-
Technical Note
Probabilistic Assessment of Serviceability Limit State
ofDiaphragm Walls for Braced Excavation in Clays
Wengang Zhang1; Anthony T. C. Goh2; and Yanmei Zhang3
Abstract: Braced excavation systems are commonly required to
ensure stability in the construction of basements for shopping
malls andunderground transportation facilities. For excavations in
deposits of soft clays, stiff retaining wall systems such as
diaphragm walls help torestrain ground movements and wall
deflections in order to prevent damage to nearby buildings and
utilities. It is quite common for designersto limit the maximumwall
deflection to 0.5% times the maximum excavation depth. However, a
review of measured diaphragm wall displace-ments from various
published case histories of successful deep excavations show that
wall deflections can be up to more than 2% times theexcavation
depth. Since the allowable threshold wall displacement depends on
various influencing parameters, wall deflections should not
belimited to an arbitrary value. This paper presents a
probabilistic framework combining a simplified wall deflection
estimation model withreliability methods to determine the
probability of serviceability limit state failure. Consequently,
using this approach, the paper presents amethodology that allows
engineers to determine the required threshold limiting hm=HeT
ratios to meet the different target serviceabilityreliability
indices. DOI: 10.1061/AJRUA6.0000827. 2015 American Society of
Civil Engineers.
Author keywords: Wall deflection; Braced excavation;
Probabilistic assessment; Polynomial regression model; Inverse
analysis.
Introduction
For excavations in ground that comprises thick soft clays
overlyingstiff clay, braced walls are usually constructed to
minimize groundmovements. The maximum wall deflection hm is an
importantconsideration in the design of the wall and in the
assessment ofits impact on the adjacent structures. The response of
the structuresto the wall displacement depends on many parameters
such as thewall type, the location, the foundation systems, and the
existingcondition of the structures. It is common to normalize
themaximum horizontal displacement of the retaining walls, hm,by
dividing it by the excavation depth, He.
It is common to use a limiting ratio hm=He of 0.5% as a
gov-erning criterion for safety of the works and adjacent
buildings.However, published case histories of successful
excavations fromvarious countries indicate that the measured wall
deflections (hm)are in the range of 0.05% to more than 2% of the
excavation depthHe (Yoo and Kim 1999, Long 2001, Moormann 2004).
Fok et al.(2012) reported that in actual practice, the performance
of retainingwalls for deep excavation depends on various factors
such as typesof soil, type of wall, ground water conditions,
support systems,construction method (top down or bottom up method),
wall instal-
lation method, and workmanship. Thus, limiting the ratio to a
sin-gle value hm=He may not be appropriate.
The soil parameters used for analysis by numerical methodsare
not necessarily the true values of these parameters, since suchdata
obtained are estimated based on limited field tests,
laboratorydata, or local experience, generally involving
considerable uncer-tainty. In addition, the wall and ground
responses in a supportedexcavation may also be influenced by
construction quality/workmanship and other environmental factors.
Thus, deterministicmodeling using numerical methods such as the
finite element orfinite difference method, even with advanced soil
constitutivemodels, may not be able to take into account all the
uncertaintiesand complexities mentioned above. To this end, the
probabilisticframework combining a simplified wall deflection
estimationmodel with reliability methods that are able to take into
accountthe soil parameter uncertainties is desirable.
In this paper, a recently developed polynomial regression
(PR)model (Zhang et al. 2015) for estimation of the
excavation-inducedwall deflection is adopted for prediction of hm.
Based on the PRmodel, design charts are proposed for preliminary
design andchecking. This paper demonstrates that with this PR
model, thefirst-order reliability method (FORM) and Monte Carlo
simulation(MCS) can be applied to determine the probability of
serviceabilitylimit state failure. Consequently, using this
approach, the paperpresents a methodology that allows engineers to
determine therequired threshold limiting hm=HeT ratios to meet the
differenttarget serviceability reliability indices.
Review of Polynomial Regression (PR) Model
The polynomial regression (PR) model is a semiempirical
modelproposed by Zhang et al. (2015) for predicting the maximum
walldeflection induced by braced excavation in clays. It is based
on morethan 1,000 FEM simulations using the hardening small strain
(HSS)model which accounts for the increased stiffness of soils at
smallstrains. The PR model consists of equations that can be used
toestimate wall deflection using the following parameters:
excavation
1Research Fellow, School of Civil and Environmental
Engineering,Nanyang Technological Univ., Singapore 639798,
Singapore (correspond-ing author). E-mail: [email protected]
2Associate Professor, School of Civil and Environmental
Engineering,Nanyang Technological Univ., Singapore 639798,
Singapore. E-mail:[email protected]
3Ph.D. Student, Nanyang Environmental and Water Research
Institute,Interdisciplinary Graduate School, Nanyang Technological
Univ.,Singapore 639798, Singapore. E-mail:
[email protected]
Note. This manuscript was submitted on November 15, 2014;
approvedon March 25, 2015; published online on May 26, 2015.
Discussion periodopen until October 26, 2015; separate discussions
must be submitted forindividual papers. This technical note is part
of the ASCE-ASME Journalof Risk and Uncertainty in Engineering
Systems, Part A: Civil Engi-neering, ASCE, 06015001(7)/$25.00.
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width B; excavation depthHe; soft clay thickness T; soil unit
weight; the system stiffness lnS [S EI=wh4avg, as defined in
Cloughand ORourke (1990), where E is the Youngs modulus of
wallmaterial, I is the moment of inertia of the wall section, w is
the unitweight of water, and havg is the average spacing of the
struts]; therelative soil shear strength ratio cu= 0v, where cu is
the undrainedshear strength and 0v denotes the vertical effective
stress; and therelative soil stiffness ratio E50=cu, where E50 is
the secant stiffnessin a standard drained triaxial test. For
illustration, Fig. 1 presents thecross-sectional soil and wall
profile considered. The maximum walldeflection estimated by the PR
model h takes the following form:
hmm a0 a1B a2B2 a3T a4T2 a5He a6H2e a7cu= 0v a8cu= 0v2 a9E50=cu
a10E50=cu2 a11 lnS a12 lnS2 a13 a142 a15 lnS a16cu= 0v lnS a17He
lnS a18HeT 1
The values of the coefficients are shown in Table 1.
Eq. (1) is applicable for the case with the ground water table
atthe ground surface, which is the most unfavorable condition.
Addi-tional analyses performed to investigate the influence of the
groundwater table indicate that the maximum wall deflections
decreasealmost linearly with decreasing ground water level and that
thewater table correction factor w can be approximated as w 1 0.1l,
where l is the depth of the ground water table belowthe ground
surface.
The developed PR model was validated through a total of
21well-documented excavation case histories in Zhang et al.(2015).
The plot of predicted wall deflections versus the measuredvalues
indicates that the PR model is able to predict reasonably wellthe
excavation-induced wall deflections.
Design Charts Based on PR Model
Based on Eq. (1) and Table 1, a series of charts relating h
toB;T=B, andHe=B for general excavation widths with common
soilrelative shear strength ratio cu= 0v and stiffness ratio E50=cu
havebeen developed as shown in Figs. 27, assuming w 1. The
pro-posed charts are potentially useful for preliminary estimation
of themaximum wall deflections for given excavation geometries,
soilparameters, and support conditions.
Example
This example considers the case of Bugis MRT in Singapore, one
ofthe 21 case histories presented in Zhang et al. (2015). Based on
thePR model, the probability density of the wall deflection can
be
Stiff Clay cu=500 kPa Eu=500cu
=20 kN/m3
2 m5 m8 m
11 m14 m17 m20 m
hFinal excavation depth H
Soft Clay
EA=3.8 106 kN
E=2.0 107kN/m2
T
d
B/2
strut
20 m
Wall
penetration depth 3~5
e
m
Fig. 1. Cross-sectional soil and wall profile
Table 1. Response Surface Coefficient for hCoefficients
Values
a0 2,816.7a1 3.2659a2 0.0255a3 2.7198a4 0.1515a5 11.634a6
0.2348a7 3,089.0a8 1,691.0a9 0.7102a10 9.8 104a11 259.91a12
5.1584a13 136.76a14 2.1079a15 7.0204a16 213.27a17 2.5553a18
0.6817
Fig. 2. Charts relating h to T=B and He=B for given soil
parametersE50=cu 200, cu= 0v 0.25, and lnS: (a) B 30 m; (b) B 40
m
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determined through MCS. The analysis involved 1,000,000
realiza-tions assuming that parameters E50=cu and cu= 0v are random
var-iables while the other parameters are deterministic and that
bothE50=cu and cu= 0v follow a lognormal distribution with a
coefficientof variation (COV) of 0.15. The values for deterministic
parametersare 16.5 kN=m3;B 21 m;He 18 m, lnS 8.18, andT 35 m. The
mean values of cu= 0v and E50=cu are 0.25 and150, respectively.
Fig. 8 shows the plot of the probability densityof h based on the
parameter statistics and the PR model. As can beseen from Fig. 8,
the variation of h best follows the normal dis-tribution, ranging
from 75 mm to 275 mm. The most probable hvalue is 136 mm, very near
the actual measured value of 150 mm.This section shows that the
combined use of the PR model andMCS can be used to determine the
distribution of the wall deflec-tions and to perform reliability
analysis.
Reliability Analysis on Serviceability Limit State
Statistical Information of Input Variables
Among the seven input variables for the PR model, the soil
unitweight , the soft clay thickness T, the excavation width B
anddepth He, and the system stiffness lnS are simply treated
asconstants, as they have rather limited influence on the results
ofprobabilistic analysis due to their low uncertainty (Hsiao et
al.2008). The two key soil parameters E50=cu and cu= 0v are
treatedas random variables and are assumed as lognormal (LN)
distribu-tion. The COV values for the two soil parameters are
assumed to be0.15. Useful guidelines on typical COVs of many common
soil
strength properties have been summarized by Phoon and
Kulhawy(1999) and Duncan (2000). The statistics of the random
variablesand values used for the deterministic variables are listed
in Table 2.
Serviceability Criterion and Threshold hm=HeTThe ratio hm=He is
adopted as the serviceability limit state cri-terion and in
assessing the serviceability limit state of diaphragmwalls for
braced excavation, the limit state function based on the PRmodel
can be expressed as
gx R S hm=HeT h=He 2in which hm=HeT is the limiting threshold
value while h=He isderived from the PR model. Serviceability limit
state failure isdeemed to occur if the predicted h=He is greater
than the limitinghm=HeT .
Reliability Assessment Methods: FORM and MCS
The probability that the serviceability limit state Pf is
exceeded canbe determined using Eq. (2). Such calculation of Pf
involves thedetermination of the joint probability distribution of
the resistanceR and the loading S and the integration of the
Probability DensityFunction (PDF) over the failure domain gx <
0. Considering thatthe PDFs of the random variables are not known
in engineeringapplications and the integration is computationally
demandingwhen multiple variables are involved, an approximate
methodknown as the first-order reliability method (Hasofer and
Lind1974), is commonly used to assess Pf. The approach involvesthe
transformation of the limit state surface into a space of
standard
Fig. 3. Charts relating h to T=B and He=B for given soil
parametersE50=cu 300, cu= 0v 0.3, and lnS: (a) B 30 m; (b) B 40
m
Fig. 4. Charts relating h to T=B and He=B for given soil
parametersE50=cu 200, cu= 0v 0.3 and lnS: (a) B 30 m; (b) B 40
m
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normal uncorrelated variables, wherein the shortest distance
fromthe transformed limit state surface to the origin of the
reduced var-iables is the reliability index (Cornell 1969). For
normal distrib-uted random variables, Pf 1 , in which =
cumulativeprobability of a standard normal distribution.
Mathematically, Lowand Tang (2004) have shown that can be computed
using
minxFxi ii
TR1
xi ii
s3
in which xi = the set of n random variables; i = the set of
meanvalues; i = the standard deviation; R = the correlation matrix;
and
Fig. 7. Charts relating h to T=B and He=B for given soil
parametersE50=cu 250, cu= 0v 0.3 and lnS: (a) B 30 m; (b) B 40
m
Fig. 8. Distribution of the predicted h values for Bugis MRTFig.
6. Charts relating h to T=B and He=B for given soil
parametersE50=cu 250, cu= 0v 0.3 and lnS: (a) B 30 m; (b) B 40
m
Fig. 5. Charts relating h to T=B and He=B for given soil
parametersE50=cu 150, cu= 0v 0.25 and lnS: (a) B 30 m; (b) B 40
m
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F = the failure region. Low (1996) has shown that a
MicrosoftExcel spreadsheet can be used to perform the minimization
anddetermine .
As an alternative to the FORM method, MCS is adopted
forcomparison. In this study, the Microsoft Excel add-in
program@RISK (http://www.palisade.com) was used.
The Developed FORM_PR and MCS_PR Frameworks
For each MCS simulation, the number of iterations is
1,000,000and Latin hypercube sampling is adopted. The MCS
procedures
and the calculation of probability failure Pf for the casewith B
30 m, 16.7 kN=m3, He 18 m, lnS 7.30,T 15 m, mean value of E50=cu
200, and average value ofcu= 0v 0.25 are illustrated in Fig. 9(a).
It can be observed thatfor a limiting threshold hm=HeT 0.5%, the
probability thatthis value is exceeded is 4.3%, while for hm=HeT
0.2%,the Pf value can be as high as 56%. The typical FORM
proceduresand the calculation of Pf are illustrated in Fig. 9(b).
Cells D3:G4are parameters which are set corresponding to the
variabledistribution types. For normal distributions, cells D3:D4
corre-spond to the mean values and cells E3:E4 correspond to
the
Table 2. Statistics of Random Variables and Values Used for
DeterministicVariables
Parameter Values considered
Random variablescu= 0v Mean 200, 300; COV 0.15E50=cu Mean 0.25,
0.29; COV 0.15
Deterministic variableskN=m3 16.7, 17.5B (m) 30, 40T (m) 15 (for
B 30 m), 30 (for B 40 m)He (m) 18 (for B 30 m), 24 (for B 40 m)lnS
7.30, 8.18
Fig. 9. The developed reliability assessment frameworks: (a)
MCS_PR; (b) FORM_PR
Table 3. Hypothetical Cases and Parameters Considered
Case number Parameters considered
1 B 30 m; T 15 m; He 18 m; lnS 7.30; 16.7 kN=m3
E50=cu LN200,30; cu= 0v LN0.25; 0.042 B 40 m; T 30 m; He 24 m;
lnS 8.18;
17.5 kN=m3E50=cu LN300,45; cu= 0v LN0.29; 0.045
3 B 30 m; T 15m; He 12m; lnS 6.10; 16.7 kN=m3
E50=cu LN200,30; cu= 0v LN0.25; 0.04
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standard deviations. For non-normal distributions, the
non-normalparameters are replaced by an equivalent normal
ellipsoid, centeredat the equivalent normal mean. The correlation
matrix R cells K3:L4 are used to define the correlations between
E50=cu and cu= 0v.The dimensionless x 0i vector in cells M3:M4
contains equations forxi ui=i. Cell I3 contains the estimations of
the maximum walldeflection based on the predictive PR model h. Cell
N3 containsthe serviceability limit state function gx hm=HeT
h=He.The design point (xi values) was obtained by using the
spread-sheets built-in optimization routine SOLVER to minimize the
cell,by changing the x 0i values, under the constraint that gxi 0.
Priorto invoking the SOLVER search algorithm, the x 0i values in
cellsM3:M4 were set equal to the mean values (0, 0),
correspondingto the mean values (200, 0.25) of the original two
random variables.Iterative numerical derivatives and directional
search for designpoint xi were automatically carried out in the
spreadsheet. It is ob-vious from Figs. 9(ab) that both methods give
the same Pf valueof 4.3%.
Probabilistic Analyses and Target Reliability Indices
Based on the MCS_PR and FORM_PR frameworks illustrated inthe
section above, three hypothetical cases were adopted to showthe
influence of the chosen threshold hm=HeT values on the Pfresults
and the determination of hm=HeT according to the
targetserviceability reliability index SLS. The assumed statistics
of theseven input variables for the three cases are listed in Table
3. Thelognormal distribution was assumed for the two random
variablessince it avoids allowing negative values in modeling
material prop-erties. Furthermore, the lognormal distribution has
the advantage inmodeling the right-hand tail of the distribution.
The right-hand tail
Fig. 10. Influence of hm=HeT on Pf and determination ofhm=HeT
according to the target SLS, Case 1
Fig. 12. Influence of hm=HeT on Pf and determination ofhm=HeT
according to the target SLS, Case 3
Fig. 11. Influence of hm=HeT on Pf and determination ofhm=HeT
according to the target SLS, Case 2
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is important because it governs the probability of excessive
defor-mations and the reliability of the serviceability limit state
(SLS)design. Typical values for reliability index for ultimate
limit state(ULS) ULS and the corresponding Pf have been proposed by
theU.S. Army Corps of Engineers (1997). Wang and Kulhawy
(2008)stated that although ULS design codes apply reliability
principals,serviceability limit state designs are still evaluated
using conven-tional approaches where the serviceability reliability
index SLS re-mains unknown. That is, there are no guidelines with
regard to thetarget SLS, though Wang and Kulhawy (2008) presented
for thefirst time a relationship between SLS and ULS and applied
itto augered cast-in-place piles. In this study, SLS values of
2.0,2.3, 2.6, and 3.1 are selected as the target reliability
indices sincetheir corresponding Pf values are 2%, 1%, 0.5%, and
0.1%, respec-tively, which are commonly used for serviceability
considerations.
Figs. 1012(a) show the significant influence of hm=HeT onPf. It
is obvious that small limiting hm=HeT will result in higherPf since
low thresholds are easily exceeded. The Pf decreases withthe
increase of the hm=HeT values. Figs. 1012(b) plot thecurves of the
threshold hm=HeT values versus the target SLSof 2.0, 2.3, 2.6 and
3.1. It can be observed that hm=HeT increaseswith the increase of
the target SLS, indicating that the choice oftarget SLS has a
significant effect on determination of the thresholdhm=HeT
values.
Conclusions and Discussions
This paper presents a probabilistic framework combining a
recentlyproposed polynomial regression model used for estimating
walldeflections induced by excavation with reliability methods
MCSand FORM to determine the probability of serviceability limit
statefailure. The main work and findings include the following:1. A
series of design charts are proposed, which are potentially
useful for preliminary estimation of the maximum wall
deflec-tions for given excavation geometries, soil parameters,
andsupport conditions. In addition the example case of BugisMRT
station shows that the combined use of the PR modeland MCS can be
used to determine the distribution of the walldeflections and to
perform reliability analysis.
2. Probabilistic frameworks FORM_PR and MCS_PR are devel-oped
for reliability assessment of the probability of service-ability
limit state failure.
3. The influence of the threshold hm=HeT values on probabil-ity
of serviceability limit state failure is significant. The choiceof
the threshold hm=HeT values can be determined throughthe target
serviceability reliability index SLS.
It should be noted that the focus of this paper is on the
proposedprobabilistic framework combining a simplified empirical
estima-tion model with reliability methods to determine the
probability ofserviceability limit state failure, and thus its
application is limited to
soil deposits that can be modeled with the set of soil
parametersused in this study. In addition, the error in the maximum
wall de-flection measurements will also influence the reliability
results.Furthermore, as the hypothetical cases to build the PR
model as-sume a single layer of soft clay for numerical
simulations, conse-quently, the PR model and the proposed framework
are notapplicable to excavations in heterogeneous or layered clay
deposits.
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