Calibration of Reliability-Based Safety Factors for Sand · Draft 1 1 CALIBRATION OF RELIABILITY-BASED SAFETY FACTORS FOR 2 SAND BOILING IN EXCAVATIONS 3 4 Ignatius Tommy Pratama1,

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Calibration of Reliability-Based Safety Factors for Sand Boiling in Excavations

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2018-0498.R3

Manuscript Type: Article

Date Submitted by the Author: 16-Jun-2019

Complete List of Authors: Pratama, Ignatius; National Taiwan University of Science and Technology, Department of Civil and Construction EngineeringOu, Chang-Yu; National Taiwan University of Science and Technology, Department of Civil and Construction EngineeringChing, Jianye; National Taiwan University, Department of Civil Engineering

Keyword: Sand boiling, Excavation, Factor of safety, Reliability analysis

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cgj-pubs

Canadian Geotechnical Journal

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1

1 CALIBRATION OF RELIABILITY-BASED SAFETY FACTORS FOR

2 SAND BOILING IN EXCAVATIONS

3

4 Ignatius Tommy Pratama1, Chang-Yu Ou2, Jianye Ching3

5

6 1Master’s Student, Dept. of Civil and Construction Engineering. National Taiwan University of Science

7 and Technology, 43, Sec. 4, Keelung Rd., Taipei, 10607, Taiwan. Email: [email protected]

8 2Professor, Dept. of Civil and Construction Engineering. National Taiwan University of Science and

9 Technology, 43, Sec. 4, Keelung Rd., Taipei, 10607, Taiwan (corresponding author). Email:

10 [email protected]

11 3Professor, Dept. of Civil Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Rd., Taipei,

12 10617, Taiwan. Email: [email protected]

13

14 Abstract: This study calibrated the required factors of safety of five analysis methods for sand

15 boiling using reliability theory. The factors of safety computed by the five analysis methods were

16 compared with the results of a series of sand boiling model tests. The comparison shows that

17 rigorous methods (Terzaghi’s and Harza’s methods) were more accurate in predicting the factors

18 of safety compared to the simplified methods (Harr’s, simplified Terzaghi’s, and simplified

19 Harza’s methods). The statistics of the model factor for each method, defined as the actual factor

20 of safety divided by the computed one, was calibrated by the model test results. These statistics

21 were then used to establish the relationship between the target probability of failure and the

22 required factor of safety by reliability theory. Verification using a full-scale sand boiling case

23 history shows that the required factor of safety calibrated by the reliability theory was more

24 reasonable than the required factors of safety in references and design codes.

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25 Key words: Sand boiling; excavation; reliability analysis; factor of safety.

26

27 Introduction

28 The phenomenon of sand boiling in a sheeted excavation was first recognized by Terzaghi

29 (1922) based on the results of model tests, where sand boiling was identified by the rise of the

30 sand-water mixture in a region close to the retaining wall, depicted as the bcde area in Fig. 1.

31 Similar observations on sand boiling in excavations were also obtained in model tests by several

32 investigators (e.g., Marsland 1953; Tanaka et al. 2012; Tanaka et al. 2016). In the literature,

33 several analysis methods have been commonly adopted to evaluate the safety factor (FS) for sand

34 boiling in excavation, such as Terzaghi’s method (Terzaghi 1943), Harza’s method (Harza 1935),

35 Marsland’s method (Marsland 1953), Harr’s method (Harr 1962), simplified Terzaghi’s method

36 (JSCE 1996), and the simplified Harza’s method (TGS 2001; Ou 2006). Due to the assumptions

37 made by these methods, the computed FS, denoted by FSC, is not the same as the actual FS,

38 denoted by FSA. The difference between the FSC and FSA has been observed in various studies

39 (e.g., Tanaka et al. 1994; Tanaka et al. 2002). As a result, it is not certainly safe to require FSC to

40 be 1. Some safety margin is required (i.e., FSC > 1), and there is an issue on how to select a

41 reasonable required FS. In the literature and existing design codes, the required FS value for sand

42 boiling ranges from 1.2 to 5 (see Table 1). In some case histories, failure still occurred even when

43 certain safety margin has been adopted. For instance, sand boiling occurred in the Tokushima

44 excavation case (Tanaka et al. 2002) although the design code requirement for the simplified

45 Terzaghi’s method (required FS = 1.5) was satisfied.

46 In the literature, many investigators only focused on the mechanism of seepage failures and

47 factors affecting it (Benmebarek et al. 2005; Zhang and Chen 2006). However, no previous

48 studies have calibrated the required FS using reliability theory. One main purpose of this study

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49 was thus to calibrate the required FS using reliability theory. First, the model uncertainty was

50 calibrated by Marsland’s model test results (Marsland 1953). Then, the required FS was

51 calibrated by reliability theory to achieve a certain threshold for the sand boiling probability (e.g.,

52 10-2). The results indicate that this calibrated required FS could potentially maintain a uniform

53 probability of failure, whereas the existing required FS in Table 1 could not achieve so.

54 The rest of the paper has the following structure. First, the six analysis methods for sand

55 boiling are reviewed. Second, Marsland’s model tests are reviewed, and the numerical analysis

56 (finite element) on these tests is presented to confirm that the sand boiling mechanisms observed

57 in Marsland’s tests. Third, the statistics for the model factor of each analysis method, defined as

58 the actual FS divided by the computed FS, is calibrated by the model test results. Fourth, these

59 statistics are used to establish the relationship between the target probability of failure and the

60 required factor of safety by reliability theory. Finally, the adequacy of the calibrated FS is

61 verified using a real case history.

62

63 Analysis Methods for Sand Boiling

64 The schematic for the seepage in excavation is illustrated in Fig. 1, where W is the

65 submerged weight of the bcde soil prism, Ue is the seepage pressure acting on the base of bcde

66 soil prism, B is the excavation width, Hp is the wall penetration depth, He is the excavation height,

67 D is the depth of impermeable layer measured from the wall toe, T is the depth to the

68 impermeable layer measured from the excavation surface (i.e., T = D + Hp), ΔHw is the head

69 difference, di is the distance from excavation surface to the groundwater level in the excavation

70 zone, and hw is the distance from ground surface to the water level in the retained zone (hw > 0

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71 means the water level is above the ground surface, while hw < 0 means the water level is below

72 the ground surface).

73 Terzaghi (1922) defined that sand boiling occurs when Ue exceeds W'. Thus, a factor of

74 safety can be computed as a ratio of W' to Ue. This analysis method is referred to as Terzaghi’s

75 method:

76 (1) ' 'cC

e prism w prism

iWFSU i i

77 where FSC is the computed factor of safety; ic is the critical hydraulic gradient, defined as the

78 hydraulic gradient necessary to cause zero effective stress; iprism is the average hydraulic gradient

79 of the bcde soil prism in Fig. 1; γ' is the submerged unit weight of soil; and γw is the unit weight

80 of water.

81 By assuming one-dimensional flow in a homogeneous soil layer, iprism in Eq. (1) can be

82 estimated as ΔHw/2Hp (JSCE 1996). Thus, Terzaghi’s method can be simplified into the

83 following equation, referred to as the simplified Terzaghi’s method:

84 (2)2' c p

Ce w

i HWFSU H

85 Harza (1935) defined that sand boiling occurs when the hydraulic gradient at the exit of flow

86 (ie) exceeds ic. This analysis method is referred to as Harza’s method:

87 (3) 'cC

e w e

iFSi i

88 The hydraulic gradients in Terzaghi’s and Harza’s methods (iprism and ie) can be evaluated by a

89 flow net or by numerical analysis (e.g., finite element).

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90 By also assuming one-dimensional flow in a homogeneous soil layer, the ie in Eq. (3) can be

91 approximated as a ratio of ΔHw to the length of the shortest flow path. Harza’s method can then

92 be simplified into the following equation, referred to as the simplified Harza’s method (TGS

93 2001; Ou 2006):

94 (4) 2c p e ic

Ce w

i H H diFSi H

95 Note only if hw < 0, hw must be added to the (2Hp + He – di) part (i.e., 2Hp + He – di + hw).

96 Marsland (1953) proposed a chart solution for FSC. Marsland (1953) distinguished the factor

97 of safety for medium sand and very dense sand. The FSC for medium sand follows the concept of

98 Terzaghi (1943), whereas it follows the concept of Harza (1935) for very dense sand.

99 Harr (1962) used the conformal mapping technique to calculate the exit gradient (ie) by

100 considering the influence of various variables, including excavation width (B), excavation height

101 (He), and depth of impermeable layer (D). However, a version of Harr’s equation, referred to as

102 Harr’s method, does not depend on B and D:

103 (5) 1c ecC

e w

i H miFSi H m

104 where m is a factor in relation to He and Hp (Harr 1962), which can be solved by trial and error

105 according to the following equation:

106 (6)2

11 cos p

e

Hm mm H

107 Table 1 shows the required FS for the aforementioned methods, summarized from various

108 references and geotechnical codes.

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109 Comparison with actual factor of safety – Marsland’s model tests

110 Marsland (1953) performed 23 seepage model tests using saturated Ham River sand with two

111 different density states: “loose” and “dense”. For the loose sand, the void ratio (e) was 0.72 and

112 the critical hydraulic gradient (ic) was 0.97. For the dense sand, e was 0.59 and ic was 1.05. Ham

113 River sand was identified with the void ratio in the loosest state (emax) varied from 0.87 to 0.92

114 and the void ratio in the densest state (emin) varied from 0.53 to 0.59 (Bishop and Green 1965;

115 Coop and Lee 1993; Jovicic and Coop 1997). This suggests that the loose sand had relative

116 density (Dr) ≈ 51%, whereas the dense sand had Dr ≈ 88%. Although Marsland (1953) referred to

117 them as “loose” and “dense”, the relative densities of 51% and 88% were classified as “medium”

118 and “very dense”, respectively, by Terzaghi and Peck (1967) and Lambe and Whitman (1969).

119 The latter classification (medium and very dense) was adopted in the current paper.

120 Marsland’s test parameters consisted of excavation width (B), wall penetration depth (Hp),

121 excavation height (He), and soil density. Table 2 lists Marsland’s seepage test parameters and

122 results. The results were reported as the head difference at the initiation of sand boiling, denoted

123 by ∆Hw,f in Table 2. Each test was represented by a series of character and numbers, as shown in

124 Table 2. For example, the series “V0076025” indicates that the test was conducted in the very

125 dense sand (“V”) with B = 76 mm (“0076”) and Hp = 25 mm (“025”). For tests conducted in the

126 medium sand, the “M” character was used. The tests were undertaken in a seepage tank with

127 2750 mm long, 610 mm high, and 152.5 mm wide. Two sheet piling conditions were considered:

128 single sheet piling and double sheet piling conditions. The schematic of the seepage tank model is

129 illustrated in Fig. 2a for the single sheet piling condition and in Fig. 2b for the double sheet piling

130 condition. The excavation width (B) was 1375×2 = 2750 mm for the single sheet piling condition,

131 whereas B = 76 mm or 152 mm for the double sheet piling condition.

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132 Marsland (1953) also classified the tests into wide excavation and narrow excavation cases.

133 A wide excavation case was defined when Hp < B, while a narrow excavation when Hp B. Table

134 2 uses “W” to indicate a wide excavation and “N” to indicate a narrow excavation.

135 Before starting the test, the sand was in a fully saturated condition. The water level inside the

136 excavation zone was then lowered slowly to induce seepage until sand boiling or a moderate

137 amount of heave was observed. Table 2 shows the head difference at the initiation of sand boiling

138 (ΔHw,f).

139 It is worth noting that the water level inside the excavation zone was above the excavation

140 surface for all Marsland’s tests. This condition is different from typical real cases, where the

141 water level is usually below or at the excavation surface. Our analysis results (not shown here)

142 indicate that for two cases with the same head difference (ΔHw) and flow path length (e.g., same

143 Hp and He), the hydraulic gradient was insensitive to whether the water level was above or at the

144 excavation surface. Based on this numerical fact, it was assumed in this study that the position of

145 water level had no effect as long as ΔHw and flow path length stayed the same. Under this

146 assumption, the Marsland’s tests, as well as the subsequent calibration results in this study, may

147 be still applicable to real cases where the water level was below or at the excavation surface.

148 The aforementioned six analysis methods were used to compute the FSC for Marsland’s

149 model tests. For Terzaghi’s and Harza’s methods, iprism and ie were evaluated by a finite

150 difference method (FDM) developed in this study using MATLAB. The FDM was developed to

151 analyze two-dimensional steady-state flow analysis (i.e., Laplace’s equation) in an isotropic-

152 homogeneous soil. To maintain acceptable accuracy, the grid size in the FDM was less than 0.2

153 m. The hydraulic gradients obtained from the FDM code were very close to those obtained from

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154 PLAXIS. For Harr’s, simplified Terzaghi’s, and simplified Harza’s methods, the FDM was

155 unnecessary, as their FSC values could be evaluated by simplified equations.

156 Fig. 3 shows the FSC values for all Marsland’s tests. Since all Marsland’s tests were failure

157 cases, the actual factor of safety (FSA) was close to unity. As shown in Fig. 3, the FSC values for

158 Harr’s, simplified Terzaghi’s, and simplified Harza’s methods deviated from unity significantly.

159 Note that these methods do not consider the effect of the excavation width (B) and the depth to

160 the impermeable layer (D). Consequently, these methods were less accurate. On the contrary, the

161 FSC values for Terzaghi’s, Harza’s, and Marsland’s methods were relatively close to unity. These

162 methods consider the effect of B and D. Therefore, they were more accurate.

163

164 Numerical Analysis for Sand Boiling

165 In this section, numerical analysis (fully-coupled flow deformation analysis in PLAXIS) was

166 adopted to simulate Marsland’s model tests to confirm the sand boiling mechanisms observed in

167 Marsland’s tests. PLAXIS was used to obtain Ue and ie for Terzaghi’s and Harza’s methods. The

168 FSC values for these two methods are subsequently computed based on Ue and ie. The effective

169 stress distribution and the heave at the middle of the excavation surface were also simulated by

170 PLAXIS. In PLAXIS, the Ham River sand was modeled as a saturated isotopic homogeneous soil

171 that followed the linear elastic-perfectly plastic (Mohr-Coulomb) constitutive law. The soil in the

172 model tests was divided into five soil layers with depth-dependent effective Young’s moduli. The

173 secant Young’s modulus corresponding to the 50% stress level (E50) was adopted as the effective

174 Young’s modulus for each soil layer. The E50 values for the five soil layers were estimated by

175 following Ohde’s (1939, 1951), von Soos’s (1980), and Schanz and Vermeer’s (1998) empirical

176 relationships. The effective Poisson’s ratio (ν) was assumed to be 0.33 for all layers. The five

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177 soil layers used the same effective friction angle (ϕ), and it was estimated by the average relative

178 density (Dr) and the dry unit weight (γd) of the Ham River sand using an empirical relationship

179 proposed by NAVFAC DM7.1 (1982). The dilation angle (ψ') was taken to be ϕ' – 30o as

180 suggested in the PLAXIS manual (Brinkgreve 2002). To model the lack of soil-wall friction

181 during sand boiling, the soil strength of soil-wall interface was relatively low. The at-rest earth

182 Jaky’s (1948) empirical equation. According to Marsland (1953), the coefficient of permeability

183 (k) for the loose Ham River sand was about 6.8 cm/min, whereas the k value for the dense Ham

184 River sand was estimated to be 1.8 cm/min using Chapuis’s (2004) equation. The constant head

185 boundary condition was applied at the upper and left boundaries of the retained zone.

186 For the dewatering simulation, the water head in the excavation zone changed with the

187 dewatering time. At each dewatering time, the effective stress distribution, FSC values for

188 Terzaghi’s and Harza’s methods, and the heave at the middle of excavation surface, denoted by

189 δv,mid, were simulated. Figs. 4a, 4b, 4c, and 4d show how the normalized heave (δv,mid/He) and FSC

190 change with the normalized head difference (Hw/Hp) for the four representative Marsland’s test

191 cases (M0076025, V0076025, M0076076, V0076076). The results show that δv,mid/He increased

192 and FSC decreased with increasing Hw/Hp. It was also found that when FSC was close to unity,

193 δv,mid/He usually started to increase more rapidly. The onsets of the observed sand boiling in

194 Marsland’s tests (see Fig. 4) matched well with the time instants for FSC 1. This indicates that

195 the numerical analysis results were qualitatively and quantitatively consistent with the

196 observations obtained in Marsland’s tests. This conclusion (the observed sand boiling matched

197 well with FSC 1) did not change even if (ϕ, E50, k) in PLAXIS were subjected to (10%, 30%,

198 50%) variations, respectively.

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199 To further illustrate the mechanism of sand boiling, Fig. 5 shows the evolution of the

200 effective stress distribution during dewatering in V0076025. The gray area in Fig. 5 indicates the

201 critical zone with zero effective stress. For this particular case, Marsland (1953) indicated that

202 sand boiling occurred when the normalized head difference (ΔHw/Hp) reached 4.56 (Fig. 4b). As

203 shown in Fig. 5, the effective stress generally decreased with increasing ΔHw/Hp. When ΔHw/Hp

204 was increased to a certain value, a large decrease of effective stress was initiated near the wall toe

205 (e.g., Fig. 5d) because a large head loss occurred here due to the fact that the length of the

206 equipotential drop was short near the wall toe. However, sand boiling did not yet appear because

207 the critical zone did not propagate to the excavation surface.

208 As ΔHw/Hp kept increasing, the critical zone propagated to the excavation surface with a

209 shape resembling a wedge with a width of approximately Hp/3 to Hp/2 (Fig. 5e). This aligned

210 with the sand boiling phenomenon observed by Harza (1935) and Terzaghi (1922, 1943). This

211 state (Fig. 5e) was generally understood as sand boiling. At this state, the Terzaghi’s and Harza’s

212 FSC values were both very close to unity (Fig. 4b). However, both Marsland’s test and numerical

213 analysis results indicated that only a moderate increase in the heave (δv,mid) was found (Fig. 4b).

214 A total collapse of the excavation did not yet occur in this state because the critical zone was

215 confined. As ΔHw/Hp was increased further (Fig. 5f), the critical zone expanded to the entire

216 excavation zone, followed by a drastic increase in δv,mid and by a total collapse of the excavation.

217

218 Required factor of safety calibrated by reliability approaches

219 As shown in Fig. 3, the computed factor of safety (FSC) was not the same as the actual factor

220 of safety (FSA). The latter should be around unity, because all Marsland’s test cases were in a

221 boiling condition. The discrepancy between FSC and FSA was assumed to be mainly due to the

222 inaccuracy of the analysis methods. For instance, Terzaghi’s method requires iprism, which can be

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223 estimated by the FDM developed in this study. However, the FDM has model error, so the

224 computed iprism may not be the same as the actual iprism. Simplified methods such as Eqs. (2), (4),

225 and (5) have model errors as well. It is worth noting that Marsland’s model tests are scaled tests,

226 and they may also have scaling error. However, to our best knowledge, full scale tests for sand

227 boiling problems did not exist in the geotechnical engineering literature. As a result, the

228 calibration results in this study was based on the assumption that the scaling error for model tests

229 was negligible compared to the model error of the analysis methods (Terzaghi, Harza, Harr,

230 simplified Terzaghi, and simplified Harza). Limited evidence that supports this assumption can

231 be found in the literature. For instance, Phoon and Kulhawy (2005) evaluated the statistics for the

232 model factor, defined as (test capacity)/(predicted capacity), for laterally loaded drilled shafts.

233 They found that the statistics are robust and appear not to be seriously affected by whether the

234 test is a laboratory-scale model test or a field test, possibly because of normalization during the

235 calculation of the model factor (Phoon and Kulhawy 2005).

236 To quantify the model error, it was customary to define the model factor M (Lesny et al.

237 2017) as the ratio of a measured (e.g., in a load test) to a calculated quantity. In the context of the

238 current paper, FSA could be considered as the measured quantity, whereas FSC could be

239 considered as the calculated quantity. Therefore,

240 (7) A CM FS FS

241 The model factor M was not a constant but a random variable. In the literature, model factors are

242 usually assumed to be lognormally distributed with a mean value (μM) and coefficient of variance

243 (COV) (δM) (e.g., Phoon and Kulhawy 2005; Juang et al. 2012). Two sets of results (medium

244 sand and very dense sand) were calibrated, because the statistics for these two sets were different.

245 Calibration for Terzaghi’s and Harza’s methods

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246 The definition of model factor in Eq. (7) was suitable for Terzaghi’s and Harza’s methods

247 because they did not exhibit systematic bias (see Fig. 3). For these two methods, M samples

248 could be obtained by assuming FSA 1 for all Marsland’s test cases:

249 (8) 1 CM FS

250 Because M was assumed lognormal, ln(M) was normal. The mean and standard deviation of ln(M)

251 could be readily estimated:

252 (9) 2

ln(M) ln(M) ln(M)1 1

1 1ln ln1

N N

i ii i

M MN N

253 where ln(M) and ln(M) are the mean and standard deviation for ln(M), and N is the sample number.

254 Table 3 shows the estimated ln(M) and ln(M) for Terzaghi’s and Harza’s methods.

255 The theorem of equivalence between reliability and factor of safety proposed in Ching (2009)

256 was adopted to calibrate the reliability-based required factor of safety, denoted by FSR. By

257 applying the theorem, it was possible to establish the relationship between the target probability

258 of failure (Pf*) and FSR. The use of this theorem requires a set of calibration cases. Table 4 shows

259 the ranges for He, Hp/He, B/He, di/Hp, D/ΔHw, mean value of ΔHw/He (μΔHw), and mean soil unit

260 weight (μγ) for the calibration cases. The definitions for all dimensions are illustrated in Fig. 1.

261 The ranges in Table 4 were summarized from the geometries of the excavation cases in our

262 database. The range for μγ was based on the common range for the sand unit weight.

263 The use of the theorem in Ching (2009) also requires the specification of the probability

264 distributions of all random variables. Table 5 lists the statistics and distribution types of all

265 random variables. Three random variables were considered: the saturated unit weight of the sand

266 (γsat), normalized head difference (ΔHw/He), and logarithm of the model factor (ln(M)). The three

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267 random variables were assumed to be statistically independent. γsat was assumed to follow the

268 normal distribution with COV equal to 0.05. ΔHw/He was assumed to follow the lognormal

269 distribution with COV equal to 0.10. ln(M) was normal because M was assumed to be lognormal.

270 The values for μln(M) and ln(M) can be found in Table 3.

271 According to Ching (2009), the relationship between the target failure probability (Pf*) and

272 FSR can be written as follows:

273 (10)

*,E ,

CR f

C

FS XP FS P

M FS X

274 where X represents the two random variables sat and ΔHw/He, and E(X) represents their mean

275 values; represents all parameters in Table 4, treated as random variables uniformly distributed

276 over the ranges defined in Table 4 (Ching 2009); FSC is the factor of safety computed by

277 Terzaghi’s or Harza’s method; M is the model factor (random). The Pf*-FSR relationships for

278 Terzaghi’s and Harza’s methods were then computed based on Eq. (10) by adopting subset

279 simulation (SubSim) (Au and Beck 2001) with sample size = 5000 per stage. Fig. 6 shows the

280 Pf*-FSR relationships for the medium sand and very dense sand. Given a target probability of

281 failure (Pf*), the corresponding FSR could be calibrated from the Pf*-FSR relationship. According

282 to Ching (2009), the requirement for failure probability, denoted as Pf Pf* is roughly equivalent

283 to the following requirement for FSC:

284 (11) C RFS FS

285 It was worth noting that when evaluating FSC in Eq. (11), the X variable should be fixed at their

286 mean values, rather than at the 5% fractiles [i.e., the characteristic value in Eurocode 7 (EN

287 2004)].

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288 USACE (1997) considered that satisfactory performance of geotechnical system corresponds

289 to Pf* ≈ 10-3. Some investigators (e.g., Wu et al. 2014; Goh et al. 2008) also adopted Pf* ≈ 10-3.

290 However, it was argued in the current study that a slightly higher Pf* may be acceptable, because

291 when sand boiling occurred, only a limited heave (Fig. 4) was observed and full-scale failure did

292 not yet occur. The requirement of Pf* = 10-3 may be too conservative for a local sand boiling. As

293 a result, three Pf* values were considered in the current study: Pf* = 10-1, 10-2, and 10-3. Table 6

294 lists the calibrated FSR values for Pf* = 10-1, 10-2, and 10-3. For instance, consider a future

295 excavation case in a medium sand with a target failure probability Pf* = 10-2, the FSR for

296 Terzaghi’s method is 1.76 (Table 6). Accordingly, the requirement of failure probability (Pf)

297 10-2 is roughly equivalent to the requirement of FSC 1.76.

298 Calibration for Harr’s, simplified Terzaghi’s, and simplified Harza’s methods

299 Harr’s, simplified Terzaghi’s, and simplified Harza’s methods do not consider the effect of

300 the excavation width (B) and the depth to the impermeable layer (D). As a result, these methods

301 exhibited systematic biases in FSC (see Fig. 3), and the model factor M defined in Eq. (7) was not

302 suitable for these methods. The systematic bias depended on certain normalized factors (e.g.,

303 ΔHw/Hp and D/ΔHw). Instead of directly adopting Eq. (7), the framework proposed by Zhang et al.

304 (2015) was adopted in the current study to handle the systematic bias. The basic idea of Zhang et

305 al. (2015) is to adopt a more sophisticated numerical model that is without systematic bias as a

306 surrogate model. In the current study, Terzaghi’s method was adopted as the surrogate model,

307 because it was without systematic bias. The model factor M for a simplified method can be

308 decomposed into two factors, M = MTC:

309 (12) , ,T A C T C T CM FS FS C FS FS

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310 where FSC,T denotes the FSC value computed by Terzaghi’s method; FSC denotes the FSC value

311 computed by a simplified method; MT is the model factor for Terzaghi’s method; C is the ratio

312 between FSC,T and FSC. The statistics of MT have been calibrated in Table 3 (ln(M) and ln(M) in

313 the first row). C is the ratio between FSC,T and FSC. In principle, C is not random, and it can be

314 expressed as the product between a regression factor f and a residual factor :

315 (13) or ln ln ln C f C f

316 where ln() is modeled as a zero-mean pseudo random variable with standard deviation = . One

317 can see that ln(FSC,T) = ln(C)+ln(FSC) = ln(f)+ln()+ln(FSC) ln(f)+ln(FSC). This implies that

318 FSC,T fFSC. Hence, fFSC can be regarded as the “corrected” FSC, because FSC of a simplified

319 method has been corrected into FSC,T by multiplying with f. The factor f quantifies the trend in

320 the ratio C = FSC,T/FSC, and this trend generally depends on X and , i.e., f = f(X,). Moreover,

321 because f(X,) is dimensionless, f(X,) depends on dimensionless factors such as Hp/He, B/He,

322 ΔHw/He, D/ΔHw, di/Hp, and ic.

323 f(X,) can be determined by regression. Ten thousand cases with X and ranges given in

324 Table 4 were randomly generated. For each case, FSC,T was computed by Eq. (1) based on the

325 iprism obtained from the FDM code, and FSC was computed by Eqs. (2), (4), or (5). The ratios

326 FSC,T/FSC for these ten thousand cases were computed. Linear regression analysis was then

327 conducted by assuming ln(FSC,T/FSC) ln(f) to be a linear function of the dimensionless factors

328 or their transforms (e.g., logarithms, squares, products). To yield an effective regression equation,

329 Bayesian information criterion (BIC) (Schwarz 1978) was adopted to select the optimal input

330 dimensionless factors. The following shows the resulting f(X,):

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331 (14)

2 2

0 1 2 3 4

5 6 7 8

9 10

ln ln ln ln

, exp ln ln ln

ln ln ln ln

p p

e e e e

w ic

e w p

p p

e e w e

H H B Ba a a a aH H H H

H dDf X a a a a iH H H

H HB Da aH H H H

332 where the coefficients (a0, a1, …, a10) are listed in Table 7 as well as the corresponding residual

333 standard deviations .

334 As a result, the model factor M for a simplified method can be expressed as

335 (15) , , , A C A C T C T C TM FS FS FS FS FS FS M f X

336 The relationship between Pf* and FSR for a (corrected) simplified method is as follows:

337 (16)

*E , E ,

, ,C

R fT C

f X FS XP FS P

M f X FS X

338 To yield a stable Pf*-FSR relationship, Monte Carlo simulation (MCS) was adopted with sample

339 size = 1.0 × 107. A large sample size was used because the computation for FSC for the simplified

340 method was fast. Fig. 7 shows the Pf*-FSR relationships for the medium sand and very dense sand

341 cases. The requirement for failure probability (Pf) Pf* is thus roughly equivalent to the

342 following requirement:

343 (17) , C Rf E X FS FS

344 where f[E(X),]FSC is the “corrected” FSC. Again, when evaluating FSC in Eq. (17), the X

345 variable should be fixed at their mean values. Table 6 lists the calibrated FSR values for the

346 simplified methods for Pf* = 10-1, 10-2, and 10-3.

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347 Note that the calibrated FSR in Table 6 is for the “corrected” FSC, not for an uncorrected FSC.

348 Accordingly, the safety against sand boiling should be verified against the (corrected FSC) =

349 f[E(X),]FSC FSR rather than against the (uncorrected FSC) FSR. For Terzaghi’s and Harza’s

350 methods, FSC required no correction, and it sufficed to verify FSC FSR. For instance, consider a

351 future excavation case in a medium sand with He = 10.27 m, di/Hp = 0.1, Hp/He = 0.74, B/He =

352 1.42, D/Hw = 1.11, mean γsat = 17.5 kN/m3 (mean ic = 0.785), mean Hw/He = 0.56 and suppose

353 the target failure probability is Pf* = 10-2. The FSR for the simplified Terzaghi’s method is 1.80

354 for a medium sand case (Table 6). The regression factor f for this particular case can be evaluated

355 using Eq. (14) with the listed a coefficients in the third column of Table 7: f = exp[-0.0782 –

356 0.378ln(Hp/He) – 0.0363ln(Hp/He)2 + 0.4994ln(B/He) – 0.0944ln(B/He)2 – 0.07ln(Hw/He) –

357 0.1714ln(D/Hw) + 1.7083(di/Hp) – 0.1165ln(ic) + 0.1884ln(B/He)ln(Hp/He) –

358 0.0578ln(D/Hw)ln(Hp/He)] 1.492. As a result, the requirement of failure probability (Pf) 10-2

359 is roughly equivalent to the requirement of 1.492FSC 1.80. In terms of uncorrected FSC, the

360 requirement is FSC 1.80/1.492 = 1.21.

361 Comparison with existing design codes

362 Note that Figs. 6 and 7 could be used to infer the target failure probability (Pf*) underlying a

363 required FS prescribed in the literature or in existing design codes. For instance, consider the

364 same future excavation case in a medium sand with He = 10.27 m, di/Hp = 0.1, Hp/He = 0.74, B/He

365 = 1.42, D/Hw = 1.11, mean γsat = 17.5 kN/m3, mean Hw/He = 0.56 and consider the simplified

366 Terzaghi’s method. The Japan design code (JSCE 1996) requires that FSC 1.2 (second column

367 in Table 1). Note the JSCE requirement is on the uncorrected FSC. The regression factor for this

368 particular case is f = 1.492. The JSCE requirement is thus equivalent to the following requirement

369 on the corrected FSC: (corrected FSC) = 1.492FSC ≥ 1.4921.2 = 1.79 = FSR. Based on Fig. 7a,

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370 the Pf* corresponding to FSR = 1.79 is about 1.05×10-2. This value is then regarded as the

371 underlying Pf* for the JSCE requirement FSC 1.2 for this particular medium sand case. The

372 rightmost column in Table 1 shows the ranges of the underlying Pf* values for different analysis

373 methods for design cases within the allowable regions in Table 4.

374 According to the above comparison, it was evident that the underlying Pf* was rather non-

375 uniform. For instance, the underlying Pf* value for Harza’s method ranged from 1.2×10-2 to

376 6.2×10-4. This suggests that the sand boiling risk could not be effectively controlled by the

377 requirement prescribed in the literature or in existing design codes. In some circumstances, the

378 underlying Pf* could be quite high (e.g., 1.00), indicating that sand boiling was likely to occur

379 even if the requirement prescribed in the literature or in existing design codes was strictly

380 followed. In contrast, the underlying Pf* for the FSR calibrated in this study (Table 6) was

381 uniform (Pf* was fixed at 10-1, 10-2, or 10-3). This showed the advantage of the reliability-based

382 FSR over the traditional required FS.

383

384 Verification of the FSR value calibrated in this study

385 In this section, the calibrated FSR in Table 6 was verified by a case history in Tokushima

386 prefecture (Tanaka et al. 2002). The excavation was located in Otsucho Daiko, Naruto City,

387 Tokushima Prefecture, Japan. Fig. 8 shows the cross-section of the excavation plan. The ground

388 condition consisted of sandy and silty soils with standard penetration test N value (NSPT) ranging

389 from 7 to 21. The groundwater was located at 0.9 m below the ground level. The excavation

390 height (He) was 5.4 m, whereas the wall penetration depth (Hp) and the excavation width (B) were,

391 respectively, 5.6 m and 5.2 m. One level of the strut was applied at GL -1.8 m (GL refers to

392 ground surface level) as a reinforcement of the type-IV sheet pile walls.

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393 Sand boiling occurred when the excavation reached GL -5.4 m. The water sprang out at

394 several locations on the excavation surface. As a result, the upper part of the silty sand layer was

395 loosened. Tanaka et al. (2002) indicated that the sand boiling in the Tokushima excavation was

396 due to a large head loss in the less permeable silty sand layer. The large head loss induced large

397 hydraulic gradient and a decrease of effective stresses in the silty sand layer. Tanaka (2002) also

398 indicated that the sand boiling was due to the two-dimensionally concentrated seepage flow

399 instead of the three-dimensional seepage flow concentration into the excavation corner. As a

400 remedial action, the jet grouting was adopted to improve the underlying sand layer, whereas the

401 voids in the loosened silty sand layer were filled with sands.

402 The FSC for the Tokushima excavation when the excavation reached GL -5.4 m (Hw/Hp =

403 0.80 and Hw/He = 0.83) was computed using the analysis methods listed in Table 1, excluding

404 Marsland’s method. For Terzaghi’s and Harza’s methods, PLAXIS was adopted to evaluate Ue

405 and ie under the “groundwater flow only” calculation option. In PLAXIS, all soil layers with

406 different hydraulic conductivities (k), as shown in Fig. 8, were modeled.

407 The simplified methods such as Harr, simplified Terzaghi, and simplified Harza methods

408 were also used in this case study. The layered soils were considered in a simplified way: the

409 saturated unit weight (γsat) for each soil layer along the wall was weighted averaged to obtain the

410 average saturated unit weight (γsat,ave), and the critical hydraulic gradient (ic) was taken to be

411 (γsat,ave-γw)/γw. The “uncorrected” FSC values for Harr’s, simplified Terzaghi’s, and Harza’s

412 methods were computed using Eqs. (2), (4), and (5), respectively.

413 The second column in Table 8 lists the uncorrected FSC value for all methods. Table 8 shows

414 that except for Harza’s and Harr’s methods, all the uncorrected FSC values were larger than the

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415 requirements in the literature (third column). However, sand boiling still occurred in the

416 Tokushima excavation case.

417 To verify whether the Tokushima case satisfies the FSR value calibrated in this study, the

418 uncorrected FSC values for the simplified methods must be converted to the corrected values by

419 multiplying them with the regression factor f. The regression factor f was evaluated based on

420 Hp/He = 1.04, B/He = 0.96, Hw/He = 0.83, D/Hw = 2.0, di/Hp = 0, and ic = 0.85. On the other

421 hand, FSC for Terzaghi’s and Harza’s methods required no correction. The fourth column in

422 Table 8 shows the corrected values and these values could be considered as the median FS. The

423 soil density state for the Tokushima case was regarded as medium sand, as described by Tanaka

424 et al. (2002). Two Pf* values were also considered in this case study which were 10-1 and 10-2.

425 Table 8 lists the FSR values for medium sand with Pf* = 10-1 (fifth column) and 10-2 (sixth

426 column). In contrast to the previous assessment using the requirements in the literature (third

427 column), all corrected FSC values were now smaller than the FSR values calibrated by this study

428 for Pf* = 10-2 (sixth column). Moreover, the corrected FSC values for Terzaghi’s and Harza’s

429 methods were smaller than the FSR values calibrated by this study for Pf* = 10-1 (fifth column).

430 For further illustration, the variation of the corrected FSC (median FS) and the failure

431 probability (Pf) with respect to the head difference (ΔHw) for the Tokushima case are shown in

432 Fig. 9. Figure 9 was obtained by the following procedure: The FSC value was first computed for

433 each varied ΔHw value. Then, the FSC value was converted into a corrected FSC by multiplying it

434 with the regression factor f which was evaluated based on Hp/He = 1.04, B/He = 0.96, D/Hw =

435 2.0, di/Hp = 0, and ic = 0.85. Finally, Figs. 6 and 7 were used to obtain the Pf value corresponding

436 to the corrected FSC. Figure 9b shows that Harza’s method predicted the largest Pf, and

437 Terzaghi’s method predicted the second largest Pf.

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438 Given that the case history was actually a failure case, the following observations were made:

439 1. The requirements in the literature (third column) were not reasonable in general. Many of the

440 requirements were satisfied (e.g., FSC in the first column > required FS in the third column

441 for Terzaghi’s, simplified Terzaghi’s, and simplified Harza’s requirements) but the case

442 actually failed.

443 2. The FSR calibrated for Terzaghi’s and Harza’s methods seemed reasonable. The corrected

444 FSC (fourth column) was less than FSR for Pf* = 10-1 (fifth column). Figure 9b suggests that

445 the Pf value at the onset of actual sand boiling (indicated by the vertical dashed line in Fig.

446 9b) was 0.16 for Terzaghi’s method and 0.82 for Harza’s method. These large Pf values

447 agreed well with the fact that this case actually failed. On the contrary, the FSR calibrated for

448 the simplified methods (Harr’s, simplified Terzaghi’s, and simplified Harza’s methods)

449 seemed less reasonable but the corrected FSC was still between the FSR values for Pf* = 10-2

450 and 10-1. Figure 9b suggests that this case history had a Pf value of about 1.55×10-2 to

451 2.12×10-2. This Pf was not very large, although this case actually failed.

452 3. Figure 9b further indicates that the Pf values predicted by Terzaghi’s and Harza’s methods

453 were not uniform.

454 Discussions

455 Why were the FSR values calibrated for the simplified methods less reasonable?

456 This discussion is for observation #2 above. The FSR values calibrated for the simplified

457 methods (Harr’s, simplified Terzaghi’s, and simplified Harza’s methods) were less reasonable

458 probably because these simplified methods were not suitable for the case history. All simplified

459 methods assumed one-dimensional seepage flow in a homogeneous soil so that the factor of

460 safety only depended on the problem geometry (see Eqs. (2), (4), and (5)). For an inhomogeneous

461 soil such as the case history, different soil layers have different hydraulic conductivities (k). Eqs.

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462 (2), (4), and (5) could not take this inhomogeneity into account. As a result, the factors of safety

463 computed by these simplified methods as well as their results in Fig. 9 were considered to be not

464 as accurate as those computed by Terzaghi’s and Harza’s methods. In the current study, the bias

465 of these simplified methods with respect to the effect of B and D has been corrected. The

466 correction was based on Marsland’s test cases, but all these cases were still homogeneous cases.

467 As a result, the simplified methods, even after corrections, could not take the inhomogeneity into

468 account and hence were not suitable for the Tokushima case. Nevertheless, the FSR values

469 calibrated for these simplified methods were still presented in this paper because some design

470 codes (e.g., Japan, Taiwan, Malaysia, Eurocode 7, etc.) adopted these simplified methods. The

471 FSR values calibrated for these simplified methods were still deemed acceptable for a relatively

472 homogeneous soil.

473 In contrast, Terzaghi’s and Harza’s methods could take the inhomogeneity into account

474 because their input variables (iprism or ie) were based on rigorous analysis such as PLAXIS, and

475 PLAXIS could take the inhomogeneity into account. The FSR values calibrated for these two

476 rigorous methods were deemed acceptable even for an inhomogeneous soil such as the

477 Tokushima case.

478 Why did Terzaghi’s and Harza’s methods exhibit non-uniform failure probabilities?

479 This discussion is for observation #3 above. Although Terzaghi’s and Harza’s FSC

480 definitions were both based on the zero-effective-stress principle, the involved soil volumes were

481 different. Terzaghi (Eq. 1) considered the seepage pressure acting on a soil mass (the prism bcde

482 in Fig. 1), whereas Harza (Eq. 3) considered the hydraulic gradient at a single point (the exit

483 point of the shortest seepage path). Because the involved soil volumes were different, it was

484 possible for Terzaghi’s and Harza’s methods to have two different Pf values. Our opinion was

485 that Terzaghi's method was more relevant to a large-scale sand boiling because it involved a large

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486 amount of soil volume, whereas Harza's method was more relevant to local sand boiling where

487 springs appeared locally on the excavation surface.

488

489 Conclusions

490 Five analysis methods for sand boiling (Terzaghi’s, Harza’s, Harr’s, simplified Terzaghi’s,

491 and simplified Harza’s methods) were considered in this study and their required factors of safety

492 (FSR) were calibrated using reliability theory based on Marsland’s model tests (Marsland 1953).

493 The results show that simplified methods (Harr’s, simplified Terzaghi’s, and simplified Harza’s

494 methods) exhibited systematic biases that mainly depended on the head difference, excavation

495 width, and the distance to impermeable soil because the flow in these methods was assumed one-

496 dimensional. In contrast, rigorous methods (Terzaghi’s and Harza’s methods) did not exhibit

497 systematic bias and were generally accurate in predicting sand boiling. The FSR value for each

498 method was calibrated by reliability theory for a target failure probability (Pf*) of 10-1, 10-2, and

499 10-3. Table 6 shows the calibrated FSR values.

500 The verification results show that the FSR calibrated in this study was more reasonable than

501 the requirement in the literature. This was verified by a case history investigated by Tanaka et al.

502 (2002). It should be noted that the use of the FSR calibrated in this study might be not suitable if a

503 future design scenario was with dimensions/settings that were outside the allowable regions in

504 Table 4. The FSR calibrated in this study was based on the Marsland’s model tests with

505 homogeneous sand. However, in real projects, the ground may be inhomogeneous or layered. For

506 Terzaghi’s and Harza’s methods, if the hydraulic gradients (iprism or ie) were computed from

507 numerical analysis which took into account the multi-layered soils, their FSR values calibrated in

508 this study were applicable. For simplified methods (Harr’s, simplified Terzaghi’s, and simplified

509 Harza’s methods), their FSR calibrated in this study were applicable to a relatively homogeneous

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510 ground. For an inhomogeneous ground, caution must be taken when the calibrated FSR values for

511 these simplified methods were applied.

512

513 Acknowledgements

514 This study was conducted under the sponsorship of the Ministry of Science and Technology

515 (MOST) of the Republic of China, under the project MOST 104-2221-E-011-110-MY3. The

516 authors would like to acknowledge this gracious support from MOST. The authors also would

517 like to thank the anonymous reviewers for their insightful comments and suggestions that have

518 led to the improvements of this paper.

519

520 References

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526 of a cohesionless soil. Géotechnique, 15(3): 243-266.

527 Brinkgreve, R.B.J. 2002. PLAXIS version 8 reference manual. Plaxis BV, Delft, the Netherlands.

528 European Committee for Standardization. 2004. Eurocode 7 Part 1: Geotechnical design: General

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531 effective diameter and void ratio. Canadian Geotechnical Journal, 41(5): 787-795.

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591 Taiwan Geotechnical Society (TGS), Taipei, Taiwan (in Chinese).

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597 Taschenbuch, Teil 1, 4 Auflage, Ernst & Sohn Verlag, Berlin, German. (in German).

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Table 1. Required Factors of Safety from Literature

Method Required FS Reference(s)Underlying Target Failure Probability

(Pf*)Terzaghi 1.2 JSCE (1996) 0.45Harza 2.0 NAVFAC DM7.1 (1982) 1.210-2 – 6.210-4

Marsland 1.5 – 2.0 Marsland (1953), NAVFAC DM7.1 (1982) -

Harr 4.0 – 5.0 Harr (1962) 3.710-9 a – 0.056Simplified Terzaghi 1.2 JSCE (1996) 0.99

Simplified Harza 1.5 – 2.0 TGS (2001), Ou (2006) 1.410-8 a – 1.00aThe small Pf

* values were evaluated by SubSim.

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Table 2. Parameters of Marsland (1953) Tests

Test Number

Name Series

B(mm)

Hp

(mm)He

(mm)T

(mm)∆Hw,f

(mm)Remarks

1 M0076025 76 25 335 101 77.5 ± 5.0 W2 M0076051 76 51 309 127 116.0 ± 4.0 W3 M0076076 76 76 284 152 145.0 ± 3.0 N4 M0076102 76 102 258 178 170.0 ± 10.0 N5 M0152025 152 25 335 101 95.0 ± 4.0 W6 M0152051 152 51 309 127 153.0 ± 3.0 W7 M0152076 152 76 284 152 191.0 ± 12.0 W8 M0152102 152 102 258 178 200.0 ± 13.0 W9 M2750025 2750 25 335 101 113.0 W10 M2750051 2750 51 309 127 208.0 W11 M2750076 2750 76 284 152 290.0 W12 V0076025 76 25 335 101 114.0 ± 4.0 W13 V0076051 76 51 309 127 138.0 ± 3.0 W14 V0076076 76 76 284 152 173.0 ± 3.0 N15 V0076102 76 102 258 178 205.0 ± 4.0 N16 V0076127 76 127 233 203 225.0 ± 4.0 N17 V0076152 76 152 208 228 251.0 ± 2.0 N18 V0152025 152 25 335 101 123.0 ± 8.0 W19 V0152051 152 51 309 127 190.0 ± 5.0 W20 V0152076 152 76 284 152 247.0 ± 5.0 W21 V2750025 2750 25 335 101 155.0 ± 2.0 W22 V2750051 2750 51 309 127 247.0 ± 4.0 W23 V2750076 2750 76 284 152 365.0 ± 5.0 W

Note: B = excavation width; Hp = wall penetration depth; He = excavation depth; T = depth to the tank base measured from the excavation surface = 76 + Hp (mm); ΔHw,f = head difference at sand boiling; W = wide excavation; N = narrow excavation.

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Table 3. Statistics for Logarithms of Model Factors for Terzaghi’s and Harza’s Methods

[ln(M) ln(M)]Method

Medium Sand Very Dense Sand

Terzaghi [-0.165 7.48%] [0.0210 5.47%]Harza [-0.222 11.13%] [-0.0290 8.11%]

Table 4. Ranges for Calibration CasesDesign Parameter Symbol Allowable Region Unit

Excavation depth He 4.27 – 16.27 mNormalized depth of unsaturated area in the excavation zone di/Hp 0.00 – 0.20 -

Normalized wall penetration depth Hp/He 0.18 – 1.30 -Normalized excavation width B/He 0.73 – 2.11 -Normalized depth of impermeable layer D/ΔHw 0.22 – 2.00 -

Mean value of γsat μγ 14.00 – 21.00 kN/m3

Mean value of ΔHw/He μΔHw 0.22 – 0.89 -

Table 5. Mean Values and Variabilities of Random VariablesRandom Variable

Mean Value

COV or Standard Deviation Unit Distribution

γsat μγ COV = 0.05 kN/m3 NormalΔHw/He μΔHw COV = 0.10 - Lognormalln(M) μln(M) St. Dev.= ln(M) - Normal

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Table 6. Required Factors of Safety Calibrated by Reliability ApproachesMedium Sand Very Dense Sand

MethodPf* = 10-1 Pf* = 10-2 Pf* = 10-3 Pf* = 10-1 Pf* = 10-2 Pf* = 10-3

Terzaghi 1.46 1.76 2.05 1.20 1.44 1.69Harza 1.62 2.04 2.45 1.30 1.61 1.93Harr 1.47 1.80 2.12 1.21 1.47 1.73Simplified Terzaghi 1.47 1.80 2.12 1.21 1.47 1.73

Simplified Harza 1.47 1.79 2.11 1.21 1.46 1.72

Note: Design requirement is f[E(X),]FSC FSR, f[E(X),] is computed based on Eq. (14).

Table 7. Coefficients for Regression factor f(X,) and Residual Standard Deviation

MethodCoefficients

Harr Simplified Terzaghi

Simplified Harza

a0 -0.6669 -0.0782 -0.5081a1 -0.2713 -0.3780 -0.1504a2 -0.0703 -0.0363 -0.1424a3 0.4994 0.4994 0.4950a4 -0.0943 -0.0944 -0.0863a5 -0.0700 -0.0700 -0.3688a6 -0.1714 -0.1714 -0.1693a7 1.7083 1.7083 2.4550a8 -0.1165 -0.1165 -0.1176a9 0.1884 0.1884 0.1869a10 -0.0578 -0.0578 -0.0543 0.0344 0.0344 0.0350

Page 31 of 42



Draft

Table 8. Verification on Safety Status for Tokushima ExcavationVerification of Reliability-based

FSR in Table 6Method Uncorrected

FSC

Requirement in the

Literature (Required FS in Table 1)

Corrected FSC

(f FSC)

FSR forPf* = 10-1

FSR forPf* = 10-2

Terzaghi 1.39 1.2 1.39 1.46 1.76Harza 1.05 2.0 1.05 1.62 2.04Harr 3.81 4.0 – 5.0 1.74 1.47 1.80

Simplified Terzaghi 2.12 1.2 1.74 1.47 1.80

Simplified Harza 2.98 1.5 – 2.0 1.69 1.47 1.79

Page 32 of 42



Draft

1

Figure 1. Schematic Diagram of an Excavation

Figure 2. Seepage Tank Model Dimensions (in millimeter) for (a) Single Sheet Piling and (b)

Double Sheet Piling Conditions

Figure 3. Comparison for FSC of (a) Medium Sand Cases and (b) Very Dense Sand Cases

Figure 4. Change of the Heave of the Excavation Surface and Factor of Safety for (a) M0076025,

(b) V0076025, (c) M0076076, and (d) V0076076 Tests Subject to Dewatering

Figure 5. Effective Stress Contour (in kPa) of V0076025 Test in the Excavation Zone at ΔHw/Hp

= (a) 0.07, (b) 0.33, (c) 0.86, (d) 2.04, (e) 4.56, and (f) 6.84. Sand boiling occurred at Fig. 5(e).

Figure 6. Relationships between Pf* and FSR for Terzaghi’s and Harza’s Methods: (a) Medium

Sand and (b) Very Dense Sand

Figure 7. Relationships between Pf* and FSR for (Corrected) Simplified Methods: (a) Medium


Figure 8. Cross-Section of the Tokushima Excavation Plan

Figure 9. Evolutions of (a) the Corrected FSC and (b) the Failure Probability (Pf) with Respect to

Head Difference (ΔHw) for Various Methods.

Page 33 of 42



Draft

hw

Hp

a

b

d e

W'

He

Ue

c

Impermeable Layer

D

CL

B/2

Dow

nward

Flow

Flo

w

Upw

ard

ExcavationZone

RetainedZone

ΔH

w

Hp/2

Wall Toe

Excavation Surface

d i

Figure 1. Schematic Diagram of an Excavation

Page 34 of 42



Draft

Ham River Sand

1375

∆H

w

610

Brass Plates 1/8"Seepage Tank

y

x Wall ToeTank Base

1375

380

Seepage Tank

Tank Base

He

Hp

76

(a)

Ham River Sand

1375B/2

∆H

w

610


CL

y

x Wall ToeTank Base

Ham River Sand

1375B/2

380


Tank Base

He

Hp

76

(b)

Figure 2. Seepage Tank Model Dimensions (in millimeter) for (a) Single Sheet Piling and (b)

Double Sheet Piling Conditions

Page 35 of 42



Draft1.50 2.00 2.50 3.00 3.50 4.00 4.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50Medium Sand

Com

pute

d F

acto

r of

Saf

ety,

FS C

ΔHw/Hp (a)

1.00 2.00 3.00 4.00 5.00 6.00 7.000.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

Com

pute

d Fa

ctor

of

Safe

ty, F

S C

ΔHw/Hp

Very Dense Sand

(b)

Simplified Terzaghi’s MethodSimplified Harza’s MethodHarr’s Method

Harza’s MethodMarsland’s MethodTerzaghi’s Method

Figure 3. Comparison for FSC of (a) Medium Sand Cases and (b) Very Dense Sand Cases

Page 36 of 42



Draft0.0 1.0 2.0 3.0 4.0 5.0

10-4

10-3

10-2

10-1

100

101

10-3

10-2

10-1

1

101

102

ΔHw/Hp

Com

puted Factor of Safety, F

SC

δ v,m

id/H

e FSA

Marsland’s Sand Boiling

t = 1.71 mins

t = 0.03 mins0.050.08

0.15

0.220.35

0.520.78

1.15

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

10-4

10-3

10-2

10-1

100

101

10-3

10-2

10-1

1

101

102

ΔHw/Hp

Com

puted Factor of Safety, F

SC

δ v,m

id/H

e FSA


t = 9.76 mins

6.57

t = 0.10 mins

0.250.47

0.771.23

1.922.92

4.38

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.010-3

10-2

10-1

100

101

10-3

10-2

10-1

1

101

102

ΔHw/Hp

Com

puted Factor of S

afety, FS

C

δ v,m

id/H

e FSA


t = 3.20 mins

t = 0.03 mins

0.08

0.15

0.27

0.420.63

0.971.45

2.15

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10-4

10-3

10-2

10-1

100

101

10-3

10-2

10-1

1

101

102

ΔHw/Hp

Com

puted Factor of S

afety, FS

C

δ v,m

id/H

e FSA


t = 0.15 mins

0.37

0.701.17

1.872.90

4.426.67

9.97

t = 14.82 mins

(c) (d)

Vertical Deformation at 0.5BFSC - Terzaghi’s MethodFSC - Harza’s Method

Figure 4. Change of the Heave of the Excavation Surface and Factor of Safety for (a) M0076025,

(b) V0076025, (c) M0076076, and (d) V0076076 Tests Subject to Dewatering

Page 37 of 42



DraftB/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

B/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

(a) (b)

B/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

B/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

(c) (d)

Critical Zone

B/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

B/2 = 38 mm

Hp

= 2

5 m

m

Hp

3Hp

2 CL

Critical Zone

0.0

(e) (f) Figure 5. Effective Stress Contour (in kPa) of V0076025 Test in the Excavation Zone at ΔHw/Hp

= (a) 0.07, (b) 0.33, (c) 0.86, (d) 2.04, (e) 4.56, and (f) 6.84. Sand boiling occurred at Fig. 5(e).

Page 38 of 42



Draft1.0 1.5 2.0 2.5

10-4

10-3

10-2

10-1

10Medium Sand

Reliability-Based Required FS, FSR

Tar

get F

ailu

re P

roba

bili

ty, P

f*

0

TerzaghiHarza

(a)

1.0 1.2 1.4 1.6 1.8 2.010-4

10-3

10-2

10-1

100

Very Dense Sand


Tar

get F

ailu

re P

roba

bili

ty, P

f*

TerzaghiHarza

(b)

Figure 6. Relationships between Pf* and FSR for Terzaghi’s and Harza’s Methods: (a) Medium


Page 39 of 42



Draft

Medium Sand


Tar

get F

ailu

re P

roba

bili

ty, P

f*

1.0 1.5 2.0 2.510-4

10-3

10-2

10-1

100

Simplified TerzaghiSimplified HarzaHarr

(a)

Very Dense Sand


Tar

get F

ailu

re P

roba

bili

ty, P

f*

1.0 1.2 1.4 1.6 1.8 2.010-4

10-3

10-2

10-1

100

Simplified TerzaghiSimplified HarzaHarr

(b)

Figure. 7. Relationships between Pf* and FSR for (Corrected) Simplified Methods: (a) Medium


Page 40 of 42



Draft

STRUT

GL -1.8 m

GL ±0.00 m

GWL -0.90 m

GL -4.05 m

GL -9.50 m

GL -11.00 m

GL -5.40 m

Hp

= 5

.6 m

He =

5.4

m

SANDY SILTγsat = 18.63 kN/m3

k = 5.19 × 10-6 cm/sic = 0.899

SILTY SANDγsat = 17.91 kN/m3

k = 1.58 × 10-3 cm/sic = 0.827

SANDγsat = 18.51 kN/m3

k = 1.17 × 10-2 cm/sic = 0.888

Sheet Pile Wall(IV-Type)

1.0 m

B/2 = 2.6 mCL

Figure 8. Cross-Section of the Tokushima Excavation Plan

Page 41 of 42



Draft

ΔHw at Failure

2 3 4 5 6 7Head Difference, ΔHw (m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Cor

rect

ed F

S C

(a)

ΔHw at Failure

2 3 4 5 6 7Head Difference, ΔHw (m)

10-4

10-3

10-2

10-1

100

Failu

re P

roba

bili

ty, P

f

(b)

Simplified Harza MethodSimplified Terzaghi MethodHarr's MethodHarza's MethodTerzaghi's Method

Figure 9. Evolutions of (a) the Corrected FSC and (b) the Failure Probability (Pf) with Respect to

Head Difference (ΔHw) for Various Methods.

Page 42 of 42



Calibration of Reliability-Based Safety Factors for Sand · Draft 1 1 CALIBRATION OF RELIABILITY-BASED SAFETY FACTORS FOR 2 SAND BOILING IN EXCAVATIONS 3 4 Ignatius Tommy Pratama1,

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