Computers and Geotechnics 59 (2014) 67–74

Contents lists available at ScienceDirect

Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Limit equilibrium based design approach for slope stabilization usingmultiple rows of drilled shafts

http://dx.doi.org/10.1016/j.compgeo.2014.03.0010266-352X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 330 972 7190; fax: +1 330 972 6020.E-mail addresses: rearlin@gmail.com (L. Li), rliang@uakron.edu (R.Y. Liang).

Lin Li, Robert Y. Liang ⇑Department of Civil Engineering, University of Akron, Akron, OH 44325-3905, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 June 2013Received in revised form 24 February 2014Accepted 1 March 2014

Keywords:LandslideDesignSlope stabilityDrilled shaftsMultiple rows

In this paper, a limiting equilibrium based methodology, incorporating the method of slices and archingeffects of the drilled shafts, is developed for optimizing the use of multiple rows of drilled shafts. Thisproposed method is focusing on the number of rows, the location of each row, the dimension and spacingof the drilled shafts. Three design criteria are used for optimization: target global factor of safety, the con-structability and service limit. A PC-based program called M-UASLOPE has been coded to allow for han-dling of complex slope geometry, soil profile, and ground water conditions. A design example is presentedto illustrate the application of the M-UASLOPE program in the optimized design of multiple rows ofdrilled shafts for stabilizing the example slope.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The practice of using drilled shafts to stabilize an unstable slopehas been widely and successfully adopted [1–12]. However, mostof the analysis and design methods are based on the limiting equi-librium approach, which considers the effects of drilled shafts to bean increase in resistance. The estimation of resistance from drilledshafts has been based on ultimate soil reaction pressure or dis-placement-based finite element analysis or an LPILE [13] type ofcomputation to determine the displacement-dependent soil pres-sures against the drilled shafts. The alternative approach, withinthe concept of limiting equilibrium method of slices, has been pro-posed and developed by several researchers [14–28], in which theeffects of the drilled shafts are considered as the reduction in thedriving forces due to the soil arching between the adjacent drilledshafts. The estimate of the reduction in the driving force is basedon empirical load transfer factor equation. The equation wasderived from regression analysis of more than 40 cases of three-dimensional (3-D) finite element simulation results. This methodhas been coded into a PC based computer program (M-UASLOPE)for design applications.

When looking at the various proposed methods for slope stabil-ization using drilled shafts, it was found that all of them wereconcerned with the design of a single row of drilled shafts. The

method proposed by Liang [17] allows for optimization of the size(diameter and length) of the drilled shafts, the spacing between theadjacent drilled shafts, and the location of the row of drilled shaftson the slope. However, in reality, the use of a single row of drilledshafts may not be feasible due to the dimensions of the failed slopeand the large earth thrust applied to the drilled shafts, whichwould render the structural design of the drilled shafts nearlyimpossible. In these difficult conditions, the use of multiple rowsof drilled shafts could be a feasible solution to both ensure thatthe global factor of safety of the stabilized slope meets the targetfactor of safety and that the size of drilled shafts and the amountof reinforcement used in the drilled shafts is constructible and eco-nomical. Specifically, multiple rows of drilled shafts are needed toarrest slope movement and enhance the safety margin of the fail-ing slope for the following two scenarios: (1) one row of drilledshafts cannot satisfy the target factor of safety (FSTarget), regardlessof the dimension of the drilled shafts and location of the drilledshafts; and (2) although the use of single row of drilled shaftscan increase the global factor of safety of the slope to the targetvalue, the net force applied to the drilled shafts is excessively large,which either precludes the design of a constructible reinforcementor produces a deflection of the shaft that may be too large to meetthe service limit requirements.

In this study, an analysis and design approach for using multiplerows of drilled shafts to stabilize an unstable slope is presented.The method is based on the limit equilibrium method of slices,incorporating the concept of soil arching in determining the

68 L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74

reduction of the driving force in the stability analysis. The formu-lation of this method, together with the empirical equations fordetermining the arching-induced reduction in the driving forcesis described. Optimization of the number of the rows, the dimen-sions of the drilled shafts, and the locations of each row of thedrilled shafts to achieve safety and service limit requirements aredescribed. An example is presented to demonstrate the use of theproposed method to optimally design multiple rows of drilledshafts for the purpose of slope stabilization.

Fig. 1. A typical slice showing all force components.

2. Multi-rows of drilled shafts for stabilizing a slope

Ideally, one would like to use only a single row of drilled shaftsto stabilize an unstable slope. However, conditions exist that mayrequire use of more than one row of drilled shafts. Usually, thereare two major circumstances where multiple rows of drilled shaftsare necessary: (1) cases where using one row of drilled shafts can-not satisfy the target FS regardless of the chosen dimension of thedrilled shafts, the spacing of drilled shafts, and location of the rowof drilled shafts, and (2) although the global factor of safety can besatisfied with the use of a single row of drilled shafts, the forcesapplied to the drilled shaft are excessively large such that eitherthe reinforcement cannot be practically provided without causingconstructability issues or the drilled shaft deflection is too exces-sive to meet the service limit requirements. Therefore, to addresssituations like these, there is a need to use multiple rows of drilledshafts.

2.1. Method of slices for two-row of drilled shaft/slope system

The global factor of safety of two or more rows in a drilled shaft/slope system can be calculated based on the limit equilibriummethod of slices. The method of slices for two rows of drilled shaftsincorporating the arching-induced reduction in driving force onthe downslope side of drilled shafts is formulated herein. The basicassumptions made in the calculation of the method of slices areenumerated below:

(1) FS was considered to be identical for all slices.(2) Normal force on the base of the slice was applied at the mid-

point of the slice base.(3) The location of the thrust line of the interslice forces was

placed at one-third of the average interslice height abovethe failure surface, as in the study by Janbu [29].

(4) The inclinations of the interslice forces were assumed as fol-

lows: the right-interslice force PRi

� �was assumed to be par-

allel to the inclination of the preceding slice base (i.e., ai�1),

and the left-interslice force PLi

� �was assumed to be parallel

to the current slice base (i.e., ai).(5) There is no group pile effect between the two adjacent rows

of drilled shafts. The group pile effect will be produced if thetwo rows or more rows of drilled shafts are too close, whichwill affect the p-y curve analysis. In this paper, group pileeffect is assumed non-existent, and the minimum spacingof two adjacent rows of drilled shafts is chosen as 5 m.

(6) Soil arching for each row of drilled shafts is considered asindependent from each other. The soil arching has beenexpressed as load transfer factor in this paper. In practicalengineering, when the two adjacent rows of drilled shaftsare quite close, the soil movement induced by the deflectionof up-slope row of drilled shafts may affect the down-sloperow of drilled shafts. Therefore, the load transfer factor ofdown-slope row of drilled shafts may be affected. In thispaper, however, the limit equilibrium method of slices has

been employed, which means no soil movement will be pro-duced. Therefore, soil arching for each row of drilled shaftscan be considered as independent from each other.

Referring to Figs. 1 and 2 and applying the force equilibrium forany slice i of the slope, the force summations in two directions, thenormal and tangential directions to the base of the slice, result inthe following two equations:

Ni �wi cos ai � PRi sinðai�1 � aiÞ ¼ 0 ð1Þ

Ti þ PLi �wi sinai � PR

i cosðai�1 � aiÞ ¼ 0 ð2Þ

Using Mohr–Coulomb strength equation of the soil to the baseof the slice, the following relationship is obtained.

Ti ¼cili

FSþ ½Ni � uili�

tan ui

FSð3Þ

Substituting Eqs. (1) and (2) into Eq. (3), one obtains the follow-ing equations:

Ti ¼cili

FSþ wi cos ai þ PR

i sinðai�1 � aiÞ � uili

h i tan ui

FSð4Þ

PLi ¼ wi sin ai �

ciliFSþ ðwi cos ai � uiliÞ

tan ui

FS

� �þ kiP

Ri ð5Þ

ki ¼ cosðai�1 � aiÞ � sinðai�1 � aiÞtan ui

FSð6Þ

where wi is weight of slice i, Ni is force normal to the base of slice i, Ti

is force parallel to the base of slice i, PLi is interslice force acting on the

left side of slice, PRi is interslice force acting on the right side of slice, ai

is inclination of slice i base, ai�1 is inclination of slice i � 1 base, ui isthe pore pressure at the base of slice i, ci is soil cohesion at the base ofslice i, ui is soil friction angle at the base of slice i. It is noted that c andu are in terms of effective stress throughout this paper.

After the drilled shafts are inserted into the slope, the intersliceforce on the downslope side of the drilled shaft will be reduced dueto soil arching, i.e., it will be reduced by a multiplier called the loadtransfer factor (g) from the previous interslice force PR

i . Referring toFigs. 1–3 and applying the force equilibrium while invoking Mohr–Coulomb’s strength criterion for any slice i of the slope, the inter-slice force shown in Fig. 3 can be written as the followingexpressions:

PLi ¼ wi sin ai �

ciliFSþ ðwi cos ai � uiliÞ

tan ui

FS

� �þ kig1PL

i�1 ð7Þ

Non-yielding base

1

i-1i

j-1j

Phreatic line

Slip surface

Slice

n

Drilled shafts row #1

Drilled shafts row #2

Fig. 2. A typical cross-section divided into slices for a slope reinforced with tworows of drilled shafts.

L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74 69

PLj ¼ wj sinaj �

cjlj

FSþ ðwj cos aj � ujljÞ

tan uj

FS

� �þ kjg2PL

j�1 ð8Þ

ki ¼ cosðai�1 � aiÞ � sinðai�1 � aiÞtan ui

FSð9Þ

kj ¼ cosðaj�1 � aiÞ � sinðaj�1 � aiÞtan uj

FSð10Þ

where i < j; PLi and PL

j are the interslice forces acting on the down-slope side of slices i and j; wi and wj are weight of slices i and j;ai and aj are inclinations of the bases of slices i and j; ci and cj denotethe soil cohesion at the bases of slices i and j; ui and uj are the soilfriction angles at the bases of slices i and j; ui and uj are the porewater pressures at slices i and j; and g1 and g2 are the load transferfactors for row 1 and row 2.

The net force applied to the two rows of drilled shafts, which isdue to the difference in the interslice forces on the upslope anddownslope sides of the drilled shaft, can be calculated as follows:

Fshaft-row1 ¼ ð1� g1ÞPLi�1S01 ð11Þ

Fshaft-row2 ¼ ð1� g2ÞPLj�1S02 ð12Þ

where S01 and S02 are the center-to-center spacing between twoadjacent drilled shafts of the two rows (Fig. 2), PL

i�1 and PLj�1 are

Fig. 3. Method of slices using t

the interslice force acting on the upslope side of slice i � 1 andj � 1 (Fig. 3) and Fshaft-row1 and Fshaft-row2 are the net forces on thedrilled shaft in row 1 and row 2, respectively. Eqs. (5), (7), and (8)relate the left-interslice force PL

i to the right-interslice force PRi for

slice i. Based on Eqs. (5)–(10), the global factor of safety for tworows in a drilled shaft/slope system can be calculated in an iterativecomputational algorithm to satisfy boundary load conditions andequilibrium requirements, along with Mohr–Coulomb strength cri-terion. More detailed explanation for this iteration methodologycan be found in [19]. For multiple rows of drilled shafts, the calcu-lation principles of global factor of safety and the net force for eachrow of shafts are the same as in the case of two rows of drilledshafts. A PC-based computer program, M-UASLOPE, has been devel-oped based on the above computational algorithm to calculate theglobal factor of safety for the multiple rows of a drilled shaft/slopesystem and the net force imparted on each drilled shaft for each rowof drilled shafts.

In Eqs. (7)–(12), the load transfer factor g is defined as the ratiobetween the horizontal force on the downslope side of the verticalplane at the interface between the drilled shaft and soil (i.e.,Pdown-slope) to the horizontal force on the upslope side of the verticalplane at the interface between the drilled shaft and soil (i.e.,Pup-slope). For the expression of the load transfer factor, Joorabchi[19] proposed a semi-empirical equation, given in Eq. (13), to com-pute the load transfer factor using the regression analysis tech-niques on more than 40 cases of 3-D finite element modelsimulation results conducted by Al Bodour [18]. Considering thata sufficient length of drilled shaft is socketed into the rock layer,the other important influencing factors on the load transfer factorconsist of the following six parameters: soil cohesion C, friction an-gle u, drilled shafts diameter D, center to center shaft spacing S0,shaft location on slope nx, and slope angle b.

g ¼ �0:272C0:153ðtan bÞ�0:429 �1:17þ 1:114S0

D

� �ðeð�0:578 tan /ÞÞ

� ð0:065þ 0:876DÞ �0:252þ 0:61nx� 0:57 n2x

� � ð13Þ

where nx = (xi � xtoe)/(xcrest � xtoe); xi = the x-coordinate of the loca-tion of drilled shaft; xtoe = the x-coordinate of the location of thetoe of the slope; and xcrest = the x-coordinate of the location of thecrest of the slope. The method for computing the forces from thefinite element simulation results and the techniques for modeling

wo rows of drilled shafts.

70 L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74

3-D effects of drilled shafts on a slope are presented in both AlBodour [18] and Joorabchi [19].

Fig. 4b. Validations of M-UASLOPE program, comparison of the net force on shaft(one row of drilled shafts).

2.2. Optimization strategy and criteria

The following principles are used for the optimization of multi-ple drilled shafts on a slope:

(1) Global FS should be equal to or bigger than FSTarget. Forexample, FSTarget = 1.5 is used in the example given below.

(2) The lateral deflection of each drilled shaft should not exceedthe prescribed service limit, say, 12.7 mm (0.5 in.) on the topof the shaft.

(3) The feasible locations of drilled shafts should be selected tofollow site accessibility and local availability of appropriateconstruction equipment.

After satisfying the above principles, further steps in the optimi-zation strategy are taken to ensure constructability for practicalreinforcement placements (such as rebar spacing and cover thick-ness) and to determine the drilled shaft with the minimum vol-ume, as suggested below:

(4) To minimize the cost of reinforcement, the net force ofdrilled shafts should be as low as possible.

(5) To minimize the total cost of concrete, the total volume ofdrilled shafts for each combination (shaft diameter (D), shaftclear spacing (S), and shaft length (L)) should be calculatedand compared to find which shaft has the smallest volume.

2.3. FEM validation for one row of drilled shaft stabilizing slope

The latest version of UASLOPE, M-UASLOPE program, was codedon the basis of the developed limiting equilibrium based method ofslices together with the load transfer factor in Eq. (13) [17,19]. Thefactor of safety and the net force computed by the finite elementsimulations and the M-UASLOPE program are presented inFigs. 4a and 4b for a total of 41 different cases. Fig. 4a shows a com-parison between the FS obtained from M-UASLOPE and the FSobtained from FEM simulations, with the correlation coefficientR2 = 0.77. Fig. 4b shows a comparison between the net forceobtained from M-UASLOPE and the net force obtained from FEMsimulation, with the correlation coefficient R2 = 0.84.

The differences between the factors of safety obtained from thetwo methods can be attributed to the following reasons:

Fig. 4a. Validations of M-UASLOPE program, comparison of FS (one row of drilledshafts).

(1) Although the load transfer factor was obtained from 3-Dfinite element simulation results, the M-UASLOPE analysiswas based on 2-D limit equilibrium analysis.

(2) The FEM analysis is based on continuum mechanics thattreats the soils in the slope as a deformable body, whilethe M-UASLOPE is based on force equilibrium principle thattreats the soil in the moving part of the soil to be a rigidbody.

In addition to the comparisons between the M-UASLOPE com-putational results and FEM simulation results for 41 cases, a com-parison was made with a full-scale field study at the Ohio ATH-124Project site. Details of the project can be referred to Liang et al.[17,21], in which computational results of M-UASLOPE and ABA-QUS are compared with field test results at different stages of slopemovement. A good result, shown in Table 1, for the computed FSand the resultant forces based on the two methods of M-UASLOPEand ABAQUS FEM analysis was obtained for the ATH-124 Project.

3. Illustrative design example

An example slope is used herein to demonstrate the designapproach for using multiple rows of drilled shafts in stabilizingthe slope to the target global factor of safety. During the designprocedure, one row of drilled shafts was analyzed first to demon-strate that it is not sufficient to stabilize the slope. Next, two rowsof drilled shafts for stabilizing slope were considered in the design.Finally, if the need is demonstrated, more rows of drilled shafts areconsidered.

The slope shown in Fig. 5 consists of six soil layers, and the soilproperties for each layer are summarized in Table 2. The criticalslip surface for the slope was determined by a conventional slopestability analysis program, STABLE, with the computed FS equalto 1.00. The identified critical slip surface is represented by con-necting seven points. It is common that in most design problemsinvolving the reinforcement of an unstable slope, the slip surfaceof the failed slope is typically identified during field monitoring

Table 1M-UASLOPE analysis versus FEM analysis results (one row of drilled shafts).

Method Resultant shaft force (kN) Global factor of safety

FEM 132.5 1.3M-UASLOPE 121.5 1.2

Shaft Row #1

Shaft Row #2

Shaft Row #3

Fig. 5. Geometry of illustrative example (unit: meter).

Table 2Soil properties of illustrative example.

Layer no. Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

C (kN/m2) 25.0 30.0 45.0 35.0 80.0 120.0u (�) 8.0 9.0 14.0 3.5 24.0 30.0c (kN/m3) 16.1 16.2 17.2 18.0 21.0 22.5

1.1

1.2

1.3

1.4

1.5

1.6

30 40 50 60 70 80 90

Glo

bal F

S

Shaft Location / m

D=0.6, S/D=1.0

D=0.6, S/D=2.0

D=0.6, S/D=3.0

D=1.2, S/D=1.0

D=1.2, S/D=2.0

D=1.2, S/D=3.0

D=1.8, S/D=1.0

D=1.8, S/D=2.0

D=1.8, S/D=3.0

D=2.4, S/D=1.0

D=2.4, S/D=2.0

D=2.4, S/D=3.0

Target FS

Fig. 6. Global FS versus shaft location for different (S, D) combinations using onerow of drilled shafts.

L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74 71

through the use an inclinometer. Therefore, in the present study,the slip surface determined by the slope stability analysis is treatedas a pre-existing failure surface and is used in the subsequentdesign of drilled shafts for stabilization of this particular slope. Arock layer (i.e. layer 6) has been provided in this example. Thegroundwater table is assumed at an elevation of�33.0 m, as shownin Fig. 5. It is noted that effective stress approach is used in the sta-bility analysis.

3.1. Step by step design procedure

Step 1: Collect data concerning the geometry of the slope, soilprofile, soil parameters, groundwater table conditions, locationof the slip surface, etc. For this example, the relevant informa-tion is presented in Fig. 5 and listed in Table 2. It is noted thatthe slip surface identified for the original slope will be usedfor the subsequent calculation and design, as this surface is con-sidered to be the weakest surface for subsequent slope move-ment to occur.Step 2: Choose a target factor of safety (FSTarget) for the drilledshaft/slope system equal to 1.5.Step 3: Select feasible locations for drilled shafts, which maydepend on the site situation and the local availability of con-struction equipment suitable for drilling shafts. The feasiblelocations for drilled shafts are between 30.0 m and 90.0 m hor-izontally from the left side of the slope (as shown in Fig. 5). Inthe current design, we analyze the location starting atX = 30 m and ending at X = 90 m, in increments of 5.0 m.Step 4: Perform analysis and design using one row of drilledshafts, considering different combinations of clear spacing Sand shaft diameter D within the permissible range. In thisexample, the range for D is selected to be between 0.6 m and2.4 m (2–8 ft), and the range of S/D is selected to be between1.0 and 3.0. The following combinations for (S, D) are selectedfor computation: (0.6, 0.6), (1.2, 0.6), (1.8, 0.6), (1.2, 1.2), (2.4,1.2), (3.6, 1.2), (1.8, 1.8), (3.6, 1.8), (5.4, 1.8), (2.4, 2.4), (4.8,2.4), and (7.2, 2.4), where the unit is in meters.Step 5: Calculate global FS and shaft net force using thedescribed method and the computer program M-UASLOPE. For

each (S, D) combination, it is necessary to plot the relationshipbetween the computed global FS and different shaft locations,as well as to plot the relationship between the net force onthe drilled shaft and the different shaft locations, as shown inFigs. 6 and 7, respectively.

As can be seen from Fig. 6, the computed global FS for all com-binations and shaft locations considered do not satisfy the target FSof 1.5. The highest FS is 1.483, corresponding to the drilled shaftlocated at 80 m and the (S, D) combination of (0.6 m, 0.6 m). Mean-while, the corresponding shaft net forces for this combination areextremely high at the location of 80 m (i.e., 838.58 kN), as shownin Fig. 7. Therefore, we need to consider the use of two rows ofdrilled shafts.

Step 6: Perform analysis and design using two rows of drilledshafts, considering different combinations of clear spacing S,shaft diameter D, and various drilled shaft locations. For the(S, D) combinations and the drilled shaft locations, we choosethe same range as was discussed in Steps 3 and 4.

At the beginning of Step 6, the location of the first row of drilledshafts is placed on the slope crest (i.e. X = 30 m shown in Fig. 5);next, the location of the second row of drilled shafts is changedfrom the slope crest (i.e. X = 30 m) to the slope toe (i.e. X = 90 m).Thereafter, the first row is placed 5.0 m further to the downslopedirection. We repeat the above procedure until the location ofthe both rows reach to the slope toe (i.e. X = 90 m).

0

100

200

300

400

500

600

700

800

900

30 40 50 60 70 80 90

Shaf

t F

orce

/ kN

Shaft Location / m

D=0.6, S/D=1.0D=0.6, S/D=2.0D=0.6, S/D=3.0D=1.2, S/D=1.0D=1.2, S/D=2.0D=1.2, S/D=3.0D=1.8, S/D=1.0D=1.8, S/D=2.0D=1.8, S/D=3.0D=2.4, S/D=1.0D=2.4, S/D=2.0D=2.4, S/D=3.0

Fig. 7. Shaft force versus shaft location for different (S, D) combinations using onerow of drilled shafts.

TabDes

N

1234

Not

TabDes

N

123

Not

72 L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74

Step 7: Calculate the global FS and the shaft net force by usingthe computer program M-UASLOPE. Since there are many possi-ble combinations and it is hard to display all of the data in onefigure, only those combinations that provide a global FS largerthan FSTarget of 1.5 and a shaft net force less than 600 kN are pre-sented in Table 3. As can be seen in this table, only five feasiblecombinations are obtained for the case of using two rows ofdrilled shafts.Step 8: Use the p–y method based software, LPILE v5.0 [13], todetermine the internal forces and moment in the drilled shaftand the shaft deflection under the net shaft force given inTable 3. The calculation results of analysis using LPILE indicatethat the lateral deflections on the top of the shaft are very high,such that the program cannot even converge if the shaft netforce exceeds 300 kN. Although the use of two rows of drilledshafts can ensure that the global FS will be greater than the tar-get FS, the LPILE analysis shows excessive deflections that donot meet the service limit requirements. Consequently, morerows of drilled shafts are considered for the subsequent analysisand design.Step 9: Perform analysis and design using more rows of drilledshafts, considering the same range of shaft location, shaft clear

le 3ign results using 2 rows of drilled shafts.

o. FS Row #1

Force (kN) Location (m) D (m)

1.53 314.01 45 0.61.57 398.74 50 0.61.54 547.86 55 0.61.52 596.05 60 0.6

e: D is the diameter of drilled shaft; S is the clear spacing between adjacent drilled s

le 4ign results using 4 rows of drilled shafts.

o. FS Row #1

Force (kN) Location (m) D (m)

1.51 110.79 35 0.61.52 205.42 40 0.61.50 227.15 40 0.6

Row #3196.32 60 0.6182.58 60 0.6191.39 60 0.6

e: D is the diameter of drilled shaft; S is the clear spacing between adjacent drilled s

spacing S, and shaft diameter D as in Step 6. In the current step,we follow the same logic in trying different combinations ofdrilled shaft locations as described before.Step 10: Calculate the global FS and shaft net force for eachdrilled shaft by using M-UASLOPE, using the same methodologydiscussed in Step 7 and outputting only the combinations thatprovide the global FS larger than FSTarget of 1.5. In the meantime,only combinations showing net forces on drilled shaft that areless than 250 kN are considered. Table 4 shows the designresults using four rows of drilled shafts. Three combinationscan satisfy the requirements on the FSTarget, and all the com-puted shaft net forces are less than 250 kN.Step 11: Optimize design results by using the strategy of mini-mum concrete volume. As shown in Table 4, all combinations(No. 1, No. 2 and No. 3) yield a similar shaft net force; however,the clear spacing for row 1 of Combination No. 3 (i.e., S = 1.2 m)is larger than that for Combination No. 1 and No. 2 (i.e.,S = 0.6 m), meaning that less concrete material is needed inCombination No. 3. Further structural analysis and total volumeof drilled shafts are analyzed in Step 12.Step 12: Soil–structure interaction analysis.

The computer program LPILE [13] was used for the structuralanalysis of the drilled shaft. Assuming that at least 20% of thelength of drilled shaft is embedded into the rock layer (layer 6)below the slip surface, the shaft parameters of the three combina-tions obtained from Step 11 with the corresponding net forces foreach drilled shaft can be used as inputs in LPILE to calculate the lat-eral deflection and the internal forces and moments on the drilledshaft. In the LPILE analysis, c–u soil is used to model the soil andReese’s method for modeling p–y curves in weak rock is used torepresent the rock. The boundary conditions at the top of thedrilled shaft are zero shear and moment. The net force is distrib-uted as a triangularly distributed shear force acting on the drilledshaft above the slip surface. The total net force is taken fromTable 4. After calculation with LPILE, the lateral deflections onthe top of the shaft for each row are shown in Table 5, and the totalshaft lengths of the three combinations are summarized in Table 5as well. All the lateral deflections on the top of the shaft can beconsidered to be within the allowable deflection (say, half an inch,or about 12.7 mm). If we perform the quantity analysis of the vol-

Row #2

S (m) Force (kN) Location (m) D (m) S (m)

0.6 581.89 75 0.6 0.60.6 532.99 75 0.6 0.61.2 548.40 85 0.6 0.61.8 585.58 75 0.6 0.6

hafts.

Row #2

S (m) Force (kN) Location (m) D (m) S (m)

0.6 245.56 50 0.6 0.60.6 169.07 50 0.6 0.61.2 218.68 50 0.6 0.6

Row #40.6 208.32 70 0.6 0.60.6 206.11 70 0.6 0.60.6 207.46 70 0.6 0.6

hafts.

Table 5Design results of the two combinations shown in Table 4.

Combination Row #1 Row #2 Row #3 Row #4

d (m) L (m) d (m) L (m) d (m) L (m) d (m) L (m)

No. 1 0.0121 54 0.0125 44 0.0114 34 0.0110 31No. 2 0.0122 47 0.0112 44 0.0111 34 0.0108 31No. 3 0.0124 47 0.0118 44 0.0113 34 0.0109 31

Note: d is the deflection on the top of drilled shaft; L is the shaft length.

Table 6Total volume per unit width for the four combinations.

Combination Row #1 (m3) Row #2 (m3) Row #3 (m3) Row #4 (m3) Sum (m3)

No. 1 12.72 10.36 8.00 7.30 38.38No. 2 11.07 10.36 8.00 7.30 36.73No. 3 7.38 10.36 8.00 7.30 33.04

L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74 73

ume of the required drilled shafts, the drilled shaft volume per unitwidth of a slope for the three combinations can be determined (theresults are shown in Table 6). Clearly, Combination No. 3 needs lesstotal concrete volume than the others. Thus, Combination No. 3 canbe chosen as the most economical design with the least amount ofdrilled shaft volume while meeting the target global factor ofsafety.

It is noted that the specific reinforcement requirements for thedrilled shaft were not discussed herein. Suitable reinforcementscan be designed by a qualified structural engineer, based on thecomputed internal forces and moments. And also, the final designresults should follow the site accessibility and constructability inpractical engineering.

4. Summary and conclusions

In this paper, a design procedure for using multiple rows ofdrilled shafts to stabilize a slope is presented. The limiting equilib-rium based slope stability analysis method, modified to incorpo-rate the induced arching effects of multiple rows of drilled shaftsthrough a semi-empirical load transfer factor, is coded into a gen-eral slope stability computer program (M-UASLOPE) for handlingcomplex slope geometry, soil profile, and groundwater table condi-tions. Optimization principles for designing multiple rows ofdrilled shafts are discussed. A step-by-step design procedure isgiven to illustrate the use of the design methodology. Specific con-clusions drawn from the illustrative design examples are asfollows:

� The need for using multiple rows of drilled shafts to stabilize alarge slope was illustrated by the design example. It was shownthat as compared to one row of drilled shafts, multiple rows ofdrilled shafts can effectively increase the global factor of safetyand at the same time reduce the net force imparted on the shaft,thus making the reinforcement design more constructible andmeeting the service limit.� Different combinations of shaft diameter, shaft spacing, and

shaft location for each row can affect the global factor of safety.However, the final design recommendation should follow theprinciple of using the least amount of reinforcement, as wellas minimizing the amount of total concrete volume.� Although the limit equilibrium method of slices is expressed as

a two-dimensional calculation, the load transfer factor wasderived from 3-D finite element simulation results, indicatingthat this method is capable of representing three-dimensionalsoil arching effects.

References

[1] Fukumoto Y. Study on the behaviour of stabilization piles for landslides. JJSSMFE 1972;12(2):61–73.

[2] Fukumoto Y. Failure condition and reaction distribution of stabilization pilesfor landslides. In: Proc, 8th annual meeting of JSSM; 1973. p. 549–62.

[3] Gudehus G, Schwarz W. Stabilization of creeping slopes by dowels. In: Proc11th international conference on soil mechanics and foundation engineering,San Francisco, vol. 3; 1985. p. 1679–700.

[4] Ito T, Matsui T. Methods to estimate lateral force acting on stabilizing piles.Soils Found 1975;15(4):43–59.

[5] Ito T, Matsui T, Hong PW. Design method for stabilizing piles againstlandslide—one row of piles. Soils Found 1981;21(1):21–37.

[6] Ito T, Matsui T, Hong PW. Extended design method for multi-row stabilizingpiles against landslide. Soils Found 1982;22(1):1–13.

[7] Morgenstern NR, Price VE. The analysis of the stability of general slip surfaces.Geotechnique 1965;15:70–93.

[8] Nethero MF. Slide control by drilled pier walls. In: Reeves RB, editor. Proc,application to landslide control problems. ASCE; 1982. p. 61–76.

[9] Poulos HG. Design of reinforcing piles to increase slope stability. Can Geotech J1995;32:808–18.

[10] Poulos HG. Design of slope stabilizing piles. In: Yagi N, Yamagami T, Jiang J-C,editors. Slope stability engineering: giotechnical and geoenvironmentalaspects: proceedings of an international symposium Is-Shohoku,Matsuyama, Shohoku, Japan. New York: Taylor and Francis; 1999.

[11] Reese LC, Wang ST, Fouse JL. Use of drilled shafts in stabilizing a slope. In: Procspecialty conference on stability and performance of slopes andembankments, Berkeley; 1992.

[12] Rollins KM, Rollins RL. Landslide stabilization using drilled shaft walls. In:Geddes JD, editor. Ground movements and structures, vol. 4. London: PentechPress; 1992. p. 755–70.

[13] Reese LC, Wang ST, Isenhower WM, Arrellaga JA. LPILE Plus 5.0 for windows,technical and user manuals. Austin (Texas): ENSOFT Inc.; 2004.

[14] Zeng S, Liang R. Stability analysis of drilled shafts reinforced slope. Soils Found,Japanese Geotechnical Society 2002;42(2):93–102.

[15] Yamin MM. Landslide stabilization using a single row of rock-socketed drilledshafts and analysis of laterally loaded shafts using shaft deflection data. Ph.D.dissertation, Department of Civil Engineering, The University of Akron, Akron,Ohio; 2007.

[16] Yamin MM, Liang R. Limiting equilibrium method for slope/drilled shaftsystem. Int J Numer Anal Meth Geomech 2010;34(10):1063–75.

[17] Liang R. Field instrumentation, monitoring of drilled shafts for landslidestabilization and development of pertinent design method. 2010 Reportfor Ohio Department of Transportation. Report number: FHWA/OH-2010/15.

[18] Al Bodour W. Development of design method for slope stabilization usingdrilled shaft. Ph.D. dissertation, Department of Civil Engineering, TheUniversity of Akron, Akron, Ohio; 2010.

[19] Joorabchi AE. Landslide stabilization using drilled shaft in static anddynamic condition. Ph.D. dissertation. The University of Akron; Akron,Ohio; 2011.

[20] Liang R, Joorabchi AE, Li L. Analysis and design method for slope stabilizationusing a row of drilled shafts. J Geotech Geoenviron Eng, ASCE 2014. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001070.

[21] Liang R, Al Bodour W, Yamin M, Joorabchi AE. Analysis method for drilledshaft-stabilized slopes using arching concept. Transport Res Rec: J TransportRes Board 2010;2:38–46.

[22] Li L, Liang R. Reliability-based optimization design for drilled shafts/slopesystem. DFI J: J Deep Found Inst 2012;6(2):48–56.

[23] Li L, Liang R. Reliability-based design for slopes reinforced with a row of drilledshafts. Int J Numer Anal Meth Geomech 2014;38(2):202–20.

74 L. Li, R.Y. Liang / Computers and Geotechnics 59 (2014) 67–74

[24] Liang R, Li L. Probabilistic analysis algorithm for UA Slope software program.Report for Ohio Department of Transportation 2013. Report number: FHWA/OH-2013/14.

[25] Liang R, Li L. Reliability based design for drilled shafts for slope stabilization.In: Geo-congress, GSP 231, Reston, VA, 1997–2006, ASCE; 2013.

[26] Joorabchi AE, Liang R, Li L, Liu H. Yield acceleration and permanentdisplacement of a slope reinforced with a row of drilled shafts. Soil DynEarthq Eng 2014;57:68–77.

[27] Liang R, Joorabchi AE, Li L. Design method for drilled shaft stabilization ofunstable slopes. In: Geo-congress, GSP 231, Reston, VA, 2017–2026, ASCE;2013.

[28] Joorabchi AE, Liang R., Li L. Yield acceleration of a slope reinforced with a rowof drilled shafts. In: Geo-congress, GSP 231, Reston, VA, 1987–1996, ASCE;2013.

[29] Janbu N. In: Hirschfeld RC, Poulos SJ, editors. Slope stability computations inembankment-dam engineering. New York: Wiley; 1973. p. 47–86.