INTRODUCTION - ir.lib.ntust.edu.tw
Transcript of INTRODUCTION - ir.lib.ntust.edu.tw
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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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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