Quantile value method versus design value method ...

1

Quantile value method versus design value method for calibration of

2

reliability-based geotechnical codes

3

Jianye Ching1 and Kok-Kwang Phoon2

4 5

ABSTRACT

6

This paper compares two methods for geotechnical reliability code calibration, namely the well

7

known design value method (DVM) based on first-order reliability method and a recently devel-

8

oped method based on quantile, called the quantile value method (QVM). The feasibility of cali-

9

brating a single partial factor to cover the wide range of coefficients of variation (COVs) commonly

10

encountered in geotechnical designs is studied. For analytical tractability, a simple design example

11

consisting of one resistance random variable and one load random variable is first examined. A re-

12

sistance factor is first calibrated using a single calibration case associated with a typical COV. The

13

objective is to evaluate the departure from the target reliability index analytically when this cali-

14

brated resistance factor is applied to validation cases associated with a range of COVs. The results

15

show that QVM is more robust than DVM in terms of achieving a more uniform reliability level

16

over a range of COVs. Two realistic geotechnical design examples are studied to demonstrate that

17

the theoretical insights garnered in the simple analytical example are applicable.

18

Key Words: geotechnical engineering; reliability-based design; quantile; first-order reliability

1

(Corresponding Author) Professor, Dept of Civil Engineering, National Taiwan University, Taipei, Taiwan. Tel: 886-2-33664328. Email: [email protected]. 2 Professor, Dept of Civil and Environmental Engineering, National University of Singapore, Singapore. 1

19

method; design codes

20 21

1. INTRODUCTION

22

The coefficients of variation (COVs) for geotechnical parameters are not constant. Taking the un-

23

drained shear strength (su) of a clay as an example, measurement errors for undrained shear

24

strengths obtained from unconfined compression (UC) tests are typically larger, compared to su ob-

25

tained from more sophisticated tests such as isotropically consolidated undrained compression

26

(CIUC) tests. Spatial averaging over a large volume of soil mass may also significantly reduce the

27

COV in su [1]. The averaging volume is problem dependent. In addition, there are various methods

28

of estimating su. For example, su can be estimated from the preconsolidation stress or from the li-

29

quidity index. Different transformation equations are needed to convert the measured parameter

30

(preconsolidation stress or liquidity index) to the desired design parameter (su). The transformation

31

uncertainties can vary significantly as well [2]. For example, the su versus preconsolidation stress

32

transformation usually is associated with less transformation uncertainty than the su versus liquidity

33

index transformation. Geotechnical models, such as the classical limit equilibrium models for pile

34

capacity, are not exact. Construction effects cannot be readily modeled or quantified to say the least.

35

Construction effects can be significant in geotechnical engineering. Model factors are needed to re-

36

late somewhat idealized calculations with actual measured capacities. It is well established that

37

model factors are random variables, typically lognormally distributed. The mean and COV of a

38

model factor are typically obtained from calibration with field measurements (e.g., pile load test 2

39

database). These statistics may change depending on the database, even for the same problem and

40

the same calculation model.

41

The issue of COVs varying over a wide range is also often encountered in geotechnical design

42

practice, because soil is a natural material and there is a diversity of testing methodologies devel-

43

oped to suit different site conditions. In contrast, concrete and steel are manufactured and testing

44

methodologies are accordingly more standardized. Hence, structural design practice does not need

45

to contend with COVs varying over a wide range. This issue must be dealt with in geotechnical re-

46

liability-based design, although it poses a significant challenge. To elaborate on this challenge, con-

47

sider a simple pile design problem involving two variables, the resistance Q and the load L. Let VQ

48

and VL be the COVs of the resistance and load, respectively. Assume that VL = 0.15 is constant, but

49

VQ is not constant. Let scenario A be a case where a detailed site investigation and extensive load

50

tests have been conducted. As a result, VQ is small and equal to 0.2. Scenario B is a case where the

51

site investigation is cursory and no load test is conducted. As a result, VQ is large and equal to 0.45.

52

It is evident that a set of constant load and resistance factors (or partial factors) cannot maintain a

53

uniform reliability level over these two disparate scenarios. The challenge is obviously non-existent

54

if one adopts a full probabilistic approach, rather than a simplified reliability-based design approach.

55

It is assumed in this paper that the former is not acceptable to practitioners at the present moment,

56

which is indeed the case in the geotechnical engineering community.

57

In this paper, a method named “quantile value method (QVM)” for calibrating partial factors is

58

presented. This method is based on the quantile-based theoretical approach developed in [3]. The 3

59

name QVM is herein selected to differentiate from the more widely known “design value method

60

(DVM)” [4,5] based on the first-order reliability method (FORM) [6]. Both DVM and QVM adopt

61

conservative design locations situated on the limit state line, but DVM adopts the FORM design

62

point, while QVM adopts a design location that has not been explored in literature thus far. In this

63

study, DVM and QVM will be compared using a simple geotechnical pile design involving only

64

two random variables. For this simple example, exact solutions for both DVM and QVM are avail-

65

able, so the comparison can be made analytically and geometric interpretations can be presented

66

visually in the standard Gaussian space. The comparison focuses on the ability to maintain a uni-

67

form reliability level over a wide range of validation design scenarios, such as different COVs, us-

68

ing a single prescribed number (resistance factor or quantile). The analytical comparison will be

69

mostly limited to the case where the prescribed number is calibrated from a single design scenario,

70

but validation would cover a number of design scenarios. Calibration involving multiple design

71

scenarios will be addressed numerically in association with two realistic geotechnical design exam-

72

ples. It will be shown that most of the issues encountered for the realistic examples can be explained

73

by the theoretical insights garnered in the simple analytical example.

74 75

2. ANALYTICAL EXAMPLE

76

The following simple example is adopted to compare DVM and QVM analytically. Consider a pile

77

with axial resistance Q and subjected to axial load L. Q and L are independent and lognormally dis-

78

tributed with mean values (Q, L) and COVs (VQ, VL). The limit state function is defined to be G = 4

79

ln(Q) – ln(L). In the standard Gaussian space,

80

g  z Q , z L    Q  Q z Q   L   L z L

81

where  and  are respectively the mean and standard deviation of the logarithm of the subscripted

82

variable, and (zQ, zL) are jointly standard Gaussian. The safety ratio can be defined as

83

SR  z Q , z L  

84

Whenever SR(zQ, zL) < 1, failure occurs, and vice versa.

  ln 1  V 2 

  ln     0.5  2

Q  exp   Q  Q z Q   L  L z L  L

(1)

(2)

85

Two cases would be considered: a calibration case and a validation case. The mean values and

86

COVs for the calibration case are (Q, L) and COVs (VQ, VL), and those for the validation case are

87

(’Q, ’L) and COVs (V’Q, V’L). Basically, the calibration case will be used to calibrate the partial

88

factors (or load and resistance factors) to achieve a prescribed target reliability index of T. The

89

validation case will be used to examine whether these partial factors indeed produce a design with

90

an actual reliability index ’A that is reasonably close to T.

91

The geometric interpretation is illustrated in Figure 1. For this simple example, the limit state

92

lines are linear in the standard Gaussian space. However, the limit state lines for the calibration case

93

(g = 0) and validation case (g’ = 0) are different and are in general not parallel to each other. The

94

reason is that VQ  V’Q and VL  V’L.

95 96

3. DESIGN VALUE METHOD AND QUANTILE VALUE METHOD

97

Reliability-based design is typically implemented in design codes using a set of partial factors

98

(or load and resistance factors) that achieves the target reliability index T for the calibration case. 5

99

However, this set of numbers is not unique. In the standard Gaussian space, any point on the “ad-

100

justed” limit state line g(z) = Q + QzQ – L – LzL = 0 can be used to derive a set of partial factors.

101

The adjusted limit state line is a limit state line with distance to the origin adjusted to T. The cho-

102

sen point on the adjusted limit state line for evaluation of partial factors will be called the “design

103

location” in this paper. The design location is not necessarily the same as the widely known FORM

104

“design point”: the design location can be anywhere on the limit state line g = 0, and the FORM de-

105

sign point is only a special case. It is the point on g = 0 nearest to the origin.

106

Before choosing a design location on the limit state line g = 0 for the calibration case, the limit

107

state line must be adjusted to a distance of T from the origin to fulfill the target reliability index.

108

This can be done by adjusting one or several of the design parameters {Q, L, VQ, VL} or, equiva-

109

lently, among {Q, L, Q, L}. It is usually impractical to adjust the COVs. The mean resistance can

110

be increased by lengthening the pile and the mean load can be reduced by distributing the column

111

load to multiple piles in a pile group. In the ensuing analysis, L is adjusted to a value equal to

112

Q–T(Q2+L2)0.5. The adjusted design parameter will be called the “pivoting design parameter” in

113

this paper. After the adjustment, the limit state line becomes

114

g  z   Q z Q   L z L  T Q2  2L  0

115

Note that this adjustment is carried out on the calibration case, not on the validation case. There are

116

many possible choices for the set of partial factors for the calibration case. The DVM and QVM are

117

two special cases. Both methods select design locations on the adjusted limit state line but impose

118

some restrictions explained below so that the resulting set is unique.

(3)

6

119

3.1 Design value method

120

The design value method (DVM) chooses the design location to be the FORM design point,

121

which is the point on the adjusted limit state line that is closest to the origin (shown as z* in the left

122

plot of Figure 2). It is also the most probable point on the adjusted limit state line for the calibration

123

case. Direct calculation shows that the FORM design point has the following coordinates:

124

z*Q 

125

The corresponding Q* and L* are

126

T Q   2 Q

2 L

z*L 

T  L

(4)

Q2  L2



Q*  exp   Q  Q z*Q   exp  Q  T Q2



L*  exp   L   L z*L   exp  L  T  2L

Q2   L2 Q2  2L





(5)

127

By definition, the algebraic design equation, Q* – L* = 0, for the calibration case will be associated

128

with a reliability index identical to T. This design equation can be expressed in the alternate form,

129

QQ – LL = 0, where (Q, L) are the resistance and load factors calibrated from the calibration

130

case

131



 Q  exp   Q  Q z*Q  Q  exp  0.5Q2  Q  z*Q   exp 0.5Q2  T Q2



 L  exp   L   L z*L   L  exp  0.52L   L  z*L   exp 0.5 2L  T 2L

Q2   L2 Q2   2L





(6)

132

The equation, QQ – LL = 0, is also called the Load and Resistance Factor Design (LRFD)

133

format. In this LRFD format, the nominal load and resistance are assumed to be equal to their re-

134

spective mean values for simplicity. Although the LRFD format is exact for the calibration case, it is

135

of limited practical interest as the purpose of recommending a design equation in a design code is to 7

136

apply it over a range of commonly encountered design scenarios (intended scope of coverage of the

137

design code). It is rarely mentioned that the LRFD format is assumed to work adequately for other

138

design scenarios without re-calibration of the resistance/load factors.

139

It is of interest to know the actual reliability index, denoted by ’A, implied by these partial

140

factors (exactly applicable only for the calibration case) when they are applied to the validation case.

141

Although it is evident that ’A  T, it is important to know the difference particularly for ’A < T

142

(unconservative design). The same design equation is applied to the validation case. The only dif-

143

ference is that the calibrated factors are applied to the mean resistance (Q) and mean load (L) for

144

the validation case

145

 QQ   LL

146

Or equivalently,

147

0.5Q2  T Q2

or

ln   Q   ln  Q   ln   L   ln  L 

Q2   L2  Q  ln  Q   0.5 L2  T L2 

Q2   L2  ln  L  

Q  0.5 Q2

148

(7)

(8)

 L  0.5 L2

The actual reliability index ’A for the validation case implied by the partial factors (Q, L) is Q  L

0.5  2L  L2   0.5  Q2  Q2   T Q2   L2

149

A 

150

Note that ’A does not depend on the mean values (Q, L) but only on the COVs (VQ, VL) in this

151

example. It is evident from Eq. (9) that ’A = T if the validation case has exactly the same (VQ, VL)

152

as the calibration case, i.e., ’Q = Q and ’L = L. However, if the validation case has very different

153

(V’Q, V’L), ’A may be far from T. The departure is theoretically quantified in Eq. (9). In particular,

154

it is possible for ’A < T.

Q2  L2



Q2  L2

(9)

8

155

Consider a scenario where VL = 0.2 and VQ = 0.3 for the calibration case, and consider V’L =

156

VL and V’Q ranges from 0.1 to 0.5 for the validation case. The target reliability index is T = 3.0

157

(corresponding to a target failure probability pT = 1.310-3). The calibrated partial factors from Eq.

158

(6) are Q = 0.462 and L = 1.367. This means that for the validation case, one needs to assure

159

0.462’Q = 1.367’L in the physical space of Q and L. The actual reliability index ’A under var-

160

ious V’Q is plotted as the thick solid line in the left plot in Figure 3. It is clear that ’A may be as

161

high as 5.0 (actual failure probability p’A = 2.910-7) when V’Q = 0.1 and as low as 2.0 (p’A =

162

2.310-2) when V’Q = 0.5. The uniform reliability level is not achieved by the calibrated partial fac-

163

tors Q = 0.462 and L = 1.367. In particular, unconservative designs could be produced.

164

3.2 Quantile value method

165

The basic idea of the quantile value method (QVM) is to reduce the resistance Q to its  quan-

166

tile ( is small) but to increase the load L to its 1- quantile. The parameter  is called the probabil-

167

ity threshold, and the same threshold  is applied to both random variables: taking  quantiles for

168

stabilizing variables and 1- quantiles for destabilizing variables. In the standard Gaussian space,

169

this is equivalent to selecting the design location z* to be a point on the adjusted limit state line for

170

the calibration case that satisfies zQ = -zL = -1() (shown as z* in the right plot of Figure 2). Ching

171

and Phoon [3] showed that the relation between  and T is as follows:

172

 SR  z Q   1   , z L   1        1    P  T   SR z , z   Q L  

173

This leads to

(10)

9

174

 exp   1      1      2   2          1    Q L T Q L Q L         T   1    P 2 2 2 2              exp z z   Q L Q Q L L T Q L    

175

As a result, the calibrated  is

176

   2   2 T Q L     Q  L 

177

Or equivalently, the design location has the following coordinates in standard normal space:

178

z 

179

The corresponding Q* and L* are

180

181

182





* Q

1

  

(11)

   

(12)

T Q2   2L Q   L

z   * L

1

  



Q*  exp   Q  Q z*Q   exp  Q  T Q Q2   L2



L*  exp   L   L z*L   exp  L  T L Q2   2L

T Q2   2L



(13)

Q   L

Q

 L 

 Q  L 





(14)

and the corresponding resistance and load factors are



 Q  exp   Q  Q z*Q   Q  exp  0.5Q2  Q z*Q   exp 0.5Q2  T Q Q2   L2



 L  exp   L   L z*L   L  exp  0.5 2L   L z*L   exp 0.5 2L  T  L Q2   2L



Q

 L 

 Q   L 





(15)

183

If these partial factors (originally calibrated for the calibration case) are applied to the validation

184

case, the actual reliability index can be derived based the steps similar to Eqs. (7)-(9). Quite sur-

185

prisingly, the actual reliability index ’A for the validation case implied by the calibrated partial

186

factors (Q, L) has the same expression as Eq. (9), i.e. exactly the same as the DVM result, although

187

the partial factors for QVM are different from those for DVM. As a result, a uniform reliability lev-

188

el cannot be achieved by QVM, either.

189 10

190

4. STRATEGIES OF VARIABLE PARTIAL FACTORS

191

Note that the above DVM and QVM results are based on the strategy of applying partial factors

192

calibrated for the calibration case to the validation case. The underlying assumption is that there

193

may be a set of constant partial factors that is suitable for both design cases. Intuitively, this is un-

194

likely to be true because partial factors should be closely related to the uncertainty level, or COVs –

195

the random variables with large COVs should be multiplied by partial factors far away from 1, and

196

vice versa. It has been observed in the Introduction that the COVs for structural materials only vary

197

in a narrow range and it is sensible to apply constant partial factors for structural reliability-based

198

design. On the other hand, the COVs of geo-materials and calculation models can vary over a wide

199

range. There is an emerging realization in the geotechnical reliability literature that the direct appli-

200

cation of LRFD or similar approaches with constant partial factors is overly simplistic. This im-

201

portant distinction between geotechnical and structural reliability has been articulated in Annex D

202

(Reliability of Geotechnical Structures) of ISO2394 (3rd edition), which is currently being drafted.

203

In this section, strategies involving variable partial factors based on DVM and QVM are studied.

204

4.1 DVM with variable partial factors

205

In DVM, instead of applying partial factors, it may be possible to apply the FORM design point

206

computed from the calibration case. In other words, we assume that the design point for the calibra-

207

tion case is applicable to the validation case, rather than assuming that the partial factors for the

208

calibration case is applicable to the validation case. The motivation for this assumption is that the

209

design point is defined in standard normal space and it is hopefully less sensitive to COVs. This ap11

210

proach of applying the FORM design point is denoted by “DVM-z*”, and the standard approach of

211

applying the partial factors is denoted by “DVM”. It will be clarified later that DVM-z* has certain

212

advantages in terms of maintaining a uniform reliability level. For standard DVM, one needs to as-

213

sure that the design constraint Q’Q = L’L is fulfilled for the validation case – this is done in

214

the physical space of Q and L. For DVM-z*, one needs to assure the FORM design point for the

215

calibration case is also on the limit state line for the validation case – this is done in the standard

216

Gaussian space. For DVM-z*, the limit state line for the validation case is forced to pass through the

217

design point in Eq. (4), i.e.,

218

Q  Q z*Q  L  L z*L  Q  L  T

219

As a result, the actual reliability index ’A for the validation case implied by the FORM design

220

point (zQ*, zL*) is

221

A 

222

where the ratio a = L/Q and a’ = ’L/’Q. Note that ’A does not depend on the mean values (Q, L)

223

but only on the COVs (VQ, VL) through the ratios (a, a’), and ’A = T if the validation case has ex-

224

actly the same ratio as the calibration case, i.e., ’L/’Q = L/Q. This is somewhat less strict than

225

standard DVM, which requires ’Q = Q and ’L = L to achieve the prescribed target reliability in-

226

dex.

Q  L Q2  L2



Q Q   L 

Q Q  L L Q2   L2  Q2  L2

Q2   L2

T 

0

1  a  a 1  a 2  1  a 2

(16)

T

(17)

227

Although ’A = T can be achieved over a wider range of scenarios for DVM-z*, a closer ex-

228

amination of Eq. (17) reveals that ’A cannot be greater than T. This implies that implementing the 12

229

FORM design point calibrated for a target level T to another case will always lead to a design with

230

’A no greater than T, i.e., the validation case is always unconservative. This issue will be referred

231

to as the “unconservative design issue” for subsequent discussion. This is because (1+aa’)2  (1+a2)

232

 (1+a’2), following the Cauchy-Swartz Inequality [7]. This phenomenon can be explained geomet-

233

rically in the left plot in Figure 4. Recall that the adjusted limit state line for the calibration case has

234

a distance exactly equal to T to the origin. For DVM-z*, the calibrated design location for the cali-

235

bration case is at the FORM design point (z* in the left plot). Substituting this design location to the

236

validation case is equivalent to enforcing the limit state line g’ = 0 in the plot to pass through z*. It is

237

clear that the distance of from the limit sate line g’ = 0 to the origin cannot be greater than T. This

238

distance is exactly the actual reliability index ’A for the validation case. Simple trigonometric cal-

239

culations show that this distance is indeed (1+aa’)/(1+a2)0.5/ (1+a’2)0.5T, as derived in Eq. (17).

240

Consider the same scenario where VL = 0.2 and VQ = 0.3 for the calibration case, and consider

241

V’L = VL and V’Q ranges from 0.1 to 0.5 for the validation case, and the target is T = 3.0. The cali-

242

brated design point from Eq. (4) is (z*Q, *L) = (-2.487, 1.678). This means that the limit state line

243

for the validation case, namely ’Q + ’Qz’Q – ’L – ’Lz’L = 0, must pass through (-2.487, 1.678),

244

i.e., one needs to comply with the algebraic design equation, ’Q – ’Q2.487 = ’L + ’L1.678 in

245

the standard Gaussian space. Equivalently, one can state the design equation in the physical space of

246

Q and L, Q’Q = L’L, where the partial factors (Q, L) now depend on the COVs for the vali-

247

dation case:

13

248

 Q  exp  Q  Q z*Q  Q  exp  0.5Q2  Q  z*Q   exp  0.5Q2  2.487Q   L  exp   L   L z*L   L  exp  0.5L2  L  z*L   exp  0.5L2  1.678L 

(18)

249

Equation (18) is very similar to Eq. (6) except that ’Q and ’L are used to compute the partial fac-

250

tors. Hence, assuming that the design point is invariant implies that the partial factors would vary

251

with the COVs for the validation case (variable partial factors). The resulting ’A under various V’Q

252

is plotted as the thick solid line in the middle plot in Figure 3. It is clear that ’A is always less than

253

or equal to 3.0. Compared to standard DVM or standard QVM results, ’A from DVM-z* is much

254

more uniform, ranging from 2.6 for V’Q = 0.1 to 2.95 for V’Q = 0.5. Hence, DVM-z* is more ad-

255

vantageous in terms of maintaining a uniform reliability level and assuming the FORM design point

256

to be invariant seems to be a better than assuming the partial factors are invariant.

257

4.3 QVM with variable partial factors

258

The strategy here is exactly the same as for DVM-z*, except that the QVM design location is con-

259

sidered as invariant. Recall that the design location for QVM is not the FORM design point but a

260

point on the 1-to-(-1) line in the standard Gaussian space, as seen in the right plot of Figure 2. This

261

approach of applying the QVM design location is denoted by “QVM-z*”. For QVM-z*, one needs to

262

assure the design location for the calibration case is also on the limit state line for the validation

263

case – this is done in the standard Gaussian space. For QVM-z*, the limit state line for the valida-

264

tion case is forced to pass through the design location in Eq. (13), i.e.,

265

Q   z  L   z  Q  L 

266

As a result, ’A for the validation case implied by the design location (zQ*, zL*) is

* Q Q

* L L

T  Q  L  Q2   L2 Q  L

0

(19)

14

267

A 

Q  L Q2  L2

           Q

Q

L

Q2   L2

L

Q2  L2

T 

1 a 1 a2



1  a 2 T 1  a

(20)

268

The above ’A can also be visualized geometrically as shown in the right plot in Figure 4. Applying

269

the QVM design location to the validation case is equivalent to enforcing the limit state line g’ = 0

270

to pass through z* located on the 1-to-(-1) line. The distance from this g’ = 0 to the origin is exactly

271

the actual reliability index ’A for the validation case. Simple trigonometric calculations show that

272

this distance is indeed (1+a)/(1+a2)0.5(1+a’2)0.5/(1+a’)T, as derived in Eq. (20).

273

Note that ’A does not depend on the mean values (Q, L) but only on the COVs (VQ, VL)

274

through the ratios (a, a’), and ’A = T if the validation case has exactly the same ratio as the cali-

275

bration case, i.e., ’L/’Q = L/Q. More interestingly, ’A = T if a’ = 1/a, i.e., ’Q/’L = L/Q. This

276

is somewhat less strict than DVM-z*, which requires a’ = a to achieve the same target reliability in-

277

dex. This peculiar scenario is shown in the left plot in Figure 5. In essence, if a’ = 1/a, the limit state

278

line for the validation case (g’ = 0) will be a mirror image of that for the calibration case (g = 0)

279

over the 1-to-(-1) line. As a result, ’A = T.

280

In addition to QVM-z* achieving the ideal validation ’A = T under a less strict condition than

281

DVM-z*, the more critical issue of unconservative design that happens in DVM-z* does not exist for

282

QVM-z*. In fact, one can easily show (partly) by the Cauchy-Swartz Inequality that the ratio

283

(1+a)/(1+a)0.5 has a lower bound = 1 and an upper bound =

284

lower bound = T/ 2 and an upper bound = T 2 . The upper bound can happen if the limit state

285

line for the calibration case is a perfectly vertical limit state line (i.e., L = 0), shown as g1 = 0 in the

2 . The consequence is that ’A has a

15

286

right plot in Figure 5, but the validation case has a limit state line inclines at 45o (i.e., Q = L) ,

287

shown as g2 = 0 in the plot. In this special scenario, it is clear that ’A = T 2 . The lower bound

288

can happen in the converse way – if the calibration case has a limit state line inclines at 45o (g2 = 0),

289

but the validation case has a vertical limit state line (g1 = 0). Surprisingly, if the calibration case has

290

a vertical limit state line (g1 = 0) but the validation case has a horizontal limit state line (g3 = 0 in

291

the right plot in Figure 5, i.e., Q = 0), perfect validation ’A = T will occur. For DVM-z*, the

292

above scenario will give ’A = 0!

293

Consider the same scenario where VL = 0.2 and VQ = 0.3 for the calibration case, and consider

294

V’L = VL and V’Q ranges from 0.1 to 0.5 for the validation case, and the target is T = 3.0. The cali-

295

brated design location from Eq. (13) is (z*Q, *L) = (-2.161, 2.161). This means that for the valida-

296

tion case, one needs to comply with the algebraic design equation, ’Q – ’Q2.161 = ’L +

297

’L2.161 in the standard Gaussian space. Equivalently, one can state the design equation in the

298

physical space of Q and L, Q’Q = L’L, where the partial factors (Q, L) now depend on the

299

COVs for the validation case:

300

 Q  exp  Q  Q z*Q  Q  exp  0.5Q2  Q  z*Q   exp  0.5Q2  2.161Q   L  exp   L   L z*L   L  exp  0.5L2  L  z*L   exp  0.5L2  2.161L 

(21)

301

Equation (21) is very similar to Eq. (15) except that ’Q and ’L are used to compute the partial fac-

302

tors. Hence, the partial factors would vary with the COVs for the validation case (variable partial

303

factors). The resulting actual reliability index ’A under various V’Q is plotted as the thick solid line

304

in the right plot in Figure 3. It is clear that ’A is fairly close to 3.0, ranging from 2.85 for V’Q = 0.1 16

305

to 2.8 for V’Q = 0.5. Also, ’A is not always less than 3.0; in fact, ’A = 3.1 reaches its maximum at

306

V’Q = 0.2.

307

4.4 Additional in-depth comparisons

308

The thick solid lines in Figure 3 only compares the robustness of standard DVM (or standard

309

QVM), DVM-z*, and QVM-z* for the case with VQ = 0.3. We say a reliability-based design equa-

310

tion is robust if it achieves the target reliability index for validation cases covering a wide range of

311

design scenarios different from the calibration case. For other VQ values, the results are shown in

312

Figure 3 as the grey lines. The number labels in the figure are the VQ values. It is now clear that

313

QVM-z* performs the best in terms of maintaining ’A to a value that is fairly close to the target

314

value T = 3.0. The standard DVM or QVM performs the worst, indicating that applying constant

315

partial factors calibrated for one case to another different case is not robust. The two strategies

316

DVM-z* and QVM-z* with variable partial factors significantly outperform the standard DVM and

317

QVM. The DVM-z* approach always gives ’A that is no greater than the target value T = 3.0. In

318

some extreme scenarios, e.g., VQ = 0.1 and V’Q = 0.5, DVM-z* performs quite poorly.

319

4.5 Most probable point versus uniform quantiles

320

The main difference between DVM-z* and QVM-z* is that the former uses the most probable point

321

for Q and L on the limit state line, while the latter uses a point with equal exceedance/

322

non-exceedance probability for Q and L, referred to as the point with uniform quantiles in subse-

323

quent discussions. The former uses a design location z* that is sensitive to the gradient of the limit

324

state line, but the latter uses a design location z* that is insensitive to the gradient. The gradient ba17

325

sically quantifies which variable (Q or L) dominates the overall uncertainty.

326

The problem with DVM-z* is that the most probable scenario for the calibration case is un-

327

likely to be the most probable design scenario for the validation case. In fact, the chance that these

328

two scenarios coincide or are very similar is rather small. As a result, it is not sensible to design the

329

validation case based on the most probable point for the calibration case. Unfortunately, under the

330

framework of FORM, the actual reliability index ’A for the validation case will be the same as the

331

target value T only if the most probable point for the calibration case coincides with the most

332

probable point for the validation case. The notion of “most probable point”, although plausible and

333

helpful in reliability analysis is counterproductive in reliability-based code calibration. It also ex-

334

plains why DVM-z* is inferior to QVM-z*.

335

On the contrary, QVM-z* does not need to use a point that is “the most” in any sense. This is

336

because the theory proposed in [3] does not require optimizing any quantity. Moreover, the notion

337

that a conservative design could be produced by reducing Q and increasing L to uniform quantiles

338

makes sense for most cases and it is in line with how an engineer thinks intuitively.

339

Compared to using the FORM design point, using a design location with uniform quantiles has

340

other advantages. In the previous example, only one case with VQ = 0.3 is taken as the calibration

341

case, and the validation case has V’Q ranging from 0.1 to 0.5. If V’Q for the validation cases covers

342

such a wide range, it will make more sense to conduct the calibration based on more than one cali-

343

bration case, e.g., use five calibration cases with VQ = 0.1, 0.2, 0.3, 0.4, and 0.5. Under DVM-z*, it

344

is not straightforward to incorporate five calibration cases. One may find the FORM design points 18

345

for the five cases and, say, take the centroid of those five design points. However, the distance be-

346

tween this centroid to the origin is not even T, so we cannot achieve the desired target value of T

347

even approximately within the calibration domain. Ditlevsen and Madsen [4] proposed a method

348

involving solving an optimization problem for obtaining a unique design location for multiple cali-

349

bration cases. A similar method proposed by [8] for obtaining the optimal partial factors will be

350

discussed in a later section.

351

In contrast, it is very simple to incorporate five calibration cases under QVM-z*: simply find

352

the calibrated  values for the five cases and average them to get ave. The resulting design location

353

is therefore z*Q = -1(ave) and z*L = --1(ave). According to Eq. (12), the calibrated  values for

354

the five calibration cases with VQ = (0.1, 0.2, 0.3, 0.4, 0.5) are (0.0127, 0.0169, 0.0153, 0.0129,

355

0.0110), so the average ave is 0.0138. The resulting design location is therefore z*Q = -1(ave) =

356

-2.203 and z*L = --1(ave) = 2.203. The QVM-z* based on the average ave is therefore to assure the

357

above point is on the limit state line for the validation case, i.e.,

358

Q  2.203Q  L  2.203L  0

359

As a result, the actual reliability index ’A for the validation case is

360

A 

361

The red line in the right plot in Figure 3 shows the resulting actual reliability index ’A under vari-

362

ous V’Q. It is clear that the resulting ’A is fairly close to the target value 3.0.

363

4.6 Why choose the 1-to-(-1) line in QVM-z*?

Q  L Q2  L2



2.203  Q  L  Q2  L2

(22)

(23)

19

364

In QVM-z*, the design location for the calibration case is the intersection of the 1-to-(-1) line

365

and the adjusted limit state line g = 0. The choice of the 1-to-(-1) line seems somewhat arbitrary. In

366

this section, the rationale for this choice will be demonstrated for the simple example. It is possible

367

to use another line with a different angle. Let us consider using a line inclining with angle  to the

368

horizontal zQ axis, i.e., the 1-to-[-tan()] line. Due to the geometric symmetry, one only needs to

369

consider the range of -45o    45o. As a result, the design location should be the intersection of

370

the 1-to-[-tan()] line and the adjusted limit state line g = 0:

371

z 

372

It can be shown that the ’A for the validation case implied by the design location (zQ*, zL*) is

373

A 

374

One can easily show (partly) by the Cauchy-Swartz Inequality that the ratio [1+atan()]/ (1+a)0.5

375

has a lower bound = tan() and an upper bound = 1/cos() (note that -45o    45o). The conse-

376

quence is that ’A has a lower bound = sin()T and an upper bound = T/sin(). The difference

377

between the two bounds is minimized by choosing  = 45o. As a result, the choice for the 1-to-(-1)

378

line not only is intuitively plausible but is also optimal in the sense that the actual reliability index

379

for a validation case has the tightest upper and lower bounds. The above results show that the

380

choice of  is quite important. In the two realistic examples presented later on, it will be seen that as

381

long as the principle “taking  quantiles for stabilizing variables and 1- quantiles for destabilizing

382

variables” is followed, QVM-z* seems to be quite robust with regards to the choice of .

* Q

T Q2   2L

Q   L tan   

1  a tan    1 a2



z  * L

T tan    Q2   2L Q  L tan   

1  a 2 T 1  a  tan   

(24)

(25)

20

383

In the next section, two geotechnical design examples will be presented and analyzed to test the

384

above theoretical insights garnered from a simple analytical example against more realistic exam-

385

ples. It will be shown that the above insights are also applicable to more realistic and complicated

386

geotechnical design examples.

387 388

6. REALISTIC DESIGN EXAMPLES

389

In this section, two geotechnical examples are presented to demonstrate the robustness of var-

390

ious reliability calibration methods, including (a) standard DVM and standard QVM; (b) an ap-

391

proach applied by [8] for finding the optimal constant partial factors; (c) DVM-z*; and (d) QVM-z*.

392

The first example is a pile foundation installed in a layered soil profile, and the second example is a

393

pad foundation supported on boulder clay.

394

The following notations are used in the two examples. Let X be a collection of random varia-

395

bles and  be a collection of design parameters for the calibration case. Let X’ be a collection of

396

random variables and ’ be a collection of design parameters for the validation case. For example,

397

in pile design, X may include the uncertain loading and soil parameters while  may include the de-

398

terministic diameter and depth of the pile. Let Z be the random variables obtained by transforming

399

X to the standard Gaussian space. Let D represents a feasible design domain in the  space, e.g., the

400

diameter is less than 2 m because of the available construction method/equipment and greater than

401

0.6 m because of typical column loadings; the depth must be greater 30 m to reach to bedrock level

402

in a particular site but must be less than 80 m due to the difficulty of maintaining vertical alignment. 21

403

Aside from pile dimensions, there may be other design parameters that are required in the design

404

process, such as the mean values and COVs of some soil parameters, surcharge pressures, and dead

405

load to live load ratio. These design parameters may also fall within a typical range commonly en-

406

countered in practice, e.g., COV for undrained shear strength typically ranges from 0.1 to 0.5; live

407

load to dead load ratio may be between 0.1 and 1.0. For clays, mean undrained shear strengths fall-

408

ing between 25 kN/m2 and 200 kN/m2 would cover soft to very stiff clays. In this paper, the domain

409

D makes precise the concept of coverage of a design code. An ideal reliability-based design code

410

achieves the target reliability index over the entire domain D. For a random variable,  denotes the

411

mean value, and V denotes the COV. For a lognormal random variable X,  is the mean value of

412

ln(X) and  is the standard deviation of ln(X). The following relationships are well known:  =

413

ln() – 0.52, and 2 = ln(1+V2).

414

The central issue examined in this section is the robustness of various code calibration methods.

415

The approach presented previously is adopted here to quantify robustness. First, the design location

416

(e.g., the FORM design point) for a calibration case satisfying a target reliability index = T is de-

417

termined. The partial factors are determined once the design location is known. Second, these par-

418

tial factors are applied to a validation case to find the actual reliability index ’A. For standard

419

DVM, standard QVM, and DVM-z*, it is not easy to incorporate multiple calibration cases. For

420

these methods, a single case randomly sampled from the design domain D will be used for calibra-

421

tion. An additional nv cases (j = 1, …, nv) are independently and randomly sampled from D for val-

422

idation. The entire exercise is repeated nc times (i = 1, …, nc) by selecting a new calibration case 22

423

randomly each time. By doing so, we hope to minimize unintended bias produced by calibration

424

and/or validation cases that would either favor or disfavor a specific reliability calibration method.

425

It is evident that there are nc  nv values of ’A – {’A,ij: i = 1, …, nc, j = 1, …, nv}. The reliability

426

calibration methods are considered robust if these ’A values are very close to T. The validation

427

results based on a single calibration case will be denoted as “cal-1”.

428

As mentioned earlier, QVM-z* can easily incorporate multiple calibration cases. In this case, a

429

set of n cases randomly sampled from D will be used for calibration, and an additional nv cases (j =

430

1, …, nv) are independently and randomly sampled from D for validation. The entire exercise is re-

431

peated nc times (i = 1, …, nc). It is evident that there are nc  nv values of ’A – {’A,ij: i = 1, …, nc, j

432

= 1, …, nv}. The reliability calibration methods are considered robust if these ’A values are very

433

close to T. These validation results based on n calibration cases will be denoted as “cal-n”.

434

6.1 Example 1: Pile foundation installed in a layered soil profile

435

Consider the pile foundation installed in a clay layer overlying a sand layer. For a deep founda-

436

tion, it is quite common to encounter multiple soil layers rather than one soil layer. It is reasonable

437

to consider a simple two-layer soil profile in this initial study. The shaft diameter B = 1 m, and the

438

pile is loaded by dead load (DL) and live load (LL). The loadings DL and LL are random and

439

lognormally distributed with mean values [DL, LL] and COVs [VDL = 10%, VLL = 20%]. The

440

mean live to dead load ratio rL/D = LL/DL.

441

The shaft resistance is provided by the sand layer and the clay layer. In the clay layer, the re-

442

sistance is (su,)suLcB (in kN/m2), where su (in kN/m2) is the undrained shear strength of the 23

443

clay, Lc (in meters) is the shaft length supported by clay, (su,) = 14.9su-0.7 is the adhesion

444

factor, and  is the transformation uncertainty term with median = 1 and COV = 32% [9]. Note that

445

 can also be viewed as a model factor for the calculation of shaft resistance in the undrained mode.

446

In the sand layer, the resistance is 2.71NNLsB (in kN/m2), where Ls (in meters) is the shaft

447

length supported in sand, N is the average SPT-N value of the sand layer, 2.71 is an empirical coef-

448

ficient for the correlation, and N is the transformation uncertainty term with median = 1 and COV =

449

70% [9]. Hence, the limit state function is

450

g  X,    14.9   s 0.3 u L c B  2.71 N  NL s B  DL  LL

451

and the safety ratio is given by:

452

SR  X,   

453

It is assumed herein that tip resistance can be neglected, because the shaft is sufficiently long. Ran-

454

dom variables X contain {su, , N, N, DL, LL}, and their distribution models are summarized in

455

Table 1. There are seven design parameters in : su, Vsu, N, DL, rL/D, total shaft length L = Lc+Ls,

456

normalized thickness of clay layer tc = Lc/L.

14.9   s 0.3 u L c B  2.71 N  NL s B DL  LL

(26)

(27)

457

One of the goals of this example is to furnish the details underlying the calibration of the par-

458

tial factors for standard DVM and standard QVM as well as to furnish the details underlying the

459

calibration of the design locations for DVM-z* and QVM-z*. The same details have been given for

460

the simple analytical example in which closed-form solutions for the calibration are available. For

461

the more realistic examples, no comparable closed-form solutions are available. Therefore, it is

462

helpful to illustrate the calibration steps. It suffices to present the calibration steps for Example 1 24

463

only. Similar calibration steps are followed in Example 2.

464

Calibration and validation based on a single case (cal-1)

465

Consider a single calibration case with the following design parameters: B = 1 m, L = 45 m,

466

rL/D = 0.5, su = 125 kN/m2, Vsu = 0.3, N = 30, and tc = 0.2. The target reliability index T is chosen

467

to be 3.0. For QVM, the following decision is made regarding the stabilizing/destabilizing random

468

variables: {su, , N, N} are stabilizing, so they will be reduced to their  quantiles; {DL, LL} are

469

destabilizing, so they will be increased to their 1- quantiles. Recall that stabilizing variables are

470

those that increase the safety ratio when they increase while destabilizing variables produce the op-

471

posite effect. Also consider a single validation case with the following design parameters: B’ = 1 m,

472

L’ = 60 m, r’L/D = 0.4, ’su = 80 kN/m2, V’su = 0.1, ’N = 30, and t’c = 0.8.

473

Step 1: Obtain the adjusted limit state line for the calibration case

474

As discussed in the simple example, it is necessary to first adjust the limit state line for the calibra-

475

tion case so that the adjusted limit state achieved a reliability index = T. This can be easily done by

476

adjusting one of the design parameters, called the pivoting design parameter, and by checking the

477

resulting reliability index after each adjustment by any reliability analysis method. The pivoting de-

478

sign parameter can be any design parameter that can be adjusted as long as the adjusted value is not

479

unrealistic, e.g., outside domain D. For Example 1, the mean value of the dead load DL is chosen to

480

be the pivoting design parameter, and a simple line search can be undertaken to find the DL value

481

producing a reliability index = T.

482

For the example at hand, an initial trial value of DL = 1500 kN is assumed. Reliability analysis 25

483

based on Monte Carlo simulation (MCS) is carried out. The reliability index is found to be 3.04,

484

which is larger than the target value 3.0, indicating DL could be increased. Several trials show that

485

DL = 1526.7 kN achieves the target of T = 3.0. Instead of MCS, FORM can also be applied to

486

evaluate the reliability index, and several trials show that DL = 1486 kN achieves the target. The

487

minor difference is due to the slight difference between MCS and FORM solutions. To be consistent,

488

for FORM-based calibration methods (standard DVM and DVM-z*), FORM will be adopted to ad-

489

just the value of DL. For MCS-based standard QVM and QVM-z*, MCS will be adopted to adjust

490

the value of DL.

491

Step 2: Find the design location

492

Once the DL value for the calibration case is determined, design locations can be readily found. For

493

standard DVM and DVM-z*, the FORM design point is the design location. By inserting DL =

494

1486 kN into Eq. (26) and converting it into standard Gaussian space, we obtain: g  Z,    14.9 exp    z    exp  0.3 su  0.3su  z su  Lc B

495

 2.71exp  N  z N   exp  0.3 N  0.3 N  z N  Ls B

(28)

 exp   DL   DL  z DL   exp   LL   LL  z LL  496

where the z’s are jointly standard Gaussian, and ’s and ’s are the means and standard deviations

497

for the corresponding log variables, e.g., su = ln(1+0.32)0.5 and su = ln(su)-0.52su. The FORM de-

498

sign point is found to be z*su = -0.317, z* = -1.152, z*N = -1.032, z*N = -2.461, z*DL = 0.468, and

499

z*LL = 0.480.

500

For standard QVM and QVM-z*, the design location satisfies z*su = z* = z*N = z*N = -z*DL =

501

-z*LL = -1(), where  is the cumulative density function (CDF) of standard Gaussian. The  value 26

502

can be found by solving [3]:

503

 14.9   s 0.3 L B  2.71  N  L B DL1  LL1     SR  X  ,      u c N s  P  1  P   1    T  0.3  SR  X,    14.9 s L B 2.71 N L B DL LL             u c N s       

504

where X (or X1-) denotes the  (or 1-) quantile of X. For lognormal random variables, X =

505

exp[+-1()]. By inserting DL = 1526.7 kN into Eq. (29) and carrying out another simple line

506

search for , the corresponding  value is found to be 0.0589. Therefore, the design location for

507

QVM is z*su = z* = z*N = z*N = -1() = -1.564 and z*DL = z*LL = 1.564.

508

Step 3: Determine the partial factors

509

For standard DVM and standard QVM, the following equations applicable to lognormal random

510

variables can be used to obtain the constant partial factors [see Eqs. (6) and (15)]:

511

X 

512

Note that all parameters in Eq. (30) refer to the calibration case (unprimed parameters). The result-

513

ing partial factors for standard DVM are su = 0.873,  = 0.692, N = 0.707, N = 0.179, DL =

514

1.043, and LL = 1.079, and those for standard QVM are su = 0.605,  = 0.606, N = 0.605, N =

515

0.335, DL = 1.163, and LL = 1.337. These partial factors calibrated using a single calibration case

516

will be applied to the validation case. Note that the partial factors are numerically different. Differ-

517

ent sets of partial factors can achieve the same target reliability index, although this is rarely em-

518

phasized in the literature.



X*  exp  0.5 X2   X  z*X  X



(29)

(30)

519

For DVM-z* and QVM-z*, the calibrated design locations rather than the partial factors are ap-

520

plied to the validation case. The following equations can be used to obtain the partial factors [see 27

521

Eqs. (18) and (21)]:

522

 X  exp  0.5X2  X  z*X 

523

Note that the ’ parameters in Eq. (31) are now for the validation case (primed parameters). It is

524

clear that the partial factors are not constant, but they change according to the COVs associated

525

with the validation case. For the validation case noted above, the resulting partial factors for

526

DVM-z* are su = 0.964,  = 0.692, N = 0.707, N = 0.179, DL = 1.043, and LL = 1.079, and

527

those for QVM-z* are su = 0.851,  = 0.606, N = 0.605, N = 0.335, DL = 1.163, and LL = 1.337.

528

For all methods (standard DVM, standard QVM, DVM-z*, and QVM-z*), the design equation

(31)

529

for the validation case has the same format:

530

  LLDL 14.9      su su  Lc B  2.71 N   N N Ls B   DLDL  rL/D

531

If the design parameters {’su, ’N, ’DL, r’L/D, L’, t’c} for the validation case satisfy Eq. (32), the

532

expectation is that the actual reliability index ’A = T = 3.0, although this expectation may or may

533

not be fulfilled depending on the robustness of the calibration method.

534

Step 4: Determine the actual reliability index ’A

535

There is certainly no theoretical guarantee that ’A = T = 3.0 for the validation case, and verifica-

536

tion can be carried out to find the actual reliability index ’A. Recall that the validation case has the

537

following design parameters: B’ = 1 m, L’ = 60 m, r’L/D = 0.4, ’su = 80 kN/m2, V’su = 0.1, ’N = 30,

538

and t’c = 0.8 (so that L’c = 12 m and L’s = 48 m). If the pivoting design parameter for the validation

539

case ’DL is at its “limit value”:

0.3

(32)

28

14.9     su su  Lc B  2.71 N   N N Ls B   LL  DL  rL/D 0.3

540

DL 

541

the corresponding design is expected to achieve an actual reliability index ’A = T = 3.0. Consider

542

the standard QVM as an example and recall that the partial factors are su = 0.605,  = 0.606, N =

543

0.605, N = 0.335, DL = 1.163, and LL = 1.337. As a result, the limit value for the pivoting design

544

parameters is

545

14.9  0.606   0.605  80  12 1  2.71 0.335  0.605  30 48 1 DL   2931.7kN 1.163  0.4 1.337

546

A reliability analysis based on MCS is carried out to determine the actual reliability index ’A under

547

the limit value ’DL = 2931.7 kN. The actual reliability index ’A is found to be 3.34. The same ver-

548

ification is carried out for the standard DVM, DVM-z*, and QVM-z*, and the resulting ’A values

549

are respectively 2.33, 2.24, and 3.05.

550

Results of the validation

551

The above detailed steps explain the reliability calibration procedure for a single calibration case

552

and a single separate validation case. In this section, the robustness of the standard DVM, standard

553

QVM, DVM-z*, and QVM-z* is evaluated in a more systematic way. The design domain D is a

554

rectangular volume formed by the ranges of typical values summarized in Table 2. The range for

555

DL is not specified as this particular design parameter is taken to be the pivoting design parameter,

556

as discussed earlier. The value of Vsu is fixed at 0.3, because Example 1 is not intended to illustrate

557

the issue associated with widely varying COVs.

(33)

0.3

558

(34)

For standard DVM, standard QVM, and DVM-z*, “cal-1” study is conducted using nc = 20 and 29

559

nv = 100 – partial factors/design locations are established using a single calibration case, these fac-

560

tors/locations are validated using 100 cases, and the entire process is repeated 20 times to reduce

561

possible biases from the selected calibration and/or validation cases. It is evident that there are

562

20100 ’A values. The mean, COV, minimum, and maximum of these values will be reported and

563

compared. The reliability calibrated design equation performs perfectly if mean = min = max = 3.0

564

and COV = 0.

565

For QVM-z*, both “cal-1” and “cal-20” studies are carried out with nc = 20 and nv = 100. For

566

the “cal-20” study, partial factors/design locations are established using 20 cases, rather than 1 case.

567

The rest of the procedure is the same as “cal-1”. Hence, there are also 20100 ’A values.

568

Besides QVM-z*, an optimization approach proposed by [8] can also incorporate multiple cal-

569

ibration cases. Hence, it forms part of the “cal-20” study. The basic idea is to find the optimal con-

570

stant partial factors by solving the following optimization problem:

571

min

 su ,   ,  N ,  N ,  DL ,  LL

20

    i 1

A,i

su

,   ,  N ,  N ,  DL ,  LL   3.0 

2

(35)

572

where A,i is the actual reliability index for the i-th calibration case. Note that the actual reliability

573

index A depends on the partial factors because A depends on the limit value of the pivoting design

574

parameter DL, which in turn depends on the partial factors [refer to Eq. (33)]. The optimal partial

575

factors are basically the ones that minimize the sum of square of the discrepancy from the target re-

576

liability index of 3.0. One can treat the performance of these optimized partial factors as the best

577

performance that can be achieved by the strategy of constant partial factors.

578

Table 3 shows the validation results for the aforementioned methods. Note that standard DVM 30

579

and DVM-z* are the same because COVs for all random variables are constant for Example 1. The

580

same reason applies for the identical performance between standard QVM and QVM-z*-cal-1. It is

581

clear that both DVM-based methods perform poorly, with mean values of ’A significantly less than

582

3.0 and the minimum value of ’A close to zero. At a cursory glance, this result contradicts the ob-

583

servation made in the simple example: standard DVM should perform well if ’resistance = resistance

584

and ’load = load. For Example 1, ’load is indeed the same as load, but ’resistance is in fact not the

585

same as resistance even though V’su = Vsu. Consider a calibration case with the normalized thickness

586

of clay layer tc = 0.1 and a validation case with tc’ = 0.9. Sand dominates the calibration case so that

587

resistance is large, but clay dominates the validation case so that ’resistance is small. As a result, stand-

588

ard DVM performs poorly in this example even though this example has not explored the more

589

typical situation where COVs of geotechnical parameters can vary over a wide range.

590

DVM-z* performs poorly for this example because it is plagued by the issue of unconservative

591

design. This can be understood by referring to Table 4, where the resulting FORM design points are

592

shown for three selected design scenarios. Basically, all design parameters for the three scenarios

593

are fixed at the center of D, except that tc takes three different values: 0.1, 0.5, and 0.9. It is clear

594

that for the two cases tc = 0.1 and 0.9, the FORM design points are very different. In particular, it is

595

correct that z*su and z* are closer to zero and z*N and z*N are far from zero for tc = 0.1, because the

596

total resistance is insensitive to su and  and is sensitive to N and N. The converse is true for tc =

597

0.9. If the case with tc = 0.1 is selected as the calibration case and the case with tc = 0.9 is selected

598

as the validation case, the resulting ’A will be far less than 3.0, because the direction cosines for 31

599

the FORM design points of these two cases are very different. Note that the direction cosines at the

600

design point are simply the values shown in Table 4 divided by the reliability index, which is equal

601

to 3.0 in the examples. In fact, the direction cosines for the FORM design point of the middle case

602

with tc = 0.5 are also obviously different from those for the cases with tc = 0.1 and 0.9.

603

As a result, even though Example 1 does not involve design scenarios covering a wide range of

604

COVs for the input parameters, design scenarios with different tc values can impact DVM-based

605

methods in two negative ways: (a) it results in ’resistance  resistance, which degrades the performance

606

of standard DVM; and (b) it results in FORM design points with very different direction cosines,

607

which degrades the performance of DVM-z*. The above conclusions are in line with the theoretical

608

observations given in the simple example.

609

When tc varies from 0 to 1, all QVM-based methods perform reasonably well, as seen in Table

610

3. The satisfactory performance of QVM-z* can be understood by referring to Table 4 – the cali-

611

brated  values for the three design scenarios are similar. As a result, the QVM-z* strategy of ap-

612

plying the  value calibrated for one case to another works well. The QVM-based methods do not

613

seem to be affected by the different direction cosine issue, which is inherent in all design codes that

614

attempt to cover a reasonably diverse group of design scenarios. The standard QVM performs well

615

simply because standard QVM is equivalent to QVM-z* for Example 1 under the critical assump-

616

tion of Vsu = 0.3. Note that in the simple example, the performances of the standard QVM and the

617

standard DVM are the same, but this is obviously not the case for Example 1. As a result, one can

618

conclude that standard QVM and standard DVM are in general different, although they may behave 32

619

exactly the same way for some special cases. Among the QVM-based methods, QVM-z*-cal-20

620

performs the best. The performance of the optimization approach proposed by [9] is comparable to

621

QVM-z*-cal-20. This shows the benefit of adopting more than one case for calibration, which is ra-

622

ther commonsensical.

623

6.2 Example 2: Pad foundation supported on boulder clay

624

The pad foundation example adopted herein was originally developed by the European Tech-

625

nical Committee 10 [10] of the International Society of Soil Mechanics and Geotechnical Engi-

626

neering. The square pad foundation has size of BB and is embedded at a depth of 0.8 m. It is re-

627

quired to support the following loads: (a) vertical dead load DL, excluding weight of foundation;

628

and (b) live load LL inclined at an angle of  degrees to the vertical. The live load is assumed to act

629

at the base of the footing. Hence, there is no load eccentricity. Both loads are assumed to be inde-

630

pendent of each other. The soil consists of boulder clay, with a bulk unit weight of 21.4 kN/m3. The

631

concrete unit weight γc is 25 kN/m3. The ground water table is assumed to be deeper than B below

632

the depth of embedment. Hence, it has no influence on the bearing capacity of the foundation.

633

This study focuses on the ultimate limit state (ULS) requirement, i.e. the total vertical load

634

(sum of dead load and the vertical component of the live load) cannot exceed the bearing capacity

635

qu multiplied by foundation area B2. In this study, we adopt the equation recommended in Eurocode

636

7 (Eq. (D1) in page 157 in BS EN 1997-1:2004 [11]) for calculating qu:

637

q u     2  s u  sc  ic  q

638

where sc = 1.2 for the square pad foundation, ic = 0.5[1+(1-[LLsin()]/B2/su)0.5], and q = total sur-

(36)

33

639

charge pressure = 0.821.4 = 17.12 kN/m2. The limit state function can be written as

640

g  X,    q u B2  DL  LL cos     Wp

641

where Wp = cB20.8 = 20B2 kN is the weight of the pad foundation. There are four random varia-

642

bles {su, DL, LL, }. Their assumed probability distributions and the statistics are summarized in

643

Table 5. There are five design parameters: B, su, Vsu, DL, and rL/D (mean live load to mean dead

644

load ratio).

(37)

645

For QVM-based methods, the following decision is made regarding the stabiliz-

646

ing/destabilizing random variables: su is stabilizing, so it is reduced to its  quantile; {DL, LL} are

647

destabilizing, so they are increased to their 1- quantiles. Note that  is neither completely stabiliz-

648

ing nor completely destabilizing. This is because when  is small, the vertical component of LL be-

649

comes larger – this seems to be destabilizing, but the inclination factor ic increases – this seems to

650

be stabilizing. As a result, three QVM scenarios are considered: (a)  is assumed to be stabilizing,

651

so it is reduced to its  quantile (case S); (a)  is assumed to be destabilizing, so it is increased to its

652

1- quantile (case D); and (a)  is assumed to be neutral and fixed at the average value of 10o (case

653

N).

654

The limit state line is adjusted based on the mean value of the dead load DL, i.e. DL is chosen

655

to be the pivoting design parameter. The design domain D is the rectangular volume formed by the

656

ranges summarized in Table 6. The range for DL is not specified as it is taken to be the pivoting de-

657

sign parameter. Note that the value of Vsu varies between 0.2 and 0.5. As a result, Example 2 in-

658

volves design scenarios with varying COVs for the input parameters. This is a critical issue for cal34

659

ibration of geotechnical reliability-based design code as explained below. The configurations of the

660

calibration and validation exercises are the same as those in Example 1.

661

Table 7 shows the validation results for Example 2. It is clear that all methods based on con-

662

stant partial factors, including standard DVM, standard QVM, and the optimization method, per-

663

form poorly. To be specific, the COVs of the actual reliability indices produced by the constant par-

664

tial factor design format are significantly larger than those for DVM-z* and QVM-z*, regardless of

665

the numerical values of the partial factors. This is consistent to the theoretical insights garnered

666

from the simple analytical example. The DVM-z* performs well with a very small COV of ’A,

667

while the mean value of ’A is slightly less than the target value. This is because the direction co-

668

sines for the FORM design points are not very different under different design scenarios. Recall that

669

for Example 1, DVM-z* performs poorly because the direction cosines of the FORM design points

670

can become very different for different choices of tc. In contrast, for Example 2 this issue does not

671

exist – su always dominates the overall uncertainty because the uncertainty level for DL+LL is al-

672

ways smaller than that for su, and mode switching does not happen, hence DVM-z* performs well.

673

“Mode switching” refers to the situation where the dominant random variable (one that dominates

674

the overall uncertainty of the response) changes with design scenarios.

675

According to Table 7, all QVM-z* methods perform well. The fact that the QVM-z*-cal-20 re-

676

sults have ’A COVs slightly less than those for QVM-z*-cal-1 shows the benefit of adopting more

677

than one case for calibration. Cases S, D, and N perform equally well, indicating that the decision of

678

stabilizing/destabilizing over variable  is not critical in this problem, probably because this varia35

679

ble is neither completely stabilizing nor completely destabilizing.

680

Illustrative design calculations for QVM-z*

681

It is useful to examine the practicality of applying QVM-z* for design, which has been shown

682

to perform the best among the methods studied in this paper. Suppose that the calibrated  value is

683

0.013 and that the design problem is to propose a footing width (B) for the following scenario: su =

684

150 kN/m2, Vsu = 0.3, DL = 1500 kN, rL/D = 0.3. The required partial factors are: 1

1    1  0.1  1

 DL

DL1  DL   DL VDL     DL  DL

 LL

1 LL1  LL   LL VLL 0.5774 6  + 6  ln   ln 1     1-     LL  LL

 0.0987   1.22  Gaussian 









=1  0.2  0.45  0.78ln   ln  0.0987    1.59  Gumbel 

685  su 



exp ln su 



(38)

1  Vsu2  ln 1  Vsu2    1     su

 exp  0.5ln 1  0.32   ln 1  0.32    1  0.013   0.50  lognormal       1    0.987  uniform 

686

Based on these partial factors, one needs to find B that satisfies the following equation:

687

q u B2   DL DL   LL LL cos     20o   20B2   LL LL sin     20o     i c  0.5 1  1    B2  su su  

(39)

688

q u     2   su su 1.2  i c  17.12

689

Simple trial-and-error produces a feasible B = 2.53 m. The actual reliability index for this design is

690

estimated to be 3.25 based on Monte Carlo simulation.

(40)

691

In general, it is evident that the theoretical insights garnered from the simple analytical example

692

are applicable to more realistic foundation examples. It is worthy to note these examples involving

693

piles in layered soils and pad foundations under inclined loadings are commonly encountered in 36

694

foundation engineering practice. They are not contrived counter-examples to highlight limitations of

695

any reliability calibration methods.

696 697

8. DISCUSSIONS AND CONCLUSIONS

698

Based on the results observed from a simple analytical example and two relatively more realistic

699

foundation examples, several useful conclusions can be drawn from this study:

700

1. Methods based on constant partial factors cannot achieve a uniform reliability level if the COVs

701

of some random variables in actual design scenarios are different from those in the calibration

702

scenarios. Such methods include standard DVM, standard QVM, and even the optimization ap-

703

proach proposed by [8]. Furthermore, even if the COVs of basic soil parameters are assumed to

704

be constant within the scope of the design code, subtle problems such as the relative thickness

705

of clay versus sand in a layered soil profile in Example 1 can produce varying COVs for the

706

overall side resistance. For design problems with varying COVs in the basic soil parameters or

707

overall resistance, it is not robust to apply partial factors calibrated for one case to another case.

708

The resulting reliability index for the latter case may be far from the target one. The implication

709

is that a design code with constant partial factors (or constant load and resistance factors) may

710

not be robust for realistic geotechnical designs, because COVs of geo-materials indeed vary

711

over a wide range and layered soil profiles are common for deep foundations. This issue may

712

not be critical for a structural design code because the COVs of structural materials typically fall

713

between 5% and 20%, but it is certainly critical for a geotechnical design code. 37

714

2. The DVM-z* and QVM-z* prove to be a more robust strategy than those based on constant par-

715

tial factors. With these two methods, the partial factors depend on the COVs of the geotechnical

716

parameters. However, DVM-z* suffers from a rather undesirable shortcoming of producing con-

717

sistently unconservative design. This shortcoming is related to the principle of FORM – the

718

FORM design point is the most probable point on the limit state line. The FORM design point

719

calibrated for one case is suitable for another case only if this point is also the most probable

720

point for the latter case. However, this is quite unlikely to happen. The QVM-z* does not have

721

this shortcoming. This unconservative issue for DVM-z* will be serious whenever the dominant

722

random variable (one that dominates the overall uncertainty of the response) changes with de-

723

sign scenarios. This phenomenon is called “mode switching” in this paper.

724

3. The QVM-z* is based on the idea of uniform quantiles – reducing stabilizing random variables

725

to their  quantiles and increasing destabilizing random variables to their 1- quantiles. More-

726

over, QVM-z* seems to be unaffected by the issue of variable direction cosines for the FORM

727

design point, which occurs when there is mode switching. This is a critical advantage, because

728

mode switching is embedded in the physical model of the problem. It is difficult to judge from

729

casual observation if mode switching exists in a particular problem and hence, if DVM-z* is ap-

730

plicable. The performance of QVM-z* does not seem to be sensitive to the decision of selecting

731

stabilizing and destabilizing variables, as long as the selection is not absurd. It is more robust to

732

adopt more than one design scenario in the calibration domain.

733 38

734

ACKNOWLEDGEMENTS

735

The name “quantile value method” (QVM) was suggested by Professor Yusuke Honjo of Gifu Uni-

736

versity. We would like to express our appreciation for his suggestion.

737 738

REFERENCES

739

[1] Vanmarcke, E.H. (1977). Probabilistic modeling of soil profiles. ASCE Journal of Geotechnical

740 741 742 743 744 745 746

Engineering Division, 103(11), 1227-1246. [2] Phoon, K. K. (1995). Reliability-based design of foundations for transmission line structures. Ph.D. Dissertation, Cornell University, Ithaca, NY.

[3] Ching, J. and Phoon, K.K. (2011). A quantile-based approach for calibrating reliability-based partial factors, Structural Safety, 33, 275-285. [4] Ditlevsen, O. and Madsen, H. (1996). Structural Reliability Methods. J. Wiley and Sons, Chichester.

747

[5] Honjo, Y., Amatya, S., Suzuki, M., and Shirato, M. (2002). Determination of partial factors for

748

vertically loaded piles for a seismic loading condition based on reliability theory. Workshop on

749

Reliability Based Code Calibration, Swiss Federal Institute of Technology, ETH Zurich, Swit-

750

zerland, March 21-22, 2002.

751 752 753

[6] Hasofer, A.M. and Lind, N.C. (1974). Exact and invariant second-moment code format. ASCE Journal of Engineering Mechanics, 100(1), 111-121. [7] Steele, J. M. (2004). The Cauchy-Schwarz Master Class. Cambridge University Press, Cam39

754

bridge.

755

[8] Sørensen, J. D. (2002). Calibration of partial safety factors in Danish Structural Codes. Work-

756

shop on Reliability Based Code Calibration, Swiss Federal Institute of Technology, ETH Zurich,

757

Switzerland, March 21-22, 2002.

758 759 760 761 762 763

[9] Chen, J.R., Ching, J., Chuang, B.Y., Yang, Z.Y., and Shiau, J.Q. (2013). “Axial behavior of dilled shafts in multiple strata – Taipei database”, Soils and Foundations (in review). [10] ETC 10. Geotechnical design ETC10 Design Example 2.2. Pad foundation with inclined eccentric load on boulder. http://www.eurocode7.com/etc10/Example%202.2/index.html. [11] British Standards Institute (2004). Eurocode 7: Geotechnical design – Part 1: General Rules. BS EN 1997-1:2004, London.

764 765

Table 1 Probability distributions for the random variables in Example 1. Variable DL LL su N  N

Description Dead load Live load Undrained shear strength Average SPT-N value of sand Model factor for clay Model factor for sand

Distribution Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

Statistics Mean = DL, COV = 10% Mean = rL/DDL, COV = 20% Mean = su, COV = Vsu Mean = N, COV = 30% Median = 1, COV = 32% Median = 1, COV = 70%

766 767

Table 2 Feasible ranges for the design parameters of Example 1. Parameter L rL/D DL su Vsu N tc

Description Pile length = Lc + Ls Mean live load/mean dead load = LL/DL Mean value of DL Mean value of su COV of su Mean value of N tc = Lc/L

Range or value [10, 80] m [0.1, 1.0] [50, 200] kN/m2 0.3 [10, 50] [0, 1]

768 40

769

Table 3 Validation results for Example 1. Calibration methods

Cal-1

Cal-20

Standard DVM /DVM-z*

Standard QVM /QVM-z*

QVM-z*

Sorenson

Mean of ’A

2.43

3.09

3.01

3.00

COV of ’A

0.29

0.17

0.10

0.11

Max. ’A

3.11

4.47

3.45

3.73

Min. ’A

0.22

2.02

2.18

1.65

770 771

Table 4 Calibrated FORM design points and  values for three design scenarios of Example 1.

z su z* z*N z*N z*DL z*LL

tc = 0.1 -0.20 -0.73 -1.10 -2.63 0.40 0.40

tc = 0.5 -0.52 -1.88 -0.82 -1.96 0.57 0.59

tc = 0.9 -0.71 -2.59 -0.38 -0.91 0.61 0.64



0.043

0.085

0.057

*

772

Table 5 Probability distributions for the random variables in Example 2. Variable DL LL su 

Description Vertical dead load Inclined live load Undrained shear strength Inclined angle of LL

Distribution Gaussian Gumbel Lognormal Uniform

Statistics Mean = DL, COV = 10% Mean = rL/DDL, COV = 20% Mean = su, COV = Vsu Min. = 0o, Max. = 20o

773 774

Table 6 Feasible ranges for the design parameters of Example 2. Parameter B DL su Vsu rL/D

Description Foundation width Mean value of DL Mean value of su COV of su Mean live load to mean dead load ratio

Range [2, 4] m [50, 200] kN/m2 [0.2, 0.5] [0.1, 1.0]

775 776

Table 7 Validation results for Example 2. Calibration

Cal-1

Cal-20 41

Standard QVM/QVM-z* S D N

QVM-z* S D N

methods

Standard DVM/ DVM-z*

Mean of ’A

3.05/2.96

3.06/2.99

3.07/3.00

3.07/2.99

3.01

3.02

3.01

2.90

COV of ’A

0.83/0.03

0.85/0.02

0.85/0.08

0.84/0.03

0.03

0.05

0.02

0.58

Max. ’A

4.75/3.12

4.75/3.46

4.75/3.76

4.75/3.45

3.35

3.51

3.34

4.75

Min. ’A

1.22/2.48

1.20/2.58

1.20/2.29

1.23/2.52

2.81

2.44

2.68

1.25

Sorenson

777 778

Figure 1 Limit state lines of the calibration case (g = 0) and validation case (g’ = 0).

779 780

Figure 2 Design locations z* for the calibration case for DVM (left) and QVM (right) (a = L/Q).

781

Both limit state lines are adjusted to a distance of T from the origin.

782

42

783 784

Figure 3 The actual reliability index ’A for the validation case.

785 786

787 788

Figure 4 Geometric interpretations for DVM-z* (left) and for QVM-z* (right).

789 790 43

791 792

Figure 5 (left) the scenario of a’ = 1/a in QVM-z*; (right) the extreme scenarios for QVM-z*.

44