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INTRODUCTION

Simplified geotechnical design equations are typically biased. For the basal heave stability of deep

excavations, the most popular design equations, such as the ones proposed by Terzaghi (1943) and

Bjerrum and Eide (1956) as well as the slip circle method (JSA 1988, TGS 2001), usually employ

reasonable assumptions to facilitate the calculation of closed-form factors of safety (FS). However,

these assumptions may introduce systematic errors (bias) into the calculated FS. For instance, based

on the more sophisticated MIT-E3 model, Hashash et al. (1996) reported that Terzaghi’s and

Bjerrum and Eide’s methods are less conservative than numerical results in deep clay deposits.

Ukritchon et al. (2003) reported that basal stability can be enhanced by incorporating the stiffness of

wall embedment, which is not modeled by Terzaghi’s and Bjerrum and Eide’s methods. Similar

results were observed by Faheem et al. (2003). Furthermore, realistic characteristics, such as the

staged construction sequence, dewatering, wall stiffness and soil-wall interaction, are typically not

modeled in the simplified design equations. All these characteristics make the quantification of the

biases in the design equations difficult. Up to date, these biases are still not well understood.

With the biases unknown, reliability-based design (RBD) may lead to recommendations that

are inconsistent to engineering practices. For instance, Goh et al. (2008) considered Terzaghi’s and

Bjerrum and Eide’s design equations for RBD. Without considering the possible biases, it was

concluded that the required FS to achieve failure probability of 0.01 is about 2.0. This required FS

of 2.0 is quite high compared to 1.5 that is required in codes (JSA 1988, NAVFAC 1982). Based on

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Goh et al.’s analysis, Wu et al. (2010) further considered the impact of spatial variability in soil

shear strengths, without considering the possible biases. Based on their conclusions, the required FS

is reduced to the range of 1.4-1.9, yet in overall still higher than the recommended 1.5. Besides

Terzaghi’s and Bjerrum and Eide’s equations, Wu et al. (2010) also studied the slip circle equation,

and the required FS to achieve 0.01 failure probability is also higher than the recommended 1.2 in

codes (JSA 1988; TGS 2001). A possible explanation for the aforementioned inconsistency between

analytical studies and engineering practices is that these design equations are more or less biased to

the conservative side.

For RBD, not only the biases of the design equations require calibration, but the uncertainties

associated with these design equations also require calibration. These uncertainties are referred as

“transformation uncertainties” by Phoon et al (1999) for the simplified equations. In essence, the

required FS for a design should not only depend on the bias but also depend on these uncertainties.

This aspect cannot be addressed by the traditional design methods. Calibrating model bias and

uncertainty have been addressed for many geotechnical design problems [e.g., Phoon et al. (2003)

for retaining wall design, Juang et al. (2005) for liquefaction, Phoon et al. (2005) for drilled shaft

design]. However, such calibration for equations of basal heave stability is quite limited in

literature.

In this paper, the bias and uncertainty associated with each of the three design equations are

calibrated by using real case histories with wide excavation where the excavation depths are less

than the excavation widths. In order to separately quantify the transformation uncertainties, efforts

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are also taken to quantify inherent variabilities in soil parameters and measurement errors for all

case histories. The entire calibration is taken under the framework of probabilistic analysis. The

calibration results could provide a basis for RBD of the basal heave stability for wide excavations in

clay.

CASE HISTORIES

In this study, fifteen case histories are collected to calibrate the model bias and uncertainties of the

design equations. The basic information for all case histories related to the analysis is summarized

in Table 1, including failure states, case locations, case geometries and references. Besides the

fifteen real cases, there are four numerical cases done by Hashash et al. (1996) with the MIT-E3 soil

model. Except cases 9 and 15, all cases are wide excavation cases where excavation depths less than

the excavation widths. For failure cases 2 and 12 listed in Table 1, serious collapse due to basal

instability did not really occur. However, the deformation of these two cases was extremely large, so

these two cases are identified as failure cases in this study. For the failure cases, the basal heave

analyses are based on the construction stage right before failures, while for the non-failure cases,

the analyses are based on the final construction stage, except for case 7 where only the documented

information is only available up to the fourth stage.

The excavation depths (He), wall embedment depths (Hd) and excavation widths (B) of the

fifteen real case histories are in the range of 5.5-19.7 m, 1.0-23.7 m and 4.6-62 m, respectively. For

most cases, the total wall length (Hw) equals He + Hd. There are exceptions (cases 2, 11, 12 and 15)

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where the ground surface behind the wall is not level with the top of the wall, so Hw may not equal

He + Hd. For these cases, He is taken to be the distance between the excavation base to the ground

surface. For many cases, the excavation lengths (L) are unknown. However, based on the available

information, most of these cases are judged to be close to plane strain condition. In Table 1, D is the

distance between the excavation base and the hard stratum, and Hs is the distance between the

lowest strut and excavation base, which are needed for the following stability analysis.

For the fifteen real cases, the undrained shear strengths were tested based on various types of

tests, including UC (unconfined compression), CK0UC (traxial K0 consolidated undrained

compression), CK0UE (traxial K0 consolidated undrained extension), DSS (direct simple shear),

FV(field vane) and CPT (cone penetration test), which are denoted by su(UC), su(TC), su(TE),

su(DSS), su(FV) and qc, respectively. The in-situ mobilized su may be different from the tested su

value since su typically depends on stress state, strain rate, sampling disturbance, etc. Based on the

available information, the mobilized su [denoted by su(mob)] is estimated from the above tested

values by following the methods described in Table 2. In Table 2, strain rate corrections are taken to

su(TC), su(TE) and su(DSS): μt is the correction factor for strain rate effect, which is summarized by

Kulhawy and Mayne (1990) as follows:

( )( )101 0.1 logt f labt tμ = − × (1)

where ft is the time to failure in the field, taken to be two weeks for all failure cases as a rough

estimate;

labt is the time to failure in laboratory, which is about 5 hours for consolidated undrained

test. The correction factors for all cases are listed in Table 1. For su(UC), su(FV) and su(CPT), the

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strain rate corrections are typically not taken, according to Mesri (2007).

Based on Table 2, the resulting depth-dependent trends for the estimated mobilized su with

depth are determined for all cases and are plotted in Fig. 1 as the dashed lines. Plotted together are

the corrected tested values of su(UC), su(TC), su(TE), su(DSS), etc. For case 11, the mobilized su is

not estimated based on the TC tests (the marker for case 11 shown in Fig. 1) but based on the

SHANSEP procedure (Ladd and Foote 1974) based on DSS tests.

DESIGN EQUATIONS FOR BASAL HEAVE STABILITY

Before studying the model bias and uncertainties, three design equations are briefly reviewed in this

section. Terzaghi’s and Bjerrum and Eide’s equations are based on bearing capacity theory, while

the slip circle equation is empirical and is widely employed in Asia countries as design codes.

Terzaghi’s method

The schematic of Terzaghi’s model is shown in Fig. 2. The original Terzaghi’s equation is for

stability calculation in clays with constant undrained shear strengths and unit weights. The

calculated FS, denoted by FSC, against basal heave proposed by Terzaghi (1943) is expressed as the

ratio of bearing resistance of saturated clay over the loadings given by the soil weight above

excavation base and surcharge pressure:

(2)

(1)

5.7( )

u TC

e s T u e

s dFSH q d s Hγ

=+ −

(2)

where (1)us is the undrained shear strength of the clay above the excavation base; (2)

us is the

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undrained shear strength of the clay below the excavation base; B is the excavation width; γ is the

unit weight of the soil above the excavation base; eH is the excavation depth; sq is the surcharge

pressure; Td is the radius of the circular arc in Fig. 2.

In reality, undrained shear strengths and unit weights of clays are often not homogeneous. In

particular, undrained shear strengths are usually depth-dependent. Although previous studies [such

as Hashash et al. (1996)] have chosen undrained shear strengths average over depth, this study uses

the undrained shear strengths average along the critical slip lines/arcs to replace (1)us and (2)

us and

uses the average unit weight above the excavation base to replace γ . This implies

5.7( )

bdeu T

C ab abe s T u e

s dFSH q d s Hγ

=+ −

(3)

where abus is the average undrained shear strength along line segment ab; bde

us is the average

undrained shear strength along arc bde; abγ is the average soil unit weight along depth ab. For

convenience of discussion, the average soil parameters along lines or arcs will be mentioned as

“arc-average”, e.g., bdeus is the arc-average su along bde.

An issue in Eq. (3) is that the denominator may be negative, especially when su near ground

surface is significantly larger. This leads to a difficulty in conducting probabilistic analysis: the

denominator may not be modeled as a lognormal random variable. Terzaghi’s equation (Eq. (3)) is

therefore modified as follows:

5.7( )

bde abu T u e

C abe s T

s d s HFSH q dγ

+=

+ (4)

By making this slight modification, both the numerator and denominator are guaranteed

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non-negative, hence the use of lognormal assumption is proper.

In the case of the distance D between the excavation base and the hard stratum is greater than

2B ( 2D B≥ ), the failure surface will be fully developed as shown in Fig. 2 (a). In this case,

Td in Eq. (4) is equal to 2B as shown in Fig. 2(a). Otherwise, the failure circular arc will

intersect with the hard stratum, and Td in Eq. (4) is replaced by D. The recommended FS in codes

is 1.5 for Terzaghi’s method (JSA 1988). For practical purposes, Terzaghi’s method may be suitable

for shallow excavation cases, i.e. He is smaller than B, because extension of the failure surface to

the ground surface is assumed.

For cases 1, 7, 15 and 19, Hp is larger than Td so that the assumed failure surface passes

through the wall. It is more logical to shift down the failure surface so that the shifted failure

surface would not intersect the wall, as shown in Fig. 3. In this case,

( )5.7( )

bde af fbu T u e u d T

C afe s T

s d s H s H dFS

H q dγ+ + −

=+

(5)

where Td remains the same value as in the original Terzaghi’s calculation.

Bjerrum and Eide’s method

The schematic of Bjerrum and Eide’s equation is shown in Fig. 4. For Bjerrum and Eide’s equation,

the FSC against basal heave is similar to that of Terzaghi’s method:

cdefc u

C abe s

N sFSH qγ⋅

=+

(6)

where Nc is the Skempton bearing capacity factor, which can be calculated by the following

equation (Skempton, 1951):

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5(1 0.2 )(1 0.2 )ec

H BNB L

= + +

(7)

where L is the excavation length. If He/B is greater than 2.5, the maximum value of 2.5 is taken.

Compared to Terzaghi’s equation where extension of the failure surface to the ground surface is

assumed, Bjerrum and Eide’s equation seems more reasonable. Similar to Terzaghi’s equation, when

the distance between hard stratum and the excavation base is less than 2B , the development of

failure circular arc is constrained by hard stratum. The recommended FS in codes for Bjerrum and

Eide’s method is 1.2 (JSA 1988).

For cases 1, 7, 15 and 19, the similar issue of failure surface passing through the wall may be

encountered. Again, it is more logical to shift down the failure surface so that the shifted failure

surface would not intersect the wall, as shown in Fig. 5. In this case, He in Eq. (7) is replaced by

He,eq shown in Fig. 5 for calculating FSC.

Slip circle method

The slip circle method has been used as codes in some Asia countries (JSA 1988, TGS 2001) for a

long period of time and is proved to be reliable based on past experiences (Hsieh et al. 2008). The

schematic of the slip circle design equation is shown in Fig. 6. The FSC for the slip circle method is

expressed as:

2 1

2 2

cos2

2 2

cbde su

rC

abde s

Hs rrMFS

r rM H q

π

γ

−⎡ ⎤⎛ ⎞⋅ + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= =⋅ ⋅ + ⋅

(8)

where rM is the resisting moment; dM is the driving moment; r is the radius of the failure

circle (as shown in Fig. 6), i.e. the vertical distance between the lowest strut and the tip of the wall;

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ω is the angle shown in the figure; sq is the surcharge pressure. The recommended FS is 1.2 for

the slip circle method (JSA 1988; TGS 2001).

MEAN VALUE OF FSC FOR ALL CASE HISTORIES

As seen in the preceding section, the calculated factors of safety FSC for various design

equations are functions of arc-average soil parameters and surcharge. Therefore, FSC is in principle

uncertain because arc-average soil parameters and surcharge are always uncertain. The

quantification of the model bias and uncertainties requires the estimation of the mean value and

coefficient of variation (c.o.v.) of the FSC for each case history. The mean value of FSC, denoted by

CFSμ , can be readily evaluated by replacing the arc-average su and unit weights in Eqs. (4), (6) and

(8) with their mean values and by replacing sq by its nominal value. Except for case 2, accurate

estimates for the surcharge pressures are not possible. From sensitivity analysis, the nominal

surcharge pressure in the range of 5-20 kPa is found to have insignificant effect on the analysis

results, so a typical value of 10 kPa for design surcharge pressure ( sq in Table 3) is adopted in this

study.

The mean value of the arc-average su(mob) is estimated based on the depth-dependent trend of

the estimated su(mob) profile (dashed line) shown in Fig. 1. For instance, for Bjerrum and Eide’s

equation, the arc-average su(mob) along arc cdef , i.e. cdefus , is desirable. Its mean value can be

obtained by first mapping the su(mob) trend values onto the arc cdef and followed by integrating

the mapped values along the arc. Similarly, the mean value of the arc-average unit weight abγ can

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be obtained by integrating the depth-dependent unit weight trend along the line ab. The resulting

mean values for the arc-average su(mob) and unit weights are summarized in Table 3 for all cases,

together with the assumed nominal values of sq . The resulting CFSμ will be summarized in a later

table for all cases.

COEFFICIENT OF VARIATION OF FSC FOR ALL CASE HISTORIES

FSC depends on the arc-average su, arc-average unit weight and surcharge. Therefore, the c.o.v. of

FSC, denoted by CFSδ , should depend on the c.o.v.s of these parameters. In the following, the

relationship between the c.o.v. of FSC and the c.o.v.s of the input parameters will be derived for

modified Terzaghi’s equation. For the other two design equations, this relationship can be readily

derived based on the same principle. For the ease of discussion, the denominator and numerator of

modified Terzaghi’s equation are treated as loading (P) and resistance (Q), respectively:

5.7 ( )bde ab abu T u e e s TQ s d s H P H q dγ= + = + (9)

By assuming independence between bdeus and ab

us , the c.o.v. of Q, denoted by δ(Q), can be

evaluated by

( )( ) ( )2 2

5.7

5.7

bde bde ab abT u u e u u

bde abu T e u

d s s H s sQ

s d H s

δ δδ

⎡ ⎤ ⎡ ⎤⋅ ⋅ ⋅ + ⋅ ⋅⎣ ⎦ ⎣ ⎦=⋅ ⋅ + ⋅

(10)

By similarly assuming independence between abγ and sq , the c.o.v of P, denoted by δ(P), can be

evaluated as

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( )( ) ( )

2 2ab abe T s T s

abe T s T

H d q d qP

H d q d

γ δ γ δδ

γ

⎡ ⎤ ⎡ ⎤⋅ ⋅ ⋅ + ⋅ ⋅⎣ ⎦⎣ ⎦=⋅ ⋅ + ⋅

(11)

By further assuming lognormality for both P and Q and independence between P and Q, the c.o.v. of

FSC, denoted by CFSδ , can be estimated by the following equation:

( ) ( )( )2 2exp log 1 log 1 1CFS Q Pδ δ δ⎡ ⎤ ⎡ ⎤= + + + −⎣ ⎦ ⎣ ⎦ (12)

Furthermore, FSC is also lognormal because the lognormal distribution is preserved under the

operation of multiplications and divisions. As a consequence, CFSδ depends on δ(Q) and δ(P),

which in turn depend on ( )bdeusδ , ( )ab

usδ , ( )abδ γ and ( )sqδ . Among them, the estimation of

( )bdeusδ , ( )ab

usδ and ( )abδ γ deserves further discussion. The discussion will be taken for the

estimation of the c.o.v. for arc-average su. For the estimation of ( )abδ γ , the same principle holds.

First of all, the observed variability in the su data points seen in Fig. 1 is not the same as the

variability for the arc-average su. Table 4 lists the c.o.v.s of the observed variabilities of su for all

cases. The observed variability in the raw su data points is composed of the point-wise inherent

variability and the measurement error, while the variability for the arc-average su involves taking

averaging of lots of point-wise su along arcs.

In fact, the variability for the arc-average su will be typically smaller than the observed

variability in the su data points because the inherent variability of the point-wise su will be reduced

during the along-arc averaging and the variability due to measurement errors will be reduced during

the data averaging. The mechanisms of variability reduction for these two sources of variabilities

are different and independent. The variability reduction by along-arc averaging applies to the

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inherent variability of su and is due to the spatial averaging [as noted by Vanmarcke (1977)] along

arcs. On the other hand, the mechanism for the variability reduction in the measurement errors is

rather statistical: e.g., the variance of the average of N data points is equal to 1/N of their original

variance. The former reduction depends on the arc length but not on the number of raw su data

points, while the latter reduction is on the contrary. As a consequence, the c.o.v. of the arc-average

su can be evaluated by the following equation:

2 2 2i i m m tR Rδ δ δ δ= ⋅ + ⋅ +

(13)

where iδ is the c.o.v. of (point-wise) inherent variability of su; iR is the variance reduction factor

for the along-arc averaging; mδ is the c.o.v. of measurement error; mR is the variance reduction

factor for measurement errors; tδ is the c.o.v. of transformation uncertainty if transformation

equations are required to convert tested values (e.g., field vane and CPT) to su values. For the

arc-average unit weights, their c.o.v.s can be estimated by using the same formula.

Estimation of the c.o.v. of arc-average su

Estimation of δm and δi

As stated above, the observed variability in the su data points is composed of the point-wise

inherent variability in su (with c.o.v. = δi) and the measurement error for su (with c.o.v. = δm).

Although it is impossible to separately estimate iδ and mδ from the observed variability, it is for

sure that mδ is less than the c.o.v. of the observed variability. The ranges and average values of

c.o.v.s for the measurement error for various types of su tests, as documented by Phoon (1995), are

listed in Table 5. Since mδ cannot be larger than the c.o.v. of the observed variability, it can be

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roughly estimated to be the closest value in Table 5 that is also smaller than the observed c.o.v. For

instance, in case 6, the c.o.v. of the observed variability is 0.54 (see the second column in Table 4),

and the su test type for case 6 is the CU test. From Table 5, it is known that the upper, mean and

lower c.o.v. values for the CU measurement errors are 0.05, 0.15 and 0.40, respectively. Among the

three c.o.v. values, 0.40 is closest to and smaller than 0.54, and therefore 0.40 is taken to be a rough

estimate of mδ for case 6. Once this mδ estimate is obtained, iδ can be found by

( )2 2c.o.v. of observed variabilityi mδ δ= − (14)

Table 4 lists the resulting iδ and mδ estimates for all cases.

Estimation of Rm and Ri

In the calculation of FS, FSC depends on the developed dashed lines in Fig. 1 [depth-dependent

trends for the estimated su(mob)] rather than on the raw data points. These dashed lines contain less

measurement errors since they are the more stable regression lines of the noisy data points.

Therefore, the measurement error does not fully but partially propagate into FSC. As a result, the

magnitude of mδ should be reduced depending on the amount of available data points. In the cases

where the dashed lines are vertical (i.e. the trend is constant with depth), the variance reduction

factor Rm should be 1/N, N is number of available data points. For dashed lines with linearly

increasing/decreasing trends, Rm can be estimated by using simple statistical analysis. The resulting

estimates for Rm are plotted in Fig. 7 and listed in Table 6 for all cases. In Fig. 7, the value of Rm

turns out to be very close to 2/N. Here, the number of available data points N is the number of data

points to evaluate the relevant dashed lines in Fig. 1. For instance, for analysis of case 6 using

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modified Terzaghi’s equation, the excavation depth is 12.2m and wall length is 19.2m. There are

nine data points to evaluate the dashed lines within the depth interval [0m, 12.2m] (see Fig. 1): this

is the depth interval for evaluating abus of modified Terzaghi’s equation, hence N = 9 for Rm of ab

us

in case 6. For the same case, there are eight data points to evaluate the dashed lines within the depth

interval [12.2m, 19.2m]: this is the depth interval for evaluating bdeus of modified Terzaghi’s

equation, hence N = 8 for Rm of bdeus in case 6.

As stated above, only a fraction of iδ propagates into the c.o.v. of the arc-average su due to

spatial averaging. The variance reduction factor Ri can be estimated by actually simulating random

fields of su profiles along the arcs and conducting averaging there. The same simulation techniques

described in Wu et al. (2010) are employed in this study for the purpose, with the assumption that

the vertical scale of fluctuation for su is 2.5m, the average scale of fluctuation for su summarized in

Phoon (1995). The estimated Ri value is simply the variance of the simulated arc-average su divided

by the inherent variance of su. Note that the variance reduction factors Ri are different for the three

design equations since the assumed failure surfaces (i.e. the arcs) are different.

The estimated Ri are also listed in Table 6 for all 15 cases and for all three design equations,

and the relation between Ri and the extent of spatial averaging is shown in Fig. 8. The so-called

extent of spatial averaging is defined differently for the three design equations: for modified

Terzaghi’s equation, the extent is dT for bdeus and is He for ab

us ; for Bjerrum and Eide’s equation,

the extent is dT for cdefus ; for the slip circle equation, the extent is r (circle radius) for cbde

us .

Estimation of δt

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For cases 4 and 5, the su(mob) profiles are estimated based on the results of field vane (FV),

while for case 15, the su(mob) profile is estimated based on the CPT results. For these cases, there

are transformation uncertainties in the estimated su(mob). For the other cases, δt are taken to be

zeros since su(mob) is directly from laboratory undrained shear strength tests. There is no consensus

on how to estimate δt for FV results to su(mob). In this study, the c.o.v. of the correction factor λ

(see method 4 in Table 2) of the historical data discussed in Bjerrum (1972) is taken to be the δt,

which is estimated to be 0.15. The δt for CPT results to su(mob) is roughly 0.35, as concluded by

Phoon (1995).

Estimation of the c.o.v. of soil unit weight

In principle, Eq. (13) still holds for the c.o.v. of the arc-average unit weights but compared to their

inherent variabilities, the measurement errors for unit weights are much smaller. As a consequence,

δm is taken to be zero without losing much accuracy. Moreover, there is no transformation needed

because unit weights are typically directly measured. Therefore, the c.o.v. of abγ is simply i iRδ .

As investigated by Phoon (1995), δi is around 0.1 for unit weights of fine grained soils, which is

adopted in this study. The variance reduction factor Ri is simply 1 divided by the number of

equivalent independent samples along line segment ab, i.e. Ri = (scale of fluctuation)/(length of ab).

The scale of fluctuation is taken as 5 m for the unit weights of fine grained soils, the average value

summarized by Phoon (1995).

Adopted c.o.v. of surcharge pressure

Equation (13) is not implemented to estimate the c.o.v. for qs because there is no spatial and data

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averaging involved in qs. Instead, its c.o.v. is directly assumed to be 0.2 for all cases in this study.

This value was taken in Goh et al. (2008) and Wu et al. (2010). Sensitivity studies show that this

choice does not significantly affect the analysis results, provided that this c.o.v. is in the range of

0.05-0.3.

Table 7 summaries the resulting c.o.v. estimates for all input parameters, including the

arc-average su, arc-average unit weights and qs, for all cases and for all design equations.

Resulting mean values and c.o.v.s of FS for all cases

Based on the mean values and c.o.v.s of all input parameters summarized in Tables 3 and 7, the

mean values of FSC (CFSμ ) for all cases can then be obtained by substituting the input parameters

listed in Table 3 into Eqs.(4),(6) and (8), while the c.o.v.s of FSC can be obtained by applying Eqs.

(9)-(12) with the c.o.v.s listed in Table 7. The resulting mean values and c.o.v.s of FSC for all cases

and for all design equations are summarized in Table 8 and plotted in Fig. 9.

Note that for the failure cases, the actual FS, denoted by FSA, should be unity. However, for

the non-failure cases, it is only certain that their FSA is greater unity. As a consequence, the

non-failure cases do not offer as much information as the failure cases do. For instance, there is one

case whose CFSμ for modified Terzaghi’s equation is greater than 4.5. This does not imply that

modified Terzaghi’s equation is terribly inaccurate for this case. In contrast, there is one case whose

CFSμ for the slip circle method is nearly 0.5, but this indeed implies that the slip circle equation is

quite conservative for this case. Although the non-failure cases do not offer as much information as

the failure cases do, they still offer certain amount of information, especially for non-failure cases

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with small CFSμ . On the other hand, the cases with smaller

CFSδ offer more information than those

with large c.o.v.s.

By comparing CFSμ for the failure cases (open circles in the figure) with the limit state FS = 1,

it is obvious that modified Terzaghi’s equation is nearly unbiased, while the other two design

equations are more or less biased to the conservative side. Modified Terzaghi’s and the slip circle

equations perform well with respect to the non-failure cases (solid circles), judging from the fact

that none of the non-failure cases has CFSμ less one. For Bjerrum and Eide’s equation, three

non-failure cases have CFSμ less than one. This may be either due to the bias in the equation or due

to the higher variability in FSC.

It is of interest to compare the results shown in Fig. 9 with well documented numerical failure

cases. One set of such examples (cases 16-19 in Table 1) are the numerical failure cases investigated

by Hashash et al. (1996) by using the MIT-E3 model. For all the failure cases, the DSS undrained

shear strength and unit weights profiles are known. The excavation stages are simulated by

incrementally removing soil layers with thickness of 2.5 m, up to the stage of numerical divergence.

As a result, the FS for the excavation geometries at the point of divergence should be close to and

slightly less than one.

With the excavation geometries at the point of divergence, the given DSS su profiles and soil

unit weights are therefore taken to obtain the depth-dependent trends for su(mob) and unit weight,

and FSC for the three design equations can then be calculated based on these trends. The resulting

FSC are plotted on the horizontal axes in Fig. 9. The FSC for these numerical failure cases should be

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directly compared to the open circles (real failure cases) in Fig. 9. However, as a first glance, the

FSC for these cases seem to have different patterns compared to the open circles. Moreover, the FSC

for the four numerical cases can be either all less than one (slip circle), all greater than one

(modified Terzaghi) or the mixture of the above two (Bjerrum and Eide), depending on which

design equation is used. In other words, the FS predicted by the numerical MIT-E3 model is on

average more conservative than modified Terzaghi’s equation, on average less conservative than the

slip circle equation and on average similar to Bjerrum and Eide’s equation. The above conclusion

for the comparisons over modified Terzaghi’s and Bjerrum and Eide’s equation is largely consistent

to the findings by Hashash et al. (1996).

CALIBRATION OF MODEL FACTOR

As discussed in the preceding section, the actual FS (FSA) for the failure cases (they should be

exactly unity) may be different from CFSμ for the three design equations. It is quite possible that

FSC from the design equations should be corrected by a model factor:

A CFS FSα= ⋅

(15)

where FSA is the actual FS; α is the model factor; FSC is the FS calculated from the design

equations. Moreover, α should not be a fixed number among all cases because the deviations

between FSA and FSC vary throughout all cases. In this study, α is modeled as a lognormal

random variable with mean value αμ and c.o.v. αδ . Note that these mean value and c.o.v. should

depend on the adopted design equation. Since FSC is lognormal, FSA is also lognormal. Moreover,

19

the mean value and c.o.v. of AFS are

( ) ( )2 21 1 1A C A CFS FS FS FSα αμ μ μ δ δ δ= = + ⋅ + −

(16)

by assuming independence between α and FSC.

Numerous methods can be taken to calibrate αμ and αδ based on the real failure cases, e.g.,

simply calculate the sample average and sample c.o.v. of 1/CFSμ for all failure cases may yield

possible estimates for αμ and αδ . However, there are only six failure cases in the database.

Therefore, it is important to incorporate the information contained in the non-failure cases as well:

these cases still possess significant information for αμ and αδ . Only few methods are able to

achieve so. One of such methods is the maximum likelihood method, which is to be described in

details in the following.

For the six failure cases, FSA is known to be one, so it is possible to write down the likelihood

function:

( ) ( )2 22

( , ) ( 1| , , , )

1 1 1exp ln2 2ln 1 1ln 1

C C

A

A AA

f A FS FS

FS

FS FSFS

L p FSα α α αμ δ μ δ μ δ

μ

π δ δδ

= =

⎧ ⎫⎡ ⎤⎛ ⎞−⎪ ⎪⎢ ⎥⎜ ⎟= ⋅ ⋅ −⎨ ⎬⎢ ⎥⎜ ⎟+ ++ ⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

(17)

where ( )p ⋅ is the probability density function.

For the nine non-failure cases, FSA are greater than one, and it is still possible to write down

the likelihood function:

( )( )

2

2

ln 1( , ) ( 1| , , , ) 1

ln 1

A A

c c

A

FS FS

nf A FS FS

FS

L P FSα α α α

μ δμ δ μ δ μ δ

δ

⎛ ⎞− +⎜ ⎟= > = − Φ ⎜ ⎟+⎜ ⎟

⎝ ⎠

(18)

20

where ( )P ⋅ is the probability, and ( )Φ ⋅ denotes the cumulative density function for the standard

Gaussian distribution. Since the fifteen cases are statistically independent, the total likelihood

function can be expressed as the multiplication of each individual likelihood function:

6 9( ) ( )

1 1

( , ) ( , ) ( , )i jf nf

i j

L L Lα α α α α αμ δ μ δ μ δ= =

= ⋅∏ ∏

(19)

Based on the principle of maximum likelihood, the ( αμ , αδ ) pair that maximizes the total likelihood

is an optimal estimate for the pair. The contour diagrams for the total likelihood are plotted in Fig.

10 for the three design equations. All of the total likelihoods seem to be uni-modal, indicating that

the optimal estimates for ( αμ , αδ ) are unique. The resulting optimal estimates for ( αμ , αδ ) are

listed in Table 9 for all three design equations.

From the calibration results, modified Terzaghi’s equation is the least biased with a reasonably

small model c.o.v., while Bjerrum and Eide’s equation is somewhat more conservative with the

smallest model c.o.v. The slip circle equation is the most conservative with the largest model c.o.v.

It is interesting to see that modified Terzaghi’s and Bjerrum and Eide’s equations, which are derived

from bearing capacity theories, seems to have less model uncertainties than the slip circle, which is

fundamentally more empirical. It is also interesting to see modified Terzaghi’s equation, which is

long believed to be suitable for cases with excavation depths less than excavation widths (e.g., wide

excavation), has the least bias.

The recommended FS in codes for the three design equations are also listed in Table 9. At the

first glance, the recommended values are quite different, 1.5 for Terzaghi’s equation and 1.2 for

21

Bjerrum-Eid’s and the slip circle equations. According to the above discussion, these recommended

FS in codes should not be directly compared without the adjustment of the biases. This adjustment

can be easily done by multiplying the recommended FS in codes by the corresponding bias αμ ,

resulting in the “recommended FS after bias correction” in Table 9. It is clear that after the bias

correction, the recommended FS is now much uniform. Note that for the slip circle equation, the

recommended FS after correction is the largest: this is because the model uncertainty for the slip

circle equation is the largest.

RELIABILITY-BASED DESIGN FOR BASAL HEAVE STABILITY

With the model factors calibrated, the basal heave failure probability can then be expressed as a

function of AFSμ and

AFSδ as follows:

( )( )

( )( )

2

2

log 11

log 1

A A

A

FS FS

A

FS

P FSμ δ

βδ

⎛ ⎞+⎜ ⎟< = Φ − = Φ −⎜ ⎟+⎜ ⎟

⎝ ⎠

(20)

where β is the reliability index, and AFSμ and

AFSδ are related to αμ , CFSμ , αδ and

CFSδ

through Eq. (16). The values of αμ and αδ can be found in Table 9. For reliability-based design,

a design engineer is given the target reliability index βT, and he/she must determine the required

CFSμ to achieve this target level of safety. If an estimate of CFSδ is available, the design engineer

can easily find the required CFSμ by inverting Eq. (20):

( ) ( )

( ) ( ) ( ) ( )

2 2

2 2 2 2

1 exp log 1

1 1 exp log 1 log 1

A A

C

c c

TFS FS

FS

TFS FS

α

α α

α

δ β δμ

μ

δ δ β δ δ

μ

⎡ ⎤+ ⋅ +⎢ ⎥⎣ ⎦=

⎡ ⎤+ ⋅ + ⋅ + + +⎢ ⎥⎣ ⎦=

(21)

22

Figures 11-13 show the relation between required CFSμ and βT for various levels of

CFSδ for

modified Terzaghi’s, Bjerrum and Eide’s and the slip circle design equations, respectively. Note that

the required design FS CFSμ is the FS based on su(mob), so the design engineer needs to make sure

all su measurement to be converted into su(mob) when calculating the FS (e.g., following Table 2).

Note that the aforementioned design requires an estimate for CFSδ . From Figs. 11-13, it is clear

that the resulting design FS (i.e. CFSμ ) is very sensitive to the

CFSδ estimate, hence the estimation

for CFSδ must be done cautiously. One will first need to estimate the c.o.v.s for all input parameters

for the chosen design equation. Taking modified Terzaghi’s equation as an example, they are

( )bdeusδ , ( )ab

usδ , ( )abδ γ and ( )sqδ . For ( )bdeusδ and ( )ab

usδ , Eq. (13) should be used to

find these c.o.v.s, given the , , , ,i m t i mR Rδ δ δ for each parameter. When iδ , mδ and tδ are not

able to be obtained from site investigation, they can be roughly estimated based on the statistics

compiled by Phoon (1995). When site investigation is already done, the steps described in the

section “coefficient of variation of FSC for all case histories” can be taken to estimate iδ , mδ and

tδ . Estimates for mR and iR can be found from Figs. 7 and 8 according to the number of real

tested data points and the dimension of the excavation. For ( )abδ γ , mδ and tδ can be taken to

be zeros, iδ can be found either from the site investigation results or from Phoon (1995), iR can

be taken to be (5m)/(excavation depth). Once the c.o.v.s for all input parameters are estimated, CFSδ

can be obtained by following Eqs. (10)-(12) for modified Terzaghi’s equation. For other two design

equations, the similar procedures as shown by Eqs. (10)-(12) can be found to obtain CFSδ estimates,

although Eqs. (9)-(11) need to be re-derived.

As mentioned, the recommended FS in current codes for modified Terzaghi’s, Bjerrum and

Eide’s and slip circle methods are 1.5, 1.2 and 1.2, respectively. It is of interest to investigate the

23

actual target reliability index for these code regulations. Obviously, the conclusion depends on CFSδ .

For the fifteen real case histories, most CFSδ ranges from 0.05 to 0.2. Under this range, the required

FS of 1.5 for modified Terzaghi’s equation corresponds to a target reliability index ranging from 1.8

to 4.6, the required FS of 1.2 for Bjerrum and Eide’s equation corresponds to a target reliability

index ranging from 2 to 5, and the required FS of 1.2 for the slip circle equation corresponds to a

target reliability index ranging from 1.6 to 2.4. Note that the narrow range of [1.6, 2.4] for the slip

circle method is because the c.o.v. of its model factor is relatively large, so the change of CFSδ does

not affect the βT range as much as the other two equations. Further investigation for Figs. 11-13

reveals that when CFSδ is in the range of [0.2, 0.3], the target reliability indices for the three design

equations corresponding to the code regulations do not differ much.

CONCLUSIONS

In this study, the three popular design equations of basal heave are calibrated based on real case

histories with excavation depths less than the excavation widths (i.e. wide excavation). The actual

FS is expressed as the FS calculated from design equations corrected by a model factor α due to

model uncertainty, and the probabilistic characterization of model factor for three design equations

are calibrated from real case histories. From the calibration results, the mean values of model factor

( αμ ) are 1.01, 1.31 and 1.39 and its corresponding c.o.v. ( αδ ) are 0.072, 0.064 and 0.21 for

modified Terzaghi’s, Bjerrum and Eide’s, and slip circle design equations, respectively. Modified

Terzaghi’s equation is the least biased with a reasonably small model c.o.v., while Bjerrum and

24

Eide’s equation is somewhat more conservative with the smallest model c.o.v. The slip circle

equation is the most conservative with the largest model c.o.v. It is interesting to see that modified

Terzaghi’s and Bjerrum and Eide’s equations, which are derived from bearing capacity theories,

seems to have less model uncertainties than the slip circle, which is fundamentally more empirical.

It should be fair to say that modified Terzaghi’s equation is the most accurate and reliable equation

for determining FS for wide excavation.

Before the bias correction, the recommended FS in current codes for the three design equations

are rather non-uniform: 1.5 for modified Terzaghi’s equation and 1.2 for Bjerrum-Eide’s and slip

circle equations. Nonetheless, after the bias correction, the resulting recommended FS are quite

uniform: 1.52, 1.57 and 1.67 for modified Terzaghi’s, Bjerrum-Eide’s and slip circle equations,

respectively. Note that for the slip circle equation, the recommended FS after correction is the

largest: this is because the model uncertainty for the slip circle equation is the largest.

Compared to previous studies, the required FS calibrated by this study is more consistent to

engineering practices. For instance, in order to achieve reliability index of 2 (failure probability =

0.0228) with modified Terzaghi’s equation, from Fig. 11 it is clear that a FSC of 1.55 is needed if

CFSδ is 0.2. From the results of Goh et al. (2008), the required FSC is roughly 1.65. In order to

achieve the same reliability index with Bjerrum-Eide’s equation, from Fig. 12 it is clear that a FSC

of 1.2 is needed if CFSδ is 0.2. From the results of Goh et al. (2008), the required FSC is nearly 1.8.

Note that the required FSC in codes is 1.5 for Terzaghi’s equation and is 1.2 for Bjerrum-Eide’s

equation. These requirements are closer to the conclusion of this study, probably because the biases

25

of these two equations have been considered in this study. In order to achieve the same reliability

index with the slip circle equation, from Fig. 13 it is clear that a FSC of 1.32 is needed if CFSδ is

0.2. From the results of Wu et al. (2010), the required FSC is around 1.5-1.65. Note that the required

FSC in codes is 1.2 for the slip circle equation. This requirement is closer to the conclusion of this

study, probably because the bias of the slip circle equation has been considered.

Furthermore, the recommended FS values in current codes for the three design equations are

verified, which shows that the required FS of 1.5 for modified Terzaghi’s equation corresponds to a

target reliability index ranging from 1.8 to 4.6, the required FS of 1.2 for Bjerrum and Eide’s

equation corresponds to a target reliability index ranging from 2 to 5, and the required FS of 1.2 for

the slip circle equation corresponds to a target reliability index ranging from 1.6 to 2.4.

For the purpose of reliability-based design, design charts that relate required FS and target

reliability index are provided for the three design equations and for various levels of FS c.o.v. With

the target reliability index prescribed, the required design FS can be obtained from these charts to

facilitate reliability-based design for basal heave stability for wide excavation. These charts are

calibrated against real cases with excavation depths less than excavation widths. For design cases

with depths greater than widths, cautions and judgments should be taken when implementing these

design charts.

26

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Engineering, 134(2), 145-153.

Hsieh, P.G., Ou, C.Y. and Liu, H.T. (2008). “Basal heave analysis of excavations with

consideration of anisotropic undrained strength of clay.” Canadian Geotechnical Journal, 45(6),

788-799.

Hashash, Y.M.A. and Whittle, A.J. (1996). “Ground movement prediction for deep excavations in

soft clay.” ASCE Journal of Geotechnical Engineering, 122(6), 474-486.

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Ph.D. Dissertation, Cornell University.

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29

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30

Table 1 Basic information of the case histories

Case number Location (country) Failure

State

Method1for

estimating

su(mob) tμ He

(m)Hd (m)

Hw2

(m) L

(m)B

(m)D

(m) Hs

(m)Reference

1 San Francisco (U.S.A) Failure 1 - 7.30 6.40 13.70 n/a3 7.60 >11 0 Clough et al. (1984)

2 Oslo (Norway) Failure 2 0.82 10.35 1.00 9.55 19 13.00 >9.65 1.35 Aas (1984)

3 Taipei (Taiwan) Failure 2 0.82 13.45 10.55 24.00 100 25.80 >31.25 3.3 Hsieh et al. (2008)

4 Singapore (Singapore) Non-failure 4 - 13.00 23.70 36.70 174 48.00 23.7 3.9 Wallace et al. (1992)

5 Taipei (Taiwan) Non-failure 4 - 19.70 15.30 35.00 80 40.00 17.8 3.2 Ou et al. (1998)

6 Chicago (U.S.A) Non-failure 1 - 12.20 7.00 19.20 n/a 12.20 10.1 2.8 Finno et al. (1989)

7 (Japan) Non-failure 1 - 12.40 23.60 36.00 n/a 16.20 23.6 4.55 Chang et al. (1980)

8 Tokyo (Japan) Non-failure 1 - 11.00 13.25 24.25 n/a 34.75 >53 2.2 Tanaka (1994)

9 San Francisco (U.S.A) Non-failure 1 - 9.20 4.50 13.70 n/a 7.60 >9.1 0 Clough et al. (1984)

10 Taipei (Taiwan) Non-failure 1 - 14.10 15.70 29.80 68 62.00 23.9 2.3 Fang (1987)

11 San Francisco (U.S.A) Non-failure 3 0.82 13.70 13.70 24.50 68 37.40 13.7 2.7 O’Rourke (1992)

12 (Norway) Failure 1 - 11.0 5.40 14.40 100 11.00 5.4 1 NGI (1962); Mana (1978);

13 Taipei (Taiwan) Failure 2 0.82 9.30 6.10 15.40 45 12.30 >20.7 3.3 Hsieh et al. (2008)

14 Taipei (Taiwan) Failure 1 - 9.20 5.80 15.00 64.8 22.50 13.9 2.45 Wang et al (1996)

15 Central Italy (Italy) Non-failure 5 - 5.5 5.50 12.00 n/a 4.60 >16.0 1 Rampello (1992)

16 - Failure - - 10 2.50 12.50 ps4 40.00 110.0 2.5 Hashash et al (1996)

31

17 - Failure - - 15 5.00 20.00 ps 40.00 105.0 2.5

18 - Failure - - 22.5 17.50 40.00 ps 40.00 97.5 2.5

19 - Failure - - 30 30.00 60.00 ps 40.00 90.0 2.5

Remarks: 1 details described in Table 2 2 Hw is the wall length 3 information not available 4 plane strain

32

Table 2 Estimation methods for the mobilized su

Method for

estimating

su(mob)

Available information Equation Reference

1 su(UC) only su(mob) ≈ su(UC) Mesri (2007) 2 su(TC), su(TE), su(DSS) su(mob) ≈ μt [su(TC)+su(TE)+su(DSS)]/3

3 su(DSS) only su(mob) ≈ μt su(DSS) 4 su(FV) only su(mob) ≈ λ su(FV) Bjerrum (1972)

5 qc only su(mob) ≈ average of su(CPT)

su(CPT) = (qc-σv1)/16

Mesri (2007)

1 σv is the total vertical stress

33

Table 3 Mean and nominal values used in calculating the mean values of FSC

Case number

sq (kPa)

abγ (kN/m3)

Modified Terzaghi(kPa)

Bjerrum-Eide (kPa)

Slip circle (kPa)

bdeus ab

us cdefus cbde

us

1 10 14.30 19.81 16.08 16.40 19.69

2 5 18.50 35.36 14.73 26.32 25.68

3 10 19.65 62.31 31.49 50.51 53.55

4 10 16.29 42.78 20.06 36.60 42.63

5 10 18.27 109.52 50.85 84.20 103.27

6 10 19.00 57.22 12.17 39.14 49.73

7 10 14.70 128.81 74.52 99.93 114.87

8 10 15.00 55.56 18.23 48.90 47.56

9 10 14.70 40.29 16.89 29.44 38.85

10 10 16.68 49.26 30.58 41.67 46.99

11 10 18.35 58.82 27.61 46.05 58.47

12 10 19.10 22.63 41.14 25.32 23.15

13 10 17.90 32.41 20.31 25.91 29.32

14 10 18.50 23.08 17.05 19.21 19.79

15 10 18.40 44.97 33.94 38.01 43.10

16 0 18.00 52.21 13.11 47.18 24.17

17 0 18.00 60.81 17.62 52.22 35.59

18 0 18.00 73.71 24.21 58.29 62.27

19 0 18.00 89.56 32.22 65.65 88.87

34

Table 4 c.o.v.s of the observed variabilities of inherent variability and measurement error

Case number

Adopted su test type

Observed c.o.v. in su(mob)

Estimated level for

measurement error

Resulting δm[su]

estimate

Resulting δi[su]

estimate 1 CU 0.11 Lower bound 0.05 0.10

2 CU 0.13 Lower bound 0.05 0.12

3 CU 0.047 - 0.001 0.047

4 FV 0.25 Upper bound 0.20 0.14

5 FV 0.12 Lower bound 0.10 0.069

6 CU 0.54 Upper bound 0.40 0.37

7 CU 0.24 Average 0.15 0.19

8 CU 0.36 Average 0.15 0.32

9 CU 0.12 Lower bound 0.05 0.11

10 CU 0.23 Average 0.15 0.18

11 CU 0.53 Upper bound 0.40 0.35

12 CU 0.21 Average 0.15 0.15

13 CU 0.041 - 0.001 0.041

14 CU 0.49 Upper bound 0.40 0.27

15 CPT 0.43 Average 0.15 0.41 1 c.o.v. in observed su(mob) is less than the lower bound of measurement error

35

Table 5 Ranges of measurement error c.o.v. of su summarized from Phoon (1995)

CU tests Field vane CPTLower bound 0.05 0.10 0.05

Average 0.15 0.15 0.10 Upper bound 0.40 0.20 0.15

36

Table 6 Summary of the Rm and Ri for three different design equations of basal heave

Case Number

mR mR iR iR

N1

Modified Terzaghi

N

Modified Terzaghi/ Bjerrum-Eide/slip circle

Modified Terzaghi

Bjerrum- Eide

Slip circle

abus bde

us / cdefus / cbde

us abus bde

us

cdefus cbde

us abγ

1 2 1 14 0.14 0.30 0.44 0.28 0.39 0.692 2 1 4 0.50 0.29 0.30 0.27 0.75 0.483 - -2 - - 0.19 0.17 0.17 0.25 0.374 27 0.073 28 0.07 0.20 0.15 0.14 0.15 0.395 2 1 5 0.40 0.13 0.18 0.16 0.20 0.256 9 0.21 8 0.24 0.23 0.34 0.31 0.38 0.417 18 0.11 9 0.22 0.11 0.24 0.17 0.15 0.408 20 0.10 40 0.05 0.23 0.14 0.14 0.22 0.469 4 0.50 6 0.33 0.29 0.45 0.36 0.52 0.5410 2 1 8 0.25 0.19 0.13 0.13 0.18 0.3611 5 0.38 7 0.27 0.18 0.23 0.20 0.23 0.3712 14 0.14 6 0.33 0.27 0.41 0.22 0.39 0.4613 - - - - 0.27 0.31 0.25 0.35 0.5414 17 0.11 20 0.094 0.27 0.21 0.20 0.37 0.5415 24 0.083 11 0.18 0.33 0.61 0.39 0.42 0.91

Remark: 1 total number of raw data points. 2 the value of mδ is set to be zero, therefore Rm is not

needed.

37

Table 7 c.o.v.s of input parameters for all cases and for all design equations

Case number

Modified Terzaghi Bjerrum-Eide Slip circle ( )abδ γ ( )sqδ

( )abusδ ( )bde

usδ ( )cdefusδ ( )cbde

usδ

1 0.075 0.070 0.057 0.066 0.083 0.20 2 0.081 0.073 0.071 0.11 0.070 0.20 3 0.021 0.020 0.020 0.024 0.061 0.20 4 0.17 0.17 0.17 0.17 0.062 0.20 5 0.18 0.17 0.17 0.17 0.050 0.20 6 0.25 0.29 0.28 0.30 0.064 0.20 7 0.079 0.12 0.11 0.10 0.064 0.20 8 0.16 0.12 0.12 0.16 0.067 0.20 9 0.068 0.078 0.071 0.083 0.074 0.20 10 0.17 0.098 0.098 0.11 0.060 0.20 11 0.29 0.27 0.26 0.27 0.060 0.20 12 0.095 0.13 0.11 0.13 0.067 0.20 13 0.021 0.023 0.021 0.024 0.073 0.20 14 0.20 0.18 0.17 0.21 0.074 0.20 15 0.42 0.48 0.44 0.44 0.095 0.20

38

Table 8 Mean values and c.o.v.s of FSC of all cases and for all design equations

Case number

Modified Terzaghi

Bjerrum-Eide Slip circle

CFSμ CFSδ

CFSμ

CFSδ CFSμ

CFSδ

1 1.21 0.097 0.87 0.096 1.08 0.10 2 0.91 0.096 0.72 0.098 0.54 0.13 3 1.13 0.062 0.87 0.062 0.93 0.064 4 1.15 0.17 0.92 0.18 1.15 0.18 5 1.84 0.16 1.38 0.17 1.66 0.17 6 1.42 0.28 0.97 0.29 1.17 0.31 7 4.65 0.11 3.39 0.12 3.56 0.12 8 1.86 0.14 1.49 0.14 1.63 0.17 9 1.78 0.10 1.26 0.10 1.68 0.11 10 1.22 0.11 1.05 0.11 1.16 0.12 11 1.14 0.26 0.86 0.27 1.09 0.28 12 0.97 0.11 0.71 0.13 0.63 0.14 13 0.96 0.073 0.73 0.073 0.76 0.074 14 0.79 0.18 0.62 0.19 0.62 0.22 15 3.03 0.39 2.29 0.45 2.32 0.45 16 1.68 - 1.38 - 0.70 - 17 1.32 - 1.04 - 0.74 - 18 1.09 - 0.80 - 0.93 - 19 1.01 - 0.70 - 1.01 -

39

Table 9 Summary of mean values and c.o.v.s of the model factor for all design equations

Modified Terzaghi Bjerrum-Eide Slip circle

αμ 1.01 1.31 1.39

αδ 0.072 0.064 0.21 Recommended FS in codes 1.5 1.2 1.2

Recommended FS after bias correction 1.52 1.57 1.67

40

0 10 20 30 40

30

25

20

15

10

5

00 20 40 60 80

20

15

10

5

00 25 50 75 100

40

35

30

25

20

15

10

5

00 20 40 60 80

40

35

30

25

20

15

10

5

00 50 100150200

50

45

40

35

30

25

20

15

10

5

0

0 25 50 75 100

30

25

20

15

10

5

00 50 100150200

40

35

30

25

20

15

10

5

00 20 40 60 80

40

35

30

25

20

15

10

5

00 20 40 60 80

20

15

10

5

00 20 40 60 80

40

35

30

25

20

15

10

5

0

0 25 50 75 100

30

25

20

15

10

5

00 25 50 75 100

20

15

10

5

00 20 40 60

30

25

20

15

10

5

00 10 20 30 40 50

30

25

20

15

10

5

00 25 50 75 100

30

25

20

15

10

5

0

Case 1 Case 2 Case 3 Case 4 Case 5

Case 6 Case 7 Case 8 Case 9 Case 10

Case 11 Case 12 Case 13 Case 14 Case 15

Wall top

Excav. depth

Gnd. surf.

Wall bottom

Wall topGnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall topGnd. surf.

Excav. depth

Wall bottom

Wall topGnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top Gnd. surf.

Excav. depth

Wall bottom

Wall top

Gnd. surf.

Excav. depth

Wall bottom

: su(TC) corrected by t ;: su(FV) corrected by ;

: Average of mobilized su

: su(TE) corrected by t ;

: su(DSS) corrected by t ;

: su(UC) ;

: su(CPT) ;

su (kPa)D

epth

(m)

Figure 1 Profiles of undrained shear strengths for all cases.

41

2/B

o45

/ 2D B≥

2/BD <

2/B

D

Figure 2 Schematic of Terzaghi’s equation

42

sq sq

He

dT

a

bc

d

f

450450e

Original failure surface

Hd

Shifted failure surface

Figure 3 Schematic of failure surface for Terzaghi’s equation in case of deeper wall embedment

43

sq sq

Figure 4 Schematic of Bjerrum and Eide’s equation [re-drawn from NAVFAC (1982)]

44

45 0

sq sq

He

a

b

c

d

B

He,eq

e

dT

f

Original failure surface

Shifted failure surface

Figure 5 Schematic of failure surface in Bjerrum and Eide’s equation for deeper wall embedment

45

O

Lowest level of strutsExcavation depth

Wall embedment

rd

a

bc

d

e

qs

He

Hd

Hs

Figure 6 Schematic of the slip circle method for basal stability analysis

46

Figure 7 Rm versus number of data points for estimating c.o.v. of su(mob)

47

( )11 1.75 0.5

1 exp 0.26iRx

⎡ ⎤= + × −⎢ ⎥

+⎢ ⎥⎣ ⎦

bdeusabuscdefus

cbdeus

Figure 8 Variance reduction factor under various extent of spatial averaging

48

0 1 2 3 4

FSC

0

0.1

0.2

0.3

0.4

0.5

FSC

FSC

=1

Code

regu

latio

n

Slip circle

0 1 2 3 4 5

FSC

0

0.1

0.2

0.3

0.4

0.5FS

C

FSC

=1

Code

regu

latio

n

Modified Terzaghi

0 1 2 3 4

FSC

0

0.1

0.2

0.3

0.4

0.5

FSC

FSC

=1

Code

regu

latio

n

Bjerrum and Eide

Failure cases (Numerical simulation)

Failure cases (Case history)

Non-failure cases (Case history)

Figure 9 Relation between mean values and c.o.v.s of FSC for all cases and for all design equations

49

Figure 10 Contour of the likelihood functions for the three design equations

50

Re

quire

dFS

C

Figure 11 Relation of Tβ and required CFSμ under various

CFSδ for modified Terzaghi’s design

equation

51

Figure 12 Relation of Tβ and required CFSμ under various

CFSδ for Bjerrum and Eide’s design

equation

52

Figure 13 Relation of Tβ and required CFSμ under various

CFSδ for slip circle design equation