PROBABILISTIC APPROACHES TO STABILITY AND DEFORMATION PROBLEMS IN BRACED EXCAVATION

Transcription

1 Clemson Uniersity TigerPrints All Dissertations Dissertations PROBABILISTIC APPROACHES TO STABILITY AND DEFORMATION PROBLEMS IN BRACED EXCAVATION Zhe Lo Clemson Uniersity, Follow this and additional works at: Part of the Ciil Engineering Commons Recommended Citation Lo, Zhe, "PROBABILISTIC APPROACHES TO STABILITY AND DEFORMATION PROBLEMS IN BRACED EXCAVATION" (2011). All Dissertations This Dissertation is broght to yo for free and open access by the Dissertations at TigerPrints. It has been accepted for inclsion in All Dissertations by an athorized administrator of TigerPrints. For more information, please contact

2 PROBABILISTIC APPROACHES TO STABILITY AND DEFORMATION PROBLEMS IN BRACED EXCAVATION A Dissertation Presented to the Gradate School of Clemson Uniersity In Partial Flfillment of the Reqirements for the Degree Doctor of Philosophy Ciil Engineering by Zhe Lo December 2011 ŀ Accepted by: Dr. C. Hsein Jang, Committee Chair Dr. Sez Atamtrktr, Co-Chair Dr. Ronald D. Andrs Dr. William C. Bridges Jr. i

3 ABSTRACT This dissertation is aimed at applying probabilistic approaches to ealating the basal-heae stability and the excaation-indced wall and grond moements for sericeability assessment of excaation in clays. The focses herein are the inflence of spatial ariability of soil parameters and small sample size on the reslts of the probabilistic analysis, and Bayesian pdating of soil parameters sing field obserations in braced excaations. Simplified approaches for reliability analysis of basal-heae in a braced excaation in clay considering the effect of spatial ariability in random fields are presented. The proposed approaches employ the ariance redction techniqe (or more precisely, eqialent ariance method) to consider the effect of spatial ariability so that the analysis for the probability of basal-heae failre can be performed sing well-established first-order reliability method (FORM). Case stdies show that simplified approaches yield reslts that are nearly identical to those obtained from the conentional random field modeling (RFM). The proposed approaches are shown to be effectie and efficient for the probabilistic analysis of basal-heae in a braced excaation considering spatial ariability. The ariance redction techniqe is then sed in the probabilistic sericeability assessment in a case stdy. To characterize the effect of ncertainty in sample statistics and its inflence on the reslts of probabilistic analysis, a simple procedre inoling bootstrapping is presented. The procedre is applied to assessing the probability of sericeability failre in a braced excaation. The analysis for the probability of failre, referred to herein as ii

4 probability of exceeding a specified limiting deformation, necessitates an ealation of the means and standard deiations of critical soil parameters. In geotechnical practice, Ǣ these means and standard deiations are often estimated from a ery limited data set, which can lead to ncertainty in the deried sample statistics. In this stdy, bootstrapping is sed to characterize the ncertainty or ariation of sample statistics and its effect on the failre probability. Throgh the bootstrapping analysis, the probability of exceedance can be presented as a confidence interal instead of a single, fixed probability. The information gained shold enable the engineers to make a more rational assessment of the risk of sericeability failre in a braced excaation. The case stdy demonstrates the potential of bootstrap method in coping with the problem of haing to ealate failre probability with ncertain sample statistics. Finally, a Bayesian framework sing field obserations for back analysis and pdating of soil parameters in a mlti-stage braced excaation is deeloped. Becase of the ncertainties in the initial estimates of soil parameters and in the analysis model and other factors sch as constrction qality, the pdated soil parameters are presented in the form of posterior distribtions. In this dissertation, these posterior distribtions are deried sing Marko chain Monte Carlo (MCMC) sampling method implemented with Metropolis-Hastings algorithm. In the proposed framework, Bayesian pdating is first realized with one type of response obseration (maximm wall deflection or maximm grond srface settlement), and then this Bayesian framework is extended to allow for simltaneos se of two types of response obserations in the pdating. The proposed iii

5 framework is illstrated with a qality excaation case and shown effectie regardless of the prior knowledge of soil parameters and type of response obserations adopted. The probabilistic approaches presented in this dissertation, ranging from probability-based design of basal heae, to probabilistic analysis of sericeability failre in a braced excaation considering spatial ariability of soil parameters, to bootstrapping for characterizing the ncertainty of sample statistics and its effect, and to MCMC-based Bayesian pdating of soil parameters dring the constrction, illstrate the potential of probability/statistics as a tool for enabling more rational soltions in geotechnical fields. The case stdies presented in this dissertation demonstrate the seflness of these tools. i

6 DEDICATION I dedicate this dissertation to my parents. This dissertation exists becase of their loe and spport.

7 ACKNOWLEDGMENTS I wold like to thank my adisors, Drs. Hsein Jang and Sez Atamtrktr, for all of their ntiring gidance and spport dring my doctoral research at Clemson Uniersity. Withot their gidance and spport, I wold not hae been able to complete my dissertation stdies. I am also ery gratefl to my other committee members: Drs. Ronald Andrs, and William Bridges, for their assistance and reiew of my dissertation. They are all spportie and actiely helped with my corse stdy and the completion of this dissertation. Throgh their contribtion my doctoral stdy and research hae been an enjoyable and rewarding experience. Finally, I wold like to thank the Glenn Department of Ciil Engineering of Clemson Uniersity for the teaching assistantship dring my doctoral stdy and research. I also wish to thank the Shrikhande family and the Glenn Department of Ciil Engineering for awarding me the Aniket Shrikhande Memorial Annal Gradate Fellowship. i

8 TABLE OF CONTENTS TITLE PAGE...i ABSTRACT...ii DEDICATION... ACKNOWLEDGMENTS...i LIST OF TABLES...x LIST OF FIGURES...xi CHAPTER I. INTRODUCTION...1 Backgrond Prpose of the Research...1 Objecties and Scope of Dissertation...6 Significance of the Research...7 Strctre of the Dissertation...8 Page II. RELIABILITY ANALYSIS OF BASAL-HEAVE IN A BRACED EXCAVATION IN A ONE-DIMENSIONAL RANDOM FIELD...10 Introdction...10 Factor of Safety against Basal-Heae Failre...13 Stationary Random Field Modeling of s /σ'...19 Spatial Aeraging Effect...22 Reliability Analysis of Basal-Heae Stability Considering Spatial Variability...24 Reliability-Based Design Considering Spatial Variability...38 Smmary...45 III. RELIABILITY ANALYSIS OF BASAL-HEAVE IN A BRACED EXCAVATION IN A TWO-DIMENSIONAL RANDOM FIELD...46 Introdction...46 ii

9 Table of Contents (Contined) Page Two-Dimensional Random Field Modeling of s /σ'...47 Simplified Reliability Method for Assessing Probability of Basal-Heae Failre in Braced Excaation in 2-D Random Field...55 Smmary...73 IV. PROBABILISTIC SERVICEABILITY ASSESSMENT IN A BRACED EXCAVATION CONSIDERING SPATIAL VARIABILITY...75 Introdction...75 Finite Element Modeling with a Small-Strain Nonlinearity Soil Model...77 Modeling Spatial Variability in Braced excaations in Clays...81 Fzzy Sets Methodology - Modeling and Processing of Uncertain Parameters...84 Case Stdy TNEC Excaation Case...93 Smmary V. EFFECT OF SMALL SAMPLE SIZE ON THE PROBABILISTIC SERVICEABILITY ASSESSMENT Introdction Performance Fnction for Probability of Exceedance Point Estimate Method for Uncertainty Propagation and Probability of Exceedance Variation of Sample Statistics Determined by Bootstrapping Case Stdy TNEC Excaation Case Frther Discssions Smmary VI. BAYESIAN UPDATING OF SOIL PARAMETERS IN BRACED EXCAVATIONS Introdction Framework of Bayesian Updating with Marko Chain Monte Carlo Simlation Example Application: TNEC Excaation Case Smmary iii

10 Table of Contents (Contined) Page VII. CONCLUSIONS AND RECOMMENDATIONS Conclsions Recommendations REFERENCES ix

11 Ǣ LIST OF TABLES Table Page 1.1 Criteria for excaation protection leels in Shanghai, China (PSCG 2000) Parameters for a basal-heae stability problem shown in Figre Inpt parameters for a basal-heae stability problem shown in Figre Propping arrangement for the excaation and the stiffness of strts and floor slabs in the FEM analysis in TNEC case (after Kng et al. 2007a) Soil profile and soil model parameters sed in FEM analysis (from Kng et al. 2007a) Coefficients for transformation of inpt ariables (Kng et al. 2007b) Small-stain triaxial test reslts for Taipei clay (adapted from Kng 2003) Probability of exceedance in the TNEC excaation sing PEM and bootstrapping method with 17 data points Probability of exceedance in the TNEC excaation sing PEM and bootstrapping method with only 8 data points Probability of exceedance in the TNEC excaation nder three excaation protection scenarios Statistics of for prior distribtions sed in the Bayesian pdating scheme x

12 LIST OF FIGURES Figre Page 1.1 Schematic diagram of basal-heae failre Schematic diagram of excaation effects (Hsiao 2007) Geometry of slip circle method for basal-heae stability analysis (Adapted from W et al. 2010a) Gamma sensitiity index at arios COVs of s /σ' based on reliability analysis Example of simlated spatial ariability of normalized ndrained shear strength s /σ' by means of random field modeling Relatie freqency of the compted factor of safety sing random field modeling (nder the scenario that mean s /σ' = 0.3, COV of s /σ' = 0.3 and θ = 2.5m) Relationship between probability of failre and factor of safety at arios scales of flctation generated with MCS-based random field modeling Mean of normalized resisting moment as a fnction of the scale of flctation and COV of s /σ' COV of resisting moment (M R ) as a fnction of the scale of flctation and COV of s /σ' Inflence of COV and scale of flctation of s /σ' on the probability of failre Flow chart for searching for the redction factor for a gien pair of standard deiation σ and scale of flctation θ Back-calclated redction factor Γ for arios scales of flctation Comparison between the redction factors back-calclated sing the MCS-based RFM and those deried based on Eq. (2.13) with different assmed characteristic lengths...37 xi

13 List of Figres (Contined) Figre Page 2.12 Reliability-based procedre for ealating failre probability of basal-heae Comparison between the MCS-based random field modeling and the simplified approach Effect of ncertainty in the scale of flctation on the probability of failre against basal-heae in a braced excaation in clay (scale of flctation = 2.5m, COV of s /σ' = 0.3) Relationship between reqired factor of safety and scale of flctation at failre probability of Geometry of slip circle method and 2-dimensional random field modeling region for basal-heae stability analysis Inflence of scale of flctation on the 2-D random field modeling of s /σ' at gien mean of 0.3 and COV of Effect of 1-dimensional spatial ariability on the relationship between failre probability and factor of safety in the reliability-based design Effect of 2-dimensional spatial ariability on the relationship between failre probability and factor of safety in the reliability-based design Comparison between the redction factors back-calclated sing the MCS-based RFM and those compted sing Eq. (3.8) with assmed characteristic lengths (L = 18m or L h = 36m) Reliability-based procedre for ealating failre probability of basal-heae Comparison between the MCS-based RFM soltions and those by the simplified approach Effect of model bias of the slip circle method on the relationship between failre probability and factor of safety deried throgh reliability analysis...72 xii

14 List of Figres (Contined) Figre Page 4.1 Maximm wall deflections and maximm grond settlements at arios stages of excaation of the TNEC case A comparison of the obsered ales with those obtained by Kng et al. (2007a) sing AFENA and in this stdy sing PlaxisTM; both FEM codes implemented with the MPP soil model An example of failre srface in braced excaation Example of trianglar fzzy nmber and α-ct interal Vertex method for fzzy FEM analysis of braced excaation Fzzy nmber that represents the model otpt. The shaded area normalized to the fll area nder the shape is the probability of exceeding the limiting response ( x lim ) Schematic of checkerboard stdy on the ariation of soil parameters in an FEM model of TNEC case Inflence of scale of flctation on wall deflection and grond settlement Fzzy inpt parameters at different scales of flctation Reslting fzzy nmbers for maximm wall deflection and grond-srface settlement Probability of exceedance compted at arios leels of limiting wall deflection and grond srface settlement Generation of one bootstrap sample from the original obserations throgh random choice with replacement (adapted from Most and Knabe 2010) Soil profile and excaation depth of TNEC case: LL,liqid limit; N, blow cont; PI, plasticity index; w, moistre content (adapted from Kng et al. 2007a) Probability distribtion of the small-strain triaxial test reslts listed in Table xiii

15 List of Figres (Contined) Figre Page 5.4 Bootstrap samples generated from the original test data listed in Table Probability distribtion of the correlation coefficients for the generated bootstrap samples in Figre Bootstrap mean and standard deiation of s /σ' and E i /σ' with respect to nmber of bootstrap simlations Probability distribtion of the mean ale and standard deiation of s /σ' Probability distribtion of the mean ale and standard deiation of E i /σ' Distribtion of the reliability indices compted with the specified limiting (a) wall deflection and (b) grond settlement (at final excaation stage) Confidence interals for probability of exceedance in the TNEC excaation compted with arios leels of limiting wall deflection and grond srface settlement Effect of scaling factor on the efficiency of Marko chain sampling: (a) s = 0.01, (b) s = 3, (c) s = Relationship between acceptance rate and scaling factor pdated with both obsered δ hm and δ m Inflence of Marko chain length on the mean ales and standard deiations of the posterior distribtions estimated from 30 different Marko chains with s = Sampling process of MCMC simlations with the obserations from the 6th excaation stage (s = 3) Histograms of posterior distribtion for s /σ' and E i /σ' with data from Figre xi

16 List of Figres (Contined) Figre Page 6.6 Maximm settlement and wall deflection predictions prior to excaation stages 3, 4, 5, 6, 7 (sing Prior distribtion 1) Updated mean ale and one standard deiation bonds of the settlement predictions at target depth of 19.7 m prior to Stages 3, 4, 5, 6, and 7 of excaations (sing Prior distribtion 1 and the maximm settlement obserations) Updated mean ale and one standard deiation bonds of wall deflection predictions at target depth of 19.7 m prior to Stages 3, 4, 5, 6, and 7 of excaations (sing Prior distribtion 1 and the maximm wall deflection obserations) Ǣ 6.9 Comparisons of pdated predictions with three pdating schemes (sing Prior distribtion 1) Distribtions of predictions prior to final stage of excaation (sing Prior distribtion 1) Updated mean of soil parameters prior to Stages 3, 4, 5, 6, and 7 of excaations sing arios prior distribtions Updated COV of soil parameters prior to Stages 3, 4, 5, 6, and 7 of excaations sing arios prior distribtions Updated COV of soil parameters prior to Stages 3, 4, 5, 6, and 7 of excaations sing mean ale of Prior distribtion 2 and arios COVs Updated distribtion of soil parameters prior to final stage of excaation sing mean ale of Prior distribtion 2 and COV = Inflence of correlation coefficient ρ between model biases on predictions pdated with obsered maximm settlement and wall deflection (sing Prior distribtion 1) x

17 CHAPTER I INTRODUCTION Backgrond Prpose of the Research In the rban enironment, deep braced excaation is a commonly-sed constrction method for high-rise bilding basements, ndergrond transportation stations, and ndergrond parking and commercial spaces, etc. In the design of a braced excaation in clays, two important isses are ineitable, namely, (1) basal-heae stability dring the constrction, and (2) the sericeability isses sch as the excaation-indced wall and grond responses. Basal-heae failre in a braced excaation in clays may be indced by insfficient shear strength, which spports the weight of soil within the critical zone arond the excaation. Dring an excaation, soil otside the excaation zone moes downward and inward becase of its own weight and srcharge; this tends to case soil inside the excaation zone to heae p, as shown in Figre 1.1. Collapse of ŀ the bracing system may occr if the amont of basal-heae moement is excessie. On the other hand, een if the basal-heae stability can be achieed in the design, the adjacent strctres may still be damaged becase of the excessie wall deflection and grond srface settlement, as shown in Figre 1.2. Failres of excaation projects hae been reported worldwide: e.g., Infopedia (2004) and Chen et al. (2007). Considering that most braced excaation projects are condcted in the nban enironment, the social, economic, and enironmental impacts cased by the failre of an excaation project can be significant. In the design of a braced excaation, both stability and sericeability reqirements shold be garanteed. 1

18 Strts Settlement Bottom heae Retaining wall Failre srface Figre 1.1: Schematic diagram of basal-heae failre. Figre 1.2: Schematic diagram of excaation effects (Hsiao 2007). 2

19 Table 1.1: Criteria for excaation protection leels in Shanghai, China (PSCG 2000). Protection leel Limiting wall deflection and grond srface settlement Reqirements of the enironmental protection 1. Maximm wall deflection 0.14% H Metro lines and important facilities sch as gas mains and water drains exist within a distance I 2. Maximm grond srface settlement 0.1% H of 0.7H from the excaation; safety has to be ensred. 3. FS (basal stability) 2.2 II III 1. Maximm wall deflection 0.3% H 2. Maximm grond srface settlement 0.2% H 3. FS (basal stability) Maximm wall deflection 0.7% H 2. Maximm grond srface settlement 0.5% H 3. FS (basal stability) 1.5 Important infrastrctres or facilities sch as gas mains and water drains exist within a distance of (1-2) H from the excaation. No important infrastrctres or facilities exist within a distance of 2 H from the excaation Note: H = final excaation depth; FS = factor ŀ of safety against basal heae, calclated sing the slip circle method. 3

20 Conentionally, a safe design may be realized by satisfying the factor of safety (FS) reqirements, as well as the wall and grond reqirements as a means of preenting the excaation failre and damage to the adjacent infrastrctres. An example of the design codes for braced excaations sed in China (PSCG 2000) is illstrated in Table 1.1. The limiting FS, wall and grond deformations to preent failre in the designs of excaations are sggested for arios protection leels, depending on the reqirements of the enironmental protection. It shold be noted that a design based on those limiting criteria in a deterministic approach may not garantee safety, since it is always difficlt to estimate the FS, wall and grond responses with certainty mainly de to the ncertainty of design soil parameters. The sorces of parameter ncertainty inclde inadeqate site inestigation, measrement errors, as well as inherent and spatial ariability of soil. Inherent ariability of soil parameters is interpreted by their probability distribtions or sample statistics (e.g., mean ale and standard deiation). Spatial ariability is generally described by the scale of flctation, which is the maximm distance within which the spatially random parameters are correlated (Akbas and Klhawy 2009). Spatial ariability may be modeled with the random field theory (Vanmarcke 1977). Recent stdies of random field modeling (RFM) based on Monte ŀ Carlo simlation (MCS) demonstrate that spatial ariability plays an important role in reliability-based design in geotechnical engineering (Griffiths and Fenton 2009; Hang et al. 2010). Neglecting spatial soil ariability in reliability analysis of geotechnical problems can lead to either oerestimation or nderestimation of the failre probability in a gien design, depending on the specified limiting FS, wall deflection or grond 4

21 settlement leels (Wang et al. 2011b). In this regard, Chapters II and III of this dissertation are deoted to deeloping simplified approaches for the basal-heae stability analysis of braced excaations with the consideration of the effect of one-dimensional (1-D) and two-dimensional (2-D) spatial ariability, respectiely. Frther, the inflence of spatial ariability on the probabilistic sericeability assessment in braced excaation is presented in Chapter IV. Small sample size can lead to large ncertainty in soil parameters. Becase of bdget constraints, the geotechnical engineers often hae to derie sample statistics from a small sample (i.e., a small data set), which can lead to ncertainty in the sample statistics: the accracy of the estimated mean and standard deiation of the ncertain soil parameters is qestionable. This isse is important becase it can significantly inflence the reslts of the reliability/probability analysis (Schweiger and Peschl 2005). Considering that the problem of small sample size of soil parameters in geotechnical projects is not ncommon, the effect of this ncertainty on the failre probability in braced excaation in clays shold be examined. Ths, Chapter V of this dissertation is deoted to deeloping a simple procedre inoling bootstrapping approach (Efron 1979) for assessing the ncertainty of sample statistics cased by small sample size. The procedre is then applied to the analysis of the probability of sericeability failre in a braced excaation. The failre probability (or the corresponding reliability index) is interpreted sing confidence interals in order to take into accont of those ncertainties cased by small sample size. Deep braced or spported excaations are generally performed with staged 5

22 constrction. The predictions of wall and grond responses may not be accrate de to the ncertainty of design soil parameters. Howeer, the soil parameters may be pdated with the obsered wall and grond responses to refine the knowledge of them. The predictions for the sbseqent stages will be then improed with the pdated soil parameters. It shold be noted that in the traditional back analysis, the focs is on finding a set of fixed ales for the parameters of concern, withot considering the ncertainty in the obserations and model bias. Becase of the high degree of ncertainty inoled in the soil-strctre interaction, the fixed parameter ales may not be feasible or physically meaningfl. Therefore, parameters of concern are preferably treated as a random ariable and the pdated parameters are expressed in terms of posterior distribtions. Throgh comparison of model predictions against obserations in field, soil parameters are pdated sing Bayes theory, which reslts in posterior distribtions of soil parameters. To this end, Chapter VI is deoted to deeloping sch a Bayesian framework sing field obserations for pdating soil parameters in braced excaation in clays. Objecties and Scope of the Research The scope of this dissertation focses on the applications of probabilistic approaches to ealate the basal-heae stability and the excaation-indced wall and grond moement in clays for sericeability assessment. The specific objecties of this dissertation are: 1. Stdy the inflence of spatial ariability on the reliability-based design against 6

23 - basal-heae stability in braced excaation in clays sing random field modeling. 2. Deelop simplified procedres for reliability-based design against basal-heae stability sing eqialent ariance techniqe. 3. Perform a probabilistic ealation of the excaation-indced wall and grond responses considering spatial ariability. 4. Stdy the inflence of small sample size of soil parameters on the probabilistic sericeability assessment in braced excaation in clays. 5. Deelop a Bayesian framework for pdating key soil parameters in braced excaation sing obsered excaation-indced wall and grond responses. Significance of the Research The spatial ariability of soil has significant inflence on the reliability-based design in geotechnical engineering. It is difficlt to apply random field modeling (RFM), which necessitates the se of Monte Carlo simlation, in a complicated soil-strctre problem sch as braced excaation. Therefore, the main contribtion of this dissertation is the deelopment of simplified approaches, which employ eqialent ariance techniqe and first-order reliability method (FORM), for the reliability-based design against basal-heae stability considering spatial ariability. Throgh properly selecting the characteristic lengths, those simplified approaches are shown to be eqialent to RFM. The simplified approaches are implemented in a spreadsheet and reqire mch less comptation effort. The simplified approaches are easy to se, and hae potential as a practical tool for reliability-based design that has to deal with spatial 7

24 ariability of soils. Another contribtion of this dissertation is the deelopment of Bayesian framework for the pdating key soil parameters. In this framework, the pdating procedre starts with an assmption for the prior distribtions for soil parameters, based on pblished opinions and engineering jdgment. After the initial excaation stage is condcted, the maximm wall deflection and maximm settlement are obsered. Those obserations are sed to pdate the distribtions of soil parameters, and the pdated soil parameters are then sed to predict the responses in the sbseqent stages. This straightforward procedre is repeated as the staged excaation proceeds, and the soil parameters are pdated accordingly. The predictions sing pdated soil parameters can reprodce the reality with improed fidelity, comparing to those obtained with the prior distribtions. Frthermore, the two types of obserations in the sericeability assessment, maximm wall deflection and maximm grond srface settlement, may be simltaneosly employed to refine the knowledge of ncertain soil parameters and the predictions of the wall and grond responses. The Strctre of the Dissertation This dissertation consists of seen chapters. In Chapter I, the crrent chapter, an introdction is presented to organize the entire dissertation. The prpose and the scope of the research and the otline of the dissertation are also presented. Chapter II throgh Chapter VI present the major contents of this dissertation and Chapter VII presents the conclsions of this dissertation. In Chapter II, a simplified approach for 8

25 reliability analysis of basal-heae in a braced excaation considering the 1-D spatial ariability of soil parameters is presented. In Chapter III, the aforementioned simplified approach is extended to consider the 2-D spatial ariability. In Chapter IV, the simplified approach sing eqialent ariance techniqe is applied in the probabilistic sericeability assessment in braced excaation. In Chapter V, the effect of small sample size of soil parameters on the probabilistic sericeability assessment is examined throgh bootstrapping approach. Chapter VI demonstrates the deelopment of the Bayesian framework for pdating soil parameters in braced excaation. Finally, in Chapter VII, the main conclsions of this dissertation are presented. 9

26 CHAPTER II RELIABILITY ANALYSIS OF BASAL-HEAVE IN A BRACED EXCAVATION IN A ONE-DIMENSIONAL RANDOM FIELD Introdction Conentionally, basal-heae stability in a braced excaation in clay is ealated with a factor of safety (FS), defined as the ratio of the resistance oer the load (e.g., Terzaghi 1943; Bjerrm and Eide 1956). In designs based on FS, soil parameters are generally considered as constant inpts for simplicity. Howeer, it is well known that FS greater than nity does not garantee basal-heae stability in clay de to the inherent ariability of soil parameters sch as ndrained shear strength and nit weight. Althogh ncertainty in the soil parameters is often dealt with by se of conseratie parameter ales, the probabilistic approach sing reliability analysis offers a more direct and consistent way to consider soil ariability explicitly. Examples of the reliability-based design for basal-heae stability of a braced excaation can be fond in Goh et al. (2008) and W et al. (2010a), in which a design chart that relates the probability of basal-heae failre ( p f ) to the factor of safety FS is proided. In traditional reliability analysis, ncertain soil parameters are interpreted as continos random ariables defined by their probability distribtions or sample statistics (e.g., mean ale and standard deiation). The soil parameters are often considered as homogeneos or spatially constant fields in sch analysis. Howeer, the ncertainty A similar form of this chapter has been pblished at the time of writing: Lo Z, Atamtrktr S, Cai Y, Jang CH. Simplified approach for reliability-based design against basal-heae failre in braced excaations considering spatial effect. Jornal of Geotechnical and Geoenironmental Engineering, doi: /(asce)gt

27 stems not only from the inherent ariability, bt also from spatial ariability. For the latter, the ariation of soil parameters may be modeled with the random field theory (Vanmarcke 1977). Spatial ariability is generally described by the scale of flctation, which is the maximm distance beyond which the spatially random parameters are ncorrelated (Akbas and Klhawy 2009). As the scale of flctation decreases, the soil parameters in the random field tend to ary more rapidly; conersely, as the scale of flctation increases, the soil parameters in the random field tend to ary less and become more niform. The effect of spatial ariations of soil properties can be significant in many geotechnical problems, as demonstrated by recent stdies of random field modeling (RFM) by Griffiths and his colleages (Fenton and Griffiths 2003; Fenton et al. 2005; Griffiths and Fenton 2009; Hang, et al. 2010). In their stdies, Griffiths and his colleages adopted local aeraging sbdiision techniqes to model the random field. Of corse, the random field can also be modeled sing other approaches sch as the Cholesky decomposition method (Fenton 1997; Haldar and Bab 2008; Sriastaa et al. 2010; Schomel and Mašín 2010). The conentional random field modeling (RFM), howeer, has to be realized with Monte Carlo simlations (MCS), and a large nmber of simlations are needed to obtain conergent reslts. As an alternatie to the conentional RFM, simplified methods that implement a proper spatial aeraging strategy hae been shown to be effectie in considering the effect of spatial ariability of soil properties (e.g., Phoon and Klhawy 1999a,b; Goh et al. 2008; Klammler et al. 2010; Most and Knabe 2010). To consider spatial aeraging in 11

28 the reliability analysis, ariances of soil parameters are redced by mltiplying a redction factor that is a fnction of scale of flctation and characteristic length (Vanmarcke 1983). Typical scales of flctation for commonly sed soil properties hae been reported by Phoon and Klhawy (1999b). The characteristic length often depends on the problem nder inestigation, and is generally assmed to be eqialent to the length of the failre srface (Schweiger and Peschl 2005; Most and Knabe 2010) or taken as the distance in the random field oer which the ariance redction is calclated (Cherbini 2000; Bab and Dasaka 2008). Recent stdies (Peschl and Schweiger 2003; Schomel and Mašín 2010) show that the ariance redction-based simplified approach can captre the oerall trend deried from the conentional RFM approach. Althogh RFM copling with Monte Carlo simlation (MCS) is a rigoros approach to accont for spatial ariability, se of this approach is qite limited in geotechnical reliability-based design for at least two reasons: (1) a rigoros simlation of the random field is ery time-consming, which is not practical, especially for complicated problems sch as braced excaations; (2) MCS is frther complicated by the lack of knowledge on spatial ariability (for example, the scale of flctation cold be ncertain). Contrarily, with the ariance redction-based simplified approach, traditional reliability methods can be adopted in lie of MCS to redce the comptational effort. Howeer, the application of sch simplified approach reqires a proper assessment of the characteristic length, which is problem-specific and may be difficlt to determine. In this chapter, a simplified approach that considers the spatial ariability of soil parameters for reliability analysis of basal-heae stability in a braced excaation in clay is 12

29 presented. This approach is deeloped and presented in 5 steps. Firstly, the conentional RFM sing the exponential correlation fnction is condcted in a stdy of basal-heae stability to proide a reference for frther deelopment. Secondly, by trial-and-error, the ariance redction factor is determined, with which the simplified approach can yield reslts that are comparable to those obtained sing the conentional RFM. Thirdly, the characteristic length for the stability analysis is back-calclated based on the deried ariance redction factors. Forthly, the proposed approach, which is first-order reliability method (FORM) implemented with the ariance redction to accont for the spatial ariability, is adopted for reliability analysis of basal-heae stability. Lastly, the effect of ncertainty of the scale of flctation (becase of the lack of knowledge) is frther ealated with the proposed approach. It concldes that the proposed simplified approach is easy to se and yields reslts that are comparable to those obtained with the comptationally expensie RFM approach. Factor of Safety against Basal-Heae Failre Slip circle method The basal-heae failre in a braced excaation in clay occrs when the shear strength of the soil cannot spport the weight of the soil within the critical zone arond the excaation. Soil otside the excaation zone moes downward and inward becase of its own weight and the soil inside the excaation zone is forced to heae. The bracing system will collapse if the amont of basal-heae moement is excessie. Traditionally, the basal-heae stability is ealated with FS sing the deterministic 13

30 approach. Semi-empirical methods (Terzaghi 1943; Bjerrm and Eide 1956; Eide et al. 1972; Chang, 2000) to estimate FS are widely sed in the traditional deterministic design. In this chapter, the slip circle method adopted by Japanese, Chinese, and Taiwanese bilding codes (JSA 1988; PSCG 2000; TGS 2001) to calclate the FS against basal-heae is sed for its simplicity to consider the increase of ndrained shear strength with depth, and for its conenience to implement random field theory. With the slip circle method, the FS is defined below (see Figre 2.1): M M R FS = (2.1) D where M R and M D are resistance moment and driing moment respectiely. The driing moment M D is cased by the weight of soil and possible srcharge: 2 r r M D = W + qs (2.2) 2 2 where W is the total weight of the soil in front of the ertical failre plane and aboe the excaation srface, q s is the srcharge, r is the radis of the slip circle, and r = H w H s in which H w is the length of diaphragm wall and H s is the depth of the final strt. The resistance moment M R comes from three arcs (bc, cd, de) along the slip srface, as show in Figre 2.1. Althogh niform ndrained shear strength may be sed in the comptation of FS, the ndrained shear strength generally increases with depth for most normally consolidated clay. Howeer, the ratio of ndrained shear 14

31 - strength oer the effectie oerbrden stress ( s / σ ) remains roghly constant (Ladd and Foott 1974). For this reason, the slip circle method can be easily adapted to consider the increase of s with depth. The total resistance moment M R is compted by smming the resistances contribted by all the small arcs: M R = r π / 2+ α 0 s r dβ (2.3) where β is the angle from ob to the crrent slice as shown in Figre 2.1. qs = 10 kpa / m Diaphragm wall a He = 20 m γ soil = 19 kn / m 3 e Final strt α o β b c H p = 24 m r dβ ds Failre srface d Figre 2.1: Geometry of slip circle method for basal-heae stability analysis (Adapted from W et al. 2010a). 15

32 In a deterministic analysis of basal-heae stability in a braced excaation in clay, the reqired FS generally depends on the method sed. The recommended minimm reqired FS is 1.2 (JSA 1988; PSCG 2000; TGS 2001) when the slip circle method is employed. In the slip circle method, the resistance is compted sing the smmation of the resistance of nmeros small arcs [Eq. (2.3)]. This formlation makes it easy to implement the random field model, which is the main reason behind the choice of the slip circle method in this stdy. Gamma sensitiity index Many factors inflence the basal-heae stability in a braced excaation in clay as reflected in Eqs. (2.2) and (2.3). The relatie importance of those inpt ariables is first examined sing the gamma sensitiity index (Der Kireghian and Ke 1985), which is a by-prodct of reliability analysis. This index is expressed as: αj y, xm γ i = (2.4) αj M y, x where γ i = gamma sensitiity index for the i th inpt ariable, α = directional cosine at the design point in the original random ariable space, J, = Jacobian matrix with y x element of y / x with T ( x) y = where ( ) T is an orthogonal transformation fnction, y i = ncorrelated standard normal random ariable, and M = diagonal matrix of the standard deiation of each parameter x i. The ncertain ariables 16

33 Ï considered in this stdy inclde x 1 = s / σ (normalized ndrained shear strength), x 2 = H e (excaation depth), x 3 = H w (length of diaphragm wall), x 4 = H s (depth of the final strt), x 5 = D (depth of grond water table), x 6 =γ (nit weight of soil), and x 7 = q s (srcharge). The gamma sensitiity index indicates the relatie contribtion of each of these inpt ariables to the compted reliability index or failre probability. A higher gamma sensitiity index ale indicates a greater inflence of the ariable of concern on the failre probability. Table 2.1: Parameters for a basal-heae stability problem shown in Figre 2.1. Parameters Notations Statistics of parameter Mean Coefficient of ariation Normalized ndrained shear strength s /σ Unit weight of soil γ 19 kn/m (1) Srcharge q s 10 kpa/m 0.2 (2) Depth of GWT D 2 m 0.05 (3) Final excaation depth H e 20 m 0.05 (3) Final strt depth H s 17 m 0.05 (3) Penetration depth H p 24 m 0.05 (3) (1) Based on the COV ales gien by Harr (1987) and DiMaggio (2008) (2) W et al. (2010a) (3) Hsiao et al. (2008) Based on the first order reliability method (FORM) analysis of basal-heae stability withot considering spatial ariability, the gamma sensitiity index for each of the seen inpt parameters is obtained with Eq. (2.4). The statistics of the ncertain parameters sed in this FORM analysis are smmarized in Table 2.1. For this gamma 17

34 sensitiity analysis, the mean of s / σ (normalized ndrained shear strength) is set at 0.3 and the coefficient of ariation (COV) of s / σ is aried between 0.1 and 0.6. As shown in Figre 2.2, the parameter s / σ is fond to hae the greatest inflence on the probability of basal-heae failre, and all other factors are relatiely insignificant. The gamma sensitiity index of s / σ is also fond to increase drastically with the coefficient of ariation (COV) of s / σ. Ths, this stdy is focsed on the effect of the spatial ariability of the normalized ndrained shear strength s / σ on the probability of basal-heae failre in a braced excaation. 1.0 Gamma sensitiity, γ i SRR1 s / HSRR2 e σ H w SRR3 SRR5 D Ï q s SRR8 H s SRR4 γ SRR COV of s / σ Figre 2.2: Gamma sensitiity index at arios COVs of s /σ' based on reliability analysis. 18

35 Stationary Random Field Modeling of s /σ' Z To proide a reference for the proposed simplified approach for reliability-based design against basal-heae failre in a braced excaation considering spatial random effect, the conentional random field modeling of the normalized ndrained shear strength / is first condcted in this stdy. Here, the parameter s σ s / σ is modeled sing a stationary lognormal random field. It shold be noted that the mean of the ndrained shear strength s of the clay often increases linearly with depth; howeer, this trend is remoed herein by adopting the normalized parameter s / σ. The stationary random field of s / σ has the characteristics of a second-order process (Baecher and Christian 2003): (1) the mean and ariance of s / σ ( z) are the same regardless the absolte location of z, and (2) the correlation coefficient between s / ( z ) σ 1 and 2 s / σ ( z ) is the same regardless of the absolte locations of z 1 and z 2 ; rather, it depends only on the distance between z 1 and z 2. All other inpt parameters are modeled as spatially-constant lognormal ariables or constants. The assmption of lognormal distribtion for inherent ariability for soil properties is not ncommon in the RFM (e.g., Akbas and Klhawy 2009; Griffiths et al. 2009). The assmption of lognormal distribtion preents negatie ales for soil parameters, and is spported by past stdies (e.g., Phoon and Klhawy 1999b). Variations of s / σ in the field are represented by its scale of flctation θ, mean ale µ ln, and coefficient of ariation COV ln (note: the sbscript in the last two terms, ln, denotes the statistic for lognormal distribtion). The standard deiation and 19

36 Z mean of the eqialent normal distribtion of s σ /, denoted as ln ( s / σ ), are expressed as: 2 ( 1 COV ) σ n = ln + ln (2.5) 1 2 µ n = ln µ ln σ n (2.6) 2 where the sbscript n denotes normal distribtion. The lognormally distribted random field of s / σ can be obtained by the transformation (Fenton et al. 2005): s ( x ) = exp{ µ + G ( x )} / σ σ (2.7) i n n n i where µ n and σ n are determined from Eqs. (2.5) and (2.6); x i is the spatial position at which s σ / is modeled; ( ) G is a normally distribted random field with zero n x i mean, nit ariance and correlation fnction ρ ( τ ), where ( τ ) ρ is defined as an exponentially decaying correlation fnction (Jaksa et al. 1999; Haldar and Bab 2008): ρ 2τ = exp (2.8) θ ( τ ) where τ = x x is the absolte distance between any two points in the random field i j and θ is the scale of flctation. The correlation matrix is bilt with the correlation fnction and can be decomposed by Cholesky decomposition (Fenton 1997; Haldar and Bab 2008; Schomel and Mašín 2010; Sriastaa et al. 2010): 20

37 T L L = ρ (2.9) With the matrix L, the correlated standard normal random field can be obtained by linearly combining the independent ariables as follows (Fenton 1997): G n ( xi ) = i j= 1 L Z ij j i = 1, 2,, M (2.10) where M is the nmber of points in the random field; Z j is the seqence of independent standard normally distribted random ariables. To begin with, two random ariables niformly distribted between 0 and 1, U j and U j+ 1, are generated first. Then two independent standard normally distribted ariables are gien by: Z ln(1 U ) cos(2π U ) (2.11) j = 2 j j+ 1 Z ln(1 U ) sin(2π U ) (2.12) j+ 1 = 2 j j+ 1 The stationary random field of the normalized ndrained shear strength s / σ at each spatial position is obtained by Eq. (2.7) for a specified mean, standard deiation, and scale of flctation. The Monte Carlo simlation (MCS) is then sed to generate samples in the lognormal random field. Each simlation of the Monte Carlo process inoles the same mean, standard deiation and scale of flctation of s / σ. Howeer, the spatial distribtion aries among these simlations. Gien a sfficient nmber of simlations, the 21

38 otpt sch as M R [Eq. (2.3)] or FS [Eq. (2.1)] can be obtained and statistically analyzed to prodce estimates of the probability density fnction of M R or FS and the failre probability p f. The failre probability p f is compted as the ratio of the nmber of simlations that yield failre (FS < 1) oer the total nmber of simlations N. The nmber of MCS samples shold be at least 10 times of the reciprocal of the target failre probability (Ang and Tang 2007; Wang et al. 2011a). In this stdy, the leel of failre probability of interest is greater than 10-4, therefore N is set at Spatial Aeraging Effect Spatial aeraging is a concept with which, the spatial ariability of the soil property is aeraged in order to approximate a random ariable that represents a soil parameter (Vanmarcke 1977). The ariability of the aeraged soil property oer a large domain is less than oer a small domain. The redced ariability of the soil properties oer a large domain can be qantified with the ariance redction techniqe. The redction is compted sing the ariance redction fnction, which is a fnction of the scale of flctation θ and characteristic length L. The form of the ariance redction fnction depends on the type of correlation fnction employed. To consider spatial aeraging in a reliability analysis, the ariances of soil parameters may be redced by mltiplying a factor known as the ariance redction factor which is compted sing the ariance redction fnction (Vanmarcke 1983). Many sccessfl applications of the ariance redction techniqe hae been reported in the literatre, e.g., constant model (Cherbini 2000; Schweiger and Peschl 2005), 22

39 trianglar model (Bab and Dasaka 2008), exponential model (Most and Knabe 2010). The exponential model, which is often employed in the RFM in geotechnical engineering, is adopted herein. The ariance redction fnction for the exponential model is gien as follows (Vanmarcke 1983): Γ 2 = 2 1 θ 2L 2L 1 + exp 2 L θ θ (2.13) where θ is the scale of flctation and L is the characteristic length. Gien the ariance redction factor Γ 2, the redced ariance σ Γ 2 can be obtained with the following eqation: σ 2 = Γ 2 Γ σ 2 (2.14) where σ is the standard deiation of the soil parameter of concern ( s / σ in this stdy). It is noted that the positie sqare root of the ariance redction factor is referred to herein as the standard deiation redction factor or simply the redction factor ( Γ ) to differentiate it from the ariance redction factor Γ 2. Unlike RFM, the FORM analysis sing the ariance redction techniqe does not reqire MCS. Therefore, this approach of sing the ariance redction techniqe reqires mch less comptational effort and is more practical than with RFM in engineering practice. Past inestigators (Peschl and Schweiger 2003; Schomel and Mašín 2010) hae shown that the reliability analysis with the ariance redction method can captre the oerall trend deried with RFM. 23

40 Howeer, the choice of characteristic length is critical to the reliability analysis with the ariance redction techniqe. For analysis of braced excaations in clay, it has been sggested that the characteristic length may be assmed to be the length of the sliding srface (Schweiger and Peschl 2005). In this stdy, an effort is made to inestigate the appropriate characteristic length to se and to examine the effect of the ariation of the characteristic length on the probability of basal-heae failre. Reliability Analysis of Basal-Heae Stability Considering Spatial Variability Random field modeling of clay for basal-heae stability analysis Past stdies (e.g., Goh et al. 2008; W et al. 2010a) on basal-heae stability hae shown that high failre probability p f can exist in a design that meets the minimm FS reqirement specified in the codes. Howeer, yielding high failre probabilities for those designs that are known to be safe raises qestions, since the codes are generally conseratie and ths exceeding the minimm FS reqirement wold indicate a safe design. One possible reason for haing a higher compted failre probability than what the experience or the code wold sggest is oerestimation of the ariation of soil parameters, which might be cased by the negligence of the effect of spatial ariability in traditional reliability analysis. In this stdy, the aboe isse is examined within the context of basal-heae stability. Here, basal-heae stability in a braced excaation is examined sing the conentional random field model with the Cholesky decomposition method. The excaation case analyzed by W et al. (2010a), illstrated in Figre 2.1 and with 24

41 additional data shown in Table 2.1, is employed in this stdy. In reference to Figre 2.1 and Table 2.1, all soil and strctral parameters, except the normalized ndrained shear strength s / σ, are treated as spatially-constant random ariables or constants. The parameter s / σ is modeled as a spatially random ariable. The COV of ndrained shear strength s can be as high as 0.8 bt typically is abot 0.3 (Phoon and Klhawy 1999a). In this stdy, the COV of s / σ is first selected as 0.3. Howeer, the effect of the ariation of this COV will be examined later. According to Phoon and Klhawy (1999a), the aerage horizontal and ertical scales of flctation for clay are 50.7m and 2.5m, respectiely. Ths, only the ertical spatial randomness is modeled in this stdy, as the horizontal scale of flctation is mch greater and its effect is far less significant. It shold be noted that for basal-heae stability Z analysis in a braced excaation, only the random field from the depth of the final strt to the bottom of the diaphragm wall (see Figre 2.1) needs to be considered since the resistance moment comes only from this region. It shold be noted that the Cholesky decomposition method is not practical if the nmber of points in the random field exceeds 500 (Fenton 1997). In this stdy, Arc bcd on the slip circle, as shown in Figre 2.1, is sbdiided by means of eqal ertical distance into 100 small arcs (elements) which are considered sfficient in both stability analysis and random field modeling. Note that Arc de is sbdiided in the same way as for Arc cd. Frther refinement with more than 100 elements (arcs) is not necessary as it yields practically the same reslts. Figre 2.3 shows an example of the simlated spatial ariability of normalized 25

42 Z ndrained shear strength s / σ with different scales of flctation (θ = 0.5m, 2.5m and 10m). As expected, the spatial ariation in the case of smaller θ is mch more significant than that for larger θ. As θ decreases toward zero, the random field s / σ tends to ary drastically from point to point; conersely, as θ increases toward infinity, the random field s / σ tends to become niform (or spatially constant) in each simlation. Traditional reliability analysis often assmes the field to be spatially constant and the effect of spatial correlation is ignored. 0 s 1/ σ Mean Vale Depth (m) 5 10 θ = 0.5m θ = 2.5m θ = 10m Figre 2.3: Example of simlated spatial ariability of normalized ndrained shear strength s /σ' by means of random field modeling. 26

43 Z Relatie freqency Histogram Lognormal fit Factor of safety Figre 2.4: Relatie freqency of the compted factor of safety sing random field modeling (nder the scenario that mean s /σ' = 0.3, COV of s /σ' = 0.3 and θ = 2.5m). For a gien mean, COV and θ of s / σ, Monte Carlo simlation (MCS) may be carried ot and in each simlation, the same mean, COV and θ are sed to generate the random field. After a sfficient nmber of simlations (10 5 in this stdy), the histogram or the probability density for the otpt ariable (for example, FS) can be obtained. Figre 2.4 shows an example of histogram of the compted FS sing the conentional RFM with 100,000 simlations nder the following scenario: mean of s / σ = 0.3, COV of s / σ = 0.3 and θ = 2.5m. The shape of the histogram sggests a lognormal 27

44 distribtion. The failre probability p f is determined by the ratio of the nmber of simlations with FS < 1. 0 oer the total nmber of simlations, which is the area nder the fitted cre for FS < Reqired factor of safety θ = 1000m θ = 100m θ = 10m θ = 5m θ = 2.5m θ = 0.5m Target failre probability 0 Figre 2.5: Relationship between probability of failre and factor of safety at arios scales of flctation generated with MCS-based random field modeling. To stdy the effect of spatial ariability, a series of scales of flctation (θ = 0.5m, 2.5m, 5m, 10m, 100m and 1000m) is selected in the reliability analysis. For each scale of flctation, 10 5 simlations sing MCS are condcted and the reslts are shown in 28

45 Figre 2.5. Note that for each data point in Figre 2.5, the exection time for 10 5 simlations is approximately 4 mintes on a laptop PC eqipped with an Intel Pentim Dal CPU T2390 rnning at 1.86GHz. The effect of the scales of flctation is qite obios: smaller scale of flctation reslts in smaller p f at the same FS. As shown in Figre 2.5, if the target p f is set at 10-3, the reqired FS is abot 1.7 at θ = 2.5m [note: this θ ale is mean of the ertical scale of flctation for clay as per Phoon and Klhawy 1999a], and is abot 2.6 at θ = 1000m (note: this θ ale is close to spatially-constant condition). On the other hand, for FS = 1.2, the minimm ale that is adopted in many codes (JSA 1988; PSCG 2000; TGS 2001) for the design of excaations against basal-heae based on the slip circle method, the failre probability p f is abot 0.32 nder the condition of θ = 1000m ( spatial constant). As basal-heae failre occrs infreqently, these codes are considered adeqate in practice; therefore, the failre probability of 0.32 obtained from the reliability analysis that does not consider spatial ariability (emlated by the case with θ = 1000m) is likely to be oer-estimated. Finally, it shold be noted that the analysis of basal-heae stability presented in this section for RFM of clay is primarily sed as a reference for the sbseqent stdy of the effect of spatial ariability sing the ariance redction-based simplified approach. Parametric stdy A series of parametric analyses are condcted to stdy the inflence of spatial ariability on the reliability-based design of braced excaation (basal-heae stability) in clay. For these analyses, only the inherent ariability and the spatial ariability of 29

46 s / σ are considered to assess the effect of the spatial correlation. All other inpt parameters are treated as constant parameters (only the mean ales listed in Table 2.1 are sed in the analysis). For s / σ, the following ranges of parameters are analyzed: COV = 0.1, 0.2,, 1.0 θ = 0.5m, 1m, 2.5m, 5m, 10m, 25m, 50m, 100m, For each pair of COV and θ, 10 5 MCS rns are exected, and the mean and COV of the reslting 10 5 resistance moments M R are obtained. Mean of normalized M R θ=0.5m θ=1m θ=2.5m θ=5m θ=10m θ=25m θ=50m θ=100m θ= COV of s / σ Figre 2.6: Mean of normalized resisting moment as a fnction of the scale of flctation and COV of s /σ'. Figre 2.6 shows how the mean of the normalized M R, defined as the ratio of the 30

47 M R obtained from MCS for a gien pair of COV and θ ales oer the M R obtained from a deterministic analysis that ses the mean ales of all inpt parameters, aries with the COV and θ of s σ /. The mean of the normalized R M is shown to be arond 1.0, indicating that the mean of the M R throgh 10 5 simlations is consistent with the deterministic soltion regardless of the inherent ariability and the spatial ariability of s / σ. This is expected as the resisting moment M R in the slip circle method is a linear fnction of ndrained shear strength s. Figre 2.7 shows how the COV of M R changes with the COV and θ of s / σ. Two obserations can be made: (1) at the same θ leel, the COV of M R increases almost linearly with the COV of s σ / ; (2) at the same COV of s / σ, the COV of M R increases with increasing θ and reaches the maximm ale at θ =. The COV of M R at θ = approaches to the 1:1 line. As shown in Figre 2.7, smaller θ reslts in smaller ariability of M R, which corresponds to smaller ariability of s / σ. This obseration is consistent with the concept of spatial aeraging effect that a smaller scale of flctation reslts in a larger ariance redction in the soil parameters, which wold yield a smaller ariation of otpt responses. Frthermore, the failre probability p f for each combination of COV and θ is shown in Figre 2.8. The p f increases with both COV and θ of s / σ. For a gien COV, the maximm p f is reached at θ = ; the implication is that the design can be too conseratie withot considering the effect of spatial ariability of soil parameters. 31

48 COV of M R θ=0.5m θ=2.5m θ=10m θ=50m θ= θ=1m θ=5m θ=25m θ=100m COV of s / σ Figre 2.7: COV of resisting moment (M R ) as a fnction of the scale of flctation and COV of s /σ'. Probability of failre θ=0.5m θ=2.5m θ=10m θ=50m θ= θ=1m θ=5m θ=25m θ=100m COV of s / σ Figre 2.8: Inflence of COV and scale of flctation of s /σ' on the probability of failre. 32

49 Gien a pair of σ and θ Random field modeling Simplified approach Generate correlation matrix for θ Cholesky decomposition Generate standard normally distribted random nmbers Assme the interal of redction factor Γ for θ: [ Γ L, Γ U ] Compte the midpoint: Γ p = (Γ L + Γ U )/2 Obtain redced σ Γ = Γ p σ Generate lognormal random field for s Generate lognormally distribted s Rn stability analysis with slip circle method sing Monte Carlo simlations Rn stability analysis with slip circle method sing Monte Carlo simlations Obtain standard deiation for M R : σ NS Obtain standard deiation for M R : σ S σ NS = σ S? Yes No If σ NS - σ S > 0, Γ L =Γ p, else Γ U =Γ p Obtain redction factor Γ for θ Figre 2.9: Flow chart for searching for the redction factor for a gien pair of standard deiation σ and scale of flctation θ. Simplified approach sing ariance redction techniqe As mentioned preiosly, the focs of this stdy is on the simplified approach 33

50 sing the ariance redction techniqe. To apply this techniqe, it is necessary to determine an appropriate characteristic length L. In this regard, the ariance redction factor Γ 2 for the simplified approach is first established by matching the soltions obtained from the ariance redction-based simplified approach with those by RFM. Again, only s / σ is modeled as a spatially random ariable herein in order to stdy the inflence of the inherent ariability and the spatial ariability of s / σ. All other parameters are treated as constants. The criterion for matching the two approaches (the simplified approach erss RFM) is to achiee the same leel of ariability of the response ( M in this case), since the mean M R is expected to be approximately the same R (as shown in Figre 2.6). Throgh this calibration, the ariance redction factor Γ 2 to be sed in the simplified approach for a gien case is obtained. Figre 2.9 shows a flowchart for searching for the redction factor Γ for a gien pair of standard deiation (σ) and scale of flctation (θ) of s / σ. In Figre 2.9, the flow seqence on the left smmarizes the procedre of the conentional RFM with the Cholesky decomposition method [Eqs. ( )]. After 10 5 simlations of the basal-heae stability analysis, the standard deiation of M R (denoted as σ NS ) is obtained. In Figre 2.9, the flow seqence on the right smmarizes the procedre of simplified approach sing the eqialent ariance techniqe. First, an interal of the redction factor, [Γ L Γ U ], is assmed for this case (with the same σ and θ of s / σ ). Γ L and Γ U are the assmed lower and pper bonds, which may be set at 0 and 1, respectiely. Then the bisection method is sed to search for the eqialent 34

51 ariance: the interal is diided into two segments by the midpoint Γ p = (Γ L + Γ U )/2, and the ariance is redced with Γ p. With the redced ariance σ Γ [obtained from Eq. (2.14)], MCS may be performed withot the Cholesky decomposition. The standard deiation of M R, denoted herein as σ S, is then obtained from 10 5 MCS of the basal-heae analysis of the same case, as in the RFM analysis (left side of the flowchart shown in Figre 2.9). If the redction factor Γ p is correct, the two standard deiations, σ NS and σ S, will be eqal to each other for the gien pair of standard deiation σ and scale of flctation θ of s σ /. In this stdy, the Γ ale at which NS σ σ S (or 3 NS S / NS 10 ) is σ σ σ the target redction factor for a gien pair of σ and θ of s / σ. As shown in Figre 2.9, 3 if the aboe stopping criterion ( σ σ / σ 10 ) is not satisfied, the interal of Γ is NS S NS shortened by setting Γ L = Γ p (for σ σ > 0 ) or Γ U = Γ p (for σ σ < 0 ). The new NS S midpoint Γ p is then compted and the aforementioned procedre is repeated ntil the final NS S redction factor for a gien pair of σ and θ of s / σ is obtained. It shold be noted that for this eqialency analysis, the simplified approach is implemented with the MCS. As will be shown later, the simplified approach can also be implemented with FORM to frther redce the comptational effort. Figre 2.10 shows the back-calclated Γ ales for arios pairs of σ (or COV) and θ of s / σ sing the MCS-based RFM approach. It is apparent that the inherent ariability rarely inflences the ariance redction at the same θ leel. The redction factor Γ depends only on θ at the same COV leel, which is consistent with the ariance redction models presented in the literatre (e.g., Vanmarcke 1983). 35

52 Redction factor, Г COV = 0.2 COV = 0.4 COV = 0.6 COV = 0.8 COV = Scale of flctation, θ (m) Figre 2.10: Back-calclated redction factor Γ for arios scales of flctation. Alternatiely, the redction factor Γ can also be determined sing the ariance redction fnction if the characteristic length is known. To find an appropriate characteristic length for the basal-heae problem, the redction factors are ealated sing the exponential model [Eq. (2.13)] with three assmed characteristic lengths: (1) L = 27m, (2) L = 39m, and (3) L = 98m. The first characteristic length L = 27m is the distance od (from the depth of the final strt to the bottom of the diaphragm wall), as shown in Figre 2.1. This length is the ertical scale of the spatially random region. The second characteristic length L = 39m is the length of Arc cd, and the third characteristic length L = 98m is the length of the sliding srface (Arc abcde). The redction factors compted with the assmed characteristic lengths are shown in Figre 2.11 and compared 36

53 with the back-calclated redction factors obtained preiosly. As shown in Figre 2.11, the assmption of L = 27m yields redction factors that are most consistent with those back-calclated sing the MCS-based RFM approach. This is reasonable as L = 27m is actally the distance between the depth of the final strt to the bottom of the diaphragm wall, which is the only region that contribtes to the resistance moment in the random field. Ths, the aboe analysis shows that the ertical aeraging length in the random field modeling of the basal-heae stability can be taken as the distance between the depth of final strt and the bottom of the diaphragm wall. 1 Redction factor, Г MCS-based soltion Eq. (2.13) with L=27m Eq. (2.13) with L=39m Eq. (2.13) with L=98m Scale of flctation, θ (m) Figre 2.11: Comparison between the redction factors back-calclated sing the MCS-based RFM and those deried based on Eq. (2.13) with different assmed characteristic lengths. 37

54 In smmary, the ariance redction-based simplified approach is sitable for basal-heae stability analysis if an appropriate characteristic length (and ths redction factor) can be determined. The ariance redction-based simplified approach yields almost identical reslts with those obtained sing the MCS-based RFM approach; howeer, the former is easy to apply, less demanding on resorces, and offers significant adantages in engineering practice. It shold be noted that the approach described aboe (Figre 2.9) for back-calclating the ariance redction factor and characteristic length is demonstrated to be effectie for the problem of basal heae that inoles a linear limit state that has an explicit form. For other geotechnical problems that inole more complicated and nonlinear limit states, frther stdy is needed to examine its general applicability. Reliability-Based Design Considering Spatial Variability With the alidated ariance redction techniqe for basal-heae stability, the reliability analysis sing FORM, in lie of MCS, can be performed, which is an effectie and efficient means to consider spatial ariability. The principle and procedre of FORM is well docmented (e.g., Ang and Tang 1984). A spreadsheet soltion implementing FORM (Low and Tang 1997) has shown to be effectie and can be a practical tool in engineering practice. Figre 2.12 shows the setp of a spreadsheet soltion for reliability analysis of the braced excaation case presented preiosly (see Figre 2.1 and Table 2.1). 38

55 Initially, enter original mean ales for x* colmn, followed by inoking Excel Soler, to atomatically approach the target reliability index β, by changing x* colmn, sbject to g(x) = 0. s /σ' γ(kn/m 3 ) q s (kn/m 2 ) θ (m) H e (m) H w (m) H s (m) D (m) γ w (kn/m ) Original inpt eqialent normal parameters Parameters at design point ŀ Mean COV η λ x* µ N σ N Spatial factors L(m) Γ Variance redction factor compted sing exponential redction fnction Correlation Matrix ρ (x* - µ N )/σ N Reslts r (m) = α (Deg) = M R = M D = FS Original = g(x) = β = P f = Figre 2.12: Reliability-based procedre for ealating failre probability of basal-heae. 39

56 Reqired factor of safety Simplified approach: θ = 2.5m (Γ = 0.298) θ = 5m (Γ = 0.471) θ = 10m (Γ = 0.550) θ = 100m (Γ = 0.918) RFM soltion (at arios θ leels) ŀ Target probability of failre Figre 2.13: Comparison between the MCS-based random field modeling and the simplified approach. Since the scale of flctation is fond to be an important parameter, and knowledge abot which is limited, it shold be of interest to also examine the effect of the possible ncertainty of this parameter. Ths, the scale of flctation θ of s / σ is treated as a lognormally distribted random ariable in the spreadsheet soltion as shown in Figre The simplified approach to consider the spatial effect of s / σ is realized sing the exponential ariance redction fnction [Eq. (2.13)]. All other inpt parameters are treated as spatially-constant random ariables or simply constants. 40

57 With the spreadsheet soltion set p as shown in Figre 2.12, the inflence of spatial ariability can easily be assessed by considering seeral scales of flctation: θ = 2.5m, 5m, 10m and 100m. For this series of analysis, the scale of flctation is considered as a constant inpt, which is typical for these kinds of stdies. The relationship between shown in Figre shown in Figre p f and FS for each scale of flctation is nmerically deried, as The redction factor deried from the spreadsheet soltion is also For comparison prposes, the reslts from the conentional RFM condcted in this stdy (presented preiosly in Figre 2.5) are re-drawn and also inclded in Figre Again, both approaches (RFM erss the simplified approach sing FORM with ariance redction) yield approximately the same reslts. Frthermore, the significant effect of spatial ariability on the compted failre probability can be obsered. Finally, the effect of ncertainty in the scale of flctation in the reliability analysis of basal-heae stability is examined. As an example in this demonstration analysis, the COV of the scale of flctation is set to 0.1, 0.3, 0.6 and 0.9; the scale of flctation is set to the typical mean ale of 2.5m and the COV of s /σ' is set to 0.3. The inflence of ncertainty (in terms of COV) in the scale of flctation on the compted failre probability is shown in Figre It is fond that for FS smaller than 1.2, the ariability of the scale of flctation has irtally no effect on the compted failre probability p f. For FS greater than 1.2, the predicted p f increases with the increasing ariability of the scale of flctation. Since the reqired minimm FS in the design against basal-heae sing the slip circle method is 1.2 (JSA 1988; PSCG 2000; 41

58 TGS 2001), the effect of the ariability of the scale of flctation is far less significant than the mean scale of flctation itself. 1.5 Reqired factor of safety_ COV = 0 COV = COV = COV = 0.6 COV = Target failre probability 0 Figre 2.14: Effect of ncertainty in the scale of flctation on the probability of failre against basal-heae in a braced excaation in clay (scale of flctation = 2.5m, COV of s /σ' = 0.3). Practical engineering application In the reliability-based design against basal-heae considering the effect of spatial ariability, it is desirable to facilitate the design procedre with design chart that relates the traditional design index (sch as the reqired factor of safety) to the degree of spatial effect (sch as the scale of flctation). In this regard, the preios analysis reslts are frther interpreted and the relationship between the reqired factor of safety and the scale of flctation at a certain failre probability can be obtained. Figre 2.15 shows sch a 42

59 relationship at failre probability of It shold be noted that 10-3 satisfies the expected performance leel of aboe aerage as classified by U.S. Army Corps of Engineers (1997). Similar design charts may be obtained at arios leels of target failre probability. Using the design chart sch as Figre 2.15, the reliability-based design may be realized by meeting the reqired factor of safety at a project site that is characterized with a scale of flctation throgh site inestigation. Reqired factor of safety_ Scale of flctation, θ Figre 2.15: Relationship between reqired factor of safety and scale of flctation at failre probability of Procedre for applying the proposed approach The proposed simplified approach for reliability analysis of basal-heae stability in a braced excaation considering the spatial ariability of soils is smmarized into the following procedre: 43

60 1. Select the analytical model for the basal-heae stability analysis of a braced excaation in clay [for example, slip circle method as per JSA (1988)]. 2. Determine the ariation of soil parameters (sch as ndrained shear strength) described with their COVs and the spatial ariability, defined by the correlation fnction and scale of flctation, based on site inestigation, soil testing, and engineering jdgment gided by pblished literatre. 3. For the basal-heae stability analysis in a braced excaation in clay, the spatial ariability of soil parameters sch as ndrained shear strength can be modeled with a one-dimensional (ertical) random field as stated preiosly. In this random field model, the characteristic length is taken as the distance from the final strt to the bottom of the diaphragm wall. The ariance redction factor ( Γ 2 ) is then ealated sing Eq. (2.13) with this characteristic length, and finally the redced ariance (σ 2 Γ ) for this spatially random soil parameter can be determined with Eq. (2.14). 4. With the redced ariance of the ndrained shear strength, reliability analysis can be performed sing traditional reliability methods sch as FORM for the probability of failre against the basal-heae. The soltion can easily be implemented in a spreadsheet as shown in Figre Reliability or probability-based design can be realized by meeting a target probability of failre against the basal-heae. 44

61 Smmary In this chapter, the inflence of one-dimensional spatial ariability of soil parameter on the reliability analysis of basal-heae stability is presented. The reslts of RFM of s / σ sing the Cholesky decomposition method shows that the model with a smaller scale of flctation wold yield a greater ariance redction in soil parameters (sch as s / σ ), which in trn wold yield a smaller ariation in the otpt responses (for example, FS against basal-heae). The compted probability of basal-heae failre can be too high if the spatial ariability is not considered in the reliability analysis. Ths, the basal-heae stability design will be too conseratie if the effect of spatial ariability is ignored. A ariance redction-based simplified approach for the reliability-based design against basal-heae failre in a braced excaation is presented. The proposed simplified approach with ariance redction techniqe is shown to be able to prodce almost identical reslts with those obtained sing the MCS-based RFM approach, proided that an appropriate characteristic length (and ths the redction factor) can be determined. For the basal-heae stability case in this stdy, the appropriate characteristic length for the exponential redction fnction is determined to be the distance from the final strt to the bottom of the diaphragm wall, which is the ertical scale of the random field in this case. This approach can be implemented in a spreadsheet and reqires far less comptational effort than the MCS-based RFM approach, is easy to se, and has potential in geotechnical reliability-based design that deals with spatial ariability of soils. 45

62 CHAPTER III RELIABILITY ANALYSIS OF BASAL-HEAVE IN A BRACED EXCAVATION IN A TWO-DIMENSIONAL RANDOM FIELD Introdction As demonstrated in Chapter II, the traditional reliability analysis that does not accont for the effect of one-dimensional (1-D) spatial ariability tends to oerestimate the failre probability in the stdy on basal-heae stability sing the slip circle method. Similar conclsions hae also been reported in the literatre, e.g., W et al. (2010a) reported in their reliability analysis of basal-heae failre, it was fond that the failre probability tends to be oerestimated if the effect of 1-D spatial ariability is neglected. Considering that none of the preios stdies consider the effect of the two-dimensional (2-D) random field in the basal-heae problem, the methodology inclding the simplified approach for the 1-D random field stdy deeloped in Chapter II, is employed and extended herein for the 2-D random field stdy. In this chapter, a simplified approach to consider the effect of spatial ariability in a 2-D random field for reliability analysis of basal-heae in a braced excaation in clay is formlated. This simplified approach is demonstrated throgh a case stdy. As the first step, the 2-D RFM analysis is performed in a stdy of basal-heae stability to proide a benchmark. Then, ariance redction factors for both ertical and horizontal directions, at which the simplified approach yields reslts that match well with those obtained with A similar form of this chapter has been pblished at the time of writing: Lo Z, Atamtrktr S, Cai Y, Jang CH. Reliability analysis of basal-heae in a braced excaation in a 2-D random field. Compters and Geotechnics, doi: /j.compgeo

63 RFM, are back-calclated. Next, assmptions of the characteristic lengths of both ertical and horizontal directions in the stability analysis are erified based on the back-calclated ariance redction factors. Finally, a simplified approach which combines the first-order reliability method (FORM) and the ariance redction techniqe to accont for spatial ariability is proposed for reliability analysis of basal-heae stability. The proposed approach is easy to se, reqires less comptational effort, and yields reslts (in terms of probability of basal-heae failre) that are nearly identical to those obtained with the MCS-based RFM method. Two-Dimensional Random Field Modeling of s /σ' Conentional random field modeling of s /σ' In this chapter, the slip circle method (JSA 1988; PSCG 2000; TGS 2001) for determining FS against basal-heae in soft clay is adopted for its simplicity and sitability for modeling the random field of ndrained shear strength. The details of this method is docmented in Chapter II [Eqs. ( )]. As reflected in the formlation [Eqs. ( )] of the slip circle method, the ndrained shear strength ( s ) of clay plays a critical role in the design of a braced excaation against basal-heae. In other words, FS is a fnction of s and other parameters. As noted preiosly, the first step toward deeloping a simplified reliability-based procedre for ealating the probability of basal-heae failre in an excaation in clay with significant spatial ariability is to establish a benchmark sing the MCS-based RFM. A two-dimensional RFM approach is deemed especially sitable for the basal-heae 47

64 problem analyzed with the slip circle method (Figre 3.1). qs = 10 kpa / m a H e = 18 m H s Diaphragm wall = 15 m γ soil = 19 kn / m 3 H p = 15 m e Final strt o α β H w = 33m r dβ ds b c Failre srface d Random field modeling region Figre 3.1: Geometry of slip circle method and 2-dimensional random field modeling region for basal-heae stability analysis. The ndrained shear strength generally increases with depth for most normally consolidated clay bt the ratio of ndrained shear strength oer the effectie oerbrden stress ( s / σ ) remains roghly constant (Ladd and Foott 1974). Ths, in this stdy the parameter s / σ is modeled sing lognormal random field, and all other inpt parameters are modeled as spatially-constant lognormal ariables or constants. The assmption of lognormal distribtion for inherent soil ariability assres positie soil parameters and has been widely adocated by past stdies (e.g., Phoon and Klhawy 48

65 1999b). In RFM, the ncertainty of s / σ is represented by its spatially-constant mean µ ln and coefficient of ariation COV ln and its scale of flctation θ. Ths, the basal heae problem here inoles a stationary random field modeling of deiation and mean of the eqialent normal distribtion of s / σ. The standard s / σ, denoted as ln ( s / σ ), are expressed as: 2 ( 1 COV ) σ = ln (3.1) n + ln 1 2 µ n = ln µ ln σ n (3.2) 2 where n denotes normal distribtion and ln denotes lognormal distribtion. The lognormally distribted random field of s / σ can be generated throgh the following transformation (Fenton et al. 2005): s ( x ) = exp{ µ + G ( x )} / σ σ (3.3) i n n n i where x i is the spatial position at which s σ / is modeled; ( ) G is a normally n x i distribted random field with zero mean, nit ariance and correlation fnction ρ ( τ ). In this stdy, the exponential correlation fnction, which is commonly sed in random field modeling, is selected (Jaksa et al. 1999): ρ 2τ = exp (3.4) θ ( τ ) 49

66 where τ is the absolte distance between any two points in the random field and θ is the scale of flctation. As shown in a preios stdy, the ertical and the horizontal scales of flctation in the field for clay are generally different (Phoon and Klhawy 1999a). In the 2-D RFM, Eq. (3.4) may be modified to consider the neqal scales of flctations (Schomel and Mašín 2011): 2 2 ( ) τ τ h ρ τ = exp 2 + (3.5) θ θh where τ and τ h are the absolte ertical and horizontal distance between any two points in the random field, respectiely; and θ and θ h are the ertical and the horizontal scales of flctation, respectiely. In this stdy, the correlation matrix bilt with the correlation fnction is decomposed by Cholesky decomposition which has been proed simple and effectie (Fenton 1997; Schomel and Mašín 2010): T L L = ρ (3.6) With the matrix L, the correlated standard normal random field can be obtained by linearly combining the independent ariables as follows (Fenton 1997): G n ( xi ) = i j= 1 L Z ij j i = 1, 2,, M (3.7) where M is the nmber of points in the random field; Z j is the seqence of independent standard normally distribted random ariables. 50

67 The normalized ndrained shear strength s / σ at each spatial position in the random field can be obtained with Eq. (3.3) for a specified mean, standard deiation, and scale of flctation sing Monte Carlo simlation. standard deiation, and scales of flctation of In each simlation, the same mean, s / σ are sed. The statistics of otpt sch as FS [Eq. (2.1)] can be obtained after a sfficient nmber of simlations are carried ot. The failre probability p f is compted as the ratio of the nmber of simlations that yield failre (M R < M D or FS < 1) oer the total nmber of simlations N. The nmber of MCS samples shold be at least 10 times of the reciprocal of the target failre probability (Ang and Tang 2007; Wang et al. 2011a). In this stdy, the leel of failre probability of interest is greater than 10-4, therefore N is set at Figre 3.2 shows the reslts of random field modeling at for combinations of θ and θ h gien as an example the mean of s / σ = 0.3 and coefficient of ariation (COV) = 0.3: (a) θ h = θ = 2.5m; (b) θ h = θ = 10m; (c) θ h = 2.5m, θ = 10m; and (d) θ h = 10 m, θ = 2.5m. The RFM region shown in Figre 3.2 incldes 36 by 18 sqare elements with element size of 1m. Considering that the aforementioned RFM procedre is defined at the point leel, the local aeraging oer the sqare element size is performed to obtain the locally aeraged statistics. The local aeraging is realized throgh mltiplying a ariance redction factor to the ariance of a normal ariable. Then the statistics of the eqialent lognormal ariable are compted (Griffiths and Fenton 2004; Schomel and Mašín 2011). As shown in Figre 3.2, the darker color represents higher s / σ and lighter color represents smaller s / σ. The effect of scales of flctation is apparent in the 2-D RFM: 51

68 - either in the ertical or horizontal direction, smaller θ corresponds to more drastic ariation of s / σ in that direction of the random field; conersely, a larger θ corresponds to more niform s / σ in that direction of the random field. In either direction, the spatial ariation in the case of smaller θ is mch more significant than that for larger θ. ( a) θ = θ = 2.5 h m ( b) θ = θ = 10 h m ( c) θ = 2.5 m, θ = 10m ( ) θ = 10, θ = 2.5 h d m m h Scale bar for s /σ' in the plots aboe: Horizontal scale Figre 3.2: Inflence of scale of flctation on the 2-D random field modeling of s /σ' at gien mean of 0.3 and COV of

69 Simplified approach based on eqialent ariance techniqe The second step toward deeloping a simplified reliability-based procedre for ealating the probability of basal-heae failre is to establish a simplified soltion that matches closely with the MCS-based RFM soltion. The simplified approach is based on the concept of spatial aeraging in which the spatial ariability of the soil property is aeraged in order to approximate a random ariable that represents a soil parameter (Vanmarcke 1977). The aeraged ariability of the soil property oer a larger domain can be qantified with an ariance redction techniqe in which the ariances of soil parameters may be redced by mltiplying a factor known as ariance redction factor 2 ( Γ ). With two inpts: scale of flctation and characteristic length, the ariance redction factor that adopts an exponential form is gien as follows (Vanmarcke 1983): Γ 2 = 2 1 θ 2L 2L 1 + exp 2 L θ θ (3.8) where θ is the scale of flctation and L is the characteristic length. In the 2-D random field, the ariance redction factor is expressed as the prodct of the ariance redction factors in the ertical and horizontal directions, compted with respectie scales of flctation and characteristic length (Vanmarcke 1977): Γ = Γ Γ h (3.9) where 2 Γ and 2 Γ h are the ertical and horizontal ariance redction factors, 53

70 respectiely. The redced ariance σ Γ 2 can be obtained with the following eqation: σ 2 = Γ 2 Γ σ 2 (3.10) where 2 σ is the ariance of the soil parameter of concern ( s / σ in this stdy). In this stdy, the positie sqare root of the ariance redction factor is referred to herein as the redction factor ( Γ ) to differentiate it from the ariance redction factor ( Γ 2 ). It is noted that if the ariance redction techniqe is sed to simplify the effect of the spatial ariability of a lognormal distribted parameter, as is the case in this stdy, only the ariance of its eqialent normal distribtion, defined preiosly, shold be redced throgh the se of ariance redction factor. For the simplified approach sing the eqialent ariance techniqe, the analysis can be condcted either with MCS or with a reliability method. The latter is preferred, as it reqires far less comptational effort and is more practical than the MCS-based RFM. Implementation of the reliability methods in a spreadsheet has been demonstrated to be a practical approach to geotechnical problems (e.g., Low and Tang 1997; Jang et al. 2006; Jang et al. 2009). Past inestigators (Peschl and Schweiger 2003; Schomel and Mašín 2010) hae shown that reliability analysis with the eqialent ariance techniqe can captre the oerall trend of the MCS-based RFM. In this chapter, the two approaches are compared within the context of 2-D random field modeling. 54

71 Simplified Reliability Method for Assessing Probability of Basal-Heae Failre in Braced Excaation in 2-D Random Field RFM approach for probability of basal-heae failre The probability of basal-heae failre in a braced excaation in clay is first analyzed herein sing the conentional random field modeling with Cholesky decomposition method. The geometry and inpt data for the excaation case employed in this stdy is illstrated in Figre 3.1 and listed in Table 3.1, respectiely. The ndrained shear strength s / σ is modeled as a spatially random ariable, and the nit weight of soil and srcharge are modeled as spatially-constant random ariables. All other geotechnical and strctral parameters are treated as constants for simplicity, since the ncertainties in these parameters are relatiely negligible. Table 3.1: Inpt parameters for a basal-heae stability problem shown in Figre 3.1. Parameters Notations Vales Mean COV Unit weight of soil γ 19 kn/m Srcharge q s 10 kpa/m 0.2 Depth of GWT D 2m N/A Final excaation depth H e 18m N/A Final strt depth H s 15m N/A Penetration depth H p 15m N/A *In this stdy, many basal-heae problems defined with this set of inpt parameters and geometry are analyzed. The difference in these problems is in the choice of the mean ale of the normalized ndrained shear strength ( s / σ ), which reslts in different factors of safety (FS). 55

72 Typical COV of the ndrained shear strength s is abot 0.3, althogh it cold be as high as 0.8 (Phoon and Klhawy 1999a). In this stdy, the COV of set at 0.3 and the effect of assming higher COVs is examined later. s / σ is first Based on a statistical stdy by (Phoon and Klhawy 1999a), the aerage ertical and horizontal scales of flctation for clay are 2.5m and 50.7m, respectiely. In a 2-D RFM, both ertical and horizontal spatial ariability are considered. It shold be noted that for basal-heae analysis, only the random field shown in Figre 3.1 (the box region) needs to be modeled since the resistance moment [Eq. (2.3) in Chapter II] comes only from this region. Here, the RFM region in Figre 3.1 is sbdiided into 36 by 18 sqare elements (horizontal direction by ertical direction). The size of the sqare elements is 1m. For this 2-D RFM, the total nmber of elements (648 elements in this case) is comparable to the sggested maximm nmber by Fenton (1997) for the Cholesky decomposition operation. Althogh a larger modeling region can be adopted, the region shown in Figre 3.1 is the minimm region that coers the slip circle where the resistance moment M R is deried. To proide a reference, the effect of ertical and horizontal scales of flctation is first examined separately (i.e., treating it like 1-D RFM). Ths, when ertical or horizontal spatial ariability is considered, the other direction is assmed to be spatially constant. To stdy the effect of spatial ariability, a series of scales of flctation (θ = 1m, 2.5m, 10m, and 100m) for each direction is inestigated. Ths, gien a set of inpt data for a braced excaation (Figre 3.1 and Table 3.1), the probability of basal-heae failre is compted for a design with a gien FS [say, FS = 2.0 as per Eq. (2.1) in Chapter II] and a gien scale of flctation that reflects the 1-D 56

73 random field of s / σ. This probability can be calclated sing the MCS. For each gien scale of flctation and design FS, 10 5 simlations are condcted. The probability of failre is determined by the ratio of the nmber of failre cases (defined here as FS < 1.0) oer the total nmber (i.e., 10 5 ). This process is repeated for each series of scale of flctation and each series of designs (signaled by a series of FS ales, which was realized by assming different mean s / σ ales while keeping the mean ales of all other parameters the same). For a single rn of MCS, the exection time for 10 5 simlations is approximately 4 mintes on a laptop PC eqipped with an Intel Pentim Dal CPU T2390 rnning at 1.86GHz sing MATLAB (MathWorks 2010). The reslts are shown in Figre 3.3(a) for ertical spatial ariability and Figre 3.3(b) for horizontal spatial ariability. The reslts presented in this figre proide the engineer a basis for selecting a factor of safety for design against basal-heae sing the slip circle method [Eqs. ( )]. This basis is the target probability of failre that considers the spatial ariability of soil parameters. The design, based on the target probability of failre is referred to herein as the probability-based (or reliability-based) design against basal-heae failre. The effect of scales of flctation in a 1-D random field is qite obios: a smaller scale of flctation reslts in a smaller p f at the same FS. As shown in Figre 3.3(a), if the target p f is set at 10-3, the reqired FS is abot 1.85 at θ = 2.5m [note: this θ ale is the mean of the ertical scale of flctation for clay as per Phoon and Klhawy (1999a)], and is abot 2.65 at θ = 100m (note: this θ ale is close to a spatially-constant condition). Similar conclsions may be drawn from Figre 3.3(b) for 57

74 Probability of failre_ ( a) Effect of ertical scale of flctation θ = 1m 1. 0 m θ = 2.5m 5 m θ = 10m m θ = 100m m Factor of safety Probability of failre_ ( b) Effect of horizontal scale of flctation θ = 1m 1. m h 0 θh = 2.5m 5 m θh = 10m m θh = 100m m Factor of safety Figre 3.3: Effect of 1-dimensional spatial ariability on the relationship between failre probability and factor of safety in the reliability-based design. 58

75 0 101 ( a) θ = 2.5 m Probability of failre_ θh = 1m 2. 5 m θ θ h = 2.5m 50 m θ = 10m θ h Factor of safety ( b) θ h = 50 m Probability of failre_ θ = 1m 2. 5 m θ = 2.5m 50 m θ = 10m θ Factor of safety Figre 3.4: Effect of 2-dimensional spatial ariability on the relationship between failre probability and factor of safety in the reliability-based design. 59

76 the effect of horizontal spatial ariability. The implication is that the reqired FS will be oerestimated for a target p f if the effect of spatial ariability is ignored. Ths, traditional reliability analysis that considers ariation of inpt soil parameters (for example, throgh COV) bt not spatial ariability exhibited in a random field can oer-estimate the probability of failre for a gien deterministic-based design (i.e., a gien FS). The effect of 2-D spatial ariability is examined next. To begin with, three different horizontal scales of flctation (θ h = 2.5m, 50m, and ) are considered simltaneosly with the aerage ertical scale of flctation [θ = 2.5m as Phoon and Klhawy (1999a)] to stdy the effect of θ h at fixed θ. The reslts are shown in Figre 3.4(a). Afterwards, three different ertical scales of flctation (θ = 2.5m, 50m, and ) are considered simltaneosly with the aerage ertical scale of flctation [θ h 50m as per Phoon and Klhawy (1999a)] to stdy the effect of θ at fixed θ h. The reslts are shown in Figre 3.4(b). The effect of the scales of flctation in a 2-D random field is also obios: at a fixed scale of flctation in one direction, a smaller scale of flctation in the other direction reslts in a smaller FS will still be oerestimated for a target p f at the same FS. Frthermore, the reqired p f if the soil parameter is modeled with only a 1-D random field, as opposed to a 2-D random field. Therefore, it is essential to consider 2-D spatial ariability in the probability-based (or reliability-based) design against basal-heae failre in a braced excaation. One concern with traditional reliability-based design in geotechnical practice in the past is that the compted failre probability is often high in a design that satisfies the 60

77 minimm FS specified in the codes, bt failre seldom occrs in sch cases. For example, this concern has been reported by Goh et al. (2008) and W et al. (2010a) in their stdy of basal-heae stability in an excaation. Oerestimation of ariation in soil parameters is often pointed ot as a possible case for haing a higher compted probability of failre. Based on the reslts presented (Figres 3.3 and 3.4), it is eident that negligence in the effect of spatial ariability of soil parameters can lead to an oerestimation, perhaps to a high degree, of the failre probability. Ths, to apply the traditional simplified reliability-based method for ealating the failre probability, an adjstment is needed to model spatial ariability. Eqialent simplified approach with ariance redction techniqe While the RFM analysis generally yields the most accrate reslts, it reqires se of the MCS. On the other hand, the simplified approach can be implemented with reliability-based methods. Past stdies hae shown that simplified approaches with a proper ariance redction can match well with the RFM soltion. In other words, for a RFM soltion, an eqialent soltion sing simplified approach is possible. Therefore, the analysis for the probability of basal-heae failre in a braced excaation in a random field can be performed sing traditional reliability-based methods, proided that an eqialent simplified approach can be established first. The desired eqialency between simplified approach and RFM soltions in this case is eqal probability of failre, or p = P[ M < M ]. For the basal-heae f R D problem analyzed herein sing the slip circle method [Eqs. ( )], the ndrained shear 61

78 strength is the only soil parameter that is treated as spatially random, while all other parameters are set as constants so that the spatially random effect of the ndrained shear strength is examined explicitly. Here, in search for the eqialency between the RFM approach and the simplified approach, M D is treated as a constant and M R is treated as a random ariable. Therefore, the eqialency in the compted failre probability can be achieed if M R, determined from the two approaches (simplified approach and RFM) agrees with each other at the same leel of soil ariability (i.e., the same leel of COV and scale of flctation). Becase the mean of M R is approximately the same regardless of which of the two approaches is employed [This is apparent since in this stdy M R is linearly correlated with s as per Eq. (2.3) and Figre 2.6], it is considered appropriate and adeqate to se the ariation (or more precisely, the standard deiation) of the compted M R as a basis for establishing the eqialency. It shold be noted that the mean of M R (and ths FS) may not be independent of the scale of flctation θ as obsered from the reslts of the slip circle method, since the shear zone may deelop throgh softer areas (Schomel and Mašín 2010). If other approaches sch as finite element method are employed for basal heae analysis, the compted mean of M R is likely to change with θ (Schomel and Mašín 2010). This effect is not acconted with the slip circle method, which is a limitation of the proposed approach. Howeer, this isse is beyond the scope of this stdy. Eqialency between simplified approach and RFM soltion may be achieed by applying ariance redction to the former, which reqires determination of the redction factor (Γ). For the basal-heae problem, Γ ales at arios leels of ariability (in terms 62

79 of standard deiation σ and scale of flctation θ of s / σ ) can be back-calclated sing the procedre illstrated in Figre 2.9 in Chapter II. As shown in Figre 2.9, the flow seqence on the left smmarizes the procedre of RFM with the Cholesky decomposition method [Eqs. ( )]. The standard deiation of M R, denoted herein as σ NS, is obtained from 10 5 MCS of the basal-heae analysis for a braced excaation in clay with a gien pair of standard deiation σ and scale of flctation θ of s / σ. In Figre 2.9, the flow seqence on the right smmarizes the procedre of simplified approach sing the eqialent ariance techniqe. First, an interal of the redction factor, [Γ L Γ U ], is assmed for this case (with the same σ and θ of s / σ ). Γ L and Γ U are the assmed lower and pper bonds, which may be set at 0 and 1, respectiely. Then the bisection method is sed to search for the eqialent ariance: the interal is diided into two segments by the midpoint Γ p = (Γ L + Γ U )/2, and the ariance is redced with Γ p. With the redced ariance σ Γ, MCS may be performed withot the Cholesky decomposition. The standard deiation of M R, denoted herein as σ S, is then obtained from 10 5 MCS of the basal-heae analysis of the same case, as in the RFM analysis (left side of the flowchart shown in Figre 2.9). If the redction factor Γ p is correct, the two standard deiations, σ NS and σ S, will be eqal to each other for the gien pair of standard deiation σ and scale of flctation θ of s / σ. In this stdy, the Γ ale at which σ NS σ S (or 3 NS S / NS 10 ) is the target redction σ σ σ factor for a gien pair of σ and θ of s / σ. As shown in Figre 2.9, if the aboe 63

80 3 stopping criterion ( σ σ / σ 10 ) is not satisfied, the interal of Γ is shortened by NS S NS setting Γ L = Γ p (for σ σ > 0 ) or Γ U = Γ p (for σ σ < 0 ). The new midpoint Γ p is NS S then compted and the aforementioned procedre is repeated ntil the final redction NS S factor for a gien pair of σ and θ of s / σ is obtained. It shold be noted that for this eqialency analysis, the simplified approach is implemented with the MCS. As will be shown later, the simplified approach can also be implemented with FORM to frther redce the comptational effort. Throgh the aboe back-calclation procedre (Figre 2.9), the redction factor Γ for an eqialent simplified approach can be obtained. Figres 3.5(a) and 3.5(b) show the back-calclated Γ ales for arios pairs of σ (or COV) and θ of s / σ, for ertical and horizontal spatial ariability, respectiely. Two obserations are made: (1) the inherent ariability rarely inflences the ariance redction at the same θ leel; (2) the redction factor Γ depends only on θ at the same COV leel. These obserations are consistent with the ariance redction models presented in literatre [e.g., Eq. (3.8) from Vanmarcke (1983)]. Finally, a comparison is made between redction factors (Γ), compted sing the ariance redction fnction [Eq. (3.8)] and those obtained throgh the back-calclation procedre discssed aboe. Note that the ealation of Eq. (3.8) reqires knowledge of the characteristic length. In this stdy, the ertical characteristic length L is assmed to be the ertical distance between the depth of the final strt and the bottom of the diaphragm wall (L = od = 18m as in Figre 3.1); horizontal characteristic length L h is 64

81 Redction factor, Г ( a) Vertical redction factor L = 18m Back-calclated redction factors at arios COV's: COV = 0.1 COV = 0.3 COV = 0.6 COV = 1.0 Exponential redction fnction Vertical scale of flctation, θ (m) Redction factor, Гh ( b) Horizontal redction factor L h = 36m Back-calclated redction factors at arios COV's: COV = 0.1 COV = 0.3 COV = 0.6 COV = 1.0 Exponential redction fnction Horizontal scale of flctation, θ h (m) Figre 3.5: Comparison between the redction factors back-calclated sing the MCS-based RFM and those compted sing Eq. (3.8) with assmed characteristic lengths (L = 18m or L h = 36m). 65

82 assmed to be the horizontal scale of the slip circle (L h ec 36m as in Figre 3.1). The rationale for these assmptions is that these lengths are the ertical and horizontal scales of the region that contribtes to the resistance moment in the random field. With the assmed characteristic lengths, redction factors are compted with Eq. (3.8), and the reslts are shown in Figre 3.5 for comparisons with those that are back-calclated based on the eqialency analysis presented preiosly. As shown in Figre 3.5, the assmptions of L = 18m and L h = 36m yield redction factors consistent with those that are back-calclated from the eqialency analysis. The implication is that for basal-heae analysis in a 2-D random field sing the slip circle method (Figre 3.1), ertical characteristic length L can be taken as the ertical distance between the depth of the final strt and the bottom of the diaphragm wall (the length od as in Figre 3.1) and horizontal characteristic length L h can be taken as the horizontal scale of the slip circle (the length ec as in Figre 3.1). This follows that ariance redction factor (Γ 2 ) or redction factor (Γ) can be compted for a gien spatial ariability leel, and with the concept of spatial aeraging, the redced ariation of the random ariable for an eqialent simplified approach is obtained. Finally, the simplified reliability analysis can be performed in this eqialent stationary random field. Practical reliability analysis of basal-heae considering 2-D spatial ariability Based on reslts presented in the preios sections, a step-by-step procedre is established for the simplified reliability analysis of basal-heae in a braced excaation in clay considering 2-D spatial ariability: 66

83 1. Select an analytical model for basal-heae analysis (for example, slip circle method). 2. Obtain spatially-constant inpt parameters and random ariables, sch as nit weight of soils and applied srcharge. 3. Characterize a spatially random ariable (in this case, the normalized ndrained shear strength) with its mean, COV and horizontal and ertical scales of flctation (θ h and θ,) for this 2-D random field. 4. Determine the ertical and horizontal characteristic lengths (L and L h ) based on the selected random field region (sch as the one shown in Figre 3.1). Apply the eqialent ariance techniqe [Eqs. ( )] to determine the redced ariance (σ 2 Γ ) of normalized ndrained shear strength. 5. When the eqialent ariance techniqe is sed to simplify the effect of the spatial ariability of a lognormal distribted parameter, the ariance of its eqialent normal distribtion shold be redced in this process. 6. Condct FORM analysis (for example, sing a spreadsheet implementation as shown in Figre 3.6) sing the redced ariance of the normalized ndrained shear strength. The reliability index and probability of failre can be determined sing FORM. 67

84 Initially, enter original mean ales for x* colmn, followed by inoking Excel Soler, to atomatically approach the target reliability index β, by changing x* colmn, sbject to g(x) = 0. s /σ' γ(kn/m 3 ) q s (kn/m 2 ) θ (m) θ h (m) H e (m) H w (m) H s (m) D (m) BF Original inpt eqialent normal parameters Parameters at design point Mean COV η λ x* µ N σ N L (m) L h (m) Γ Γ h Spatial factors Variance redction factors compted sing exponential Model bias factor E-04 redction fnction Correlation Matrix ρ (x* - µ N )/σ N Reslts ŀ r (m) = α (Deg) = M R = M D = FS Original = g(x) = β = P f = Figre 3.6: Reliability-based procedre for ealating failre probability of basal-heae. 68

85 Probability of failre_ ( a) θ = 2.5 m, θ h = 2.5 m Random field modeling Simplified approach: Γ = 0.36, Γ h = 0.26 Probability of failre_ ( b) θ = 2.5 m, θ h = 50 m Random field modeling Simplified approach: Γ = 0.36, Γ h = Factor of safety Factor of safety Probability of failre_ ( c) θ = 50 m, θ h = 50 m Random field modeling Simplified approach: Γ = 0.89, Γ h = 0.81 Probability of failre_ ( d) θ = 50 m, θ h = 2.5 m Random field modeling Simplified approach: Γ = 0.89, Γ h = Factor of safety Factor of safety Figre 3.7: Comparison between the MCS-based RFM soltions and those by the simplified approach. 69

86 As an example, basal-heae in a braced excaation shown in Figre 3.1 with inpt parameters listed in Table 3.1 is analyzed. All parameters except s / σ are treated as spatially-constant random ariables or simply constants. The parameter s / σ is modeled with a 2-D random field, and characterized by a mean ale of 0.30, a COV of 0.3, and scales of flctation θ h = 2.5 m and θ = 50 m. As shown in Figre 3.7, the eqialent simplified approach that considers the spatial ariability of s / σ is realized sing the ariance redction fnction [Eq. (3.8)] with characteristic lengths L = 18 m and L h = 36 m. The redced ariance of s / σ in the eqialent simplified approach is ths obtained. FORM analysis is then performed sing the spreadsheet as shown in Figre 3.6, which yields reliability index β = and probability of failre p f = It shold be noted that the model bias (BF) of the deterministic slip circle method is ignored at this point [as shown in Figre 3.6, mean ale of model bias (denoted as µ BF ) is set as 1.0 and the COV of the model bias (denoted as COV BF bias is examined later]. ) is set as 0.0); the effect of model ŀ To frther examine the capability of the spreadsheet that implements the FORM procedre with the eqialent ariance techniqe, the basal-heae problems that were analyzed with the MCS-based RFM approach (Figre 3.4), are re-analyzed. Comparisons with the preios reslts from Figre 3.4 are shown in Figre 3.7 for for different scenarios of constant ertical and horizontal scales of flctation: (a) θ = θ h = 2.5m; (b) θ = 2.5m, θ h = 50m; (c) θ = θ h = 50m; (d) θ = 50 m, θ h = 2.5m. Note that Case (b) represents the mean ales for θ and θ h sggested by Phoon and Klhawy (1999a). As in 70

87 Figre 3.4, a series of basal-heae problems (signaled by different FS ales, which were realized by assming different mean s / σ ales while keeping the mean ales of all other parameters the same) are analyzed. In Figre 3.7, the reslts show that regardless of the chosen scales of flctation and design safety leel (FS ale), the probabilities of basal-heae failre obtained from the spreadsheet agree ery well with those determined with the MCS-based RFM approach. At any gien target probability of failre (Figre 3.7), the maximm difference in the reqired FS between the two approaches (RFM s. simplified approach) is less than 5%. Ths, the simplified approach is deemed effectie. Effect of model bias on the compted probability of failre The analyses of basal-heae failre presented so far are performed assming no model bias in the adopted slip circle method. In the reliability analysis shown in Figre 3.6, the model bias is implemented with a bias factor (BF), which is treated as a random ariable. To implement the assmption of no model bias, the mean of the model bias factor is taken as nity ( µ =1. 0 ) with no ariance ( COV = 0. 0 ). Most geotechnical BF analysis models are biased one way or the other becase they often represent a conseratie approximation of the actal conditions. If the model bias exists bt is not acconted for, the compted probability of failre may be either nderestimated or oerestimated. This is an additional sorce of ncertainty that cold inflence the selection of a reqired factor of safety for a target failre probability in a reliability-based design. BF 71

88 Probability of failre_ Withot µ BF = 1.0, model COVBF bias = 0.0 With µ BF = model 1.39, COV bias BF = Factor of safety Figre 3.8: Effect of model bias of the slip circle method on the relationship between failre probability and factor of safety deried throgh reliability analysis. A recent calibration stdy on the slip circle method (W et al. 2011) reports that the mean ale ( µ BF ) and the COV of model bias ( COV BF ) of the method are 1.39 and 0.21, respectiely. To examine the effect of this model bias on the failre probability determined sing the FORM-based spreadsheet soltion, Case (b) in Figre 3.7(b), in which θ = 2.5m, θ h = 50m, is reanalyzed with this model bias factor. The reslts with and withot this model bias factor are compared in Figre 3.8. It is obsered that at the same FS leel, the failre probability is smaller if the model bias of the slip circle method is considered. Ths, negligence of the model bias in the reliability analysis of basal-heae failre can lead to an oerestimation of the failre probability. As shown in Figre 3.8, howeer, the effect of the model bias is less significant at smaller target probability leel (e.g., p f = 10-3 to 10-4 ). For example, at the target probability of failre p f = 10-4, the 72

89 difference in the reqired FS for the basal-heae design between the two conditions (with and withot model bias consideration) is less than 1.5%. As a reference, at this target probability of failre, the difference in the reqired FS for the basal-heae design between the two conditions of spatial ariability (with and withot considerations of spatial ariability of the ndrained shear strength of clay) is approximately at 50%. Ths, at the generally accepted leel of failre probability (p f = 10-3 to 10-4 ), the need to consider spatial ariability in the analysis is clearly demonstrated while the effect of model bias of the slip circle method is relatiely insignificant. Smmary In this chapter, the simplified approach deeloped in Chapter II to consider the effect of 1-D spatial ariability on the reliability analysis of basal-heae in braced excaation in clays, is extended for a 2-D random field. This simplified approach is demonstrated throgh a case stdy. As the first step, the 2-D RFM analysis is performed in a stdy of basal-heae stability to proide a benchmark. The reslts show that negligence of 1-D or 2-D spatial ariability in reliability analysis can significantly oerestimate the probability of basal-heae failre for a gien deterministic design with a certain factor of safety (e.g., Figres 3.3 and 3.4). Then, ariance redction factors for both ertical and horizontal directions, at which the simplified approach yields reslts that match well with those obtained with RFM, are back-calclated. The assmptions of the characteristic lengths of both ertical and horizontal directions in the stability analysis are erified based on the back-calclated 73

90 ariance redction factors. This case stdy fond that for basal-heae analysis in a 2-D random field sing the slip circle method (Figre 3.1), the ertical characteristic length L can be taken as the ertical distance between the depth of the final strt and the bottom of the diaphragm wall (length od in Figre 3.1) and the horizontal characteristic length L h can be taken as the horizontal scale of the slip circle (length ec in Figre 3.1). Finally, a simplified approach which combines the first-order reliability method (FORM) and the ariance redction techniqe to accont for spatial ariability is proposed for reliability analysis of basal-heae stability. The proposed approach is implemented in a spreadsheet and ths is easy to se, reqires less comptational effort. The simplified approach yields reslts (in terms of probability of basal-heae failre) that are nearly identical to those obtained with the MCS-based RFM method. 74

91 CHAPTER IV PROBABILISTIC SERVICEABILITY ASSESSMENT IN A BRACED EXCAVATION CONSIDERING SPATIAL VARIABILITY Introdction One of the main concerns in a braced excaation in an rban area is the risk of damage to adjacent infrastrctres cased by the excaation-indced wall deflections and grond moements. Damage to the adjacent infrastrctres cased by grond moements is referred to herein as the sericeability failre in a braced excaation. In many excaation projects, the owners or reglatory agencies establish the limiting wall and grond responses as a means of preenting excaation failre and damage to adjacent infrastrctres. Table 1.1 shows an example of sch limiting response criteria from China (PSCG 2000). Ths, it is essential to hae the ability to accrately predict the maximm wall deflection and grond settlement dring the design of braced excaations. Past experience has shown that constrction details can hae a great effect on the wall deflection and grond settlement that actally occr in the field. In this stdy, the effect of constrction seqence is simlated in the finite element analysis, as braced excaations are carried ot in stages. Howeer, good workmanship is assmed in the excaation and no other constrction related effect is considered in the analysis. To accrately predict the maximm wall deflection and grond settlement, it is essential to properly characterize the site conditions. An appropriate site inestigation A similar form of this chapter has been pblished at the time of writing: Lo Z, Atamtrktr S, Jang CH, Hang H, Lin PS. Probability of sericeability failre in a braced excaation in a spatially random field: Fzzy finite element approach. Compters and Geotechnics, doi: /j.compgeo

92 program is needed; in particlar, design soil parameters mst be properly ealated and selected. For braced excaations in clays, Hsiao et al. (2008) pointed ot that normalized ndrained shear strength ( s / σ ) and normalized initial tangent modls ( i E / σ ) are the most important soil factors. As in many geotechnical projects, howeer, it is difficlt to determine the ales of these parameters with certainty, especially with limited test data. Uncertainty in these parameters leads to ncertainty in the compted maximm wall deflection and grond settlement, which makes it more difficlt to assess whether the predicted responses are excessie as compared to the specified limiting wall and grond responses. This problem cold be complicated frther with the inherent ariability of soils and the spatial correlation. A sensible approach wold be to consider all these ncertainties, derie the probabilities of exceeding the limiting wall and grond responses, then make the design decisions based on these exceedance probabilities. The effect of inherent spatial ariation of soil properties has been demonstrated in many geotechnical problems, and modeling of this ariation with random field theory has already been reported (Griffiths and Fenton 2009). Howeer, a rigoros simlation of the random field sing the finite element method (FEM) soltion demands a large amont of comptation time, which is not practical for analyzing complicated problems sch as wall and grond responses in a braced excaation (Schweiger and Peschl 2005). To this end, the proposed approach, consisting of sing fzzy sets (Zadeh 1965) and a ariance redction techniqe (Vanmarcke 1977) to approximate the effect of a random field, appears to be a feasible alternatie for analysis for the probability of sericeability failre. 76

93 In smmary, this chapter focses on a simplified approach for estimating the probability of sericeability failre (i.e., exceeding the limiting wall and grond responses) in a braced excaation in a spatially random field. Uncertain soil parameters are represented by fzzy sets and spatial ariability is considered by means of ariance redction. Propagation of the ncertainty of these soil parameters throgh finite element soltion is carried ot by the alpha-ct method (Jang et al. 1998). The FEM analysis of the braced excaation is condcted sing a finite-element compter code with a constittie model that can effectiely model the small-strain nonlinear soil behaior. The reslts of the FEM-based fzzy set approach are fzzy nmbers that represent wall and grond responses. The probability of exceeding a specified response is then compted from the reslting fzzy nmbers. A case stdy is presented to demonstrate the proposed simplified approach, which is shown to be effectie and simple to se. Finite Element Modeling with a Small-Strain Nonlinearity Soil Model Nmerical methods sch as the finite difference method or finite element method are often sed to analyze wall deflection and grond settlement in a braced excaation (Whittle and Hashash 1994; O et al. 1998; Hsieh et al. 2003; Kng et al. 2007a). In this stdy, wall and grond responses in a braced excaation in clays are analyzed sing a commercially aailable finite element code, Plaxis TM (Brinkgree and Vermeer 2002). It shold be noted that se of this software in this stdy does not represent an endorsement of the software; other FEM codes can be employed. Kng et al. (2007a) indicated that wall deflection was relatiely easier to predict 77

94 accrately than grond settlement and the accracy of the prediction often depended on the capability of tilized soil models. They showed that proper modeling of small-strain nonlinearity soil behaior is essential for accrate predictions of grond settlement in a braced excaation sing FEM. This iew was shared by many preios inestigators (Hsieh and O 1997). In the work by Kng et al. (2007a), a small-strain nonlinear soil model, known as the Modified Psedo Plasticity (MPP) model, was implemented in the compter code AFENA (Hsieh and O 1997) for the analysis of braced excaations. The MPP model was deeloped by Hsieh et al. (2003) for clays, with considerations for anisotropic properties, high stiffness at small-strain, and degradation behaiors. Here, the range of small strain is from 0 to Throgh a series of analyses of laboratory tests (the conentional and small-strain CK 0 UC tests) and well-docmented case histories of braced excaation, Kng et al. (2007a) demonstrated the alidity of the MPP model and the accracy of the FEM predictions of wall and grond responses in braced excaations sing AFENA with the MPP soil model. Plaxis TM (Brinkgree and Vermeer 2002) is a proprietary FEM code, bt it prescribes a programming format that the ser can follow to implement a constittie law of soils. To follow p on the preios work (Kng et al. 2007a), it is desirable to implement the MPP model as a ser-defined model in Plaxis TM. Ths, Plaxis TM with the MPP soil model (implemented by Dang 2009) is sed in this stdy. To erify the accracy of the FEM code, we re-analyze the excaation case at Taipei National Enterprise Center (TNEC) that was docmented by O et al. (1998). The reslts are compared with those reported preiosly by Kng et al. (2007a), who analyzed 78

95 the same TNEC case sing the AFENA code with the MPP model. In that preios stdy, the capability of the AFENA code with the MPP model for predicting wall and grond responses in a braced excaation was demonstrated. In the present stdy, we compare the maximm wall deflections and the maximm grond settlements at arios stages of excaation of TNEC obtained by the two FEM codes (Plaxis TM with MPP erss AFENA with MPP). Figres 4.1(a) and 4.1(b) compare the reslts of FEM predictions of the maximm wall deflections and the maximm grond settlements, respectiely. The reslts show that the Plaxis TM soltions in this stdy are as accrate as those obtained by Kng et al. (2007a) sing AFENA, and both agree well with field obserations. Ths, Plaxis TM code with the MPP soil model is fond to be satisfactory for predicting the wall and grond responses in a braced excaation. Frthermore, in this stdy, Plaxis TM code with the MPP soil model is frther sed to stdy the effect of the spatial ariability of soils on the probability of exceeding the limiting responses in a braced excaation. For conenience, the software Plaxis TM implemented with the MPP soil model is referred to hereinafter as the FEM code. 79

96 160 (a) Maximm wall deflection 100 (b) Maximm grond settlement δhm(mm) - Kng et al. (2007a) or this stdy _ Kng et al. (2007a) This stdy Stage 1 Stage 2 Stage 3 Stage 4 Stage δ hm (mm) - Obseration Stage 7 Stage 6 ŀ δm(mm) - Kng et al. (2007a) or this stdy _ Kng et al. (2007a) This stdy Stage 3 Stage 2 Stage 1 Stage 4 Stage 5 Stage 7 Stage δ m (mm) - Obseration Figre 4.1: Maximm wall deflections and maximm grond settlements at arios stages of excaation of the TNEC case A comparison of the obsered ales with those obtained by Kng et al. (2007a) sing AFENA and in this stdy sing PlaxisTM; both FEM codes implemented with the MPP soil model. 80

97 Modeling Spatial Variability in Braced excaations in Clays Spatial ariability In a traditional deterministic approach, the FEM soltion generally assmes soil parameters to be spatially constant. In recent stdies sing random FEM to consider spatial ariability, Griffiths et al. (2009) fond the effect of inherent spatial ariations of soil properties can be significant in FEM soltions of many geotechnical problems. The random FEM approach, howeer, is comptationally intensie (Schweiger and Peschl 2005). For the complex problem of staged, braced excaations, it can be considered appropriate to se a simplified model of spatial ariability. Among the methods dealing with the spatial ariability of soil, the spatial aeraging of the ariation of the soil properties has shown to be an effectie tool (for example see, Phoon and Klhawy 1999a,b; Phoon et al. 2003; Klammler et al. 2010). In this stdy, the focs is to examine the effect of spatial ariability of soil parameters on the wall and grond responses in a braced excaation sing the FEM code; and to this end, the spatial aeraging approach is adopted. ŀ81 Spatial aeraging The concept of spatial aeraging was described by Vanmarcke (1977) as follows: the ariability of the aerage soil properties oer a large domain is less than that oer a small domain. The redced ariability of soil properties oer a large domain can be characterized by the ariance fnction, which is related to the atocorrelation fnction. The exponential model that is widely sed in the stdy of spatial ariability is selected

98 herein: ρ z exp (4.1) ( z) = 2 θ where z is the distance between any two points in the field; θ is the scale of flctation that is sed to normalize z. To consider spatial aeraging in a reliability analysis, ariances of soil parameters are redced by mltiplying a factor that depends on the scale of flctation (Vanmarcke 1983). This factor is the ale of the ariance redction fnction that can be obtained by integration of an atocorrelation fnction sch as Eq. (4.1). Ths, the ariance redction fnction may be expressed as (Vanmarcke 1983): θ 2L 2L Γ = f ( L, θ ) = 1 + exp 2 L θ θ (4.2) where L is the characteristic length with respect to a potential failre srface. In general, the ale of the ariance redction fnction is less than 1, and as sch, this ale is often referred to as the ariance redction factor ( Γ 2 ). The characteristic length may be assmed to be the length of the sliding srface (failre srface) in the stability analysis of a braced excaation, as sggested by Schweiger and Peschl (2005). A similar assmption was made by Most and Knabe (2010) in their stdy of bearing capacity of footings sing ariance redction techniqe. Figre 4.2 shows an example of the sliding srface based on the slip circle method (JSA 1988). 82

99 In this stdy, the characteristic length L is taken as the total length of the arc length bcd and the ertical length ab. a H d Final spport o r b Failre srface c Figre 4.2: An example of failre srface in braced excaation. With the known characteristic length (L) and the scale of flctation (θ ), the redced ariance σ Γ 2 can then be obtained with the following eqation: σ 2 = Γ 2 Γ σ 2 (4.3) where 2 σ is the ariance of the soil parameter, and Γ 2 is the ariance redction factor. The spatial aeraging approach has been shown to be an effectie simplification 83

100 of the real random field. As pointed ot by Schweiger and Peschl (2005), a more rigoros simlation of the random field demands a large amont of comptational time, which may not be practical for complicated nmerical simlations sch as the problem of braced excaation. In this stdy, the spatial aeraging effect is examined sing the checkerboard approach (Griffiths and Fenton 2009) and the feasibility of the ariance redction techniqe to model the spatially randomness within the context of a braced excaation is demonstrated. Fzzy Sets Methodology - Modeling and Processing of Uncertain Parameters Uncertainty modeling with fzzy sets Geotechnical engineers almost always hae to deal with ncertainty, whether it is formally acknowledged or not (Jang and Elton 1996). When inpt soil parameters cannot be ascertained de to limited data aailability, engineering jdgment is often exercised to select a conseratie design parameter. Uncertainty in soil parameters may be dealt with by sing an appropriate factor of safety. Howeer, in many cases, it is adantageos to assess this ncertainty and to inclde it in the analysis so that a more informed design decision can be made. Fzzy set theory (Zadeh 1965) has been shown effectie and sitable for modeling ncertainty in soil parameters (Jang et al. 1991; Jang et al. 1992a,b; Jang and Elton 1996; Jang et al. 1998) when data are insfficient to flly define a probability distribtion. A fzzy set is a set of paired ales [x, µ A (x)], where an element x belongs to the set A to a degree defined by its membership fnction µ A (x). The membership grade, 84

101 ranging from 0 to 1, is sed to characterize the degree of belief that x belongs to A. The fzzy set theory has been widely sed in many engineering and non-engineering fields. Examples of the application of fzzy sets in geotechnical engineering can be fond in literatre (e.g., Jang et al. 1991; Jang et al. 1992a,b; Valliappan and Pham 1995; Chen and Jang 1996; Jang and Elton 1996; Jang et al. 1998; Dodagodar and Venkatachalam 2000; Peschl and Schweiger 2003; Saboya et al. 2006). For rotine geotechnical ncertainty modeling, se of a sbset of a fzzy set, called fzzy nmber, to represent an ncertain soil parameter may be sfficient (Jang and Elton 1996). A fzzy nmber is a fzzy set that is normal and conex the shape of the membership fnction is single hmped and has at least one ale whose membership grade (or degree of belief) is 1. Figre 4.3(a) shows an example of a fzzy nmber. If there is no reason to sggest otherwise (becase of lack of data), the shape of the membership fnction may be assmed to be trianglar, as shown in Figre 4.3(a), becase of its simplicity in formlation and ease of comptation (Jang et al. 1998). The trianglar fzzy nmber has been shown to be sefl in many engineering applications (Elton et al. 2000). A trianglar fzzy nmber is characterized by three ales: a lower bond, an pper bond, and a mode (most probable ale). The mode has a membership grade of 1, the highest possibility, to represent ncertain soil parameter. As the ale of the parameter departs from the mode, the degree of belief for this ale to represent the soil parameter decreases, and when the ale reaches the lower bond (or the pper bond), the degree of belief is redced to zero. 85

102 1.0 (a) Trianglar fzzy nmber Membership, µ A(x)_ a m Parameter x b Membership, µ A(x )_ 1.0 α i 0.0 (b) α-ct interal α -ct interal a m + x α i Parameter x x αi b Figre 4.3: Example of trianglar fzzy nmber and α-ct interal. The concept of a simple representation of an ncertain soil parameter is not new. In a widely cited paper, Dncan (2000) proposed the concept of the highest conceiable 86

103 ale and lowest conceiable ale as a way to estimate the ncertainty of a soil parameter. He initially sggested that the standard deiation (σ ) of a soil parameter may be estimated by taking the difference between the highest conceiable ale and the lowest conceiable ale and diiding it by 6. Howeer, in the field of geotechnical engineering, lack of sfficient nmber of obserations is often a norm rather than exception; as sch, the ariation of soil parameters can often be nderestimated. Ths, it wold be more appropriate to adopt a diider of less than 6 [for example, 4, as later recommended by Dncan (2001)]. In many cases, the standard deiation may also be estimated by adopting the pblished coefficients of ariation (COV) on a gien soil parameter (e.g., Harr 1987; Phoon and Klhawy 1999a). Of corse, these COV ales may not be accrate for local soils and some adjstment may be needed. Frthermore, the mean ( µ ) of the parameter ɽ87 of concern may be compted from a limited nmber of data points or simply estimated as the most probable ale. With knowledge of the standard deiation and mean, simple reliability methods sch as the first order second moment (FOSM) method can be sed to compte the probability of failre. In this stdy, howeer, a different approach is employed. Assming the engineer can estimate a soil parameter with three ales, the highest conceiable ale (pper bond), the lowest conceiable ale (lower bond), and the most probable ale (mode), then a trianglar fzzy nmber as illstrated in Figre 4.3(a) can be readily defined. Generally, the most probable ale (mode) can be fairly accrately estimated by taking the mean of the aailable data (een with only a few data points). The pper and lower

104 bonds can be estimated based on pblished coefficient of ariation (which yields standard deiation), for example, by taking ±2 standard deiations from the mean. The estimate of the mode and the pper and lower bonds with ery limited data shold be gided by local experience and engineering jdgment. With the estimate of the mode, lower bond, and pper bond, the soil parameter can be modeled with a trianglar fzzy nmber. If the data are lacking or insfficient to flly define a probability distribtion, as in many geotechnical engineering projects de to cost constraints, then as an approximation, se of a trianglar fzzy nmber to model ncertain soil parameter is considered appropriate. Sch se of a fzzy nmber allows s to analyze the effect of ncertainty and compte the probability of sericeability failre in an efficient way. Of corse, probabilistic analysis of braced excaations can also be effectiely analyzed sing the probability theory (e.g., Baroth and Malécot 2010). Fzzy data processing by means of ertex method Almost all rotine geotechnical analyses are performed with deterministic models. If the inpt soil parameters are ncertain and take fzzy nmbers as their ales, the otpt of the deterministic model will be a fzzy nmber (or fzzy nmbers). In this case, ncertainties in the inpt parameters are propagated throgh the soltion processes; and in this stdy, the processes primarily inole the finite element soltion of wall and grond responses in braced excaations. The fzzy finite element approach (FFEA) is taken in this stdy to handle the 88

105 ncertainty propagation throgh the finite element soltion. To propagate fzzy inpt throgh the FEM code, the ertex method proposed by Dong and Wong (1987) is adopted. This method is based on the α-ct concept. As shown in Figre 4.3(b), at a membership grade of α i, an interal with a lower bond of x α i and an pper bond of x α + i can be formed. Mathematically, it can be shown that any fzzy nmber can be represented by a set of α-ct (or α-leel) interals with α ranging from 0 to 1. Ths, to propagate fzzy inpt throgh the FEM code, fzzy nmbers are first discretized into a set of α-ct interals (for example, taking α at 0.2 for α ranging from 0 to 1 yields 6 different leels, α = 0, 0.2, 0.4, 0.6, 0.8, and 1.0). This changes the analysis from the operation of fzzy nmbers into the operation of interals. Howeer, the traditional interal analysis cannot handle complex comptation processes that are inoled in finite element soltions. The ertex method remoes this difficlty with an effectie sampling techniqe. At each α-leel, the interals of the fzzy inpt ariables are obtained and the combinations of ertexes (i.e., the lower bonds and the pper bonds of the α-ct interals of all fzzy inpt) are determined. Gien n fzzy inpt ariables, the nmber of combinations of ertexes is 2 n. Each ertex represents a set of fixed ales of inpt ariables that can be readily entered into the FEM code for a deterministic analysis. Each of the 2 n combinations of ertexes are sed one-by-one in the FEM analysis, which yields a set of 2 n soltions. Taking the minimm and the maximm of these soltions, an interal is obtained at the specified α-leel. Dong and Wong (1987) has proen mathematically that at a gien α-leel, the interal soltion obtained with this ertex method is an exact soltion. Repeating the aboe process for a set of α ales, a set of 89

106 interal soltions are obtained. Recalling that a fzzy set is defined by a set of paired ales [x, µ A (x)], ths the lower bonds and the pper bonds of these interals along with the corresponding α ales define a fzzy nmber that represents the otcome of the fzzy FEM analysis. Define fzzy nmber s /σ' Define fzzy nmber E i /σ' Non-fzzy parameters Vertex method α -leels: 0, 0.2, 0.4, 0.6, 0.8, and ertexes for eachα -leel (4 combinations of s /σ' and E i /σ' ) FEM analysis for wall deflection and grond settlement sing PLAXIS α -ct interals of maximm wall deflection, δ hm α -ct interals of maximm grond settlement, δ m Fzzy nmber δ hm Fzzy nmber δ m Figre 4.4: Vertex method for fzzy FEM analysis of braced excaation. 90

107 It is noted that the otpt fzzy nmber is generally not sensitie to the nmber of α-leels, which depends on the magnitde of α, adopted for discretization of fzzy inpt ariables. For most geotechnical problems, se of α = 0.2 is adeqate (see Figre 4.4). If in dobt, howeer, a sensitiity analysis can be performed sing smaller α ales to confirm the conergence in the soltion. In preios stdies by Hsiao et al. (2008), wall and grond responses are reported to be strongly affected by ariation in the normalized ndrained shear strength ( s / σ ) and the normalized initial tangent modls ( E / σ ). In this stdy, to deal with ncertain parameters sing the proposed methodology, these two parameters are treated as fzzy parameters and all other factors sch as the stiffness of wall and strt, the excaation depth and width, etc. are treated as non-fzzy parameters. Figre 4.4 shows a flowchart depicting the process of fzzy data propagation throgh the FEM code by means of the ertex method. i Interpretation of the reslting fzzy nmber The reslting fzzy nmber, obtained by applying the ertex method to a deterministic approach (sch as Plaxis TM soltion), reflects the ncertainty in the model otpt. In this stdy, the model otpt is the maximm wall deflection and the maximm grond settlement in a braced excaation. An important design consideration is to ensre the probability of exceeding the maximm wall deflection (or grond settlement) is less than a threshold ale. To this end, a simple way to compte sch probability from the reslting fzzy nmber is needed. Using the reslting fzzy nmber shown in Figre

108 as an example, the probability of exceeding a limiting ale can be compted as follows: ( x > x ) p = A E E = p lim (4.4) AF where A E is the shade area depending on the limiting ale x lim and A F is the entire area nder the cre of the fzzy nmber. Membership, µ A(x) m A E 0.0 a Parameter x x lim b Figre 4.5: Fzzy nmber that represents the model otpt. The shaded area normalized to the fll area nder the shape is the probability of exceeding the limiting response (x lim ). It is noted that in Figre 4.5, normalization of the shade area with respect to the entire area nder the cre is needed as the latter is not necessarily eqal to 1. Althogh a more elegant formlation of the failre probability can be fond in the literatre (for example, Go and L 2003), Eq. (4.4) is easy to follow and implement. 92

109 Case Stdy TNEC Excaation Case The TNEC excaation case in Taiwan (O et al. 1998) is sed herein as an example to illstrate the fzzy finite element approach (FFEA) for the analysis of wall and grond responses in braced excaations in clay with a consideration for its spatial ariability. In this case, the excaation width is 41.2 m and the length of the diaphragm wall is 35 m. The excaation is carried ot sing a top-down constrction method in seen stages with spport proided by steel strts and floor slabs. Excaation depths and spport locations are detailed in Table 4.1. Soil parameters sed in the FEM code are shown in Table 4.2. In this stdy, the ndrained condition for clay layers is modeled. It is noted that the second layer (8 m 33m) in the soil profile is a clay layer that dominates the maximm wall and grond responses in this excaation. Hsiao et al. (2008) estimated the coefficient of ariation (COV) of the strength and stiffness parameters of Taipei clay as In the present stdy, we consider the ncertainty in soil parameters as well as their spatial ariability in the FEM soltion. To model the ncertainty of this clay soil, the normalized ndrained shear strength ( s / σ ) and normalized initial tangent modls ( E / σ ) are treated as fzzy parameters and all other factors sch as the i stiffness of the wall and strt, the excaation depth and width, etc. are treated as constant parameters in the analysis. 93

110 Table 4.1: Propping arrangement for the excaation and the stiffness of strts and floor slabs in the FEM analysis in TNEC case (after Kng et al. 2007a). Stage Excaation depth H (m) Depth of strts H p (m) * **, 0 ** ** ** ** * ** Note: *Steel strt; **Floor slab. stiffness of strts and slab floor, EA [kn/(m m -1 )] Table 4.2: Soil profile and soil model parameters sed in FEM analysis (from Kng et al. 2007a). Soil γ Depth(m) Soil model type (kn/m 3 s / σ Ei / σ φ (º) K=K ) r n CL MPP SM Dncan-Chang CL MPP CL MPP SM Dncan-Chang SM Dncan-Chang Note: φ = effectie friction angle; K r = elastic modls of nloading-reloading stages; n = elastic modls exponent; MPP = Modified Psedo Plasticity model. Aeraging the effect of spatial ariation A checkerboard stdy The effect of spatial ariation of soil parameters may be analyzed with a checkerboard analysis (Griffiths and Fenton 2009). In this approach, the random field is 94

111 meshed to many small sqare areas where the pper bond and the lower bond of the soil parameters alternate in the two-dimensional array. The soil parameters in the horizontal direction hae mch larger scales of flctation and are generally spatially correlated when compared with the ertical direction (Phoon and Klhawy 1999a). Therefore, for simplicity, only the ariability throgh the excaation depth is considered in this stdy. The pper and lower bonds of the soil parameters ( s / σ and Ei / σ ) are assmed to be the mean, pls and mins one standard deiation respectiely, and they only ary ertically on the checkerboard, as shown in Figre 4.6. Fie scenarios are examined in the checkerboard stdy sing the FEM code. In the base scenario (denoted as S0), the entire clay layer is assigned the mean soil parameters ( s / σ and Ei / σ ) as in a deterministic analysis. In the first scenario (S1), the scale of flctation is assmed to be infinite and the two soil parameters, s / σ and E / σ, are taken as the mean, mins and pls one standard deiation [S1(a) and S1(b), i respectiely]. In the second scenario (S2), the clay layer is sbdiided into for sb-layers. The pper bond and lower bond of s σ / and i E / σ alternate on the checkerboard and ths there are two sb-scenarios in the following seqence: pper-lower-pper-lower bond seqence [S2(a)] and lower-pper-lower-pper bond seqence [S2(b)]. In the third and forth scenarios [S3 and S4], the clay layer is sbdiided into eight and sixteen sb-layers respectiely, with a similar alternating seqence of lower and pper bond. Examples of scenarios S3(a) and S4(a) are shown in Figres. 4.6(a) and 4.6(b), respectiely, to illstrate the schematic of the checkerboard 95

112 stdy on the TNEC excaation case. (a) Scenario Three: 8 sb-layers, Case (a) [denoted as S3(a)] Mean + standard deiation Mean - standard deiation Strts or floor slabs Diaphragm wall CL SM CL 0.0m 5.6m 8.0m SM CL 33.0m 35.0m 37.5m SM 46.0m (b) Scenario For: 16 sb-layers, Case (a) [denoted as S4(a)] Mean + standard deiation Mean - standard deiation Strts or floor slabs Diaphragm wall CL SM CL 0.0m 5.6m 8.0m SM CL 33.0m 35.0m 37.5m SM 46.0m Figre 4.6: Schematic of checkerboard stdy on the ariation of soil parameters in an FEM model of TNEC case. 96