1

Robust Geotechnical Design of Drilled Shafts in Clay Using Spreadsheet 1

2

Wenping Gong, S.M.ASCE1, Sara Khoshnevisan

2, C. Hsein Juang, F.ASCE

3, 3

Sez Atamturktur, M.ASCE3 4

5

Abstract 6

This paper presents an efficient robust geotechnical design (RGD) approach that 7

considers performance requirements, design robustness, and cost efficiency simultaneously. 8

In this paper, the design robustness is measured with the variation of the performance function 9

of concern that is established for a reliability analysis. In the proposed RGD approach, the 10

performance requirement of the system is evaluated using reliability analysis. Thus, the 11

evaluation of the design robustness and the evaluation of the performance requirement share 12

common computational steps, referred to herein as computational “coupling.” This coupling 13

for computational efficiency is a significant feature of the proposed RGD approach. Within 14

the framework of the proposed RGD, the design robustness, cost efficiency, and performance 15

requirements are considered simultaneously by means of multi-objective optimization. 16

Furthermore, in this paper a practical procedure is developed for such optimization using 17

Solver, an optimization feature of the Excel spreadsheet software. Through an example of the 18

design of a drilled shaft in clay, the effectiveness and significance of this new RGD approach 19

is demonstrated. The results show that the hard-to-control variability resulting from 20

construction variation, loading conditions, model errors, and geotechnical parameters in the 21

design of drilled shafts in clay can be effectively considered with the proposed RGD approach. 22

23 Keywords: Drilled shaft; optimization; Pareto front; performance function; reliability analysis; 24

robust geotechnical design. 25

2

26

_______________ 27

1Research Assistant, Glenn Department of Civil Engineering, Clemson University, Clemson, 28

SC 29634; and Department of Geotechnical Engineering, Tongji University, Shanghai 200092, 29

China. 30

31 2Research Assistant, Glenn Department of Civil Engineering, Clemson University, Clemson, 32

SC 29634; and Department of Geotechnical Engineering, Tongji University, Shanghai 200092, 33

China. 34

35 3Glenn Professor, Glenn Department of Civil Engineering, Clemson University, Clemson, SC 36

29634. 37

38 4Associate Professor, Glenn Department of Civil Engineering, Clemson University, Clemson, 39

SC 29634. 40

41

42

3

Introduction 43

Performance or response of a geotechnical system under loading can be either over- or 44

under-estimated due to the uncertainty in soil parameters, loading conditions, analysis model 45

errors, and construction variation. To cope with the uncertainty in the predicted system 46

response, the engineer often selects a conservative design that may be cost inefficient. While 47

the reliability-based methods are capable of considering explicitly the uncertainty in input 48

parameters and solution models (e.g., Wu et al. 1989; Christian et al. 1994; Baecher and 49

Christian 2003; Phoon et al. 2003; Fenton and Griffiths 2008), reliability analysis and design 50

requires an accurate statistical characterization of such uncertainty, which is often challenging 51

in practice for various reasons. The results of a reliability-based design (RBD) might be 52

questionable if an accurate statistical characterization of the uncertainty cannot be obtained, 53

e.g., the under-estimated variability of soil parameters can lead to an under-design while the 54

over-estimated variability of soil parameters can lead to an over-design (Juang et al. 2013a). 55

To deal with the uncertainty in the estimated statistics of soil parameters (called noise 56

factors herein), Juang et al. (2013a&b&c) proposed a reliability-based robust geotechnical 57

design (RGD) approach. The goal of RGD is to reduce the effect of the uncertainty in the 58

noise factors on the predicted performance of the geotechnical system. Robust design is 59

achieved through adjusting the easy-to-control design parameters (i.e., geometry parameters) 60

so that the variation of the system performance or response is reduced to an extent that the 61

designed system is considered insensitive to, or robust against, the variability in the hard-to-62

control noise factors (i.e., estimated statistics of soil parameters). Here, robust is interpreted 63

as a kind of capability that measures the insensitivity of the system response to the variability 64

4

in the noise factors; accordingly, robustness is referred to as an ability of the system to resist 65

the inevitable and unforeseen variation of noise factors. 66

RGD is not a new concept in engineering; rather, it is adapted from the well-67

established robust design methodology, initiated by Taguchi (1986) in the field of quality 68

engineering. In fact, numerous applications of this concept have been reported in various 69

engineering fields (e.g., Tsui 1992; Chen et al. 1996; Doltsinis et al. 2005; Zhang et al. 2005; 70

Park et al. 2006; Beyer and Sendhoff 2007; Marano et al. 2008; Lee et al. 2010). Adapting the 71

concept of robust design to the field of geotechnical engineering, however, requires some 72

refinements as the level of uncertainty in input parameters, solution model, and construction 73

variation is generally much higher than that in other fields of engineering. Therefore, how to 74

select the most effective and efficient approach is a critical issue that must be resolved for a 75

successful implementation of robust design principles in geotechnical engineering. 76

In the previous study, Juang et al. (2013a) presented a reliability-based RGD approach, 77

in which the performance of drilled shafts is analyzed using reliability-based methods, and the 78

variation of the system performance, in terms of failure probability, is utilized as a measure of 79

the design robustness. The goal of RGD is to optimize the design robustness and the cost 80

simultaneously, while satisfying the safety or performance requirements. To this end, a multi-81

objective optimization is often implemented in the RGD. The reliability-based RGD approach 82

by Juang et al. (2013a) involves three loops, one for computing failure probability using 83

reliability methods, one for computing the variation of the failure probability considering the 84

uncertainty in the estimated statistics of input parameters, and the last one for conducting 85

multi-objective optimization. Although this reliability-based RGD approach is shown as an 86

effective design tool in geotechnical engineering (i.e., Juang et al. 2013a; Juang and Wang 87

5

2013; Wang et al. 2013), its multi-objective optimization-based implementation is quite 88

computationally demanding. A more efficient RGD approach is desirable. 89

The efficiency of a robust design is affected by how the system response is measured 90

and computed, and how the design robustness is defined. For example, Juang et al. (2013c) 91

described a RGD approach with the deformation, in lieu of failure probability, as the system 92

response so that the evaluation of the variation of the system response involves only one loop 93

of iterative procedure rather than two, which yields significant savings in computation time. 94

Indeed, the type of chosen system response also dictates the choice and efficiency of a 95

particular robustness measure. In literature, the commonly adopted robust design approaches 96

include Taguchi’s method (e.g., Taguchi 1986; Phadke 1989), worst-case approach (e.g., 97

Nagy and Braatz 2004; Zhu and Fukushima 2009), mean-variation compromising approach 98

(e.g., Chen et al. 1996; Park et al. 2006), and feasibility robustness approach (e.g., Parkinson 99

1993; Du and Chen 2000). Interested readers are referred to the literature for detailed 100

information, and to Juang et al. (2013b), Wang et al. (2013), and Gong et al. (2014) for 101

example applications in geotechnical engineering. 102

In this paper, the RGD approach proposed by Juang et al. (2013a) is further refined. 103

The main objective is to enhance the RGD approach as an effective and efficient design tool. 104

Several new features are introduced into the refined RGD framework. First, a fundamentally 105

sound and intuitive measure of the design robustness based on the variation of the 106

performance function is presented. Second, greater computation efficiency is achieved by 107

“coupling” the evaluation of design robustness with the evaluation of the performance 108

requirement using reliability methods (note: the word “coupling” here means that the two 109

evaluation procedures share common computational steps). Third, an effective procedure to 110

6

transform the multi-objective optimization into a series of single-objective optimizations is 111

developed so that the entire RGD process can be easily implemented in the user friendly 112

Excel spreadsheet environment. The improved RGD approach presented in this paper is not 113

only efficient but also user friendly and practical. 114

This paper is organized as follows. First, the formulation of a robustness measure, in 115

the form of the variation of the system performance function, is presented. Next, a simplified 116

reliability-based approach to couple the evaluation of design robustness with the evaluation of 117

the performance requirement is discussed. Then, the framework of the proposed efficient 118

RGD approach is formulated. Finally, an application of the proposed RGD approach to the 119

design of a drilled shaft in clay is demonstrated, where all formulations are implemented in 120

the Excel spreadsheet environment. 121

122

Variation of the Performance Function as a Robustness Measure 123

Robust design is aimed at deriving an optimal design, represented by a set of design 124

parameters (d), the system response of which is insensitive to, or robust against, the variation 125

of noise factors (θ) (e.g., Taguchi 1986; Tsui 1992; Chen et al. 1996; Beyer and Sendhoff 126

2007; Juang et al. 2013a&b&c; Gong et al. 2014). In principle, the design robustness can be 127

raised by reducing the variation of the system response that arises from the variation of noise 128

factors. In this paper, the variation of the response or performance of a geotechnical system, 129

which may be obtained through a reliability analysis, is used as a robustness measure. Use of 130

such robustness measure enables a “coupling” of the evaluation of design robustness with the 131

evaluation of the system performance. 132

Formulation of the performance function in the context of robust design 133

7

Within the context of robust design, the input parameters are classified into two 134

categories, one is the easy-to-control design parameters (d) and the other is the hard-to-135

control noise factors (θ). For a geotechnical system with design parameters of d and noise 136

factors of θ as inputs, the performance function, denoted as g(d, θ), can be set up as: 137

( , ) ( , ) ( , )g R S d d d (1) 138

where R(d, θ) and S(d, θ) are the resistance term and load term, respectively. The performance 139

of a design (d) is considered unsatisfactory if g(d, θ) < 0. Noise factors such as uncertain soil 140

parameters, loading conditions, model errors, and construction variation are treated as random 141

variables in a reliability analysis. To meet the performance requirement, the evaluated failure 142

probability (Pf) of a design, referred to herein as the probability of unsatisfactory performance, 143

must be less than a specified target value (PT). 144

Variation of the performance function as a robustness measure 145

With the performance function g(d, θ) defined in Eq. (1), the design robustness can be 146

intuitively measured by its variation (i.e., standard deviation). Fundamentally, a greater 147

variation of the performance function indicates a higher level of sensitivity of the system 148

response to the variation of noise factors. Using Taylor series expansions, the design 149

robustness of a design (d), in the form of the standard deviation of the performance function, 150

can be approximated as (e.g., Ang and Tang 2004; Zhang et al. 2011): 151

[ ] Tg CG G (2) 152

where Cθ is the covariance matrix among noise factors (θ), and G is the gradient vector of the 153

performance function to noise factors (θ), formulated as: 154

8

1 2

( , ) ( , ) ( , ), , ,

n

g g g

μ μ μ

d d dG (3) 155

where θ presents the mean of noise factors (θ). Accordingly, a lower standard deviation of 156

the performance function ([g]) signals a higher degree of design robustness. 157

Two possible approaches to improve design robustness 158

The measure of design robustness expressed in Eq. (2) is affected by two terms, the 159

gradient vector (G) and the covariance (Cθ). Hence, two approaches are possible to improve 160

the design robustness, either by minimizing the gradient (referred to as Approach 1 in Figure 161

1) or by minimizing the variation of noise factors (referred to as Approach 2 in Figure 1). 162

The first approach is to tune the design parameters (d) such that the gradient (G) is 163

reduced for a given variation of noise factors (). This is precisely the approach taken in the 164

current robust design (e.g., Taguchi 1986; Chen et al. 1996; Nagy and Braatz 2004; Juang et 165

al. 2013a&b&c; Liu et al. 2013; Gong et al. 2014). The second approach seeks to raise the 166

design robustness by reducing the variation of noise factors (θ). 167

168

MFOSM-Based Procedure for Reliability Analysis of System Performance 169

Given noise factors, the question of whether the performance requirement of a system 170

is satisfied or not is difficult to evaluate with certainty using deterministic methods. Thus, it is 171

appropriate to evaluate the system performance using probabilistic approaches. Although such 172

evaluation can be done using rigorous methods such as Monte Carlo simulation (MCS) and 173

first order reliability method (FORM), the computational demand of the RGD optimization 174

that involves MCS or FORM is often prohibitive. Therefore, the mean first order second 175

9

moment method (MFOSM), a much simpler reliability method, is employed herein to 176

evaluate the system performance within the proposed RGD framework. 177

Mean first order second moment method (MFOSM) 178

Within the context of MFOSM, the mean (E[g]) of the performance function g(d, θ) 179

can be approximated as follows (e.g., Ang and Tang 2004; Zhang et al. 2011): 180

[ ] ( , )E g g μ d (4) 181

While the standard deviation ([g]) of the performance function g(d, θ) is readily available 182

from the formulation of design robustness (see Eq. 2), the reliability index () of the system 183

performance, g(d, θ) > 0, can be computed as follows, with an assumption that g(d, θ) follows 184

a normal distribution (e.g., Zhao and Ono 2001; Ang and Tang 2004; Zhang et al. 2011): 185

[ ]

[ ]

E g

g

(5) 186

If desired, the failure probability (Pf) with respect to the system performance 187

requirement, or the probability of unsatisfactory performance, can be computed according to 188

the obtained reliability index (), which is expressed as (e.g., Zhao and Ono 2001; Ang and 189

Tang 2004; Zhang et al. 2011): 190

( )fP (6) 191

It is noted that through the common formulation of the variation of the performance 192

function, the evaluation of the design robustness and the evaluation of the system 193

performance, two main procedures in the RGD, share common computational steps or are 194

“coupled” from the perspective of computational effort. This “coupling” for computational 195

efficiency is a significant feature of the efficient RGD approach proposed herein. 196

Evaluation of system performance using MFOSM-based procedure 197

10

Though MFOSM is often favored for its simplicity over more rigorous methods such 198

as MCS and FORM, the accuracy of MFOSM might be questionable in many cases. To 199

overcome the dilemma between accuracy and computation efficiency, the procedure presented 200

by Zhang et al. (2011) is employed here. Zhang et al. (2011) pointed out that the reliability 201

index (M) that is computed using MFOSM is usually highly correlated with the reliability 202

index (T) that is computed using more sophisticate and accurate reliability methods such as 203

MCS and FORM. Thus, a mapping function can be established for a given problem: 204

M T( )f (7) 205

Note that the reliability index obtained using MCS or FORM is considered a close 206

approximation of the “true” reliability index, and thus the superscript “T” (for “true”) is used 207

for the reliability index computed with MCS or FORM. Similarly, the superscript “M” is for 208

the reliability index computed using MFOSM. The mapping function defined in Eq. (7) can 209

be constructed through the following procedures. First, compute the values of M using 210

MFOSM and T using MCS or FORM for a series of designs in a given design space of a 211

given problem. Second, carry out the least-square regression analysis to develop a mapping 212

function of M = f(T

). Third, validate the obtained mapping function. 213

Once the mapping function shown in Eq. (7) is successfully established in a design 214

space of a given problem, the target reliability index (M

T ) for system performance 215

requirement, within the context of MFOSM analysis, can be set up as M

T = f(T), where T is 216

the specified target reliability index (Zhang et al. 2011). Thus, the MFOSM analysis along 217

with the calculated M

T can readily be used to analyze the system performance requirement 218

without the need of performing MCS or FORM analysis. 219

11

Framework for Efficient Robust Geotechnical Design 220

Figure 2(a) illustrates the robust design implemented as a multi-objective optimization 221

problem that considers design robustness and cost as objectives while the system performance 222

requirement and the design space are treated as constraints. As noted, the robust design seeks 223

an optimal design in the design space that the design robustness is maximized and the cost (C) 224

is minimized simultaneously, while the system performance requirements are satisfied. 225

A simplified optimization procedure 226

Typically, the design robustness and the cost efficiency are two conflicting objectives. 227

Thus, a single best solution that is optimal with respect to both objectives simultaneously is 228

not attainable; rather, a set of non-dominated optimal solutions often exist that are superior to 229

all others in the design space, but within which, none of them are superior or inferior to others. 230

These non-dominated solutions form a Pareto front. As can be seen later, all these non-231

dominated optimal designs on the Pareto front are equally optimal in the sense that no 232

improvement can be achieved in one objective without worsening in the other objective. 233

Figure 3 plots a conceptual illustration of the non-dominated solution and Pareto front in a 234

robust design optimization. The multi-objective optimization problem shown in Figure 2(a) 235

may be solved using genetic algorithms such as Non-dominated Sorting Genetic Algorithm 236

version II, NSGA-II (Deb et al. 2002). Although NSGA-II is a fast algorithm, such multi-237

objective optimization requires the knowledge of genetic algorithms and considerable 238

programming skills, and is usually computationally demanding since it also requires 239

reliability analysis within the framework of multi-objective optimization. 240

In this paper, the multi-objective optimization problem shown in Figure 2(a) is solved 241

through a series of single-objective optimizations, as shown in Figure 2(b). Unlike the 242

12

weighted-sum-of-objectives approach (e.g., Kim and De Weck 2005; Marler and Arora 2010), 243

the cost, in the proposed approach, is modeled as a constraint instead of an objective (along 244

with the constraints of design space and system performance requirements). Using the single-245

objective optimization setting in Figure 2(b), the most robust design within a given acceptable 246

cost level (CTj) can readily be screened out using an Excel spreadsheet. As depicted in Figure 247

3, the resulting optimal design, denoted as (Cj, j[g]), would be a non-dominated solution on 248

the Pareto front that would have been obtained using multi-objective optimization in Figure 249

2(a). Through this approach, the Pareto front can be collectively formed by a series of optimal 250

designs obtained from single-objective optimizations. 251

The approach of transforming multi-objective optimization into a series of single-252

objective optimizations is inspired by previous studies (e.g., Haimes et al. 1971; Hämäläinen 253

and Mäntysaari 2002). While algorithms such as NSGA-II are generally more suitable for 254

multi-objective optimization, it is equally efficient to implement it through a series of single-255

objective optimizations when the optimization involves only two objectives (in our case, cost 256

and robustness). Since the geotechnical problems such as the design of drilled shafts can 257

generally be modeled as a bi-objective optimization problem, the proposed approach is 258

equally efficient. The proposed approach, however, has a tremendous advantage as it can be 259

easily implemented in an Excel spreadsheet without the knowledge of genetic algorithms and 260

considerable programming skills. As will be shown later, the proposed approach is efficient 261

and able to yield practically the same results as those obtained with NSGA-II multi-objective 262

optimization algorithm. Thus, the proposed RGD approach has a potential as a practical 263

design tool for engineering applications. 264

Procedures to implement the RGD approach 265

13

The procedures to implement the proposed efficient robust geotechnical design (RGD) 266

approach are summarized in the following four main steps: 267

Step 1: Characterize the problem of concern. Here, the system performance of concern, 268

noise factors, and design parameters are identified; meanwhile, the design space, requirements 269

of system performance, design robustness, and cost are formulated. 270

Step 2: Construct the mapping function that relates T (computed using MCS or 271

FORM) to M (computed using MFOSM) for the domain problem characterized in Step 1. 272

This step enables an accurate evaluation of the system performance using MFOSM analysis. 273

Step 3: Carry out the robust design optimization considering the design robustness, 274

cost efficiency, and design constraints using the optimization procedure shown in Figure 2(b). 275

Here, the most robust design within a given cost level is identified using Excel “Solver”. 276

Step 4: Construct the Pareto front to reveal the tradeoff between design robustness and 277

cost efficiency in the given design space. The Pareto front may be constructed through 278

following steps. First, determine the least cost design and the most robust design among all 279

feasible designs (i.e., those that satisfy the performance requirements) in the design space and 280

denote the costs for these designs as CL and CR, respectively. Second, divide the resulting cost 281

interval of [CL, CR] into a series of cost levels, denoted as CT = {CT1, CT2, CT3, …, CTn}. Third, 282

determine the most robust design within each cost level. Fourth and lastly, plot all the 283

resulting designs, in terms of design robustness versus cost, which yields a Pareto front. 284

285

Illustrative Example Robust Design of A Drilled Shaft in Clay 286

To demonstrate the proposed efficient robust geotechnical design (RGD) approach, a 287

design example published by ETC10 (2009) for the design of a drilled shaft in clay, shown in 288

14

Figure 4, is adopted here. The RGD optimization of this example drilled shaft is accomplished 289

with the following steps. 290

Step 1: Model characterization and parameters setting 291

Two fundamental requirements of system performance in the design of drilled shafts 292

in clay are bearing capacity (safety requirement) and settlement (serviceability requirement). 293

These performance requirements are generally specified in terms of ultimate limit state (ULS) 294

and serviceability limit state (SLS), respectively. The performance functions with respect to 295

ULS and SLS, in terms of g1(d, θ) and g2(d, θ), are formulated respectively as follows: 296

1 ULS d l( , )g Q F F d (8) 297

2 SLS d l( , )g Q F F d (9) 298

where QULS represents the ULS bearing capacity; QSLS is the SLS bearing capacity that is 299

derived based on a specified maximum allowable settlement of sa (e.g., taken as 20 mm in this 300

example); and, Fd and Fl represent the dead load and live load on the drilled shaft, 301

respectively. For drilled shafts in clay, the ULS bearing capacity, QULS, can be assessed using 302

the formulations proposed by Phoon and Kulhawy (2005): 303

ULS s bQ Q Q W (10) 304

where Qs and Qb are the side resistance and tip resistance, respectively, and W is the weight of 305

drilled shaft. The side resistance, Qs, in Eq. (10) is estimated as (Phoon and Kulhawy 2005): 306

s u1 u10.33 0.17 ( p ) πaQ c BDc (11) 307

where cu1 is the average undrained shear strength over the shaft length; pa is the atmosphere 308

pressure (taken as 100 kPa); B and D are the diameter and length of drilled shaft, respectively; 309

15

and, is a normal random variable with a mean of 0 and a standard deviation of 0.12. The tip 310

resistance, Qb, in Eq. (10) is computed as (Phoon and Kulhawy 2005): 311

1.12

2

b tc u2 cd cr6.17 π 4Q M c q B

(12) 312

where cu2 is the average undrained shear strength within a depth of B below the tip; cd and 313

cr are modifiers for the shaft depth and soil rigidity, respectively; q is the total vertical stress 314

at shaft dip, taken as D ( is the total unit weight of clay); and, Mtc is a lognormal random 315

variable with a mean of 0.45 and a standard deviation of 0.13. The depth modifier in Eq. (12) 316

is evaluated as (Phoon and Kulhawy 2005): 317

1

cd 1 0.33tan D B (13) 318

The rigidity modifier in Eq. (12), cr , is generally taken as 1.0. Finally, the SLS bearing 319

capacity of drilled shafts in clay, QSLS, can be estimated as (Dithinde et al. 2010): 320

SLS a

ULS a

Q s

Q a bs

(14) 321

where a and b are two hyperbolic curve-fitting parameters for the normalized load-settlement 322

curve, which can be statistically represented with lognormal variables; and, sa is the maximum 323

allowable settlement of drilled shafts. The mean of a and b are 2.79 mm and 0.82, 324

respectively, while the standard deviations are 2.04 mm and 0.09, respectively; and the 325

correlation between these two fitting parameters is -0.801. It is noted that the model errors 326

with respect to both ULS bearing capacity and SLS bearing capacity are explicitly considered 327

through the model parameters of , Mtc, a, and b. While the aforementioned empirical models 328

are employed here to estimate the ULS and SLS bearing capability of drilled shafts in clay, it 329

is not the limitation of the proposed RGD approach. Indeed, the more sophisticated finite 330

16

element method (FEM) that considers the non-linear soil-structure interaction and non-linear 331

and time-dependent behavior of soil can also be used. In which, the response surface method 332

(e.g., Faravelli 1989; Li et al. 2011) might be adopted to predict the ULS and SLS bearing 333

capability of drilled shafts in clay and to calibrate the associated model uncertainty. 334

For this drilled shaft example, a well-documented and well-studied example, the water 335

table is at the ground surface and the unit weight of clay () is 20 kN/m3. Hence, the 336

uncertainty of geotechnical parameters mainly comes from the undrained shear strength (cu). 337

In reference to Figure 5, the undrained shear strength of clay, obtained from undrained triaxial 338

tests, increases linearly with depth; the normalized undrained shear strength, cn = cu / z, is 339

sufficiently represented by a lognormal variable with a mean of 13.04 and a standard 340

deviation of 4.15. The adopted lognormal distribution of cn is validated, with 95% confidence 341

level, using Chi-Square test (2) and the statistics are estimated using the maximum likelihood 342

principle. The unit weight of concrete, an essential input parameter for evaluating QULS and 343

QSLS, is treated as a constant at 24 kN/m3. To account for the uncertainty in loading conditions, 344

both the dead load (Fd) and live load (Fl) are represented as lognormal variables. Given the 345

characteristic value of load F (from Eurocode 7: Fdc = 300 kN and Flc = 150 kN), the mean of 346

load F, denoted as (E[F]), can be approximated as follows (Wang et al. 2011): 347

c

F

[ ]1 1.645

FE F

(15) 348

where Fc represents the characteristic value of load F, and F is the coefficient of variation of 349

load F. With an assumption that the coefficients of variation of the dead load (Fd) and live 350

load (Fl) are 0.1 and 0.18, respectively (Zhang et al. 2011), the mean of the dead load (Fd) and 351

live load (Fl) are calculated as 258 kN and 116 kN, respectively. 352

17

Due to construction imperfections and unforeseen geological conditions, differences 353

between the designed geometry and the constructed geometry of a drilled shaft should be 354

expected. The effect of such discrepancy on the pile capacity can be significant (O’Neill 2001; 355

Iskander et al. 2003; Poulos 2005). Yet, a quantitative index measuring such differences is 356

rarely reported in literature. In the context of robust design, the as-built geometry parameters 357

of a drilled shaft (i.e., BT and DT) are herein modeled as lognormal variables. The mean of 358

these parameters (i.e., BT and DT) are assumed to be the same as the parameters of the 359

designed geometry (i.e., B and D), while the standard deviations of these parameters are both 360

set to 0.05 m; further, the correlation coefficient between the two geometry parameters is 361

assumed as T T, 0.5B D . The standard deviations and the correlation coefficient assumed 362

above are just an example for the purpose of illustrating the effect of construction noise. In 363

practice, other distribution type and different statistics can be adopted based on local 364

experience to yield a more preferred design. Additional discussion on the effect of the level of 365

construction variation is presented later. 366

In summary, the geotechnical parameters, loading parameters, construction variations, 367

and model errors are explicitly included in the robust design with noise factors of cn, Fd, Fl, 368

BT, DT, , Mtc, a, and b. The designed geometry parameters (i.e., B and D) are treated as 369

design parameters. For illustration purpose, the design space is set up as a continuous space, 370

DS = {(B, D) | B [0.3 m, 0.6 m], D [10.0 m, 25.0 m]}. Within the framework of robust 371

design, the target reliability indexes (i.e., T1 and T2) with respect to ULS performance and 372

SLS performance are set as 3.2 and 2.6, respectively, to ensure an adequate system 373

performance. These target reliability indexes correspond to the target failure probabilities (i.e., 374

PT1 and PT2) of 6.910-4

and 4.710-3

, respectively. For this design example, the study 375

18

performed by Orr et al. (2011) found that the critical factor that controls the design is the ULS 376

performance requirement. Thus, the variation of the performance function with respect to 377

ULS performance, [g1(d, )], is adopted herein to measure the design robustness. In this 378

paper, the cost of the drilled shaft (C) is simply represented with the volume of concrete 379

(B2D/4), although other, more sophisticated cost estimate models may be adopted. Figure 6 380

illustrates all the elements considered in the robust design of the drilled shaft in clay. 381

Step 2: Mapping from FORM to MFOSM 382

To establish the mapping function that relates T (computed using MCS or FORM) to 383

M (computed using MFOSM), 18 designs, equally distributed in the design space (DS), are 384

analyzed. The analysis is carried out with respect to both ULS and SLS performance 385

requirements, in terms of 1 and 2, respectively. FORM is adopted herein to evaluate the true 386

reliability indexes of the system performances, as it can be implemented in the Excel 387

spreadsheet environment and yields the results of the similar accuracy as that obtained using 388

MCS. Shown in Figure 7 is an example of FORM analysis of a drilled shaft in clay, with 389

respect to ULS performance, in the spreadsheet environment following the procedures 390

advanced by Low and Tang (2007). Note that for simplicity, the correlation matrix is assumed 391

to retain the original unmodified correlation matrix. Shown in Figure 8 is the reliability 392

analysis of drilled shafts in clay, with respect to both ULS performance and SLS performance, 393

using MFOSM in the spreadsheet environment. As reflected in Figure 9(a) & 9(b), the results 394

are fitted well with a third-order polynomial function, yielding the following mapping 395

functions: 396

M T 3 T 2 T

1 1 1 10.0163( ) 0.1453( ) 0.8862( ) 0.1438 (16) 397

19

M T 3 T 2 T

2 2 2 20.0099( ) 0.1024( ) 0.8164( ) 0.0890 (17) 398

where

and

are the reliability indexes with respect to ULS and SLS, respectively, 399

computed using MFOSM; and, and

are the reliability indexes with respect to ULS and 400

SLS, respectively, computed using FORM. 401

As shown in Figure 9, the clustering is observed at higher reliability index levels (or 402

lower failure probability levels) of drilled shaft performance. Indeed, the reliability analysis of 403

a system with a higher safety level always requires more computational effort and more 404

sophisticated reliability methods; in other words, more computation error might be expected 405

at higher reliability index levels of the system performance due to the less chance of failure 406

(Zhao and Ono 2001). Although higher variability is associated with the reliability analysis 407

results of the drilled shaft at higher reliability index levels, the computed coefficients of 408

determination (R2), with respect to ULS and SLS, are 0.9909 and 0.9839, respectively, 409

indicating the true reliability indexes of the drilled shaft performance can be accurately 410

captured through MFOSM analysis and the corresponding mapping functions. 411

Based upon the obtained mapping functions (see Eqs. 16 & 17), the target reliability 412

indexes (i.e., M

1T and M

2T ) with respect to ULS and SLS, within the context of MFOSM 413

analysis, are set as 2.03 and 1.69, respectively. The resulting target reliability indexes (i.e., 414

M

1T and M

2T ) are readily applicable to the MFOSM-based procedure for evaluating the 415

system performance requirements. Thus, no FORM or MCS is required in the subsequent 416

robust design optimization process, which is a significant feature of the proposed RGD 417

approach that reduces the computational effort. 418

Step 3: Efficient multi-objective optimization 419

20

As formulated previously, the proposed efficient robust geotechnical design (RGD) 420

can be implemented through a series of single-objective optimization problems in which the 421

cost is treated as an additional constraint (see Figure 2b). The single objective optimization-422

based RGD can be efficiently implemented in the spreadsheet environment using Excel’s 423

built-in optimization routine “Solver”, as shown in Figure 10. It is noted that the evaluation of 424

design robustness, in terms of the standard deviation of the performance function with respect 425

to ULS performance, and the evaluation of the system performance requirements, in terms of 426

the reliability indexes with respect to both ULS and SLS, share common computational steps 427

(i.e., they are “coupled” computationally). With the spreadsheet shown in Figure 10, the most 428

robust design within a given cost level (CT) can readily be identified. For example, the most 429

robust designs for three arbitrarily specified cost levels are identified and listed in Table 1. 430

Step 4: Pareto front tradeoff between design robustness and cost efficiency 431

Although the most robust design within a given cost level (CT) can readily be 432

identified using the spreadsheet shown in Figure 10, it should be noted that the optimal cost 433

level is generally unknown in prior, and thus the resulting robust design may be a biased 434

design. To guide the design decision, it is desirable to construct a complete Pareto front that 435

shows the tradeoff between design robustness and cost efficiency. 436

For the given drilled shaft problem, the costs of the least cost and the most robust 437

designs are 2.04 m3 and 2.91 m

3, respectively. The resulting cost interval of [2.04 m

3, 2.91 m

3] 438

is then divided into 10 cost levels, denoted as CT = {2.04 m3, 2.14 m

3, 2.24 m

3, 2.34 m

3, 2.44 439

m3, 2.54 m

3, 2.64 m

3, 2.74 m

3, 2.84 m

3, 2.91 m

3}. For each cost level in CT, the most robust 440

design is identified using the spreadsheet shown in Figure 10. Thereafter, all the resulting 441

optimal designs are plotted as shown in Figure 11. The resulting Pareto front shows a global 442

21

tradeoff between design robustness and cost efficiency in the design space. The Pareto front 443

may be used as a decision aid in selecting the most preferred design at a desired cost level. 444

Also shown in Figure 11 is the Pareto front obtained using the genetic algorithm of 445

NSGA-II (Deb et al. 2002). In NSGA-II, the population size is set at 200 and the generation 446

number is set at 500. At convergence, this multi-objective optimization algorithm (see Figure 447

2a) yields the Pareto front shown in Figure 11. The results show that the proposed RGD 448

approach (see Figure 2b) using Excel can derive practically the same Pareto front as that 449

obtained using more sophisticated multi-objective optimization algorithms such as NSGA-II. 450

As mentioned previously, all the designs on the Pareto front are equally optimal in the 451

sense that no improvement can be achieved in one objective without worsening in the other 452

objective. Nevertheless, if one wishes to select the most preferred design on a Pareto front, the 453

“knee point” concept (Branke et al. 2004; Deb and Gupta 2011) may be used. The knee point 454

on the Pareto front conceptually yields the best compromise among conflicting objectives in 455

the design space. According to Branke et al. (2004), the knee point may be determined using 456

the following marginal utility function: 457

, min , , ( )i j iU' U U i j d d d (18) 458

where U(d, ) is a linear utility function defined as (Branke et al. 2004): 459

, i iU f d d (19) 460

where d are design parameters, in this example: d = (B, D); i is random parameter with a 461

value ranging from 0.0 to 1.0, and i = 1.0; and, fi(d) = objective functions to be minimized, 462

in this case: f1(d) = ([g1] min[g1]) / (max[g1] min[g1]), and f2(d) = (C Cmin) / (Cmax 463

Cmin), where max[g1] and min[g1] represent the maximum value and minimum value of [g1] 464

22

on the Pareto front, respectively; and, Cmax and Cmin represent the maximum value and 465

minimum value of C on the Pareto front, respectively. 466

By means of Monte Carlo simulation (MCS), different random values of 1 and 2 are 467

generated with an assumption of uniform distribution, and the marginal utility function is 468

computed for each of the 10 non-dominated optimal designs on the Pareto front. The design 469

with the maximum expected marginal utility is then taken as the knee point on the Pareto front. 470

Based on the calculated expected marginal utility, Design 2 (see Table 1) is identified as the 471

most preferred solution (or knee point). 472

As can be observed in Figure 11, on the left side of the knee point, a slight reduction 473

in the cost can lead to a large increase in the variation of the ULS performance function 474

(indicating a drastic reduction in the design robustness); on the right side of the knee point, a 475

slight reduction in the variation of the ULS performance function (indicating a slight 476

improvement of design robustness) requires a large increase in cost. Therefore, the knee point 477

on the Pareto front yields the best compromise between design robustness and cost efficiency 478

and can be treated as the most preferred design in the design space 479

480

Further Discussions on the Proposed RGD Approach 481

482

Effectiveness of the variation of the performance function as a robustness measure 483

Figure 12 shows the variation or distribution of the system performance with respect 484

to ULS performance for the three designs listed in Table 1. Through 1,000,000 MCS runs, the 485

distribution of the system performance (defined in Eq. 8) caused by the variation of noise 486

factors can be obtained for each design. The results show that a design with a higher variation 487

of the performance function (i.e., higher robustness) has a wider distribution of system 488

23

performance, which implies a higher degree of uncertainty as to whether the system 489

performance can satisfy the pre-defined performance requirement. 490

Figure 13(a) & 13(b) show the gradient of the system performance to noise factors and 491

the variation of the system performance that arises from noise factors, respectively, for the 492

three designs listed in Table 1. Figure 13(a) depicts that the system performance is most 493

sensitive to the variation of the as-built diameter of drilled shafts (T) and model parameters 494

of QULS (i.e., and Mtc). Figure 13 (b) illustrates that the variation of the system performance is 495

mainly affected by the variation of the normalized undrained shear strength (cn), as-built 496

diameter of drilled shafts (T), and model parameters of QULS (i.e., and Mtc). Compared with 497

the existing robust measures, the presented robustness measure in this paper shows a 498

significant advantage as it is an integration of the gradient (or sensitivity) and the variation of 499

noise factors (see Eq. 2). Thus, the knowledge of how the variation of the system performance 500

is affected by the variation of each noise factor is effectively acknowledged through the 501

formulation of the proposed robustness measure. 502

Figure 14 shows the distribution of the system performance for the three designs listed 503

in Table 1, which is similar to Figure 12, but in Figure 14, the uncertainty in the estimated 504

statistics of noise factors is included. As in most geotechnical problems, it is often challenging 505

to determine the statistics of key geotechnical parameters with certainty; such uncertainty in 506

the estimated statistics can lead to another layer of uncertainty in the system performance. For 507

illustration purposes, the standard deviations of noise factors are treated as independent 508

lognormal variables with a coefficient of variation (COV) of 20%. Through 1,000,000 MCS 509

runs, the distribution of the system performance can be achieved as shown in Figure 14. The 510

results indicate that the design with a lower variation of the performance function (i.e., higher 511

24

robustness) can tolerate a greater uncertainty in the estimated statistics of noise factors, as it 512

yields a shallower distribution of system performance and a lower failure probability, defined 513

as probability of unsatisfactory performance (see Table 2). Therefore, the design with higher 514

design robustness (i.e., a lower variation of the performance function) has an additional 515

benefit of being able to guard against the uncertainty in the estimated statistics of noise factors. 516

Comparison between the RGD and the RBD 517

The Pareto front shown in Figure 11 is employed herein to illustrate the difference 518

between the robust geotechnical design (RGD) and the reliability-based design (RBD). Since 519

all the designs on the Pareto front are feasible designs (i.e., the requirements of system 520

performances are satisfied), the design that yields the least cost (in this case, Design 1) can be 521

readily selected as the final design through the RBD. This least cost design using the RBD 522

approach (Design 1), however, is associated with the highest variation of the system 523

performance (see Figure 12), highest sensitivity of the system performance to the variation of 524

noise factors (see Figure 13), and least tolerance of the system performance to the uncertainty 525

in the estimated statistics of noise factors (see Figure 14). Thus, the least-cost RBD design is 526

not necessarily the most preferred design since a modest variation in the noise factors could 527

lead to a dramatic variation of the system performance. On the other hand, the RGD design 528

approach considers explicitly the safety (performance requirements), cost efficiency, and 529

design robustness, which leads to a Pareto front that offers a tradeoff between design 530

robustness and cost efficiency. Thus, the effect of the uncertain noise factors is realistically 531

managed in the design decision using the proposed RGD approach. 532

Improving design robustness by reducing the variation of noise factors 533

25

So far the discussion focuses on improving the design robustness by adjusting only the 534

design parameters, assuming that the uncertainty in noise factors cannot be reduced or it is not 535

economical to do so. An alternative is to improve the design robustness by reducing the 536

variation of noise factors, referred to as Approach 2 shown in Figure 1. In this section, the 537

construction variation as a noise factor is used as an example to illustrate the robust design 538

using Approach 2. 539

The three designs listed in Table 1 are analyzed for their design robustness, in terms of 540

the variation of the performance function, at three levels of construction variation. Here, the 541

level of construction variation is represented with the standard deviation of as-built geometry 542

(applied to both BT and DT). Three levels of standard deviation, 0.05 m, 0.10 m, and 0.15 m, 543

are adopted for illustration purposes. The results are plotted in Figure 15. As expected, the 544

design robustness can be raised (meaning that the variation of the system performance can be 545

lowered) by reducing the variation of the as-built geometry of drilled shafts that caused by the 546

construction variation. Thus, Approach 2 shown in Figure 1 is demonstrated as an effective 547

alternative for improving the design robustness, although an additional cost that is associated 548

with noise variation reduction may be incurred. On the other hand, cost savings may be 549

achieved from the resulting optimal design at a lower level of construction noise. Although 550

further study to determine the net effect on the overall cost is needed, the above analysis 551

demonstrates that reducing the variation of noise factors might be a cost-effective step in a 552

robust design of a geotechnical system. 553

554

Summary and Conclusions 555

26

This paper presents an efficient robust geotechnical design (RGD) approach, in which 556

the design robustness is represented with the variation of the performance function that is 557

readily available in a reliability analysis. Through the coupling of the evaluation of design 558

robustness with the reliability analysis of system performance, additional savings in the 559

computational effort are achieved. Finally, the multi-objective optimization, typically 560

implemented within the robust design framework, is effectively solved with a series of single 561

objective optimizations using Excel Solver. In short, an efficient and practical robust 562

geotechnical design (RGD) approach is developed. 563

The following conclusions are reached based on the results presented: 564

1) The proposed efficient robust geotechnical design (RGD) approach is shown to 565

be effective and efficient in producing an optimal design that yields the best 566

compromise between design robustness and cost efficiency, while satisfying all 567

design safety and performance requirements. 568

2) The developed robustness measure, in terms of the variation of the performance 569

function, is demonstrated to be effective, intuitive, and fundamentally sound. 570

Higher variation of the performance function signals lower design robustness, 571

which implies a higher degree of uncertainty as to whether the system can satisfy 572

the pre-defined performance requirement. 573

3) Within the context of the proposed RGD approach, the evaluation of design 574

robustness and the evaluation of system performance, two essential procedures 575

in the robust design, share computational steps, as both can be analyzed using 576

MFOSM with a common formulation. Thus, the computational efficiency is 577

greatly improved over other existing reliability-based robust design approaches. 578

27

4) Although the RGD is presented as a multi-objective optimization problem, it can 579

be efficiently solved using a series of single-objective optimizations that does 580

not require the more sophisticated genetic algorithms and programming skills. 581

The Pareto front obtained from a series of single-objective optimizations 582

implemented with Excel Solver is shown to be identical to the one obtained with 583

the more sophisticated multi-objective optimization algorithm. 584

5) While the robust design using Approach 1 as shown in Figure 1 has been shown 585

effective, it is possible that Approach 2, which allows for consideration of some 586

reduction in the variation of noise factors within the framework of the robust 587

design, can yield more cost-efficient designs while improving the design 588

robustness. 589

6) As shown in Figure 6, the proposed RGD approach considers the noises from 590

construction variation, loading conditions, model errors, and geotechnical 591

parameters in the design of drilled shafts in clay. The proposed approach is 592

flexible in allowing use of either Approach 1 or Approach 2 or both for the more 593

preferred design with respect to cost efficiency, design robustness, and 594

performance requirements. In light of the recognition that the capacity and 595

performance of a drilled shaft are strongly affected by the construction quality, 596

the proposed RGD approach offers a practical and effective tool to examine 597

possible alternatives. 598

7) While the proposed efficient RGD approach is demonstrated only through an 599

example of a drilled shaft design, the formulation and solution algorithm are set 600

up in a way that is suitable for general applications to most engineering problems. 601

28

Thus, the proposed RGD approach can be easily adapted for design of other 602

geotechnical systems. 603

604

Acknowledgments 605

The study on which this paper is based was supported in part by National Science 606

Foundation through Grant CMMI-1200117 (“Transforming Robust Design Concept into a 607

Novel Geotechnical Design Tool”) and the Glenn Department of Civil Engineering, Clemson 608

University. The results and opinions expressed in this paper do not necessarily reflect the 609

views and policies of the National Science Foundation. 610

29

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35

Lists of Tables

Table 1. Examples of most robust design at various cost levels

Table 2. Probability of unsatisfactory performance of each of the three designs at each of the

three levels of variation of noise factors

36

Table 1. Examples of most robust design at various cost levels

Specified

cost level,

CT (m3)

Most robust design

within the cost level

Design parameters Actual

cost,

C (m3)

Resulting

robustness,

[g1(d, θ)] (kN)

Diameter,

B (m)

Depth,

D (m)

2.04 Design 1 0.32 25.00 2.04 800.46

2.24 Design 2 0.42 16.19 2.24 476.26

2.91 Design 3 0.60 10.29 2.91 347.27

37

Table 2. Probability of unsatisfactory performance of each of the three designs at each of the

three levels of variation of noise factors

Design

Level of variation of noise factors (in terms of COV of

the standard deviation of each of noise factors)

COV = 10% COV = 20% COV = 30%

Design 1 1.169 10-3

2.135 10-3

3.937 10-3

Design 2 0.949 10-3

1.667 10-3

3.044 10-3

Design 3 0.530 10-3

0.917 10-3

1.843 10-3

38

Lists of Figures

Figure 1. Illustration of possible approaches to improve design robustness

Figure 2. Optimization setting of robust design

(a) Robust design through multi-objective optimization

(b) Robust design through proposed optimization procedure

Figure 3. Illustration of non-dominated solution and Pareto in a robust design optimization

(note: lower [g] signals higher design robustness)

Figure 4. Schematic diagram of a drilled shaft in clay (after ETC 10)

Figure 5. Characterization of undrained shear strength from test data

Figure 6. Illustration of robust design of a drilled shaft in clay

Figure 7. Layout of a spreadsheet for FORM analysis of a drilled shaft based on ULS

Figure 8. Layout of a spreadsheet for MFOSM analysis of a drilled shaft based on ULS and

SLS

Figure 9. Mapping from FORM to MFOSM

(a) Mapping function of reliability index based on ULS

(b) Mapping function of reliability index based on SLS

Figure 10. Layout of a spreadsheet for robust design of a drilled shaft

Figure 11. Pareto front showing tradeoff between design robustness and cost efficiency (note:

lower [g1(d, )] signals higher design robustness)

Figure 12. Distribution of ULS performance of three drilled shaft designs (standard deviation

of each and every noise factor is a constant)

Figure 13. Sensitivity of performance function with respect to noise factors

(a) Gradient of performance function to each noise factor

(b) Effect of each noise factor on the variation of performance function

Figure 14. Distribution of ULS performance of three drilled shaft designs (standard deviation

of each and every noise factor is a random variable with 20% COV)

Figure 15. Robustness of three designs at various levels of construction variation (in terms of

standard deviation of as-built geometry)

39

Table 1. Examples of most robust design at various cost levels

Specified

cost level,

CT (m3)

Most robust design

within the cost level

Design parameters Actual

cost,

C (m3)

Resulting

robustness,

[g1(d, θ)] (kN)

Diameter,

B (m)

Depth,

D (m)

2.04 Design 1 0.32 25.00 2.04 800.46

2.24 Design 2 0.42 16.19 2.24 476.26

2.91 Design 3 0.60 10.29 2.91 347.27

40

Table 2. Probability of unsatisfactory performance of each of the three designs at each of the

three levels of variation of noise factors

Design

Level of variation of noise factors (in terms of COV of

the standard deviation of each of noise factors)

COV = 10% COV = 20% COV = 30%

Design 1 1.169 10-3

2.135 10-3

3.937 10-3

Design 2 0.949 10-3

1.667 10-3

3.044 10-3

Design 3 0.530 10-3

0.917 10-3

1.843 10-3

41

Figure 1. Illustration of possible approaches to improve design robustness

Noise factors, 1

Sy

stem

res

po

nse

, g

(d1,

1)

Robust design Noise factors, 1

Sy

stem

res

po

nse

, g

(d2,

1)

Noise factors, 2

Sy

stem

res

po

nse

, g

(d1,

2)

Approach 2

Approach 1

42

(a) Robust design through multi-objective optimization

(b) Robust design through proposed optimization procedure

Figure 2. Optimization setting of robust design

Find: d (Design parameters)

Subjected to: d DS (Design space)

M

i M

iT (Requirements of system performances)

Objectives: min [g] (Maximizing the design robustness)

min C (Minimizing the cost)

Find: d (Design parameters)

Subjected to: d DS (Design space)

M

i M

iT (Requirements of system performances)

C < CTj (Cost level is dealt as an additional constraint)

Objectives: min [g] (Maximizing the design robustness)

43

Figure 3. Illustration of non-dominated solution and Pareto front in a robust design

optimization (note: lower [g] signals higher design robustness)

(CTj,

j[g])

Infeasible domain

Pareto front

Ro

bu

stn

ess

mea

sure

,

[g]

Cost, C

Feasible domain

Non-dominated solution

44

Figure 4. Schematic diagram of a drilled shaft in clay (after ETC 10)

Fd + Fl

D = ?

B = ?

Stiff clay

= 20 kN/m3

cu is obtained from

undrained triaxial test

sa = 20 mm

45

Figure 5. Characterization of undrained shear strength from test data

30

25

20

15

10

5

0 0 100 200 300 400 500 600

E[cu] [cu]

E[cu] + 2[cu]

Data from test

Undrained shear strength, cu (kPa)

Dep

th,

z (m

)

E[cu]

46

Figure 6. Illustration of robust design of a drilled shaft in clay

Robust design

of drilled shafts in clay

Design objectives

Maximize design robustness

Minimize cost

Satisfy design constraints

Noise on

geotechnical parameters

(cn)

Noise on

model errors

(, Mtc, a, and b)

Noise on

loading conditions

(Fd, Fl)

Noise on

as-built geometry

(BT and DT)

47

Figure 7. Layout of a spreadsheet for FORM analysis of a drilled shaft based on ULS

Distribution n n n *

cn (kPa/m) Lognormal 13.04 4.15 2.520 0.311 -1.008 9.09 B (m) 0.45

B T (m) Lognormal 0.45 0.05 -0.805 0.111 -0.893 0.41 D (m) 10

D T (m) Lognormal 10.00 0.05 2.303 0.005 -0.478 9.98

Normal 0.00 0.12 0.000 0.120 -1.141 -0.14

Mtc Lognormal 0.45 0.13 -0.839 0.283 -0.439 0.38 g 1(d , * 0.00

a (mm) Lognormal 2.79 2.04 0.812 0.654 0.000 2.25

b Lognormal 0.82 0.09 -0.204 0.109 0.000 0.82

F d (kN) Lognormal 258.00 25.80 5.548 0.100 0.442 268.29

F l (kN) Lognormal 116.00 20.88 4.738 0.179 0.359 121.72

1 (ULS) 1.906

PT

f 1 (ULS) 2.83E-02

1 0 0 0 0 0 0 0 0

0 1 0.5 0 0 0 0 0 0

0 0.5 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 -0.8 0 0

0 0 0 0 0 -0.8 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

Notes:

For this case, in solver's option, use automatic scaling

Quadratic for Estimates, Central for Derivatives and Newton for Search

Others as default options.

Initially, enter the mean , standard deviation , and correlation matrix [R ] of noise factors and the geometry of drilled shaft (in terms of B

and D ), followed by invoking Excel Solver to automatically minimize reliability index by changing column n , subjected to g 1(d , ) = 0.

Correlation matrix [R ]

Results

n i = 1

[F ( i )]Statistical characterization of noise

factors

Equivalent normal

parameters

Performance function

Design parameters (d )

1

T 1

1( , *) 0

T T

1 1

min [ ] [ ] [ ]

( )

T

g

f

R

P

dn n

See Eq. (8)

48

Figure 8. Layout of a spreadsheet for MFOSM analysis of a drilled shaft based on ULS and SLS

g d , i i g d , i i

cn (kPa/m) 13.04 4.15 37.94 39.54 E [g 1 (d , )] B (m) 0.45

T (m) 0.45 0.05 2033.75 2119.59 329.66 D (m) 10.00

D T (m) 10.00 0.05 102.94 107.28 [g 1 (d , )]

0.00 0.12 921.72 960.62 228.62

Mtc 0.45 0.13 438.53 457.04

1 (ULS) 1.442

a (mm) 2.79 2.04 0.00 -37.81 E [g 2 (d , )]

2 (SLS) 1.481

b 0.82 0.09 0.00 -757.21 359.36 PM

f 1 (ULS) 7.47E-02

F d (kN) 258.00 25.80 -1.00 -1.00 [g 2 (d , )] PM

f 2 (SLS) 6.93E-02

F l (kN) 116.00 20.88 -1.00 -1.00 242.6106849

Covariance Matrix [C ]

17.223 0 0 0 0 0 0 0 0

0 0.003 0.001 0 0 0 0 0 0

0 0.001 0.003 0 0 0 0 0 0

0 0 0 0.014 0 0 0 0 0

0 0 0 0 0.017 0 0 0 0

0 0 0 0 0 4.162 -0.147 0 0

0 0 0 0 0 -0.147 0.008 0 0

0 0 0 0 0 0 0 665.640 0

0 0 0 0 0 0 0 0 435.974

Notes:

The partial derivative of performance function is approximated as g / i =Dg /D i , where D i = 0.1 i

Initially, enter the mean , standard deviation , and covariance matrix [C ] of noise factors and the geometry of drilled shaft (in terms of B and D ),

followed by automatocal computation of the reliability indexes (in terms of

1 and M

2 ).

Partial derivatives of performance

fucntions G

Statistics characterization

of noise factors

Statistics of performance

functions g 1 and g 2

Design parameters (d )

Results

ULS

behavior

SLS

behavior

See Eq. (5)

See Eqs. (2) & (4)

49

(a) Mapping function of reliability index based on ULS

(b) Mapping function of reliability index based on SLS

Figure 9. Mapping from FORM to MFOSM

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

M 1

fro

m M

FO

SM

T

1 from FORM

Eq. (16), R2 = 0.9909

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

M 2

fro

m M

FO

SM

T

2 from FORM

Eq. (17), R2 = 0.9839

50

Figure 10. Layout of a spreadsheet for robust design of a drilled shaft

g d , i i g d , i i

cn (kPa/m) 13.04 4.15 78.34 81.65 E [g 1 (d , )] Cost level C T (m

3) 2.24

T (m) 0.42 0.05 3932.32 4098.30 966.81

T 1 (ULS) 2.03

D T (m) 16.19 0.05 130.78 136.30 [g 1 (d , )]

T 2 (SLS) 1.69

0.00 0.12 2253.12 2348.22 476.26

Mtc 0.45 0.13 640.33 667.36

a (mm) 2.79 2.04 0.00 -72.05 E [g 2 (d , )]

b 0.82 0.09 0.00 -1442.85 1023.40

F d (kN) 258.00 25.80 -1.00 -1.00 [g 2 (d , )] B (m) 0.42

F l (kN) 116.00 20.88 -1.00 -1.00 504.19184 D (m) 16.19

1 (ULS) 2.03

Covariance Matrix [C ]

2 (SLS) 2.03

17.223 0 0 0 0 0 0 0 0

0 0.003 0.001 0 0 0 0 0 0 PM

f 2 (SLS) 2.12E-02

0 0.001 0.003 0 0 0 0 0 0 Cost C (m3) 2.24

0 0 0 0.014 0 0 0 0 0 [g 1 (d , )] 476.26

0 0 0 0 0.017 0 0 0 0

0 0 0 0 0 4.162 -0.147 0 0

0 0 0 0 0 -0.147 0.008 0 0

0 0 0 0 0 0 0 665.640 0

0 0 0 0 0 0 0 0 435.974

Notes:

The partial derivative of performance function is approximated as g / i =Dg /D i , where D i = 0.1 i

For this case, in solver's option, use automatic scaling

Quadratic for Estimates, Central for Derivatives and Newton for Search

Others as default options.

Initially, enter the mean , standard deviation , and covariance matrix [C ] of noise factors , the geometry of drilled shaft (in terms of B and D ), and the cost level C T ,

followed by invoking Excel Solver to automatically minimize the standard deviation of performance fucntion [g 1(d , )] by changing design parameters B and D ,

subjected to Bm, 0.6m], D[10.0m, 25.0m], M

1 > M

T 1, M

2 > M

T 2, and C < C T

Partial derivatives of performance

fucntions G

Statistics characterization

of noise factors

Statistics of performance

functions g 1 and g 2

Robust optimizatin constraint

ULS

behavior

SLS

behavior

Performance

requirements

Most robust design within given cost level

Design

parameters

Objectives

2.12E-02

Product

performance PM

f 1 (ULS)

See Eqs. (2) & (4) M

T Tf

51

Figure 11. Pareto front showing tradeoff between design robustness and cost efficiency

(note: lower [g1(d, )] signals higher design robustness)

2.0 2.2 2.4 2.6 2.8 3.00

200

400

600

800

1000

NSGA-II solution

Robust

nes

s m

easu

re,

[g1(d

,)]

(kN

)

Cost, C (m3)

Design 1

Design 2

(knee point) Design 3

Spreadsheet solution

52

Figure 12. Distribution of ULS performance of three drilled shaft designs

(standard deviation of each and every noise factor is a constant)

-1000 0 1000 2000 3000 4000 50000.0000

0.0003

0.0006

0.0009

0.0012

0.0015

Design 3

Design 2

(knee point)

Pro

bab

ilit

y d

ensi

ty f

un

ctio

n,

PD

F

Performance function of ULS, g1 (kN)

Design 1

53

(a) Gradient of performance function to each noise factor

(b) Effect of each noise factor on the variation of performance function

Figure 13. Sensitivity of performance function with respect to noise factors

-2000

0

2000

4000

6000

8000

baMTc

Fl

DT

BT

Fd

Design 1

Design 2 (knee point)

Design 3

Gra

die

nt

to n

ois

e fa

cto

rs

cn

-200

0

200

400

600

800

baMTc

Fl

DT

BT

Fd

Design 1

Design 2 (knee point)

Design 3

Var

iati

on

fro

m n

ois

e fa

cto

rs

cn

54

Figure 14. Distribution of ULS performance of three drilled shaft designs

(standard deviation of each and every noise factor is a random variable with 20% COV)

-1000 0 1000 2000 3000 4000 50000.0000

0.0003

0.0006

0.0009

0.0012

0.0015

Design 3

Design 2

(knee point)

Pro

bab

ilit

y d

ensi

ty f

un

ctio

n,

PD

F

Performance fucntion of ULS, g1 (kN)

Design 1

55

Figure 15. Robustness of three designs at various levels of construction variation (in terms of

standard deviation of as-built geometry)

0.00 0.05 0.10 0.15 0.200

300

600

900

1200

1500

Robust

nes

s m

easu

re,

[g1(d

,)]

(kN

)

Standard deviation of as-built geometry (m)

Design 1

Design 2 (knee point)

Design 3