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Reliability Based Design for Slope Stabilization Using Drilled Shafts

Lin Li1

and Robert Y. Liang2

1Research Assistant, Department of Civil Engineering, University of Akron, Akron, OH

44325-3905; Email: [email protected], Tel: 234-738-6084.2Distinguished Professor (Corresponding Author), Department of Civil Engineering, University

of Akron, Akron, OH 44325-3905; Email: [email protected]; Tel: 330-972-7190; Fax:

330-972-6020.

Re-Submission date: 11/08/2012

Word count: 4750

Figure count: 10

Table count: 1

TRB 2013 Annual Meeting Paper revised from original submittal

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Lin Li and Robert Y. Liang 1

ABSTRACT1

In this paper, a reliability-based computational algorithm was developed and coded into a2

computer program, P - UASLOPE, for design of a row of equally spaced drilled shafts to achieve3

target reliability index for the drilled shaft-reinforced slope system. The Monte Carlo simulation4

technique was used in the previously developed deterministic computational program, in which5

the limiting equilibrium method of slices was modified to incorporate arching effects of the6

drilled shafts in a slope. Uncertainties of soil parameters for each soil layer in the slope were7

considered by statistical descriptors, including mean, coefficient of variance (c.o.v.), and8

distribution function. Model errors involving the semi-empirical predictive equation for the load9

transfer factor for characterizing the soil arching effects were considered by statistics of bias. A10

total of 41 cases of 3-D finite element simulation results were used to determine the statistics of11

bias. A design example was given to demonstrate the use of P-UASLOPE program for optimized12

design of a drilled shaft-reinforced slope system for achieving the most economic combination of13

design variables (i.e., location, spacing, diameter, and length of drilled shafts) while satisfying14

the design requirements in terms of target reliability index of the drilled shafts/slope system and15

the structural performance of the drilled shafts. Sensitivity analysis of the influence of bias of16

model errors on the computed probability of failure for the design example indicates the need for17

more cases of 3-D finite element simulation results for obtaining a more accurate semi-empirical18

predictive equation for the load transfer factor.19

20


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INTRODUCTION21

Slope stabilization of un-stable slopes or man-made embankments of highways has been an22

important geotechnical issue that needs to be addressed to ensure operational safety of roadways.23

A vast variety of slope stabilization methods have been presented by numerous researchers in the24

past. Among these methods, a concept of using a row of drilled shafts to stabilize unstable slopes25

has been used successfully by the Ohio DOT (Liang, 2010). Analysis and design of the drilled26

shafts to stabilize an unstable slope has been a research topic since 1980s. In general, the analysis27

methodologies for treating the effects of drilled shafts on slope could be categorized into two28

approaches: a) an increase in the resistance due to the added shear strength of the reinforced29

concrete drilled shafts (e.g., Ito, et al. 1981; Hassiotis, et al. 1997; Reese, et al. 1992; Poulos,30

1995), and b) a decrease in the driving force due to soil arching from the inclusions of rigid31

structural elements. The later approach was pioneered by Liang and his associates (Liang and32

Zeng 2002; Yamin 2007; Al Bouder 2010; Liang, 2010 and Joorabchi 2011).33

Soil arching phenomena in a drilled shaft-reinforced slope system has been studied in34

Liang and Zeng (2002) by means of 2-D finite element simulations and in Yamin (2007) and Al35

Bouder (2010) by detailed and comprehensive 3-D finite element simulation results. Liang and36

Zeng (2002) and Liang (2010) have shown the procedure of incorporating the soil arching effects37

in the drilled shaft-reinforced slope system in a framework of a limiting equilibrium method of38

slices methodologies for determining the global factor of safety of the drilled shafts/slope system.39

The resulting computer program, UA SLOPE was available from Ohio Department of40

Transportation web site for open access. However, the UA SLOPE program is a deterministic41

program which cannot systematically take into account of the uncertainties of soil parameters42

and the semi-empirical equation for quantifying the soil arching effects. There is a need for a43

probability based version of UA SLOPE program for reliability based analysis and design of a44row of drilled shafts to stabilize an unstable slope. Presented in this paper is the theoretical45

basis of the probabilistic computational algorithms for determining the probability of failure (or46

reliability) of a slope reinforced with a row of equally spaced drilled shafts, considering all47

important sources of uncertainties of the soil parameters input and any bias introduced by the48

semi-empirical equations for the soil arching effects. The details of finite element simulation49

techniques for deriving the semi-empirical equations for soil arching are presented, together with50

statistical analysis to derive mean and variance of bias of the semi-empirical equation. Finally, a51

design example of a two-layer soil slope is presented to illustrate the use of the probabilistic52

computing algorithm for optimization of the design of a row of drilled shafts on slope to achieve53

the required target reliability of the global factor of safety as well as economy of construction54

cost.55

56

LIMIT EQUILIBRIUM METHOD WITH SOIL ARCHING57

58

Global Factor of Safety59


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The method for incorporating the effects of drilled shafts in a drilled shafts/slope system within the60

framework of limit equilibrium approach can be done in two ways. In the first approach, as given61

in Equation (1), the resistance to the slope sliding is increased after installing a row of the drilled62

shaft, thus enhancing the safety factor (Ito, et al., 1981; Reese et al, 1992 and Poulos, 1995). In63

contrast, as given in Equation (2), the increase of factor of safety is due to soil arching induced64

reduction on the driving force (Liang and Zeng, 2002; Yamin, 2007; Liang and Al Bouder, 2010).65

D

sha ftRR

F

FFFS

)( (1)66

archingDD

R

FF

FFS

)( (2)67

where, FS is global factor of safety of a slope/shaft system, FR is the resistance Force,68

(FR)shaft is additional resistance due to drilled shafts, FD is driving force, (FD)archingis drilled69

shaft induced arching effect on the driving force.70

71

3-Dimensional Finite Element Analysis on Arching Effect72

To interpret the arching effects in a drilled shafts/slope system, a load transfer factor has been73

introduced as shown in Figure 1, which is defined as the ratio between the horizontal force on the74

down-slope side of the vertical plane at the interface between the drilled shaft and soil (i.e.,75

Pdown-slope) to the horizontal force on the up-slope side of the vertical plane at the interface76

between the drilled shaft and soil (i.e., Pup-slope). Mathematically, the load transfer factor is77

expressed as:78

=Pdown-slope/Pup-slope (3)79

80

81

82

FIGURE 1 Definition of the load transfer factor.83

84

The resultant force in the soil upslope and downslope is estimated by integrating the85


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horizontal soil stresses from the top of the shaft down to the failure surface, as described in86

equation (4) and (5).87

/2

0 0

f D

up xxF dsdz

(4)

88

/2

0 0

'

fL D

down xxF dsdz

(5)89

where, D is the diameter of the drilled shaft; Lfis the distance from the top of the shaft90

down to the failure surface; 11xx S , which is the horizontal soil stresses on the upslope side of91

the shaft; 11' 'xx S , which is the horizontal soil stresses on the downslope side of the shaft; ds is92

the integration increment along the periphery of the shaft; dz is the depth of increment.93

To analyze the slide force acting on the pile, Vessely et al. (2007) has provided a case94

history to deal with a discrete, deep-seated shear zones. To quantify the soil arching effects and95

the load transfer factor in a drilled shaft/slope system, Al Bouder (2010) constructed a 3-D finite96

element model using ABAQUS program (version 6.6) for studying the soil structure interaction97

behavior of the drilled shafts on a slope under the effect of shear strength reduction. The strength98

reduction method in finite element simulation was first proposed by Zienkiewicz et al. (1975) and99

then developed by Qianjun et al. (2009) to study the slope stability problems. The concept of the100

strength reduction method in the finite element method for determining the FS of a slope is to101

gradually decrease the soil strength parameters (C and ) until the condition of slope failure (FS=1)102

is reached. The initial soil strength parameters [C and tan ()] are reduced incrementally by103

dividing them with a Reduction Factor (RF). Therefore, the reduced cohesion, CRand internal104friction angle, Rare given as:105

RC C RF (6)106

[tan( )] tan( )R

RF (7)107

In Al Bouders (2010) 3-D model, soil is modeled as a linear elastic-perfectly plastic108

material characterized by the angle of internal friction, cohesion, elastic modulus and Poissons109

ratio. The mesh of the 3-D model is depicted in Figure 2, which consists of 7, 696 hexahedral110elements for the soil body, 23, 600 similar elements for the rock, and 420 similar elements for the111

drilled shafts. The mesh of the shaft and the adjacent area was finer than that of the other zones112

because this region was expected to be a high-stress zone. The mesh was refined on the basis of the113

convergence of the numerically computed FS. At the beginning, a trial mesh was made, and the114

corresponding FS was found; then the mesh was refined incrementally, and the FS was computed.115

When the obtained value of the FS becomes stable (almost constant value), then the mesh with the116


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minimum number of elements corresponding to this safety factor is used.117

Based on the arching effect (Liang, 2010), the single row of drilled shafts in a slope cause118

the reduction of the driving stresses in the soil. The effect of the shaft is observed in the changes119

occurring in the horizontal stresses in the soil mass on the up-slope and down-slope sides of the120

drilled shafts. The variation of the horizontal soil stresses and the soil arching effects can be seen121

in Figure 3, which represents a horizontal stress in a horizontal cross section of the drilled122

shaft/slope system.123

Joorabchi (2011) proposed a semi-empirical equation, given in Equation (8), to compute124

the load transfer factor using the regression analysis techniques on a total of 41 cases of 3-D finite125

element model simulation results conducted by Al Boulder (2010). Considering that sufficient126

length of drilled shaft is socked into rock layer, the most important influencing factors consist of127

six parameters: soil cohesion C, friction angle , drilled shafts diameter D, center to center shaft128

spacing S0, shaft location on slope , and slope angle . The stiffness of the shaft will be evaluated129

in the final design step using LPILE program.130

))0.57(-0.61+(-0.252

0.876D)+)(0.065)(eD

S1.114(-1.17)(tan-0.272C=

2

)(-0.578tan00.429-0.153

xx

(8)

131

where, x= (xi-xtoe)/(xcrest-xtoe), xi= the x-coordinate of the location of drilled shaft, xtoe=132

the x-coordinate of the location of the toe of the slope, x crest= the x-coordinate of the location of133

the crest of the slope.134

135

136

137

FIGURE 2 3-D finite element model developed by Al Bouder (2010).138

139

To check the validity of the developed semi-empirical formula, Joorabchi (2011)140

calculated the load transfer factor using Equation (8) for all cases of the FE parametric studies. The141

comparisons between the results of Equation (8) and the FE results are shown in Figure 4,142

indicating that there is bias of the semi-empirical equation predictions.143


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144

145

FIGURE 3 Soil arching as observed from the horizontal soil stresses in the direction of the146

soil movement (after Al Bouder, 2010).147

148

149

150

FIGURE 4 Comparison of load transfer factor computed by semi-empirical equation and151

FEM (after Joorabchi 2011).152

153

Limiting Equilibrium Method of Slices Incorporating Soil Arching Effects154

The limiting equilibrium method incorporating the arching effects of the drilled shafts on the slope155

was developed by Liang and Zeng (2002). Figure 5a shows a typical slice with all force156

components acting on the slice. The load in the drilled shafts/slope system will be generated by157

gravity and friction and then be transferred slice by slice through the interslice force P i. Without158

y = 0.9783xR = 0.7869

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

C

omputed

FEM

Series1Linear (Series1)


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drilled shafts on the slope, the interslice force Pi can be expressed as an equilibrium equation159

given in Equation (9). However, with the insertion of drilled shafts on the slope, the interslice160

force on the down-slope side of the drilled shaftwill experience a reduction due to arching, i.e., it161

will be reduced by a multiplier called the load transfer factor () from the previous interslice162

force Pi-1, shown in Equation (10) and (11):163

ta (9)164 ta (10)165

cos sin (11)166The net force applied to the drilled shaft due to the difference in the interslice forces on the167

up-slope and down-slope sides of the drilled shaft can be calculated as follows:168 1 (12)169where, Pi is the interslice force acting on the down-slope side of slice, Wi is weight of170

slice i, iis inclination of slice i base, ciis soil cohesion at the base of slice i, iis soil friction171

angle at the base of slice i, uiis the pore water pressure at slice i, is the load transfer factor, P i-1172

is the interslice force acting on the up-slope side of slice (Figure 5b). Fshaft is the force on the173

drilled shaft, and S0is the center-to-center spacing between two adjacent drilled shafts. Based on174

Equations (9) to (11), factor of safety for a drilled shafts/slope system can be calculated in an175

iterative computational algorithm by satisfying boundary load conditions and equilibrium176

requirements, along with Mohr-Coulomb strength criterion. A PC based computer program, UA177SLOPE, has been developed (Liang 2010) based on the above computational algorithm to178

calculate the factor of safety and the shaft net force for a drilled shaft- reinforced slope system.179

180

181

182

FIGURE 5a A typical slice showing all FIGURE 5b Slice force change183

force components. due to arching.184

185


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RELIABILITY-BASED DESIGN FOR A DRILLED SHAFTS/SLOPE SYSTEM186

187

Uncertain Parameters in a Drilled Shafts/Slope System188

The influencing parameters in the drilled shafts/slope system can be divided into two major189

categories: soil properties (cohesion C, friction angle , and unit weight ) and drilled shaft190

related parameters (shaft diameter D, clear spacing S between the adjacent drilled shafts, the191

location of the shaft on the slope x). In this paper, the drilled shaft related parameters are treated192

as certain, while the soil properties are considered as uncertain in the developed reliability based193

analysis method for a drilled shaft-reinforced slope system. The independent lognormal194

distributions are assumed for these soil parameters. The mean and c.o.v. of soil parameters can be195

obtained from in-situ test data. In addition, the Ziggurat algorithm (Marsaglia and Tsang, 2000),196

an algorithm for pseudo-random number sampling based on the Accept-Reject algorithm, is used197

in the developed probability based computational program, P-UA SLOPE, as a pseudo-random198

number generator.199

200

Bias of Load Transfer Factor201

The semi-empirical load transfer factor () function given in Equation (8) contains bias as202

compared to the truevalue from the results of 41 cases of 3-D finite element simulations. The203

load transfer factor bias () is considered as a random variable with the mean and variance204statistically analyzed by comparing the finite element simulation results and the predictions of205

the semi-empirical equation. The mean and c.o.v. is 1.01 and 0.15, respectively. As indicated in206

Equation (13), the load transfer factor is randomly generated through the randomly generated207

bias:208

,,,, , (13)209where, is the randomly generated load transfer factor; is the randomly generated210

bias of load transfer factor; and are the randomly generated soil cohesion and friction211angle, respectively; , D, xand S0are considered as deterministic parameters.212

213

Monte-Carlo Simulation (MCS)214

The probability of failure for the drilled shafts/slope system is computed by means of Monte215

Carlo simulation method, as expressed by Equations (14) to (15). A probabilistic version of the216

UASLOPE computer program, P-UA SLOPE, based on the described algorithm was coded for217

reliability analysis of a drilled shafts stabilized slope.218

[] [ < 1]= + < 0= (14)219() + < 0

2= (15)220


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1 () (16)221where, Pf is the probability of failure for the drilled shafts/slope system, is the222

coefficient of variance (c.o.v.) of Pf, I[FS

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A STEP BY STEP DESIGN EXAMPLE243

The slope shown in Figure 7 was considered by Joorabchi (2011) as an example of the244

deterministic method of computation. It consists of two soil layers with soil properties for each245

layer summaried in Table 1, in which the range of coefficient of variance of each soil parameter246

is taken from Phoon and Kulhawy (1999) and the soil parameters for the two soil layers follow247

independent identical distribution. The critical slip surface for the slope without the drilled shafts248

was determined by a conventional slope stability analysis program, STABLE, with the computed249

FS equal to 0.78. The identified critical slip surface is represented by connecting 14 points. The250

ground water table is assumed at elevation of (-7.93m). It is noted that effective stress approach251

is used in the analysis. After obtaining the critical slip surface using STABLE program, the252

reliability index for the drilled shaft-reinforced slope system will be calculated by PUASLOPE253

program.254

During Monte Carlo Simulation (MCS), the relationship between probability of failure (Pf)255

and the corresponding c.o.v. can be expressed as follows:256

() ( 1 )/ 1/ (17)257In the present design example, we generated 100,000 samples for each variable: cohesion258

C, friction angle and unit weight as well as thebias of load transfer factor.259

260

261

262

FIGURE 7 Slope geometry for two-layer slope (x= 0.42).263

264

9.15m 9.15m 6.71m 8.54m

7.01m

7.93m

X


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TABLE 1 Soil Properties of the Two-Layer Slope265

266

Layer No.

Meanc.o.v.

Layer 1 Layer 2

C (kN/M2

) 0.958 2.394 0.2(degree) 11 18 0.1

(kN/M3) 21.21 22.78 0.01

267

Step by Step Design Procedure268

Step 1: Data concerning the geometry of the slope, soil paremeters, ground water table,269

critical slip surface, etc. are collected and presented in Figure 7 and Table 1.270

Step 2: Choose a target reliability index. As recommended by Abramson et al. (2002), the271

target reliability index Targetcan be seleced as 3.0.272

Step 3: Select different drilled shafts locations. The feasible locations for drilled shafts273

are between 12.3m and 21.8m (x= 0.2~0.8) horizontally from the crest of the slope to the toe of274

the slope (shown in Figure 7). In the current design, we analyze the location starting from275

X=12.3m and ending at X=21.8m, with an increment equal to 1.9m.276

Step 4: Select different pairs of clear spacing S and shaft diameter D combinations within277

the permissible range. Usually, this may depend on the site situations, and local availability of278

drilled shaft construciton equipment. In this example, the range for D is selected to be between279

0.6m to 2.4m (2ft~8ft), and the range of S/D is selected to be between 1.0~3.0. The following280

combinations for (S, D) were selected: (0.6, 0.6), (1.2, 0.6), (1.8, 0.6), (1.2, 1.2), (2.4, 1.2), (1.8,281

1.8), (2.4, 2.4). All units in the parenthesis are in meter.282

Step 5: Calculate reliability index using P - UASLOPE. For each (S, D) combination, plot283

the relationship between computed reliability index and shaft location, as shown in Figure 8.284

Step 6: Calculate shaft force using P - UASLOPE. For each (S, D) combination, plot the285

relationship between the computed net shaft force and shaft location, as shown in Figure 9.286

Step 7: Optimize the design for achieving the target reliability index, while requiring the287

least amount of drilled shaft volume for the project. As can be seen in Figure 8, the reliability288

index tends to increase with shaft location (measured by the distance from the crest of slope)289

and then decrease after reaching the middle location of the slope. The location of 16m provides290

the highest reliability index for the given shaft diameter and spacing. However, as shown in291

Figure 9, the drilled shafts at the location equals to 16m are also subject to the largest net forces,292which would have resulted in higher internal moments and shears in the shaft and higher293

required reinforcement ratio as well. It appears that the force on the shafts would decrease after294

the location 18m, and the computed reliability index would still satisfy the target reliability295

index. Furthermore, the required length of drilled shafts could be reduced if the location of the296

drilled shafts is moved further downslope. With the objective of minimizing the net shaft force297

while satisfying the target reliability index , three combinations are selected: (a) location298


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X=21.8m, D=0.6m, S/D=1.0; (b) X=19.9m, D=0.6, S/D=2.0; (c) X=19.9m, D=1.2m, S/D=1.0.299

300

301

FIGURE 8 Reliability index of the shaft/slope system versus shaft location for different (S, D)302

combinations.303

304

305

306

FIGURE 9 Shaft force versus shaft location for different (S, D) combinations.307

-2

-1

0

1

2

3

4

5

6

12 14 16 18 20 22

D=0.6m, S/D=1 D=0.6m, S/D=2D=0.6m, S/D=3 D=1.2m, S/D=1D=1.2m, S/D=2 D=1.8m, S/D=1D=2.4m, S/D=1

ReliabilityIndex()

Shaft Location (m)

Target

100

250

400

550

700

850

1000

12 14 16 18 20 22

D=0.6m, S/D=1 D=0.6m, S/D=2D=0.6m, S/D=3 D=1.2m, S/D=1D=1.2m, S/D=2 D=1.8m, S/D=1D=2.4m, S/D=1

ShaftNetForce(kN

)

Shafts Location (m)


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Step 8: The software LPILE (Reese and Wang, 1989) was used for the structural analysis308

of the shaft. Assuming that at least 20% length of drilled shaft is embedded into rock layer below309

slip surface, the three combinations obtained from step 7 (i.e., (a) D=0.6m, S/D=1.0, F=351KN,310

X=21.8m; (b) D=0.6m, S/D=2.0, F=423KN, X=19.9m; (c) D=1.2m, S/D=1.0, F=583KN,311

X=19.9m) with the corresponding net forces can be used as input in LPILE program to calculate312

the lateral deflection and the internal forces and moments in the drilled shaft. The computer313

program, LPILE (Reese et al., 2004), is used to analyze the three combinations. In LPILE314

analysis c- soil is used to model the soil and weak rock-Reese model is used to represent the315

rock. In LPILE simulation, zero shear and moment at the shaft head are set as the boundary316

conditions. The net force is distributed as a triangularly distributed shear force acting on the317

drilled shaft above the slip surface. The total net force is taken from step 7. After calculation with318

LPILE, the lateral deflection on the top of the shaft is 0.0035m, 0.0052m, 0.0081m respectively319

for the design combinations of (a), (b) and (c), and the total shaft lengths of the three320

combinations are 2.5m, 3.5m, 3.5m respectively. All the lateral deflections on the top of the shaft321

can be considered to be within the allowable deflection (say, half an inch, or equivalently322

12.7mm). If we perform the quantitative analysis of the required drilled shafts volume, the323

drilled shaft volume per unit width of a slope for three combinations are 0.588m3, 0.550m

3,324

1.649m3 respectively. Thus, the combination (b) (D=0.6m, S/D=2.0, F=423KN, X=19.9m) can325

be chosen as the most economical design with the least amount of drilled shaft volume required326

to meet the required target reliability index. The computed reliability index is 3.4 for the327

combination (b).328

329

Sensitivity Analysis of Soil Arching Model Errors330

This section presents a sensitivity analysis on the influence of model errors associated with the331empirical equation for the load transfer factor. This is because the statistics of bias of the predictive332

equation for load transfer factor could be different from current values, once additional finite333

element simulation results are available, or even high quality field monitoring data becomes334

available. The combination b (D=0.6m, S/D=2.0, F=423KN, X=19.9m) in last section is chosen as335

a model in the sensitivity analysis to demonstrate how the model errors will affect the design336

results. Figure 10 presents the relationship between the computed probability of failure and the337

mean value of bias of the load transfer factor, while keeping the other values to be the same as in338

the example run. SD in Figure 10 is standard deviation of bias of the predictive equation for load339

transfer factor. It can be seen that the computed probability of failure is sensitive to the statistics of340

the bias of eta. Therefore, it is important to note that more 3-D finite element simulation results341

would be desirable for improving the developed semi-empirical equation for load transfer factor.342

343


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344

345

FIGURE 10 Relationship between Pfand mean of bias of for different standard deviation.346

347

SUMMARY AND CONCLUSIONS348

In this paper, a reliability-based computational algorithm for design of drilled shafts to stabilize an349

unstable slope is presented. The computational algorithm is based on the deterministic approach350

using a modified limiting equilibrium method of slices incorporating the drilled shafts induced soil351

arching effects in calculating the global factor of safety of a drilled shaft-reinforced soil slope352

system. The uncertainties of soil parameters and the semi-empirical equation for predicting the353

load transfer factor in characterizing the soil arching effects are taken into consideration in the new354

computer program, P - UASLOPE, by using Monte Carlo simulation techniques. The Ziggurat355

algorithm is used for pseudo- random generation of random number for the uncertain variables of356the system, including soil cohesion, soil friction angle, soil unit weight, and bias of the load357

transfer factor. The P - UASLOPE program is used in a design example to illustrate the design358

procedure to achieve the required target reliability index while using the most economical drilled359

shafts combinations in terms of location of the drilled shafts on the slope, the diameter, the length,360

and the spacing of the drilled shafts. Finally, a sensitivity analysis of model error on the computed361

probability of failure of the drilled shaft/slope system is presented to demonstrate the need to362

continue to improve the predictive model of the semi-empirical equation for the load transfer363

factor. Specific conclusions based on the design example and sensitivity study can be made as364

follows.365

366

1. Design of a single row of equally spaced drilled shafts for stabilizing an unstable slope367

involves consideration of both geotechnical global factor of safety (or reliability index) as368

well as structural performance of the drilled shafts in terms of meeting the requirements of369

limiting the drilled shaft deflection and the necessary reinforcements for sustaining the370

1.00E-05

1.00E-04

1.00E-03

0.7 0.8 0.9 1.0 1.1 1.2

SD=0.11

SD=0.13

SD=0.15

SD=0.17

ProbabilityofFailure

Mean of Load Transfer Factor Bias


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internal forces and moments. This paper has presented methodologies for addressing both371

design issues.372

2. The location, spacing, diameter, and length of the drilled shafts can be varied in different373

combinations while achieving the target reliability index. The final selection of the design374

combination should be based on economic analysis and consideration of constructability of375

the drilled shafts.376

3. The bias of the load transfer prediction equation, or the model error, can affect the377

computed probability of failure of a drilled shaft-reinforced slope system. There is an378

incentive for conducting more 3-D finite element simulations covering more simulation379

conditions so that the bias of the semi-empirical predictive equation can be reduced.380

381

REFERENCES382

1. ABAQUS. ABAQUS Standard Users Manual, Version 6.6.Hibbitt, Karlsson, and Sorensen,383

Inc., Palo Alto, Calif., 2006.384

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