation Division, ASCE 1970;96(SM4):1131–334.

[20] Finn W. Application of limit plasticity in soil mechanics. Journal of Soil Mechanics and Foundation

Division, ASCE 1967;89(SM5):101–19.

[21] Chen WF. Limit analysis and soil plasticity. Amsterdam (The Netherlands): Elsevier Science, 1975.

[22] Karal K. Energy method for soil stability analyses. Journal of the Geotechnical Engineering Divi-

sion, ASCE 1977;103(GT5):431–45.

[23] Michalowski RL. Three dimensional analysis of locally loaded slopes. Geotechnique 1989;39(1):27–

38.

[24] Michalowski RL. Slope stability analysis: a kinematical approach. Geotechnique 1995;45(2):283–93.

[25] Donald IB, Chen Z. Slope stability analysis by the upper bound approach: fundamentals and meth-

ods. Canadian Geotechnical Journal 1997;34:853–62.

[26] Taylor DW. Fundamentals of soil mechanics. New York: John Wiley and Sons, 1948.

[27] Cao J, Zaman MM. Analytical method for analysis of slope stability. International Journal for

Numerical and Analytical Methods in Geomechanics 1999;23:439–49.

[28] Huang SL, Yamasaki K. Slope failure analysis using local minimum factor-of-safety approach.

Journal of Geotechnical Engineering, ASCE 1993;119(12):1974–87.

[29] Janbu N. Stability analysis of slopes with dimensionless parameters. Harvard Soil Mechanics Series

no. 46, 1954.

[30] Hull TS, Poulos HG. Design method for stabilization of slopes with piles (discussion). Journal of

Geotechnical and Geoenvironmental Engineering, ASCE 1999;125(10):911–3.

[31] De Beer E. Piles subjected to static lateral loads. In: Proc. 9th International Conference on Soil

Mechanics and Foundation Engineering, Tokyo, 1977. p. 1–14.

[32] Viggiani C. Ultimate lateral load on piles used to stabilise landslides. In: Proc. 10th International

Conference on Soil Mechanics and Foundation Engineering, Stockholm, 1981. p. 555–60.

[33] Ito T, Matsui T, Hong WP. Design method for stabilizing piles against landslide — one row of piles.


[34] Ito T, Matsui T, Hong WP. Extended design method for multi-row stabilizing piles against landslide.


[35] Maugeri M, Motta E. Stresses on piles used to stabilize landslides. In: Proc. 6th International Sym-

posium on Landslides, Christchurch, 1992. p. 785–90.

[36] Chen LT, Poulos HG. Piles subjected to lateral soil movements. Journal of Geotechnical and

Geoenvironmental Engineering, ASCE 1997;123(9):802–11.

[37] Duncan JM, Wrigth SG. The accuracy of equilibrium methods of slope stability analysis. Engineer-

ing Geology 1980;16:5–17.


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