Computers and Geotechnics 44 (2012) 1–8

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Bayesian updating of KJHH model for prediction of maximum groundsettlement in braced excavations using centrifuge data

Lei Wang a,⇑, Nadarajah Ravichandran a, C. Hsein Juang a,b

a Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USAb National Central University, Jhongli City, Taoyuan County 32001, Taiwan

a r t i c l e i n f o

Article history:Received 2 December 2011Received in revised form 11 February 2012Accepted 7 March 2012Available online 6 April 2012

Keywords:ProbabilityBayesian updatingParameter uncertaintyModel uncertaintyBraced excavationGround settlementCentrifuge test

0266-352X/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compgeo.2012.03.003

⇑ Correspondent author.E-mail address: lwang6@clemson.edu (L. Wang).

a b s t r a c t

In this paper, a Bayesian approach for updating a semi-empirical model for predicting excavation-induced maximum ground settlement using centrifuge test data is presented. The Bayesian approachinvolves three steps: (1) prior estimate of the maximum ground settlement and model bias factor, (2)establishment of the likelihood function and posterior distribution of the model bias factor using thesettlement measurement in the centrifuge test, and (3) development of posterior distribution of thepredicted maximum settlement. This Bayesian approach is demonstrated with a case study of a well-documented braced excavation, and the results show that the accuracy of the maximum settlementprediction can be improved and the model uncertainty can be reduced with Bayesian updating.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

One of the main concerns in a braced excavation in an urbanarea is the risk of damages to adjacent infrastructures caused bythe excavation-induced ground movements. In many excavationprojects, the owners or regulatory agencies often establish a limitfor the maximum ground settlement, referred to herein as the lim-iting ground settlement, as a means of preventing damages to adja-cent infrastructures. Table 1 shows an example of such limitingresponse criteria used in China [1]. However, a binary assessmentof whether the maximum excavation-induced ground settlementwill exceed the specified limiting value may not be meaningfulfor two reasons. First, it is generally difficult to predict accuratelythe excavation-induced ground responses [2]. Thus, model uncer-tainty must be considered when evaluating the potential of dam-age to the adjacent structures caused by the maximum groundsettlement. Second, uncertainty often exists in the input parame-ters, which can also lead to the uncertainty in the computed max-imum settlement. Thus, it appears more meaningful to assess thedamage potential by computing the probability of exceeding aspecified limiting ground settlement, taking into account bothmodel uncertainty and parameter uncertainty.

ll rights reserved.

Determination of the probability of exceedance (i.e., the probabil-ity of exceeding a limiting ground settlement) is an important stepin the damage potential assessment of the adjacent infrastructures.A more informed design decision for a satisfactory braced excava-tion project may be achieved by considering the probability ofexceedance and the consequence of such event (in terms of dam-ages to the adjacent infrastructures).

An accurate calculation of the probability of exceedance re-quires an accurate analysis model for the maximum ground settle-ment (which can provide an accurate and well-defined limit stateor performance function) and an accurate statistical characteriza-tion of the input soil parameters. Many researchers have demon-strated finite element method (FEM) with proper soil model as avalid tool for computing the excavation-induced maximum groundsettlement [3–9]. However, combining FEM-based solutions withreliability theory for computing the probability of exceedancecan be a major undertaking. Although the random finite elementmethod (RFEM) has been introduced in recent years [10–12], useof RFEM in a complex braced excavation problem can be computa-tionally challenging. As an alternative to the more sophisticatedRFEM solution, reliability analysis based on a limit state or perfor-mance function defined by response surface method or artificialneural network has been proposed [13,14]. The latter approachallows for consideration of model and parameter uncertainties ina computationally more efficient manner.

Table 1Criteria for excavation protection levels in Shanghai, China [1].

Excavationprotection level

Infrastructures and facilities to be protected at the site Requirements for maximum wall deflection andmaximum ground surface settlement

I Metro lines and important facilities such as gas mains and water drains exist within adistance of 0:7He from the excavation; safety has to be ensured

1. Maximum wall deflection 6 0:14%He

2. Maximum ground surface settlement 6 0:1%He

II Important infrastructures or facilities such as gas mains and water drains exist within adistance of (1–2)He from the excavation

1. Maximum wall deflection 6 0:3%He

2. Maximum ground surface settlement 6 0:2%He

III No important infrastructures or facilities exist within a distance of 2He from the excavation 1. Maximum wall deflection 6 0:7%He

2. Maximum ground surface settlement 6 0:5%He

Note: He = final excavation depth.

2 L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8

Instead of the response surface method, which still relies on well-placed FEM solutions, the KJHH model [2] is adopted in this study forcomputing the maximum ground settlement in a braced excavation.The KJHH model, summarized in Appendix A, is a semi-empiricalmodel that was generated with hundreds of FEM simulations andvalidated with many well-documented case histories. The maxi-mum ground settlement (y) computed with the KJHH model maybe used to define the limit state or performance function:

gðÞ ¼ ylim � y ð1Þ

where ylim is the specified limiting ground settlement.Methods for computing the reliability index and the probability

of exceedance based on the performance function as expressed inEq. (1) are readily available [15]. Example for such reliability anal-ysis using a well-documented case history is presented later. Theaccuracy of the computed probability of exceedance is of coursedependent upon the accuracy of the KJHH model. To this end,the KJHH model can be updated or calibrated with knowledgelearned from full-scale testing or well-controlled centrifuge test-ing. Thus, the focus of this paper is to demonstrate how the KJHHmodel can be updated with the maximum ground displacementobserved in the well-controlled centrifuge testing, how the up-dated KJHH model can be used in the prediction of the maximumground settlement in a future excavation, and how the updated re-sults can be used in a reliability analysis for the probability ofexceedance.

2. Bayesian updating methodology

A Bayesian approach is taken in this paper to calibrate the KJHHmodel with knowledge learned from well-controlled centrifugetesting. Bayesian methodology has been used to calibrate or updatemodel bias of geotechnical models [16–18]. In the work of Zhanget al. [18], Bayes’ theorem was employed to update model bias ina slope stability analysis using observations from centrifuge test-ing. In that study, the performance of the slope in the centrifugetesting is observed in terms of failure or no-failure; in other words,the outcome (system response) is of ‘‘yes or no’’ type. This Bayesianupdating methodology [18] is adapted herein for the study of exca-vation problems. Instead of using the ‘‘yes-or-no’’ type of observa-tions, however, in this study the maximum ground settlementmeasured in the centrifuge testing of a supported excavation isused to update the model bias factor of the KJHH model. Further-more, as excavation is conducted in multiple stages, the groundsettlements from multiple stages allow for multiple updating ofthe model bias factor in a single centrifuge testing.

The Bayesian methodology along with detailed formulations forupdating the model bias of the KJHH model for predicting the max-imum ground settlement is presented in the sub-sections thatfollow.

2.1. Updating model factor that relates KJHH model to centrifugetesting

The maximum ground settlement in a braced excavation pre-dicted by the KJHH model, denoted as Ym (note: the subscript mstands for modelprediction), can be related to the maximum groundsettlement measured in the centrifuge test, denoted as Yc (note:the subscript c stands for centrifuge testing), through a model factorNc defined as follows [18]:

Yc ¼ NcYm ð2Þ

Considering the existence of model uncertainty and inputparameter uncertainty, the maximum ground settlement Ym pre-dicted by the KJHH model should be treated as a random variable.The probability density function of Ym, denoted as fYmðymÞ, may beobtained for example by Monte Carlo simulation. The settlementprediction by the KJHH model may be updated, if the goal is tomatch Ym to Yc, by updating the factor Nc. Assume that the priordistribution function of Nc is f 0Nc

ðncÞ and the likelihood functioncan be obtained based on centrifuge testing, then the posterior dis-tribution of Nc can be determined using Bayes’ Theorem. In thiscase, the likelihood function is a conditional probability of observ-ing yc in the centrifuge testing (i.e., Yc = yc), given that Nc = nc. Inother words, the likelihood learned from the centrifuge testingcan be denoted as P[(Yc = yc)|nc]. In the context of Eq. (2), this like-lihood may be expressed as:

P½ðYc ¼ ycÞjnc� ¼ P½ðNcYm ¼ ycÞjnc� ¼ P Ym ¼yc

Nc

� �����nc

� �

¼ fYm

yc

nc

� �dym ð3Þ

Here, the conditional probability P[(Yc = yc)|nc] defined in thedomain of Yc has been transformed into the probability definedin the domain of Ym. Thus, for a given Nc = nc, the likelihood ofobserving a maximum ground settlement Yc = yc in a centrifugetesting is a probability that is a product of the functional valuefYm

ycnc

� �and a small increment dym. For a given yc value, this likeli-

hood is a function of nc. Thus, the posterior distribution function ofNc, denoted as f 00Nc

ðncÞ, can be expressed as [19]:

f 00NcðncÞ ¼ kcfYm

yc

nc

� �f 0NcðncÞ ð4Þ

A normalization constant kc in Eq. (4) is introduced to ensurethat the area under the curve of the probability density functionf 00NcðncÞ is equal to 1.

2.2. Updating model factor that relates the KJHH model to realperformance

The KJHH settlement prediction Ym can also be related to thereal performance (the actual maximum ground settlement caused

L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8 3

by a braced excavation), denoted as Yr, in the same way as Eq. (2).Using a model factor Nr, this relationship can be expressed as [18]:

Yr ¼ NrYm ð5Þ

Direct calibration or updating of Nr requires data from well-con-trolled full-scale field tests. Because such field tests are rare, theupdating of Nr for real performance prediction is carried out byrelating the real performance to the centrifuge measurement. Tothis end, it is essential to introduce a ‘‘similarity’’ factor Qc that re-lates the centrifuge result to the real performance [18]:

Yr ¼ Q cYc ð6Þ

The similarity factor Qc is of course a random variable and it isoften assumed to follow a lognormal distribution with a unitymean and a certain coefficient of variation (COV) for well-equippedhigh-quality centrifuge tests. The factor Qc is further discussedlater.

Combining Eq. (5) and Eq. (6), the following equation is ob-tained [18]:

Yc ¼ NrYm

Q c

� �¼ NrY

� ð7Þ

Eq. (7) turns the required data for calibration of Nr from the pair of(Yr,Ym) to the pair of (Yc,Y�) where Y� is equal to Ym/Qc. Thus, the cal-ibration or updating of Nr can be carried out using the procedurerepresented by Eqs. (3) and (4). Assume that the prior probabilitydistribution function of Nr is f 00Nr

ðnrÞ. Next, the likelihood functionis a conditional probability of observing yc in the centrifuge testing(i.e., Yc = yc), given that Nr = nr. In a manner similar to Eq. (3), thelikelihood can be expressed as:

P½ðYc ¼ ycÞjnr � ¼ P ðNrYast ¼ ycÞjnr

¼ P Y� ¼ yc

Nr

� �����nr

� �

¼ fY�yc

nr

� �dy� ð8Þ

The posterior distribution function of Nr, denoted as f 00NrðnrÞ, can be

expressed as [19]:

f 00NrðnrÞ ¼ krfY�

yc

nr

� �f 0NrðnrÞ ð9Þ

Again, the normalization constant kr in Eq. (9) is introduced toensure that the area under the curve of the probability densityfunction f 00Nr

ðnrÞ is equal to 1.It should be noted that if information of more than one centri-

fuge tests is available and the observations in these tests are inde-pendent, these observations can be used to update the biasedfactor Nr. The likelihood function in the case of multiple observa-tions is obtained as the multiplication of the likelihood functionsof individual observations [19]. Thus, the posterior distributionfunction of Nr can be expressed as:

f 00NrðnrÞ ¼ kr

Qmc

i¼1fY�

yci

nr

� �f 0NrðnrÞ ð10Þ

where kr is the normalization constant; mc is the number of centri-fuge tests; yci is the maximum ground settlement observation of ithtest; f 0Nr

ðnrÞ = prior distribution function of Nr.It should be noted that the above procedure to update Nr using

centrifuge tests can easily be adapted for updating Nr using fieldobservations by assuming the COV of Qc to be zero.

2.3. Predicting real performance using the KJHH model and updated Nr

For a given case, the real performance, in terms of the maximumground settlement caused by a braced excavation, can be predictedusing the KJHH model and the updated Nr. Assume that the

maximum ground settlement in a braced excavation computedby the KJHH model is Ym and the posterior distribution functionof Nr is f 00Nr

ðnrÞ, then the distribution of Yr can be expressed as[20]:

fYr ðyrÞ ¼Z 1

�1f 00Nr

yr

ym

� �� fYm ðymÞ �

1ym

� �� �dym ð11Þ

3. Updating Nc with centrifuge testing

3.1. Centrifuge testing at Cambridge Schofield Centre

High-quality centrifuge testing that can simulate the true or realperformance in the field is crucial for any Bayesian updating. Thecentrifuge test used in the calibration of Nc herein was carried outby Lam [21] at Cambridge Schofield Centre. A two-axis in-flightexcavator and propping system were developed for the TurnerBeam Centrifuge at Cambridge University, which can simulate theexcavation and propping process in the real excavation [22]. Thisequipment is capable of simulating the complex stress-path andboundary conditions in the excavation process.The soil in the centri-fuge test is mainly the Speswhite Kaolin with a very thin slice offraction-E sand in the bottom of the model. The triaxial undrainedcompression test with local strain measurement is conducted tomeasure mechanical properties of core samples from the centrifugesoil model; the normalized undrained shear strength su=r0v is esti-mated as 0.2 and the normalized initial modulus Ei=r0v is estimatedas 218.75 [21]. Fig. 1 shows a sketch of the centrifuge model. The to-tal height of the model is 300 mm and the test was carried out at60 g, which represents the prototype soil of 18 m height [21]. Thedetailed data of excavation geometry and soil parameters are listedin Table 2. The excavation in the centrifuge was carried out after themodel was spun to 60 g and the consolidation settlement was stea-dy. The scrape was used to cut into and scrape off the soil, and afterfinishing each excavation, the prop was installed and pressurized tosupport the retaining wall. There were totally three props in thiscentrifuge testing with a spacing of 32 mm in model scale [22].The observed ground settlement in prototype scale is 35 mm atthe final stage (He = 5.4 m) [21]. The reader is referred to Lam [21]for details of the experimental set-up and the testing procedureand results.

3.2. Results of Bayesian updating of Nc

The proposed approach for updating model factor Nc requires anestimate of the variation of input parameters that are required forpredicting or computing the maximum ground settlement usingthe KJHH model. These parameters include the excavation depth(H), excavation width (B), system stiffness (EI=cwh4

avg), normalizedundrained shear strength (su=r0v ), and normalized initial modulusat small strain (Ei=r0v). For demonstration purpose, these parame-ters are assumed to follow Normal distribution and the coefficientof variation (COV) of each input parameter is assumed based onpublished literature [15]. Thus, soil parameters su=r0v and Ei=r0vare assumed to have 16% variation and all other parameters areassumed to have 5% variation in this example. The coefficient ofcorrelation between parameters su=r0v and Ei=r0v is assumed to be0.3 based on the statistical analysis of limited data presented inKung [23]. It is further assumed that there is no correlation be-tween any other pair of input parameters. It should be noted thatthe assumed values of variation are used herein for demonstrationpurpose only.

Given the input data listed in Table 2 which was derived fromthe centrifuge testing and the assumed variation and correlationdata, the maximum ground settlement Ym of this ‘‘excavation case’’

Gate wall

Open space

Speswhite Kaolin Clay

Load cell

Scraper

Porous plastic

Bearing stratum:fraction E sand

mm

Model wall

Prop of strut

mm

Fig. 1. Sketch of centrifuge test at the last stage of excavation (modified after Lam[21]).

Table 2Summary of geometry and initial conditions of the centrifuge test (model scale).

Source of information Lam [21]

Test number SYL04Design g-level (g) 60Depth of clay in model (mm) 300Soil type Kaolin (clay)Prop stiffness (kN/mm) 1.66Excavation depth at final stage He (mm) 90Excavation width B (mm) 120Normalized undrained strength su=r0v 0.2Normalized initial modulus Ei=r0v 218.75

System stiffness EI=cwh4avg

2860

0 10 20 30 40 50 600

2

4

6

8

10

12

Ground settlement (mm)

His

togr

am (

×10

3 )

Fig. 2. Histogram of the predicted maximum ground settlement obtained byapplying KJHH model to the centrifuge test conducted by Lam.

4 L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8

(or more precisely, a centrifuge model) may be computed using anappropriate method such as the KJHH model. Although simplifiedmethods such as first order second moment (FOSM) or point esti-mate method (PEM) may be used to compute the statistics (such asmean and standard deviation) of Ym, the Monte Carlo simulationmethod is used herein, as the objective is to obtain a distributionof Ym. Fig. 2 shows the histogram of the computed Ym values. Acurve-fitting is performed with data shown in Fig. 2, which yieldsa Lognormal distribution, ln Nð3:39; 0:46Þ, which has a mean of26.64 mm and a standard deviation of 13.05 mm. This representsa prior distribution of Ym.The prior distribution of Ym can be up-dated using the likelihood function obtained from the centrifugetest. As formulated previously, the updating is carried out throughthe updating of Nc. To begin with, the prior distribution of Nc is as-sumed to be a standard Normal distribution Nðl; rÞ with meanl = 1.0 and standard deviation r = 0.30 (or COV = 0.30). Thisassumption is based on an estimate of approximately 30% of errorin the computed maximum settlement using the KJHH model byKung et al. [2]. The effect of this assumption of prior distributionis examined later. In the centrifuge test [21], the maximum groundsettlement at the final stage of excavation is yc ¼ 35 mm. Thus, fora given value of Nc ¼ nc , the likelihood of Yc ¼ yc can be computedwith Eq. (3), and the resulting likelihood function is shown inFig. 3. By integrating the prior distribution and the likelihood func-tion, as per Eq. (4), the posterior distribution, an updated distribu-tion of Nc can be obtained. Note that in this example, thenormalized constant kc is calculated through numerical integra-tion, which yields kc = 48.75. The posterior distribution, shown inFig. 4, is fitted well with a Normal distribution Nðl; rÞ with mean

l = 1.136 and standard deviation r = 0.229. From Fig. 4, it can beseen that the variation of Nc is reduced after updating.

The influence of the prior estimate of COV of Nc on the posteriormean and standard deviation of Nc is analyzed herein. Fig. 5 showsthat the posterior mean and standard deviation of Nc increase withthe prior estimate of the COV of Nc. Thus, it is essential to adopt anaccurate model for predicting the maximum ground settlement.Nevertheless, the variation of Nc (see Fig. 4), as well as the KJHHprediction, is reduced after updating with the centrifuge measure-ment regardless of the prior estimate of the COV of Nc.

4. Updating Nr and maximum settlement prediction

4.1. Updating Nr with an assumed knowledge of Qc

The bias factor Nr can also be updated using centrifuge data.Similar to the updating of Nc, a prior distribution of Nr is assumedto be a Normal distribution Nðl; rÞ with mean l = 1.0 and stan-dard deviation r = 0.34 (or COV = 0.34). This assumption is mainlybased on the estimate of model uncertainty of the KJHH model byKung et al. [2]. However, a slightly higher COV is assumed for Nr

because the discrepancy between the real (field) response andmodel prediction is likely greater than that between the centrifugetesting and model prediction.

For updating the bias factor Nr, an estimate of the similarity fac-tor Qc is needed. The factor Qc is of course difficult to estimate as itis affected not only by measurement errors but also by spatial var-iability [24]. Popescu and Prevost [25] estimated the result of cen-trifuge measurement lies within a ±10% error of the ‘‘true’’ (field)result. In a model bias calibration using centrifuge testing of slopes,Zhang et al. [18] assumed Qc with a mean of 1 and COV of 0.2. Thisassumption is adopted herein. The similarity factor Qc is assumedto be lognormally distributed following a previous study of Bayes-ian updating of slope stability [18]. The effect of this assumption ischecked by repeating the analysis with an assumption of normaldistribution, and the updated results are practically identical inthis case regardless of whether normal or lognormal distributionis assumed. Furthermore, the response (the maximum ground set-tlement prediction by the KJHH model) Ym in the excavation is as-sumed to follow a Lognormal distribution; and Ym and Qc arefurther assumed to be independent. Given the distribution of Ym

and Qc, the probability density function of Y� ¼ Ym=Q c can beobtained. Next, the likelihood function, which is a conditionalprobability of observing yc in the centrifuge testing (i.e., Yc = yc),for a given Nr = nr, is obtained with Eq. (8). The resulting likelihood

0.0 1.0 2.0 3.0

Bias factor, nc

0.00

0.01

0.02

0.03

0.04

Lik

elih

ood

func

tion

Fig. 3. Likelihood function of Nc using data from centrifuge test.

0.0 1.0 2.0 3.0Bias factor, nc

0.0

0.5

1.0

1.5

2.0

Prob

abili

ty d

ensi

ty, f

(n

c)

PosteriorPrior

Fig. 4. Prior and Posterior distribution of Nc.

0.0 0.1 0.2 0.3 0.4 0.5

Prior COV of Nc

1.0

1.1

1.2

1.3

1.4

Post

erio

r m

ean

of N

c (a)

Prior COV of Nc

0.0

0.1

0.2

0.3

0.4

Post

erio

r st

d. d

ev. o

f N

c (b)

0.0 0.1 0.2 0.3 0.4 0.5

Fig. 5. Effect of prior COV of Nc on the posterior mean and variation of Nc.

0.0 1.0 2.0 3.0Bias factor, nr

0.00

0.01

0.02

0.03

0.04

Lik

elih

ood

func

tion

Fig. 6. Likelihood function of Nr using data from centrifuge test.

L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8 5

function is shown in Fig. 6. Finally, the posterior distribution of Nr

updated with centrifuge data can be obtained using Eq. (9); theresulting posterior distribution, which can be approximated wellwith a Normal distribution N(1.134,0.260), is shown in Fig. 7 alongwith the prior distribution.

The accuracy of the posterior distribution of Nr updated withcentrifuge data is likely affected by the knowledge of similarity fac-tor Qc, which can best be characterized by a large set of high qualityexcavation case histories with corresponding well-controlled,well-run centrifuge tests. Lacking such data set, a sensitivityanalysis is performed to examine the effect of the assumed COVof Qc (assuming that the mean is equal to 1) on the posterior meanand standard deviation of Nr. The results are shown in Fig. 8. As canbe seen from Fig. 8, the variation in the resulting posterior meanand standard deviation of Nr is quite limited when the COV of Qc

increases from 0 to 0.2, but the variation becomes more significantwhen the COV increases from 0.2 to 0.3. Thus, for the excavationproblem examined, the effect is likely insignificant if the COV ofQc is less than 0.2.

4.2. Prediction of the maximum ground settlement in TNEC excavation

In the previous sections, the KJHH model has been updatedusing a well-controlled centrifuge test within the proposed Bayes-ian framework. In this section, the updated KJHH model is used to‘‘predict’’ (more precisely, to compute) the maximum ground set-tlement in a well-documented excavation case history.

Taipei National Enterprise Center (TNEC) is located in Taipei Ba-sin, and the excavation for construction of this building is mainly indeposits of soft to medium clay. The TNEC excavation case [26] hasa rather complete set of high-quality field observations of groundsurface settlement, which is well suited for the validation of thedeveloped Bayesian updating framework.

The TNEC excavation was carried out using the top-down con-struction method, in which the diaphragm wall was supportedby 150 mm thick solid concrete floor slabs. The diaphragm wall,which was 0.9 m thick and 35 m deep, was used as the earth-retaining structure [26]. The excavation was completed in sevenstages to a maximum depth of 19.7 m. Fig. 9 shows the excavationdepths at different stages and the soil profile. The soil profile indi-cates that it is a clay-dominated site [2], and the wall and excava-tion are basically in clay. The input parameters that are requiredfor computing the maximum ground settlement in the TNEC caseusing the KJHH model (Appendix A) are listed in Table 3.

The proposed approach is implemented in two steps. First, theresponse (maximum ground settlement) Ym at the final excavationdepth of 19.7 m is predicted (computed) using the original KJHHmodel. This yields a prior estimate of Ym. Note that to considerexplicitly the uncertainty in the input parameters, Monte Carlosimulation is adopted in conjunction with the KJHH model for esti-mate of Ym. For this TNEC case, the prior Ym is a Lognormal distri-bution, ln Nð4:25; 0:41Þ, which has a mean of 64.2 mm and astandard deviation of 27.6 mm. Secondly, this prior distributionof Ym is then multiplied with the updated bias factor Nr, as perEq. (11), which yields the posterior distribution of the predicted

0.0 1.0 2.0 3.0Bias factor, nr

0.0

0.5

1.0

1.5

2.0

Prob

abili

ty d

ensi

ty, f

(n

r)

PosteriorPrior

Fig. 7. Prior and posterior distribution of Nr.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

COV of Q c

0.0

0.5

1.0

1.5

Post

erio

r m

ean

or s

td. d

ev. o

f N

r

Posterior meanPosterior std. dev.

Fig. 8. Effect of COV of Qc on the posterior distribution of Nr.

5

45

40

35

30

25

20

15

10

Depth(m)

Soil ProfileExcavationDepth (m)

CL, N = 9 - 11SM, N = 22 - 24

CL, PI = 13 - 16 LL = 33 - 36

SM, N = 4 - 11

CL, w = 32 - 40% PI = 13 - 16 LL = 33 - 36

SM, N = 14 -37

Gravel, N > 100

2.84.9

8.611.8

15.217.319.7

(Stage 1)(Stage 2)

(Stage 3)(Stage 4)

(Stage 5)(Stage 6)(Stage 7)

Fig. 9. Soil profile and excavation depth of TNEC (adapted from Kung et al. [9]).

Table 3Summary of information for final stage of TNEC excavation.

KJHH model input parameter Mean value COVa

System stiffness EI=cwh4avg

1294 0.05

Excavation depth at final stage He (m) 19.7 0.05Excavation width B (m) 41.2 0.05Normalized undrained strength su=r0v 0.31 0.16Normalized initial modulus Ei=r0v 650 0.16Normalized clay-layer thickness

PHclay=Hwall 0.87 0.05

a COV values estimated by Hsiao et al. [15].

0 50 100 150 200 250

Predicted ground settlement, yr (mm)

0.000

0.005

0.010

0.015

Prob

abili

ty d

ensi

ty, f

(y r

)

Predicted distributionObserved settlement

78 mm

Fig. 10. Posterior distribution of predicted ground settlement Yr.

6 L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8

response Yr. The results are shown in Fig. 10. The resulting poster-ior distribution of Yr can be approximated with a Lognormal distri-bution, ln Nð4:18; 0:47Þ. The mean of this updated distribution ofthe maximum ground settlement is 72.8 mm, which is close tothe observed maximum settlement of 78 mm [26]. It should benoted that in the above analysis, the COV of Qc is assumed to be0.20. The assumption of 0.20 is considered appropriate for awell-controlled centrifuge test as the updated prediction of themean of the maximum ground settlement agrees well with thefield observation.

For future cases, when a candidate design obtained from anumerical or empirical model for a critical excavation project isproposed, the centrifuge test may be conducted to update the pre-diction of the maximum ground settlement using the proposedframework to obtain a more accurate prediction. The assumed nor-mal distribution of similarity factor Qc with a mean of 1 and stan-dard deviation of 0.20 may be used along with the formulationspresented in this paper, provided that the centrifuge test is welldesigned, controlled and run. The updated maximum settlementcan confirm the candidate design before the project constructionis carried out.

However, it is generally not feasible to conduct a centrifuge testfor each excavation project. In such cases, the updated Nr presentedin this paper may be used in conjunction with the original KJHHmodel to predict the maximum ground settlement in a bracedexcavation. The procedure is the same as presented previously inthe study of TNEC case.

5. Reliability analysis for probability of exceedance

Accurate characterization of the posterior distribution of themaximum ground settlement in a braced excavation allows foraccurate calculation of the probability of exceedance (i.e., the prob-ability of exceeding a specified limiting ground settlement), whichcan be used to assess the risk of damages to the adjacent infra-structures. To compute the probability of exceedance with reliabil-ity analysis, the limit state defined previously in Eq. (1) can beused. Note that the variable y in Eq. (1) is now replaced by yr.For

Table A-1Coefficients for linear transformation of five variables [2].

Variables x Linear transformation equation X ¼ tðxÞ ¼ b1x2 þ b2xþ b3

b1 b2 b3

He (m) �0.4 24 �50

lnðEI=cwh4avgÞ 11.5 �295 2000

B/2 (m) �0.04 4 90su=r0v 3225 �2882 730Ei=r0v 0.00041 �1 500

L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8 7

demonstration purpose, the limiting tolerable settlement ylim is as-sumed to be a Lognormal distribution with a mean ðlylim

Þ of 75 mmand a standard deviation ðrylim

Þ of 38 mm [16,27]. For the TNECexcavation examined previously, the updated maximum settle-ment yr computed using the updated distribution of Nr, is foundto be a Lognormal distribution with a mean ðlyr

Þof 72.8 mm anda standard deviation ðryr

Þ of 36.2 mm. Thus, the reliability indexcan be computed as (assuming ylim and yr are independent):

b ¼ln

lylimlyr

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þCOV2

yr

1þCOV2ylim

r� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln½ð1þ COV2

ylimÞð1þ COV2

yr�

q ¼ 0:0384 ð12Þ

And thus the probability of exceedance is: pf ¼ 1�UðbÞ ¼0:485.

Obviously, the computed probability is influenced not only bythe means of the limiting tolerable settlement and the predictedmaximum ground settlement but also by the variations in thesesettlements. Thus, for a more informed design decision, a seriesof sensitivity analysis considering all possible uncertainty scenar-ios will be needed. However, this is not within the scope of thispaper.

6. Concluding remarks

A Bayesian framework for calibration of bias factors of a semi-empirical model for predicting the excavation-induced maximumground settlement using centrifuge test data is presented. TheBayesian approach involves three steps: (1) a prior estimate ofthe maximum ground settlement is carried out using the KJHHmodel, (2) a likelihood function of the KJHH model bias factor isestablished using the settlement measurement in a centrifuge test,followed by the development of the posterior distribution of themodel bias factor, and (3) the posterior distribution of the pre-dicted maximum settlement is the product of the prior estimateof the maximum ground settlement and the posterior distributionof the model bias factor. This Bayesian approach has been showneffective in improving the accuracy of the maximum settlementprediction and reducing the model uncertainty.

Thus, for a large and critical project, well-controlled centrifugetests may be conducted to calibrate bias factor of an adopted pre-diction model using the Bayes’ theorem, and the calibrated poster-ior distribution of the bias factor can be used to improve theaccuracy of the maximum settlement prediction made with theadopted model. In the absence of a centrifuge test for a given futureproject, however, the updated bias factor Nr, in the form of a Nor-mal distribution N(1.134,0.260), may be used in conjunction withthe original KJHH model to predict the maximum ground settle-ment in a braced excavation.

Improved knowledge or characterization of the posterior distri-bution of the maximum ground settlement in a braced excavationallows for more accurate calculation of the probability of exceedinga specified limiting ground settlement, which leads to more accu-

rate assessment of damage potential and risk to the adjacentinfrastructures.

Acknowledgments

The anonymous reviewers are thanked for their constructivecomments that have helped sharpen the work presented in this pa-per. The authors also wish to thank Dr. Lulu Zhang of Shanghai JiaoTong University, and Dr. Jie Zhang of Tongji University for helpfuldiscussions of the work presented. Finally, the first author wishesto thank Glenn Department of Civil Engineering, Clemson Univer-sity for its financial support.

Appendix A. An overview of the KJHH model for excavation-induced settlement [2]

In general, an evaluation of excavation-induced wall andground movements using empirical methods may proceed asfollows:

(1) Determine the maximum lateral wall deflection dhm.(2) Estimate the deformation ratio R (=dvm/dhm).(3) Calculate the maximum surface settlement dvm.

The KJHH model [2] follows this general approach to predict thewall deflection and ground movement in an excavation in soft tomedium clays. For simplicity, the effect of the presence of hardstratum is excluded here, although the presence of hard stratummight restrain the displacement of soil beneath and around thebottom of excavation as indicated in Kung et al. [2]. Thus, five basicparameters are considered essential for predicting the maximumwall deflection (dhm) caused by excavation in soft to medium clays.These parameters include the excavation depth (He), the excava-tion width (B), the system stiffness [EI=cwh4

avg , E is the Young’smodulus of wall material, I is the moment of inertia of the wall sec-tion, cw is the unit weight of water, and havg is the average supportspacing], the ratio of shear strength over vertical effective stress(su=r0v ), and the ratio of initial Young’s modulus over vertical effec-tive stress (Ei=r0v ). The maximum lateral wall deflection dhm is cal-culated as [2]:

dhm ¼ a0 þ a1X1 þ a2X2 þ a3X3 þ a4X4 þ a5X5 þ a6X1X2

þ a7X1X3 þ a8X1X5 ðA-1Þ

where X1 ¼ tðHeÞ; X2 ¼ t½lnðEI=cwh4avgÞ�; X3 ¼ tðB=2Þ; X4 ¼ tðsu=r0vÞ;

and X5 ¼ tðEi=r0vÞ; and t is a transformation function defined inEq. (A-2). The coefficients are as follows: a0 = �13.41973, a1 =�0.49351, a2 = �0.09872, a3 = 0.06025, a4 = 0.23766, a5 =�0.15406, a6 = 0.00093, a7 = 0.00285, and a8 = 0.00198. It is notedthat variables Xi (i = 1, 5) are the transformed variables of the fivebasic input variables defined as [2]:

X ¼ tðxÞ ¼ b1x2 þ b2xþ b3 ðA-2Þ

where x is each of the input variables (He, lnðEI=cwh4avgÞ, B=2, su=r0v ,

and Ei=r0v ), X is the transformed variable, and the coefficients, b1, b2,and b3 are listed in Table A-1.Kung et al. [2] found that the deforma-tion ratio R, the ratio of the maximum ground settlement over themaximum wall deflection, in clay-dominant sites is mainly influ-enced by three parameters,

PHclay=Hwall, su=r0v , and Ei=1000r0v .

The parameterP

Hclay=Hwall is normalized clay-layer thickness (ra-tio) with respect to the length of the diaphragm wall in the bracedexcavation. For an excavation in a clay site, in which the entire dia-phragm wall is in the clay layers, this normalized clay-layer thick-ness is equal to 1. The deformation ratio R is determined as [2]:

8 L. Wang et al. / Computers and Geotechnics 44 (2012) 1–8

R ¼ c0 þ c1Y1 þ c2Y2 þ c3Y3 þ c4Y1Y2 þ c5Y1Y3 þ c6Y2Y3

þ c7Y33 þ c8Y1Y2Y3 ðA-3Þ

where Y1 ¼P

Hclay=Hwall, Y2 ¼ su=r0v , Y3 ¼ Ei=1000r0v , and thecoefficients for Eq. (A-3) determined through the least-squareregression are as follows: c0 = 4.55622, c1 = �3.40151, c2 =�7.37697, c3 = �4.99407, c4 = 7.14106, c5 = 4.60055, c6 = 8.74863,c7 = 0.38092, and c8 = �10.58958.

The excavation-induced maximum ground settlement dvm canbe obtained by multiplying dhm with R.

dvm ¼ R � dhm ðA-4Þ

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