Reliability-Based Design and Quality Control of Bored Pile ...

Reliability-Based Design and Quality Controlof Bored Piles

Reliability-Based Design and Quality Controlof Bored Piles

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op woensdag 3 december 2014 om 10:00 uur

door

BACH DuongMaster of Engineering

National University of Civil Engineeringgeboren te Hanoi, Vietnam

mailto:[email protected]

Dit proefschrift is goedgekeurd door de promotoren:

Prof.drs.ir. J.K. VrijlingProf.dr.ir. P.H.A.J.M. van Gelder

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf.drs.ir. J.K. Vrijling Technische Universiteit Delft, promotorProf.dr.ir. P.H.A.J.M. van Gelder Technische Universiteit Delft, promotorProf.dr. M.A. Hicks Technische Universiteit DelftProf.ir. T. Vellinga Technische Universiteit DelftDr.ir. K.J. Bakker Technische Universiteit DelftDr. T.V. Cuong Vietnam Institute for Building Science and TechnologyDr.ir. W.M.G. Courage TNO, the NetherlandsProf.dr.ir. S.N. Jonkman Technische Universiteit Delft, reservelid

This research has been financially supported by the Ministry of Education and Trainingof Vietnam and Delft University of Technology.

Keywords: Bored piles, reliability-based design, quality control, resistance factor calibra-tion, set-up, Bayesian inference, Prob2B-Plaxis.

This thesis should be referred to as: Bach, D. (2014). Reliability-based design and qualitycontrol of bored piles. Ph.D. thesis, Delft University of Technology.

Cover layout by Tran Trong AnCover image: Probability Density Functions of Load and ResistancePrinted by Sieca Repro, Delft, the Netherlands

ISBN 978-94-6186-379-9

Copyright c© 2014 by BACH Duong

All rights reserved. No part of the material protected by this copyright notice may bereproduced or utilized in any form or by any means, electronic or mechanical, includingphotocopying, recording or by any information storage and retrieval system, withoutwritten permission of the author.

mailto:[email protected]

To my family

Contents

Summary vii

Samenvatting xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Introduction to bored piles . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Design approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Quality control approaches . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Problem outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Objective and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Study approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Design approaches 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Parameter uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Design approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Allowable stress design . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Limit state design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Load and resistance factor design . . . . . . . . . . . . . . . . . . . 202.3.4 Reliability-based design . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Reliability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Level II reliability methods . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Level III reliability method . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Quality control approaches 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Types of defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Integrity testing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Sonic Echo (SE) method . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Cross-hole Sonic Logging (CSL) method . . . . . . . . . . . . . . . . 37

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3.3.3 Drilling and Coring examination . . . . . . . . . . . . . . . . . . . . 403.4 Reliability evaluation for CSL method . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Number of access tubes . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.3 Inspection probability . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.4 Encountered probability . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.5 Detection probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.6 Analysis of inspection probability . . . . . . . . . . . . . . . . . . . 56

3.5 Recommended number of access tubes . . . . . . . . . . . . . . . . . . . . . 593.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Calibrating resistance factors 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Resistance factor calibration using FOSM . . . . . . . . . . . . . . . . . . . 644.3 Resistance factor calibration using FORM . . . . . . . . . . . . . . . . . . . 664.4 Resistance factor calibration using MCS (proposed) . . . . . . . . . . . . . 684.5 Calibration of a common resistance factor - Part 1 . . . . . . . . . . . . . . 69

4.5.1 Database for calibration . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.2 Statistical parameters and probability distributions of the resistance

bias factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.5.3 Calibrated resistance factors . . . . . . . . . . . . . . . . . . . . . . . 724.5.4 Correlation analyses between calibrated resistance factors and sta-

tistical parameters of resistance bias factors . . . . . . . . . . . . . . 754.5.5 Correlation analyses between resistance factors using different re-

liability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5.6 Validation of code calibration . . . . . . . . . . . . . . . . . . . . . . 78

4.6 Calibration of shaft and base resistance factors - Part 2 . . . . . . . . . . . . 784.6.1 Shaft and base resistance factor calibration using MCS . . . . . . . 794.6.2 Osterberg cell test (O-cell test) . . . . . . . . . . . . . . . . . . . . . 804.6.3 Data set for calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6.4 Statistical parameters and probability distributions of the shaft and

base resistance bias factors . . . . . . . . . . . . . . . . . . . . . . . 824.6.5 Calibrated shaft and base resistance factors . . . . . . . . . . . . . . 844.6.6 Sensitivity analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.6.7 Benefit of using shaft and base resistance factors compared to using

a common resistance factor . . . . . . . . . . . . . . . . . . . . . . . 864.6.8 Regarding regional calibration . . . . . . . . . . . . . . . . . . . . . 88

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Incorporating set-up into LRFD 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Causes leading to set-up effect . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Formulation for set-up effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 Incorporating Set-up into LRFD . . . . . . . . . . . . . . . . . . . . . . . . . 945.5 Reference and set-up resistance factor calibration . . . . . . . . . . . . . . . 94

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5.5.1 Reference and set-up resistance factor calibration based on experi-ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5.2 Reference and set-up resistance factor calibration using FORM . . 955.5.3 Reference and set-up resistance factor calibration using MCS . . . 96

5.6 A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.6.1 Data set for calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 975.6.2 Side shear set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6.3 Statistical parameters and probability distributions of the reference

resistance bias factors . . . . . . . . . . . . . . . . . . . . . . . . . . 995.6.4 Statistical parameters and probability distributions of the set-up

resistance bias factors . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.6.5 Calibrated reference and set-up resistance factors . . . . . . . . . . 1035.6.6 Incorporating set-up into LRFD procedure for the SR20 Bridge . . 106

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Updating resistance factors based on Bayesian inference 1096.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Within-site variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.4 Updating pile capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4.1 Survival test loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.2 Failure test loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.4.3 Multiple test loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.4.4 Multiple type of test . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Updating procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.6 A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.6.1 A data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.6.2 Initial prior distribution and initial resistance factors . . . . . . . . 1186.6.3 Updating resistance factors based on dynamic tests . . . . . . . . . 1186.6.4 Updating resistance factors based on static loading tests . . . . . . 1206.6.5 Regarding the updating order . . . . . . . . . . . . . . . . . . . . . . 1216.6.6 Effect of predicted resistance on the updated resistance factors . . . 1236.6.7 Likelihood function with static test pile behaviour . . . . . . . . . . 124

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Reliability-based design 1297.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Coupling calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.2.1 Reliability methods in Prob2B . . . . . . . . . . . . . . . . . . . . . . 1307.2.2 Coupling Prob2B-Plaxis . . . . . . . . . . . . . . . . . . . . . . . . . 1317.2.3 Limit state functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3 Models in Plaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.3.2 Geometry model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3.3 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3.4 Calculation types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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7.4 A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.4.1 Project description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.4.2 Loads on a pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.4.3 Material models and soil properties . . . . . . . . . . . . . . . . . . 1407.4.4 Soil parameter uncertainties . . . . . . . . . . . . . . . . . . . . . . . 1427.4.5 Reliability of intact bored pile . . . . . . . . . . . . . . . . . . . . . . 1457.4.6 Reliability of defect bored pile . . . . . . . . . . . . . . . . . . . . . 150

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 Conclusions and recommendations 1578.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.1.2 Regarding the reliability of the CSL method (Chapter 3) . . . . . . 1588.1.3 Regarding the resistance factor calibrations (Chapter 4) . . . . . . . 1598.1.4 Regarding the set-up effect (Chapter 5) . . . . . . . . . . . . . . . . 1618.1.5 Regarding updated resistance factors based on the Bayesian infer-

ence (Chapter 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.1.6 Regarding the reliability-based design model (Chapter 7) . . . . . . 162

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Bibliography 167

A Resistance bias factors 179A.1 Empirical distributions of the sixteen calibration cases . . . . . . . . . . . . 179A.2 Shaft and base resistance bias factors . . . . . . . . . . . . . . . . . . . . . . 181A.3 Factored total resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

B Set-up effect 185B.1 Set-up resistance bias factors for Ability 1 . . . . . . . . . . . . . . . . . . . 185B.2 Set-up resistance bias factors for Ability 2 . . . . . . . . . . . . . . . . . . . 186

C Prob2B-Plaxis results 187C.1 Reliability analysis results for intact bored piles . . . . . . . . . . . . . . . . 187

C.1.1 Nine calculation cases under GF mode . . . . . . . . . . . . . . . . 187C.1.2 Nine calculation cases under SF mode . . . . . . . . . . . . . . . . . 191C.1.3 Working pile subjected to working load under SF mode . . . . . . . 196

C.2 Reliability analysis results for defect bored piles . . . . . . . . . . . . . . . 197C.2.1 The effect of necking near the pile top . . . . . . . . . . . . . . . . . 197C.2.2 The effect of poor concrete zone . . . . . . . . . . . . . . . . . . . . 199C.2.3 The effect of soft bottom . . . . . . . . . . . . . . . . . . . . . . . . . 202

List of symbols 205

List of figures 212

List of tables 217

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Acknowledgements 221

Curriculum Vitae 223

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vi

Summary

Bored piles are a type of deep foundations which have been and are being widely usedin construction engineering such as high-rise buildings, bridges, jetties, and so on. Al-though bored piles have remarkable advantages over driven piles, the quality of boredpiles is frequently affected by many causes of imperfection, which mainly come from theinadequate ground investigation and construction procedures. It can be said that designand quality control of bored piles are two closely related stages. A quality control proce-dure has to be clearly addressed in the design stage; and decision making in the design,for many cases, has to be based on the testing results of a quality control procedure.

Therefore, two major objectives need to be solved in this thesis as: (1) Propose models tocalibrate resistance factors for the Load and Resistance Factor Design (LRFD) approachand find a suitable model aiming to directly determine reliability of a bored pile consid-ering some types of defect that may occur in the bored pile. (2) Select a quality controlmethod and evaluate its reliability when applied to bored piles.

The thesis consists of chapters, in each of which a new model is proposed, and then isapplied for a specific case study. The logicality and succession of the theoretical issuesbetween chapters are systematically presented.

In Chapter 2, a history of the development of design approaches is presented, includingthe Allowable Stress Design (ASD), the Limit State Design (LSD), and the Reliability-Based Design (RBD). Advantages and limitations of each design approach are discussedin detail. This chapter focuses on analyzing the LSD with the use of partial safety factorsfollowing the ultimate limit state. In which, the calibration of resistance factors underthe framework of the LRFD is one of the main objectives of this thesis. The level II andlevel III reliability methods are used to calibrate these resistance factors.

In Chapter 3, the quality control approaches of bored piles are briefly introduced as animportant part of the design and construction process. The post-construction tests com-prise planned and unplanned tests, in which planned tests are typically non-destructivetest methods. Of these methods, the Cross-hole Sonic Logging (CSL) method, the mostwidely used method for testing the integrity of bored pile concrete, is chosen aiming toevaluate its reliability. The inspection probability, which is used as a measurement of re-liability for the CSL method, was formulated based on the encountered probability andthe detection probability. For an assigned target inspection probability, the magnitudeof a defect that can be detected is a function of the pile diameter and the number of ac-cess tubes arranged. A necessary number of access tubes is recommended in this study.This finding is a good reference associated with design engineers and project managers

vii

Summary Summary

in making decisions for the design of bored piles.

In Chapter 4, the calibration models of resistance factors, following the framework of theLRFD, are proposed and presented with respect to some technical aspects of bored piles.The calibrated resistance factors aim at achieving target reliability levels for a set of loadfactors that were already specified in the structure code. For a calibration procedure, themodel uncertainty is considered and represented through the resistance bias factor.

At first, calibration models of a common resistance factor using three reliability meth-ods are presented. The reliability methods consist of the First Order Second Moment(FOSM) method, the First Order Reliability Method (FORM), and Monte Carlo Simula-tion (MCS). In this study, the calibration model using MCS is proposed, aiming to gain amore precise resistance factor and to reduce the calibration time. Sixteen calibration casesare considered; each calibration case is represented by a soil type, a prediction method,and a construction method. The resistance factors obtained from the proposed calibra-tion procedure have a good correlation with those from other calibration procedures thatwere officially accepted in practice. This confirms that the proposed calibration modelis valid and applicable. One interesting finding is that the calibrated resistance factorstrongly depends on the ratio of the coefficient of variation to the mean of the resistancebias factor with a linear relation. This is an important basis for calibrating the shaft andbase resistance factors separately.

Next, a calibration procedure for separate shaft and base resistance factors is proposed,because the degrees of uncertainty of shaft and base resistances are different. The use ofa common resistance factor as mentioned above clearly does not reflect this difference.In order to calibrate shaft and base resistance factors separately, the shaft and base resis-tance bias factors need to be determined. By the proposed calibration procedure, manycouples of values for the shaft and base resistance factors would be derived; all of whichsatisfy the target reliability levels. Therefore, a ”correlation ratio” is proposed aiming torepresent the correlation between uncertainty degrees of shaft and base resistance biasfactors. To which, a unique couple of values for the shaft and base resistance factors isfinally obtained. Through a case study at the site of the Los Angeles Memorial Coliseum(the US), using shaft and base resistance factors may lead to a more economical designthan a design using a common resistance factor.

In Chapter 5, the increase of pile resistance with time, compared to the initial resistance,is usually referred to as ”set-up” effect. The initial resistance is also called the referenceresistance; and the portion of increasing resistance with time is called the set-up resis-tance. Although the bored pile set-up effect is not as dramatic as the driven pile set-upeffect, incorporating the set-up effect into the LRFD for bored piles is more or less nec-essary. By this incorporation, an economical design can also be gained. Therefore, acalibration procedure for the reference and set-up resistance factors is presented and ap-plied for a case study at the site of the new SR20 eastbound bridge in Florida (the US).Due to the compatibility in the calibration algorithm, the calibration procedure used forthe set-up effect is completely the same as that for the shaft and base resistance factors.

The calibration model of a common resistance factor as mentioned in Chapter 4 is nor-mally based on the initial empirical distributions of the resistance bias factor. In general,these distributions have been built up by a large amount of data collected from many

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Summary Summary

different sites. Therefore, applying a common resistance factor, which is calibrated fromthe initial empirical distribution, for a specific site may not be completely consistent.Through the experimental outcomes of pile loading tests at a designed site, the Bayesianinference enables to reduce uncertainty with respect to the initial empirical distributionin terms of load test results within a site. To which, a posterior distribution of the resis-tance bias factor is then derived. A re-calibration process of a common resistance factoris subsequently carried out and an updating resistance factor is obtained. As a result, amore precise design using the updating resistance factor can be reached. The Bayesianinference is applied for a case study at the site of the 330 MW Uong Bi Extension No. 2Thermal Power Plant in Quang Ninh province (Vietnam). The derived results and somecomments are presented in Chapter 6.

It can be seen that, the LRFD uses the resistance factors that were obtained through thecalibration process and satisfies the specified target reliability levels. This approach doesnot require the explicit use of the probabilistic description of random variables and there-fore it has been familiar to design engineers in terms of its simplicity. In current practice,however, clients and project managers are more and more interested in the reached re-liability level or the probability of failure of a pile foundation. Therefore, applying theRBD aiming to directly estimate reliability levels for a specific bored pile foundation isof interest.

In Chapter 7, the reliability of a single bored pile is directly determined by the use of acoupling calculation between the finite element package (Plaxis version 9.0) and the nu-merical probabilistic toolbox (Prob2B). The reliability is assessed, not only for an intactbored pile but also for a defect bored pile, by assuming different types and magnitudesof defect that may occur within the pile body. Two failure modes, the Geotechnical Fail-ure (GF) mode and the Structural Failure (SF) mode, are proposed in this study. TheGF mode pertains to the geotechnical resistance of bored piles and the SF mode is re-lated to the compressive stress in bored pile concrete. Both modes are evaluated throughreached reliability levels for bored piles that are subjected to a specified combination ofloads from the superstructure. Based on which, the reliability of an axially loaded pileis comprehensively assessed. Through a case study at Pier T10 of the An Dong bridgein Ninh Thuan province (Vietnam), some findings and comments are presented in thischapter.

Finally, the conclusions and recommendations are stated in Chapter 8. Some modelsproposed in Chapters 4, 5, and 6 can be well applied for driven piles.

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x

Samenvatting

Boorpalen zijn diepe funderingen die veel en wijdverbreid in de bouwtechniek gebruiktworden, zoals bij hoogbouw, bruggen en steigers. Hoewel boorpalen een aanzienlijkvoordeel hebben ten opzichte van geheide palen, wordt de kwaliteit van boorpalen vaakbeınvloed door velerlei oorzaken van onvolmaaktheid, voornamelijk als gevolg van on-toereikend grondonderzoek en de constructieprocedures. Ontwerp en kwaliteitscontrolevan boorpalen zijn daarom twee nauw verwante fasen. Een kwaliteitscontroleproceduremoet in de ontwerpfase worden aangepakt; en besluitvorming in het ontwerp moet inveel gevallen worden gebaseerd op de testresultaten van een kwaliteitscontroleproce-dure.

Twee belangrijke doelstellingen worden in deze thesis opgelost: (1) Het voorstellen vanmodellen om de weerstandsfactoren voor de belasting en weerstand van de ”Load andResistance Factor Design” (LRFD) benadering te kalibreren, en een geschikt model tevinden dat is gericht op de betrouwbaarheid van boorpalen, met inachtname van ver-schillende soorten defecten die zich in boorpalen kunnen voordoen. (2) Het selecterenvan een methode voor kwaliteitscontrole, en het evalueren van de betrouwbaarheid vandeze methode bij toepassing op boorpalen.

In ieder hoofdstuk in dit proefschrift wordt een nieuw model voorgesteld, hetgeen ver-volgens wordt toegepast op een specifieke case study. De opeenvolging van de theore-tische kwesties in de hoofdstukken wordt op die manier systematisch en logisch gepre-senteerd.

In hoofdstuk 2 wordt de geschiedenis van de ontwikkeling van ontwerpbenaderingengepresenteerd, met inbegrip van ”Allowable Stress Design” (ASD), ”Limit State Design”(LSD), en ”Reliability-Based Design” (RBD). Voordelen en beperkingen van elke ont-werpbenadering worden in detail besproken. Dit hoofdstuk is vooral gericht op de ana-lyse van de LSD met gebruik van partiele veiligheidsfactoren volgens de ”Ultimate Li-mit State”. De kalibratie van weerstandsfactoren in het kader van de LRFD is een vande belangrijkste doelstellingen van dit proefschrift. Niveau II en niveau III betrouwbaar-heidsmethoden zijn gebruikt om deze weerstandsfactoren te kalibreren.

In hoofdstuk 3 worden diverse benaderingen van kwaliteitscontrole van boorpalen inhet kort geıntroduceerd als een belangrijk onderdeel van het ontwerp en het bouwpro-ces. De testen na de bouw omvatten geplande en ongeplande proeven, waarin de ge-plande proeven typische, niet-destructieve testmethoden zijn. Van deze methodes is de”Cross-hole Sonic Logging” (CSL) methode de meest gebruikte methode voor het testen

xi

Samenvatting Samenvatting

van de integriteit van betonnen boorpalen. De betrouwbaarheid van deze methode isgeevalueerd. De inspectiekans, die wordt gebruikt als een maat voor de betrouwbaar-heid van de CSL methode, werd geformuleerd op basis van de kans op het stuiten opeen defect en de kans op het daadwerkelijk detecteren van een defect, de detectiekans.Voor een bepaalde streefwaarde voor de inspectiekans is de omvang van een defect datgedetecteerd kan worden een functie van de diameter van de paal en het aantal toegangs-buisjes. In deze studie wordt een noodzakelijk aantal toegangsbuizen aanbevolen. Dezebevinding is een goede referentie voor ingenieurs en projectmanagers die zich bezig hou-den met ontwerp en met het nemen van beslissingen voor het ontwerp van boorpalen.

In hoofdstuk 4 worden de kalibratiemodellen van weerstandsfactoren volgens het kadervan de LRFD voorgesteld en gepresenteerd met betrekking tot bepaalde technische as-pecten van boorpalen. De gekalibreerde weerstandsfactoren zijn gericht op het bereikenvan streefwaarden van betrouwbaarheid voor een reeks belastingsfactoren die al in deconstructiecode gespecificeerd zijn. Bij een kalibratieprocedure wordt de onzekerheid inhet model beschouwd, en vertegenwoordigd door de weerstand-bias-factor.

Allereerst worden de kalibratiemodellen van een gemeenschappelijke weerstandsfactormet behulp van drie betrouwbaarheidsmethoden gepresenteerd. De betrouwbaarheids-methoden bestaan uit de ”First Order Second Moment” (FOSM) methode, de ”First Or-der Reliability Method” (FORM) en Monte Carlo Simulatie (MCS). In deze studie wordthet kalibratiemodel met behulp van MCS voorgesteld, met als doel een meer preciezeweerstandsfactor te verkrijgen en om de tijdsduur van kalibratie te verminderen. Zes-tien studies van kalibratie worden beschouwd; ieder geval van kalibratie wordt ge-karakteriseerd door een bodemtype, een voorspellingsmethode en een bouwmethode.De weerstandsfactoren, die uit de voorgestelde kalibratieprocedure verkregen zijn, heb-ben een goede correlatie met die uit andere kalibratieprocedures, welke in de praktijkreeds officieel aanvaard zijn. Dit bevestigt dat het voorgestelde kalibratiemodel geldigen toepasbaar is. Een interessante bevinding is dat de gekalibreerde weerstandsfactorsterk afhangt van de verhouding tussen de variatiecoefficient en het gemiddelde vande weerstand-bias-factor, met een lineaire relatie. Dit is een belangrijke basis voor hetafzonderlijk kalibreren van de schacht- en de puntweerstandsfactoren.

Vervolgens wordt een kalibratieprocedure voor afzonderlijke schacht- en puntweerstands-factoren voorgesteld, omdat de mate van onzekerheid van schacht- en puntweerstandverschillend is. Het gebruik van een gemeenschappelijke weerstandsfactor, zoals hierbo-ven genoemd, reflecteert dit verschil niet. Om de schacht- en puntweerstandsfactoren af-zonderlijk te kalibreren, moeten de schacht- en puntweerstands-bias-factoren worden be-paald. In de voorgestelde kalibratieprocedure worden vele paren van waarden voor deschacht- en puntweerstandsfactoren afgeleid; welke allen voldoen aan de betrouwbaar-heid van de streefwaarden. Een ”correlatieverhouding” wordt daarom voorgesteld, diede correlatie tussen onzekerheidswaarden van schacht- en puntweerstands-bias-factorenvertegenwoordigt. Tenslotte wordt een uniek paar waarden voor de schacht- en punt-weerstandsfactoren verkregen. Het gebruik van schacht en puntweerstandsfactoren lei-den voor een casestudy op het terrein van de Los Angeles Memorial Coliseum (V.S.) toteen economischer ontwerp dan een ontwerp met behulp van een gemeenschappelijkeweerstandsfactor.

xii


In hoofdstuk 5 wordt het zogenaamde ”set-up”-effect beschouwd. Dit is de verhogingvan de weerstand van de boorpaal in de tijd ten opzichte van de aanvankelijke weer-stand. De aanvankelijke weerstand wordt ook wel de referentieweerstand genoemd; enhet gedeelte van de in de tijd toenemende weerstand wordt de ”set-up”-weerstand ge-noemd. Hoewel het ”set-up” effect bij boorpalen niet zo sterk is als bij geheide palen, ishet meenemen van het ”set-up”-effect in de LRFD voor boorpalen min of meer een nood-zaak. Door dit effect mee te nemen, is het ook mogelijk om een economisch ontwerp teverkrijgen. Een kalibratieprocedure voor de referentie- en ”set-up”-weerstandsfactorenwordt vervolgens gepresenteerd en toegepast op een casestudy op de locatie van denieuwe SR20 eastbound bridge in Florida (V.S.). Vanwege de compatibiliteit in het kali-bratie-algoritme is de kalibratieprocedure die gebruikt is voor het effect van de ”set-up”volledig hetzelfde als die voor de schacht- en puntweerstandsfactoren.

Het kalibratiemodel met een gemeenschappelijke weerstandsfactor, zoals vermeld inhoofdstuk 4, is gewoonlijk gebaseerd op de eerste empirische verdelingen van de weer-stands-bias-factor. In het algemeen zijn deze verdelingen opgebouwd door een grotehoeveelheid gegevens die verzameld zijn op vele verschillende plekken. Dus, het toe-passen van een gemeenschappelijke weerstandsfactor voor een bepaalde plek, die uit deeerste empirische verdeling is gekalibreerd, is mogelijk niet geheel consistent. Dankzij deexperimentele resultaten van de belasting van boorpalen op een testlocatie, maakt Bay-esiaanse gevolgtrekking het mogelijk de onzekerheid met betrekking tot de eerste em-pirische verdeling in termen van de experimentele belastingsresultaten voor een locatiete verminderen. Naar aanleiding hiervan kan een verdeling van de weerstands-bias-factor achteraf afgeleid worden. Een herkalibratieproces van een gemeenschappelijkefactor van weerstand wordt vervolgens uitgevoerd en een aangepaste weerstandsfactorwordt verkregen. Op deze wijze kan met behulp van de aangepaste weerstandsfactoreen nauwkeuriger ontwerp worden verkregen. De Bayesiaanse gevolgtrekking is toe-gepast op een casestudy voor de locatie van de 330 MW Uong Bi uitbreiding van detweede thermische krachtcentrale in de provincie Quang Ninh (Vietnam). De afgeleideresultaten en toepasselijke opmerkingen worden gepresenteerd in hoofdstuk 6.

Het blijkt dat, met het gebruik van de LRFD, de weerstandsfactoren die zijn verkregenin het kalibratieproces voldoen aan de opgegeven streefwaarden van betrouwbaarheid.Deze aanpak vereist geen expliciet gebruik van de probabilistische beschrijving van sto-chastische variabelen, en is door haar eenvoud een vertrouwde benadering geweest voortechnische ontwerpers. Echter, in de huidige praktijk zijn klanten en projectmanagerssteeds meer geınteresseerd in het te bereiken betrouwbaarheidsniveau of de waarschijn-lijkheid van falen van een paalfundering. Het is dus van belang om RBD toe te passenvoor een directe schatting van de betrouwbaarheidsniveaus van een specifieke boorpaal-fundering.

In hoofdstuk 7 wordt de betrouwbaarheid van een enkele boorpaal rechtstreeks bepaalddoor het gebruik van een berekening, die de koppeling maakt tussen het eindige elemen-ten pakket (Plaxis versie 9.0) en het numerieke probabilistische gereedschap (Prob2B).De betrouwbaarheid is niet alleen voor een intacte boorpaal, maar ook voor een defecteboorpaal beoordeeld, door uit te gaan van verschillende typen en groottes van defectendie zich bij de boorpalen kunnen voordoen. Twee wijzen van falen, de Geotechnische

xiii


faal (GF) modus en de Structurele faal (SF) modus, worden in deze studie voorgesteld.De GF modus heeft betrekking op de geotechnische weerstand van boorpalen en de SFmodus is gerelateerd aan de drukspanning in betonnen boorpalen. Beide modi wordengeevalueerd via de bereikte betrouwbaarheidsniveaus voor boorpalen, die onderworpenworden aan een specifieke combinatie van belasting van de bovenbouw. Op basis hier-van wordt de betrouwbaarheid van een axiaal belaste paal uitgebreid beoordeeld. Aande hand van een casestudy bij Pier T10 van de brug van An Dong, in de provincie NinhThuan (Vietnam), worden de bevindingen en opmerkingen gepresenteerd.

De conclusies en aanbevelingen zijn tot slot vermeld in hoofdstuk 8. Sommige van demodellen, zoals voorgesteld in hoofdstukken 4, 5 en 6, kunnen ook zeer goed wordentoegepast voor geheide palen.

xiv

Chapter 1

Introduction

1.1 Background

1.1.1 Introduction to bored piles

The usual role of a deep foundation is to transfer vertical loads through weak, near-surface soils to rock or strong soil layers at a certain depth. There are many types of deepfoundations, and a classification can be made in various ways. Several factors that canbe used in classifying deep foundations are given below (O’Neill and Reese, 1999):

• Materials: Steel; concrete-plain, reinforced, or pre-stressed; timber; or some combi-nation of these materials.

• Methods of transferring load to the soil or rock: Principally in end-bearing, princi-pally in skin friction, or in some combination of the two methods.

• Methods of installation: Impact hammers, vibratory hammers, drilling an openhole; or by use of some special method.

• Influence of installation on soil or rock: Displacement piles, such as a closed-endedsteel pile pipe, that displace a large volume of soil as the piles are driven; or non-displacement piles, such as H-pile or open-ended steel piles, that displace a rela-tively small volume of soil during driving, or bored piles, which result in essen-tially no displacement of the soil or rock.

A bored pile is a deep foundation that is constructed by placing fluid concrete in an opendrilled hole, which is typically from 1.0 to 2.5 m in diameter, and up to 60 m in length,but which can extend to depths of as much as 90 m or more in special cases (Brownet al., 2010). A reinforcing steel cage can be installed in the excavation, if desired, priorto placing the concrete. Bored piles are also referred to as drilled piers, caissons, cast-in-dry-hole piles, and drilled shafts (the United States). The bored pile, as constructed,mainly supports axial loads through a combination of shaft and base resistances. The

1

1.1 Background Introduction

Figure 1.1: Scheme of a bored pile subjected to loads.

large diameter bored pile is also capable of providing substantial resistance to lateraland flexural loads as illustrated in Figure 1.1.

The methods of bored pile construction, which normally use rotary drilling rigs, can beclassified in three main categories: (1) The dry method, (2) the casing method, and (3) theslurry method. The ”mixed” method is given when two of those methods are appliedthe same time for a bored pile. The method of construction that is selected depends onthe subsurface conditions at a site. Because elements of the bored pile design can dependon the method of construction; consideration of the construction method is a part of thedesign process.

The dry method is applicable to soil and rock that are above the groundwater level andthat will not cave or slump when the hole is drilled to its full depth. A geomaterial thatmeets this requirement is a homogeneous, stiff clay. The dry method can sometimes beused for soils under the groundwater level if the soils are low in permeability and thehole is excavated and concreted quickly, so that only a small amount of water will seepinto the hole during the time the excavation is open.

The casing method is applicable to sites where the caving or excessive soil or rock de-formation can occur when a bore hole is excavated. Sometimes, soils or rocks that arestable when they are cut but which will slough soon afterwards. In such a case, a cas-ing (a simple steel pipe) needs to be used to keep the bore hole stable. Another notablecase, in which a casing can be used, is a clean sand layer below the groundwater levelunderlain by a layer of impermeable limestone, into which the bored pile will penetrate.Since the overlying sand layer is water bearing, it is necessary to seal the bottom of thecasing into the limestone layer to prevent flow of water into the bore hole. Most casingis recovered as the concrete is being placed and is called as temporary casing. In special

2

Introduction 1.1 Background

cases, the casing which remains and becomes a permanent part of bored pile is called thepermanent casing.

Figure 1.2 describes the principle construction steps of a bored pile under the slurrymethod. The steps consist of drilling a hole into the soil with slurry, placing a reinforcingsteel cage, and placing concrete. The slurry is utilized to keep the bore hole stable forthe entire depth of the excavation. The soil conditions for which the slurry is used couldbe any of the conditions described for the casing method. The slurry method can be afeasible option in a permeable, water bearing soil, if it is impossible to seal casing into astratum of soil or rock with low permeability. It can also be used in very deep holes witha narrow construction space where casing is hardly used since the difficulty of handlinga very long casing.

(a) Drilling with slurry

(c) Placing concrete (d) Completed bored pile

(b) Placing reinforcing cage

Figure 1.2: Slurry method of construction (O’Neill and Reese, 1999).

It is also noted that there are two types of drilled piles which differ from the bored pile:Micropiles and Continuous Flight Auger (CFA) piles.

Micropiles are drilled piles which are typically less than 0.3 m in diameter and con-structed using a high-strength steel rod or pipe which is considered as a hard core of thepile. These piles can even be drilled into hard rock and achieve very high axial resistancefor a very small structural component. Micropiles are favored in conditions where thesmall size is an advantage, and where lightweight, mobile drilling equipment must beemployed (Armour et al., 2000).

3


CFA piles are typically 0.3 to 0.9 m in diameter. These piles are distinguished from boredpiles in that the pile is formed by screwing the continuous auger or displacement toolinto the ground and then grouting or concreting through the hollow center of the auger;thus there is not an open hole at any time during the construction process. Guidelinesfor the design and construction of this type of pile are provided by Brown et al. (2007).

Bored piles can be installed in a variety of soil and rock profiles, and are most efficientlyutilized where a strong bearing layer is present. When the pile toe is founded within oron rock, extremely large axial resistance can be achieved with a small footprint. Boredpiles can also be installed into hard, scour-resistant soil and rock formations to be foundbelow scourable soil in conditions where installation of driven piles might be impracti-cal or impossible. Bored piles have increasing uses for highway bridges in seismicallyactive areas because of the flexural strength of a large diameter column of reinforcingconcrete. Furthermore, bored piles may be used as foundations for other applicationssuch as jetties, high-rise buildings, retaining walls, etc.

The most significant limitations of bored piles are related to the sensitivity of the con-struction procedure, to ground conditions and to the influence of ground conditions onbored pile performance. A summary of advantages and limitations of bored piles com-pared to other types of deep foundation is addressed by Brown et al. (2010) in Table 1.1.

Table 1.1: Advantages and limitations of bored piles

Advantage Limitation. Easy construction in cohesive materials,even rock

. Construction is sensitive to groundwa-ter or difficult drilling conditions

. Suitable to a wide range of ground con-ditions

. Performance of the bored pile may beinfluenced by the construction procedure

. Visual inspection of bearing stratum . No direct measurement of axial resis-tance during installation

. Possible to have extremely high axial re-sistance

. Load testing of high axial resistance maybe challenging and expensive

. Excellent strength in flexure . Structural integrity requires careful con-struction

. A single bored pile foundation withoutthe need for a pile cap

. A single bored pile foundation lacks re-dundancy and must have a high degreeof reliability

. Low noise and vibration and thereforewell suited to use in urban areas and nearexisting structures

. Requires an experienced, capable con-structor

. Can penetrate below scour zone into sta-ble, scour-resistant formation

. May not be efficient in deep soft soilswithout suitable bearing formation

. Can be easily adjusted to accommodatevariable conditions encountered in pro-duction

. Requires thorough site investigationwith evaluation of conditions affectingconstruction

4

Introduction 1.1 Background

1.1.2 Design approaches

The Allowable Stress Design (ASD), also called the Working Stress Design (WSD), hasbeen used in civil engineering since the early 1800’s. Under the ASD, a design load,which consists of the actual forces applied to piles, has to be less than the resistance di-vided by a global Factor of Safety (FS). This approach has several shortcomings, the mostsignificant of which is that it does not provide a consistent framework for incorporatingthe sources of uncertainty into the design. In fact, each component of the load and theresistance has a different level of variability and uncertainty.

In the 1950s, the demand for a more economical design of piles brought about the use ofthe Limit State Design (LSD). Two types of limit states are usually considered, UltimateLimit States (ULS), and Serviceability Limit States (SLS). The ULS pertains to structuralsafety and involves structural collapse or, in relation to piles, the ultimate resistance ofsoils. The SLS pertains to conditions, such as excessive deformations and settlements ordeterioration of the structure that would affect the performance of the structure underexpected working loads. The format of the LSD involves the application of Partial SafetyFactors (PSFs) to increase the loads (factored loads) and to decrease the resistance (fac-tored resistance). This approach represents a fundamental improvement over the globalfactor of safety in the ASD, because the partial safety factors are applied directly to theuncertain quantities of load and resistance.

The PSFs in the LSD are determined subjectively based on two criteria: (1) A larger PSFshould be applied to a more uncertain quantity. (2) The PSFs should result in approxi-mately the same dimensions as those obtained from traditional practice. This approachdid not satisfy one of the basic requirements of limit state design, because the occurrenceof each limit state is sufficiently improbable (Brown et al., 2010).

During the past two decades, the next step in the advanced LSD methodology has beento apply reliability analysis to establish the PSFs, in order to account for the uncertaintyand variability for loads and resistances. One of the advantages of this approach is thatall components of the structure, including the foundations, can be designed to a uni-form level of safety. However, the LSD has developed differently in North America andin Europe, mainly in the manner for calculating factored resistance for the ULS. Thisproblem will be further discussed in Chapter 2. In North America, Japan, South Korea,Hong Kong China, and recently in Vietnam, the LSD based on the reliability analysishas increasingly been used with a new name as the Load and Resistance Factor Design(LRFD), in which the PSFs applied to loads are termed load factors and those appliedto resistances are termed resistance factors. Here, each resistance factor is the productof a calibration study in which a Limit State Function (LSF) is evaluated to predict aspecific component of resistance (e.g., shaft or base or both types of resistance) to a spec-ified target reliability level. These efforts have led to a number of design codes aroundthe world: For highway structure foundations (e.g., Barker et al., 1991; Nowak, 1999;Paikowsky et al., 2004), for transmission line structure foundations in the United States(Phoon et al., 1995; Phoon et al., 2003), the National Building Code in Canada (Becker,1996), and the Geo-Code 21 in Japan (Honjo and Kusakabe, 2002).

In design, the ”trial-and-error” way in the LRFD is similar to that in the ASD, with the FS

5


being replaced by the PSFs. Therefore, there is some degree of compatibility between theLRFD and the ASD. However, the calibration processes are often invisible to design engi-neers, many assumptions and simplifications (e.g., probability distributions of loads andresistances) and model uncertainties adopted in the calibration processes are unknownto the designer. This situation can lead to potential misuse of the PSFs that are only validfor the assumptions and simplifications adopted in a certain calibration case. Designengineers have no flexibility in changing any of these assumptions or simplifications be-cause works of re-calibration are usually unfamiliar to them. In addition, the PSFs inthe LRFD are calibrated for some specified target reliability levels. A change in requiredreliability levels will cause a designer considerable difficulty in finding or selecting suit-able PSFs. Therefore, in current practice, designers as well as clients are becoming moreand more attracted to a new calculation procedure, called the Reliability-Based Design(RBD), aiming to directly estimate reliability levels for pile foundations. In the RBD, it ispossible to accommodate specific needs of a particular project when considering param-eter uncertainties (e.g., Orr and Breysse, 2008; Wang et al., 2011) or model uncertainties(e.g., Teixeira, 2012).

1.1.3 Quality control approaches

Most bored piles are constructed routinely and without difficulty, and are sound struc-tural elements. However, unexpected defects in a completed bored pile can arise duringthe construction process through errors in handling of slurry, reinforcing steel cages, con-crete, casings, and other factors. Therefore, tests to evaluate the structural soundness, or”integrity”, of completed bored piles are an important part of bored pile quality control.This is especially important where non-redundant piles are installed or where construc-tion procedures are employed in which visual inspection of the concreting process isimpossible, such as underwater or under slurry concrete placement.

From a management perspective, post-construction tests on completed bored piles canbe placed into two categories (Brown et al., 2010):

• Planned tests that are included as a part of the quality control procedure.

• Unplanned tests that are performed as part of a forensic investigation in responseto observations made by an inspector or constructor that indicates a defect mightexist within a pile.

Planned tests for quality control typically are Non-Destructive Tests (NDT) and are rel-atively inexpensive; such tests are performed routinely on bored piles. Meanwhile, un-planned tests performed as part of a forensic investigation will normally be more time-consuming and expensive, and the result can be more ambiguous than those of properlyperformed and planned tests.

The most common NDT methods are the Cross-hole Sonic Logging (CSL) method, theGamma-Gamma Logging (GGL) method, and the Sonic Echo (SE) method. Of thesemethods, the CSL method is currently the most widely used test for quality assurance ofbored pile concrete. For this method, vertical access tubes are cast into the pile prior to

6

Introduction 1.2 Problem outline

concrete placement. The tubes are normally placed inside the reinforcing steel cage andmust be filled with water to facilitate the transmission of high frequency compressionalsonic waves between a transmitter probe and a receiver one, which are lowered the sametime into each access tube. Acoustic signals are measured providing evaluation of con-crete quality between the tubes. This method has advantages that are relatively accurateand relatively low cost. The limitation of this method is that it is difficult to locate defectsoutside the line of sight between tubes. The physical description for this method will bepresented in detail in Chapter 3.

A frequent response to concerns about the integrity of a bored pile, usually as a result ofa problem observed during placement of concrete and identified by the inspector, or asa result of significant anomalies detected from non-destructive tests, is to institute a pro-gram of drilling and/or coring. Core sampling provides a direct visual examination ofconcrete and the opportunity to conduct strength tests on production concrete. However,drilling and coring are time-consuming and expensive, and which fall into the categoryof unplanned tests.

1.2 Problem outline

1.2.1 Objective and scope

Through some features regarding the development history of the design models and thecurrent quality control approaches for bored pile foundations as mentioned above, thisstudy will focus on the objectives as follows:

1. Evaluating the reliability of the CSL method, the most widely used method, in testi-fying the concrete quality of bored piles.

2. Calibrating resistance factors for the design of bored pile foundations, which followsthe framework of the LRFD. Resistance factor calibrations are conducted for differ-ent resistance prediction methods considering different construction methods. Thecalibrated resistance factors have to meet specified target reliability indices.

3. Applying a model of the reliability-based design aiming to directly estimate the re-liability of bored pile foundations. Parameter uncertainties of soils are included insoil models. The reliability of bored pile is evaluated by considering the situationsof pile with and without defects in light of geotechnical and structural failure modesproposed.

1.2.2 Research questions

In order to clarify the objectives of this study as stated above, the following questionshave to be answered. The chapter number, in which the corresponding question is an-swered, is shown in brackets.

7

1.3 Thesis outline Introduction

• In order to evaluate the reliability of the CSL method, some terms are given, whichare the encountered probability, the detection probability, and the inspection prob-ability. So, what are the encountered, detection, and inspection probabilities? Howto determine the optimal number of access tubes for the CSL method? (Chapter 3)

• Until now, resistance factors have usually been calibrated through some reliabilitymethods, which are the First Order Second Moment (FOSM) method and the FirstOrder Reliability Method (FORM). A question given is: How is a common resis-tance factor calibrated following the LRFD through the Monte Carlo Simulation(MCS)? (Chapter 4)

• By the Monte Carlo Simulation (MCS), how are the shaft and base resistance factorscalibrated separately? What is the correlation between them? (Chapter 4)

• The set-up effect is represented as the increase in resistance of piles with time. Howis the set-up effect incorporated into the LRFD aiming to reach an economic design?(Chapter 5)

• For a specific site, based on the Bayesian inference, how do the pile loading testresults affect the initial calibrated resistance factors? (Chapter 6)

• How will the reliability of a bored pile change when considering the influenceof various types of defect? To evaluate this problem, a model of the reliability-based design is applied and the parameter uncertainties of soils are included inthis model. (Chapter 7)

1.2.3 Study approach

Most of chapters of the thesis follow the modelling approach using the knowledge ofprobabilistic methods. For each proposed model there will be an accompanied case studyaiming to apply the proposed model. The data sets of bored pile foundations used in casestudies are collected from many sources, mainly from the United States and Vietnam,including superstructures such as highway bridges and buildings.

The validation of the models is quite difficult due to the lack of physical experiments aswell as actual proofs of past projects. Therefore, the model results are compared to thoseof the models of other authors that are widely accepted, or those have been accepted intothe current design standards.

This study is dedicated to axially loaded bored pile foundations under the ultimate limitstate. Other types of loads following the serviceability limit state are not mentionedherein; they are beyond the scope of this study.

1.3 Thesis outline

The thesis is structured as follows. Chapter 2 provides an overview on the developmenthistory of design approaches for bored pile foundations and probabilistic methods used.

8

Introduction 1.3 Thesis outline

In Chapter 3, the quality control approaches for the bored pile concrete are introduced.The cross-hole sonic logging method is chosen as a study object in which, the concepts as”encountered, detection, and inspection probabilities” are presented. Through these con-cepts, the reliability of the cross-hole sonic logging method will sufficiently be evaluated.Next, in Chapter 4, a model is used to calibrate a common resistance factor. In addition,a new model is proposed to calibrate shaft and base resistance factors separately aimingto reflect exactly the difference between shaft and base resistance uncertainties. Chapter5 presents the set-up phenomenon of piles. A model is proposed aiming to incorporatethe set-up into the LRFD. A procedure for the LRFD-based design with the set-up ispresented and an economic design may be attained. The Bayesian inference is used inChapter 6 to update resistance factors as some pile loading tests are conducted on a site.Based on this information, the use of the Bayesian inference will reduce uncertainty ofthe model that was initially used to calibrate resistance factors; then new values of re-sistance factors can be obtained through the re-calibration process. Chapter 7 describesa model of the reliability-based design which is used to directly determine reliability ofbored pile foundations with and without defects. Also, parameter uncertainties of soilsand the influence of types of defects on reliability of piles are discussed in this chapter.Finally, conclusions and recommendations from the present study are drawn in Chapter8. A schematic outline of this thesis is depicted in Figure 1.3.

9

1.3 Thesis outline Introduction

Figure 1.3: Overview of thesis outline.

10

Chapter 2

Design approaches

2.1 Introduction

In this chapter the design approaches, as introduced in Chapter 1, will be discussed inmore detail. The methodology, the advantages and the limitations of each approach willbe presented and the logical succession from approach to approach will be discussed.Tracking the history of approaches, we can recognize that the probabilistic method or thereliability method has played a very important role in developing and improving designapproaches. Current knowledge on probabilistic theory is broad, thus this chapter doesnot intend to thoroughly represent entire concepts as well as methods in this theory, butto briefly introduce those concepts and methods directly related to the study.

The outline of this chapter is as follows: Section 2.2 introduces the concept ”uncertainty”,one of the most basic concepts of the probabilistic theory. The design approaches aresystematically presented in Section 2.3. The reliability methods applied to the designapproaches are described in Section 2.4. Finally, the chapter ends with conclusions inSection 2.5.

2.2 Uncertainty

In van Gelder (2000), uncertainties in all aspects related to the civil engineering werecomprehensively described, and can primarily be divided into two categories:

• Inherent uncertainties, which stem from variability in known (or observable) popu-lations and, therefore, represent randomness in samples. It is impossible to reduceinherent uncertainties.

• Epistemic uncertainties, which come from a basic lack of knowledge of fundamen-tal phenomena and which may change as knowledge increases.

From the two categories of uncertainties mentioned above, van Gelder (2000) proposedsubdividing the inherent uncertainty and epistemic uncertainty into five types of un-

11

2.2 Uncertainty Design approaches

certainty: Inherent uncertainty in time and in space, parameter uncertainty, distributiontype uncertainty and, subsequently, model uncertainty as shown in Figure 2.1.

Figure 2.1: Types of uncertainty (van Gelder, 2000).

Two types of uncertainty, which are parameter uncertainty and model uncertainty, aredirectly related to the contents of the thesis. They are briefly introduced below.

2.2.1 Parameter uncertainty

This uncertainty occurs when the parameters of a distribution are determined by a lim-ited number of data. The smaller the number of data, the larger the parameter uncer-tainty. A parameter of a distribution function is estimated from the data and is thus arandom variable. The parameter uncertainty can be described by the distribution func-tion of the parameter (van Gelder, 2000).

2.2.2 Model uncertainty

Many design models aiming to estimate the design factor, like loads and resistances,are imperfect. They can be imperfect, because the physical phenomena are not knownor insufficiently considered, or some variables of lesser importance are omitted in thedesign model for purpose of simplicity in calculations.

To construct a framework in which model uncertainty can be determined in the environ-ment of parameter and observation uncertainties, the parameter, the observation, andthe model uncertainties should be defined and their relationships need to be clarified(Zhang, 2009). Let g(θ) denote a scalar prediction model to simulate the behaviour ofa system, where θ is a vector denoting uncertain parameters. Due to the presence ofmodel uncertainty, g(θ) may not be the same as the actual system response, y. Therefore,a model correction factor is used to model the effect of model uncertainty. The exactvalue of the model correction factor may be system-dependent; but it is reasonable toassume that the value of the model correction factor may follow a common probabilisticdistribution. Thus the model correction factor is described as a random variable (Zhang,2009).

The model correction factor can be applied in either an additive way (e.g., Christian

12

Design approaches 2.3 Design approaches

et al., 1994; Moss et al., 2006) or in a multiplicative way (e.g., Ang and Tang, 1990; Barkeret al., 1991; Zhang et al., 2001; Paikowsky et al., 2004; Allen, 2005; Abu-Farsakh and Yu,2010). The additive model correction factor, ε, is defined as the difference between actualperformance, y, and prediction model, g(θ):

y = g(θ) + ε (2.1)

and the multiplicative model correction factor, λ, is defined as:

y = λg(θ) (2.2)

Hereafter, the multiplicative model correction factor, λ, will be used throughout thestudy by a new name ”bias factor” aiming to be compatible with other works of authorsthat are cited in this study.

2.3 Design approaches

The primary objectives of engineering design are safety, serviceability, and economy.Safety and serviceability can be improved by increasing the safety margins or levelsof safety in order to reduce the probability of failure. However, this will increase thecost of the structure. Consideration of overall economy in design involves balancing theincreased cost against the potential losses (i.e., failure). Regardless of the design philos-ophy and approach used, the basic design criterion is that the capacity or resistance ofthe system must be greater than the demand or loads on the system for an acceptable orrequired level of safety. In equation format, the design criterion is given by:

Resistance (R) > Load e f f ects (Q) (2.3)

The design approaches have not remained stagnant, but have changed over the years inresponse to a changing social environment, higher public awareness and expectations,and advancements in technology (Becker, 1996). The comparison between the load ef-fects and resistance for an assessment of safety can be conducted in various ways, in-cluding the following: (1) A single global factor of safety as embodied in allowable stressdesign; (2) partial factors of safety as embodied in limit state design; and (3) reliability-based design.

2.3.1 Allowable stress design

The Allowable Stress Design (ASD), also called the Working Stress Design (WSD), hasbeen the traditional design basis in civil engineering since it was first introduced in theearly 1800’s. Under the ASD, an applied load, Qa, which consists of actual forces appliedto piles, has to be less than the ultimate resistance, Ru, divided by a global Factor ofSafety (FS):

Qa ≤Ru

FS(2.4)

13

2.3 Design approaches Design approaches

A global factor of safety is used, which lumps all uncertainties associated with the de-sign process into a single value with no distinction made as to whether it is applied toresistance or to load effects. The assessment of the level of safety of the structure is madeon the basis of global factors of safety, which were developed from previous experiencewith similar structures in similar environments or under similar conditions. The valuesof the FS selected for design reflect past experience and the consequence of failure. Thehigher the uncertainty, the larger the FS. Table 2.1 presents FS values used in the StandardSpeci f ications f or Highway Bridges (AASHTO, 1997) in conjunction with different levelsof control in design analysis and during construction. Presumably, when a more reliableand consistent level of control is used, a smaller FS can be used, which leads to a moreeconomical design.

Table 2.1: Factors of safety on ultimate axial geotechnicalresistance based on level of construction control (AASHTO,1997)

Basis for design Increasing design/constructionand control level control

Subsurface exploration√ √ √ √ √

Static calculation√ √ √ √ √

Dynamic formula√

Wave equation√ √ √ √

CAPWAP(*)analysis√ √

Static loading test√ √

FS 3.50 2.75 2.25 2.00 1.90(*) CAse Pile Wave Analysis Program (Rausche et al., 1985).

In the ASD, both the load and resistance are generally considered to be deterministicand characterized in calculation by a single value. In Figure 2.2(a), Eq. 2.4 implies thatboth load and resistance are well defined, each with a unique value. However, loadsand resistances are dependent upon a number of variables. In reality, ranges in loadsand resistances exist following certain frequency distribution diagrams as shown in Fig-ure 2.2(b), which are called Probability Density Functions (PDF) of load and resistance,respectively. Thus, unique values do not exist for loads and resistances.

The probability density functions, as shown in Figure 2.2(b), can be assigned specificvalues, such as the mean of the distribution curves, Q and R, or nominal values, Qnand Rn, to assist in characterizing the frequency diagrams. The design values do notnecessarily need to be taken as the mean values, although this is common geotechnicaldesign practice. The design process may also involve overestimating the mean load ef-fects (i.e., Qn≥Q) and underestimating the mean resistance (i.e., Rn≤R) as presented inBecker (1996).

Two alternative definitions of the FS can be defined as follows:

Mean FS =RQ

(2.5)

14


0

1.0

(a) In ASD

Load or resistance (Q,R)

Pro

babi

lity

of o

ccur

ence

0

(b) In reality


Fre

quen

cy o

f occ

uren

ce

Safety margin =

Safety margin =

PDF of

PDF of

Qa Ru RQnQ Rn

Ru −Qa

R−Q

Q

R

Figure 2.2: Definitions of load and resistance in ASD and in reality.

Nominal FS =Rn

Qn(2.6)

The values for FS as defined in Eqs. 2.5 and 2.6 do not equal each other; the mean FSis larger than the nominal FS. The intersection of the Q and R curves, depicted by theshaded region in Figure 2.2(b), represents a condition where, under some combinationsof loads and resistances, the resistance is less than the load and failure can occur. Thisintersection indicates that a probability of failure exists for some combinations of loadsand resistances. For given distributions of load and resistance, different values of FS canbe calculated, meanwhile the actual level of safety or probability of failure remains thesame. Therefore, the global FS in the ASD does not provide a consistent measure of thelevel of safety or probability of failure.

0

(a) Well−defined Q and R


Fre

quen

cy o

f occ

uren

ce

0

(b) Poorly−defined R


Fre

quen

cy o

f occ

uren

ce

0

(c) Poorly−defined Q and R


Fre

quen

cy o

f occ

uren

ce

RQRQRQ

Q

R

R

Q

Q

R

Figure 2.3: Possible load and resistance distributions.

The shortcoming of the ASD is described in Figure 2.3 through consideration of the dis-tribution curves. Figure 2.3(a) represents the case where the load and resistance are welldefined and controlled. There is a relative low probability of failure as shown by thesmall overlap area of the Q and R distribution curves. The case, where the load is welldefined, but the resistance is not, is illustrated in Figure 2.3(b). In this case both the load

15


and resistance are not well defined, as represented by the broad distribution curves asshown in Figure 2.3(c). It can be seen that, although the value of Q and R are the sameand, therefore, FS is the same for all three cases, the overlap area is much larger; respec-tively the probability of failure is much higher with respect to the second and third cases.It is noted that the overlap area denoted by the shaded area is not equal to the probabilityof failure; however, it is related to the probability of failure.

2.3.2 Limit state design

Limit states are defined as conditions under which a structure or its component membersno longer perform their intended functions. Whenever a structure or part of a structurefails to satisfy one of its intended performance criteria, it is said to have reached a limitstate (Becker, 1996).

In the Limit State Design (LSD), two types of limit states are usually considered: UltimateLimit States (ULS) and Serviceability Limit States (SLS).

The ULS pertains to structural safety and to defining components that are dangerous(Allen, 1994); they involve the total or partial collapse of the structure, e.g., strength, ul-timate bearing capacity, overturning, sliding, and so on. The ULS conditions are usuallychecked using separate Partial Safety Factors (PSFs) on loads and resistances. Because oftheir relationship to safety, the ULS conditions have a low probability of occurrence forwell-designed structures (Duncan et al., 1989).

The SLS represents conditions which affect the function or service requirements of thestructure under expected service or working loads. The SLS includes conditions that mayrestrict the intended use of the structure such as deformation, cracking, excessive total ordifferential settlement, excessive vibrations, local damage, and deterioration. Deforma-tion or settlement of foundations could also cause loss of serviceability in the building.The SLS may be viewed as those things that ”make life difficult” (Allen, 1994). The SLShas a higher probability of occurrence than the ULS (Duncan et al., 1989).

Note that the SLS conditions are checked using unfactored loads and geotechnical prop-erties. A partial factor of one is used on all specified load effects and on the characteristicvalues of deformation and compressibility properties of soils, which are generally basedon conservative mean values obtained from in situ or laboratory tests. In this sense, themethodology of calculation in connection with the SLS in the LSD is virtually identicalto that of the ASD (Becker, 1996).

In geotechnical design, a serviceability condition or settlement criterion frequently con-stitutes the primary limit state. The design would be based on specific SLS; the ULSwould be checked subsequently. Regardless of the complexity of calculation, all limitstates designs are carried out to satisfy the following criteria:

• The ULS:

Factored resistance ≥ Factored load e f f ects (2.7)

16


• The SLS:

De f ormation ≤ Tolerance de f ormation to remain serviceable (2.8)

Becker (1996) featured the development history of geotechnical limit states design as fol-lows: The first uses of limit state concepts in geotechnical engineering include the workof Coulomb in 1773 who, based on limit states consideration, derived the critical heightof a vertical embankment in a cohesive soil. In 1857, Rankine established limit statesof active and passive earth pressure. Therefore, limit state problems in soil mechanicssuch as theories of earth pressure and bearing capacity, stability of slopes, and seepagewere already treated in the 18th and 19th centuries. In 1943, Terzaghi pointed out twoprinciple groups of problems in geotechnical limit states, namely, stability problems andelasticity problems. The above two groups of problems coincide with the ULS and SLS,respectively, in the LSD. Taylor (1948) introduced partial safety factors for the cohesiveand frictional components (i.e., c′ and tanϕ′) of the shear strength of soil for the stabilityanalysis of slopes. This approach was subsequently formalized by Brinch Hansen (1953,1956) who established a philosophy of geotechnical design based on separately applyingpartial safety factors to loads and strength.

In this approach, the characteristic load effects are multiplied by their respective par-tial factors to obtain design loads, and the strength parameters are divided by their re-spective partial factors to arrive at the design strength parameters for the calculation ofgeotechnical resistance:

Design load = Characteristic load× Load f actor (γ) (2.9)

Design strength =Characteristic strength

Partial strength f actor( fc or fϕ)(2.10)

The values of the partial factors are summarized in Table 2.2. Meyerhof (1995) notedthat the partial factors, from Brinch Hansen (1953, 1956) until Eurocode 7 (CEN, 1992),were chosen to give the same design as the conventional ASD. Scrutinizing some partialfactors in Eurocode 7 (CEN, 2004), it can be seen that these values have undergone onlyminor changes during the past 50 years.

The use of the LSD with partial safety factors has developed differently in North Americaand in Europe, mainly in the manner of calculating factored resistances for the ULS. Thisproblem is presented following (Becker, 1996):

In the f actored strength (European) approach, partial factors are applied directly to onlythe strength parameters that contribute to overall resistance for each applicable limitstate. In particular, specified partial factors are applied to the individual soil strengthproperties of angle internal friction (tanϕ′) and cohesion (c′) prior to using them in cal-culating design resistance. In the f actored resistance (North American) approach, anoverall resistance factor is applied to the resistance for each applicable limit state. Withthis approach, the calculated ultimate resistance is firstly calculated using characteristicstrength parameters (unfactored strength parameters); the calculated ultimate resistanceis then multiplied by a single resistance factor to obtain the factored resistance for design.

17


Table 2.2: Summary of partial factors for foundation design

Item Brinch Hansen DS 415 Eurocode 7 Eurocode 7

(1953) (1956) (DI 1965) (CEN 1992)(*) (CEN 2004)(*)

Loads.Dead loads 1.0 1.0 1.0 1.1 (0.9) 1.1 (0.9).Live loads 1.5 1.5 1.5 1.5 (0) 1.5 (0)

Soil strength.Friction (tanϕ′) 1.25 1.2 1.25 1.25 1.25.Cohesion (c′)

Spread foundations - 1.7 1.75 1.4 - 1.6 1.0 - 1.4Pile foundations - 2.0 2.0 1.4 - 1.6 1.0 - 1.5

Ultimate pile capacity.Load tests - 1.6 1.6 1.7 - 2.4 1.0 - 1.4

(*) Values in parentheses indicate minimum factors for certain load combinations.

A comparison of the European and North American approaches is shown in Figure 2.4.For both approaches, the design (factored) resistance must be greater than or equal to thedesign (factored) load effects. The primary difference in the conceptualization lies in theresistance side. The load effects side is identical for both approaches; the characteristicload effects are multiplied by appropriate load factors to produce the design (factored)load effects for design as indicated in Eq. 2.9. However, different values of load factorsare used in the two approaches.

Figure 2.4: Comparison of limit states design approaches for the ULS (Ovesen and Orr,1991).

18


In the factored strength approach, the characteristic strength parameters are divided bypartial factors, fc and fϕ, to produce design strength parameters according to Eq. 2.10.The design strength parameters are then used directly in theoretically predicted modelsto calculate the design resistance, Rd. The value of Rd must be greater than or equal tothe design load effects, Qd, for design.

In the factored resistance approach, the characteristic strength parameters are used di-rectly in similar theoretically predicted models to calculate a nominal resistance, Rn, sim-ilar to calculating ultimate resistance in the ASD. The nominal resistance, Rn, is thenmultiplied by a resistance factor, φ, to calculate the factored resistance, φRn, for design,which must be greater than or equal to the factored load effects, γQn (see Figure 2.4). Thefactored resistance (North American) approach is called the Load and Resistance FactorDesign (LRFD), which is described in detail in Subsection 2.3.3.

The factored resistance approach combines all uncertainties when calculating resistanceinto one term, a resistance factor, φ (generally, φ<1.0). The value of a resistance factorreflects the probability that the actual resistance may be smaller than the calculated nom-inal resistance using characteristic values of soil strength as a result of the uncertaintiesassociated with the strength parameters, the theoretical models used to calculated re-sistance, geometry, construction methods, and the consequence of failure. Different φvalues are used for different types of resistance and failure modes. Selection of the valueof φ depends on the quality of data and the method of calculation. The major advan-tages of the factored resistance approach are its simplicity in calculation and familiarityto geotechnical engineers. Only one factor needs to be considered instead of the manypartial factors of material strength, geometry, theoretical models, and others (Becker,1996).

With respect to the factored strength approach, a potential advantage is that it may allowfor a more precise calibration over a wide range of soil types; therefore, it should lead toa more uniform reliability (Meyerhof, 1995). A key disadvantage of the factored strengthapproach is that it does not allow for the sources of other uncertainties to be taken intoaccount in the calculation of design resistance. That means that only the sources of un-certainties associated with soil strength parameters are considered; there is no explicitmeans to account for other factors that affect resistance, such as the calculation models,geometry, construction procedures, and types of failure. The use of a unique set of ma-terial strength factors, fc and fϕ, has been critically questioned by many geotechnicalengineers, both researchers and practitioners. A common theme of criticism emerges inthat design strength parameters alone do not capture all sources of uncertainties in thecalculation of resistance (Becker, 1996).

In geotechnical engineering design, empirical correlations are usually used. The mostcommon correlations involve the Standard Penetration Test (SPT) and the Cone Penetra-tion Test (CPT). Geotechnical resistances, such as bearing capacity of footings and pile re-sistances, have been correlated directly to these in-situ tests. In fact, the design equationsfor the calculation of resistance, in many cases, do not contain the strength parameterstanϕ′ and c′. This is embarrassing to design engineers in calculation when applying thefactored strength approach. To overcome this limitation, the European approach uses a”performance” factor. The nominal resistance based on empirical correlations is modi-

19


fied by multiplication with a performance factor that has a value less than unity aimingto account for uncertainties associated with the design process and methodology. Essen-tially, this modified way drives the European approach to a new format that is the sameas the factored resistance approach. Therefore, the factored resistance approach is a con-sistent manner in geotechnical design regardless of whether the calculation method usessoil strength parameters of ϕ′ and c′, empirical correlations with SPT, CPT, and otherin-situ tests, or performance testing such as pile loading tests (Becker, 1996).

2.3.3 Load and resistance factor design

As mentioned above, the factored resistance (North American) approach is also calledthe Load and Resistance Factor Design (LRFD), in which the PSFs applied to loads aretermed load factors and those applied to resistances are resistance factors. The use ofseparate load and resistance factors is logical because loads and resistance have differ-ent sources of uncertainty. In North America, Japan, South Korea, Hong Kong China,and recently in Vietnam, the LRFD has been used increasingly in many applications offoundation design.

A remarkable point of the LRFD is that it makes use of the concept of PSFs, which areusually based on, or calibrated, using probabilistic methods. PSFs can also be evaluatedusing judgment and comparison or calibration with the ASD to ensure that consistentresults will be obtained between the two design procedures. It can be seen that the LRFDis a logical development step of the ASD. Probabilistic methods and statistical data areused as background tools to derive PSFs; however, such probabilistic methods are notused explicitly in the design procedure developed and specified in the various codes.Therefore, the LRFD is also called the semi-probabilistic method in design.

The applied loads and resistances are generally treated as random variables in the deriva-tion of the respective partial factors. The loads and resistances are characterized throughProbability Density Functions (PDF) as shown in Figure 2.5. The LRFD examines theprobability of failure or reliability indices of a structure by underestimating its resistanceand overestimating the load effects, to provide a factored resistance that is greater thanor equal to the factored load effects. The level of safety is defined in terms of acceptableprobability of failure or specified reliability indices. PSFs on loads and resistances areused to obtain such an acceptable probability of failure or specified reliability indices.

The LRFD is expressed in the following format:

φRn ≥n

∑i=1

γiQni (2.11)

where φ is the resistance factor; Rn is the nominal resistance; γi is the load factor for theith load component; and Qni is the ith nominal load component. The load factors, γi,are usually greater than one; they account for uncertainties in loads and their probabil-ity of occurrence. The resistance factors, φ, are generally less than one and account forvariabilities in the geotechnical parameters and the model when calculating resistance.

The LRFD can be characterized by examining the interaction of the PDF curves for resis-tance and load effects as shown in Figure 2.5. Assume that the resistance and load effects

20


are independent variables. The nominal values for resistance, Rn, and load effects, Qn,do not necessarily need to be taken as the mean values of resistance and load effects,respectively. The nominal values for design are related to the mean values as follows(Becker, 1996):

Rn =RkR

(2.12)

Qn =QkQ

(2.13)

where kR and kQ are the ratios of mean value to nominal value for resistance and loadeffects, respectively. Typically, kR values are greater than one (i.e., Rn≤R) and kQ valuesare smaller than one (i.e., Qn≥Q).

0


Fre

quen

cy o

f occ

uren

ce

PDF of R

PDF of Q

Q Rn RQn

φRn = γQn

Rn(1 − φ)Qn(γ − 1)

Figure 2.5: Load and Resistance Factor Design (LRFD).

In practice, values for γ and φ are usually specified in codes corresponding to targetvalues of the reliability index or the acceptable probability of failure, selected to be con-sistent with the current state of practice (Becker, 1996). In general, values of γ may differbetween codes in various countries, as well as between types of superstructure; load fac-tors are typically in the range of 0.83÷1.3 for dead loads and in the range of 1.5÷2.0 forlive and environmental loads (Ellingwood and Galambos, 1982; Nowak, 1999). Typicalvalues of the resistance factor, φ, range from 0.2÷0.9 depending on soil types, methodsof calculating resistance, and types of foundation structure (AASHTO, 2007). The cali-bration of resistance factors have led to a number of design codes around the world, forhighway structure foundations (e.g., Barker et al., 1991; Nowak, 1999; Paikowsky et al.,2004) and for transmission line structure foundations (Phoon et al., 1995; Phoon et al.,2003) in the United States, the National Building Code (Becker, 1996) in Canada, and theGeo-Code 21 (Honjo and Kusakabe, 2002) in Japan.

Although the LRFD may be viewed as an extension to the ASD, the main advantageof the LRFD over the ASD is that it can provide a more consistent and uniform level ofsafety for all load combinations, different types of materials, structures, and foundations.

21


This is achieved through the use of several different partial factors instead of one globalfactor of safety (Meyerhof, 1970).

2.3.4 Reliability-based design

In fact, the ”trial-and-error” way in the LRFD is similar to that in the ASD, with the FSbeing replaced by PSFs. However, the calibration processes of PSFs are often invisibleto design engineers, as many assumptions, simplifications (e.g., probability distributionsof loads and resistances) and model uncertainties adopted in the calibration processesare unknown to the designer. This situation can lead to potential misuse of the PSFs,that are only valid for the assumptions and simplifications adopted in a certain calibra-tion case. Design engineers have no flexibility in changing any of these assumptions orsimplifications, because works of re-calibration are usually unfamiliar to them. In ad-dition, the PSFs in the LRFD are calibrated for some specified target reliability levels.A change in required reliability levels will cause a designer considerable difficulty infinding or selecting suitable PSFs. Therefore, in current practice, designers as well asclients are becoming more and more interested in a new calculation procedure, calledthe Reliability-Based Design (RBD), aiming to directly estimate reliability levels for pilefoundations. In the RBD, it is possible to accommodate specific needs of a particularproject, when considering parameter uncertainties.

In the last two decades, there has been considerable interest and an increasing trendtowards the use of the RBD (e.g., Tang, 1993; Phoon et al., 1995; Christian, 2004; Orr andBreysse, 2008; Wang et al., 2011; Teixeira, 2012; and Juang et al., 2013). In the RBD, theparameters are treated as random variables instead of constant deterministic values. Themeasure of safety is the probability of failure that can be computed directly if the actualprobability density functions of parameters are known.

0

(a) PDF of Q and R


Fre

quen

cy o

f occ

uren

ce

0

(b) Overlap region is scaled up


Fre

quen

cy o

f occ

uren

ce

Overlapregion

PDF of Q

PDF of R PDF of R

x x+ dx

P (R < x)

PDF of Q(fQ)

Figure 2.6: Definition of probability of failure (Withman, 1983).

As mentioned above, the probability of failure is related to, but not equal to, the shadedarea, representing the overlap between the load and resistance PDF curves as shown inFigures 2.2(b), 2.3, and 2.5. If the actual load is x, then there is some probability that theactual resistance is less than x, which is indicated by the shaded area in Figure 2.6(b),

22


implying a failure. According to Withman (1983), if load and resistance are independent,the overall contribution to the probability of failure from the event x≤Q≤x + dx is:

P(x ≤ Q ≤ x + dx)P(R < x) = fQ(x)P(R < x)dx (2.14)

Integrating all possible levels of load gives the total probability of failure, Pf , as:

Pf =∫

fQ(x)P(R < x)dx (2.15)

It can be seen that applying Eq. 2.15 to determine the probability of failure is relativelydifficult. For more practical purposes, Limit State Functions (LSF), which correspond tolimit states for significant failure modes, are required. A general form of a limit statefunction for a structural component is given by:

g = R−Q (2.16)

where g is the limit state function; R is the resistance (or strength); and Q is the loadeffect on the structural component. The failure is defined in the domain where g is lessthan zero or R is less than Q, that is: g < 0 or R < Q. Whereas, the safety is defined inthe domain where g is greater than zero or R is greater than Q, that is: g > 0 or R > Q,as shown in Figure 2.7.

Limit state function,

Fre

quen

cy

g = 0

g

Failure domain Safe domain

β.σg

(fg)PDF of

Failure limit

Pf

g

g

X0

Figure 2.7: Definition of limit state function.

The RBD assumes the resistance, R, and the load effects, Q, to be random variables. If Ris greater than Q, there will be a safe domain. However, unless R is greater than Q by alarge amount, there is always a probability that Q may exceed R. This is illustrated by ashaded area in Figure 2.7, where R is really smaller than Q. Because of the variability inboth resistance and load effects, the probability of failure, Pf , can be defined as:

Pf = P(g < 0) = P(R < Q) =∫ 0

−∞fg(x)dx (2.17)

23


The probability of safety (or reliability), Ps, is given by the following expression:

Ps = P(g > 0) = P(R > Q) = 1− Pf (2.18)

The probability of failure can also be expressed conveniently in terms of a reliabilityindex, denoted by β, which represents the number of standard deviations, σg, betweenthe mean of the limit state function, g, and the failure limit g = 0 (see Figure 2.7).

The design of any structural component must provide for adequate safety regardlessof what philosophy of design is used. Reliability and risk measures can be consideredas performance measures, specified as target reliability levels denoted by the target re-liability index, βT. The selected reliability level of a structural component reflects theprobability of failure of that component as listed in Table 2.3. These levels can be setbased on implied levels in the currently used design practice with some calibration, orbased on the cost-benefit analysis.

Table 2.3: Relationship between target reliability index and probability of failure

Target reliability index, βT Probability of failure, Pf2.0 0.02282.5 0.00623.0 0.00133.5 0.00023264.0 0.00003174.5 0.0000034

If the resistance and load effects follow normal distributions, then the probability of fail-ure, Pf , in Eq. 2.15 or in Eq. 2.17 may be conveniently evaluated as:

β =R−Q√σ2

R + σ2Q

(2.19)

Pf = 1−Φ(β) (2.20)

where R and Q are the mean value of the resistance and load effects, respectively; σR andσQ are the standard deviation of the resistance and load effects, respectively; Φ denotesthe standard normal Cumulative Distribution Function (CDF).

In case, both the resistance and load effects are log-normally distributed, then Eq. 2.20 isstill valid, but β is given as:

β =

ln

(RQ

√1+COV2

Q

1+COV2R

)√

ln[(1 + COV2R)(1 + COV2

Q)](2.21)

here COVR and COVQ are the coefficient of variation for the resistance and load effects,respectively.

24

Design approaches 2.4 Reliability methods

In practice, the resistance and load effects are permanently constituted by many quanti-ties which follow different distributions. Therefore, in many cases, applying Eq. 2.19 orEq. 2.21 would become impossible. To solve this problem, we need to use other meansaiming to estimate probability of failure or reliability index through reliability methods,which will be presented in the next section.

The RBD has potential advantages compared to the deterministic method, such as be-ing more realistic, rational, consistent, and widely applicable. Most design parameterspossess a significant statistical uncertainty that is not explicitly considered in the globalfactor of safety. However, lack of the appropriate data to implement reliability based de-sign, these potential theoretical advantages cannot be realized in practical design. Thisis the most basic disadvantage in using the RBD.

2.4 Reliability methods

The LRFD as presented in Subsection 2.3.3 is called the Level I reliability method orsemi-probabilistic method. The Level I reliability method uses PSFs that are reliabilitybased, but the method does not require explicit use of the probabilistic description of thevariables.

The RBD as discussed above requires both Level II and/or Level III reliability methods.Level II reliability methods are based on the moments (mean and variance) of randomvariables, and sometimes, include a linear approximation of non-linear limit state func-tions. Meanwhile, Level III reliability methods use the fully probabilistic characteristicsof the random variables. Some features of the representative Level II and Level III relia-bility methods will be presented below.

2.4.1 Level II reliability methods

Generalization

The reliability of an engineering system may involve multiple variables. In particular, theresistance and load effects may be functions of several other variables. For such cases,the LSF in Eq. 2.16 must be generalized. A general LSF is defined as:

g(X) = g(X1, X2, ..., Xn) (2.22)

where X = (X1, X2, ..., Xn) is a vector of basic variables of the system, and the functiong(X) determines the state of the system.

Geometrically, the LSF, g(X)=0, is a n-dimensional surface that may be called the ”failuresurface”. One side of the failure surface is the safe domain, g(X)>0, whereas the otherside of the failure surface is the failure domain, g(X)<0. Expand the LSF, g(X), in a

25

2.4 Reliability methods Design approaches

Taylor series at a point x∗, which is on the failure surface g(x∗)=0; that is:

g(X1, X2, ..., Xn) = g(x∗1 , x∗2 , ..., x∗n) +n

∑i=1

(Xi − x∗i )(

∂g∂Xi

)∗

+n

∑j=1

n

∑i=1

(Xi − x∗i )(Xj − x∗j )(

∂2g∂Xi∂Xj

)∗+ . . . (2.23)

where the derivatives are evaluated at (x∗1 , x∗2 , ..., x∗n). But g(x∗1 , x∗2 , ..., x∗n)=0 on the failuresurface; therefore,

g(X1, X2, ..., Xn) =n

∑i=1

(Xi − x∗i )(

∂g∂Xi

)∗+

n

∑j=1

n

∑i=1

(Xi − x∗i )(Xj − x∗j )(

∂2g∂Xi∂Xj

)∗+ . . .

(2.24)

Introduce the set of uncorrelated reduced variables (Freudenthal, 1956):

X′i =Xi − µXi

σXi

(2.25)

where X′i is the reduced variable in the reduced coordinate system corresponding tovariable Xi in the regular coordinate system; µXi and σXi are the mean and standarddeviation of variable Xi in the regular coordinate system, respectively.

So,Xi − x∗i = (σXi X

′i + µXi)− (σXi x

′∗i + µXi) = σXi(X′i − x

′∗i ) (2.26)

and (∂g∂Xi

)∗=

(∂g

∂X′i

)∗

(dX′idXi

)∗=

1σXi

(∂g

∂X′i

)∗

(2.27)

Then,

g(X1, X2, ..., Xn) =n

∑i=1

(X′i − x′∗i )

(∂g

∂X′i

)∗+ . . . (2.28)

It can be seen that the LSF, g(X), is now represented through the reduced variables andtheir partial derivatives in the reduced coordinate system.

The First Order Reliability Method (FORM)

In first order approximation, truncate the above series at the first order term, the LSF isnow approximated as:

g(X1, X2, ..., Xn) ≈n

∑i=1

(X′i − x′∗i )

(∂g

∂X′i

)∗

(2.29)

According to Ang and Tang (1990), the mean value, µg, and variance, σ2g , of the LSF are

approximated at x∗:

µg ≈ −n

∑i=1

x′∗i

(∂g

∂X′i

)∗

(2.30)

26


σ2g ≈

n

∑i=1

(∂g

∂X′i

)2

∗(2.31)

From Eqs. 2.30 and 2.31, the ratio of µg to σg is:

µg

σg=−∑n

i=1 x′∗i

(∂g

∂X′i

)∗√

∑ni=1

(∂g

∂X′i

)2

∗

(2.32)

The ratio given by Eq. 2.32 is the distance, d, from the tangent plane of the failure sur-face at x∗ to the origin of the reduced coordinate system. This distance may not be theminimum distance, dmin, from the origin to a point on the failure surface, which wasfirst introduced by Freudenthal (1956) with the name ”most probable failure point” asshown in Figure 2.8. Hasofer and Lind (1974) stated that the position of the failure sur-face may be represented by the minimum distance from the failure surface to the origin,and Shinozuka (1983) also addressed that the point on the failure surface with minimumdistance to the origin is the ”most probable failure point”. Thus, in some approximatesense, this minimum distance may be used as a measure of reliability. In present works,the term ”most probable failure point” is re-named as ”design point” for convenience.

Figure 2.8: Definition of design point on the failure surface.

Obviously, using Eq. 2.32 alone is insufficient to determine the design point, i.e., dmin. Inaddition, variables Xi are assumed to be normal distributions. Therefore, two questionsarising herein: (1) How to determine the design point, and (2) how to transform initialnon-normal distributed variables to normal ones. An iterative procedure proposed byRackwitz and Fiessler (1978) is as follows:

27


• Step 1: In the regular coordinate system, assume a trial design point, x∗i , and in thereduced coordinate system, obtain corresponding point, x

′∗i , using Eq. 2.25:

x′∗i =

x∗i − µXi

σXi

(2.33)

where µXi and σXi are the mean value and the standard deviation of the basic ran-dom variable Xi, respectively. The mean value of basic random variables is oftenused as an initial value for the design point.

• Step 2: If the distribution of basic random variables is non-normal, approximatethis distribution with an equivalent normal distribution at the design point, hav-ing the same tail area and ordinate of the probability density function, that is theRosenblatt transformation (Rosenblatt, 1952) for equivalent mean:

µNXi

= x∗i −Φ−1[FXi(x∗i )]σNXi

(2.34)

and for equivalent standard deviation:

σNXi

=φ{Φ−1[FXi(x∗i )]}

fXi(x∗i )(2.35)

where µNXi

and σNXi

are the mean and standard deviation of the equivalent normaldistribution for variable Xi; FXi(x∗i ) and fXi(x∗i ) are the original Cumulative Distri-bution Function (CDF) and original Probability Density Function (PDF) of variable,Xi, evaluated at the design point x∗i ; Φ and φ are the CDF and PDF of the standardnormal distribution, respectively.

• Step 3: Set x′∗i = −α∗i β, in which α∗i is the direction cosine determined as follows:

α∗i =

(∂g

∂X′i

)∗√

∑ni=1

(∂g

∂X′i

)2

∗

(2.36)

here, (∂g∂X′i

)∗=

(∂g∂Xi

)∗

σNXi

(2.37)

• Step 4: The following equation is solved for β

g[(µNX1− α∗1σN

X1β), (µN

X2− α∗2σN

X2β), ..., (µN

Xn− α∗nσN

Xnβ)] = 0 (2.38)

• Step 5: A new design point obtained,

x∗i = µNXi− α∗i βσN

Xi(2.39)

28


Repeat Steps 1 to 5 until a convergence of β is achieved. This reliability index, β, isthe minimum distance, dmin, from the origin of reduced coordinate system to the failuresurface at the design point. The values of design point in the regular coordinate systemis determined by Step 5. The probability of failure, Pf , is obtained as:

Pf = 1−Φ(β) = Φ(−β) (2.40)

Note that, the iterative procedure described above for evaluating the probability of safetyor failure is based on the assumption that the random variables, Xi, are uncorrelated.For random variables that are correlated, the initial variables may be transformed to aset of uncorrelated variables, the above iterative procedure may then be applied to theuncorrelated set of transformed variables. The transformation process is not presentedherein and can be referred to in Ang and Tang (1990) in more detail.

The First Order Second Moment (FOSM) method

Similar to the FORM, the First Order Second Moment (FOSM) method also uses the firstterm of a Taylor series to estimate the mean value and variance of the LSF, g(X). It iscalled the Second Moment method because the variance is a form of the second momentand is the highest order statistical result used in the analysis. The mean value and vari-ance of the LSF, g(X), can be approximated as:

µg ≈ g(µX1 , µX2 , ..., µXn) (2.41)

and

σ2g ≈

n

∑i=1

n

∑j=1

∂g∂Xi

∂g∂Xj

Cov(Xi, Xj) =n

∑i=1

n

∑j=1

∂g∂Xi

∂g∂Xj

ρXi ,Xj σXi σXj (2.42)

where Cov(Xi, Xj) and ρXi ,Xj are the covariance and correlation coefficient between thepair of variables Xi and Xj, respectively. Once the mean and variance of the LSF, g(X),have been calculated, the reliability index, β, is easily given:

β =µg

σg(2.43)

We can see that the approximations used in the FOSM method may not be acceptable.First, the assumption, that the moments of the LSF, g(X), can be estimated by startingwith the mean values of the variables, is not exact. Second, the probabilistic distributionof g(X) is not known, thus applying Eq. 2.43 directly to determine the reliability indexis not consistent, unless all random variables are clearly known being either normal orlog-normal, i.e., the same as cases using Eqs 2.19 or 2.21 as mentioned in Subsection 2.3.4.

2.4.2 Level III reliability method

Level III reliability methods, such as the numerical integration or Monte Carlo Simula-tion (MCS), use the fully probabilistic characteristics of the random variables. MCS isbest suited for cases in which there are many independent variables or where the LSF is

29


strongly non-linear. Therefore, this subsection is served for describing some importantfeatures of MCS.

Simulation is the process of replicating the real world based on a set of assumptionsand conceived models of reality. It may be performed theoretically or experimentally.In practice, theoretical simulation is usually performed numerically; this has become amuch more practical tool since the advent of computers (Ang and Tang, 1990).

For a LSF involving random variables with known probability distributions, MCS is re-quired. This involves repeating a simulation process, using in each simulation a particu-lar set of values of the random variables generated in accordance with the correspondingprobability distributions. One of the main tasks in a MCS is the generation of randomnumbers from prescribed probability distributions; for a given set of generated randomnumbers, the simulation process is deterministic (Ang and Tang, 1990).

The generation of random numbers can be accomplished for each variable in the LSF byfirst generating a uniformly distributed random number between 0 and 1.0, and throughappropriate transformations obtaining the corresponding random number with the spec-ified probability distribution of that variable. The transformation is as follows (Ang andTang, 1990):

Suppose a random variable X with CDF as FX(x). At a given cumulative probabilityFX(x) = u, the value of X is:

x = F−1X (u) (2.44)

Now suppose that u is a value of the standard uniform variable, U, with a uniform PDFbetween 0 and 1.0; then,

FU(u) = u (2.45)

that is, the cumulative probability of U≤u is equal to u. Therefore, if u is a value of U, thecorresponding value of X obtained through Eq. 2.44 will have a cumulative probability,

P(X ≤ x) = P[F−1X (U) ≤ x] = P[U ≤ FX(x)] = FU [FX(x)] = FX(x) (2.46)

which means that if (u1, u2, ..., un) is a set of values from U, the corresponding set ofvalues obtained through Eq. 2.44, that is,

xi = F−1X (ui) (2.47)

will have the desired CDF FX(x). The relationship between u and x is depicted in Figure2.9.

So, after each simulation a set of values of the random variables, Xi, is generated andtherefore a value of the LSF, g(X), is obtained which may be greater than, or less than,or equal to zero. By repeating this procedure with a large number of simulations, theprobability of failure, Pf , can be estimated as:

Pf ≈n f

ns(2.48)

where ns is the total number of simulations; n f is the number of simulations, for whichg(X)<0.

30

Design approaches 2.5 Conclusions

0u

u

1.0

x XU

FX(X)FU (U)

FU (U);FX(X)

Figure 2.9: Relation between u and x.

A question arising herein, that is, how many simulations are necessary to ensure a de-sired level of accuracy in the results. According to Vrijling and Van Gelder (2002), fora 95% confidence level with a maximum relative error of 0.1, the required number ofsimulations amounts to:

ns ≥ 400(

1Pf− 1)

(2.49)

2.5 Conclusions

In this chapter, a rather detailed development history of design approaches has been pre-sented. The concept and categorization of uncertainty were introduced, with a focus onparameter and model uncertainties. The design approaches including the ASD, the LSD,and the RBD, have been described. The advantages and limitations of each design ap-proach were discussed in detail. Through the development history of design approaches,it can be seen that reliability methods have played an important role aiming to improveand advance the design approaches.

Applying a global factor of safety in the ASD is simple and familiar to engineers in de-sign. However, it does not provide a pertinent measure or indication of level of safety orprobability of failure like the LSD or the RBD.

Limit states are conditions under which a structure no longer performs its intended func-tion consisting of the ULS and SLS. The ULS pertains to structural safety and to a defini-tion of things that are dangerous; they involve a total or partial collapse of the structure,the ultimate bearing capacity, overturning, sliding, and so on. The SLS represents condi-tions which affect the function or service requirements of the structure under expectedservice or working loads. The SLS includes conditions that may restrict the intended useof the structure such as deformation, cracking, excessive total or differential settlement,excessive vibrations, local damage, and deterioration.

31

2.5 Conclusions Design approaches

The LSD is an evolution of the ASD. The nature of the LSD is not in the definition oflimit state conditions, but in how the level of safety is achieved (Becker, 1996). Thelevel of safety may be provided through partial safety factors, which are mainly obtainedthrough expert’s judgments or the calibration process against the ASD.

The concept of LSD with the use of partial safety factors has developed differently inEurope and in North America, mainly in the manner of calculation of factored resistanceat ULS, i.e., the factored strength approach (European) and the factored resistance ap-proach (North American). The factored resistance approach is also called the load andresistance factor design (LRFD).

The LRFD uses partial safety factors, which are load factors and resistance factors giventhrough calibration against the ASD or reliability theory. The use of separate load andresistance factors is logical for the design of foundations, because load effects and resis-tances have largely separate and unrelated sources of uncertainty. Therefore, the resis-tance factor calibration under the framework of the LRFD constitutes the main part ofthis study and will be presented in Chapters 4, 5, and 6 of this thesis.

The RBD is a probabilistic design approach, in which the parameters to calculate loadeffects and resistance are treated as random variables, and the measure of safety is thereliability index, or the probability of failure. The RBD can be computed directly if theactual PDF of the parameters are known. To do this, reliability methods are required. InChapter 7 an example of the application of the RDB will be presented.

The LRFD is called the Level I reliability or semi-probabilistic method. The FORM andFOSM method, which belong to the Level II reliability methods, were introduced basedon the moments (mean and variance) of random variables with a linear approximationof non-linear LSFs. Meanwhile, Level III reliability methods use the fully probabilisticcharacteristics of the random variables; MCS is chosen and introduced in this chapterbecause of its usefulness. The FORM will be used in Chapter 4 and 7; a part of Chapter 4will utilize the FOSM method; and especially the MCS will be used throughout Chapters4, 5, and 6.

32

Chapter 3

Quality control approaches

3.1 Introduction

As introduced in Chapter 1, most bored piles are constructed routinely and without dif-ficulty. However, unexpected defects in a completed bored pile can arise during theconstruction stage through errors in handling of slurry, reinforcing steel cages, concrete,casings, and other factors. Therefore, post-construction testing of bored piles is an im-portant part of the design and construction process.

The most common purpose of post-construction testing is quality assurance of concreteplacement, in which integrity of the production concrete is measured and evaluated.From a management perspective, post-construction tests on completed bored piles canbe categorized into two types as follows (Brown et al., 2010):

• Planned tests that are included as a part of the quality control procedure.

• Unplanned tests that are performed as part of a forensic investigation in responseto observations made by an inspector or constructor that indicate a defect mightexist within a pile.

Planned tests for quality control typically are Non-Destructive Tests (NDT) and are rel-atively inexpensive; such tests are performed routinely on bored piles. Meanwhile, un-planned tests performed as part of a forensic investigation will normally be more time-consuming and expensive. The objective of unplanned tests is to address a potentialproblem and to obtain more information that can be used if a remediation plan is needed.These tests may include both non-destructive and destructive (e.g., coring examination)methods.

The most common NDT methods are the Cross-hole Sonic Logging (CSL) method, theGamma-Gamma Logging (GGL) method, and the Sonic Echo (SE) method. Of thesemethods, the CSL method is currently the most widely used test for quality assurance ofbored pile concrete. Thus, in this chapter, the CSL method is chosen aiming to evaluateits reliability based on the probabilistic approach.

33

3.2 Types of defect Quality control approaches

The outline of this chapter is as follows. Section 3.2 introduces the types of defect thatmay occur in the bored pile with defects. The integrity testing methods to control thequality of bored pile concrete are presented systematically in Section 3.3. Reliabilityevaluation is applied to the Cross-hole Sonic Logging (CSL) method, which is the mostwidely used method in practice. The concepts, the encountered probability, the detec-tion probability, and the inspection probability will be presented and explained in detailin Section 3.4. From that, the recommended number of access tubes used for the CSLmethod is synthesized in Section 3.5. Finally, the chapter ends with conclusions in Sec-tion 3.6.

3.2 Types of defect

Poulos (2005) presented extensively the causes of imperfections in real pile foundations:Natural imperfections caused by geological circumstances and imperfections related tothe construction of the piles. The imperfections that may impact pile foundation perfor-mance may arise from a number of causes, including natural sources, inadequate groundinvestigation, construction, pile load testing, and loading during operation. In this chap-ter, only the influence of construction is considered, other sources can be referred to inChen et al. (1999) and in Poulos (2000).

In general, construction-related imperfections in bored piles can be classified into twomain categories, namely structural defects and geotechnical defects. Structural defectsresult in a reduced size and strength of bored piles than assumed in the design, including:

• ”Necking” of the shaft of bored piles, leading to a reduction of the cross-sectionalarea along part of the pile (see Figure 3.1(a)).

• Poor concrete, ”honeycomb”, or soil inclusion zones, leading to some parts of theshaft to have a lower strength than assumed in the design (see Figure 3.1(b)).

• Tensile cracking of large diameter bored piles caused by the contraction of concreteduring a hydration process, leading to reduced strength and stiffness of parts ofthe piles.

Geotechnical defects usually arise either from a faulty assessment of the in situ condi-tions during the design, or from construction-related problems (Poulos, 2005). The de-crease in shaft and base resistances of bored piles may arise from localized weaker orsofter geotechnical conditions in the vicinity of the pile shaft or pile base, respectively.Brown (2002) and Mullins et al. (2005) supposed that the use of drilling slurry, eithermineral or polymer, may have significant effects on the shaft resistance of bored piles.”Soft bottom” arising from inadequate cleaning of the base of bored piles, is one of themost common concerns in bored pile construction. It leads to a reduction in the stiffnessof the soil below the base of the pile. Therefore, the base resistance may require a largedisplacement to be fully mobilized (Poulos, 2005).

34

Quality control approaches 3.3 Integrity testing methods

(a) Necking (source : www.ulrichengineers.com) (b) Honeycomb (source : www.concrete.org)

Figure 3.1: Some types of structural defect.

3.3 Integrity testing methods

Technologies for testing have been developed and have evolved rapidly over the past 30years, leading to a steady improvement in the quality of bored pile foundations. Due tothe fact that the cost to repair defective bored piles at a later stage in construction may bequite high, it is very cost effective to detect potential problems early on using a suitableintegrity test.

In fact, there have been plenty of integrity testing methods commonly used for boredpiles. However, within the scope of this section, some of the most representative methodsare chosen and presented herein:

• Sonic Echo (SE) method;

• Cross-hole Sonic Logging (CSL) method;

• Drilling and coring examination.

3.3.1 Sonic Echo (SE) method

The SE test procedure was first developed by TNO Dynamics Laboratory in Delft, theNetherlands, as illustrated in Figure 3.2. This method has also been taken into the ASTMstandards, marked as ASTM D5882 (1996). The head of the bored pile is struck with ahand-held hammer. A sonic (compression) wave, which is generated and travels downthe bored pile, is reflected from the base of the shaft (or from a defect within the shaft) asa tension wave, and is picked up by a surface-mounted accelerometer at the head of theshaft.

Changes in impedance, Z, are inferred by identifying the arrival times, direction, andamplitudes of the reflected signals (see Figure 3.2). If there is a defect in the shaft, thevalue of LD obtained from the first reflection will be less than the constructed length, L,of the shaft and will reflect the depth to the defect. The depth at which the defect may

35

3.3 Integrity testing methods Quality control approaches

occur is easily estimated by multiplying the recorded value of time by C/2. Currently,operators of the test use experimental evidence that correlates to the shape of the curvethat is found with various kinds of defects.

The advantage of the SE method is that a test can be performed rapidly and inexpen-sively and without any internal intervention in the shaft (O’Neill and Reese, 1999). Themethod can be useful even if a cap has already been cast over the head of a bored pile(Finno, 1995).

Hammer

2LC

Bored pile

Time

Reflection from base of bored pile

Start of blow

Z = impedance;E = elastic modulus of concrete;A = cross-sectional area of pile;C = stress wave velocity in concrete (approx. 4,000 m/s).

Accelerometer

Test signal D

ispl

.Data acquisition

LWave down

Wave up Z =EAC

Figure 3.2: Scheme of sonic echo method (adapted from Sliwinski and Fleming, 1983).

According to Brown et al. (2010), the SE method has some limitations as follows:

• The strength of the echo depends on the surrounding soil or rock. The interpretermust have access to data of the bored hole and will examine the subsurface condi-tions at the site of the test shaft.

• Echo signals are frequently too weak when the ratios of pile length to diameterexceed approximately 10:1 in rock, 20:1 in stiff or hard soils, 40:1 in medium-stiffsoils, or 60:1 in very soft soils.

• At present, the smallest size of detectable anomalies is approximately 10 percentof the cross-sectional area of bored piles. However, the ability to detect anomaliesstrongly depends on the skill as well as the knowledge of operators and interpreters(Iskander et al., 2001).

• Defects, located below the uppermost major defect, exhibit weaker reflections andthe ability to identify multiple defects is limited; a major defect will make it impos-sible to detect anything below it.

36


• Defects at, or near, the pile base in general cannot be identified because of the un-certainty in the wave speed and the inability to distinguish between a reflectionfrom a normal base and a reflection caused by a defect in the same location.

In summary, the sonic echo test should be considered as a screening method that is ca-pable of locating anomalies covering at least 10 percent of the cross-sectional area. TheSE method based on the analysis of stress wave reflections was used widely in the earlyyears to control the quality of bored piles, but has been gradually replaced by the CSLmethod as a tool for routine quality assurance. However, the method is still useful forinvestigating potential defects in shafts that are not instrumented with access tubes.

3.3.2 Cross-hole Sonic Logging (CSL) method

The CSL method is currently the most widely used test for quality control of bored pileconcrete. This method has also been taken into the ASTM standards, marked as ASTMD6760 (2008). A scheme of the method is shown in Figure 3.3.

Received Signal

Winch

Acoustic profile

Shaft surface

Longitud inal steel bar

Access t ube

Shaft with Four Access Tubes

Electrical Impulse

Transmitter Receiver

Impulse generator

Data acquisition

Shaft suface

Access tube

Figure 3.3: Scheme of cross-hole sonic logging method (adapted from Brown et al., 2010).

Vertical access tubes are cast into the shaft during construction. The tubes are placed in-side the reinforcing steel cage and must be filled with water to facilitate the transmissionof high frequency compressional sonic waves (see Figure 3.4). An acoustic transmit-ter is lowered to the bottom of one access tube and a receiver is lowered to the samedepth in one of the other tubes. The transmitter emits an acoustic impulse at an assignedfrequency usually of 30 to 50 kHz (Brown et al., 2010). The signal travels through theconcrete and is gathered by the receiver. The probe cables are pulled upward simulta-neously so that the transmitter and receiver are always at the same elevation. Logginginvolves measuring and recording the emitted and received signals at specified incre-ments of depth (typically 5 cm). The ability to obtain acoustic profiles between any pairs

37


of tubes makes it possible to characterize the position of an anomaly relative to the cen-terline of the shaft. Analysis of profiles can also provide an idea of the size of a potentialdefect.

Figure 3.4: Access tubes placed inside the reinforcing steel cage (Brown et al., 2010).

Integrity of the concrete within acoustic profiles is evaluated through two test results,signal velocity and signal strength. Measured signal velocity, v, is calculated by:

v =dt

t(3.1)

where dt is the distance from center to center for each pair of access tubes; t is the mea-sured travel time, i.e., First Arrival Time (FAT). Signal strength is typically evaluated interms of relative energy, obtained by integrating the absolute value of the signal ampli-tude for a defined time period.

The measured velocity of the acoustic signal is compared to the theoretical velocity,which is also called the baseline velocity vb. The degree to which the measured veloc-ity deviates from the baseline velocity can be evaluated in terms of a Velocity Reduction(VR), expressed as:

VR =

(1− v

vb

)100% (3.2)

here vb is the baseline velocity; experience demonstrates that normal quality concretewill exhibit a signal velocity which is close to an average value of 4,000 m/s (O’Neill andReese, 1999; Brown et al., 2010). Note that, the signal velocity in water is approximately1,500 m/s and in the air is approximately 300 m/s. The qualitative rating of bored pileconcrete, which is based on the velocity reduction and energy reduction, is classified inTable 3.1.

An example regarding a CSL-tested result for an acoustic profile is shown in Figure 3.5.The left side of the figure respectively indicates the measured velocity and the relativeenergy graphs; the right side is a sonic map or a ”waterfall diagram” representing arrivaltime signals versus measured depths. The left edge of the sonic map represents the FATwith low intensity, i.e., decreased signal strength, at depths of 15 m and 16.6 m. At these

38


Table 3.1: Rating criteria for concrete quality (adapted from Brown et al., 2010)

Velocity reduction (%) Energy reduction (dB) Concrete rating0 ÷ 10 ≤6.0 Good

10 ÷ 20 6.1 ÷ 9.0 Questionable> 20 > 9.0 Poor/defect

No signal None No signal≈60 ≥12 Water

two locations, the velocity reduction and the energy reduction are greater than 20% and9 dB, respectively. According to the criteria in Table 3.1, the anomalies at depths of 15 mand 16.6 m are rated as defects.

Figure 3.5: A CSL result for an acoustic profile (ADCOM, 2011).

The advantage of the CSL method is that it is relatively accurate and relatively low cost(Brown et al., 2010). By using a suitable number of access tubes, the major portion of pile

39


shaft may be inspected. In addition, the testing performance for each acoustic profile isalso relatively rapid at about 500 to 1,000 m a day (Fleming et al., 2009).

Some limitations of the CSL method are:

• Access tubes must be installed prior to concrete placement. The bent tubes maylead to an incorrect result.

• It is difficult to locate defects outside the acoustic profile.

• The test procedure requires some strict rules: The cables have to be held in tensionbefore gathering data; the probes must have to be at the same elevation duringpulling up; always to have the top up access tubes with water associated with longpiles during pulling the probes up; and not pulling up the probes too fast (Williamsand Jones, 2009).

3.3.3 Drilling and Coring examination

As mentioned above, the SE and CSL methods are non-destructive and planned tests.A program of drilling and/or coring examination is the result of significant anomaliesdetected from non-destructive tests or identified by the inspector during placement ofthe concrete. Core sampling provides a direct visual examination of concrete and anopportunity to conduct strength tests on the production concrete. However, drilling andcoring are time-consuming and expensive, and therefore they fall into the category ofunplanned tests.

O’Neill and Reese (1999) noted a problem which arises when drilling or coring is the con-trol of the direction during drilling. The drilled hole sometimes runs out of the side of thebored pile or encounters the reinforcing cage. Therefore, experience and appropriate rigsare required to ensure that the drilling is carried out correctly into the direction intended.Jones and Wu (2005) suggested that drilled holes should be located at a minimum dis-tance of 15 cm away from the reinforcing cage and access tubes to avoid interferencewhile drilling.

There are two types of coring examination: Full-length coring and interface coring. Afull-length coring is simply drilling a hole through the entire length of the pile shaft,while an interface coring is drilling through a pile toe interface (Weltman, 1977). In mostcases, it is not necessary to core the entire length of the shaft. An effective way to employa coring program is to limit core sampling to target zones in the shaft where concretequality is questionable. In zones that are not cored, the quality of concrete can sometimesbe inferred from the drilling rate. Drilling may also reveal defects, for example, if a soil-filled cavity is encountered and the drill drops a significant distance. In addition, coringcan reveal the poor concrete or soft bottom locally at the base of bored piles. Repairingdefects within the pile shaft, or enhancing the strength for the soft soil at the pile base,can easily be performed through these coring holes. Figure 3.6 illustrates the concretecores taken from two bored piles, one in a good condition and another one with defects.

40

Quality control approaches 3.4 Reliability evaluation for CSL method

(a) Shaft in good condition (b) Shaft with defects

Figure 3.6: Concrete cores taken from two bored pile shafts (O’Neill and Reese, 1999).

3.4 Reliability evaluation for CSL method

3.4.1 Problem statement

In this section, the CSL method, the most widely used method in testifying the concretequality of bored piles, is chosen, aiming to evaluate its reliability in use. Li et al. (2005)proposed a methodology to evaluate the reliability of the CSL method, in which theconcept of inspection probability was proposed as a quantitative measure of reliabilityof the CSL method. Then, the encountered probability and the detection probability wereformulated to facilitate the evaluation of the inspection probability.

In order to determine three probabilities as mentioned above, three assumptions wereused in Li et al. (2005): (1) The defect is spherical and randomly located within the pilecross section such that each position is equally likely. (2) The detection probability isdetermined based on that for the crack inspection in the metal as proposed by Yang andTrapp (1975). (3) The sonic waves propagate in straight lines.

However, Amir (2007), based on the practical testing results and studies of other authors,pointed out some differences compared to the assumptions used in Li et al. (2005) asfollows:

1. Li et al. (2005) do not mention the first arrival time (FAT) and the relative energy indiscussing the inspection probability.

2. O’Neill (1991) and Fleming et al. (1992) addressed that defects are mostly located atthe periphery of piles. The widespread usage of the terms ”necking” or ”waisting” indescribing pile integrity attests to this phenomenon.

3. Li et al. (2005) do not mention the effect of defect positions on the encountered prob-ability, and that defect positions are not less and may be more important than its size.

41

3.4 Reliability evaluation for CSL method Quality control approaches

4. The detection probability, that is based on the study by Yang and Trapp (1975) for thecrack inspection in the metal, is not suitable to be applied to the concrete of boredpiles; concrete in pile, unlike metal, is far from homogeneous.

5. The sonic waves actually propagate in curved paths, not in straight lines as illustratedin Li et al. (2005).

Therefore, issues number 2, 3, and 4, as mentioned above, will be resolved in this section.This means that the methodology of this study will follow the one as proposed by Li et al.(2005), but the encountered probability and the detection probability will be determinedby other approaches. The first arrival time (FAT) and the relative energy are the technicalfeatures of the CSL method. It is very hard to incorporate these features into the inspec-tion probability, because they depend on, not only the size and the position of the defect,but also on its characteristic. For simplification purposes, the assumption that the sonicwaves propagate in straight lines is still used in this study.

3.4.2 Number of access tubes

To detect potential defects in a pile by the CSL method, a required number of accesstubes has to be installed prior to the placement of concrete (see Figure 3.4). In Table 3.2,Li et al. (2005) synthesized the recommended number of access tubes for different boredpile diameters according to different authors and technical codes.

Table 3.2: Recommended number of access tubes for different bored pile diameters (syn-thesized by Li et al., 2005)

Pile diameter(mm)

Tijou(1984)

Turner(1997)

O’Neill andReese (1999)

Thasnanipanet al.

(2000)

WorkBureau(2000)

MOC(2003)

600 ÷ 750 2 3 2 2 3 to 4 2750 ÷ 1,000 2 to 3 3 to 4 2 to 3 3 3 to 4 2 to 3

1,000 ÷ 1,500 4 4 to 5 4 to 5 4 3 to 4 31,500 ÷ 2,000 4 4 to 5 5 to 7 6 3 to 4 32,000 ÷ 2,500 4 4 to 5 7 to 8 6 3 to 4 42,500 ÷ 3,000 4 4 to 5 8 8 3 to 4 4

It can be seen that there is a general trend in which the number of access tubes increaseswith the pile diameter, except for Work Bureau (2000). O’Neill and Reese (1999) pre-sented, as a rule of thumb employed by several agencies to determine the number ofaccess tubes, is based on one access tube for each 0.3 m of pile diameter. Li et al. (2005)stated that there exists an inconsistency in the number of access tubes for the same pile di-ameter adopted by the current practice and no probabilistic analysis has been performedto suggest the number of access tubes in a rational manner.

42


Theoretically, the more the number of access tubes, the more precise the CSL measure-ment. However, the overly increasing number of access tubes leads to a higher cost andmay impede the flow of concrete during pile construction. Therefore, a pertinent numberof access tubes to ensure the reliability of CSL measurements corresponding to a targetprobability is very important. Li et al. (2005) recommended a number of access tubeswith a target encountered probability, PE=0.9, for given defect sizes as shown in Table3.3.

Through Table 3.3, it is recognized that the recommended number of access tubes ac-cording to Li et al. (2005) is quite similar to that as recommended by Tijou (1984), WorkBureau (2000), and MOC (2003), but that it is much smaller than the number stated byO’Neill and Reese (1999) and Thasnanipan et al. (2000). However, a remarkable advan-tage in the determination of the number of access tubes is that the authors formulatedin a relatively rational and comprehensive manner, considering both the defect sizes andthe target encountered probability. If the target encountered probability is taken as 0.9,three to four access tubes will be sufficient to encounter defects larger than 5% of thecross-sectional area of the pile with a diameter ranging from 600 to 3,000 mm. Note that,Li et al. (2005) used the target encountered probability instead of the inspection prob-ability aiming to determine the number of access tubes. Furthermore, the shape of thedefect is assumed to be spherical and the defect is equally likely located within the pilecross section. This may lead to an over-prediction of the encountered probability and,therefore, the number of access tubes trends to be small.

Table 3.3: Recommended number of access tubes for given defect sizes as a percentageof pile cross-sectional area, p(%), with target encountered probability, PE=0.9 (extractedfrom Li et al., 2005)

Pile diameter (mm) p=5% p=10% p=15%600 ÷ 750 3 3 3750 ÷ 1,000 3 to 4 3 3

1,000 ÷ 1,500 4 3 31,500 ÷ 2,000 4 3 32,000 ÷ 2,500 4 3 to 4 32,500 ÷ 3,000 4 4 3

3.4.3 Inspection probability

According to Li et al. (2005), the reliability of the CSL method can be described by theinspection probability, which is expressed as a product of the encountered probabilityand the detection probability:

PI(xd) = PE(Ee|xd)PD(Ed|Ee, xd) (3.3)

where PI(xd) is the inspection probability for a given defect size xd; Ee is the event thata defect with a given size xd is encountered; Ed is the event that a defect with a given

43


size xd is detected if it is indeed encountered; PE(Ee|xd) is the encountered probabilitythat a defect is encountered by an inspection of a given inspection plan if a defect indeedexists; and PD(Ed|Ee, xd) is the detection probability that an inspection detects a defect ifa defect is indeed encountered.

In the next subsections, the encountered probability and the detection probability will bepresented on the basis of comparison between the approaches of Li et al. (2005) and theapproaches proposed in this study.

3.4.4 Encountered probability

The approach according to Li et al. (2005)

Figure 3.7: Geometrical diagram determining encountered probability (Li et al., 2005).

Li et al. (2005) assume the defect to be spherical, which is randomly located within thepile cross section such that each position is equally likely. A cross section of pile withtwo access tubes is first taken as an example. A defect with a diameter, xd, is present in apile as shown in Figure 3.7. Theoretically, a CSL test can scan a path between two accesstubes with a width of about 2λw, where λw is the wavelength of the ultrasonic signal.The frequency of the ultrasonic signal, f , can vary from 40 to 60 kHz (O’Neill and Reese,1999). A frequency of 50 kHz is adopted for analysis (Li et al., 2005). The signal velocity,v, is close to an average value of 4,000 m/s (O’Neill and Reese, 1999; Brown et al., 2010).So, the wavelength of the ultrasonic signal is λw=v/ f =0.08 m. Therefore, if the signalpath can encounter a defect, the center of the defect must be located inside the shadedarea. The encountered probability can be taken as the ratio of the shaded area to thecross-sectional area of the reinforcing steel cage.

PE(Ee|xd) =AS

A0(3.4)

44


where AS and A0 are the shaded area and the cross-sectional area of the reinforcing steelcage. Similarly, the shaded area for other numbers of access tubes can be obtained.

Herein, the shaded area is determined when the border of the defect is only tangent withthe edge of the signal path and the shaded area also covers both sides of the signal path.Actually, the shaded area should have covered only one side of the signal path where thedefect is located. Further, Li et al. (2005) suggested that the encountered probability beadopted as an index to determine the required number of access tubes. Obviously, theseassumptions lead to an over-prediction of the encountered probability and therefore thenumber of access tubes tends to be small as mentioned in Subsection 3.4.2.

The approach of this study

In this study, assume that the defects are randomly located at the periphery of piles. Thedefect shape normally is observed with some types, which are the annulus, sector, orcircular segment, as depicted in Figure 3.8(a), (b), and (c), respectively. The possibility ofoccurrence of these types is equally likely. However, it can be seen that the encounteredprobability of the first two types is certainly greater than that of the last type, the circularsegment, because the first two types of defect readily intersect with the signal path asdemonstrated in Figure 3.8(a) and (b). Therefore, for a more conservative purpose, thedefect with the shape of circular segment is chosen as the examined object.

Figure 3.8: The shapes of defect located at the periphery of pile.

Two pictures in Figure 3.9 show a defect in shape of the circular segment. In the firstpicture, the soil occupied a major proportion of the pile shaft, and in the second picturethe defect is exposed after digging the soil out of the pile shape. The defect herein has atypically shape of the circular segment.

Consider a general case, where a pile has nt access tubes installed inside the reinforcingsteel cage as shown in Figure 3.10. A defect, which is indicated by the shaded area, hasa shape of the circular segment at the periphery of pile. The defect is located by thechord EF and its magnitude is represented by the height of circular segment, a. Considertwo adjacent access tubes, i and i + 1, being in the vicinity with the defect. AB is thechord going through the centers of the access tubes i and i + 1. M is the middle pointof the chord AB. The radius ON goes through the middle point, M, and is therefore

45


(a) Soil occupied the pile shaft (b) Shaft with defect in shape of circular segment

Figure 3.9: Pile shaft with defect in shape of circular segment (ADCOM, 2008).

perpendicular to the chord AB.

Figure 3.10: Geometrical diagram determining encountered probability in this study.

The probability of an event that the defect can be encountered by the signal path betweenthe access tubes, i and i + 1, can be determined as a ratio:

PE(Ee|a) =AD

AT(3.5)

where PE(Ee|a) is the encountered probability; AD is the cross-sectional area of the defectindicated by the shaded area in Figure 3.10; AT is the area of the circular segment locatedby the chord AB, i.e., the chord goes through the centers of two adjacent access tubes.

46


The cross-sectional area of the defect, AD, is defined as:

AD =D2

8

{2arccos

(0.5D− a

0.5D

)− sin

[2arccos

(0.5D− a

0.5D

)]}, 150mm ≤ a ≤ MN

(3.6)and the area of the circular segment located by the chord going through the centers oftwo adjacent access tubes, AT, is given:

AT =D2

8

{2arcsin

(AM0.5D

)− sin

[2arcsin

(AM0.5D

)]}(3.7)

in which,

AM =√

MN(D−MN) (3.8)

and,MN = 0.5D− (0.5D− 150)cos

π

nt(3.9)

where D is the pile diameter; the number of 150 in Eq. 3.9 represents the shortest distancein millimeters from the center of the access tube to the pile shaft perimeter.

200 400 600 800 1,0000

0.2

0.4

0.6

0.8

1.0

Magnitude of defect, a (mm)

Enc

ount

ered

pro

babi

lity,

PE

nt=2

nt=3

nt=4

Target PE=0.9

Figure 3.11: Encountered probability for bored pile D=1,000 mm.

Figures 3.11 and 3.12 show the encountered probability for different magnitudes of thedefects with a given number of access tubes for D=1,000 and 2,500 mm bored piles, re-spectively. Figure 3.13 indicates the relationship between the magnitude of the defectsand the number of access tubes for different pile diameters with the target encounteredprobability, PE=0.9. Some comments can be given as follows:

1. The encountered probability increases with the magnitude of the defect. We takebored pile D=1,000 mm in Figure 3.11 as an example. If the number of access tubesis three, the encountered probability increases from 0.34 to 1.0, as the magnitude ofdefect increases from 150 to 325 mm.

47


500 1,000 1,500 2,000 2,5000

0.2

0.4

0.6

0.8

1.0


Enc

ount

ered

pro

babi

lity,

PE

nt=2

nt=3

nt=4

nt=5

nt=6

Target PE=0.9

Figure 3.12: Encountered probability for bored pile D=2,500 mm.

2 3 4 5 6 7 8 90

200

400

600

800

1000

1200

1400

Number of access tubes

Mag

nitu

de o

f def

ect (

mm

)

D=1000 mmD=1500 mmD=2000 mmD=2500 mm

Figure 3.13: Relationship between magnitude of defect and the number of access tubesfor different pile diameters with target encountered probability, PE=0.9.

2. For a given magnitude of the defect and a given encountered probability, a pile witha greater diameter requires a larger number of access tubes to be able to encounterthe same magnitude of the defect. From Figures 3.11 and 3.12, for a defect with amagnitude of 300 mm and a target encountered probability of 0.9, a bored pile D=1,000mm needs 3 access tubes, meanwhile a bored pile D=2,500 mm needs up to 6 accesstubes.

3. For a given pile diameter and a given encountered probability, the magnitude of thedefect that can be encountered decreases as the number of access tubes increases.However, the margin of decrease in the magnitude of the defect diminishes as thenumber of access tubes increases. Bored pile D=2,500 mm in Figure 3.13 is taken as

48


an example. If the target encountered probability is 0.9, the magnitude of the defectthat is encountered gradually decreases from 1,152 down to 202 mm as the number ofaccess tubes increases from 2 up to 9. However, the magnitude of the defect tends tobe tangent with a certain value. This hints that, for a given pile diameter and a givenencountered probability, the required number of access tubes should be limited at acertain value, over which it would be less efficient.

3.4.5 Detection probability

The approach according to Li et al. (2005)

Li et al. (2005) addressed that a defect may go undetected during inspection although itis indeed encountered during a CSL test. Therefore, the detection probability should beevaluated.

Based on experimental results of crack inspection using an ultrasonic method, Yang andTrapp (1975) proposed the following detection probability function:

PD(c) =

0, 0 < c ≤ c1(

c−c1c2−c1

)p, c1 < c ≤ c2

1.0, c > c2

(3.10)

where PD(c) is the detection probability for a given crack size c; c1 is the minimum de-tectable crack size below which a crack cannot be detected; c2 is the detectable crack size,beyond which a crack can be detected with certainty; and p is the parameter. If p isassumed to be one (e.g., Ang and Tang, 1975), then the detection probability is a linearfunction of crack size.

Li et al. (2005) used Eq. 3.10 in combination with the CSL-measured experimental resultsof several authors such as Baker et al. (1993), Hassan and ONeill (1998), Chernauskasand Paikowsky (2000), Iskander et al. (2001), and Sarhan et al. (2002) to design a methoddetermining the detection probability. Li et al. (2005) suggested that the minimum de-tectable defect diameter and the detectable defect diameter with certainty be taken as 200and 300 mm, respectively. Based on the relative position of the defect compared to thesignal path, Li et al. (2005) presented detection probability functions for the general case

PD(xd) =

0, 0 < xd ≤ 200mmxd−200

100 , 200 < xd ≤ 300mm1.0, xd > 300mm

(3.11)

and for the special case given as

PD(xd) =

0, 0 < xd ≤ 133.3mmxd−133.3

66.7 , 133.3 < xd ≤ 200mm1.0, xd > 200mm

(3.12)

The detection probability for two calculation cases using Eqs. 3.11 and 3.12 is demon-strated in Figure 3.14. The detection probability for a general case in which a defect is

49


uniformly distributed over the cross section of the pile is lower than that for a specialcase in which the signal path passes through the center of the defect. The detection prob-ability will be overestimated if only the special case is taken into account (Li et al., 2005).

Note that the CSL-measured experimental results that are used in Li et al. (2005) weregiven for several specific cases, in which, the experimental pile diameter varies onlyfrom 760 mm (e.g., Sarhan et al., 2002) to 914 mm (e.g., Chernauskas and Paikowsky,2000), and the number of access tubes in the experimental piles is only 3 or 4 tubes.There was not any large diameter bored pile being carried out in these experiments.Therefore, it is concluded that: (1) The detection probability proposed by Li et al. (2005)is overestimated. (2) Although Li et al. (2005) have considered the defect size and theposition of the defect compared to the signal path, they have not yet mentioned the effectof the pile diameter as well as the number of access tubes on the detection probability.(3) It is necessary to find a more generalized method aiming to closely determine thedetection probability.

50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1.0

Defect diameter, xd (mm)

Det

ectio

n pr

obab

ility

, PD

General caseSpecial case

Figure 3.14: Comparison between the detection probabilities for a general case and aspecial case (Li et al., 2005).

The approach of this study

Once again, we consider a general case where a pile has nt access tubes installed and adefect indicated by a shaded area has a position as shown in Figure 3.15. Let point H bethe middle point of the chord EF. The segment OL going through the middle point His perpendicular to the chord EF and divides the defect into two equal parts. Therefore,the segment OL can be used as a location segment of the defect position, it represents therelative position of the defect compared to the two adjacent access tubes i and i + 1. Letpoint S be the intersection of the chord EF and the chord AB, and point T be the centerof access tube i. It can be seen that the segment ST represents the length of the secantbetween the defect and the sonic signal path, which is formed from the center to centerof two access tubes i and i + 1. Obviously, when the magnitude or the position of thedefect changes, the secant ST changes correspondingly. This hints that the length of the

50


secant ST can be used as a parameter representing the detection capability of defect withrespect to the CSL method. Thus, we will use the detection length instead of the lengthof the secant.

Let point K be the intersection of the segment OT and the perimeter of the pile. The angleω, determined by the segment OL and the segment OK, is used as a parameter locatingthe defect position; hereafter it is called the location angle of the defect. Since the sym-metric performance of the circular cross section of pile and the access tubes are equallyarranged along the reinforcing cage, the variation of the location angle ω from zero to anangle of π/nt radians is sufficient to describe all positions of the defect compared to thatof the access tubes i and i + 1. This also means that the location segment OL of the defectposition correspondingly rotates from the position of the segment OK to the one of thesegment ON as depicted in Figure 3.15.

Figure 3.15: Geometrical diagram determining detection probability in this study.

The detection length ST can be determined as follows:

Detection length =(0.5D− 150)

[cos π

nt+ sinωsin

(πnt−ω

)]− (0.5D− a)cos

(πnt−ω

)sin(

πnt−ω

)cos(

πnt−ω

)(3.13)

here D is the diameter of pile; a is the magnitude of the defect (see Figure 3.10), a=HL;the remaining parameters are already mentioned above.

Through Eq. 3.13, it can be seen that the detection length ST is now a function of fourparameters which are the pile diameter (D), the magnitude of defect (a), the number ofaccess tubes (nt), and the location angle of the defect (ω).

In order to validate Eq. 3.13, we consider two special cases associated with the location

51


angle, ω=0 and π/nt radians. When ω=0, the location segment OL coincides with thesegment OK, the detection length is now given as:

Detection length =a− 150sin π

nt

(3.14)

It can be seen that the detection length will be significant when a>150 mm, i.e., themagnitude of defect has to be larger than the shortest distance from the center of theaccess tube to the perimeter of the pile. This condition is also true with regard to thedetermination of the encountered probability. See Figures 3.11 and 3.12, the encounteredprobability is determined when, and only when, the magnitude of defect has to be largerthan 150 mm.

When ω=π/nt radians, the location segment OL now coincides with the segment ON,the detection length has a form:

Detection length =(a− 150)cos π

nt+ a− 0.5D

0=

OM + a− 0.5D0

= ±∞ (3.15)

See Eq. 3.15, the detection length = +∞ when OM + a≥0.5D, the defect covers the wholecircular segment located by the chord AB, i.e., a CSL test detects the defect with certainty.Conversely, the detection length = -∞ when OM + a<0.5D, the defect does not cover thecircular segment located by the chord AB, and the chord EF is now parallel to the chordAB, i.e., a CSL test does not completely detect the defect.

0.2 0.4 0.6 0.8 1.0 1.2 1.40

100

200

300

400

500

600

Location angle, ω (radian)

Det

ectio

n le

ngth

, (m

m) Detection threshold=300 mm

ω=0.350 ω=0.985 ω=π/3

Figure 3.16: Variation of detection length with location angle.

For demonstration purposes, Figure 3.16 shows the variation of the detection length withthe location angle of the defect for a given bored pile. Here, the pile has a diameter of1,200 mm, the number of access tubes is assumed as 3, and the magnitude of defect issupposed to be 370 mm. As a result, when the location angle, ω, varies from zero to π/3radians, the detection length gradually increases from a value of 254 mm and reaches

52


a maximum value of 342 mm, and then decreases down to -∞ as the location angle ap-proaches the value of π/3 radians. This means that a CSL test does not completely detectthe defect when ω=π/3 radians, as explained for a special case above.

For a more practical side, we assume that there exists a detection threshold, under whicha CSL test may not detect a defect. In this case, a detection threshold is assigned, forinstance, as 300 mm. In Figure 3.16 we see that when the location angle lies in the rangefrom 0.350 to 0.985 radians, the detection length is greater than or equal to the detectionthreshold. When the location angle lies outside this range, a CSL test may not detectthe defect. This issue hints at a way to determine the detection probability for a givenmagnitude of defect as follows:

PD(Ed|Ee, a) ≈ nD

nω(3.16)

where PD(Ed|Ee, a) is the detection probability; nD is the number of values of ω, forwhich the detection length is greater than or equal to the detection threshold; nω is thetotal number of values of ω being taken from the range of zero to π/nt. Obviously, themore the value of nω, the more precise the detection probability. By this definition, forthe case demonstrated in Figure 3.16, the detection probability for the defect with themagnitude of 370 mm is 0.62 if the detection threshold is assumed as 300 mm.

A question arising herein is, how much is the detection threshold, so that a CSL test re-ally detects defects. Numerous authors presented different detectable defect diameters,based on which a suitable detection threshold may be obtained. Li et al. (2005) synthe-sized the experimental results of authors regarding the defect sizes detected by the CSLmethod as shown in Table 3.4.

Table 3.4: Summary of the minimum detectable defect size and the detectable defect sizewith certainty (synthesized by Li et al., 2005)

Experimental factors

Bakeret al.

(1993)

Hassanand

ONeill(1998)

Chernauskasand

Paikowsky(2000)

Iskanderet al.

(2001)

Sarhanet al.(2002)

. Pile diameter (mm) 882 762 914 900 760

. Number of access tubes 4 3 4 3 or 4 -

. Percentage of detectable de-fect area with certainty (%)

15 16.6 20 10 9

. Percentage of minimum de-tectable defect area (%)

- 10.7 - 5 -

. Detectable defect diameterwith certainty (mm)

342 310 409 285 228

. Minimum detectable defectdiameter (mm)

- 249 - 201 -

In Table 3.4, the minimum detectable defect diameter is 249 mm (e.g., Hassan and ONeill,

53


1998) and 201 mm (e.g., Iskander et al., 2001). Amir (2007) addressed that a defecthalfway between the tubes is hardly detectable, while an identical one located close tothe emitter (or in fact to the receiver) is easily detected. In this study, except the casewhere the location angle ω=π/nt, the defect always has a position close to either theemitter or the receiver, when the defect indeed intersects with the signal path. Amir andAmir (2008) presented detection thresholds with respect to different emitter frequenciesand wavelengths of the ultrasonic signal as shown in Table 3.5.

Table 3.5: Detection threshold of CSL test (Amir and Amir, 2008)

Frequency (kHz) 20 30 50 100Wavelength (mm) 210 140 84 42Detection threshold (mm) 420 280 168 84

In Table 3.5, if a frequency of 50 kHz and a wavelength of 84 mm are adopted, the de-tection threshold is obtained as 168 mm. This detection threshold is clearly smaller thanthat presented above by Hassan and ONeill (1998) and Iskander et al. (2001). For conser-vative purposes, a detection threshold of 200 mm is adopted for this study.

0.5 1.0 1.50

100

200

300

400

500

600(a) Using 2 access tubes


Det

ectio

n le

ngth

, (m

m)

0.5 1.0 1.50

100

200

300

400

500

600(b) Using 3 access tubes


Det

ectio

n le

ngth

, (m

m)

0.5 1.0 1.50

100

200

300

400

500

600(c) Using 4 access tubes


Det

ectio

n le

ngth

, (m

m)

0.5 1.0 1.50

100

200

300

400

500

600(d) Using 5 access tubes


Det

ectio

n le

ngth

, (m

m)

PD=0 P

D=0.38

PD=0.97 P

D=1.0

π/5π/4

π/3π/2

Detection threshold=200 mm

Figure 3.17: Variation of detection length with location angle and number of access tubes.

Figure 3.17 shows the variation of the detection length with the location angle and the

54


number of access tubes. In this example, the pile has a diameter of 1,500 mm; the numberof access tubes is used in turn as 2, 3, 4, and 5 tubes. The detection threshold is assignedto be 200 mm. The magnitude of defect of 320 mm is taken for the evaluation. It canbe seen that if the pile uses only 2 access tubes, the curve of the relationship betweenthe detection length and the location angle lies below the detection threshold line, i.e.,the detection probability is equal to zero for this case (Figure 3.17(a)). Alternatively,the CSL test cannot detect this defect if only 2 access tubes are used. When 3 accesstubes are used, the detection probability is equal to 0.38 (Figure 3.17(b)) and increasesdramatically up to 0.97 when 4 access tubes are used (Figure 3.17(c)). Especially, if 5access tubes are used, the whole curve lies over the detection threshold line, this meansthat the detection probability is now equal to unity and the CSL test detects the defectwith certainty (Figure 3.17(d)).

200 400 600 800 1,0000

0.2

0.4

0.6

0.8

1.0


Det

ectio

n pr

obab

ility

, PD

nt=2

nt=3

nt=4

nt=5

Target PD

=0.9

Figure 3.18: Detection probability for bored pile D=1,500 mm.

Figure 3.18 shows the detection probability for different magnitudes of defect with agiven number of access tubes for a D=1,500 mm bored pile. Some comments can bedrawn as follows:

1. Similarly to the encountered probability, the detection probability increases with themagnitude of defect. In Figure 3.18, if the number of access tubes is three, the detectionprobability increases from zero to 1.0, as the magnitude of defect increases from 311to 443 mm.

2. For a given target detection probability, the magnitude of defect that can be detecteddecreases as the number of access tubes increases. From Figure 3.18, for a target de-tection probability of 0.9, the magnitude of defect that can be detected decreases from690 down to 260 mm as the number of access tubes increases from 2 to 5 tubes.

55


3.4.6 Analysis of inspection probability

The encountered probability and the detection probability are analyzed separately inthe previous subsections for both approaches, the approach of Li et al. (2005) and theapproach proposed in this study. In this subsection, a combination of two probabilitymeasures proposed in this study is considered, aiming to determine the inspection prob-ability using Eq. 3.3.

200 400 600 800 1,000 1,200 1,4000

0.2

0.4

0.6

0.8

1.0


Insp

ectio

n pr

obab

ility

, PI

nt=2

nt=3

nt=4

nt=5

nt=6

Target PI=0.9

Figure 3.19: Inspection probability for bored pile D=2,000 mm.

10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1.0

Magnitude of defect as a percentage of cross−sectional area (%)

Insp

ectio

n pr

obab

ility

, PI

nt=2

nt=3

nt=4

nt=5

nt=6

Target PI=0.9

Figure 3.20: Inspection probability with magnitude of defect as a percentage of cross-sectional area for bored pile D=2,000 mm .

Figure 3.19 shows the inspection probability for different magnitudes of defect with agiven number of access tubes for a D=2,000 mm bored pile. Figure 3.20 also shows the

56


inspection probability for different magnitudes of defect, which is represented as a per-centage of a cross-sectional area of pile, with a given number of access tubes for a D=2,000mm bored pile. Basically, comments with respect to the inspection probability are thesame as those for the encountered probability and the detection probability as discussedin the previous subsections. However, it needs to be emphasized that the inspectionprobability is an important factor, based on which the required number of access tubesis determined.

According to Baker et al. (1993), Hassan and ONeill (1998), and Sarhan et al. (2002); adefect occupying 15% of the pile cross-sectional area could be taken as a minor defect.Therefore, from Figures 3.19 and 3.20, for a D=2,000 mm bored pile, 4 access tubes will besufficient to detect a defect having an occupation of 15% of the pile cross-sectional area(415 mm in magnitude) and satisfy a target inspection probability of 0.9.

For illustrative purposes, two field cases are considered. The first case was conductedby ADCOM (2008), a D=1,400 mm bored pile with 4 arranged access tubes was testedat a bored pile foundation of a collective building in Hanoi, Vietnam. A fatal defectwas detected by the CSL method at a depth of about 3.0 m, and then the constructorexcavated the soil surrounding the pile to the depth of the suspected defect aiming toperform a visually-checked work. As a result, a defect in a typical shape of circularsegment with a magnitude of about 400 mm was exposed (see Figure 3.9). Figure 3.21shows the inspection probability proposed by this study for a bored pile D=1,400 mm,which has the same diameter as that of the pile tested in the field. It can be seen that, fora magnitude of defect of 400 mm, if 3 access tubes are used, the inspection probabilityreaches 0.85. Meanwhile, if 4 access tubes are used, the inspection probability is obtainedas 1.0, i.e., the defect is detected with certainty. This is true with respect to the caseconsidered.

200 400 600 800 1,0000

0.2

0.4

0.6

0.8

1.0


Insp

ectio

n pr

obab

ility

, PI

nt=2

nt=3

nt=4

Target PI=0.9

PI=0.85

Figure 3.21: Inspection probability for bored pile D=1,400 mm.

The second case was performed on the two D=915 mm bored piles using 4 access tubes,built at the National Geotechnical Experimentation Sites at Texas A&M University, the

57


United States (Briaud et al., 2002). Pile No. 8 was placed by a planned defect of cave-intype at the depth of 6.5 m below the pile top; and similarly, pile No. 9 was also placedby a planned defect of cave-in type at the depth of 3.0 m below the pile top. The defectswere created by attaching the sand-filled plastic bags to the reinforcing cage, having anoccupation of 12% and 50% of the cross-sectional area of the pile with respect to piles No.8 and 9, respectively. Two companies A and C were invited to attend and to evaluatethe prediction ability of defect through the CSL method. As a result, both companiesdetected these defects relatively successfully. However, the predicted position error ofdefect by company A was within ±0.6 m; and the predicted position errors of defect bycompany C was within ±0.6 m for pile No. 8, and within ±1.8 m for pile No. 9.

Figure 3.22 shows the inspection probability proposed by this study for a bored pileD=915 mm, which has a diameter the same as that of the two piles tested in the secondcase. It can be seen that, for the 50%-occupying defect, the CSL test easily detects thisdefect with certainty, even with only 2 access tubes used. The 12%-occupying defect,according to the approach proposed in this study, escapes to be detected. Meanwhile,as mentioned above, the 12%-occupying defect was indeed detected by both companiesA and C in the field. Why is there a such difference? This may be explained as follows.In fact, the 12%-occupying defect was planned in a type of virtual circle, so it has adiameter of about 317 mm. Therefore, this defect was relatively easy to be detected bythe CSL test. While the 12%-occupying defect, according to the assumption of this study,has an assumed shape of a circular segment with a magnitude of only 162 mm, its sizeis a half of the actual defect. The CSL test, based on the theory proposed in this study,therefore cannot detect such a small defect. See again Figure 3.22, for a D=915 mm boredpile with 4 access tubes and a target inspection probability of 0.9, the CSL test based onthe approach proposed in this study can only detect a defect that occupies at least 24.6%of the cross-sectional area of the pile with a magnitude of at least 270 mm.

10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1.0

Magnitude of defect as a percentage of cross−sectional area (%)

Insp

ectio

n pr

obab

ility

, PI

nt=2

nt=3

nt=4

Target PI=0.9

12% 24.6% 50%

Figure 3.22: Inspection probability for bored pile D=915 mm.

58

Quality control approaches 3.5 Recommended number of access tubes

3.5 Recommended number of access tubes

Based on the analyses in Section 3.4, it can be seen that the number of access tubes isa very important factor and strongly affects, not only on the measurement results ofthe CSL method, but also the construction costs of bored pile foundations. Particularly,in cases where there is a very large number of bored piles to be used in foundations.Thus, the number of access tubes needs to be addressed pertinently, so that they assuretechnical-economic requirements in the stage of design.

This section is used to synthesize the recommended number of access tubes for differ-ent diameters of bored piles and different magnitudes of defect. The target inspectionprobability is assigned as 0.99. The recommended number of access tubes is indicated inTable 3.6.

It can be seen that the recommended number of access tubes in this study is greater thanthe number proposed by Li et al. (2005), as shown in Table 3.3, with the same magnitudeof defect as a percentage of pile cross-sectional area. However, it is noted that, Li et al.(2005) assumed the defect in shape of circle arranged randomly within the pile cross sec-tion, while the defect in this study is assumed to be a circular segment located at theperiphery of pile. In general, the recommended number of access tubes in this study iscompatible with that as stated by O’Neill and Reese (1999) and Thasnanipan et al. (2000),as shown in Table 3.2. For more convenient purposes, Table 3.7 shows the detectable min-imum magnitude of defect for different pile diameters and different numbers of accesstubes used with respect to a target inspection probability, PI=0.99.

Table 3.6: Recommended number of access tubes according to pile diameters and de-tectable magnitudes of defect as a percentage of cross-sectional area of pile with targettarget inspection probability of 0.99

Pile diameter (mm) p=5% p=10% p=15% p=20% p=30% p=40% p=50%

600 ND(*) ND ND ND ND ND 3750 ND ND ND ND 5 3 2

1,000 ND ND ND 5 3 3 21,200 ND ND 6 4 3 3 21,500 ND 7 5 4 3 3 22,000 ND 5 4 4 3 3 22,500 8 5 4 4 3 3 23,000 7 5 4 4 3 3 2

(*) ND - Not Determined.

From Tables 3.6 and 3.7, some comments can be drawn as follows:

1. For a given pile diameter and the target inspection probability of 0.99, the requirednumber of access tubes decreases, as the magnitude of defect needed to be deter-mined, increases. With respect to the cases denoted by the sign ”ND” in Table 3.6, i.e.,the number of access tubes is not determined due to two reasons. First, the required

59

3.6 Conclusions Quality control approaches

Table 3.7: Detectable minimum magnitudes of defect (in millimeter) according to pilediameters and number of access tubes with target inspection probability of 0.99

Pile diameter (mm) nt=2 nt=3 nt=4 nt=5 nt=6 nt=7 nt=8600 349 293 257 237 225 215 209750 375 305 265 242 228 216 209

1,000 497 324 279 251 234 221 2131,200 596 372 288 257 239 225 2141,500 744 447 324 268 245 229 2182,000 992 571 397 312 262 237 2232,500 1,241 695 469 359 296 257 2323,000 1,489 819 541 406 329 282 251

number of access tubes is too large compared to the cross-sectional area of pile, thisleads to a difficulty in concreting and a considerable decrease in resistance of pile.Second, the defect is too small that cannot be detected by a CSL test, regardless of thenumber of access tubes used.

2. For a given magnitude of defect as a percentage of the cross-sectional area of pile, therequired number of access tubes decreases as the pile diameter increases.

3. For the target inspection probability of 0.99, the detectable minimum magnitude ofdefect decreases with the increase of the number of access tubes to be used. However,the magnitude of defect tends to be tangent with a value of approximately 200 mm,regardless of the pile diameters. This value can be considered as a minimum magni-tude of defect, under which the CSL test cannot detect the defect (see more in Figure3.13).

4. With respect to the pile diameter in the range from 600 to 3,000 mm and the target in-spection probability of 0.99, eight (8) access tubes can be considered as the maximumnumber of access tubes that can be used when the CSL method is required.

5. Through Table 3.7, for a given pile diameter, a suitable number of access tubes can beselected based on the detectable minimum magnitude of defect, if a designer supposesthat this magnitude of defect may adversely affect the safety degree of bored pilefoundations.

3.6 Conclusions

In this chapter the quality control approaches of bored piles have been introduced asan important part of the design and construction process. The post-construction testsinclude planned and unplanned tests, in which planned tests for quality control typicallyare NDT methods. Of these methods, the CSL method, the most widely used methodfor testing the integrity of bored pile concrete, has been chosen aiming to evaluate itsreliability.

60

Quality control approaches 3.6 Conclusions

According to Li et al. (2005), the reliability of the CSL method can be quantitatively eval-uated using a probabilistic analysis procedure. The inspection probability, which is usedas a measurement of reliability for the CSL method, directly depends on the encounteredprobability and the detection probability. The approach of Li et al. (2005) overpredictsthe encountered probability and therefore the required number of access tubes trends tobe small.

Based on the general approach established by Li et al. (2005), a new approach for deter-mining the encountered probability, detection probability, and, therefore, the inspectionprobability of the CSL method, has been formulated in this study. The new approachovercome the limitations in the study of Li et al. (2005) as pointed out by Amir (2007);the advantages of the new approaches consist of: (1) Defects are located at the peripheryof piles based on the discussion of O’Neill (1991) and Fleming et al. (1992). (2) The shapeof the circular segment is used to describe the defect, this assumption creates a moreconservative approach for the CSL method. (3) The detection probability is directly de-termined in terms of the detection length, a new concept is proposed in this study. Thedetection length further includes influential parameters such as the pile diameter, thenumber of access tubes, and the position of the defect. Meanwhile, these parametershave not yet been mentioned in the study of Li et al. (2005).

With a given target inspection probability, the required number of access tubes has beendetermined. The number of access tubes is an important factor, which strongly affectsnot only on the measurement results of the CSL method, but also on the constructioncosts of bored pile foundations. The required number of access tubes proposed in thisstudy is greater than that proposed by Li et al. (2005) and is compatible with that asstated by O’Neill and Reese (1999) and Thasnanipan et al. (2000).

The detectable minimum magnitude of defect decreases with the increase of the numberof access tubes to be used. For the target inspection probability of 0.99, the magnitudeof defect tends to be tangent with a value of approximately 200 mm, regardless of thepile diameters. This value can be considered as a minimum magnitude of defect, underwhich the CSL test cannot detect the defect.

For pile diameters in the range from 600 to 3,000 mm and the target inspection probabilityof 0.99, eight (8) access tubes can be considered as the maximum number of access tubesthat can be used when the CSL method is required. Using a larger number of accesstubes is considered less efficient.

61

3.6 Conclusions Quality control approaches

62

Chapter 4

Calibrating resistance factors

4.1 Introduction

In Chapter 2, the design approaches have been described in detail. In which, the LSDhas been systematically discussed with the key role of the Partial Safety Factors (PSFs).The use of the PSFs aims to deal with uncertainties for loads and resistances. The LSDhas developed differently in Europe and in North America, mainly in the manner of cal-culating designed resistances for the ULS. In the factored strength (European) approach,partial factors are applied directly to only the strength parameters that contribute to theoverall resistance. While in the factored resistance (North American) approach, whichis normally called the LRFD, a common resistance factor is applied to the resistance foreach applicable limit state.

As stated in Chapter 1, the objective of this chapter is to calibrate the resistance factorunder the framework of the LRFD. According to Brown et al. (2010), calibrating commonresistance factors can be carried out by the use of: (1) Judgment; (2) fitting to other codesor past practice; (3) reliability-based analysis; or (4) a combination of approaches. Onlythe third approach, the reliability-based analysis, satisfies the objective of the LRFD andcreates resistance factors to achieve a specified target reliability level. Several compre-hensive works based on the different reliability methods were conducted for the resis-tance factor calibration, for example, FOSM was used by Barker et al. (1991), Yoon andONeill (1997), Withiam et al. (1997), Chaney et al. (2000), McVay et al. (2002), and Kuoet al. (2002); FORM was employed by Honjo et al. (2002) and Paikowsky et al. (2003,2004); and MCS was utilized in the works of Allen (2005), Abu-Farsakh and Yu (2010),and Bach et al. (2012).

In the first part of this chapter, the calibration process of a common resistance factor willbe carried out for sixteen calibration cases of bridge bored pile foundations. In which,each case is represented by a soil type, a resistance prediction method, and a constructionmethod. The resistance factors are calibrated associated with specified target reliabilitylevels and different span lengths of bridges. From the obtained results, the correlationbetween the resistance factor and statistical parameters of the resistance bias factor is for-mulated. In addition, the correlations between resistance factors calibrated by different

63

4.2 Resistance factor calibration using FOSM Calibrating resistance factors

reliability methods are presented and discussed.

The common resistance factor calibrated in the studies mentioned above was appliedto the total resistance of piles. This means that all uncertainties of the shaft and baseresistances were lumped into a unique resistance factor. In fact, uncertainty degrees ofpredicted resistances are very different for the two responses, for example, the construc-tion methods affect the shaft and the base differently. The derived calibration resultsfor the common resistance factor have not fully satisfied the requirement for the designof pile foundations following the LRFD. Therefore, the shaft and base resistance factorsneed to be calibrated separately. This calibration process will be presented in the secondpart of this chapter based on the work of Bach et al. (2013).

The outline of this chapter is as follows. Sections 4.2, 4.3, and 4.4 respectively introducethe reliability methods applied to calibrate a common resistance factor, including FOSM,FORM, and MCS. In the first part, the calibration process of a common resistance factoris presented in Section 4.5. In the second part, the calibration process of separate shaftand base resistance factors and the benefit when using shaft and base resistance factors indesign are described in Section 4.6. Finally, the chapter ends with conclusions in Section4.7.

4.2 Resistance factor calibration using FOSM

In Chapter 2, the LRFD is expressed in the following format:

φRn ≥n

∑i=1

γiQni (4.1)

where φ is the common resistance factor; Rn is the nominal resistance, which is deter-mined depending on the resistance prediction method used; γi is the load factor for theith load component; and Qni is the ith nominal load component. The load factors, γi,are usually greater than one; they account for uncertainties in loads and their probabil-ity of occurrence. The resistance factors, φ, are generally less than one and account forvariabilities in the geotechnical parameters and the resistance prediction method.

Based on FOSM and assumed log-normal distributions for the resistance, Barker et al.(1991) determined the resistance factor, φ, as follows:

φ =λR(∑ γiQni)

√1+COV2

Q

1+COV2R

∑ Qni exp{βT

√ln[(1 + COV2

R)(1 + COV2Q)]}

(4.2)

in which λR is the mean of the resistance bias factor; COVQ and COVR are the coefficientof variation of the load and resistance bias factors, respectively; βT is the target reliabilityindex. When just the dead load, QD, and the live load, QL, are considered, Eq. 4.2 can be

64

Calibrating resistance factors 4.2 Resistance factor calibration using FOSM

rewritten as

φ =λR(

γDQDQL

+ γL)

√1+COV2

QD+COV2QL

1+COV2R

(λQDQD

QL+ λQL) exp{βT

√ln[(1 + COV2

R)(1 + COV2QD + COV2

QL)]}(4.3)

where γD and γL are the dead load and live load factors, respectively; QD/QL is the deadload to live load ratio; λQD and λQL are the mean of dead load and live load bias factors,respectively. The dead load to live load ratio varies with the span length of bridges.These ratios were determined by Hansell and Viest (1971) as indicated in Table 4.1.

Table 4.1: Relationship between dead load to live load ratio and span length of bridges

Span length (m) 9 18 27 36 45 60 75Ratio QD/QL 0.52 1.04 1.56 2.07 2.59 3.46 4.32

The actual loads transferred from the superstructure to the foundations are, by and large,unknown. The load uncertainties are taken, therefore, as those used for the superstruc-ture analysis. The probabilistic characteristics of the dead load, QD, and live load, QL,are assumed to be those used by Nowak (1999) using the load factors and the log-normaldistribution as shown in Table 4.2.

Table 4.2: Load factors and probabilistic characteristics for the dead load and live loadbias factors

Type of load Load factor Mean of bias factor, λQ COVQ DistributionDead load 1.25 1.05 0.10 Log-normalLive load 1.75 1.15 0.20 Log-normal

In Eqs. 4.2 and 4.3, we meet the term ”bias factor”, denoted by letter ”λ”, which isapplied to the resistance, dead load, and live load. This term has been presented inSubsection 2.2.2 in Chapter 2. It is also called the multiplicative model correction f actoraiming to represent the model uncertainty.

For the resistance, the mean, λR, and the coefficient of variation, COVR, of the resis-tance bias factor are computed through the theoretical prediction resistance, RP, andthe measured nominal resistance RM. The theoretical prediction resistance is estimatedaccording to the selected resistance prediction method. While the measured nominalresistance is defined as the load corresponding to a displacement that is equal to 5%diameter of bored piles or the plunging load in static loading tests (O’Neill and Reese,1999), whichever comes first. Paikowsky et al. (2004) evaluated that this criterion pro-vides a reliable and simple failure interpretation. The mean of resistance bias factor isdetermined as follows:

λR =∑N

i=1 λRi

N(4.4)

65

4.3 Resistance factor calibration using FORM Calibrating resistance factors

here λRi = RMi/RPi; in which RPi and RMi are the theoretical prediction resistance andthe measured nominal resistance of ith considered bored pile, respectively; N is the num-ber of considered bored piles in the calibration process. The standard deviation of resis-tance bias factor, σR, is obtained as:

σR =

√∑N

i=1(λRi − λR)2

N − 1(4.5)

Finally, the coefficient of variation of resistance bias factor, COVR, is given as:

COVR =σR

λR(4.6)

4.3 Resistance factor calibration using FORM

FORM is a relatively convenient tool to assess the reliability of a structural component.Based on the judgments of Freudenthal (1956), Hasofer and Lind (1974), and Shinozuka(1983), the current studies using FORM provide a means for calibrating the resistancefactor, φ, and the load factors, γi, for a target reliability index βT.

Figure 4.1: Determination scheme of partial safety factors.

We consider again the procedure of FORM as presented in Subsection 2.4.1 of Chapter 2.In the general case, the LSF has a form:

g(X) = g(X1, X2, ..., Xn) (4.7)

66

Calibrating resistance factors 4.3 Resistance factor calibration using FORM

where X = (X1, X2, ..., Xn) is a vector of basic variables, which represent the resistanceand loads. The target reliability index, βT, is now defined to be the shortest distance fromthe origin of the reduced coordinate system to the failure surface at the design point asindicated in Figure 4.1. The most general format of the calibration process is to applythe partial safety factors, γi, to the basic variables; these factors may be applied to therespective mean values of the basic variables (Ang and Tang, 1990). Thus, the LSF in Eq.4.7 can be rewritten as:

g(γ1µX1 , γ2µX2 , ..., γnµXn) = 0 (4.8)

From Eq. 4.8, γiµXi should be on the failure surface; in particular, it may be at the designpoint. Hence, the required partial safety factors, γi, are given:

γi =x∗iµXi

(4.9)

It can be seen that, the determination of the required partial safety factors is also a prob-lem of determining the design point x∗i . However, the calibration procedure has a minordifference compared to the procedure of reliability calculation. Instead of finding thereliability index as a final result, the target reliability index, βT, is now anticipated. Theiterative computation steps to determine the partial safety factors for a specified targetreliability index, βT, are as follows:

• Step 1: The same Step 1 in Subsection 2.4.1 of Chapter 2.

• Step 2: The same Step 2 in Subsection 2.4.1 of Chapter 2.

• Step 3: x′∗i and βT are known; the direction cosine, α∗i , is obtained:

α∗i =

(∂g

∂X′i

)∗√

∑ni=1

(∂g

∂X′i

)2

∗

(4.10)

here, (∂g∂X′i

)∗=

(∂g∂Xi

)∗

σNXi

(4.11)

• Step 4: A new design point obtained,

x∗i = µNXi− α∗i βTσN

Xi(4.12)

Repeat Steps 1 to 4 until a convergence of all α∗i is achieved. Through Eq. 4.9, the resis-tance factor, φ, and the load factors, γi, are derived:

φ =r∗

µR(4.13)

67

4.4 Resistance factor calibration using MCS (proposed) Calibrating resistance factors

and:

γi =q∗i

µQi

(4.14)

where r∗ and q∗i are the value of the resistance and loads at the design point, respectively.µR and µQi are the mean value of the resistance and loads, respectively. The resistancefactor is generally less than one, whereas the load factors are greater than one.

As specified by Paikowsky et al. (2004), for a given target reliability index and proba-bility distributions of the resistance and loads, the partial safety factors determined byFORM may differ with the failure mode. For this reason, the calibration of the partialsafety factors is to maintain the same values for all loads at different failure modes. Ingeotechnical codes, the resistance factor calibration is performed for a set of load factorsthat were already specified in the structural code as stated in Section 4.2. Therefore, theload factors are fixed, the following algorithm is used to determine the resistance factoronly:

• For a given target reliability index, statistical parameters, and probability distri-butions of loads and resistance variables, compute the loads and resistance at thedesign point using FORM as presented above.

• The resistance factor, φ, can be revised for a given set of load factors, γi, as follows:

φ =∑n

i=1γiq∗iλQi

r∗λR

(4.15)

here λQi and λR are the mean of loads and resistance bias factors, respectively.When just the dead load, QD, and the live load, QL, are considered, Eq. 4.15 can berewritten as:

φ =

γDq∗DλQD

+γLq∗LλQL

r∗λR

(4.16)

in which q∗D and q∗L are the value of the dead load and live load at the design point,respectively. Other quantities, such as γD, γL, λQD, and λQL have been presentedin Section 4.2.

4.4 Resistance factor calibration using MCS (proposed)

Based on the calibration procedure using MCS as recommended by Allen et al. (2005),only the dead load and the live load are considered, the limit state function can be writtenas:

g = RM −QMD −QML (4.17)

where RM, QMD, and QML are the measured nominal resistance, the measured dead loadand the measured live load, respectively. If all measured terms in Eq. 4.17 are converted

68

Calibrating resistance factors 4.5 Calibration of a common resistance factor - Part 1

to predicted terms using the bias factors of the resistance λR, the dead load λQD, and thelive load λQL; Eq. 4.17 can be rewritten as:

g = λRRP − λQDQD − λQLQL (4.18)

Combine Eq. 4.1 with Eq. 4.18, after transformation, the limit state function is nowobtained:

g = QL

(γD

QDQL

+ γL

φλR − λQD

QD

QL− λQL

)(4.19)

where terms in Eq. 4.19 are the same as those mentioned in Section 4.2.

MCS is used to generate random numbers that are needed to independently extrapolatethe cumulative distribution function value for each random variable in the calibrationprocess. There are three random variables which are the bias factors of the resistance,dead load, and live load. The computation steps are as follows:

• Step 1: Assign a target reliability index, βT

• Step 2: Select a trial resistance factor, φ

• Step 3: Generate random numbers for each set of bias factors λR, λQD, and λQL

• Step 4: Define the limit state function, g, as described in Eq. 4.19. Find the numberof cases, at which g ≤ 0. The probability of failure is then computed as:

Pf =count(g ≤ 0)

ns(4.20)

where ns is the total number of simulations, which is determined according to Eq.2.49 in Chapter 2. The corresponding calculated reliability index, β, is then definedas:

β = −Φ−1(Pf ) (4.21)

here Φ−1 is the inverse CDF of the standard normal distribution. If the calculatedreliability index, β, is different from the specified target reliability index, βT, thetrial resistance factor, φ, in Step 2 should be changed and a new iteration needs tobe repeated until |β− βT| ≤ tolerance.

4.5 Calibration of a common resistance factor - Part 1

4.5.1 Database for calibration

A database of axially loaded bored piles is extracted from Report NCHRP 507 (Paikowskyet al., 2004). Sixteen calibration cases are categorized in Table 4.3. In which each caseis represented by the number of considered pile cases, soil type, resistance predictionmethod, and construction method.

69

4.5 Calibration of a common resistance factor - Part 1 Calibrating resistance factors

Table 4.3: Sixteen calibration cases for bored pile foundations

No. No. of piles Soil type Prediction method Construction method(3)

1 12 Sand FHWA(1) Casing2 9 Sand FHWA Slurry3 12 Sand Reese and Wright (1977) Casing4 9 Sand Reese and Wright (1977) Slurry5 13 Clay FHWA Casing6 36 Clay FHWA Dry7 21 Sand + Clay FHWA Casing8 11 Sand + Clay FHWA Dry9 9 Sand + Clay FHWA Slurry10 21 Sand + Clay Reese and Wright (1977) Casing11 11 Sand + Clay Reese and Wright (1977) Dry12 9 Sand + Clay Reese and Wright (1977) Slurry13 46 Rock Carter and Kulhawy (1988) Mixed14 30 Rock Carter and Kulhawy (1988) Dry15 46 Rock IGM(2) Mixed16 30 Rock IGM Dry(1),(2) FHWA method is based on Reese and O’Neill (1988), IGM method is based on O’Neill and Reese

(1999).(3) Construction methods can be referred to O’Neill and Reese (1999).

In order to calibrate resistance factors, statistical parameters and probability distribu-tions of the resistance, dead load, and live load have to be determined. As mentionedin Section 4.2, the statistical parameters and probability distributions of the dead loadand the live load bias factors were already specified by Nowak (1999) as shown in Ta-ble 4.2. Therefore, the next subsection will focus on the determination of the statisticalparameters and probability distributions associated with the resistance bias factors. Toavoid redundancy, hereafter sixteen calibration cases will be denoted briefly by a groupof words. For example, a denotation of ”Sand-RW-Slurry”, i.e., the soil type is the sand,the resistance prediction method is the Reese and Wright method, and the constructionmethod is the slurry method.

4.5.2 Statistical parameters and probability distributions of the resistance biasfactors

Based on the given database, apply Eqs. 4.4, 4.5, and 4.6 to calculate the mean, standarddeviation, and the coefficient of variation of the resistance bias factors, respectively. As-sume that the probability distribution of the resistance bias factors is log-normal. Then,the Kolgomorov-Smirnov test (K-S test) is used to verify the fitness of the empirical cu-mulative distribution function (empirical CDF) against the fitted cumulative distributionfunction (fitted CDF). The K-S test seeks how close the fitted CDF is to the empirical CDFthrough the K-S test statistic DN , which is simply the largest (vertical) distance between

70


the fitted CDF and the empirical CDF graphs across all values of the resistance bias factor.According to Fenton and Griffiths (2008), the K-S test will be satisfactory as the adjustedK-S test statistics, AKS, is less than the critical value CV:

AKS =

(√N + 0.12 +

0.11√N

)DN ≤ CV (4.22)

where N is the number of considered resistance bias factors, it is also the number ofconsidered piles in each calibration case; DN is the K-S test statistic; CV is the criticalvalue which has a value of 1.358, corresponding to a significance level of 5%.

The result of the case of Rock-IGM-Mixed is indicated in Fig. 4.2 for the demonstra-tion purpose. The graphs in Figure 4.2(a) are the histogram and the probability densityfunction (PDF) of the resistance bias factors, which are assumed to follow the log-normaldistribution. The graphs in Figure 4.2(b) are the empirical CDF and the fitted CDF. In thiscase, the adjusted K-S test statistics is determined as 0.562, which is much less than 1.358,and therefore it satisfies the requirement of the K-S test. Hence, the log-normal distribu-tion attributed to the resistance bias factors is acceptable. The results of the remainingfifteen calibration cases also satisfy the K-S test with the log-normal distribution as listedin Table 4.4. The greatest value of the AKS is 0.763 for the case of Sand-FHWA-Slurryand the smallest one is 0.342 for the case of Sand+Clay-FHWA-Casing.

1 2 3 40

0.2

0.4

0.6

0.8

1.0

1.2

Resistance bias factor, λR

Pro

babi

lity

dens

ity

(a) Histogram & PDF

1 2 3 40

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive

prob

abili

ty

(b) Empirical CDF and fitted CDF

Empirical CDFFitted CDF

Log−normal PDF

Histogram

Fitted CDF

Empirical CDF

Figure 4.2: The K-S test for the case of Rock-IGM-Mixed.

Regarding the number of considered pile cases, Fenton and Griffiths (2008) suggestedthat if sufficient data are available, generally, at least 20 observations are needed for eachdata set. If so, there are nine out of the sixteen calibration cases in Table 4.3, that havea number of pile cases of less than 20. Paikowsky et al. (2004) evaluated that, one ofthe major difficulties with respect to the calibration process is the lack of data. There-fore, we consider this problem as a type of uncertainty when using probabilistic analysismethods.

71


Table 4.4: Statistical parameters and K-S test results for resistance bias factors

No. Calibration case No. of piles λR σR COVR DN AKS

1 Sand-FHWA-Casing 12 2.270 1.000 0.441 0.165 0.5952 Sand-FHWA-Slurry 9 1.614 1.122 0.695 0.242 0.7633 Sand-RW-Casing 12 1.650 0.944 0.572 0.117 0.4224 Sand-RW-Slurry 9 1.033 0.718 0.695 0.161 0.5075 Clay-FHWA-Casing 13 0.835 0.399 0.478 0.143 0.5396 Clay-FHWA-Dry 36 0.797 0.296 0.371 0.079 0.4827 Sand+Clay-FHWA-Casing 21 1.039 0.297 0.286 0.072 0.3428 Sand+Clay-FHWA-Dry 11 1.322 0.359 0.272 0.158 0.5499 Sand+Clay-FHWA-Slurry 9 1.288 0.333 0.259 0.209 0.66010 Sand+Clay-RW-Casing 21 0.951 0.325 0.342 0.105 0.49611 Sand+Clay-RW-Dry 11 1.206 0.365 0.303 0.149 0.51712 Sand+Clay-RW-Slurry 9 1.158 0.268 0.231 0.199 0.62813 Rock-CK-Mixed 46 1.229 0.504 0.410 0.108 0.74614 Rock-CK-Dry 30 1.350 0.584 0.433 0.112 0.62615 Rock-IGM-Mixed 46 1.298 0.437 0.337 0.081 0.56216 Rock-IGM-Dry 30 1.400 0.478 0.341 0.082 0.462

Also, the mean of bias factor, λR, represents a different level between the measured nom-inal resistance and the theoretical prediction resistance. If a resistance prediction methodhas a bias factor greater than one, i.e., this method is an underpredicted method, andvice versa, a method is overpredicted when its bias factor is smaller than one. In addi-tion, the coefficient of variation of bias factor, COVR, is an important parameter whichrepresents a level of uncertainty in the modelling of a resistance prediction method. Amethod which has a bias factor of one and a coefficient of variation of zero is a perfectprediction method. In Table 4.4, the FHWA prediction method for sandy soil with thecasing construction (Sand-FHWA-Casing) is the underpredicted method with λR= 2.270.Conversely, the FHWA prediction method used for the clayey soil with the dry con-struction (Clay-FHWA-Dry) becomes the overpredicted method with λR=0.797. It can beseen that, the RW method has COVR=0.231 when it is applied to the sandy+clayey soilwith the slurry construction (Sand+Clay-RW-Slurry), but this method itself would haveCOVR=0.695 for the sandy soil with the same construction method (Sand-RW-Slurry).For a resistance prediction method, the values of λR and COVR will vary in conjunctionwith the quantity as well as the quality of the data collected, i.e., the better data, the moreprecise the resistance bias factor.

4.5.3 Calibrated resistance factors

The calibration process is performed for the sixteen cases using three reliability methodsas FOSM, FORM, and MCS. In order to evaluate the variation of the calibrated resistancefactors, the target reliability indices are assigned as 2.0, 2.5, 3.0, and 3.5 correspondingto the target probabilities of failure as 0.0228, 0.0062, 0.0013, and 0.0002; the dead load

72


to live load ratio, QD/QL, varies from 1.04 to 4.32 corresponding to the span lengths ofbridges varying from 18 to 75 m (see Table 4.1).

The statistical parameters and the probability distributions of the dead load and liveload bias factors are taken from Table 4.2. The statistical parameters of the resistance biasfactors are taken from Table 4.4; the probability distribution of the resistance bias factors,through the K-S test as presented above, is log-normal.

For demonstration and comparison purposes, the case of Sand+Clay-RW-Slurry is se-lected. The results are shown in Figures 4.3, 4.4, and 4.5 according to the used reliabilitymethods as FOSM, FORM, and MCS, respectively. Several general observations can bemade as follows:

1 2 3 4 50

0.2

0.4

0.6

0.8

1.0

Ratio of dead load to live load, QD

/QL

Res

ista

nce

fact

or, φ

βT=2.0

βT=2.5

βT=3.0

βT=3.5

Figure 4.3: Calibrated resistance factors using FOSM

1 2 3 4 50

0.2

0.4

0.6

0.8

1.0


/QL

Res

ista

nce

fact

or, φ

βT=2.0

βT=2.5

βT=3.0

βT=3.5

Figure 4.4: Calibrated resistance factors using FORM

73


1 2 3 4 50

0.2

0.4

0.6

0.8

1.0


/QL

Res

ista

nce

fact

or, φ

βT=2.0

βT=2.5

βT=3.0

βT=3.5

Figure 4.5: Calibrated resistance factors using MCS

• In general, different reliability methods will lead to different resistance factors.

• The greater the target reliability index, the smaller the resistance factor.

• For a target reliability index, the value of the resistance factors gradually decreaseswith the increase of the dead load to live load ratio and reaches a stable value whenthis ratio is greater than 3.0.

One issue arising herein is, which target reliability index will be selected for the practicaldesigns. Based on the review of the studies, the survey of common practice, and theevaluation of several authors, Paikowsky et al. (2004) recommended the use of the targetreliability indices for single pile designs as follows:

• For redundant piles, defined as 5 or more piles per pile cap, the recommendedprobability of failure is 0.0099, corresponding to a target reliability index of 2.33.

• For non-redundant piles, defined as 4 or fewer piles per pile cap, the recommendedprobability of failure is 0.0013, corresponding to a target reliability index of 3.0.

Therefore, the values of the calibrated resistance factors for all the sixteen calibrationcases are synthesized in Table 4.5 with the target reliability indices as 2.5 and 3.0; thedead load to live load ratio is selected herein as 3.46 (i.e., greater than 3.0).

Based on a set of results shown in Table 4.5, we first realize that the correlation betweenthe values of the resistance factor using FORM and MCS is better than that betweenFOSM and FORM or between FOSM and MCS. Second, we can evaluate the reliabilitylevel of a prediction method in a combination of a certain soil type and a certain construc-tion method. If we choose the results using MCS with βT = 2.5 as an example, it can beseen that the FHWA method is consistent for the sandy soil and the casing constructionmethod (φ = 0.89) in the case of Sand-FHWA-Casing, but it is not really suitable for the

74


Table 4.5: Calibrated resistance factors for target reliability indices, βT = 2.5 and 3.0

FOSM FORM MCSCalibration case COVR/λR βT = 2.5 βT = 3.0 βT = 2.5 βT = 3.0 βT = 2.5 βT = 3.0

Sand-FHWA-Casing 0.194 0.82 0.65 0.93 0.76 0.89 0.71Sand-FHWA-Slurry 0.430 0.33 0.23 0.36 0.26 0.34 0.25Sand-RW-Casing 0.347 0.44 0.33 0.49 0.38 0.47 0.36Sand-RW-Slurry 0.673 0.21 0.15 0.23 0.17 0.22 0.16Clay-FHWA-Casing 0.573 0.28 0.22 0.31 0.25 0.30 0.24Clay-FHWA-Dry 0.465 0.34 0.27 0.39 0.33 0.38 0.31Sand+Clay-FHWA-Casing 0.275 0.53 0.45 0.62 0.55 0.61 0.52Sand+Clay-FHWA-Dry 0.206 0.70 0.59 0.82 0.73 0.79 0.69Sand+Clay-FHWA-Slurry 0.201 0.70 0.59 0.83 0.74 0.80 0.70Sand+Clay-RW-Casing 0.359 0.43 0.35 0.50 0.42 0.48 0.40Sand+Clay-RW-Dry 0.251 0.60 0.50 0.69 0.60 0.67 0.58Sand+Clay-RW-Slurry 0.200 0.66 0.57 0.79 0.72 0.78 0.68Rock-CK-Mixed 0.334 0.48 0.38 0.54 0.45 0.53 0.43Rock-CK-Dry 0.321 0.50 0.39 0.56 0.46 0.54 0.44Rock-IGM-Mixed 0.260 0.60 0.49 0.69 0.59 0.67 0.56Rock-IGM-Dry 0.244 0.64 0.52 0.73 0.63 0.71 0.60

slurry construction method (φ = 0.34) in the case of Sand-FHWA-Slurry. Similarly, usingthe IGM method in the case of Rock-IGM-Dry (φ = 0.71) clearly has a higher reliabilitylevel than the use of the CK method for the same conditions of soil and construction(φ = 0.54) in the case of Rock-CK-Dry.

4.5.4 Correlation analyses between calibrated resistance factors and statisti-cal parameters of resistance bias factors

Since the statistical parameters and the probability distributions of the dead load andlive load bias factors are fixed and commonly applied to the sixteen calibration cases;the question is to find the correlation between the calibrated resistance factors and thestatistical parameters of the resistance bias factors. Aside from the statistical parametersof the resistance bias factor such as λR, σR, and COVR that are already mentioned; aratio of COVR/λR is proposed and listed in Table 4.5 aiming to evaluate the correlationbetween the calibrated resistance factors and this ratio.

The correlation analyses between the given resistance factors, φ, versus λR, σR, COVR,and ratio COVR/λR are described in Figures 4.6(a), (b), (c), and (d), respectively. Here,the values of the resistance factor are taken from Table 4.5 when using MCS with the tar-get reliability index of 2.5. The statistical parameters, λR, σR, COVR, and ratio COVR/λR,are taken from Tables 4.4 and 4.5. It can be seen that, the correlations between φ versusλR, σR, and COVR are generally logical. φ increases with the increase of λR; and φ de-creases with the increase of σR and COVR. The correlation coefficients between φ and λR,σR, and COVR are R=0.483, -0.232, and -0.732, respectively. In particular, φ and the ratioCOVR/λR have a good correlation with the correlation coefficient R=-0.942, and a clearlylinear relation with the determination coefficient R2=0.887. To which, φ decreases with

75


0.5 1.0 1.5 2.0 2.50

0.2

0.4

0.6

0.8

1.0(a) φ vs.

Mean of resistance bias factor,

Res

ista

nce

fact

or, φ

Scatter resultsRegression line

0 0.2 0.4 0.6 0.8 1.0 1.20

0.2

0.4

0.6

0.8

1.0

(b) φ vs. σR

Standard deviation of resistance bias factor, σR

Res

ista

nce

fact

or, φ


0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

(c) φ vs. COVR

Coefficient of variation of resistance bias factor, COVR

Res

ista

nce

fact

or, φ


0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1.0

(d) φ vs. ratio COVR/

Ratio COVR

/

Res

ista

nce

fact

or, φ


R=0.483R2=0.234

R=−0.232

R=−0.732 R=−0.942

R2= 0.054

R2= 0.536 R2= 0.887

λR

λR

λR

λR

Figure 4.6: Correlation analyses between calibrated resistance factors and statistical pa-rameters of the resistance bias factors.

the increase of the ratio COVR/λR, and vice versa.

0.5 1.00

0.2

0.4

0.6

0.8

1.0(a) Using FOSM

Ratio COVR

/

Res

ista

nce

fact

or, φ


0.5 1.00

0.2

0.4

0.6

0.8

1.0(b) Using FORM

Ratio COVR

/

Res

ista

nce

fact

or, φ


0.5 1.00

0.2

0.4

0.6

0.8

1.0(c) Using MCS

Ratio COVR

/

Res

ista

nce

fact

or, φ


R2= 0.885R=−0.941 R=−0.938

R2= 0.880R=−0.941

R2= 0.885

λRλRλR

Figure 4.7: Correlation analyses between φ and ratio COVR/λR when using differentreliability methods.

For comparison and evaluation purposes, the correlation analyses between φ and theratio COVR/λR are further considered associated with FOSM and FORM. The values ofφ obtained from FOSM, FORM, and MCS are now taken from Table 4.5 with the targetreliability index of 3.0. The correlation analysis results are described in Figures 4.7(a), (b),and (c) corresponding to the use of FOSM, FORM, and MCS. It can be seen that, FOSM,

76


FORM, and MCS produce the resistance factors which have a good correlation with theratio COVR/λR; the correlation coefficients are correspondingly obtained as R=-0.941, -0.938, and -0.941. The linear relations with the determination coefficients are respectivelyderived as R2=0.885, 0.880, and 0.885.

Scrutinize the values of φ and the values of the ratio COVR/λR in Table 4.5, we rec-ognize that, the calibrated value of φ is closely dependent on the value of COVR/λR,regardless of the values of λR, σR, or COVR. The first example with βT=2.5, the cases ofSand-RW-Casing and Sand+Clay-RW-Casing have the same ratios, COVR/λR=0.347 and0.359, respectively. The obtained values of φ for these two cases are 0.44 and 0.43 (usingFOSM), 0.49 and 0.50 (using FORM), 0.47 and 0.48 (using MCS). The second examplewith βT=3.0, cases of Sand+Clay-RW-Dry and Rock-IGM-Mixed also have the same ra-tios, COVR/λR=0.251 and 0.260, respectively. The obtained values of φ for these twocases are 0.50 and 0.49 (using FOSM), 0.60 and 0.59 (using FORM), 0.58 and 0.56 (usingMCS). So, it can be concluded that: (1) The value of φ is completely decided by the ratioCOVR/λR, the same ratio COVR/λR will result in the same φ; and (2) FORM and MCSproduce the values of φ, which have more or less the same values.

4.5.5 Correlation analyses between resistance factors using different reliabil-ity methods

The correlations between the resistance factors obtained by the different reliability meth-ods need to be checked. These correlations are quantified and expressed in terms of thecorrelation coefficients and functional relations. The values of φ using FOSM, FORM,and MCS are taken from Table 4.5 with the target reliability indices of 2.5 and 3.0. Thecorrelation analysis results are shown in Figure 4.8.

0.5 1.00

0.2

0.4

0.6

0.8

1.0

(a) Correlation φFORM

− φFOSM

φFOSM

φ FO

RM


0.5 1.00

0.2

0.4

0.6

0.8

1.0

(b) Correlation φMCS

− φFOSM

φFOSM

φ MC

S


0.5 1.00

0.2

0.4

0.6

0.8

1.0

(c) Correlation φMCS

− φFORM

φFORM

φ MC

S


R=0.995R2=0.990

R=0.996R2=0.992

R=0.999R2=0.998

φMCS

=φFORM

φFORM

=φFOSM

φMCS

=φFOSM

Figure 4.8: Correlation analyses between resistance factors using different reliabilitymethods.

Based on the results described in Figure 4.8, some comments are drawn as follows:

• The correlation coefficients between the resistance factors for couples of FORM-FOSM, MCS-FOSM, and MCS-FORM are R=0.995, 0.996, and 0.999, respectively.These three reliability methods produce the resistance factors which have good

77

4.6 Calibration of shaft and base resistance factors - Part 2 Calibrating resistance factors

correlations. It can be seen that, the resistance factors obtained from MCS andFORM have a somewhat better correlation than those between FORM and FOSM,or between MCS and FOSM.

• Based on the regression analyses, FORM produces resistance factors somewhatgreater than those obtained by FOSM or MCS. The functional relations betweenthe resistance factors when using FOSM, FORM, and MCS are given as:

φFORM ≈ 1.19φFOSM (4.23)

φMCS ≈ 1.16φFOSM (4.24)

φMCS ≈ 0.96φFORM (4.25)

4.5.6 Validation of code calibration

A procedure to rationally determine partial safety factors in the design verification equa-tions based on the reliability methods, is termed code calibration (Honjo and Amatya,2005). So, it is necessary to address the validation of code calibration using MCS as pro-posed in this chapter.

The code calibration using FOSM was first proposed by Barker et al. (1991), which pro-duced resistance factors for the AASHTO bridge foundation code published in 1994(AASHTO, 1994). Then, the resistance factors on the deep foundations in the AASHTOspecification were revised by Paikowsky et al. (2004). In which a large database wasanalyzed by FORM; the obtained resistance factors from the calibration process, in com-bination with the fitting to the ASD, have been used in the AASHTO bridge foundationcode published in 2007 (AASHTO, 2007). Hence, the code calibrations using FOSM andFORM have been confirmed and used to calibrate resistance factors for the LRFD bridgedesign specifications.

Based on the correlation analyses presented in Subsection 4.5.5, it can be confirmed thatthe code calibration using MCS as proposed in Section 4.4 is acceptable. In which, thecorrelation coefficients between the resistance factors obtained from the code calibrationsusing MCS and FOSM, MCS and FORM, are 0.996 and 0.999, respectively.

4.6 Calibration of shaft and base resistance factors - Part 2

The calibration process presented in Section 4.5 is to calibrate a common resistance factor.This means that all uncertainties of shaft and base resistances were lumped into a uniqueresistance factor. Due to the fact that, for example, uncertainty degrees of predictedresistances are very different for the two responses, the construction methods affect theshaft and the base differently. Therefore, the derived calibration results for a commonresistance factor have not fully taken advantage of the LRFD, as opposed to when shaftand base resistance factors are calibrated separately.

78

Calibrating resistance factors 4.6 Calibration of shaft and base resistance factors - Part 2

In this section, a new procedure to calibrate shaft and base resistance factors separatelyis formulated. A data set, which comprises 26 bored piles tested by the Osterberg cell (O-cell) method, is used to calibrate resistance factors separately for the shaft and the baseresponses. The O-cell measured resistances and the theoretical prediction resistancesare compared through the resistance bias factors. MCS is used to calibrate shaft andbase resistance factors separately in the framework of the LRFD, aiming to satisfy targetreliability indices. The benefit from the use of separate shaft and base resistance factorsare then presented.

4.6.1 Shaft and base resistance factor calibration using MCS

In case of separating the shaft and base resistances from a total resistance Rn, Eq. 4.1 canbe rewritten as:

φSRS + φBRB ≥n

∑i=1

γiQni (4.26)

here φS and φB are the shaft and base resistance factors, respectively; RS and RB are thenominal shaft and base resistances, respectively.

Based on the calibration procedure using MCS as presented in Section 4.4, only the deadload and the live load are considered, the limit state function can be written:

g = RMS + RMB −QMD −QML (4.27)

where RMS and RMB are the measured nominal shaft and base resistances, respectively;QMD and QML are the measured dead and live loads, respectively. If all measured termsin Eq. 4.27 are converted to predicted terms using the bias factors of the shaft resistanceλRS, the base resistance λRB, the dead load λQD, and the live load λQL; Eq. 4.27 can berewritten as:

g = λRSRS + λRBRB − λQDQD − λQLQL (4.28)

Combine Eq. 4.26 with Eq. 4.28, after transformation, the limit state function is nowobtained:

g = QL

(

γDQDQL

+ γL

) (λRB

RBRS

+ λRS

)φB

RBRS

+ φS− λQD

QD

QL− λQL

(4.29)

Similar to the iteration procedure presented in Section 4.4, in this case there are four ran-dom variables, which are the bias factors of the shaft resistance, base resistance, deadload, and live load. The computation steps aiming to calibrate the shaft and base resis-tance factors are as follows:


• Step 2: Select a trial shaft resistance factor, φS

• Step 3: Select a trial base resistance factor, φB

79


• Step 4: Generate random numbers for each set of bias factors λRS, λRB, λQD, andλQL


Pf =count(g ≤ 0)

ns(4.30)


β = −Φ−1(Pf ) (4.31)

here Φ−1 is the inverse CDF of the standard normal distribution. If the calculatedreliability index, β, is different from the specified target reliability index, βT, thetrial base resistance factor, φB, in Step 3 should be changed and a new iterationneeds to be repeated until |β− βT| ≤ tolerance. Repeat Step 3 to Step 5 for the nextset of trial shaft resistance factor.

• Step 6: From Step 2 to Step 5, a series of couples of values for φS and φB may be de-rived that satisfies the specified target reliability index βT. One issue arising hereinis, which couple should be selected as the best result of the calibration process?This will be analyzed in Subsection 4.6.5.

4.6.2 Osterberg cell test (O-cell test)

The high capacity of large diameter bored piles in combination with the high cost andthe large size required for reaction systems sometimes make head-down compressivetests too costly or impractical (Figure 4.9(a)). The O-cell test often provides high capacitytesting at a more affordable cost, and therefore has become an attractive alternative fortesting bored piles. The O-cell is a hydraulic jack placed within the shaft (sometimes ator near the pile bottom) to induce upward and downward vertical loads (Figure 4.9(c)).

For the case where the O-cell is placed at the pile bottom, as the cell is pressurized, thebottom of the cell moves downward, causing a reaction force at the lower plate of the cell(base resistance, RB), while the upper plate of the cell moves upward, mobilizing shearforces along the pile shaft (shaft resistance, RS). They are measured separately from eachother (Figure 4.9(d)).

An equivalent load-displacement curve as head-down compressive tests (Figure 4.9(b))can be obtained by using test results from the O-cell tests. At an arbitrary displacement,di, there is an upward load, Qui, and a downward load Qdi. An equivalent load at the topof pile is the sum of the upward load and the downward load. Note that, the buoyantweight of pile must be subtracted from the measured upward load in order to obtainthe shaft resistance values. In case the slenderness ratio of the shaft is larger than 20;the displacement, di, needs to be added to the elastic shortening, corresponding to theaxial load in the pile (Russo et al., 2003). If the displacement, di, being considered is

80


Figure 4.9: Comparison of head-down compressive test and O-cell test.

beyond the measured maximum displacement of any resistance component (Dumax orDdmax) and does not correspond to the ultimate resistance value, the corresponding loadis conservatively assumed constant at the value of the maximum load applied (Qmax).

4.6.3 Data set for calibration

A data set of 26 O-cell equipped bored piles was collected from the Reports compiled byLAW/CRANDALL (1994a,b,c) for the remediation Project for the Los Angeles MemorialColiseum after the Northridge earthquake in Los Angeles in January 1994. The O-cellgraphs were analyzed by Fellenius (2011) and Schmertmann and Schmertmann (2012).The piles are 1.32 m in diameter and 16.5 through 20.1 m long. They were drilled through7.2 to 10.5 m of artificial fill consisting of sand and silt and into the underlying naturalsand with gravel. The mineral slurry construction method was used to keep the walls ofthe shaft stable during drilling.

The measured shaft and base resistances through the 26 O-cell testing bored piles will beused to determine shaft and base resistance bias factors in Subsection 4.6.4.

Figure 4.10: Overview of the Los Angeles coliseum (www.wildnatureimages.com).

81


4.6.4 Statistical parameters and probability distributions of the shaft and baseresistance bias factors

As presented in Section 4.2, shaft and base resistance bias factors are determined bymeans of the ratios of the measured nominal resistances to the predicted resistances atthe shaft and the base of piles, respectively.

The measured nominal shaft resistance is defined as the load corresponding to a dis-placement of 20 mm for sand or about 5 to 15 mm for clays (Osterberg, 2001) or theplunging load in the O-cell tests, whichever comes first. Similarly, the measured nomi-nal base resistance is defined as the load corresponding to a displacement that is equalto 5% of the pile diameter (O’Neill and Reese, 1999; Reese et al., 2006). Paikowsky et al.(2004) concluded that this criterion provides a reliable and simple failure interpretation.In case the nominal displacements exceed the measured maximum displacements, theload-displacement curve of the O-cell tests will be extrapolated to the nominal displace-ments using the stability plot method developed by Chin and Vail (1973) and discussedby Neely (1991).

The predicted resistance is calculated using the theoretical prediction method based onspecific ground conditions. At the remediation project for the Los Angeles MemorialColiseum, the soil condition is mainly sand. Therefore, the applied resistance predictionmethod is the FHWA method (O’Neill and Reese, 1999) for the sandy soil.

For demonstration purposes, the load-displacement curves of the O-cell test for pile 10Bis depicted in Figure 4.11. Based on the load-displacement curves, the measured nominalshaft resistance is defined as 5,801 kN corresponding to the upward displacement of 20mm. After subtracting the self-weigh of the pile, the remaining measured nominal shaftresistance is 5,259 kN. The downward load-displacement curve for the base stopped atthe displacement of 49.3 mm, which is less than the nominal displacement for the baseto be 65 mm. Therefore, this curve is extrapolated to the displacement of 65 mm. As aresult, the ”measured” nominal base resistance after extrapolation is 6,855 kN.

0 2000 4000 6000 8000 10000−80

−60

−40

−20

0

20

40

Load (kN)

Dis

plac

emen

t (m

m)

Upward load−displacement curveDownward load−displacement curve

25.6 mm

Plunging occurs

The curve needs to be extraplated

Nominal displacement for base 5%D=65 mm

Figure 4.11: Load-displacement curves of the O-cell test for pile 10B.

82


Similarly, the load-displacement curves of the O-cell test for pile 23B is depicted in Fig-ure 4.12. The ”measured” nominal shaft resistance, after extrapolation to the nominaldisplacement for the shaft of 20 mm, is obtained as 10,049 kN. Note that the self-weightof the pile must be also subtracted from the ”measured” nominal shaft resistance; theremaining ”measured” nominal shaft resistance, therefore, is 9,458 kN. The measurednominal base resistance of this pile is easily determined as 8,243 kN.

0 2000 4000 6000 8000 10000−80

−60

−40

−20

0

20

40

Load (kN)

Dis

plac

emen

t (m

m)


Norminal displacement for base 5%D= 65 mm

Norminal displacement for shaft 20 mm

The curve needs to be extrapolated

Figure 4.12: Load-displacement curves of the O-cell test for pile 23B.

The statistical parameters of the shaft and base resistance bias factors for the 26 boredpiles are determined based on Eqs. 4.4, 4.5, and 4.6. The detailed calculation results arepresented in Appendix A. The probability distributions of the shaft and base resistancebias factors are checked by the K-S test. The obtained results are indicated in Table 4.6.

Table 4.6: Statistical parameters and probability distributions of the shaft and base resis-tance bias factors for 26 bored piles

Type of resistance Mean of bias factor, λR σR COVR DistributionShaft resistance 1.138 0.299 0.263 Log-normalBase resistance 1.945 0.682 0.351 Log-normal

From Table 4.6, it can be seen that the FHWA method is much underestimated whencalculating the base resistance of the 26 bored piles with the mean of the base resistancebias factor as 1.945; while the predicted shaft resistances are relatively suitable with the”measured” nominal shaft resistances with the mean of the shaft resistance bias factor tobe 1.138.

83


4.6.5 Calibrated shaft and base resistance factors

The limit state function, g, in Eq. 4.29 is used to calibrate shaft and base resistance factorswith a procedure including the six steps as described in Subsection 4.6.1. Target reliabil-ity indices are assigned as βT=2.5 and 3.0; which correspond to the target probabilities offailure as Pf =0.0062 and 0.0013.

Since the Los Angeles Memorial Coliseum has the structure of a stadium, therefore thestatistical parameters and the probability distributions of the dead load and live load biasfactors are taken and applied to building structures as indicated in Table 4.7. The ratio ofthe dead load to live load, QD/QL, is selected as 2.5 based on the study by Ellingwoodand Galambos (1982).

Table 4.7: Load factors and probabilistic characteristics for the dead load and live loadbias factors (Ellingwood and Galambos, 1982)

Type of load Load factor Mean of bias factor, λQ COVQ DistributionDead load 1.20 1.05 0.10 NormalLive load 1.60 1.00 0.25 Type I

The statistical parameters of the shaft and base resistance bias factors are taken fromTable 4.6; the probability distributions for both are log-normal. The average ratio of thepredicted base resistance to the predicted shaft resistance, RB/RS, is given as 0.438.

Figure 4.13 demonstrates the correlation between the calibrated shaft resistance factor,φS, and the calibrated base resistance factor, φB, with respect to target reliability indices as2.5 and 3.0. It can be seen that there are numerous couples of values of φS and φB derivedthat satisfy the target reliability indices. An inverse linear relation between φS and φB isobserved; φS increases when φB decreases, and vice versa. One issue arising herein is,which couple of values of φS and φB should be selected to provide the most consistentresult for the calibration process. Based on the finding regarding the correlation betweenthe resistance factor, φ, and the ratio COVR/λR as presented in Subsection 4.5.4, i.e., thiscorrelation is linear, the larger the ratio COVR/λR, the smaller the resistance factor, andvice versa. Based on which, a couple of φS and φB should be selected when the ratio ofφS to φB approximates a ratio, temporarily called ”Correlation Ratio” - CR, as follows:

φS

φB≈ CR =

COVRBλRB

COVRSλRS

=COVRBλRS

COVRSλRB(4.32)

where λRS and λRB are the mean of the shaft and base resistance bias factors, respectively;COVRS and COVRB are the coefficient of variation of the shaft and base resistance biasfactors, respectively. These parameters are taken from Table 4.6.

Apply Eq. 4.32 to the data in Table 4.6, the ratio, CR, derived in this case is 0.78. Thevalues of φS and φB, therefore, are 0.93 and 1.20 for βT= 2.5, and 0.83 and 1.07 for βT= 3.0,respectively. Here, we recognize that the values of φB are quite large and even greater

84


0.5 1.0 1.5 2.0 2.50

0.5

1.0

1.5

Base resistance factor, φB

Sha

ft re

sist

ance

fact

or, φ

S

βT = 2.5

βT = 3.0

0.83

0.93

1.201.07

Figure 4.13: Correlation between φS and φB.

than one. This can be explained, because the mean of the base resistance bias factor islarge with the value of 1.945. Alternatively, the FHWA method is much underestimatedwhen calculating the base resistance for the 26 bored piles. Therefore, a large value per-tained to the base resistance factor, in this specific case, is consistent.

4.6.6 Sensitivity analyses

In order to independently evaluate the influence of the ratios, QD/QL and RB/RS, onthe value of the shaft and base resistance factors according to Eq. 4.29, it is necessary toperform sensitivity analyses. First, the ratio RB/RS is fixed with the value of 0.438, andthe ratio QD/QL is varied with values of 1.0, 1.5, 2.0, 2.5 and 3.0; the variation of φS andφB with the ratio QD/QL is observed. Then, the ratio RB/RS is taken as 0.25, 0.438, 0.5,0.75 and 1.0, while the ratio QD/QL is fixed as 2.5. The sensitivity analysis results aredescribed in Figure 4.14.

In Figure 4.14(a), when the ratio RB/RS is fixed as 0.438, the resistance factors, φS andφB, slightly decrease with the increase of ratio QD/QL and reach a stable value when thisratio lies in a range from 2.5 to 3.0. In Figure 4.14(b), the ratio QD/QL is fixed as 2.5,and φS and φB, with an inverse tendency, slightly increasing with the increase of the ratioRB/RS, and also reaching a stable value as this ratio lies in a range from 0.5 to 0.75. Afterthat, the resistance factors have a tendency of a slight decrease when the ratio RB/RS islarger than 0.75.

Based on the obtained results, it can be concluded that the shaft and base resistancefactors are less sensitive with respect to the ratio QD/QL as well as the ratio RB/RS.Therefore, the proposed limit state function, g, as expressed in Eq. 4.29, assures thestability for the calibration of shaft and base resistance factors.

85


0.5 1.0 1.5 2.0 2.5 3.0 3.50.5

0.75

1.0

1.25

1.5

(a) Ratio RB/R

S is fixed as 0.438

Ratio QD

/QL

Res

ista

nce

fact

ors,

φS a

nd φ

B

0 0.2 0.4 0.6 0.8 1.0 1.20.5

0.75

1.0

1.25

1.5

(b) Ratio QD/Q

L is fixed as 2.50

Ratio RB/R

S

Res

ista

nce

fact

ors,

φS a

nd φ

B

φS,β

T=2.5 φ

B,β

T=2.5 φ

S,β

T=3.0 φ

B,β

T=3.0

Figure 4.14: Sensitivity analyses associated with shaft and base resistance factors.

4.6.7 Benefit of using shaft and base resistance factors compared to using acommon resistance factor

Currently, studies have mainly focused on calibrating a unique resistance factor. Thereare two reasons leading to this selection. First, the data of measured resistances arecollected from different types of testing, i.e., from the O-cell tests and/or from the head-down compressive tests. Normally, one would transform all the O-cell test results tothose under the head-down compressive tests. Second, until now there has not yet beenan approach which is able to harmonically deal with the correlation between shaft andbase resistance factors for the calibration. As a result, the calibration for a common resis-tance factor still continues to be carried out. Note that in AASHTO (2007), the shaft andbase resistance factors have already been used separately for different resistance predic-tion methods. However, the value of these factors were primarily obtained from experts’judgments and/or fitted to the ASD.

In order to evaluate the difference in calculation results of the total resistance, two cal-culation approaches are considered, one using a common resistance factor, and anotherusing shaft and base resistance factors. First of all, the 26 O-cell test results are trans-formed to the equivalent head-down compressive test results using the transformationmanner as presented in Subsection 4.6.2. For demonstration, pile 4A is selected as an ex-ample. The O-cell test graphs are depicted in Figure 4.15(a); the equivalent head-downcompressive test graph, obtained from the O-cell test after transformation, is shown inFigure 4.15(b).

In case of using the equivalent head-down compressive tests, the statistical parameters ofthe total resistance bias factors for the 26 bored piles are now: λR=1.101 and COVR=0.184.The probability distribution of the total resistance bias factors is log-normal under the K-S test. Based on the calibration procedure using MCS described in Section 4.4, the com-mon resistance factors calibrated are 0.84 and 0.76, corresponding to the target reliabilityindices, βT=2.5 and 3.0.

For comparison, pile 4A is again taken as an example; the nominal shaft and base re-

86


0 2000 4000 6000 8000 10000−40

−30

−20

−10

0

10

20(a) O−cell test

Load (kN)

Dis

plac

emen

t (m

m)


0 5000 10000 15000 20000

0

10

20

30

40

50

(b) Equivalent head−down compessive test

Load (kN)

Dis

plac

emen

t (m

m)

Figure 4.15: Load-displacement curves by O-cell and equivalent head-down compressivetests for pile 4A.

sistances calculated according to the FHWA method are 9,692 kN and 3,969 kN, re-spectively. If we use a common resistance factor, the factored total resistance is givenas: (9, 692 + 3, 969)x0.76 = 10, 382 kN (with βT=3.0). Meanwhile, if we use the shaftand base resistance factors separately, the factored total resistance is now: 9, 692x0.83 +3, 969x1.07 = 12, 291 kN (with βT =3.0). The factored total resistance using separate re-sistance factors for pile 4A is greater than that using a common resistance factor by anamount of 1,909 kN. Therefore, the use of the separate shaft and base resistance factorsmay achieve a more economical design. For a more comprehensive view, the factoredtotal resistance results applying a common resistance factor and separate shaft and baseresistance factors for all the 26 bored piles are demonstrated in Figure 4.16. The detailedcalculation results of the factored resistance of the 26 bored piles are presented in Ap-pendix A.

4A 4B 5A 5B 6A 6B 7A 8A 8B 9A 10A 10B 19A 19B 20A 20B 21A 21B 22A 22B 23A 23B 24A 24B 25A 25B0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

4

Pile name

Fac

tore

d to

tal r

esis

tanc

e (k

N)

Using a common resistance factorUsing shaft and base resistance factors

Figure 4.16: Factored total resistance results applying a common resistance factor andseparate shaft and base resistance factors for 26 bored piles with βT=3.0.

87


4.6.8 Regarding regional calibration

Assume that it is necessary to calibrate a common resistance factor for bridge bored pilefoundations placed at the site of the Los Angeles Memorial Coliseum. Once again, thecalibration process of a common resistance factor is applied for this case study, aiming tocompare the obtained resistance factor with that of the calibration case of Sand-FHWA-Slurry in Section 4.5.

Table 4.8: Comparison of calibrated resistance factors of calibration case in Section 4.5with those of this case study

φ using MCS

Calibration case λR COVR COVR/λR βT = 2.5 βT = 3.0 Source of data

Sand-FHWA-Slurry 1.614 0.695 0.430 0.34 0.25 Paikowsky et al. (2004)Sand-FHWA-Slurry 1.101 0.184 0.167 0.83 0.74 This case study

Both cases have the same condition of soil type, resistance prediction method, and con-struction method. Note that, the statistical parameters and the probability distributionsof the dead load and live load bias factors are now taken from Table 4.2 (Nowak, 1999).The statistical parameters of the total resistance bias factors and the calibrated resistancefactors for both cases are shown in Table 4.8.

It can be seen that, although the same calibration case (i.e., Sand-FHWA-Slurry) and thesame calibration procedure using MCS, there is a big difference between the values ofthe common resistance factor given from this case study and those from the databaseof Paikowsky et al. (2004). In this case study, the common resistance factor are givenas 0.83 and 0.74 with βT=2.5 and 3.0, respectively. While, based on the database ofPaikowsky et al. (2004), the common resistance factor were only obtained as 0.34 and0.25 with βT=2.5 and 3.0, respectively. A question is why there is a such considerable dif-ference in the given common resistance factors. This issue can be explained as follows.

Paikowsky et al. (2004) used the data that were collected from many different regions.Therefore, soil parameters, construction techniques, and loading tests were certainly dif-ferent. Further, the dimensions of tested bored piles also varied considerably from regionto region. The variability or uncertainty degree of the data was rather high, which willtherefore result in small resistance factors. We see that, based on the data of Paikowskyet al. (2004), the mean and COV of the resistance bias factors are large, which are 1.614and 0.695, respectively. Conversely, in this case study, the 26 bored piles were locatedat a site; and they were constructed by the same manner with the same dimension ina relatively stable geological condition. As a result, the variability in the gathered datais not large and thus the calibrated resistance factors are large. In this case study, themean and COV of the resistance bias factors are quite small, which are 1.101 and 0.184,respectively. So, the calibrated resistance factors are large as indicated in Table 4.8.

In the United States, several Departments of Transportation (DOTs) approved and usedcalibrated resistance factors in certain regions with the terminology as ”regionally cal-

88

Calibrating resistance factors 4.7 Conclusions

ibrated resistance factors”. AbdelSalam et al. (2010) showed that the LRFD regionallycalibrated resistance factors reported for sands and clays are either equal to or greaterthan the AASHTO recommended values. In sand, the resistance factors are as much as50% above those recommended by AASHTO, while values are as much as 100% abovethe recommended values used for clay. Such large increases in resistance factors willlikely reduce the overall cost of bridge deep foundations.

4.7 Conclusions

In this chapter three reliability methods consisting of FOSM, FORM, and MCS have beenpresented and used to calibrate resistance factors following the framework of the LRFD.The chapter was then divided into two main parts. The first part presented the cali-bration procedure for a common resistance factors. The second part demonstrated thecalibration process for separate shaft and base resistance factors.

In the first part, a database including the sixteen calibration cases were considered. Ineach case, a combination of the soil type, resistance prediction methods, and the con-struction method was comprehensively addressed. The limit state function used for thecalibration process includes the total resistance, dead load, and live load bias factors.The statistical parameters and the probability distribution of the dead load and live loadbias factors were taken from the design criterion for bridge superstructures with the as-signed values (Nowak, 1999). For the resistance bias factors, the statistical parameterswere calculated and the probability distribution was checked by the K-S test. The com-mon resistance factors were calibrated using three reliability methods, according to thedifferent target reliability indices, and the different ratios of the dead to live load.

Based on the obtained calibration results, some findings are drawn as follows: (1) Thegreater the target reliability index, the smaller the resistance factor. (2) For a target reli-ability index, the value of the resistance factors gradually decreases with the increase ofthe dead load to live load ratio and reaches a stable value when this ratio is greater than3.0. (3) Each resistance prediction method should be used, depending not only on the soiltype, but also on the construction method. To which, common resistance factors will beconsistently selected (see Table 4.5). (4) FOSM, FORM, and MCS produce the resistancefactors which have the good correlations and linear relations with the ratio COVR/λR.(5) The resistance factors obtained from MCS and FORM have a somewhat better cor-relation than those between FORM and FOSM or between MCS and FOSM. (6) FORMproduces somewhat greater resistance factors than those obtained by FOSM or MCS.

In the second part, a data set including the 26 O-cell tested bored piles was considered.A limit state function was proposed and used for the calibration process, which includesthe shaft resistance, base resistance, dead load, and live load bias factors. The statisticalparameters and the probability distribution of the dead load and live load bias factorswere taken from the design criterion for building superstructures with the assigned val-ues (Ellingwood and Galambos, 1982). For the resistance bias factors, the statistical pa-rameters were calculated and the probability distribution was checked by the K-S test.The shaft and base resistance factors were calibrated using MCS, according to the target

89

4.7 Conclusions Calibrating resistance factors

reliability indices as 2.5 and 3.0, and the dead to live load ratio of 2.5.

Based on the obtained calibration result in the second part, findings are drawn as fol-lows: (1) The O-cell test is a good method to separately measure the shaft and baseresistances of bored piles. From which, it is easy to determine the shaft and base resis-tance bias factors that are employed for the calibration process. (2) An inverse linearrelation between the shaft resistance factor and the base resistance factor was observed;i.e., the shaft resistance factor increases when the base resistance factor decreases, andvice versa. All couples of values of the shaft and base resistance factors satisfy a speci-fied target reliability index. (3) A ratio, temporarily called ”Correlation Ratio”-CR, wasproposed aiming to determine a unique couple of values of the shaft and base resistancefactors. By using this ratio, the correlation between the uncertainties of the shaft andbase resistances would be solved harmonically; and obtained values of the shaft andbase resistance factors are therefore consistent. (4) The shaft and base resistance factorsare less sensitive with respect to the ratio QD/QL as well as the ratio RB/RS. Therefore,the proposed limit state function, g, in Eq. 4.29 assures the stability for the calibration ofshaft and base resistance factors. (5) Compared to the use of a common resistance factor,the use of the shaft and base resistance factors separately can result in a greater factoredtotal resistance and, therefore, a more economical design may be achieved.

Calibrating resistance factors within a certain region has been increasingly approved andused by several Departments of Transportation (DOTs) in the United States. By thisapproach, the regionally calibrated resistance factors are either equal to or greater thanthose as recommended in AASHTO. The comparison presented in Subsection 4.6.8 is aproof for this suggestion.

90

Chapter 5

Incorporating set-up into LRFD

5.1 Introduction

The increase of pile resistance with time is usually referred to as ”set-up” or ”freeze”;inversely, the decrease of pile resistance over time is often termed as ”negative set-up”or ”relaxation”.

The axial resistance of driven piles which may change over time after initial pile drivinghas been reported by a number of researchers for many years. Bartolomey and Yushkov(1985) reported a 70-80% side shear increase over 6 to 45 days after the driving of singlepiles into clay. Skov and Denver (1988) presented four case histories of static and dy-namic loading tests on driven piles and showed that set-up took place almost at the pileshaft, and that no set-up was observed at the pile toe with respect to different soil types.Fellenius et al. (1989) showed a 50% increase from 1 to 20 days in sandy clay and siltysand. Svinkin et al. (1994) studied pile capacity as a function of time in clayey and sandysoils; they also proposed functional relationships for the set-up in saturated sandy soils.Bullock (1999) indicated that set-up, based on his own experiment and on studies of oth-ers, primarily occurs with respect to side shear. The results in his study also showed thatset-up is not significantly dependent on soil type or depth. Axelsson (2000) performedseveral experiments on actual piles and steel rods in sandy soils; he concluded that set-up behavior mainly took place at the pile shaft and that the soil aging effect contributedconsiderably to this behavior in sand.

In contrast to driven piles, the available literature contains few examples of bored pileset-up. Finno et al. (1989) performed a study on both driven piles and bored piles for anASCE Pile Prediction Symposium at the Lakefill site on the Evanston Campus of North-western University. This research included two bored piles, 15.24 m long with 0.457 and0.61 m in diameter; one was constructed using slurry and one with slurry plus a tem-porary casing. They showed a 40-50% increase in resistance, from 14 to 301 days, afterinstallation in dense sand and stiff clay layers. The newsletter of LOADTEST (1998) re-ported a 20-33% increase in the side shear of a bored pile in a Vietnamese river depositover a 53 day study using an O-cell test. Bullock and others. (2003) carried out O-celltests again in 2002 on five bored piles at the site of the new SR20 eastbound bridge to

91

5.2 Causes leading to set-up effect Incorporating set-up into LRFD

evaluate set-up effect of those piles since the initial tests were conducted in 1996. Theyconcluded that the average side shear set-up identified for the 30 shaft segments in clay,sand, mixed sand and clay, and limestone was about 35% of that measured by Bullock(1999) for driven piles on the same site.

With the accumulation of experience and knowledge on set-up effect, several researchershave suggested that set-up effect should be incorporated into resistance prediction meth-ods to determine the total pile resistance. Bullock et al. (2005a,b) proposed a conservativemethod for incorporating the side shear set-up into the total pile resistance. The set-upresistance was assumed to have the same degree of uncertainty as the measured refer-ence resistance and a single safety factor was used to account for all uncertainties of loadsand resistances. Because the degree of uncertainty associated with the measured resis-tance and the predicted set-up resistance was different, Komurka et al. (2005) proposeda method to apply separate safety factors to the End of Driving (EOD) and the set-upcomponents of driven pile resistance. Yang (2006) used FORM to separately calibrate afactor for the set-up resistance. In the last few years, MCS has been gaining increasingpopularity in calibrating resistance factors for pile foundations such as Allen et al. (2005),Abu-Farsakh and Yu (2010), Bach et al. (2012), Bach et al. (2013), and so on. However,these studies only focused on calibrating a common resistance factor or on separate shaftand base resistance factors, and the set-up effect has not been yet taken into account.

In this chapter, the reference and the set-up resistance factors are separately calibrated forseveral theoretical prediction methods in some different soil types. Therefore, the out-line of this chapter is as follows. First, the causes leading to the set-up effect are brieflyintroduced in Section 5.2. Section 5.3 presents the formulation for the set-up effect. Incor-porating the set-up resistance into the LRFD is stated in Section 5.4. Calibration methodswith set-up are presented in Section 5.5. In Section 5.6, a case study involving five O-celltested bored piles is collected and processed. MCS is used to calibrate separate refer-ence and set-up resistance factors, satisfying specified target reliability levels followingthe framework of the LRFD. A procedure for the LRFD-based design of bored pile foun-dations considering the set-up is given. Finally, the chapter ends with conclusions inSection 5.7. Parts of this chapter were published as Bach and van Gelder (2014).

5.2 Causes leading to set-up effect

Until now, the causes leading to the set-up effect have been explained by many re-searchers, mostly for driven piles. Skov and Denver (1988) supposed that, the mainreasons for the set-up effect are: (1) The equalization of the pore-water pressure (re-consolidation), and (2) the re-establishment of the internal bonds in soils (re-generationin cohesive soils). Schmertmann (1991) addressed that set-up may result from, not onlya pore-pressure dissipation effect, but also involves the aging of the disrupted soils sur-rounding the piles. Axelsson (1998) concluded that for the long-term set-up of pilesdriven in sand, the effect of increasing dilatant behaviour (due to soil aging) over timewas by far the predominant cause. Chow et al. (1998) hypothesized that, the soil dis-placed during pile penetration might form a temporary radial arch around the pile,which then relaxed with time, increasing the horizontal effective stress.

92

Incorporating set-up into LRFD 5.3 Formulation for set-up effect

In contrast to driven piles, there have not been any explanations regarding the set-upeffect for bored piles, although the bored pile set-up effect has been observed by someresearchers as mentioned above. In fact, the bored pile set-up effect is not as dramaticas driven piles due to the performance of construction method; drilling pile holes virtu-ally does not affect the state of the surrounding soil layers. Therefore, the influence ofincrease in concrete strength with time and irregular shapes of pile shafts on the set-upeffect needs to be further studied.

This chapter does not intend to survey the mechanism of set-up effect for bored piles.The aim is to incorporate the set-up resistance into the total resistance under the frame-work of the LRFD. To which, the reference and the set-up resistance factors are calibratedseparately. Furthermore, the calibration model proposed in this chapter can be well ap-plied to driven piles.

5.3 Formulation for set-up effect

0 2 5 10 20 50 1001

2

3

4

5

6

t/t0

Nor

mal

ized

res

ista

nce,

RT/R

0

Predicted line for sand (1)Measured data for sandPredicted line for clay (2)Measured data for clayPredicted line for chalk (3)Measured data for chalk

(1)

(3)

(2)

Figure 5.1: Time-dependent development of pile resistance (adapted from Skov and Den-ver, 1988).

There have been formulae established for the set-up effect by several researchers, ofwhich the one proposed by Skov and Denver (1988) has been the most widely used.They presented a linear relationship between the pile set-up and the logarithm of timebased on the three case histories of static and dynamic loading tests on the driven pilesin sandy soil, clayey soil, and chalk as shown in Figure 5.1. The mathematical relationbetween them is expressed:

RT

R0= Alog10

(tt0

)+ 1 (5.1)

where R0 and RT are the reference resistance and the pile resistance at the reference time,t0, and at the elapsed time, t, respectively; factor A, which is called the set-up factor,is the relative increase in resistance per log cycle of time. Through the experimental

93

5.4 Incorporating Set-up into LRFD Incorporating set-up into LRFD

results, Skov and Denver (1988) introduced the reference time, t0, and the set-up factor,A, as shown in Table 5.1.

Table 5.1: Reference time, t0, and set-up factor A

Soil type t0 (days) Set-up factor, ASand 0.5 0.2Clay 1.0 0.6Chalk 5.0 5.0

Skov and Denver (1988) defined reference time, t0, as the time elapsed at the onset ofincreasing resistance (since the End of Driving), before which no increase in resistance isobserved. In general, the determination of this reference time is difficult, and it affectsthe value of A by changing the reference resistance R0. Based on Eq. 5.1, the relationshipbetween R0, RT, and set-up resistance, RSE, is:

RT = R0 + RSE (5.2)

therefore,

RSE = R0Alog10

(tt0

)(5.3)

5.4 Incorporating Set-up into LRFD

In order to incorporate the set-up resistance into the total resistance, combine Eq. 5.2with Eq. 4.1 in Chapter 4, and apply separately the reference resistance factor, φ0, andthe set-up resistance factor, φSE, to the reference resistance, R0, and the set-up resistance,RSE, respectively. A formula for considering the set-up effect under the framework ofthe LRFD is given:

φ0R0 + φSERSE ≥n

∑i=1

γiQni (5.4)

The reference resistance, R0, can be estimated according to the different resistance pre-diction methods as presented in Chapter 4. Note that, the set-up effect mainly takes placeat the pile shaft as observed by many researchers through several experiments. To assurethe suitableness for the calibration process, the resistance prediction methods chosen inthis study only consider the shaft resistance in calculation; the base resistance, therefore,is excluded.

5.5 Reference and set-up resistance factor calibration

5.5.1 Reference and set-up resistance factor calibration based on experience

Komurka et al. (2005) proposed a method to apply separate safety factors to the resis-

94

Incorporating set-up into LRFD 5.5 Reference and set-up resistance factor calibration

tances at End of Driving (EOD) and to the set-up resistance of driven piles as follows:

Qa =RE

FSE+

RSE

FSSE(5.5)

where Qa is the allowable pile load; RE and RSE are the reference resistance at EODand the set-up resistance, respectively; FSE and FSSE are the safety factor applied to thereference resistance and the set-up resistance, respectively.

It can be seen that, the methodology used herein is the same as that of the ASD. In which,the safety factors have been adopted by the design team and owner as shown in Table5.2. Here, the safety factors for the set-up resistance have values greater than those ofsafety factors for the reference resistance at EOD. The authors implied that the degree ofuncertainty of the set-up resistance is higher than that of the reference resistance at EOD.

Table 5.2: Safety factors used for design and pile installation (Komurka et al., 2005)

Pile diameter (m) FSE FSSE0.32 2.25 2.500.35 2.25 2.500.41 2.25 2.75

5.5.2 Reference and set-up resistance factor calibration using FORM

When just the dead load, QD, and the live load, QL, are considered; Yang (2006) usedFORM to calibrate the set-up resistance factor as follows:

φSE =

γDµQD

λQD+

γLµQL

λQL− φ0µR0

λR0µRSEλRSE

(5.6)

where γD and γL are the dead load and live load factors, respectively; µQD, µQL, µR0,and µRSE correspond to the mean values of random variables QD, QL, R0, and RSE, givenfrom FORM approach; λQD, λQL, λR0, and λRSE correspond to the mean of bias factorsof random variables QD, QL, R0, and RSE; φ0 is the reference resistance factor.

Since Eq. 5.6 contains two unknown terms as φSE and φ0; in order to find φSE, Yang(2006) had to assign an anticipated value for φ0, which was taken from the calibrationresult already performed by Paikowsky et al. (2004). Concretely, he used φ0 with thevalues of 0.65 and 0.50 corresponding to the target reliability indices as 2.33 and 3.0 forthe dynamic test of driven piles at Beginning of Re-strike (BOR). From that, the values ofφSE were obtained as 0.3 and zero corresponding to target reliability indices as 2.33 and3.0. Obviously, taking the values of φ0 from the previous calibration process without theset-up consideration, aiming to determine the set-up resistance factors, is not really con-sistent. Both the reference and set-up resistance factors need to be calibrated in the sameprocess following the framework of the LRFD and satisfying target reliability levels.

95

5.5 Reference and set-up resistance factor calibration Incorporating set-up into LRFD

5.5.3 Reference and set-up resistance factor calibration using MCS

Based on the calibration procedure as proposed in Subsection 4.6.1 of Chapter 4, if justthe dead load and live load are considered, a limit state function can be written as:

g = RM0 + RMSE −QMD −QML (5.7)

where RM0 and RMSE are the measured nominal reference resistance and the measuredset-up resistance, respectively; QMD and QML are the measured dead and live loads,respectively. If all measured terms in Eq. 5.7 are converted to predicted terms using thebias factors of the reference resistance λR0, the set-up resistance λRSE, the dead load λQD,and the live load λQL; Eq. 5.7 can be rewritten:

g = λR0R0 + λRSERSE − λQDQD − λQLQL (5.8)

Combine Eq. 5.4 with Eq. 5.8, after transformation, the limit state function is now ob-tained:

g = QL

(

γDQDQL

+ γL

) (λRSE

RSER0

+ λR0

)φSE

RSER0

+ φ0− λQD

QD

QL− λQL

(5.9)

Similar to the iteration procedure presented in Subsection 4.6.1 of Chapter 4, in this case,there are four random variables, which are the bias factors of the reference resistance,set-up resistance, dead load, and live load. The computation steps aiming to calibratethe reference and set-up resistance factors are as follows:


• Step 2: Select a trial reference resistance factor, φ0

• Step 3: Select a trial set-up resistance factor, φSE

• Step 4: Generate random numbers for each set of bias factors λR0, λRSE, λQD, andλQL


Pf =count(g ≤ 0)

ns(5.10)


β = −Φ−1(Pf ) (5.11)

here Φ−1 is the inverse CDF of the standard normal distribution. If the calculatedreliability index, β, is different from the specified target reliability index, βT, thetrial set-up resistance factor, φSE, in Step 3 should be changed and a new iterationneeds to be repeated until |β− βT| ≤ tolerance. Repeat Step 3 to Step 5 for the nextset of trial reference resistance factor.

96

Incorporating set-up into LRFD 5.6 A case study

• Step 6: From Step 2 to Step 5, a series of couples of value for φ0 and φSE may bederived that satisfies the specified target reliability index βT. The most consistentcouple of values for the reference and set-up resistance factors is selected accordingto the approach proposed in Subsection 4.6.1 of Chapter 4 through the ratio CR.

5.6 A case study

5.6.1 Data set for calibration

Figure 5.2: Soil stratigraphy at location of bored pile 7.

A data set of five O-cell tested bored piles is extracted from Report No. 4910-4504-798-12,which was compiled by Bullock and others. (2003). The Florida Department of Trans-portation set aside these piles at the site of the new SR20 eastbound bridge for futuretests following the initial tests in 1996; they were tested again in 2002, aiming to eval-uate the set-up effect after a long period of time. The piles ranged from 1.52 to 2.13 min diameter, with 25.9 to 31.7 m long with the rock socket in limestone as deep as 5.5 to10.7 m long. They were constructed using temporary casing and mineral slurry, throughoverburden soils included sand, clay and clayey sand.

For demonstration, a cross section of soil stratigraphy at the location of bored pile 7 ischosen and depicted in Figure 5.2. There were two Osterberg cells working as hydraulicjacks positioned within the limestone layer, one near the pile toe, another near the sur-face of this soil layer. The strain gauges (denoted by the plus signs in the figure) wereinstalled in radial asymmetric pairs along the pile shaft. These gauges provided a shaftload profile, from which the side shear stress on each shaft segment could be estimated.The test results of side shear stress in the two stages of the test (in 1996 and in 2002)

97

5.6 A case study Incorporating set-up into LRFD

are summarized in Table 5.3. Because the strain gauges in the clayey sand layer showednegative set-up, these test results are discarded and not considered in this study.

In Table 5.3, t0 is the first day of bored pile installation; f0 is the predicted side shear stressfor the first day, which is backward-extrapolated by Bullock and others. (2003) from themeasured data, f1 and f2; t1, f1 and t2, f2 are the elapsed time after bored pile installationand the measured side shear stress for the shaft segments conducted in 1996 and in 2002,respectively.

Table 5.3: Summary of side shear stress over time for the SR20 tested bored piles

Segment Predicted Measured in 1996 Measured in 2002

Soil type Pile No. elevation t0 f0 t1 f1 t2 f2(m) (days) (kN/m2) (days) (kN/m2) (days) (kN/m2)

Sand 11 4.88 1 50.14 7 50.14 2,021 50.1411 1.52 1 57.35 7 67.47 2,021 96.8211 -3.66 1 73.76 7 89.35 2,021 134.662 5.91 1 6.11 6 14.95 2,064 43.942 -0.18 1 48.50 6 54.61 2,064 74.492 -3.23 1 48.23 6 54.34 2,064 74.21

10 -0.76 1 -1.09 11 2.92 2,038 11.675 -2.87 1 42.03 9 43.76 1,900 48.147 0.03 1 23.61 11 23.89 1,918 24.52

Clay 5 7.20 1 3.01 9 4.92 1,900 9.665 3.38 1 1.73 9 3.28 1,900 7.115 0.03 1 18.96 9 19.15 1,900 19.60

Limestone 11 -5.64 1 182.07 7 239.14 2,021 405.622 -5.98 1 183.53 6 219.26 2,064 335.69

10 -7.01 1 63.27 11 124.17 2,038 256.5510 -10.59 1 679.58 11 731.27 2,038 843.78

5.6.2 Side shear set-up

Instead of a total resistance as presented by Skov and Denver (1988), Bullock and others.(2003) proposed using t0=1 day and limiting the set-up effect to reflect only the changein side shear stress. They divided both total resistance, RT, and reference resistance, R0,in Eq. 5.1 by the side area of pile, ASI . A new relationship for the side shear stress wasobtained as:

RT

R0=

RTASIR0ASI

=fT

f0= Alog10

(tt0

)+ 1 (5.12)

here fT and f0 are the side shear stress at the elapsed time, t, and t0, respectively. Thistransformation is significant, because the set-up only takes place along the pile shaft,not at the pile toe, as addressed by numerous researchers earlier. In fact, Bullock (1999)already verified the application of Eq. 5.12 for five piles driven into the Florida soils.If shaft set-up is a function of stress changes and consolidation, similar to driven pileset-up, then Eq. 5.12 may apply for bored pile set-up also.

98


From Eq. 5.12, in order to evaluate the set-up effect for each soil type at the site of thenew SR20 eastbound bridge, reference time, t0, and reference side shear stress, f0, haveto be defined. For convenience, Bullock and others. (2003) selected t0=1 day, and bya ”backward-extrapolation” manner, the values of f0 were obtained from f1 and f2 asindicated in Table 5.3. This extrapolation manner is completely based on the hypothesisthat the change in the side shear stress is linear with the logarithm of elapsed time. Thisseems to be convenient for calculation and suitable in the context of the method proposedby Skov and Denver (1988). However, this manner led to two limitations. First, at timet0=1 day, the bored pile concrete strength had not been fully developed; the interactionforce between the shaft surface and the surrounding soil layers could therefore not beclearly formed. Second, by this manner, there was an unreal value of f0, which wassmaller than zero associated with pile 10 in sand ( f0=-1.09 kN/m2) as shown in Table5.3. Therefore, it is necessary to have another approach to describe the set-up effectconsidering the performance of bored piles.

With respect to bored piles, loading tests are normally carried out after few days of in-stallation as the pile concrete has developed relatively sufficient strength. For example,in this case study, the time t1 after the pile installation is the time, at which the loadingtests had been conducted. Obviously, before the time t1, we do not know whether theset-up effect had been really taking place or not. For the conservative side, it should beassumed that the set-up effect had started taking place from time t1 afterward. Therefore,Eq. 5.12 is modified as:

RT

R0=

fT

f1= Alog10(t− t1) + 1 (5.13)

therefore, the set-up resistance is given,

RSE = R0Alog10(t− t1) (5.14)

here the set-up factor, A, is taken from Table 5.4. The determination of the elapsed time,t, will be presented in detail in Subsection 5.6.4. This modification overcomes two lim-itations as mentioned above. Instead of using the extrapolation-based results, we onlyanalyze the set-up effect completely based on the measured data.

Table 5.4: Set-up factor A used in this case study

Soil type Set-up factor, A ReferenceSand 0.2 Skov and Denver (1988)Clay 0.6 Skov and Denver (1988)Limestone 0.4 Assumed in this study

5.6.3 Statistical parameters and probability distributions of the reference re-sistance bias factors

Five calculation cases consisting of soil type, resistance prediction method, and construc-tion method are extracted from Report NCHRP-507 (Paikowsky et al., 2004) and catego-

99


rized in Table 5.5. The resistance prediction methods are used to determine the referenceresistance. Note that, the resistance prediction methods selected herein only consider theshaft resistance, the base resistance is excluded from these methods.

Table 5.5: Five calculation cases of the reference resistance

No. No. of piles Soil type Prediction method Construction method

1 32 Sand FHWA Mixed2 32 Sand Reese and Wright (1977) Mixed3 53 Clay FHWA Mixed4 46 Rock Carter and Kulhawy (1988) Mixed5 46 Rock IGM Mixed

To avoid redundancy, the calculation cases will be denoted briefly by a group of words.For example, a denotation of ”Clay-FHWA-Mixed”, i.e., the soil type is the clay, the resis-tance prediction method is the FHWA method, and the construction method is the mixedmethod.

Statistical parameters for the reference resistance bias factors are calculated based on Eqs.4.4, 4.5, and 4.6 in Chapter 4. Probability distributions for the reference resistance biasfactors are checked under the K-S test. The obtained results are indicated in Table 5.6.

Table 5.6: Statistical parameters and probability distributions of the reference resistancebias factors

Calculation case Mean of bias factor, λR0 σR0 COVR0 DistributionSand-FHWA-Mixed 1.094 0.538 0.492 Log-normal

Sand-RW-Mixed 0.833 0.430 0.515 Log-normalClay-FHWA-Mixed 0.873 0.310 0.355 Log-normal

Rock-CK-Mixed 1.229 0.504 0.410 Log-normalRock-IGM-Mixed 1.298 0.437 0.337 Log-normal

5.6.4 Statistical parameters and probability distributions of the set-up resis-tance bias factors

At the new SR20 eastbound bridge, three calculation cases consisting of soil type, set-upresistance prediction method, and construction method are considered and categorizedin Table 5.7. The set-up resistance prediction method proposed by Skov and Denver(1988) is used to estimate the set-up resistance.

Statistical parameters of the set-up resistance bias factors are also calculated by Eqs. 4.4,4.5, and 4.6 in Chapter 4. In which, the measured set-up resistances are taken from Table5.3; the predicted set-up resistances are calculated according to Eq. 5.13 using the set-upfactors addressed in Table 5.4. Since there were only the data measured in 1996 and in

100


Table 5.7: Three calculation cases of the set-up resistance

No. No. of data Soil type Prediction method Construction method

1 9 Sand Skov and Denver (1988) Mixed2 3 Clay Skov and Denver (1988) Mixed3 4 Limestone Skov and Denver (1988) Mixed

2002, no further data at all were recorded in the time between 1996 and 2002. Therefore,there are two Abilities to be given as follows:

Ability 1

Figure 5.3: Set-up side shear stress vs. time (Ability 1).

Assume that, the set-up side shear stress had been gradually and linearly taking placewith the logarithm of time from 1996 until 2002 as depicted in Figure 5.3. The elapsedtime t, therefore, is:

t = t2 (5.15)

where t2 is the elapsed time after bored pile installation in 2002 as shown in Table 5.3.Figure 5.4 shows the measured and predicted normalized set-up side shear stresses forAbility 1.

101


1 10 100 1,000 10,0000

1

2

3

4

Elapsed time, t − t1 (days)

Nor

mal

ized

sid

e sh

ear

stre

ss, f

T/f 1

Skov & Denver predicted line for sand (1)Measured data for sandSkov & Denver predicted line for clay (2)Measured data for claySkov & Denver predicted line for limestone (3)Measured data for limestone

(2)

(3)

(1)

Figure 5.4: Measured and predicted normalized set-up side shear stresses for Ability 1.

Ability 2

Figure 5.5: Set-up side shear stress vs. time (Ability 2).

For driven piles, many researchers showed that the time duration of 100 days for clayand 30 days for sand after installation may be considered as the point, after which theset-up effect would be diminished (Yang, 2006). So, for bored piles, assume that after28 days of installation, the time needed for sufficiently-developed pile concrete strength,the set-up side shear stress had been linearly continuing to take place with the logarithmof time for a short time duration tSE, after that it reached a stable value till 2002, i.e., f2(see Figure 5.5). Therefore, the elapsed time, t, is now given:

t = t28 + tSE (5.16)

where t28 is the 28 days of age of concrete after installation; tSE is taken as 100 days forclay, and 30 days for sand and limestone, applying to bored piles in this study. Figure 5.6shows the measured and predicted normalized set-up side shear stresses for Ability 2.

102


1 10 100 1,0000

1

2

3

4

Elapsed time, t − t1 (days)

Nor

mal

ized

sid

e sh

ear

stre

ss, f

T/f 1

Skov & Denver predicted line for sand (1)Measured data for sandSkov & Denver predicted line for clay (2)Measured data for claySkov & Denver predicted line for limestone (3)Measured data for limestone

(2)

(3)

(1)

Figure 5.6: Measured and predicted normalized set-up side shear stresses for Ability 2.

To avoid redundancy, the calculation cases will be denoted briefly by a group of words.For example, a denotation of ”Clay-SD-Mixed”, i.e., the soil type is the clay, the set-upresistance prediction method is the Skov & Denver method, and the construction methodis the mixed method.

Statistical parameters for the set-up resistance bias factors are calculated based on Eqs.4.4, 4.5, and 4.6 in Chapter 4. The detailed calculation results are presented in AppendixB. Due to the lack of available measured data, probability distributions for the set-upresistance bias factors are assumed to be log-normal. The obtained results are indicatedin Table 5.8.

Table 5.8: Statistical parameters and probability distributions of the set-up resistance biasfactors

Ability 1 Ability 2

Calculation case λRSE σRSE COVRSE λRSE σRSE COVRSE Distribution

Sand-SD-Mixed 1.109 0.593 0.535 1.094 0.434 0.397 Log-normalClay-SD-Mixed 0.579 0.168 0.290 0.765 0.222 0.290 Log-normal

Limestone-SD-Mixed 0.694 0.141 0.203 0.961 0.196 0.204 Log-normal

5.6.5 Calibrated reference and set-up resistance factors

The limit state function, g, in Eq. 5.9 is used to calibrate the reference and set-up resis-tance factors under a procedure including the six steps as described in Subsection 5.5.3.Target reliability indices are assigned as 2.5 and 3.0, which correspond to the target prob-abilities of failure as 0.00621 and 0.00135.

The statistical parameters and the probability distributions of the dead load and the live

103


load bias factors are taken from Table 4.2 in Chapter 4. The ratio of the dead load to thelive load, QD/QL, is assigned as 3.46.

The statistical parameters of the reference and set-up resistance bias factors are takenfrom Tables 5.6 and 5.8, respectively. The probability distribution for both is log-normal.The ratios of the predicted set-up resistance to the predicted reference resistance RSE/R0are shown in Table 5.9.

Table 5.9: Ratio of the predicted set-up resistance to the predicted reference resistance,RSE/R0

Calculation case Ability 1 Ability 2Sand-SD-Mixed 0.66 0.34Clay-SD-Mixed 1.97 1.25

Limestone-SD-Mixed 1.32 0.67

Figures 5.7(a), (b), and (c) demonstrate the correlation between the calibrated referenceresistance factor, φ0, and the calibrated set-up resistance factor, φSE, for three represen-tative calibration cases with respect to target reliability indices as 2.5 and 3.0. Here, forexample, a denotation of ”Clay-FHWA-SD-Mixed”, i.e., the soil type is the clay, the refer-ence resistance prediction method is the FHWA method, the set-up resistance predictionmethod is the Skov & Denver method, and the construction method is the mixed method.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Set−up resistance factor, φSE

Ref

eren

ce r

esis

tanc

e fa

ctor

, φ0 (a) Sand−RW−SD−Mixed

βT = 2.5

βT = 3.0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


Ref

eren

ce r

esis

tanc

e fa

ctor

, φ0 (b) Clay−FHWA−SD−Mixed

βT = 2.5

βT = 3.0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


Ref

eren

ce r

esis

tanc

e fa

ctor

, φ0 (c) Limestone−CK−SD−Mixed

β

T = 2.5

βT = 3.0

0.66

0.33

0.58

0.47

0.59

0.52

0.51 0.57 0.83 0.94

0.39

0.53

Figure 5.7: Correlation between φ0 and φSE.

It can be seen that, there are also numerous couples of values of φ0 and φSE derivedthat satisfy the target reliability indices. An inverse linear relation between φ0 and φSEis observed; φ0 increases when φSE decreases, and vice versa. In order to find the mostconsistent couple of values of φ0 and φSE, we use the approach as proposed in Subsection4.6.5 of Chapter 4. First of all, the correlation ratio, CR, between φ0 and φSE needs to be

104


defined:φ0

φSE≈ CR =

COVRSEλRSE

COVR0λR0

=COVRSEλR0

COVR0λRSE(5.17)

where λR0 and λRSE are the mean of the reference and set-up resistance bias factors,respectively; COVR0 and COVRSE are the coefficient of variation of the reference and set-up resistance bias factors, respectively. The obtained values of the ratio, CR, are indicatedin Table 5.10.

Table 5.10: Ratio CR for calibration cases

COVR0/λR0 COVRSE/λRSE Ratio, CRCalibration case Ability 1 Ability 2 Ability 1 Ability 2

Sand-FHWA-SD-Mixed 0.450 0.483 0.363 1.07 0.81Sand-RW-SD-Mixed 0.618 0.483 0.363 0.78 0.59

Clay-FHWA-SD-Mixed 0.407 0.501 0.379 1.23 0.93Limestone-CK-SD-Mixed 0.334 0.293 0.213 0.88 0.64

Limestone-IGM-SD-Mixed 0.260 0.293 0.213 1.13 0.82

From the ratios, CR, in Table 5.10, the obtained couples of values of φ0 and φSE for thecalibration cases are addressed in Table 5.11.

Table 5.11: Calibration results for φ0 and φSE

Ability 1 Ability 2

βT = 2.5 βT = 3.0 βT = 2.5 βT = 3.0

Calibration case φ0 φSE φ0 φSE φ0 φSE φ0 φSE

Sand-FHWA-SD-Mixed 0.55 0.52 0.47 0.43 0.52 0.64 0.43 0.53Sand-RW-SD-Mixed 0.41 0.53 0.34 0.43 0.39 0.66 0.33 0.58

Clay-FHWA-SD-Mixed 0.53 0.43 0.47 0.38 0.53 0.57 0.47 0.51Limestone-CK-SD-Mixed 0.58 0.66 0.52 0.59 0.59 0.94 0.52 0.83

Limestone-IGM-SD-Mixed 0.73 0.65 0.66 0.58 0.74 0.91 0.67 0.81

The derived values of φ0 according to both Abilities are quite the same for all calibrationcases, since the statistical parameters for the reference resistance bias factors are com-monly used.

With respect to the sandy and clayey soil types, the calibrated set-up resistance factorsof Ability 2 are greater than those of Ability 1. It can be seen that, applying the set-upfactors of 0.2 for sand, and 0.6 for clay, as recommended by Skov and Denver (1988), tothe SR20 bridge would result in a slight underprediction associated with the sandy soil,and overprediction associated with the clayey soil, compared to the measured data.

Associated with limestone (rock) at the site, the set-up resistance factors calibrated ofAbility 2 are improved considerably compared to those of Ability 1. For example, with

105

5.7 Conclusions Incorporating set-up into LRFD

respect to Ability 2, the set-up resistance factors reach a value of up to 0.94 (βT=2.5) and0.83 (βT=3.0) for the calibration case of ”Limestone-CK-SD-Mixed”, and 0.91 (βT=2.5)and 0.81 (βT=3.0) for the calibration case of ”Limestone-IGM-SD-Mixed”. Therefore, theset-up factor of 0.4 for limestone as suggested in this study is acceptable.

5.6.6 Incorporating set-up into LRFD procedure for the SR20 Bridge

Assume that, at the site of the SR20 Bridge, the set-up effect had been taking place fol-lowing Ability 2; and the target reliability index is assigned as 3.0. The LRFD procedurefor the SR20 Bridge when incorporating the set-up effect is illustrated as follows:

• Step 1: Assign a pile diameter and a length for a preliminary design.

• Step 2: Determine the reference resistance, R0, using the well-known resistanceprediction methods as stated in Table 5.5. For example, the Reese & Wright method,the FHWA method, and the IGM method are chosen and applied to the sandy,clayey, and limestone soil layers, respectively.

• Step 3: Estimate the set-up resistance, RSE, according to Eq. 5.14. In which, theset-up factors, A, are referred to those in Table 5.4 corresponding to soil types. Theelapsed time, t, should be taken as 45 to 115 days, depending on the soil types, andthe construction schedule of the project. Note that, the elapsed time, t, is countedafter pile installation. The time, t1, is assigned as 14 days.

• Step 4: Use Eq. 5.4 with the reference and set-up resistance factors obtained inTable 5.11. Here, the reference resistance factors are taken as 0.33, 0.47, and 0.67for the sandy, clayey, and limestone soil layers, respectively. The set-up resistancefactors are taken as 0.58, 0.51, and 0.81 for the sandy, clayey, and limestone soillayers, respectively. Subsequently, the factored total resistance is obtained, aimingto evaluate the applicability of the preliminary design.

• Step 5: Based on the result from Step 4, change the pile diameter or the length asrequired.

5.7 Conclusions

In this chapter, a calibration procedure of the reference and set-up resistance factors hasbeen presented following the framework of the LRFD. The reference resistances are cal-culated according to the well-known resistance prediction methods; and the set-up resis-tances are estimated in terms of the prediction method of Skov and Denver (1988), whenincorporating the set-up effect into the LRFD-based design.

A data set of the five O-cell tested bored piles was compiled and used for the set-upeffect study for the three soil types: Clay, sand, and limestone. Five calibration caseswere considered including three soil types, four reference resistance prediction methods,one set-up resistance prediction method, and one construction method. The reference

106

Incorporating set-up into LRFD 5.7 Conclusions

and set-up resistance factors were calibrated according to two target reliability indices as2.5 and 3.0.

An alternative, aiming to determine the elapsed time, was applied to the bored piles set-up only. Instead of using the ratio of t/t0 as for driven piles, we can use the differencet− t1. In which, t1 ≈14 days, the time is needed to sufficiently development the boredpile concretes strength. From that, the set-up resistance could be easily determined.

For both Abilities 1 and 2, applying a set-up factor of 0.2 for the sandy soil, and 0.6 forthe clayey soil, as recommended by Skov and Denver (1988) to the set-up study at theSR20 site would result in a slight underprediction associated with the sandy soil, andoverprediction associated with the clayey soil, compared to the measured data. Usinga set-up factor of 0.4, as suggested in this study for the limestone at this site, providedrather reasonable calibration results. This can be seen through the set-up resistance fac-tors obtained with respect to Ability 2.

An easily understandable procedure was presented in order to incorporate the set-upeffect into the LRFD, aiming to reach an economical design for bored piles foundations.

The measured data at the SR20 site was insufficient and discontinuous, but the set-upeffect was also recorded. This proved that the set-up effect really exists with respect tobored piles, although the bored pile set-up effect is not as dramatic as for driven piles.Therefore, a comprehensive study on the set-up effect of bored piles will be necessary.Effects of construction methods as well as irregular shapes of bored piles on the set-upshould be taken into account in future research.

107

5.7 Conclusions Incorporating set-up into LRFD

108

Chapter 6

Updating resistance factors based onBayesian inference

6.1 Introduction

In Part 1 of Chapter 4, the common resistance factors, obtained through the calibrationprocess in the framework of the LRFD, satisfied the specified target reliability levels.Each calibrated resistance factor should be applied to a certain combination of a soil type,a resistance prediction method, and a construction method. The most substantial factorsutilized for the calibration process are the statistical parameters and the probability dis-tributions of the resistance bias factors. It can be seen that, the values of the calibratedresistance factors reflect the different reliability levels with respect to the selected resis-tance prediction methods for a specific condition of soil and construction method.

In Part 2 of Chapter 4, the calibration of the resistance factor has been further developed.In which, the shaft and base resistance factors have been calibrated separately and theuncertainties of resistance at the shaft and the base of the pile have been consideredindependently. As a result, using separate shaft and base resistance factors might lead toa more economical design than a design using a common resistance factor.

In addition, calibrating a common resistance factor within a certain region has been in-creasingly approved and used by several Departments of Transportation in the UnitedStates (AbdelSalam et al., 2010). With this approach, the regionally calibrated resistancefactors are either equal to or greater than those as recommended in AASHTO; and there-fore the use of the regionally calibrated resistance factors can bring down the cost of pilefoundations.

From the correlation analyses in Chapter 4, we see that the resistance factors depend onthe statistical parameters of the resistance bias factors, especially the ratio of the coeffi-cient of variation to the mean of the resistance bias factor COVR/λR. The value of theresistance factor is completely decided by the ratio COVR/λR; the same ratio COVR/λRwill result in the same resistance factor, regardless of the values of other individual sta-tistical parameters of the resistance bias factor.

109

6.2 Within-site variability Updating resistance factors based on Bayesian inference

We know that, pile loading tests are an important means to deal with uncertainties inthe design and construction of pile foundations. Normally, a pile foundation is designedby the use of resistance factors, to which a pile quantity and a pile size are preliminarilydetermined. Subsequently, based on the loading test results within a site, the preliminarydesign may be changed, aiming to assure a more economic-technical efficiency for theadjusted design. The question is, whether, through the loading test results, the initialresistance factors used in the design can be re-calibrated, i.e., the resistance factors areupdated in terms of a re-calibration process. Based on which, for other projects that arelocated in the same site, we can apply the updated resistance factors to the calculations.

In order to update resistance factors through loading test results at a site, the Bayesianinference is used to determine the posterior distribution of the resistance bias factor fromthe initial empirical distribution of the resistance bias factor. Therefore, the outline ofthis chapter is as follows. The within-site variability of pile capacity is considered inSection 6.2. The general Bayesian inference is introduced in Section 6.3, from which thedetermination of the likelihood functions with pile loading tests to update pile capacityis presented in Section 6.4. Section 6.5 will present a procedure for updating resistancefactors based on the obtained posterior distribution in combination with the calibrationprocess proposed in Chapter 4. A case study applying the theories stated in previoussections is described in Section 6.6. Finally, the chapter ends with conclusions in Section6.7.

6.2 Within-site variability

The term of ”within-site variability” of pile capacity was first introduced by Baecher andRackwitz (1982) in determining safety factors when considering the pile loading tests.In Phoon and others. (2008), Zhang addressed that, in addition to uncertainties with siteinvestigation, laboratory testing, and prediction models, the values of capacity of sup-posedly identical piles within one site also vary. Suppose, several ”identical” test pilesare constructed at a seemingly uniform site and are load-tested following an ”identical”procedure. The measured values of the capacity of the piles would usually be differentdue to the so-called ”within-site variability”.

It is also in Phoon and others. (2008), that Zhang, based on the works of Kay (1976) andEvangelista et al. (1977), collected the values of the coefficient of variation (COV) of thecapacity of driven piles from the loading tests in nine sites; these values range from 0.12to 0.28.

Evangelista et al. (1977) tested 22 bored piles in one site. The piles were 0.8 m in diameter,and 20 m in length; bentonite slurry was used for pile construction. The loading testsrevealed that the COV of the settlement of these piles at the intended working load was0.21, and the COV of the applied loads at the mean settlement corresponding to theintended load was 0.13.

We consider again the data set of the 26 O-cell equipped bored piles in Subsection 4.6.3of Chapter 4. These piles were located in the site of the remediation project for the LosAngeles Memorial Coliseum. The piles are 1.32 m in diameter and 16.5 through 20.1 m

110

Updating resistance factors based on Bayesian inference 6.2 Within-site variability

long. They were drilled from 7.2 to 10.5 m of artificial fill consisting of sand and silt andinto the underlying natural sand with gravel. The mineral slurry construction methodwas used to keep the walls of the shaft stable during drilling. The statistical parametersof measured resistances and resistance bias factors are categorized in Table 6.1. Based onthe O-cell test results, it can be seen that, the variability of shaft resistance is lower thanthat of base resistance. For the identical piles, the COVs of shaft resistance are smallerthan 0.20; while, the COVs of base resistance are somewhat greater than 0.30. For allthe 26 bored piles, the COVs of shaft and base resistances slightly increase compared tothose of the identical piles. Based on the equivalent head-down compressive test results,the COVs of total resistance are more or less the value of 0.20 with respect to the identicalbored piles and the 26 bored piles. It is interesting, the COVs of resistance bias factorsare also quite the same as those of resistances.

Table 6.1: Statistical parameters of measured resistances and resistance bias factors at thesite of the Los Angeles Memorial Coliseum

Item Unit Mean value Standard deviation COV

Based on O-cell tests

For 11 identical piles, length=18 m. Measured nominal shaft resistance RMS kN 10,582 1,630 0.154. Measured nominal base resistance RMB kN 6,917 2,085 0.301. Shaft resistance bias factor λRS - 1.172 0.208 0.178. Base resistance bias factor λRB - 1.743 0.525 0.301

For 9 identical piles, length=19.2÷19.5 m. Measured nominal shaft resistance RMS kN 9,934 1,657 0.167. Measured nominal base resistance RMB kN 9,052 2,946 0.325. Shaft resistance bias factor λRS - 1.060 0.165 0.155. Base resistance bias factor λRB - 2.281 0.742 0.325

For all 26 piles, length=16.2÷20.1 m. Measured nominal shaft resistance RMS kN 10,339 2,542 0.246. Measured nominal base resistance RMB kN 7,719 2,708 0.351. Shaft resistance bias factor λRS - 1.138 0.299 0.263. Base resistance bias factor λRB - 1.945 0.682 0.351

Based on equivalent head-down compressive tests

For 11 identical piles, length=18 m. Measured nominal total resistance RM kN 14,185 3,076 0.217. Resistance bias factor λR - 1.088 0.240 0.220

For 9 identical piles, length=19.2÷19.5 m. Measured nominal total resistance RM kN 15,310 1,720 0.112. Resistance bias factor λR - 1.147 0.111 0.097

For all 26 piles, length=16.2÷20.1 m. Measured nominal total resistance RM kN 14,390 2,633 0.183. Resistance bias factor λR - 1.101 0.202 0.184

Zhang also emphasized that the within-site variability is inherent in a particular geo-logical setting and a geotechnical construction procedure at a specific site. The within-site variability represents the minimum variability for a construction procedure at a site,which cannot be reduced using load tests. Therefore, in this chapter, a value of COV=0.20

111

6.3 Bayesian inference Updating resistance factors based on Bayesian inference

is adopted for the analysis of the within-site variability of pile capacity.

6.3 Bayesian inference

Let Θ be the random variable for the parameter of a distribution with a prior PDF f ′(θ).If ε is an observed experimental outcome, the prior PDF f ′(θ) can be revised in the lightof ε using the Bayes’ theorem, obtaining the posterior distribution f ′′(θ) as follows (Angand Tang, 2007):

f ′′(θ) =P(ε|θ) f ′(θ)

∞∫−∞

P(ε|θ) f ′(θ)dθ

(6.1)

where the term P(ε|θ) is the conditional probability, or likelihood, of observing the ex-perimental outcome ε assuming that Θ=θ; P(ε|θ) is a function of θ and is commonlyreferred to as the likelihood function of θ and denoted L(θ).

Eq. 6.1 can also be written in a simplified form with the inverse of the denominatorrepresented by a normalizing constant k:

f ′′(θ) = kL(θ) f ′(θ) (6.2)

Obviously, in order to determine the posterior distribution f ′′(θ), we have to address theprior distribution, f ′(θ), and to define the likelihood function, L(θ), of the parameter θ.They will be presented in the subsequent sections.

6.4 Updating pile capacity

The target of this chapter is to update the resistance factors when having pile loading testresults at a site; this requires the statistical parameters and the probability distributionsof the resistance bias factors as performed in Chapters 4 and 5. In order to determine theposterior distribution of the resistance bias factor, the prior distribution of the resistancebias factor and the likelihood function need to be defined. Therefore, in the context ofthe Bayesian inference, the considered variable is the resistance bias factor λR, and theobserved experimental outcome is the test load QT. Eq. 6.1 can be re-written as:

f ′′(λR) =P(QT|λR) f ′(λR)

∞∫−∞

P(QT|λR) f ′(λR)dλR

(6.3)

and Eq. 6.2 therefore becomes

f ′′(λR) = kL(λR) f ′(λR) (6.4)

The Bayesian inference enables us to deal with survival and failure representing a certaintest load that was survived or led to failure. For bored piles, O’Neill and Reese (1999)

112

Updating resistance factors based on Bayesian inference 6.4 Updating pile capacity

suggested that the failure test load is a load corresponding to a displacement of 5% di-ameter of piles or the plunging load in static loading tests, whichever comes first. In thisway, we can infer that the survival test load is a non-plunging load, which produces asmaller displacement than 5% diameter of piles.

In Eqs. 6.3 and 6.4, the prior distribution of the resistance bias factor f ′(λR), which is alsocalled the empirical distribution of the resistance bias factors, was already determined.We can take the PDF of the resistance bias factor from one of the sixteen calibration casespresented in Chapter 4 to serve as a prior distribution of the resistance bias factor. In thenext subsections, we will focus on the determination of the likelihood function and thenormalizing constant.

6.4.1 Survival test loads

For loading tests that are not conducted to failure, the pile capacity values are not deter-mined, although they are greater than the maximum test load. Let λ be the ratio of themeasured test load, QT, to the predicted resistance of pile RP. Note that, λ is somewhatdifferent from the bias factor λR, since λR is always determined by the failure test load.At a particular site, λ can be assumed to follow a log-normal distribution (e.g., Barkeret al., 1991; Paikowsky et al., 2004; and Bach et al., 2012). In which, the mean and stan-dard deviation of λ are λ and σλ, and those of ln(λ) are η and ξ, respectively. Here, σλ

and ξ represent the within-site variability of pile capacity. In Section 6.2, the COV ofpile capacity is adopted as 0.20, i.e., we can take the coefficient of variation of λ to beCOVλ=0.20, and therefore ξ is given as 0.198. Assume that the measured maximum testload corresponds to a value of λ=λT. From the log-normal PDF, the probability that thetest pile does not fail at the measured maximum test load (i.e., λ ≥ λT) is:

P(λ ≥ λT) =

∞∫λT

1√2πξλ

exp

{−1

2

[ln(λ)− η

ξ

]2}

dλ = 1−Φ[

ln(λT)− η

ξ

](6.5)

where Φ is the CDF of the standard normal distribution. Suppose that N test piles areconducted and none of the test piles fails at λT. If the standard deviation, ξ, of ln(λ) isknown (i.e., ξ=0.198) but its mean, η, is a variable; then the probability of the N test piles,that do not fail at λT, is a product of N probabilities of the N test piles not to be failed:

L(QT|λ) =N

∏i=1

Pi(λ ≥ λTi) =N

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]}(6.6)

here L(QT|λ) is the likelihood function of N test loads, QTi , assuming that the valueof the parameter is λ. Let X = ln(λ), given L(QT|λ), the posterior distribution of theresistance bias factors is,

f ′′(X) = k f ′(X)N

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]}(6.7)

113

6.4 Updating pile capacity Updating resistance factors based on Bayesian inference

with the normalizing constant k,

k =

∞∫−∞

f ′(X)N

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]}dX

−1

(6.8)

6.4.2 Failure test loads

With respect to failure test loads, we have to consider two cases: (1) A plunging load, and(2) a non-plunging load, but a respective displacement reaches 5% pile diameter. In thefirst case, we know with certainty that the failure load is the plunging load; we cannotfind other loads being greater than the plunging load. In the second case, the problem issomewhat complicated, because we can find other loads that produce a greater displace-ment than 5% pile diameter. However, the load-displacement curve of a certain staticloading test generally has a slope, which gradually increases with the displacement. As-sume that a displacement of 5% pile diameter is sufficiently large; a large increase in dis-placement will be obtained by a small increase in load. Therefore, the load correspondingto a displacement of 5% pile diameter is considered to be a failure load as proposed byO’Neill and Reese (1999). In contrast to the survival test loads, the likelihood functionassociated with a test pile with a failure test load is estimated through the probabilitythat the test pile fails at the measured maximum test load (i.e., λ ≤ λT):

P(λ ≤ λT) = 1− P(λ ≥ λT) = Φ[

ln(λT)− η

ξ

](6.9)

Analogous to the procedure of test piles with survival test loads, the likelihood function,the normalizing constant, and the posterior distribution for N test piles with failure testloads are respectively given as follows:

L(QT|λ) =N

∏i=1

Pi(λ ≤ λTi) =N

∏i=1

Φ[

ln(λTi)− η

ξ

](6.10)

k =

∞∫−∞

f ′(X)N

∏i=1

Φ[

ln(λTi)− η

ξ

]dX

−1

(6.11)

f ′′(X) = k f ′(X)N

∏i=1

Φ[

ln(λTi)− η

ξ

](6.12)

6.4.3 Multiple test loads

More generally, suppose only M out of N test piles do not fail at λ=λT. By the combina-tion, the probability that this event occurs now becomes:

L(QT|λ) =M

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]} N−M

∏j=1

Φ

[ln(λTj)− η

ξ

](6.13)

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Updating resistance factors based on Bayesian inference 6.5 Updating procedure

and the normalizing constant is,

k =

∞∫−∞

f ′(X)M

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]} N−M

∏j=1

Φ

[ln(λTj)− η

ξ

]dX

−1

(6.14)

the posterior distribution is obtained as,

f ′′(X) = k f ′(X)M

∏i=1

{1−Φ

[ln(λTi)− η

ξ

]} N−M

∏j=1

Φ

[ln(λTj)− η

ξ

](6.15)

6.4.4 Multiple type of test

In many cases, multiple test methods including dynamic tests and static loading testsare used for construction quality assurance at a single site. In Phoon and others. (2008),Zhang stated that, about 10% of working piles may be tested using high-strain dynamictests, e.g., using a Pile Driving Analyzer (PDA), and one-half of these PDA tests shouldbe analyzed by a wave equation analysis such as the CAse Pile Wave Analysis Program(CAPWAP). Finally, at the end of construction, only 1% of working piles should be testedusing static loading tests.

According to Zhang, an order for updating pile capacity when having multiple test meth-ods carried out on a site is as follows. Taking the posterior distribution after the PDAtests as a prior, the pile performance is further updated based on the outcome of theCAPWAP analyses, and finally updated based on the outcome of the static loading tests.This exercise can be repeated if more indirect or direct verification tests are involved.

6.5 Updating procedure

In order to perform a comprehensive procedure for updating resistance factors at a singlesite, calculation steps are drawn as follows:

• Step 1: Based on the soil type and the construction method, select a suitable resis-tance prediction method by referring to the sixteen calibration cases presented inChapter 4.

• Step 2: For a selected case, the prior log-normal distribution of the resistance biasfactor was known with the initial statistical parameters such as λR, σR, and COVR(see Table 4.4). Transform the prior log-normal distribution to a prior normal dis-tribution having the mean, η′, and the standard deviation, ξ ′, as follows:

η′ = ln

λR√1 + COV2

R

(6.16)

and,

ξ ′ =√

ln(1 + COV2R) (6.17)

115

6.6 A case study Updating resistance factors based on Bayesian inference

• Step 3: Depend on the type of test loads, using some of Eqs. 6.6 through 6.15. Inwhich, the ratio, λTi , is determined as:

λTi =QTi

RPi

(6.18)

where QTi and RPi are the ith test load and the ith predicted resistance of the ithconsidered test pile, respectively. The predicted resistance follows the resistanceprediction method addressed in Step 1. The posterior normal distribution of theresistance bias factor is then determined by Eq. 6.4; its mean, η′′, and standarddeviation, ξ ′′, are therefore obtained.

• Step 4: The posterior normal distribution of the resistance bias factor obtained inStep 3 is now converted back to the posterior log-normal distribution. The poste-rior statistical parameters including λR

′′, σ′′R, and COV

′′R of the posterior log-normal

distribution are determined as:

λR′′= exp(η′′ +

12

ξ ′′2) (6.19)

and,

σ′′R = exp(η′′ +12

ξ ′′2)√

exp(ξ ′′2)− 1 (6.20)

therefore,

COV ′′R =√

exp(ξ ′′2)− 1 (6.21)

• Step 5: Based on the statistical parameters from Step 4, the re-calibration process ofthe resistance factor is performed according to the procedure presented in Section4.4 of Chapter 4.

6.6 A case study

6.6.1 A data set

The 330 MW Uong Bi Extension No. 2 Thermal Power Plant is located in Quang NinhProvince, Vietnam. The owner is Vietnam Electricity (EVN), and the prime contractor isthe Chengda Engineering Corporation of China. At this project, bored pile foundationswere used to support the superstructures for the items of the Plant. In order to evaluatethe reliability of bored pile foundations, six bored piles were tested in terms of some testmethods, including the Sonic Echo (SE) method, the Pile Driving analyzer (PDA), andthe axial and horizontal static loading tests. The contractor carrying out the tests is theVietnam Geotechnical Institute (VGI). Due to the scope of this chapter, only the PDA testresults and the axial static loading test results are considered, the information related tothe test piles is summarized in Table 6.2.

The dynamic test (PDA) was applied to the six piles, from S1 to S6. Especially, the staticloading tests were further applied to the three piles, S4 to S6. The static test load of

116

Updating resistance factors based on Bayesian inference 6.6 A case study

Table 6.2: Testing bored piles at the site of the Uong Bi Extension No. 2 Thermal PowerPlant (VGI, 2008)

Pile Diameter Length Test Date of Test load Displacement Loadname (m) (m) method test (kN) (mm) type

S1 0.8 21 PDA 22/10/08 6,464 11.50 SurvivalS2 0.8 21 PDA 22/10/08 6,934 11.10 SurvivalS3 0.8 21 PDA 22/10/08 6,008 9.20 SurvivalS4 0.8 21 PDA 22/10/08 6,780 11.80 SurvivalS5 0.8 21 PDA 22/10/08 7,016 8.30 SurvivalS6 0.8 21 PDA 22/10/08 6,821 12.10 SurvivalS4 0.8 21 Static test 11/10/08 5,886 27.72 SurvivalS5 0.8 21 Static test 08/10/08 5,886 47.60 FailureS6 0.8 21 Static test 04/10/08 5,886 25.49 Survival

0 40

-500.00

0.00

500.00

1000.00

ms

tons

5 L/c

Force MsdForce Cpt

0 40

-500.00

0.00

500.00

1000.00

ms

tons

5 L/c

For. MsdVel. Msd

20

40

60

80

tons

/m

00

175

350

525

700

tons

Shaft Resistance Distribution

Pile Forcesat Ru

175.00 350.00 525.00 700.000.00

4.00

8.00

12.00

16.00

Load (tons)

Dis

plac

emen

t (m

m) Pile Top

Bottom

Ru = 695.3 tons

Rb = 187.3 tonsDy = 12.1 mmDmx = 12.2 mm

(a) Measured and computed forces (b) Measured force and velocity

(c) Load-displacement curves (d) Shaft resistance distribution

Rs = 508.0 tons

Figure 6.1: CAPWAP analysis results for pile S6 (VGI, 2008).

5,886 kN was used, which is equal to twice the working load. For PDA test piles, themaximum displacement is extracted from the load-displacement curve in the CAPWAPanalysis results, which is imitated as a static test curve with an assumption that thiscurve may reflects an actual relation between the load and the displacement as shownin Figure 6.1(c). Zhang et al. (2006) suggested that the updating exercise is considered ifthe CAPWAP capacity is greater than the required capacity. This requirement is satisfiedwith respect to the six PDA test piles, since all the CAPWAP capacities are greater thanthe required capacity of 5,886 kN (see Table 6.2). All piles are 0.8 m in diameter, the

117


displacement criterion of 5% pile diameter therefore is 40 mm. So, only pile S5, by thestatic loading test, exposed a failure test load of 5,886 kN with a displacement of 47.60mm, which is greater than the displacement criterion of 5% pile diameter.

6.6.2 Initial prior distribution and initial resistance factors

The geological condition at the site of the Plant is layered from the top to the bottom asfollows: Earth fill, silty clay, fine sand, gravel, and weathered bed rock (SWEPDI, 2008).The pile toe of the six test piles were founded in the weathered bed rock. The percus-sion method was used with the slurry to drill holes (VGI, 2008). Therefore, from Table4.4, the calibration case is selected as Rock-IGM-Mixed. The initial prior distributionof the resistance bias factor is log-normal with the initial prior statistical parameters as:λR=1.298 and COVR=0.337. The initial prior log-normal PDF of the resistance bias factoris depicted in Figure 6.2. Based on the calibration procedure presented in Section 4.4 ofChapter 4, the initial resistance factors calibrated are φ=0.52 and 0.44, corresponding tothe target reliability indices, βT=2.5 and 3.0.

1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4


Pro

babi

lity

dens

ity, P

DF

Figure 6.2: Initial prior log-normal PDF for calibration case of Rock-IGM-Mixed.

6.6.3 Updating resistance factors based on dynamic tests

According to Zhang, an order for updating pile capacity when having multiple test meth-ods carried out on a site is as follows. Take the posterior distribution after the PDA tests,based on the outcome of the CAPWAP analyses, as a prior; and final updating is basedon the outcome of the static loading tests.

In order to assure the independence of the derived test results. The three PDA test resultsof piles S4, S5, and S6 are ignored. Updating resistance factors is only carried out basedon three PDA test piles S1, S2, and S3 in advance. These tests had the survival testloads; the likelihood function, the normalizing constant, and the posterior normal PDF

118


therefore are determined through Eqs. 6.6, 6.8, and 6.7, respectively. Figure 6.3 shows theinitial prior normal PDF, the likelihood function, and the posterior normal PDF based onthe three PDA test piles. In order to determine ratios λTi , Eq. 6.18 is used. In which,the predicted resistance of the three piles is calculated according to the IGM methodpresented by O’Neill and Reese (1999). Since the test piles were arranged in each pilegroup; piles S1 through S3 were in a pile group having the same predicted resistance as9,360 kN; while piles S4 through S6 were in another pile group having the same predictedresistance as 9,348 kN.

−1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

X

Initial prior normal PDFLikelihood functionPosterior PDF, scatterPosterior normal PDF, fitted

Figure 6.3: Initial prior normal PDF, likelihood function, and posterior normal PDF basedon the three PDA test piles.

Subsequently, the posterior normal PDF is converted back to the posterior log-normalPDF through Eqs. 6.19, 6.20, and 6.21. Figure 6.4 shows the initial prior log-normalPDF and the posterior log-normal PDF after considering the three PDA test piles. Theupdated resistance factors are indicated in Table 6.3.

Table 6.3: Updated resistance factors based on three PDA test piles

Statistical parameters Resistance factor, φ

Calculation case Distribution λR σR COVR βT = 2.5 βT = 3.0

Initial empirical distribution Prior 1.298 0.437 0.337 0.52 0.44Three PDA test piles considered Posterior 1.377 0.392 0.285 0.63 0.54

Based on Figure 6.4 and Table 6.3, it can be seen that, by the Bayesian inference, thethree PDA test piles with the survival test loads would lead to a distribution, whichhas a somewhat greater mean value and a somewhat smaller standard deviation of theresistance bias factor distribution than those of the initial distribution. The resistancefactors therefore are improved from 0.52 up to 0.63 for βT = 2.5, and from 0.44 up to 0.54for βT = 3.0.

119


1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4


Pro

babi

lity

dens

ity, P

DF

Initial prior log−normal PDFPosterior log−normal PDF

Figure 6.4: Initial prior log-normal PDF and posterior log-normal PDF after consideringthree PDA test piles.

6.6.4 Updating resistance factors based on static loading tests

In this subsection, the posterior distribution given in Subsection 6.6.3 is used as a priordistribution, from that the updating is performed again with respect to the three remain-ing static test piles. Note that, of the three static test piles, only pile S5 exposed a failuretest load. The likelihood function, the normalizing constant and the posterior normalPDF therefore are determined in terms of Eqs. 6.13, 6.14, and 6.15, respectively. Figure6.5 shows the prior normal PDF, the likelihood function, and the posterior normal PDFbased on the three static test piles.

Figure 6.6 shows the prior log-normal PDF and the posterior log-normal PDF after con-sidering three static test piles. The updated resistance factors are indicated in Table 6.4.

Table 6.4: Updated resistance factors based on three static test piles



Three PDA test piles considered Prior 1.377 0.392 0.285 0.63 0.54Three static test piles considered Posterior 0.819 0.118 0.145 0.53 0.48

Based on Figure 6.6 and Table 6.4, it can be seen that, due to the presence of a unique fail-ure test load with respect to pile S5, the three static test piles would lead to a distribution,which has a considerably decrease in the mean value and in the standard deviation of theresistance bias factor distribution compared to those of the prior distribution. However,the resistance factors slightly decrease from 0.63 down to 0.53 for βT = 2.5, and from0.54 down to 0.48 for βT = 3.0. These obtained results have a good agreement with those

120


−1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

X

Prior normal PDFLikelihood functionPosterior PDF, scatterPosterior normal PDF, fitted

Figure 6.5: Prior normal PDF, likelihood function, and posterior normal PDF based onthree static test piles.

1 2 3 40

0.5

1

1.5

2

2.5

3

3.5


Pro

babi

lity

dens

ity, P

DF

Prior log−normal PDFPosterior log−normal PDF

Figure 6.6: Prior log-normal PDF and posterior log-normal PDF after considering threestatic test piles.

given in Kanning (2012). To which, the survivals have a relatively high contribution tothe mean of the model factor, whereas the failures have a relatively high contribution tothe standard deviation.

6.6.5 Regarding the updating order

In Subsections 6.6.3 and 6.6.4, updating the resistance factors has been performed ac-cording to the order suggested by Zhang (Phoon and others., 2008), i.e., the dynamictests have been carried out in advance, and then being followed by the static tests. In or-

121


der to have a closer scrutiny on the variation of the statistical parameters of the posteriordistribution as well as the updated resistance factors, the updating will be conductedfollowing combinations of the test piles. In which, the test piles are sorted according toan assigned order; one more test pile will be taken into account for the next combination.Figure 6.7 shows the initial prior log-normal PDF and the posterior log-normal PDFsafter considering such combinations. The updated results are summarized in Table 6.5.

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4


Pro

babi

lity

dens

ity, P

DF

Initial prior log−normal PDFPosterior PDF, S1dPosterior PDF, S1d,S2dPosterior PDF, S1d,S2d,S3dPosterior PDF, S1d,S2d,S3d,S4sPosterior PDF, S1d,S2d,S3d,S4s,S5sPosterior PDF, S1d,S2d,S3d,S4s,S5s,S6s

Figure 6.7: Initial prior log-normal PDF and posterior log-normal PDFs based on testpiles with different combinations.

Table 6.5: Updated resistance factors based on test piles with different combinations


Combination case Distribution λR σR COVR βT = 2.5 βT = 3.0

Initial empirical distribution Prior 1.298 0.437 0.337 0.52 0.44S1d Posterior1 1.337 0.410 0.307 0.58 0.50S1d+S2d Posterior2 1.370 0.395 0.289 0.62 0.54S1d+S2d+S3d Posterior3 1.377 0.392 0.285 0.63 0.54S1d+S2d+S3d+S4s Posterior4 1.382 0.389 0.282 0.64 0.55S1d+S2d+S3d+S4s+S5s Posterior5 0.850 0.107 0.126 0.55 0.53S1d+S2d+S3d+S4s+S5s+S6s Posterior6 0.861 0.105 0.122 0.58 0.54

In Table 6.5, it can be seen that, the mean value of the resistance bias factor graduallyincreases with the number of the survival test piles. Conversely, the standard deviationand the COV gradually decrease with the number of the survival test piles. The updatedresistance factors therefore respectively increase from 0.52 up to 0.64 for βT = 2.5, andfrom 0.44 up to 0.55 for βT = 3.0. However, when static test pile S5s with the failure testload is included, the statistical parameters suddenly decrease from 1.382 down to 0.850,0.389 to 0.107, and 0.282 to 0.126 associated with λR, σR, and COVR, respectively. The

122


resistance factors also decrease from 0.64 down to 0.55 for βT = 2.5, and from 0.55 downto 0.53 for βT = 3.0. In the last combination, although test pile S6s is the survival testpile, the updated resistance factors slightly increase.

We also further consider an updating order. For which, the three static pile tests areconsidered in advance, and then being the three PDA test piles. The results are indicatedin Table 6.6.

Table 6.6: Updated resistance factors based on three static test piles in advance, and thenbeing the three PDA test piles



Initial empirical distribution Prior 1.298 0.437 0.337 0.52 0.443 static tests⇒ 3 PDA tests Posterior 0.859 0.101 0.118 0.59 0.54

From Tables 6.4, 6.5 and 6.6, we can conclude that, by different updating orders, theobtained statistical parameters and the updated resistance factors have more or less thesame results; i.e., the updated results are considered to be independent with the updatingorder. This is very convenient for the continuously-updated process, regardless of anytype of test loads; as long as all test piles are taken into account.

6.6.6 Effect of predicted resistance on the updated resistance factors

In the previous subsections, the predicted resistances RP, which have been calculatedaccording to the IGM method, had values of 9,360 kN for piles S1 through S3, and 9,348kN for piles S4 through S6. In order to evaluate the effect of the predicted resistancevalues on the updated resistance factors, assume that the predicted resistance may havedifferent values. The value of the predicted resistance is assigned in a relatively widerange, from under-prediction values to over-prediction ones, as shown in Table 6.7. Theupdating process is then conducted and resistance factors are then given.

The variation of the statistical parameters of the resistance bias factors, and of the up-dated resistance factors, are summarized in Table 6.7, and described in Figure 6.8.

Based on the results in Table 6.7 and Figure 6.8, we observe the following relations. λRand σR decrease with the increase of the predicted resistance. In contrast, COVR andratio, COVR/λR, increase with the increase of the predicted resistance. The updated re-sistance factors gradually decrease with the increase of the predicted resistance. Basedon the measured resistances shown in Table 6.2, if a value of 7,000 kN is taken as a limitvalue, under which the predicted resistance can be seen as underprediction resistance;and conversely, beyond which the predicted resistance can be seen as overprediction re-sistance. It can be seen that the decreasing rate of the updated resistance factor associatedwith the underprediction resistances is greater than that of the updated resistance factorassociated with the overprediction resistances. The updated resistance factor graduallydecreases and tends to be tangent with a certain value.

123


Table 6.7: Updated resistance factors based on assumed prediction resistances


Predicted resistance, RP (kN) Distribution λR σR COVR βT = 2.5 βT = 3.0

3,000 Posterior 2.339 0.250 0.107 1.63 1.524,000 Posterior 1.810 0.199 0.110 1.25 1.165,000 Posterior 1.486 0.168 0.113 1.02 0.956,000 Posterior 1.266 0.146 0.115 0.86 0.817,000 Posterior 1.106 0.130 0.118 0.75 0.708,000 Posterior 0.985 0.118 0.120 0.67 0.629,000 Posterior 0.890 0.108 0.122 0.60 0.56

10,000 Posterior 0.813 0.100 0.124 0.55 0.5111,000 Posterior 0.750 0.094 0.125 0.50 0.4712,000 Posterior 0.695 0.088 0.127 0.46 0.43

2000 4000 6000 8000 10000 120000

0.5

1

1.5

2

2.5

3

Predicted resistance (kN)

Updated resistance factor, βT=2.5

Updated resistance factor, βT=3.0

σR

COVR

Ratio, COVR

/

Over−predictionUnder−prediction

λR

λR

Figure 6.8: Statistical parameters of resistance bias factors and updated resistance factorsvs. different predicted resistances.

6.6.7 Likelihood function with static test pile behaviour

We consider again the determination of the likelihood functions in Eqs 6.6, 6.10, and6.13. In which, the mean, η, plays a role to be a variable; the standard deviation, ξ,has been assigned a value of 0.198 representing the within-site variability of the pileresistance. Therefore, the likelihood function depends only on the term ln(λT). Here, λTis the ratio of the measured test load, QT, to the predicted resistance RP. The effect ofpredicted resistance on the updated resistance factors has been considered in Subsection6.6.6. Hence, in this subsection, we will focus on the measured test load QT.

In a site, the predicted resistance, RP, is more or less the same value to all piles, unlessthere is a major difference in soil stratigraphy or in pile dimensions. This means that thelikelihood function may be only dependent on the value of the measured test load QT.

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From that view point, as in Table 6.2, static test piles S4 and S6 have the same likelihoodfunctions, because simply they have the same measured test load, QT=5,886 kN (twicethe working load). However, we can recognize that their load-displacement behaviouris quite different, for example, at QT=5,886 kN pile S4 has a displacement of 27.72 mm,meanwhile pile S6 has a displacement of 25.49 mm. Both displacements are smallerthan the displacement criterion of 5% pile diameter as 40 mm. This hints that, sincethe difference in the load-displacement behaviour of the static test piles, their likelihoodfunction should not be the same. To do so, we need to determine the load correspondingto the displacement criterion by extrapolating the load-displacement curve to 40 mm.The extrapolation is carried out using the stability plot method proposed by Chin andVail (1973). The load obtained from the extrapolation can be seen as the ”failure” testload.

For demonstration purposes, the load-displacement curve of pile S4 is extrapolated asshown in Figure 6.9. The obtained ”failure” test load for pile S4 is now QC=6,344 kN. Bythis way, the ”failure” test load for pile S6 is given as QC=6,830 kN.

0 2000 4000 6000 8000

0

10

20

30

40

50

(a) Load−Displacement curve

Load (kN)

Dis

plac

emen

t (m

m)

0 2 4 6 8

x 10−3

0

10

20

30

40

50

(b) Chin & Vail extrapolation

Ratio of Displacement to Load (mm/kN)

Dis

plac

emen

t (m

m)

Scatter dataRegression line

R2=0.962

Dc=40 mm Dc=40 mm

Qc Dc/Qc

Figure 6.9: Extrapolation of load-displacement curve for pile S4.

The updating of resistance factors is again conducted with all the test piles. Note that, atthis moment, there are the three PDA test piles with the survival test loads and the threestatic test piles with the ”failure” test loads. The results are shown in Figure 6.10 andTable 6.8.

Table 6.8: Updated resistance factors with considering ”failure” test loads



Initial empirical distribution Prior 1.298 0.437 0.337 0.52 0.44Without considering ”failure” test loads Posterior1 0.861 0.105 0.122 0.58 0.54With considering ”failure” test loads Posterior2 0.724 0.071 0.099 0.51 0.48

From Table 6.8, it can be seen that, by considering the static test pile behaviour, i.e., the”failure” test loads, the updated resistance factors are somewhat smaller than those of

125

6.7 Conclusions Updating resistance factors based on Bayesian inference

0.5 1 1.5 2 2.5 30

1

2

3

4

5

6


Pro

babi

lity

dens

ity, P

DF

Initial prior log−normal PDFPosterior PDF without considering "failure" test loadsPosterior PDF with considering "failure" test loads

Figure 6.10: Initial prior log-normal PDF and posterior log-normal PDFs with and with-out considering ”failure” test loads.

the case without considering the static test pile behaviour, e.g., φ=0.51 and 0.48 com-pared to φ=0.58 and 0.54 for βT=2.5 and 3.0, respectively. This means that the updatingof the resistance factors considering the static test pile behaviour tends to move to theconservative side.

Note that the three PDA test piles are not be considered to extrapolate the load-displacementcurves, because these imitated curves were created from the CAPWAP analyses, whichwere not directly measured curves. In addition, care should be taken when carrying outan extrapolation for actual load-displacement curves in static loading tests. According toO’Neill and Reese (1999), in cohesive soils, the shaft resistance reaches an ultimate valuewhen the shaft displacement is only about 0.6% pile diameter and subsequently the shaftresistance rapidly decreases; while the base resistance can reach an ultimate value whenthe base displacement is about 4 to 5% pile diameter. This means that the plunging phe-nomenon in cohesive soils may occur early prior to the pile displacement gaining a valueof 5% pile diameter. Therefore, extrapolating a load-displacement curve to 5% pile di-ameter may lead to an unduly overestimated resistance. Hence, it should not extrapolateload-displacement curves in cases of cohesive soils. In this case study, the soil stratigra-phy almost consists of gravel and weathered bed rock and thus, applying extrapolationto the load-displacement curves is acceptable.

6.7 Conclusions

In this chapter, an updating procedure of the resistance factors has been presented basedon the Bayesian inference through the experimental outcomes of pile loading tests at asite. Aside from the regional calibration of resistance factors as introduced by Abdel-Salam et al. (2010) in Chapter 4, the Bayesian inference enables to reduce uncertaintywith respect to initial empirical distributions in terms of load test results within a certain

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Updating resistance factors based on Bayesian inference 6.7 Conclusions

site. The resistance factors therefore may be improved and a more precise design may beaccomplished.

It can be seen that the within-site variability of pile resistance, represented through thecoefficient of variation (COV), is an important reference factor, based on which, the likeli-hood functions are determined. In this chapter, a value of COV=0.2 was used; this valuehas been adopted by several authors, e.g., Kay (1976), Zhang (2004), and Su (2006). Thestandard deviation, ξ, in the likelihood functions therefore was given as 0.198.

A general procedure based on the Bayesian inference in combination with the calibrationprocess for updating resistance factors has been proposed. In which, a selected predic-tion method has to pertain to a certain soil type and a certain construction method. Aninitial empirical distribution is taken from the list of the calibration cases in Chapter4; the likelihood functions are established through pile loading tests on a site, and fi-nally, the posterior distribution of the resistance bias factor is given. Subsequently, there-calibration is carried out, and updated resistance factors are then derived.

A case study with the multiple type of tests has been presented. Based on the analysisresults, some findings can be drawn as follows:

• The posterior statistical parameters and the updated resistance factors are consid-ered to be independent with the updating order, regardless of any type of test loads;as long as all test piles are taken into account.

• By the Bayesian inference, the uncertainty of the posterior distribution decreasescompared to the prior distribution, this can be seen through σR and COVR of theposterior distributions. They decrease with the increase of the number of the testpiles. Furthermore, the survival test load will result in the increase in the λR andin the updated resistance factors. In contrast, the failure test load will lead to thedecrease in the λR as well as in the updated resistance factors.

• The updated resistance factors gradually decrease with the increase of the pre-dicted resistance and tends to be tangent with a certain value. However, the de-creasing rate of the updated resistance factor associated with the under-predictionresistances is greater than that of the updated resistance factor associated with theover-prediction resistances.

• When considering the static test pile behaviour with respect to the likelihood func-tions, the updated resistance factors are somewhat smaller than those of the casewithout considering the static test pile behaviour. This means that the updating ofresistance factors, considering the static test pile behaviour, tends to move to theconservative side.

127

6.7 Conclusions Updating resistance factors based on Bayesian inference

128

Chapter 7

Reliability-based design

7.1 Introduction

In Chapter 4, the LRFD uses the resistance factors that were obtained through the calibra-tion process and satisfied the specified target reliability levels. This approach does notrequire the explicit use of the probabilistic description of random variables and thereforeit has been familiar to design engineers in terms of its simplicity. In current practice, how-ever, clients and project managers are more and more interested in the reached reliabilitylevel or the probability of failure of a pile foundation designed. Therefore, this chapterwill present a calculation procedure, normally called the Reliability-Based Design (RBD),aiming to directly estimate reliability levels for a specific bored pile foundation.

From Chapters 4 to 6, the model uncertainties were considered to calibrate the resistancefactors. In this chapter, the parameter uncertainties will be treated in soil models. Besidereliability evaluation for an intact bored pile, the influence of possible defect types andthe soft bottom situation at the pile base, caused by an imperfect construction procedureon the safety degree of a single bored pile, will be analyzed in detail.

The commercial software, Plaxis, has been widely used in the world. It is a powerful toolto efficiently solve problems in the field of geotechnical engineering based on the FiniteElement Method (FEM). At present the constitutive soil models in Plaxis are still usingsoil properties as deterministic parameters, meanwhile those actually are random vari-ables following certain probability distributions (Baecher and Christian, 2003). So, theoutcomes from Plaxis calculations, for example as resistances or displacements, shouldbe demonstrated to be random values under certain probability distributions. To do so,we need another software that enables us to control the random soil properties and tohandle Plaxis running with the specified property values of soils. The Prob2B software,formerly named Probox (Courage and Steenbergen, 2005), can meet such a requirement.Prob2B is a numerical probabilistic toolbox which is capable of being coupled to anddriving other softwares. It performs reliability analyses by steering stochastically ob-tained input to the deterministic external programs, and processing their output usingprobabilistic techniques (Courage, 2007).

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7.2 Coupling calculation Reliability-based design

The outline of this chapter is as follows. In Section 7.2, a coupling calculation procedureby Prob2B and Plaxis is described. The models in Plaxis, which consist of geometry andmaterial models for a bored pile and soil layers, are described in Section 7.3. A case studyis applied in Section 7.4 and finally the chapter ends with conclusions in Section 7.5.

7.2 Coupling calculation

7.2.1 Reliability methods in Prob2B

TNO Building and Construction Research has developed Prob2B, a toolbox for the reli-ability analysis. The limit state functions are defined in advance, in which all randomvariables are directly described in Prob2B or some of them may be given from the outputof the external programs, such as Matlab, MathCad, Excel, FEM, and so on. It meansthat Prob2B can run well in combination with other programs. In this chapter, the Plaxissoftware version 9.0 is used as an external program aiming to model the interaction be-tween a single bored pile and surrounding soil layers. The available reliability methodsin Prob2B are:

• First Order Reliability Method (FORM);

• Second Order Reliability Method (SORM);

• Directional Adaptive Response Surface sampling (DARS);

• Directional Sampling (DS);

• Monte Carlo Simulation (MCS);

• Increased Variance sampling (IV);

• Numerical Integration (NI).

In this chapter, FORM is used for the reliability analysis due to its facility. The level IIImethods are either used to confirm the applicability of FORM or for limit state functionsthat are non-linear and include system effects (Schweckendiek et al., 2007). The use ofMCS is not always available when performing a coupling calculation. Its calculationeffort depends on the probability of failure, to which the expected number of simulationscan be calculated by Eq. 2.49 (Vrijling and Van Gelder, 2002). For example, a probabilityof failure of 0.001 is expected, and a Plaxis realization spending only, e.g., 10 seconds ofcalculation time, will lead to a required amount of time of at least 1.5 months. It is reallyvery time-consuming, and in cases involving more time for the Plaxis realization, thecoupling calculation may be impossible.

In principle all random variables can be assigned probability distributions to accountfor uncertainties. Prob2B contains 14 distribution types and allows the definition of dis-tributions by means of tables. Especially, the correlations between the variables can beintroduced in the form of a correlation matrix.

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Reliability-based design 7.2 Coupling calculation

7.2.2 Coupling Prob2B-Plaxis

The coupling calculation procedure between Prob2B and Plaxis is described in Figure 7.1.In the input stage, a Plaxis model has to be first defined. The variables in the Plaxis modelare now deterministic ones given mean values. Prob2B then loads the Plaxis model as anexternal model, i.e., the Plaxis model and its input as well as output variables are knownto Prob2B. The variables which are assigned as random variables will be declared againin the Prob2B program in addition to their respective statistical parameters. Note thatthe variables that do not belong to the Plaxis model earlier, but that will be present inthe limit state functions, are also declared at this stage. The limit state functions arethen defined; all random variables are synthesized in a table and a correlation matrixbetween them is defined as well. Finally, a reliability method used in Prob2B is specifiedwith some setup parameters.

In the calculation loop stage, corresponding Plaxis input files are amended and a Plaxiscalculation is performed. After each calculation the relevant outcomes are read fromthe corresponding Plaxis output files and the limit state function is evaluated. This pro-cess is repeated until the pre-defined convergence criteria are reached and the reliabilityanalysis results are presented including: Reliability index, β; probability of failure, Pf ;influence coefficients (direction cosines), α∗i ; and design point values, x∗i .

Figure 7.1: Coupling scheme Prob2B-Plaxis (Schweckendiek et al., 2007).

131

7.2 Coupling calculation Reliability-based design

7.2.3 Limit state functions

Two types of limit state are usually considered in design, the Ultimate Limit State (ULS),and the Serviceability Limit State (SLS). In this chapter, only the ULS is considered, theSLS is beyond the scope of the thesis. The definition of the failure modes under the ULSis crucial for reliability analyses, to which limit state functions will be established. Twofailure modes are proposed, which are the Geotechnical Failure (GF) mode and the Struc-tural Failure (SF) mode. The GF mode pertains to the geotechnical resistance of boredpiles and the SF mode is related to the compressive stress in bored pile concrete. Bothmodes are evaluated through reached reliability levels for bored piles that are subjectedto a specified combination of loads from the superstructure.

For the GF mode, if only the dead load and the live load are considered, the limit statefunction is defined as:

gG = R− λQDQD − λQLQL (7.1)

where R is the axially compressive geotechnical resistance given from the Plaxis calcula-tion result, hereafter it is briefly called the resistance; QD and QL are the dead load andthe live load from the superstructure, respectively; λQD and λQL are the bias factor forthe dead load and the live load, respectively (see Table 4.2).

For the SF mode, the limit state function is:

gS = λ f n fn − σc (7.2)

in which fn is the norminal compressive strength of concrete; λ f n is the bias factor ofcompressive strength of concrete. The probabilistic characteristics of λ f n are indicatedin Table 7.1 (Nowak and Rakoczy, 2010). σc is the maximum compression stress in thepile concrete caused by external loads; the value of σc is also obtained from the Plaxiscalculation result.

Table 7.1: Concrete strength and probabilistic characteristics of λ f n

Type of concrete fn (kN/m2) Mean of bias factor, λ f n COVf n DistributionGood concrete 30,000 1.189 0.159 NormalPoor concrete 15,000 1.333 0.145 Normal

By FORM, the convergence criterion with respect to the limit state functions, g, is deter-mined:

|g| < tolerance (7.3)

and then, the convergence criterion associated with the reliability index, β, has to satisfy

|βi − βi−1| < tolerance (7.4)

here βi and βi−1 are the reliability index given at the ith and (i-1)th reliability calculationsteps.

132

Reliability-based design 7.3 Models in Plaxis

Fault trees can be used for the demonstration of failure modes. In Figure 7.2, a fault treeis presented for a case study of a bored pile. The ULS is used with respect to either anintact bored pile or a defect one. Unlike driven piles, the quality of bored piles is affectedby the construction procedure. An imperfect construction procedure may result in sometypes of defect such as necking, poor concrete or soft bottom as mentioned in Chapter3. Therefore, aside from the intact pile, reliability analyses under the GF and SF modesare also conducted for a defect pile with some types of defects that may occur in the pilebody or at the pile base (soft bottom). The analysis results will be presented in the casestudy of Section 7.4.

Figure 7.2: Fault tree under failure modes.

7.3 Models in Plaxis

7.3.1 Simulation procedure

Finite element analyses are conducted using the Plaxis software version 9.0, since Prob2Bis compatible with this version at this time. In Plaxis, a phased construction scheme isavailable, which allows users to simulate construction procedures; at each phase, thegeometry of the model, the prescribed displacement, and the groundwater level can bemodified. Figure 7.3 shows a construction procedure of a bored pile. In Phase 1, the shafthole is augered from the ground surface. In Phase 2, the concrete placement is conducted.

133

7.3 Models in Plaxis Reliability-based design

Finally, in Phase 3, the pile top is vertically displaced with a prescribed displacement, andresistance-displacement curves are obtained.

Figure 7.3: A bored pile construction procedure simulated in Plaxis.

7.3.2 Geometry model

Since a bored pile has a shape of a circular cylinder, Plaxis 2D may be used to carryout two-dimensional finite element analyses with the axisymmetric model. The 15-nodetriangle is a very accurate element that produces high quality stress results for difficultproblems. However, the use of 15-node triangles leads to relatively high memory con-sumption and relatively slow calculation and operation performance. The 6-node trian-gle is a fairly accurate element that gives good results in standard deformation analyses.In reliability analyses, the number of iterative calculations is quite large; this requires ahuge amount of time for calculation. Therefore, in this chapter, the 6-node triangle ele-ments are used to model geometry. In order to partly overcome the shortcoming of usingthe 6-node triangle elements as mentioned above, the finite element mesh at the vicinityof bored pile surface is set to be very fine.

Interface elements are used to model the soil-pile interaction; the shear strength in theinterface could be reduced by a strength reduction factor, Rinter. This factor appears tovary widely depending on the quality control of the construction site (Kulhawy, 1991).The interface properties are calculated from the soil properties and the strength reductionfactor by applying the following rules (Wehnert and Vermeer, 2004, Brinkgreve et al.,2008):

ci = Rinterc′ (7.5)

and,tanϕi = Rintertanϕ′ (7.6)

134


where ci and c′ are the cohesion of interface and soil, respectively; ϕi and ϕ′ are thefriction angle of interface and soil, respectively.

The strength of interface elements is governed by the Coulomb criterion:

τ = σntanϕi + ci (7.7)

in which τ and σn are the shear stress and the normal stress in the interface element,respectively.

7.3.3 Material models

In finite element analyses, the bored pile concrete is assumed to be a linear elastic non-porous material. For the subsoils, three constitutive models are used, i.e., the linearelastic perfectly-plastic Mohr-Coulomb model, the Hardening-Soil model, and the Soft-Soil model. These three models are briefly introduced below.

The linear elastic perfectly-plastic Mohr-Coulomb model (MC)

The Mohr-Coulomb model as shown in Figure 7.4 is a linear elastic perfectly-plasticmodel which is often used to model soil behaviour in general and serves as a first-ordermodel. The model’s stress-strain behaves linearly in the elastic range with two parame-ters defined from Hooke’s law, that are the Young’s modulus, E′, and the Poisson’s ratio,ν. There are two parameters, which define the failure criterion, the friction angle, ϕ′,and the cohesion c′. Also, another parameter, the dilatancy angle, ψ, is used to model arealistic irreversible change in volume due to shearing.

Figure 7.4: Basic idea of a linear elastic perfectly-plastic model.

The Hardening-Soil model (HS)

The Hardening-Soil model is an advanced model to simulate the behaviour of differ-ent soil types. When subjected to primary deviator loading q; soil shows a decreasingstiffness and irreversible plastic strains develop. In the special case of a drained triaxial

135


test, the relationship between the axial strain, ε1, and the deviator stress, q, can be ap-proximated by a hyperbola. Such a relationship was first formulated by Kondner (1963)and later used in the well-known hyperbolic model of Duncan and Chang (1970). TheHardening-Soil model, however, supersedes the hyperbolic model by far: (1) Using thetheory of plasticity rather than the theory of elasticity, (2) including soil dilatancy, and(3) introducing a yield cap (Brinkgreve et al., 2008).

A basic feature of the Hardening-Soil model is the stress dependency of soil stiffness. Foroedometer conditions of stress and strain, this relationship is demonstrated as:

Eoed = Ere foed(

σ′1pre f )

m (7.8)

where Eoed is the oedometer stiffness being defined as a tangent stiffness modulus; pre f

is the reference pressure, in Plaxis, pre f is taken as 100 kN/m2; σ′1 is the effective com-pressive pressure in the oedometer test; Ere f

oed is the reference tangent stiffness moduluscorresponding to the reference pressure pre f ; m is the power, in the special case of softsoil it is realistic to use m = 1.

Figure 7.5: Hyperbolic stress-strain relation in primary loading for a standard drainedtriaxial test.

Figure 7.5 describes the hyperbolic relationship between vertical strain, ε1, and the devi-ator stress, q, in a primary triaxial loading. Here standard drained triaxial tests tend toyield curves that can be describes by:

ε1 =q

Ei(1− qqa)

(7.9)

where qa is the asymptotic value of the shear strength and Ei is the initial stiffness. Ei isrelated to the secant stiffness, E50, by

Ei =2E50

2− R f(7.10)

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in which R f is the failure ratio, in Plaxis R f = 0.9 is chosen as a suitable default setting.qa and the ultimate deviator stress, q f , have a relation

qa =q f

R f(7.11)

here, q f is determined as

q f =2sinϕ′

1− sinϕ′(c′cotϕ′ − σ′3) (7.12)

where c′ and ϕ′ are the effective cohesion and the effective friction angle of soil, respec-tively; σ′3 is the confining effective pressure in a triaxial test.

The parameter E50 is the confining stress dependent secant stiffness modulus for primaryloading and is given as:

E50 = Ere f50 (

c′cosϕ′ + σ′3sinϕ′

c′cosϕ′ + pre f sinϕ′)m (7.13)

where Ere f50 is the reference secant stiffness modulus corresponding to the reference con-

fining pressure pre f . The amount of stress dependency is given by the power m. For softclays, the power should be taken as 1.0 (Brinkgreve et al., 2008). Janbu (1963) reported avalue of m of around 0.5 for Norwegian sands and silts, while von Soos (1990) reportedvarious values in the range 0.5< m <1.0.

For unloading and reloading stress paths, another stress-dependent stiffness modulus,Eur, is used:

Eur = Ere fur (

c′cosϕ′ + σ′3sinϕ′

c′cosϕ′ + pre f sinϕ′)m (7.14)

here Ere fur is the reference Young’s modulus for unloading and reloading, corresponding

to the reference pressure pre f . In many practical cases it is appropriate to set Ere fur equal

to 3Ere f50 ; this is the default setting used in Plaxis (Brinkgreve et al., 2008).

The Soft-Soil model (SS)

In the Soft-Soil model, it is assumed that there is a logarithmic relation between the volu-metric strain, εv, and the mean effective stress, p′, which can be formulated for the virgincompression as:

εv − ε0v = −λ∗ln

(p′

p0

)(7.15)

where λ∗ is the modified compression index, which determines the compressibility of thematerial in primary loading. Note that λ∗ differs from the index, λ, as used by Burland(1965). Since Eq. 7.15 is a function of volumetric strain instead of void ratio. Plotting Eq.7.15 gives a straight line as shown in Figure 7.6.

During isotropic unloading and reloading a different path is followed, which can beformulated as:

εev − εe0

v = −κ∗ln(

p′

p0

)(7.16)

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Figure 7.6: Logarithmic relation between volumetric strain and mean stress.

here κ∗ is the modified swelling index, which determines the compressibility of the ma-terial in unloading and subsequent reloading. Note that κ∗ differs from the index κ asused by Burland (1965). However, the ratio λ∗/κ∗ is equal to Burland’s ratio λ/κ. Thesoil response during unloading and reloading is assumed to be elastic as denoted by thesuperscript, e, in Eq. 7.16.

7.3.4 Calculation types

There are three basic types of calculation: Plastic calculation, Consolidation analysis andPhi − c reduction (safety analysis). In addition, Dynamic calculation is also available,which is an extension to Plaxis 2D (Brinkgreve et al., 2008). With respect to the prob-lem of a single bored pile subjected to axially-acting loads, it is consistent to select thePlastic calculation. Reliability analyses are subsequently conducted, aiming to predict aprobability of failure or a reliability index for a designed bored pile.

The plastic calculation carries out an elastic-plastic deformation analysis in which it isnot necessary to take the decay of excess pore pressure with time into account. If theupdated mesh option in the advanced general setting has not been selected, the calcu-lation is performed according to the small deformation theory. The stiffness matrix in anormal plastic calculation is based on the original undeformed geometry. This type ofcalculations is appropriate in most practical geotechnical applications.

In Plaxis, the resistance of pile is estimated corresponding to a prescribed displacement.It is necessary to address a displacement; at which the obtained resistance may be consid-ered as the ultimate resistance. Unlike driven piles, the resistance of bored piles dependsnot only on the soil conditions but also the inspection quality during construction. Dueto the construction approaches, the soil surrounding bored pile is disturbed; especiallythe softening of soil caused by a drilling auger or a thick deposit of drilling slurry willresult in a considerable decrease in resistance at the pile base. So, for bored piles, a dis-placement criterion of up to 5%D (D is the pile diameter) was adopted by O’Neill andReese (1999) and Reese et al. (2006). Paikowsky et al. (2004) also evaluated that this cri-

138

Reliability-based design 7.4 A case study

terion provides a reliable and simple failure interpretation. Therefore, a displacement of5%D is used as a maximum prescribed displacement for the case study in this chapter.

7.4 A case study

7.4.1 Project description

The An Dong bridge spans the Dinh river in Ninh Thuan province, Vietnam. The bridgeis a component of the road construction Project along the Ninh Thuan provincial coastline, connecting An Hai commune with Ninh Phuoc district in Phan Rang-Thap Chamcity. The Project was launched in May 2011 and is expected to be completed within threeyears.

Figure 7.7: Pier T10 layout and stratigraphy.

The total length of the route is 3,518 m, in which the two approaching roads of the bridgeare 2,500 m long, the main part of bridge is 1,018 m long and 20.5 m wide for four lanes.The bridge consists of 15 piers (T1 to T15), 2 abutments (M1, M2) with 16 spans, in whichthere are three 140 m long main spans. Bored pile foundations are used to support thesuperstructure with a diameter varying from 1.2 to 2.0 m and a length ranging from 30through 32 m. All the designed pile toes were founded in a weathered rock layer.

In this case study, pier T10 is taken as a calculation example; the layout of pier T10 andthe stratigraphy at its location are shown in Figure 7.7. The foundation of pier T10 wasdivided into two separate ones, each has six 1.2 m diameter bored piles with a length of30 m. They were first drilled through six layers of soil, which are the fill sand, fine sand,silty clay, sandstone, clayey sand, and weathered rock layers. The pile toes were 2.0 m

139

7.4 A case study Reliability-based design

embedded in the weathered rock layer and reached a level of -33.00. The pile head levelof the constructed bored pile was at +3.45. After the static loading test completed, thepile head was then cut off at the level of -2.90; that is the pile head level of the workingpile later; the pile cap bottom level was designed at -3.00.

7.4.2 Loads on a pile

The axially compressive loads (working loads) exerting on a pile consist of dead loadand live load. The dead load is determined from the self-weight of superstructure’scomponents, such as desk slabs, barriers, longitudinal and transverse girders, joints, apier cap, a pier body, a pile cap, and so on. The self-weight of bored pile is included inPlaxis program. The live load is determined from vehicles and pedestrians. The lateralloads in general, and the wind load in particular, may be the restrictive loading conditionfor deep foundation design. Lateral capacity of piles, which is usually controlled by theSLS, is excluded from the scope of this chapter.

Based on the given calculations, the dead load and the live load for a single bored pileare used in the case study as 2,204 kN and 272 kN, respectively. The bias factors for thedead load, λQD, and the live load, λQL, in Eq. 7.1 are taken from Table 4.2.

7.4.3 Material models and soil properties

Substantial material parameters, used for calculation models, are frequently obtainedfrom the data of geological surveys, or are sometimes based on the experimental corre-lations between parameters, or expert judgments in cases of lacking information.

The bored pile concrete was assumed to be a linear elastic non-porous material. Thestrength class for the designed concrete was C30, corresponding to the nominal compres-sive strength with respect to the cube specimen as fn=30 MPa, and the Young’s modulusas 31 GPa; the Poisson’s ratio, ν, was assigned as 0.1.

The HS model is an advanced model for simulating the behaviour of different soil types,both soft soils and stiff soils (Schanz, 1998). In comparison with a static loading testresult of a bored pile in clay, Wehnert and Vermeer (2004) supposed that the HS modelseems to make the best fit result. Youn and Tonon (2010) used the HS model to analyzethe behaviour of a post-grouted bored pile in a layered stratigraphy and received a quitesuitable result compared to a STATNAMIC result. In this case study, the HS model istherefore applied to the filled sand, fine sand, silty clay, sandstone, and clayey sandlayers.

The MC model is a consistent model to describe the stress-strain relation for weatheredrocks as indicated in Vallejo and Ferrer (2011). Kim et al. (1999), by using the MC model,created the best matched results with respect to the measured results of nine instru-mented bored piles aiming to determine the load-transfer mechanism at the interfacebetween the shafts and surrounding weathered rocks. Therefore, for the weathered rocklayer surrounding the pile toe, the MC model is used to simulate the behaviour of thislayer.

140


Apart from the soil properties that were already synthesized in the geological surveyreport, the additional soil properties and other input parameters used in soil models,would be back-calculated by matching the load-displacement curve of the Plaxis anal-ysis with that of the static loading test on the site. In Figure 7.8, the load-displacementcurve obtained from the Plaxis analysis is plotted to a displacement of 12 mm, which isequal to one percent of the pile diameter (1%D). Meanwhile, the load-displacement curvegiven from the static loading test only reached a displacement of 8 mm, it is even smallerthan 1%D and very far from the required displacement of 5%D=60 mm as stated in Sub-section 7.3.4. This may be explained as follows, although the measured displacementwas small, the measured load exerting on the pile head also reached a value of 7,848kN, which is two times greater than the working load as 4,952 kN. Therefore, consul-tant engineers might make a decision to stop the test at this displacement. In fact, staticloading tests are permanently carried out to the load being equal to two times of theworking load, following specifications such as ASTM D1143 (1994), TCXDVN269:2002(2002), and Buildings Department (2002). Thus, other additional parameters are chosen,based on the back-calculation, so that the load of the Plaxis analysis is equal to that ofthe static loading test at the measured displacement of 8 mm as shown in Figure 7.8.

0 2000 4000 6000 8000 10000 12000

0

2

4

6

8

10

12

14

Load (kN)

Dis

plac

emen

t (m

m)

Static loading testPlaxis model

1%Diameter

Figure 7.8: Load-displacement curves obtained from static loading test and Plaxis analy-sis.

By the default setting in Plaxis, the reference secant stiffness in triaxial testing is set tobe equal to the reference tangent stiffness in oedometer testing; the reference unload-ing/reloading stiffness is set to be equal to three times of the reference secant stiffness;i.e., Ere f

50 = Ere foed and Ere f

ur = 3Ere f50 (Brinkgreve et al., 2008). The stiffness parameters of the

fine sand, silty clay, and clayey sand layers were directly determined from the oedome-ter tests. The stiffness of the sandstone layer is determined based on the experimentcorrelation addressed by Kulhawy and Mayne (1990):

E′

Pa≈ 15N60 (7.17)

141


where E′ is the elastic modulus; Pa is the atmospheric pressure, Pa=101.3 kPa; N60 is theSPT index corrected for field procedure to an average energy ratio of 60 percent. Here,the average corrected SPT index of the sandstone layer is greater than 100. Therefore, theelastic modulus of this layer is obtained as 152 MPa. In general, Ere f

50 lies in a range of(0.9÷ 1.0)E′, herein Ere f

50 is then taken as 150 MPa.

By the back-calculation, the elastic modulus of the weathered rock layer is given as 1,100MPa with the dilatancy angle, Ψ, assumed as 5◦. Kim et al. (1999) obtained a valueof 1,000 MPa for the elastic modulus of the weathered rock based on site investigationand pile loading tests. Even a value of up to 2,400 MPa for the elastic modulus of theweathered rock was used by Ooi et al. (2010), aiming to evaluate the working state ofa bridge supported by bored piles founded on a weathered rock layer. Therefore, thevalue of 1,100 MPa for the elastic modulus of the weathered rock layer used in this studyis acceptable. All initial and back-calculated parameters of the soil layers are listed inTable 7.2.

Table 7.2: Initial and back-calculated soil parameters

Parameter Unit Fill Fine Silty Sandstone Clayey Weatheredsand sand clay sand rock

Soil model - HS HS HS HS HS MCE′ MPa - - - - - 1,100Ere f

oed MPa 10 2.75 3.48 150 7.19 -Ere f

50 MPa 10 2.75 3.48 150 7.19 -Ere f

ur MPa 30 8.26 10.45 450 21.58 -ν - 0.2 0.2 0.3 0.2 0.25 0.2γsat

(1) kN/m3 20 20 18.6 19.4 20.2 22γunsat

(2) kN/m3 18 16 13.7 15.1 16.1 19ϕ′ degree 30 25 21 19 21 43c′ kN/m2 - - 11.2 40.9 47.7 -m - 0.5 0.5 1.0 0.5 1.0 -Ψ degree - - - - - 5Rinter - 1.0 0.9 0.8 0.8 0.7 0.85(1),(2) Saturated and unsaturated unit weights, respectively.

7.4.4 Soil parameter uncertainties

In Table 7.2, the soil parameters were considered to be deterministic and to have meanvalues. In fact, soil parameters have a variability, which is a complex attribute that re-sults from many disparate sources of uncertainties. There are three primary sources ofgeotechnical uncertainties, which are inherent variability, measurement error, and trans-formation uncertainty, as illustrated in Figure 7.9 by Kulhawy (1992). The first resultsprimarily from natural geologic processes that produced and continually modify the soilmass in situ. The second is caused by equipment, procedural operator, and random

142


testing effects. The third source of uncertainty is introduced when field or laboratorymeasurements are transformed into design soil properties using empirical or other cor-relation models (Phoon and Kulhawy, 1999). Therefore, the soil parameters need to becharacterized in terms of random variables, their statistical parameters and probabilitydistributions will be discussed below.

Figure 7.9: Uncertainty in soil property estimates (Kulhawy, 1992).

Since the fine sand and the silty clay are two weak soil layers with small thickness; theircontribution to the resistance of pile is inconsiderable. Therefore, the soil parameters ofthese two layers are considered to be deterministic with the values as indicated in Table7.2. Three remaining soil layers, the sandstone, the clayey sand, and the weathered rockare hard soil layers and play an important role in deciding the resistance of pile. Thus,the soil parameters of these three layers are treated as random variables.

Youn and Tonon (2010) addressed three essential points which affect the Plaxis results:(1) Base resistance is controlled by c′ and ϕ′ of the underlying soil layers; (2) side shearstrength is controlled by strength reduction factors, Rinter, of the interface elements; and(3) the load-displacement curve slopes are determined by secant stiffness moduli, Ere f

50 .Of the soil parameters listed in Table 7.2, the saturated unit weights, the stiffness parame-ters, the strength parameters, and the strength reduction factors of the interface elementsare chosen and treated as random variables.

One of the statistical parameters of random variable is described in terms of coefficientof variation (COV). For a specific site of the case study, the soil variability is evaluatedas within-site variability, i.e., the variability within similar subsurface conditions of thesame site. Paikowsky et al. (2004) categorized the site variability in the following ways:(1) COV<0.25 - low, (2) 0.25≤COV<0.40 - medium, and (3) COV≥0.40 - high. Phoonand Kulhawy (1999) proposed a range of COV=0.02÷0.13 for the unit volumetric weightof fine grained soil type. Schweckendiek et al. (2007) used the COVs of 0.05, 0.25, 0.20,and 0.20 for the saturated unit weights, stiffness parameters, strength parameters, andstrength reduction factors, respectively. Orr and Breysse (2008) listed the ranges of COVsas 0.01÷0.10, 0.05÷0.15, and 0.20÷0.40 for the unit weights, effective friction angles,

143


and cohesions, respectively. Partly based on NEN-6740 (1993) and JCSS (2000), Wolters(2012) once again utilized COVs of 0.05, 0.30, 0.20, and 0.20 for the unit weights, stiffnessparameters, strength parameters, and strength reduction factors, respectively.

In the geological survey Report for the An Dong bridge Project, unfortunately, no speci-men of the weathered rock layer was given. There were few specimens of the sandstonelayer taken from the bore holes; the experimental data of this layer was insufficient tocarry out statistical analyses. With respect to the clayey sand layer, some representativeparameters were taken and statistically analyzed; the COVs of γunsat, Ere f

oed, c′, and ϕ′ areobtained as 0.08, 0.25, 0.37, and 0.15, respectively. Hence, the statistical parameters forthe sandstone and weathered rock layers will accordingly refer to those in the studies ofthe authors as mentioned above. The selected statistical parameters of the soil layers arecharacterized in Table 7.3.

Table 7.3: Statistical parameters and probability distributions of soil parameters

Soil layer, Unit Mean COV DistributionparameterSandstoneEre f

oed MPa 150 0.20 Log-normalEre f

50 MPa 150 0.20 Log-normalEre f

ur MPa 450 0.20 Log-normalγsat kN/m3 19.4 0.05 Normalϕ′ degree 19 0.20 Normalc′ kN/m2 40.9 0.20 Log-normalRinter - 0.8 0.20 NormalClayey sandEre f

oed MPa 7.19 0.25 Log-normalEre f

50 MPa 7.19 0.25 Log-normalEre f

ur MPa 21.58 0.25 Log-normalγsat kN/m3 20.2 0.08 Normalϕ′ degree 21 0.15 Normalc′ kN/m2 47.7 0.37 Log-normalRinter - 0.7 0.20 NormalWeathered rockE′ MPa 1,100 0.20 Log-normalγsat kN/m3 22 0.05 Normalϕ′ degree 21 0.20 NormalRinter - 0.85 0.10 Normal

In principle, probability distributions of soil parameters, have a normal distribution.Unit weights and friction angles of soils follow the normal distribution (Baecher andChristian, 2003). For stiffness parameters, a log-normal distribution is used, according totheir derivation from CUR-166 (2005). With regard to cohesion c′, because it has a large

144


scatter and negative values of c′ could occur in the lower tail of the normal distribution,a log-normal distribution is used instead of a normal distribution (Schweckendiek et al.,2007; Wolters, 2012). Due to lack of information, a normal distribution was applied tothe strength reduction factors of interface. The selected probability distributions of thesoil layers are listed in Table 7.3.

The correlation between stochastic soil parameters is evaluated through correlation co-efficients, which are summarized in a matrix in Table 7.4. The values of these correlationcoefficients are referred to in Gemeentewerken-Rotterdam (2003).

Table 7.4: Correlation matrix of stochastic soil parameters

Parameter E′ Ere foed Ere f

50 Ere fur γsat ϕ′ c′

E′ 1.0 - - - 0.5 - -Ere f

oed - 1.0 1.0 1.0 0.5 - -Ere f

50 - 1.0 1.0 1.0 0.5 - -Ere f

ur - 1.0 1.0 1.0 0.5 - -γsat 0.5 0.5 0.5 0.5 0.5 - 0.5ϕ′ - - - - - 1.0 -0.65c′ - - - - 0.5 -0.65 1.0

7.4.5 Reliability of intact bored pile

Introduction

In this case study, reliability analyses are first conducted for an intact bored pile. For pierT10, the number of piles per pile cap is 6.0; the pile diameter is 1.2 m; and the designedpile head level is at -3.00. The design task herein is to evaluate the reliability of a singlebored pile when the pile toe is shifted at different levels. From that a unique pile toelevel will be chosen to satisfy a required reliability level and other essential requirements.We consider nine calculation cases corresponding to nine different levels of the pile toewhich are founded in the sandstone, clayey sand, and weathered rock layers as describedin Figure 7.10.

Regarding the prescribed displacement applied to the Plaxis calculations, the displace-ment of 5%D is first used as mentioned in Subsection 7.3.4. Since the static loading testwas completed at a measured displacement of 8 mm, which is even smaller than 1%D=12mm. Thus, a displacement of 1%D is also used in calculations. In addition, a displace-ment of 3%D is also taken into account aiming to assess the sensitivity of obtained results.

General results

Figure 7.11 shows the resistance of the nine calculation cases with the mean values of thesoil parameters following the three prescribed displacements as 1%D, 3%D, and 5%D. Ingeneral, the resistance increases as the depth, at which the pile toe is founded, increases.From Case 1 to Case 6, when the pile toe is within the sandstone and clayey sand layers,

145


Figure 7.10: Nine calculation cases considered with different pile toe levels.

the increase in resistance between the displacements of 3%D and 5%D is inconsiderable.The pile probably approaches the ultimate state as the displacement reaches 3%D. Con-versely, from Case 7 to Case 9, as the pile toe is founded on/in the weathered rock layer,the resistance increases dramatically with the increase of the displacement. Care shouldbe taken when the pile toe is founded in the sandstone layer (Cases 1 to 3); under thesandstone layer is the clayey sand having a secant stiffness of 7.19 MPa, which is muchsmaller than that of the sandstone layer with a secant stiffness of up to 150 MPa. In thesecases, the lower pile toe level results in the smaller resistance; as the remaining thicknessof the sandstone layer under the pile toe becomes smaller (see Figure 7.10). This problemwas forewarned in a study of Poulos (2005), when considering compressible strata belowa founding stratum. So, Case 1 to Case 3 do not assure reliability. It can be seen that theadvantage of a FEM analysis is to clearly reflect this problem in terms of the stress-strainrelation through used stiffness parameters, meanwhile other theoretical analytical meth-ods are hardly able to address such a problem in calculations.

Similarly, the maximum compressive stress occurring in the pile concrete for the ninecalculation cases reveals the same behaviour as that of the resistance. All cases show thecalculated maximum compressive stresses which are smaller than an allowable compres-sive stress fn=30,000 kN/m2, as indicated in Figure 7.12.

Geotechnical failure mode

To analyze reliability according to the GF mode, the limit state function in Eq. 7.1 isused. In which the load effect consists of the dead load, QD, and live load, QL, as statedin Subsection 7.4.2; the bias factors for the dead load, λQD, and for the live load, λQL, aretaken from Table 4.2 of Chapter 4. The resistance, R, is given from the Plaxis output with

146


1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

x 104

Calculation case

Res

ista

nce

(kN

)

Displacement 1%DDisplacement 3%DDisplacement 5%D

Figure 7.11: Pile resistance of nine calculation cases.

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

x 104

Calculation case

Max

. com

pres

sive

str

ess

(kN

/m2 )


Allowable compressive stress, fn=30,000 kN/m2

Figure 7.12: Maximum compressive stress in pile of nine calculation cases.

the stochastic soil parameters being driven by Prob2B.

The reliability analysis results for the nine calculation cases are shown in Figure 7.13and Table C.1 of Appendix C. The number of pile per pile cap is 6.0, the designed boredpile therefore belongs to the redundant pile, and the target reliability index is βT=2.5(Paikowsky et al., 2004). So, Cases 4, 5, 6 with a displacement being larger than or equalto 3%D and Cases 7, 8, 9 satisfy the requirement. If the reliability level is upgraded toβT=3.0 for the conservative side, these cases still satisfy this requirement.

For comparison, the geotechnical safety degree of the pile is also evaluated through theASD, the results are indicated in Figure 7.14. The required factor of safety in the case ofhaving a static loading test conducted on site is FS=2.0 (Paikowsky et al., 2004). Cases 4,5, 6 with a displacement being larger than or equal to 3%D and Cases 7, 8, 9 also meet

147


1 2 3 4 5 6 7 8 9−4

−2

0

2

4

6

8

Calculation case

Rel

iabi

lity

inde

x, β


βT=3.0

βT=2.5

Figure 7.13: Reliability of nine calculation cases under GF mode.

1 2 3 4 5 6 7 8 90

2

4

6

8

10

12

14

Calculation case

Fac

tor

of S

afet

y, F

S


FS=2.0

Figure 7.14: Factors of safety of nine calculation cases under GF mode.

the required safety degree.

So, Cases 4 to 9 satisfy the requirement when both the ASD and RBD applied. However,it is necessary to further consider some technical aspects aiming to select a final case.First, the pile toe levels of Cases 4,5, and 6 are in the clayey sand layer, which is not anideal founding stratum for pile foundations as mentioned above. Second, the calculationapproaches are only applied to the axially loaded piles under the ULS. To design morecomprehensively, checking the overall stability and performing calculations accordingto the SLS need to be done. Since these requirements are beyond the scope of the thesis,the pile therefore should be socketed in the weathered rock layer with a supposition thatall requirements mentioned herein will be automatically satisfied. Third, the rock socketlength should be about 3.0 pile diameter (3D) in the absence of any data of soils (Ran-

148


dolph and Houlsby, 1984; Carter and Kulhawy, 1990). In addition, Yang and Liang (2006)addressed that the lateral response of a bored pile is mainly controlled by the propertiesof the soil near (1÷2)D at the rock surface. Based on the three analyses above, it can beseen that Case 9 is the most suitable, and therefore Case 9 is the final choice. Hereafter,Case 9 will be called the ”working pile” for the next calculations.

Structural failure mode

The SF mode is first considered for the working pile based on the RBD. The limit statefunction in Eq. 7.2 is used. In which the maximum compressive stress, σc, in pile con-crete caused by the prescribed displacements is given from the Plaxis output. The nom-inal compressive strength of concrete is 30,000 kN/m2. The bias factors of the nominalcompressive strength of concrete, λ f n, is taken from Table 7.1. The results are shown inFigure 7.15(a) and Table C.2 of Appendix C for three displacements as 1%D, 3%D, and5%D. For comparison, the structural safety degree of the working pile is also evaluatedthrough the ASD and the results are indicated in Figure 7.15(b).

1 2.2 3 50

1

2

3

4

5

6(a) Apply the RBD

Displacement (%D)

Rel

iabi

lity

inde

x, β

1 2.2 3 50

1

2

3

4(b) Apply the ASD

Displacement (%D)

Fac

tor

of S

afet

y, F

Sβ=3.517FS=2.011

βT=3.5

FS=2.0

Figure 7.15: Reliability and factor of safety of working pile under SF mode.

Under the SF mode, the target reliability index is chosen as βT=3.5 for the RBD (Elling-wood and Galambos, 1982); and the required factor of safety, FS=2.0, is applied to theASD (AASHTO, 2007). Both approaches show that the working pile is only safe whenthe displacement of 1%D applied. Although, this displacement is really far from therequired displacement of 5%D as proposed by O’Neill and Reese (1999). An issue aris-ing herein is, whether the SF mode is stricter than the GF mode. Alternatively, the risk,when considering the SF mode, is higher than that when considering the GF mode in thesame condition of displacement. For justification, the RBD is used aiming to re-valuatethe reliability for all cases from Case 1 to Case 9 in light of the SF mode. Besides, a dis-placement of 2.2%D is further considered. As a result, Cases 1 to 9 satisfy the requiredreliability with the displacements of 1%D and 2.2%D, while Cases 7, 8, and 9 do not sat-isfy the required reliability if displacements of greater than 2.2%D are applied, as shownin Figure 7.16 and Table C.3 of Appendix C. This issue may be explained that foundingthe pile toe on/in the weathered rock layer is one of the causes leading to the suddenincrease in the maximum compressive stress, and therefore a prescribed displacement of

149


1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

Calculation case

Rel

iabi

lity

inde

x, β

Displacement 1%DDisplacement 2.2%DDisplacement 3%DDisplacement 5%D

β=3.517β

T=3.5

Figure 7.16: Reliability of nine calculation cases under SF mode.

2.2%D would be available. Thus, with respect to the working pile, the displacement of2.2%D=26.4 mm is a reasonable choice and will be applied to throughout upcoming cal-culations. This value is also greater than the required displacement of 20 mm, to whichthe shaft resistance of pile is sufficiently mobilized (Osterberg, 2001).

The reliability of the working pile subjected to the working load is also considered. Asstated in Subsection 7.4.2, the dead load and the live load exerting on the pile are 2,204kN and 272 kN, respectively. The dead load factor and the live load factor are takenfrom Table 4.2 as 1.25 and 1.75, respectively. The calculated working load is now given:2, 204x1.25+ 272x1.75 = 3, 231 kN. The maximum compressive stress in the pile concretecaused by the working load as 2,904 kN/m2, which is much smaller than that caused bythe prescribed displacement of 2.2%D as 14,919 kN/m2. The given reliability index of theworking pile is large with the value of 5.779 (see Table C.4 of Appendix C); the workingpile is completely safe under the SF mode with respect to the working load.

7.4.6 Reliability of defect bored pile

In Chapter 3, some types of defect were presented as necking, poor concrete, and softbottom. The reliability analyses have been conducted for the bored piles without defects(the intact bored piles) in Subsection 7.4.5. In this subsection, the reliability analyses areonce again carried out for the working pile with different types of defect (defect boredpiles) that may occur during construction, aiming to quantify the effect of these defectson the reliability of the working pile. The defects are characterized through their shape,size, and location along the pile shaft.

The effect of necking

The shape of necking may range from an elliptical shape to a wedge form, but a simplerectangular defect is used as shown in Figure 7.17(a). This shape is chosen because itproduces the greatest loss of the shaft cross-sectional area, which produces the biggest

150


effect on pile behaviour. A 1.0 m height of necking is assumed for calculations. The neck-ing is arranged at three locations along the pile shaft corresponding to three calculationcases, those locations are near the pile top, in the middle, and near the pile toe. Each lo-cation is specified by the top and bottom levels of necking as shown in Figure 7.17. Thecross-sectional area reduction in percentage, Ar, due to necking is calculated as:

Ar =

(1− Rr

Rde

)100% (7.18)

where Rde is the radius of the designed pile; Rr is the remaining radius of the pile dueto the defect or the radius of necking. The cross-sectional area reductions of 25%, 50%,75%, and 95% are considered in reliability analyses.

Figure 7.17: Description of defects in axisymmetric Plaxis model.

The Plaxis calculation results show that the necking located near the pile top leads to a re-sistance reduction which is greater than that due to the necking in the middle or near thepile toe as shown in Figure 7.18(a). These results also have a good agreement with thoseobtained from the analyses of Petek et al. (2002). Especially, in case of necking locatednear the pile toe, the resistance does not decrease, it even slightly increases compared tothe resistance of the intact pile (see Figure 7.18(a)). This issue can be explained by the factthat the weathered rock is a hard soil layer; a necking near the pile toe unintentionallymakes a second ”pile toe” and this leads to an inconsiderable increment in resistance.

In Figure 7.18(b), the maximum compressive stress is still smaller than the allowablecompressive stress, fn=30,000 kN/m2, when the cross-sectional area reduction is 25%.This does not mean that the working pile is safe at this reduction. When the cross-sectional area reduction grows up to 50%, the working pile fails with certainty, if a neck-ing is located near the pile top or in the middle.

Based on the results obtained above, reliability analyses are only applied to the workingpile with a necking located near the pile top, the most adverse case. The results are indi-

151


0 25 50 75 951.2

1.4

1.6

1.8

2x 10

4 (a) Geotechnical failure mode

Cross−sectional area reduction, Ar (%)

Res

ista

nce

(kN

)

Necking near pile topNecking in middleNecking near pile toe

0 25 50 75 95

2

4

6

8

10

12

14x 10

4 (b) Structural failure mode


Max

. com

pres

sive

str

ess

(kN

/m2 )

Necking near pile topNecking in middleNecking near pile toe

210,686 kN/m2

248,306 kN/m2

fn=30,000 kN/m2

(intact pile) (intact pile)

Figure 7.18: Effect of necking size and its location on pile behaviour.

cated in Figure 7.19 and Tables C.5 and C.6 of Appendix C. According to the GF mode,the reliability index of the working pile virtually does not change with the values around5.56 when the cross-sectional area reduction varies from 25% to 95%. However, underthe SF mode, the reliability index strongly decreases from 3.517, associated with the in-tact pile, to 1.81 associated with the defect pile having the cross-sectional area reductionof 25%. The displacement of 2.2%D can be considered as an ultimate displacement withrespect to the SF mode; the pile will fail if there exists any defect in type of necking thatits location is near the pile top (see Figure 7.19(b)).

0 25 50 75 951

2

3

4

5

6

7(a) Geotechnical failure mode


Rel

iabi

lity

inde

x, β

0 25 50 75 95−8

−6

−4

−2

0

2

4

6(b) Structural failure mode


Rel

iabi

lity

inde

x, β


βT=3.0

βT=3.5

Figure 7.19: Effect of necking near the pile top on reliability of pile.

The reliability analysis under the SF mode associated with the working load is also con-sidered and shown in Figure 7.20 and Table C.7 of Appendix C. If the cross-sectionalarea reduction caused by necking increases up to 75%, corresponding to the pile havinga remaining diameter of 0.6 m, the working pile is still safe with the obtained reliabilityindex of 3.861. It can be seen that, the working pile will be failed if the cross-sectionalarea reduction is greater than 75% (see Figure 7.20(b)).

In short, reliability analyses need to consider both the GF and SF modes at the same timewith respect to bored piles having defects in types of necking, and the pile toe foundedon/in a very hard soil layer.

152


0 25 50 75 950

1

2

3

4

5

6

7x 10

4 (a) Stress caused by working load


Max

. com

pres

sive

str

ess

(kN

/m2 )

0 25 50 75 95−6

−4

−2

0

2

4

6(b) Given reliability index


Rel

iabi

lity

inde

x, β

fn=30,000 kN/m2


βT=3.5

Figure 7.20: Effect of necking near the pile top on reliability of pile subjected to workingload.

The effect of poor concrete

A poor concrete zone, which has an actual compressive strength being much smaller thanthat of a designed compressive strength, is considered in reliability analyses. Assumethat this zone occupies the whole cross section of the working pile with the height of 1.0m as shown in Figure 7.17(b). The poor concrete zone is arranged at four locations alongthe pile shaft corresponding to four calculation cases that are near the pile top, in themiddle, near the pile toe, and at the pile toe. The poor concrete is supposed to have acompressive strength of 15,000 kN/m2 with the bias factor as indicated in Table 7.1.

1.2

1.4

1.6

1.8

2x 10

4 (a) Geotechnical failure mode

Location of poor concrete zone

Res

ista

nce

(kN

)

0.8

1

1.2

1.4

1.6

1.8x 10

4 (b) Structural failure mode


Max

. com

pres

sive

str

ess

(kN

/m2 )

Max. stress in good concrete zoneMax. stress in poor concrete zone

atpile toe

nearpile toe

middlenearpile top

intactpile

intactpile

nearpile top

middle nearpile toe

atpile toe

fn=15,000 kN/m2

Figure 7.21: Effect of poor concrete zone on pile behaviour.

The Plaxis calculation results show that the poor concrete zone virtually does not affectthe resistance of pile. Alternatively, each case has more or less the same resistance asthat of the intact pile as indicated in Figure 7.21(a). Similarly, the maximum compressivestress in the good concrete zone is not much affected by the poor concrete zone locations,and lies in the relatively safe domain. The maximum compressive stress in the poorconcrete zone reaches a maximum value of 15,559 kN/m2, when this zone is locatednear the pile top, and has a minimum value of 10,005 kN/m2 when this zone is located

153


at the pile toe as shown in Figure 7.21(b).

2

3

4

5

6

7

8(a) Geotechnical failure mode


Rel

iabi

lity

inde

x, β

0

1

2

3

4

5

6(b) Structural failure mode


Rel

iabi

lity

inde

x, β

Good concrete zonePoor concrete zone

atpile toe

nearpile toe

middlenearpile top

intactpile

atpile toe

nearpile toe

middlenearpile top

intactpile

βT=3.0

βT=3.5

Figure 7.22: Effect of poor concrete zone on reliability of pile.

The reliability analysis results are described in Figure 7.22 and Tables C.8, C.9, and C.10of Appendix C. According to the GF mode, the reliability index of the working pilehardly changes when the location of the poor concrete zone shifts from the pile top downto the pile toe with values of reliability index around 5.60. With regard to the SF mode,the reliability analysis is applied to the maximum stress points within the good concreteand the poor concrete zones. It can be seen that, the reliability index of the maximumstress point in the good concrete zone is more or less the value of 3.50, and generallysatisfies the requirement with respect to the SF mode, regardless of the poor concretezone locations. Meanwhile, the reliability index of the maximum stress point in the poorconcrete zone has less values than βT=3.5, which increase from 1.53, associated with thepoor concrete zone near the pile top, to 2.53 associated with the poor concrete zone at thepile toe. This means that the poor concrete zone has to be improved or replaced throughconducting suitable methods on sites, if the displacement of 2.2%D is adopted.

The reliability analysis of the working pile is again considered associated with the work-ing load. In general, the maximum compressive stresses caused by the working load inthe good concrete and the poor concrete zones are small with the values less than 3,000kN/m2. The obtained reliability indices are around 5.779 for the good concrete zoneand around 6.130 for the poor concrete zone, regardless of the locations of the poor con-crete zone. The detailed results are shown in Tables C.11 and C.12 of Appendix C. Theworking pile is still safe under the SF mode, though the poor concrete zone exists in thepile.

The effect of soft bottom

The soft bottom situation at the pile toe is a consequence of an inadequate base cleaningduring construction and was described in Chapter 3. A slim layer, formed due to thedeposition of the drilling slurry material, is considered in the calculation. In this case,the soft soil model is applied to the slim layer with its calculation parameters describedin Table 7.5. Assume that the slim layer has a thickness of 0.1 m and is distributed entirelyunder the pile toe.

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Reliability-based design 7.5 Conclusions

Table 7.5: Statistical parameters and probability distributions for slim layer

Parameter Unit Mean COV Distributionλ∗ - 0.3 - Deterministicκ∗ - 0.1 - Deterministicγsat kN/m3 13 0.05 Normalν - 0.15 - Deterministicϕ′ degree 5 - Deterministic

As described by Woo and Moh (1990), the slim layer is unlikely to fail, but it will beexpected to deform considerably. This results in a decrease in base resistance comparedto that of a pile which has no slim layer at the bottom in the same displacement. Indeed,for the prescribed displacement of 2.2%D, the total resistance of the working pile withthe presence of the slim layer decreases by 17.1% compared to that of the working pilewithout the slim layer, i.e., from 17,553 kN down to 14,554 kN. The reliability of theworking pile slightly decreases from 5.619 down to 5.593 according to the GF mode. Incontrast, the reliability index increases from 3.517 up to 4.511 under the SF mode (seeTables C.13 and C.14 of Appendix C).

In order to consider again the effect of the weathered rock layer on the reliability of theworking pile under the SF mode, two prescribed displacements of 5%D=60 mm and10%D=120 mm (greater than the thickness of the slim layer assumed above as 100 mm)are more taken into account. The large displacements are given, aiming to mobilize theresistance of soil under the pile base. As a result, the maximum compressive stress inpile concrete strongly increases from 12,897 kN/m2 (2.2%D) up to 23,416 kN/m2 (5%D),and reaches 37,617 kN/m2 (10%D). This leads to a decrease in the reliability of pile from4.511 (2.2%D) to 2.292 (5%D) and 0.704 (10%D).

For working load, the working pile is safe under the SF mode with the reliability indexderived is 5.780 (see Table C.15 of Appendix C).

7.5 Conclusions

In this chapter a coupling calculation between a finite element package (Plaxis version9.0) and a numerical probabilistic toolbox (Prob2B) has been used as an efficiently com-bined tool aiming to design bored pile foundations in light of the RBD. The reliabilityanalyses were applied to a case study following the ULS. Two failure states, the GF modeand the SF mode, were proposed in order to comprehensively assess the reliability of anaxially loaded pile.

In this study, the parameter uncertainty was considered through the use of statistical pa-rameters and probability distributions for the material parameters utilized in soil mod-els. The soil parameters were treated as random variables including unit weight, stiffnessparameters, and strength parameters. The geometry parameters of pile were used as de-terministic quantities, because a change in pile shape in the calculation process requires

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7.5 Conclusions Reliability-based design

establishing a new mesh, which is now impossible with regard to Plaxis. The nominalcompressive strength of pile concrete was treated as a random variable in terms of thebias factor, λ f n, when performing reliability analyses under the SF mode. The interactiveperformance between pile surface and surrounding soils was also randomly evaluatedvia strength reduction factors, Rinter, of interface elements because the shear strength inthe interface varies, and strongly depends on the soil types as well as the surface state ofbored piles.

The mean values and COVs of stochastic calculation parameters in the case study wereselected from three sources: (1) The geological survey report, (2) the re-calculated pa-rameters by matching the load-displacement curve of the Plaxis analysis with that ofthe static loading test, (3) the studies by other authors. In order to reduce unnecessarycalculations, only the soil parameters of soil layers, which considerably affect pile resis-tance, were treated as random variables. Those of other soil layers were considered to bedeterministic.

Through the nine calculation cases to design the pile toe level, it is concluded that: (1)The RBD and the ASD gave the results which are compatible with each other. However,the RBD results are more precise and stricter than those resulting from the ASD, sinceall uncertainties of the loads and the resistance were included in the reliability analy-ses. (2) It is necessary to consider both the geotechnical and structural failure modes indesign for bored pile foundations, especially with respect to bored piles with defects.(3) The required displacement of 5% pile diameter seems to be suitable when the piletoe is founded in sandy or clayey soil layers. In case the pile toe is founded on/in veryhard soil layers (e.g., gravel, weathered rock or intact rock) and the bottom of drillingholes was well cleaned before placing the concrete, it may be necessary to reduce thisdisplacement. However, in most cases, care must be taken when lacking of data fromstatic loading tests, because test results are very important and considered to be essentialbases for selecting soil parameters in calculations.

Based on the calculation results when assuming the presence of different types of defectin bored piles and an defined displacement of 2.2%D, it is recognized that: (1) A neckingnear the pile top resulted in a greater reduction in pile resistance than that caused bya necking in the middle or near the pile toe. (2) With regard to neckings near the piletop, although the calculated reliability of the pile satisfied the required reliability whenthe cross-sectional area reduction is even up to 95% according to the GF mode, the pilewould fail as soon as the cross-sectional area is reduced under the SF mode. This revealedthat the SF mode has to be of especial interest when considering a bored pile with anecking-in type defect. (3) The poor concrete zone had less effects on pile resistance.The reliability of pile might be strongly reduced with respect to the stress state in thepoor concrete zone. (4) The slim layer would lead to a considerable reduction in pileresistance. However, the reliability of the pile still satisfied the requirement with respectto both failure modes.

For the working load, the working pile is completely safe under both the GF and SFmodes, when the poor concrete zone as well as the soft bottom really exists. However,the working pile would be failed if cross-sectional area reduction is greater than 75%under the SF mode.

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Chapter 8

Conclusions and recommendations

8.1 Conclusions

8.1.1 General

For bored pile foundations, the steps: A preliminary design, a quality control procedure,a modified design, a construction procedure, create a closed process. A preliminary de-sign normally follows a certain design approach, for which one or a few design modelsmay be chosen to predict pile resistance and deformation. Then the pile has to be con-structed. A quality control procedure with post-construction tests on completed boredpiles is subsequently conducted. An existing defect in a defect bored pile may be de-tected, which may be a minor or a major defect. Engineers will have to come to a de-cision regarding the defect pile; the defect is either ignored or repaired, or if necessary,the defect pile will have to be replaced by a new one. In many cases, through test re-sults, a modified design may be given; the number of piles or the pile dimensions maybe changed aiming to gain a more economical-technical design prior to constructing thebored piles.

In Chapter 2, a history of the development of design approaches was presented, includ-ing the Allowable Stress Design (ASD), the Limit State Design (LSD), and the Reliability-Based Design (RBD). Advantages and limitations of each design approach were dis-cussed in detail. This thesis focused on analyzing the LSD with the use of partial safetyfactors following the Ultimate Limit State (ULS). The LSD has developed differently inEurope and in North America, mainly in the manner of calculation of the factored re-sistance at the ULS, i.e., the factored strength approach (European) and the factored re-sistance approach (North American). The factored resistance approach is also called theLoad and Resistance Factor Design (LRFD). The calibration of resistance factors follow-ing the framework of the LRFD was one of the main tasks in this thesis. To which, thelevel II and level III reliability methods were used to calibrate these resistance factors.

In Chapter 3, the quality control approaches of bored piles were introduced as an impor-tant part of the design and construction process. The post-construction tests compriseplanned and unplanned tests, in which planned tests are typically Non-Destructive Test

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8.1 Conclusions Conclusions and recommendations

(NDT) methods. Of these methods, the Cross-hole Sonic Logging (CSL) method, themost widely used method for testing the integrity of bored pile concrete, was chosenaiming to evaluate its reliability. The inspection probability, which is used as a mea-surement of reliability for the CSL method, was formulated based on the encounteredprobability and the detection probability. For an assigned target probability, the magni-tude of a defect that can be detected is a function of the pile diameter and the number ofaccess tubes arranged.

From Chapters 4 to 6, the calibration models of resistance factors, following the frame-work of the LRFD, were proposed and presented with respect to some technical aspectsof bored piles. In Part 1 of Chapter 4, the calibration procedure of a common resistancefactor for different prediction methods was presented. In Part 2 of Chapter 4, the cali-bration procedure for separate shaft and base resistance factors was proposed. The lattercalibration procedure has an advantage over the earlier one, since the uncertainties ofshaft and base resistances have been considered separately, and therefore an economicaldesign using shaft and base resistance factors may be achieved. In Chapter 5, the calibra-tion procedure for the reference and set-up resistance factors was proposed, aiming toincorporate the set-up effect into the LRFD. Through this incorporation, an economicaldesign can also be gained. In Chapter 6, the updating procedure of resistance factorswas given, based on the Bayesian inference and the experimental outcomes of pile load-ing tests at a site. The Bayesian inference enables to reduce uncertainty with respect toinitial empirical distributions in terms of load test results within a site. The resistancefactors therefore may be improved and a more precise design can be reached.

In Chapter 7, the reliability of a single bored pile was directly determined by the use ofa coupling calculation between the finite element package (Plaxis version 9.0) and thenumerical probabilistic toolbox (Prob2B). The reliability was assessed, not only for theintact bored pile but also for the defect bored pile, by assuming different types and de-grees of defect that may occur within the pile body. Two failure modes, the GeotechnicalFailure (GF) mode and the Structural Failure (SF) mode, were proposed in this chapter.Based on which, the reliability of an axially loaded pile was comprehensively assessed.

The conclusions corresponding to the research questions as stated in Chapter 1 are drawnbelow.

8.1.2 Regarding the reliability of the CSL method (Chapter 3)

The reliability of the CSL method can be quantitatively evaluated using a probabilisticanalysis procedure. The inspection probability, which was first used by Li et al. (2005)as a measurement of reliability for the CSL method, was determined as a product of theencountered probability and the detection probability. However, the approach accordingto Li et al. (2005) would lead to an over-prediction of the encountered probability, and therequired number of access tubes therefore tends to be small compared to that suggestedby O’Neill and Reese (1999) and Thasnanipan et al. (2000).

Based on the general methodology established by Li et al. (2005), the new approaches fordetermining the encountered probability and the detection probability was formulated

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Conclusions and recommendations 8.1 Conclusions

in this study. The new approaches basically overcome the limitations of the study of Liet al. (2005) in some aspects as follows: (1) Defects were located at the periphery of pilesfollowing the discussion of O’Neill (1991) and Fleming et al. (1992). (2) The shape ofthe circular segment was used to describe the defect, this assumption is close to actualdefects and creates a more conservative approach for the CSL method. (3) The detectionprobability was directly determined through the detection length, the new concept wasproposed in this study. The detection length further included key parameters such asthe pile diameter, the number of access tubes, and especially the relative position of thedefect compared to the adjacent access tubes.

For a given pile diameter, the encountered probability and the detection probability in-crease with the magnitude of the defect and the number of access tubes. It can be seenthat, the encountered probability and the detection probability strongly depend on thenumber of access tubes and slightly depend on the distance from the center of the ac-cess tube to the pile shaft perimeter. In addition, the detection probability also varieswith the selected detection threshold. The detection threshold varies depending on thefrequency and the wavelength of the sonic wave created between the transmitter and re-ceiver. According to Amir and Amir (2008), a detection threshold of 168 mm was offeredcorresponding to a frequency of 50 kHz and a wavelength of 84 mm. In this study, adetection threshold of 200 mm was adopted for conservative purposes.

For an assigned target inspection probability, the required number of access tubes wasdetermined. The number of access tubes is an important factor, which strongly affectsnot only on the measurement results of the CSL method, but also on the constructioncosts of bored pile foundations. The required number of access tubes proposed in thisstudy is greater than that proposed by Li et al. (2005), but it is compatible with that asstated by O’Neill and Reese (1999) and Thasnanipan et al. (2000).

The detectable minimum magnitude of a defect decreases with the increase of the num-ber of access tubes to be used. However, the magnitude of a defect tends to be tangentwith a value of approximately 200 mm, irrespective of the pile diameters as well as thenumber of access tubes used. This value can be considered as a minimum magnitude ofdefect, under which the CSL test cannot detect the defect. For pile diameters lying in therange from 600 to 3,000 mm, 8 access tubes can be considered as the maximum numberof access tubes that can be used when the CSL method is applied. Using a larger numberof access tubes is really less efficient.

8.1.3 Regarding the resistance factor calibrations (Chapter 4)

Three reliability methods, First Order Second Moment (FOSM), First Order ReliabilityMethod (FORM) and Monte Carlo Simulation (MCS), were used to calibrate resistancefactors following the framework of the LRFD with model uncertainties. This chapterwas divided into two main parts, the first part presented the calibration procedure fora common resistance factor, and the second part described the calibration procedure forseparate shaft and base resistance factors.

In the first part, the sixteen calibration cases represented by a combination of the soil

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type, resistance prediction method, and the construction method were considered. Thelimit state function proposed for the calibration procedure included the total resistance,dead load, and live load bias factors. The bias factors represent the model uncertain-ties, in which the statistical parameters and the probability distributions of the dead loadand live load bias factors were taken from the design criterion for bridge superstructures(Nowak, 1999). The resistance bias factors were represented by the ratio of the measuredresistance to the predicted resistance (Paikowsky et al., 2004). The common resistancefactors were calibrated according to the different target reliability indices and the differ-ent ratios of the dead load to the live load.

For a target reliability index, the value of the resistance factor is less sensitive with thedead load to live load ratio and reaches a stable value when this ratio is greater than3.0. The target reliability indices, βT=2.5 and 3.0, corresponding to the redundant andnon-redundant piles were adopted for entire calculations in this thesis. The resistancefactor and the ratio of the coefficient of variation to the mean of resistance bias factor,COVR/λR, have a better correlation than the one between the resistance factor and otherstatistical parameters such as λR, σR, or COVR. The good correlation between the resis-tance factor and the ratio COVR/λR is an important basis for calibrating separate shaftand base resistance factors later. FOSM, FORM, and MCS produce the resistance factorswhich have the good correlations and linear relations with the ratio COVR/λR. The re-sistance factors obtained from MCS and FORM have a somewhat better correlation thanthose between FORM and FOSM or between MCS and FOSM. FORM produces some-what greater resistance factors than those obtained by FOSM or MCS.

In the second part, only MCS was used to calibrate the shaft and base resistance factorsseparately, aiming to reflect the different degrees of uncertainty at the pile shaft andthe pile base. The limit state function proposed for the calibration procedure includedthe shaft resistance, base resistance, dead load, and live load bias factors. In order toillustrate the theory proposed, a case study including the 26 O-cell test bored piles at thesite of the Los Angeles Coliseum (the US) was considered. Through the O-cell test, a goodmethod to separately measure the shaft and base resistances of bored piles, the shaft andbase resistance bias factors were easily determined. The statistical parameters and theprobability distributions of the dead load and live load bias factors were taken under thedesign criterion for building superstructures (Ellingwood and Galambos, 1982).

By using the ”Correlation Ratio”-(CR), suggested in this study, the correlation betweenthe uncertainties of the shaft and base resistances could be solved harmonically. Fromthat, a unique couple of values for the shaft and base resistance factors was obtained.Through the sensitivity analysis, the shaft and base resistance factors are less sensitivewith respect to the ratio QD/QL and the ratio RB/RS. Thus, the proposed limit statefunction in Eq. 4.29 assured the stability for the calibration of shaft and base resistancefactors. It is interesting that, when a comparison was carried out for the 26 bored pilesin the case study, the factored total resistances using the shaft and base resistance factorswere greater than those using a common resistance factor. Therefore, using shaft andbase resistance factors can result in a more economical design.

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Conclusions and recommendations 8.1 Conclusions

8.1.4 Regarding the set-up effect (Chapter 5)

A calibration procedure of the reference and set-up resistance factors was presented fol-lowing the framework of the LRFD. When incorporating the set-up effect into the de-sign, the reference resistance was calculated according to a specified resistance predic-tion method; and the set-up resistance was estimated through the prediction methodproposed by Skov and Denver (1988). Note that, the set-up effect only takes place atthe pile shaft, not at the pile base. Thus, only the shaft resistance was considered in theselected reference resistance prediction methods; the base resistance was ignored. Dueto the compatibility in the calibration algorithm, the calibration procedure used in thischapter is completely the same as the procedure in the second part of Chapter 4.

Due to the performance of bored pile constructions, a small modification in the set-upresistance prediction formula of Skov and Denver (1988) was given, aiming to facilitatethe determination of the elapsed time. Instead of using the ratio of t/t0 for driven piles,we can use the difference, t− t1. In which, t1 ≈14 days, is the time needed to sufficientlydevelop the bored pile concrete strength. The set-up resistance therefore could be easilyand consistently determined.

Through the measured data at the SR20 site, it can be seen that the set-up effect had beentaking place very differently between the soil types. Even in the same soil layer, the set-up effect at the different measurement points also demonstrated to be rather different, asshown in Table 5.3. This means that the set-up effect uncertainty is quite high. Therefore,care should be taken when using the same set-up factors as well as the same set-up resis-tance factors in calculations at different sites; this may lead to a wrong set-up resistanceprediction. It is necessary to perform a calibration process of set-up resistance factors foreach specific site; this work, of course, will require some set-up experiments on the site.

A problem with respect to the bored set-up effect is the lack of data. Therefore, a compre-hensive study on the set-up effect of bored piles will be necessary. Effects of constructionmethods as well as irregular shapes of bored piles on the set-up should be taken intoaccount in future research.

8.1.5 Regarding updated resistance factors based on the Bayesian inference(Chapter 6)

In this chapter, an updating procedure of the resistance factors was presented based onBayesian inference through the pile loading test results at a site. The Bayesian inferenceenables us to reduce uncertainty with respect to the initial empirical distributions of theresistance bias factor. From given posterior distributions of the resistance bias factors,the resistance factors therefore were improved and a more precise design may be accom-plished.

It can be seen that the within-site variability of pile resistance, represented through thecoefficient of variation (COV), is an important reference parameter, based on which thelikelihood functions are determined. A value of COV=0.2 was adopted based on severalstudies, e.g., Kay (1976), Zhang (2004), Su (2006), and on the data in the case study of

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the second part of Chapter 4. The likelihood functions were formulated for differentcombinations of test loads, which may consist of survival test loads, or failure test loadsor multiple test loads. In addition to conventional static loading tests, dynamic testswere also taken into account. The failure load is a load, which is a plunging load, ora load corresponding to a displacement of 5% pile diameter (O’Neill and Reese, 1999),whichever comes first. Other loads are called survival loads.

Through the case study, some findings can be addressed: (1) The posterior statistical pa-rameters and the updated resistance factors are considered to be independent from theupdating order, regardless of any type of test loads, as long as all test piles are takeninto account. (2) By the Bayesian inference, the uncertainty of the posterior distributiondecreases compared to the prior distribution, σR and COVR of the posterior distributionsdecrease with the increase of the number of the test piles. Furthermore, the survival testload would result in the increase in the updated resistance factors. In contrast, the failuretest load would lead to the decrease in the updated resistance factors. (3) The updatedresistance factors gradually decrease with the increase of the prediction resistance andtends to be tangent with a certain value. However, the decreasing rate of the updatedresistance factor associated with the under-prediction resistances is greater than that ofthe updated resistance factor associated with the over-prediction resistances. (4) Whenconsidering the static test pile behaviour with respect to the likelihood functions, the up-dated resistance factors are somewhat smaller than those of the case without consideringthe static test pile behaviour.

In order to close the conclusions for this chapter, is necessary to repeat the idea of Chris-tian (2004) that ”Probability in geotechnical engineering is not a property of the worldbut a state of mind. Thus, geotechnical and geological uncertainty is belief-based andnecessarily Bayesian; and the current challenge to the geotechnical engineering profes-sion is how to use probabilistic methods in practice”.

8.1.6 Regarding the reliability-based design model (Chapter 7)

A coupling calculation between the finite element package (Plaxis version 9.0) and thenumerical probabilistic toolbox (Prob2B) was used to design bored pile foundations inlight of the RBD. The reliability analyses were applied to a case study following theUltimate Limit State (ULS). Two failure modes, the Geotechnical Failure (GF) mode andthe Structural Failure (SF) mode, were proposed in order to comprehensively assess thereliability of an axially loaded pile.

In this chapter, the parameter uncertainty was considered through the use of statisticalparameters and probability distributions for material parameters in soil models. Thesoil parameters were treated as random variables. The geometry parameters of pile wereused as deterministic quantities, because a change in pile shape in the calculation processrequires establishing a new mesh, which is now impossible with regard to the Plaxissoftware. The nominal compressive strength of pile concrete was treated as a randomvariable in terms of the bias factor, λ f n, when performing reliability analyses under theSF mode. The interactive performance between the pile surface and surrounding soilswas also randomly evaluated via strength reduction factors, Rinter, of interface elements

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Conclusions and recommendations 8.2 Recommendations

because the shear strength in the interface varies, and strongly depends on the soil types,as well as on the surface state of bored piles.

The statistical parameters of stochastic calculation variables were selected based on thegeological survey report, the re-calculated parameters by matching the load-displacementcurve of the Plaxis analysis with that of the static loading test, and on the studies by otherauthors. In order to reduce unnecessary calculations, only the soil parameters of soil lay-ers, which considerably affect pile resistance, were treated as random variables. Thoseof other soil layers were considered to be deterministic.

For comparison purposes in the case study, the ASD was also used for the nine calcula-tion cases. It can be seen that, the RBD and the ASD gave results which are compatiblewith each other. However, the RBD’s results are more precise and stricter than thosegiven from the ASD, because all uncertainties of the loads and the soil parameters weretaken into account in the reliability analyses. It is important that both the GF and SFmodes be considered in the design for bored piles, especially with respect to bored pileswith defects.

The displacement criterion of 5% pile diameter seems to be suitable when the pile toeis founded in sandy or clayey soil layers. In case the pile toe is founded on/in veryhard soil layers (e.g., gravel, weathered rock or intact rock) and the bottom of drillingholes is carefully cleaned before placing the concrete, it may be necessary to reduce thisdisplacement criterion. However, in most cases, care must be taken when lacking datafrom static loading tests, because test results are very important and are considered to beessential bases for selecting soil parameters in calculations.

The calculation results, when assuming the presence of different types of defects in boredpiles, revealed that: (1) A necking near the pile top resulted in a greater reduction in pileresistance than that caused by a necking in the middle or near the pile toe. (2) With regardto neckings near the pile top, the calculated reliability of the pile satisfied the requiredreliability when the cross-sectional area reduction even grew up to 95% according to theGF mode, but the pile would fail as soon as the cross-sectional is reduced under the SFmode. This confirms that the SF mode has to be of especial interest when consideringbored piles with necking-in type defects. (3) The poor concrete zone had less effects onpile resistance. However, reliability of the pile might be strongly reduced with respectto the stress state in the poor concrete zone. (4) The slim layer of soil would lead to aconsiderable reduction in pile resistance. However, the reliability of the pile still satisfiedthe requirement with respect to both failure modes.

8.2 Recommendations

To objectively evaluate the defect-detecting ability of the CSL method, some experimentsusing this method need to be carried out on bored piles with planned defects (e.g., Iskan-der et al., 2001; Briaud et al., 2002). The planned defects in the previous experimentswere only made in the shapes of sphere, cylinder, or rectangular cuboid. In fact, therehave been few, if any, of these kinds of defects existing in defect bored piles; mean-while defects normally have the shape of necking or circular segment. Planned defects

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8.2 Recommendations Conclusions and recommendations

with shapes of circular segments as proposed in this study, unfortunately, have not beenmade in any previous experiment. In order to verify the reliability of the CSL methodwith the proposed approach in this study, it is recommended to do some experimentson planned-defect bored piles. Planned defects should have a shape of circular segmentwith different magnitudes and should be located at different positions compared to ad-jacent access tubes.

The calibration model for a common resistance factor using MCS as proposed in thisstudy has been validated through the correlation analyses between the resistance factorsobtained from MCS and FOSM as well as from MCS and FORM. In which, FOSM wasused by Barker et al. (1991) to calibrate resistance factors, which were already taken intothe AASHTO bridge foundation code published in 1994 (AASHTO, 1994). The resistancefactors on the deep foundations in the AASHTO specification were revised again byPaikowsky et al. (2004) using FORM; the obtained resistance factors from the calibrationprocess, in combination with the fitting to the ASD, have been used in the AASHTObridge foundation code published in 2007 (AASHTO, 2007).

Regarding the calibration model for separate shaft and base resistance factors, validatingthis model is still a problem due to the lack of proof test piles. To do this, a validationprocedure is recommended as follows: (1) For a site, at least 20 Osterberg-cell (O-cell)instrumented identical bored piles are constructed by the same construction method andthe same monitoring procedure during construction. (2) The O-cell tests are conductedfor these piles and measured resistances are collected. (3) Shaft and base resistance fac-tors are calibrated according to the procedure proposed in this study based on these testresults. (4) A proof test pile is designed using shaft and base resistance factors obtainedabove. Note that, the proof test pile has to have the same diameter as the O-cell test piles.(5) The proof test pile is loaded with a static loading test to failure; this static test resultwill be an important basis to confirm the applicability of the model.

Although the bored pile set-up effect is not as dramatic as for driven piles, incorporatingthe set-up effect into the bored pile design is necessary, aiming to reach a more economicdesign. In fact, the available literature contains few examples of bored pile set-up, orthe measured data are discontinuous (the data at the SR20 site in the case study is anexample). Therefore, more research on the bored pile set-up considering effects of con-struction methods and irregular shapes of bored piles on the set-up is recommended infuture research. For a big project or at a particular region, it is recommended to producean O-cell instrumented bored pile, and to conduct the O-cell test according to an assignedschedule. Measured data will be a valuable reference for making a decision, whether aset-up effect occurs, whether a set-up effect should be or should not be incorporated intothe LRFD.

The coupling of the finite element method (Plaxis version 9.0) and the numerical proba-bilistic toolbox (Prob2B) can directly estimate the reliability index of a single bored pilesubjected to random loads. In addition, this coupling calculation also deals with theparameters in soil models as stochastic quantities. However, the geometric parametersof bored pile are still used as deterministic quantities, because a change in pile shaperequires establishing a new mesh, which is now impossible with regard to the Plaxissoftware. Different from driven piles, the bored pile shape is very irregular, caused by

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Conclusions and recommendations 8.2 Recommendations

construction procedures and other influence factors. Another limitation of the Plaxis isthat it cannot solve a defect pile having a shape of circular segment, due to its axisym-metric performance. Therefore, improvements to the Plaxis software to overcome thesetwo limitations are recommended.

The resistance factor calibration models, which have been proposed in Chapters from 4to 6, can be well applied to driven piles.

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8.2 Recommendations Conclusions and recommendations

166

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Zhang, L., Tang, W. H., and Ng, C. W. W. (2001). Reliability of axially loaded driven pilegroups. Journal of Geotechnical and Geoenvironmental Engineering, 127(12):1051–1060.

Zhang, L. M. (2004). Reliability verification using proof pile load tests. Journal of Geotech-nical and Geoenvironmental Engineering, 130(11):1203–1213.

Zhang, L. M., Li, D. Q., and Tang, W. H. (2006). Level of construction control and safety ofdriven piles. Soils and Foundations: Journal of the Japanese Geotechnical Society, 46(4):415–425.

178

Appendix A

Resistance bias factors

A.1 Empirical distributions of the sixteen calibration cases

Figures A.1 through A.4 show the histograms and the PDFs of the resistance bias factorsfor the sixteen calibration cases as presented in Section 4.5 of Chapter 4.

2 4 6 80

0.2

0.4

0.6

0.8

1


Pro

babi

lity

dens

ity

(1) Sand−FHWA−Casing

2 4 6 80

0.2

0.4

0.6

0.8

1


Pro

babi

lity

dens

ity

(2) Sand−FHWA−Slurry

2 4 6 80

0.2

0.4

0.6

0.8


Pro

babi

lity

dens

ity

(3) Sand−RW−Casing

2 4 6 80

0.2

0.4

0.6

0.8

1


Pro

babi

lity

dens

ity

(4) Sand−RW−Slurry

No. of piles=9

=1.614

No. of piles=12

=2.270

No. of piles=12

=1.650

σR =1.000 σ

R =1.122

λRλR

λR

σR =0.944 σ

R =0.718

=1.033λR

No. of piles=9

Figure A.1: Empirical distributions of calibration cases 1 through 4.

179

A.1 Empirical distributions of the sixteen calibration cases Resistance bias factors

1 2 3 40

0.5

1

1.5

2


Pro

babi

lity

dens

ity

(5) Clay−FHWA−Casing

1 2 3 40

0.5

1

1.5


Pro

babi

lity

dens

ity

(6) Clay−FHWA−Dry

1 2 3 40

0.5

1

1.5


Pro

babi

lity

dens

ity

(7) Sand+Clay−FHWA−Casing

1 2 3 40

0.2

0.4

0.6

0.8

1

1.2


Pro

babi

lity

dens

ity

(8) Sand+Clay−FHWA−Dry

=0.835λR

σR =0.399

No. of piles=13

λR

λR

No. of piles=36

=0.797

σR =0.296

No. of piles=21

=1.039

σR =0.297

λR

No. of piles=11

=1.322

σR =0.359

Figure A.2: Empirical distributions of calibration cases 5 through 8 (continue).

1 2 3 40

0.5

1

1.5


Pro

babi

lity

dens

ity

(9) Sand+Clay−FHWA−Slurry

1 2 3 40

0.5

1

1.5


Pro

babi

lity

dens

ity

(10) Sand+Clay−RW−Casing

1 2 3 40

0.2

0.4

0.6

0.8

1

1.2


Pro

babi

lity

dens

ity

(11) Sand+Clay−RW−Dry

1 2 3 40

0.5

1

1.5


Pro

babi

lity

dens

ity

(12) Sand+Clay−RW−Slurry

λR

No. of piles=9λR =1.288σ

R =0.333

No. of piles=21

=0.951

σR =0.325

λR

No. of piles=11

=1.206σ

R =0.365

λR

No. of piles=9

=1.158

σR =0.268


180

Resistance bias factors A.2 Shaft and base resistance bias factors

1 2 3 40

0.2

0.4

0.6

0.8

1


Pro

babi

lity

dens

ity

(13) Rock−CK−Mixed

1 2 3 40

0.2

0.4

0.6

0.8

1


Pro

babi

lity

dens

ity

(14) Rock−CK−Dry

1 2 3 40

0.2

0.4

0.6

0.8

1

1.2


Pro

babi

lity

dens

ity

(15) Rock−IGM−Mixed

1 2 3 40

0.2

0.4

0.6

0.8

1

1.2


Pro

babi

lity

dens

ity

(16) Rock−IGM−Dry

λR

No. of piles=30No. of piles=46λR =1.229 =1.350

σR =0.584σ

R =0.504

No. of piles=46

=1.298λR λR

σR =0.437

No. of piles=30

=1.400σ

R =0.478


A.2 Shaft and base resistance bias factors

Since the soil conditions at the site of the Los Angeles Coliseum are predominantly sandysoils, the predicted shaft resistance, RS, for a certain soil layer is calculated according tothe β method (O’Neill and Reese, 1999):

RS =

z2∫z1

βσ′zdASI (A.1)

where σ′z is the vertical effective stress in soil at depth z; dASI is the differential circum-

ferential area along the shaft; z1 and z2 are the depth of the top and the bottom of the soillayer considered; β is the parameter which is determined as follows:

• For the sand and silt layer:

β =N60

15(1.5− 0.245

√z) and 0.25 ≤ β ≤ 1.20 (A.2)

• For the sand and silt layer:

β = 2.0− 0.15z0.75 and 0.25 ≤ β ≤ 1.80 (A.3)

here N60 is the uncorrected SPT index.

181

A.2 Shaft and base resistance bias factors Resistance bias factors

According to Reese et al. (2006), the recommended value of unit base resistance, rB, inMPa is predicted:

rB = 0.0575N60 (A.4)

and the base resistance is then:RB = ABrB (A.5)

here AB is the cross-sectional area at the base of the bored pile.

Since the uncorrected SPT index of the soil layers under the base of the 26 bored piles isgreater than 60, the unit base resistance, rB, is taken as 2,900 kN/m2 (Reese et al., 2006).

Table A.1 shows the results of the measured and predicted resistances at the shaft andthe base of the 26 bored piles at the L.A. Coliseum. Correspondingly, the shaft and baseresistance bias factors are obtained. The statistical parameters of these bias factors arealso indicated in the table.

Table A.1: Shaft and base resistance bias factors of the 26 bored piles at the L.A. Coliseum

Shaft resistance Base resistance

Pile name Measured Predicted Bias factor, Measured Predicted Bias factor,(kN) (kN) λRS (kN) (kN) λRB

4A 9,796 9,692 1.011 10,620 3,969 2.6764B 8,972 9,652 0.930 10,909 3,969 2.7495A 9,773 9,652 1.012 14,910 3,969 3.7575B 10,042 9,652 1.040 8,534 3,969 2.1506A 6,647 9,652 0.689 4,573 3,969 1.1526B 9,484 9,808 0.967 10,375 3,969 2.6147A 9,970 10,455 0.954 10,378 3,969 2.6158A 12,506 9,920 1.216 9,836 3,969 2.4788B 9,441 9,083 1.039 8,901 3,969 2.2439A 8,462 8,118 1.042 7,411 3,969 1.86710A 8,774 7,345 1.195 7,322 3,969 1.84510B 5,259 6,990 0.752 6,855 3,969 1.72719A 14,111 9,544 1.479 6,546 3,969 1.64919B 8,871 9,602 0.924 4,585 3,969 1.15520A 12,427 8,730 1.423 6,279 3,969 1.58220B 7,929 9,228 0.859 3,298 3,969 0.83121A 11,550 8,730 1.323 7,305 3,969 1.84021B 10,817 8,730 1.239 7,523 3,969 1.89522A 19,010 8,566 2.219 4,108 3,969 1.03522B 12,922 8,846 1.461 1,692 3,969 0.42623A 9,816 9,431 1.041 7,534 3,969 1.89823B 9,458 9,320 1.015 8,243 3,969 2.07724A 11,460 8,985 1.275 9,888 3,969 2.49124B 9,583 8,985 1.067 8,578 3,969 2.16125A 11,091 8,985 1.234 6,940 3,969 1.74925B 10,632 9,431 1.127 7,536 3,969 1.899

λRS 1.138 λRB 1.945σRS 0.299 σRB 0.682

COVRS 0.263 COVRB 0.351

182

Resistance bias factors A.3 Factored total resistances

A.3 Factored total resistances

Table A.2 shows a comparison of the factored total resistances when applying a commonresistance factor and applying the shaft and base resistance factors to the 26 bored pilesat the L.A. Coliseum. The target reliability index was chosen as βT=3.0. Based on the cal-ibration process in Section 4.6 of Chapter 4, the common resistance factor was obtainedas φ=0.83; and the shaft and base resistance factors were derived, φS=0.84 and φB=1.46.The factored total resistances, RF, are calculated as follows:

• When applying the common resistance factor:

RF = φ(RS + RB) = 0.76(RS + RB) (A.6)

• When applying the shaft and base resistance factors:

RF = φSRS + φBRB = 0.83RS + 1.07RB (A.7)

Table A.2: Comparison of factored total resistances when using a common resistancefactor and using shaft and base resistance factors for the 26 bored piles with βT=3.0

Factored total resistance (kN)

Pile name Using a common φ Using φS and φB Increasing rate (%)

4A 10,382 12,291 184B 10,352 12,258 185A 10,352 12,258 185B 10,352 12,258 186A 10,352 12,258 186B 10,471 12,387 187A 10,962 12,924 188A 10,556 12,480 188B 9,920 11,786 199A 9,186 10,985 2010A 8,599 10,343 2010B 8,329 10,049 2119A 10,270 12,168 1819B 10,314 12,216 1820A 9,651 11,493 1920B 10,030 11,906 1921A 9,651 11,493 1921B 9,651 11,493 1922A 9,527 11,357 1922B 9,739 11,589 1923A 10,184 12,075 1923B 10,100 11,982 1924A 9,845 11,704 1924B 9,845 11,704 1925A 9,845 11,704 1925B 10,184 12,075 19

183

A.3 Factored total resistances Resistance bias factors

184

Appendix B

Set-up effect

The measured normalized side shear stress, SMN , is determined as:

SMN =fT

f1=

f2

f1(B.1)

where fT is the measured side shear stress at the elapsed time t; f1 and f2 are the mea-sured side shear stress in 1996 and in 2002, respectively.

The predicted normalized side shear stress, SPN , is estimated using Eq. 5.13 in Chapter5:

SPN = Alog10(t− t1) + 1 (B.2)

in which A is the set-up factor; t is the elapsed time after bored pile installation, at whichthe set-up effect may be considered as termination; t1 is the elapsed time after bored pileinstallation, at which the side shear stresses were first measured in 1996.

The set-up resistance bias factors are therefore given:

λRSE =SMN

SPN(B.3)

B.1 Set-up resistance bias factors for Ability 1

185

B.2 Set-up resistance bias factors for Ability 2 Set-up effect

Table B.1: Set-up resistance bias factors for Ability 1

Soil type Pile No. Segment Set-up t1 f1 t f2 SMN SPN λRSEelevation (m) factor, A (days) (kN) (days) (kN)

Sand 11 4.88 0.2 7 50.14 2,021 50.14 1.000 1.661 0.60211 1.52 0.2 7 67.47 2,021 96.82 1.435 1.661 0.86411 -3.66 0.2 7 89.35 2,021 134.66 1.507 1.661 0.9072 5.91 0.2 6 14.95 2,064 43.94 2.939 1.663 1.7682 -0.18 0.2 6 54.61 2,064 74.49 1.364 1.663 0.8202 -3.23 0.2 6 54.34 2,064 74.21 1.366 1.663 0.821

10 -0.76 0.2 11 2.92 2,038 11.67 3.997 1.661 2.4065 -2.87 0.2 9 43.76 1,900 48.14 1.100 1.655 0.6657 0.03 0.2 11 23.89 1,918 24.52 1.026 1.656 0.620

Clay 5 7.20 0.6 9 4.92 1,900 9.66 1.963 2.966 0.6625 3.38 0.6 9 3.28 1,900 7.11 2.168 2.966 0.7315 0.03 0.6 9 19.15 1,900 19.60 1.023 2.966 0.345

Limestone 11 -5.64 0.4 7 239.14 2,021 405.62 1.696 2.322 0.7312 -5.98 0.4 6 219.26 2,064 335.69 1.531 2.325 0.658

10 -7.01 0.4 11 124.17 2,038 256.55 2.066 2.323 0.89010 -10.59 0.4 11 731.27 2,038 843.78 1.154 2.323 0.497

Sand λRSE : 1.109 σRSE : 0.593 COVRSE : 0.535Clay λRSE : 0.579 σRSE : 0.168 COVRSE : 0.290Limestone λRSE : 0.694 σRSE : 0.141 COVRSE : 0.203

B.2 Set-up resistance bias factors for Ability 2

Table B.2: Set-up resistance bias factors for Ability 2

Soil type Pile No. Segment Set-up t1 f1 t f2 SMN SPN λRSEelevation (m) factor, A (days) (kN) (days) (kN)

Sand 11 4.88 0.2 7 50.14 51 50.14 1.000 1.342 0.74511 1.52 0.2 7 67.47 51 96.82 1.435 1.342 1.07011 -3.66 0.2 7 89.35 51 134.66 1.507 1.342 1.1232 5.91 0.2 6 14.95 52 43.94 2.939 1.343 2.1882 -0.18 0.2 6 54.61 52 74.49 1.364 1.343 1.0162 -3.23 0.2 6 54.34 52 74.21 1.366 1.343 1.017

10 -0.76 0.2 11 2.92 47 11.67 3.997 1.334 2.9955 -2.87 0.2 9 43.76 49 48.14 1.100 1.338 0.8227 0.03 0.2 11 23.89 47 24.52 1.026 1.334 0.769

Clay 5 7.20 0.6 9 4.92 119 9.66 1.963 2.245 0.8745 3.38 0.6 9 3.28 119 7.11 2.168 2.245 0.9655 0.03 0.6 9 19.15 119 19.60 1.023 2.245 0.456

Limestone 11 -5.64 0.4 7 239.14 51 405.62 1.696 1.683 1.0082 -5.98 0.4 6 219.26 52 335.69 1.531 1.686 0.908

10 -7.01 0.4 11 124.17 47 256.55 2.066 1.669 1.23810 -10.59 0.4 11 731.27 47 843.78 1.154 1.669 0.691

Sand λRSE : 1.094 σRSE : 0.434 COVRSE : 0.397Clay λRSE : 0.765 σRSE : 0.222 COVRSE : 0.290Limestone λRSE : 0.961 σRSE : 0.196 COVRSE : 0.204

186

Appendix C

Prob2B-Plaxis results

C.1 Reliability analysis results for intact bored piles

C.1.1 Nine calculation cases under GF mode

Table C.1: Reliability analysis results for 9 calculation cases under GF mode

Displacement criteriaCal. Item Variable, Unit 1%D 3%D 5%Dcase Xi α∗i x∗i α∗i x∗i α∗i x∗iCase 1 Dead load λQD - -0.262 0.940 -0.179 1.064 -0.076 1.091

Live load λQL - -0.450 1.099 -0.337 1.132 -0.260 1.145Sandstone E50 kN/m2 -0.459 1.38E5 -0.477 1.48E5 -0.427 1.48E5

γsat kN/m3 -0.399 19.72 -0.284 19.26 -0.177 18.89ϕ′ degree 0.248 21.75 0.428 18.22 0.594 16.20c′ kN/m2 -0.135 41.14 -0.082 40.37 -0.117 42.47

pile Rinter - 0.370 0.850 0.474 1.132 0.451 0.74710.5 m Clayey E50 kN/m2 -0.145 6,331 -0.007 7,014 -0.002 6,954long sand γsat kN/m3 -0.195 19.76 -0.118 20.21 -0.016 20.16

ϕ′ degree -0.049 20.75 0.210 20.96 0.299 21.02c′ kN/m2 -0.119 46.66 -0.033 45.11 0.051 45.11Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 0.183 1.01E6 0.239 1.08E6 0.223 1.08E6rock γsat kN/m3 -0.107 21.66 -0.097 22.02 7.95E-4 22.02

ϕ′ degree -0.149 43.81 -0.021 42.85 -0.022 43.05ψ degree 0.000 5.00 0.000 5.00 0.000 5.00

Number of calculations (FORM): 154 171 171β -1.731 0.328 0.961Pf 0.958 0.371 0.168

187

C.1 Reliability analysis results for intact bored piles Prob2B-Plaxis results

Case 2 Dead load λQD - 0.115 1.014 0.111 1.078 0.075 1.101Live load λQL - -0.290 1.104 -0.258 1.147 -0.309 1.165Sandstone E50 kN/m2 -0.303 1.47E5 -0.383 1.47E5 -0.418 1.46E5

γsat kN/m3 -0.217 19.58 -0.214 19.13 -0.190 18.88ϕ′ degree 0.267 19.91 0.306 17.84 0.388 16.77c′ kN/m2 0.013 40.00 0.056 40.06 0.054 40.48

pile Rinter - 0.745 0.868 0.734 0.722 0.658 0.69112.5 m Clayey E50 kN/m2 -0.014 6,984 -0.041 6,979 -0.043 6,960long sand γsat kN/m3 -0.086 20.21 -0.081 20.17 -0.082 20.16

ϕ′ degree 0.149 21.04 0.166 20.96 0.201 20.96c′ kN/m2 -0.015 44.83 -0.010 45.04 0.004 44.84Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 0.309 1.08E6 0.235 1.08E6 0.224 1.09E6rock γsat kN/m3 -0.054 21.93 -0.042 22.01 0.019 22.05

ϕ′ degree -0.094 42.61 -0.006 42.92 -0.050 43.09ψ degree 0.000 5.00 0.000 5.00 0.000 5.00


Case 3 Dead load λQD - -0.118 1.015 0.131 1.053 0.162 1.067Live load λQL - -0.349 1.128 -0.265 1.132 -0.299 1.144Sandstone E50 kN/m2 -0.460 1.46E5 -0.321 1.47E5 -0.331 1.47E5


pile Rinter - 0.714 0.844 0.802 0.780 0.809 0.74313.5 m Clayey E50 kN/m2 0.005 6,858 -0.046 6,989 -0.014 6,971long sand γsat kN/m3 -0.075 20.06 -0.143 20.20 -0.035 20.18

ϕ′ degree -0.052 20.94 0.053 20.99 0.089 20.95c′ kN/m2 -0.185 44.66 -0.064 45.02 -0.047 45.29Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 0.159 1.06E6 0.235 1.08E6 0.244 1.08E6rock γsat kN/m3 0.007 21.92 0.019 22.00 -0.059 22.03

ϕ′ degree -0.157 42.93 -0.002 43.01 -0.030 42.97ψ degree 0.000 5.00 0.000 5.00 0.000 5.00



γsat kN/m3 -0.091 19.26 -0.021 19.14 -0.016 19.15ϕ′ degree 0.024 18.63 7.06E-4 18.68 -0.015 18.83c′ kN/m2 0.050 38.96 0.099 37.84 0.100 37.42


ϕ′ degree 0.023 20.94 0.032 20.54 0.038 20.32c′ kN/m2 -0.038 45.59 0.098 38.23 0.085 38.39Rinter - 0.548 0.598 0.726 0.356 0.744 0.321

Weathered E′ kN/m2 0.214 1.08E6 0.180 1.08E6 0.183 1.08E6rock γsat kN/m3 -0.018 22.04 0.011 21.96 0.006 21.96

ϕ′ degree -0.025 43.14 0.007 43.00 0.007 42.93ψ degree 0.000 5.00 0.000 5.00 1.57E-8 5.00

Number of calculations (FORM): 154 171 171β 1.329 3.389 3.639Pf 0.092 3.5E-04 1.37E-04

188

Prob2B-Plaxis results C.1 Reliability analysis results for intact bored piles


γsat kN/m3 -0.087 19.24 -0.015 19.16 -0.009 19.16ϕ′ degree 0.015 18.67 -0.014 18.78 -0.017 18.85c′ kN/m2 0.059 38.62 0.089 37.54 0.088 37.49



Weathered E′ kN/m2 0.207 1.08E6 0.161 1.08E6 0.161 1.08E6rock γsat kN/m3 -0.006 22.03 -0.004 22.01 9.88E-4 22.00

ϕ′ degree -0.014 43.05 -2.31E-4 42.89 -0.002 43.04ψ degree 0.000 5.00 6.36E-6 5.00 1.08E-5 5.00



γsat kN/m3 -0.093 19.28 -0.014 19.14 -0.016 19.17ϕ′ degree -0.006 18.72 -0.015 18.83 -0.017 18.74c′ kN/m2 0.045 38.38 0.099 37.29 0.086 37.44



Weathered E′ kN/m2 0.184 1.09E6 0.159 1.08E6 0.165 1.09E6rock γsat kN/m3 -0.009 22.06 -0.003 22.02 -0.002 22.03

ϕ′ degree -0.025 42.91 -0.004 42.99 -0.006 42.99ψ degree 8.22E-5 5.00 3.28E-5 5.00 9.71E-05 5.00


Case 7 Dead load λQD - -0.019 1.184 -0.016 1.058 0.056 1.132Live load λQL - -0.427 1.143 -0.379 1.033 -0.388 1.178Sandstone E50 kN/m2 -0.045 1.49E5 -0.338 1.62E5 -0.037 1.44E5

γsat kN/m3 -0.055 19.40 -0.054 18.49 -0.004 19.11ϕ′ degree -0.025 18.98 -0.013 16.34 0.009 18.31c′ kN/m2 0.021 38.26 0.077 33.88 0.067 38.97


ϕ′ degree 0.010 21.03 0.159 18.17 0.052 20.28c′ kN/m2 0.017 46.70 0.045 43.17 0.118 35.46Rinter - 0.359 0.456 0.327 0.467 0.500 0.307

Weathered E′ kN/m2 -0.233 9.26E5 -0.515 1.21E6 -0.275 9.65E5rock γsat kN/m3 -0.253 19.38 -0.177 20.06 -0.174 19.09

ϕ′ degree 0.637 - 0.448 8.31 0.583 6.54ψ degree 0.033 4.84 -0.029 5.15 0.025 4.86

Number of calculations (FORM): 171 171 171β 4.852 5.090 5.609Pf 6.11E-07 1.79E-07 1.02E-08

189


Case 8 Dead load λQD - -0.007 1.145 0.015 1.111 0.015 1.091Live load λQL - -0.511 1.178 -0.414 1.155 -0.444 1.132Sandstone E50 kN/m2 -0.063 1.48E5 -0.033 1.22E5 -0.037 1.38E5


pile Rinter - 0.258 0.618 0.218 0.625 0.196 0.63529.0 m Clayey E50 kN/m2 -0.011 7,131 0.049 5,739 -0.034 5,690long sand γsat kN/m3 -0.016 20.27 -0.025 19.26 -0.050 18.99

ϕ′ degree -0.020 21.42 0.313 18.79 0.205 19.40c′ kN/m2 -0.006 45.82 0.109 49.40 0.193 44.00Rinter - 0.252 0.520 0.422 0.377 0.366 0.412


ϕ′ degree 0.627 2.69 0.499 7.21 0.534 1.74ψ degree 0.025 4.87 0.001 4.99 0.008 4.95Rinter - -8.48E-5 0.850 3.28E-4 0.850 -1.51E-6 0.850



γsat kN/m3 -0.051 19.17 -0.003 19.04 -0.004 19.02ϕ′ degree 0.002 18.77 -0.005 18.32 -0.008 18.53c′ kN/m2 0.042 36.95 0.059 38.97 0.067 38.16


ϕ′ degree -0.020 21.38 0.102 18.87 0.077 19.29c′ kN/m2 -0.018 46.61 0.036 46.07 0.044 45.17Rinter - 0.325 0.470 0.306 0.472 0.432 0.352


ϕ′ degree 0.677 7.49 0.703 2.23 0.600 1.08ψ degree 0.062 4.69 0.084 4.55 0.019 4.89Rinter - 0.054 0.827 0.050 0.827 0.017 0.842


190


C.1.2 Nine calculation cases under SF mode

Table C.2: Reliability analysis results for 9 calculation cases under SF mode

Displacement criteriaCal. Item Variable, Unit 1%D 3%D 5%Dcase Xi α∗i x∗i α∗i x∗i α∗i x∗iCase 1 Concrete λ f n - 0.997 0.069 0.997 0.117 0.996 0.146

Sandstone E50 kN/m2 0.001 1.46E5 0.013 1.49E5 0.044 1.47E5γsat kN/m3 0.001 19.46 0.006 19.64 0.038 19.66ϕ′ degree -0.034 20.05 -0.053 20.43 -0.054 20.63c′ kN/m2 0.026 38.34 0.010 38.28 0.012 39.13

pile Rinter - -0.011 0.813 -0.011 0.809 -0.006 0.80610.5 m Clayey E50 kN/m2 0.022 6,923 0.023 6,987 0.012 6,917long sand γsat kN/m3 0.037 20.24 0.008 20.39 0.009 20.25

ϕ′ degree -0.018 21.25 -0.038 21.23 -0.031 21.13c′ kN/m2 -0.009 46.35 -0.012 45.80 0.002 44.76Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 0.023 1.05E6 -0.006 1.09E6 0.004 1.08E6rock γsat kN/m3 0.012 21.84 -0.004 22.06 0.003 21.99

ϕ′ degree 0.022 43.21 -0.009 42.99 0.001 43.09ψ degree 0.000 5.00 0.000 5.00 0.000 5.00Rinter - 0.000 0.850 0.000 0.850 0.000 0.850


Case 2 Concrete λ f n - 0.998 0.074 0.997 0.095 0.997 0.104Sandstone E50 kN/m2 0.025 1.47E5 0.042 1.47E5 0.039 1.49E5

γsat kN/m3 0.018 19.54 0.017 19.62 0.021 19.67ϕ′ degree -0.023 19.61 -0.027 19.83 -0.039 20.06c′ kN/m2 -0.005 40.27 -0.007 40.38 -0.012 40.06

pile Rinter - -0.048 0.846 -0.050 0.846 -0.053 0.84912.5 m Clayey E50 kN/m2 0.001 6,949 -0.001 6,987 -0.004 6,995long sand γsat kN/m3 0.010 20.13 0.008 20.17 0.007 20.22

ϕ′ degree -0.002 20.88 -0.013 20.99 -0.019 20.99c′ kN/m2 0.002 45.50 0.004 44.82 7.47E-4 45.16Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 0.004 1.08E6 0.004 1.08E6 0.002 1.08E6rock γsat kN/m3 0.002 22.00 6.08E-4 21.99 -2.36E-4 22.00

ϕ′ degree -9.85E-4 43.04 8.59E-4 42.96 6.81E-4 42.85ψ degree 0.000 5.00 0.000 5.00 0.000 5.00Rinter - 0.000 0.850 0.000 0.850 0.000 0.850


191



γsat kN/m3 0.010 19.54 0.018 19.59 0.020 19.48ϕ′ degree -0.008 19.37 -0.021 19.50 -0.006 19.29c′ kN/m2 -0.010 40.93 -0.023 40.66 -0.006 41.08


ϕ′ degree -0.009 21.26 -0.009 20.90 0.006 20.98c′ kN/m2 0.013 44.86 -0.006 45.70 0.009 45.48Rinter - 0.000 0.700 0.011 0.709 0.000 0.700

Weathered E′ kN/m2 0.003 1.08E6 2.44E-4 1.08E6 0.009 1.08E6rock γsat kN/m3 0.009 21.96 -0.008 22.04 0.015 21.95

ϕ′ degree 0.003 42.97 -0.003 42.74 0.002 43.25ψ degree 0.000 5.00 0.000 5.00 0.000 5.00Rinter - 0.000 0.850 0.000 0.850 0.000 0.850



γsat kN/m3 0.006 19.50 0.013 19.55 -3.16E-4 19.54ϕ′ degree -0.006 19.31 -0.016 19.48 -0.015 19.43c′ kN/m2 -0.005 40.47 -0.017 40.64 -0.017 40.24

pile Rinter - -0.075 0.869 -0.087 0.876 -0.088 0.87624.0 m Clayey E50 kN/m2 -0.002 7,019 0.041 6,914 0.035 7,043long sand γsat kN/m3 0.001 20.23 0.032 20.37 0.029 20.43

ϕ′ degree -0.009 21.07 -0.010 21.21 -0.016 21.20c′ kN/m2 0.001 44.57 -0.009 47.35 -0.021 47.43Rinter - -0.040 0.732 -0.047 0.736 -0.072 0.754

Weathered E′ kN/m2 0.003 1.09E6 0.010 1.08E6 0.004 1.08E6rock γsat kN/m3 -0.003 22.05 0.004 21.97 -0.012 22.06

ϕ′ degree -0.007 43.02 0.004 42.98 -0.007 42.90ψ degree 0.000 5.00 0.000 5.00 -1.25E-9 5.00Rinter - 0.000 0.850 0.000 0.850 0.000 0.850



γsat kN/m3 -3.16E-4 19.54 -0.007 19.60 0.002 19.50ϕ′ degree -0.015 19.43 -0.009 19.30 -7.93E-4 19.14c′ kN/m2 -0.017 40.24 -0.031 40.73 -0.017 41.05


ϕ′ degree -0.016 21.20 -0.028 21.44 -0.003 21.22c′ kN/m2 -0.021 47.43 -0.013 46.61 -0.009 47.40Rinter - -0.072 0.754 -0.063 0.747 -0.082 0.760

Weathered E′ kN/m2 0.004 1.08E6 0.007 1.08E6 0.002 1.09E6rock γsat kN/m3 -0.012 22.06 -0.009 22.06 0.012 21.98

ϕ′ degree -0.007 42.90 -0.008 43.07 -0.001 42.99ψ degree 0.000 5.00 0.000 5.00 -0.004 5.02Rinter - 0.000 0.850 0.000 0.850 0.000 0.850


192



γsat kN/m3 0.012 19.49 0.010 19.49 0.012 19.43ϕ′ degree -0.006 19.28 -0.003 19.27 0.008 19.14c′ kN/m2 -0.005 40.51 -0.017 40.88 -0.013 41.11

pile Rinter - -0.071 0.864 -0.071 0.861 -0.073 0.86327.0 m Clayey E50 kN/m2 -0.003 6,963 0.066 6,883 0.070 6,910long sand γsat kN/m3 -0.002 20.19 0.038 20.50 0.051 20.53


Weathered E′ kN/m2 0.010 1.08E6 0.017 1.07E6 0.026 1.06E6rock γsat kN/m3 2.21E-4 22.04 -0.005 21.99 0.016 21.90

ϕ′ degree -0.006 43.32 0.005 43.10 0.012 43.45ψ degree -7.67E-6 5.00 0.009 4.95 -0.002 5.01Rinter - 0.000 0.850 0.000 0.850 0.000 0.850



γsat kN/m3 -0.003 19.51 0.013 19.41 0.008 19.44ϕ′ degree -0.012 19.17 0.008 19.08 9.48E-4 19.09c′ kN/m2 -0.015 40.30 0.001 40.28 -0.022 40.20

pile Rinter - -0.060 0.849 -0.066 0.836 -0.047 0.81728.0 m Clayey E50 kN/m2 -0.008 7,013 0.012 6,999 0.032 7,029long sand γsat kN/m3 5.80E-4 20.18 0.008 20.28 -0.010 20.33


Weathered E′ kN/m2 0.027 1.17E6 0.123 1.23E6 0.268 1.17E6rock γsat kN/m3 0.045 22.56 0.180 22.96 0.257 22.77

ϕ′ degree -0.131 47.75 -0.405 52.71 -0.624 51.29ψ degree -0.014 5.07 -0.028 5.08 -0.079 5.12Rinter - -1.46E-6 0.850 -7.40E-5 0.850 -0.021 0.853

Number of calculations (FORM): 171 171 171β 4.893 2.857 1.456Pf 4.97E-07 2.14E-03 0.073


γsat kN/m3 0.009 19.43 -0.009 19.34 0.001 19.42ϕ′ degree -0.006 19.23 0.037 18.93 0.005 19.03c′ kN/m2 0.004 40.21 0.023 40.54 -0.016 40.24

pile Rinter - -0.053 0.843 -0.050 0.827 -0.035 0.81129.0 m Clayey E50 kN/m2 -1.49E-4 6,960 0.051 6,974 0.037 6,988long sand γsat kN/m3 0.005 20.17 0.044 20.34 0.015 20.26



ϕ′ degree -0.129 48.17 -0.362 51.90 -0.616 49.28ψ degree -0.016 5.08 -0.059 5.15 -0.103 5.12Rinter - -3.05E-5 0.850 0.014 0.847 0.007 0.849


193



γsat kN/m3 0.016 19.49 0.059 19.24 0.044 19.41ϕ′ degree -0.003 19.15 0.043 18.94 0.025 19.03c′ kN/m2 -0.015 41.05 0.044 40.28 -0.014 40.63

pile Rinter - -0.034 0.831 -0.008 0.811 0.012 0.80830.0 m Clayey E50 kN/m2 -0.002 7,006 0.062 6,881 0.011 7,019long sand γsat kN/m3 0.007 20.19 0.003 20.37 0.008 20.25

ϕ′ degree 0.000 20.96 0.006 21.17 -0.006 21.02c′ kN/m2 0.002 45.06 -0.019 45.79 -0.019 45.09Rinter - -0.028 0.719 -0.020 0.708 -0.071 0.712


ϕ′ degree -0.148 49.58 -0.405 52.93 -0.539 51.21ψ degree -0.019 5.09 -0.011 5.03 -0.101 5.12Rinter - -0.050 0.870 -0.098 0.872 -0.151 0.865


Table C.3: Reliability analysis results for 9 calculation cases under SF mode with dis-placement of 2.2%D

Calculation casesItem Variable, Unit Case 1 Case 2 Case 3

Xi α∗i x∗i α∗i x∗i α∗i x∗iConcrete λ f n - 0.997 0.117 0.998 0.089 0.996 0.085Sandstone E50 kN/m2 0.028 1.41E5 0.034 1.47E5 0.027 1.49E5

γsat kN/m3 0.018 19.42 0.017 19.57 0.017 19.61ϕ′ degree -0.032 20.36 -0.023 19.61 -0.020 19.56c′ kN/m2 0.059 38.17 -0.009 40.57 -0.023 40.57Rinter - -0.004 0.804 -0.045 0.843 -0.071 0.866

Clayey E50 kN/m2 0.007 6,858 0.007 6,901 -0.006 7,157sand γsat kN/m3 0.004 20.17 0.010 20.13 0.002 20.30

ϕ′ degree -0.017 21.13 -0.002 20.92 -0.017 20.96c′ kN/m2 -0.001 44.38 0.006 45.53 -0.009 45.15Rinter - 0.000 0.700 0.000 0.700 0.000 0.700

Weathered E′ kN/m2 -1.39E-4 1.08E6 0.009 1.07E6 -0.004 1.09E6rock γsat kN/m3 -5.66E-4 22.00 0.001 21.99 -0.015 22.08

ϕ′ degree 6.11E-4 42.95 0.002 43.10 -0.007 42.62ψ degree 0.000 5.00 0.000 5.00 0.000 5.00Rinter - 0.000 0.850 0.000 0.850 0.000 0.850


194




γsat kN/m3 0.012 19.49 0.007 19.52 0.005 19.55ϕ′ degree 5.45E-4 19.32 0.003 19.17 -0.001 19.29c′ kN/m2 -0.004 40.69 -0.013 41.05 -0.014 40.75Rinter - -0.079 0.870 -0.087 0.876 -0.082 0.870

Clayey E50 kN/m2 0.019 6,915 0.017 6,853 0.006 6,975sand γsat kN/m3 0.022 20.30 0.012 20.26 0.006 20.31

ϕ′ degree -0.021 21.48 -0.020 21.49 -0.030 21.50c′ kN/m2 0.012 44.32 0.024 43.67 0.017 43.21Rinter - -0.029 0.723 -0.039 0.729 -0.050 0.738

Weathered E′ kN/m2 0.010 1.08E6 0.010 1.08E6 0.011 1.07E6rock γsat kN/m3 0.009 21.98 0.001 22.02 -0.004 21.99

ϕ′ degree -4.74E-4 43.18 -0.004 43.15 0.003 42.93ψ degree 0.000 5.00 -8.03E-4 5.00 -0.006 5.03Rinter - 0.000 0.850 0.000 0.850 0.000 0.850




γsat kN/m3 -0.010 19.44 0.017 19.40 0.004 19.37ϕ′ degree -0.007 19.10 0.002 19.10 0.032 18.97c′ kN/m2 1.37E-4 40.02 0.005 40.72 0.009 40.74Rinter - -0.070 0.846 -0.066 0.842 -0.039 0.829

Clayey E50 kN/m2 -0.010 7,025 0.026 6,909 0.029 7,039sand γsat kN/m3 -0.014 20.19 0.015 20.20 0.017 20.41

ϕ′ degree -0.003 20.93 0.017 20.92 -0.021 21.39c′ kN/m2 -0.004 44.82 -0.008 46.32 -0.013 44.95Rinter - -0.047 0.724 -0.038 0.719 -0.031 0.715


ϕ′ degree -0.305 51.99 -0.269 51.24 -0.307 52.85ψ degree 0.002 4.99 -0.033 5.12 -0.038 5.13Rinter - 1.23E-6 0.850 -1.45E-5 0.850 -0.054 0.866


195


C.1.3 Working pile subjected to working load under SF mode

Table C.4: Reliability analysis result for working pile subjected to working load under SFmode

Item Variable, Xi Unit α∗i x∗iConcrete λ f n - 1.000 0.097Sandstone E50 kN/m2 4.07E-5 1.47E5

γsat kN/m3 -1.29E-6 19.40ϕ′ degree -4.33E-5 19.00c′ kN/m2 -4.89E-5 40.11Rinter - -2.75E-5 0.800

Clayey E50 kN/m2 2.45E-5 6,981sand γsat kN/m3 -1.21E-5 20.20

ϕ′ degree -7.25E-5 21.00c′ kN/m2 -2.33E-5 45.02Rinter - -1.70E-5 0.700

Weathered E′ kN/m2 4.56E-5 1.08E6rock γsat kN/m3 3.89E-5 22.00

ϕ′ degree -8.94E-5 43.00ψ degree -2.25E-6 5.00Rinter - -3.02E-5 0.850

Number of calculations (FORM): 171β 5.779Pf 3.77E-09

196

Prob2B-Plaxis results C.2 Reliability analysis results for defect bored piles

C.2 Reliability analysis results for defect bored piles

C.2.1 The effect of necking near the pile top

Table C.5: Reliability analysis results for working pile with necking under GF mode

Cross-sectional reduction, ArItem Variable, 25% 50% 75% 95%

Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λQD 0.013 1.130 0.013 1.130 0.015 1.131 0.007 1.141Live load λQL -0.419 1.174 -0.419 1.172 -0.417 1.174 -0.437 1.174Sandstone E50 -0.053 1.47E5 -0.052 1.46E5 -0.050 1.46E5 -0.057 1.47E5

γsat -0.031 19.20 -0.029 19.19 -0.028 19.17 -0.030 19.18ϕ′ 0.004 18.66 0.006 18.66 0.007 18.65 0.005 18.62c′ 0.029 38.41 0.034 38.43 0.037 38.47 0.033 38.36Rinter 0.239 0.607 0.241 0.606 0.244 0.6026 0.255 0.599

Clayey E50 6.83E-4 6,886 0.002 6,854 0.004 6,803 0.002 6,845sand γsat 0.002 20.11 0.002 20.09 0.004 20.05 0.002 20.07

ϕ′ 0.009 21.02 0.012 21.02 0.015 21.01 0.015 20.96c′ 0.001 45.22 0.003 45.20 0.007 45.19 0.002 45.46Rinter 0.569 0.256 0.563 0.261 0.564 0.260 0.530 0.289

Weathered E′ -0.308 1.02E6 -0.309 1.01E6 -0.306 1.01E6 -0.312 1.02E6rock γsat -0.212 19.40 -0.211 19.37 -0.209 19.36 -0.217 19.35

ϕ′ 0.546 5.86 0.552 5.66 0.553 5.80 0.560 5.08ψ 0.018 4.90 0.018 4.90 0.020 4.89 0.016 4.91Rinter 0.012 0.844 0.014 0.844 0.014 0.843 0.013 0.844

No. of cal. (FORM): 181 181 181 181β 5.577 5.577 5.574 5.543Pf 1.23E-08 1.23E-08 1.25E-08 1.49E-08

Table C.6: Reliability analysis results for working pile with necking under SF mode


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iConcrete λ f n 0.802 0.915 0.574 1.180 0.541 1.405 0.186 1.400Sandstone E50 -0.037 1.49E5 0.024 1.47E5 0.052 1.47E5 0.054 1.47E5

γsat -0.069 19.41 -0.018 19.40 0.035 19.31 0.029 19.17ϕ′ 0.007 18.45 -0.009 19.00 -0.007 18.75 -0.005 18.72c′ -0.070 40.46 -0.019 40.11 -0.026 39.63 -0.036 38.08Rinter -0.026 0.813 -0.063 0.801 -0.149 0.738 -0.173 0.582

Clayey E50 -0.041 7,157 0.024 6,980 0.026 6,951 -0.003 6,868sand γsat -0.098 20.19 -0.030 20.20 0.006 20.07 -0.001 20.13

ϕ′ 0.065 20.27 -0.030 21.01 -0.035 20.84 -0.006 21.09c′ -0.050 47.62 0.011 45.01 -0.006 45.09 -0.001 45.05Rinter -0.103 0.726 -0.129 0.702 -0.202 0.640 -0.533 0.254

Weathered E′ 0.204 1.16E6 0.405 1.08E6 0.364 9.71E5 0.559 1.05E6rock γsat 0.237 22.58 0.252 22.04 0.247 20.80 0.225 19.34

ϕ′ -0.411 50.38 -0.606 43.56 -0.658 29.48 -0.532 2.39ψ -0.071 5.13 -0.065 5.01 -0.075 4.84 0.000 5.00Rinter -0.212 0.883 -0.216 0.852 -0.076 0.836 -0.010 0.845

No. of cal. (FORM): 171 171 171 171β 1.809 0.087 -2.116 -5.979Pf 0.035 0.465 0.983 1.000

197

C.2 Reliability analysis results for defect bored piles Prob2B-Plaxis results

Table C.7: Reliability analysis results for working pile subjected to working load underSF mode


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iConcrete λ f n 1.000 0.143 1.000 0.248 1.000 0.459 1.000 2.088Sandstone E50 1.07E-4 1.47E5 2.12E-4 1.47E5 4.08E-4 1.47E5 0.001 1.47E5

γsat -3.51E-6 19.40 -7.14E-6 19.40 -1.28E-5 19.40 6.82E-6 19.38ϕ′ -1.09E-4 19.00 -2.28E-4 19.01 -4.53E-4 19.01 -0.002 18.95c′ -1.23E-4 40.11 -2.54E-4 40.11 -4.91E-4 40.10 -0.002 40.12Rinter -6.79E-5 0.800 -1.35E-4 0.800 -2.51E-4 0.800 -8.30E-4 0.799

Clayey E50 6.18E-5 6,982 1.28E-4 6,983 2.52E-4 6,983 9.99E-4 6,969sand γsat -3.05E-5 20.20 -6.26E-5 20.20 -1.21E-4 20.20 -4.50E-4 20.18

ϕ′ -1.79E-4 21.00 -3.75E-4 21.00 -7.42E-4 21.00 -0.003 20.98c′ -5.62E-5 45.02 -1.19E-4 45.02 -2.38E-4 45.02 -0.001 45.02Rinter -3.90E-5 0.700 -8.01E-5 0.700 -1.57E-4 0.700 -6.71E-4 0.700

Weathered E′ 1.12E-4 1.08E6 2.35E-4 1.08E6 4.65E-4 1.08E6 0.002 1.08E6rock γsat 9.66E-5 22.00 2.01E-4 22.00 3.99E-4 22.00 0.002 21.98

ϕ′ -2.24E-4 43.01 -4.65E-4 43.02 -9.19E-4 43.03 -0.004 42.83ψ -5.71E-6 5.00 -1.16E-5 5.00 -2.31E-5 5.00 -1.42E-4 5.00Rinter -7.55E-5 0.850 -1.57E-4 0.850 -3.11E-4 0.850 -0.001 0.850

No. of cal. (FORM): 171 171 171 171β 5.535 4.979 3.861 -4.755Pf 1.56E-08 3.21E-07 5.64E-05 1.000

198


C.2.2 The effect of poor concrete zone

Table C.8: Reliability analysis results for working pile with poor concrete zone under GFmode

Poor concrete zone locationItem Variable, At the pile top In the middle Near the pile toe At the pile toe

Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λQD 0.014 1.132 0.014 1.132 0.014 1.128 0.014 1.126Live load λQL -0.413 1.172 -0.417 1.174 -0.429 1.171 -0.423 1.170Sandstone E50 -0.052 1.47E5 -0.052 1.47E5 -0.050 1.47E5 -0.050 1.47E5

γsat -0.031 19.19 -0.032 19.20 -0.030 19.20 -0.030 19.20ϕ′ 0.009 18.67 0.008 18.68 0.009 18.67 0.008 18.67c′ 0.033 38.38 0.031 38.40 0.030 38.47 0.031 38.47Rinter 0.243 0.603 0.244 0.602 0.239 0.604 0.234 0.609

Clayey E50 0.006 6,860 0.007 6,886 0.007 6,886 0.006 6,872sand γsat 0.003 20.15 0.003 20.18 0.003 20.17 0.004 20.16

ϕ′ 0.003 21.19 0.001 21.19 0.003 21.17 0.003 21.18c′ 0.004 44.81 0.002 44.92 0.001 45.02 0.004 44.86Rinter 0.586 0.240 0.587 0.239 0.567 0.252 0.568 0.251

Weathered E′ -0.302 1.02E6 -0.304 1.03E6 -0.327 1.05E6 -0.324 1.03E6rock γsat -0.206 19.43 -0.209 19.47 -0.207 19.44 -0.207 19.40

ϕ′ 0.536 6.32 0.530 6.30 0.532 5.19 0.540 5.19ψ 0.017 4.90 0.016 4.91 0.013 4.93 0.015 4.91Rinter 0.013 0.844 0.010 0.845 0.010 0.845 0.011 0.845

Number ofcalculations (FORM): 181 181 181 181

β 5.611 5.614 5.648 5.643Pf 1.01E-08 9.92E-09 8.13E-09 8.34E-09

Table C.9: Reliability analysis results for working pile with poor concrete zone under SFmode, considering stress state in good concrete zone


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λ f n 0.924 0.590 0.938 0.568 0.950 0.555 0.923 0.58Sandstone E50 -0.029 1.55E5 0.011 1.46E5 0.028 1.46E5 0.038 1.47E5

γsat 0.051 19.51 0.034 19.35 0.022 19.41 0.025 19.44ϕ′ 0.007 19.23 0.015 19.08 0.008 18.94 0.013 18.88c′ -0.051 40.39 0.014 40.45 -0.007 40.96 -0.018 41.45Rinter -0.010 0.811 -0.074 0.849 -0.038 0.827 -0.058 0.841

Clayey E50 0.050 7,032 0.033 6,837 0.007 6,882 0.022 6,790sand γsat 0.036 20.63 0.020 20.26 -7.49E-4 20.15 0.004 20.14

ϕ′ -0.091 21.99 -0.008 21.31 0.013 20.97 0.013 21.08c′ 0.016 43.34 0.016 44.70 0.015 44.92 0.031 44.78Rinter -0.024 0.712 -0.021 0.710 -0.024 0.712 -0.044 0.722

Weathered E′ 0.098 1.21E6 0.138 1.17E6 0.101 1.18E6 0.153 1.18E6rock γsat 0.158 22.78 0.126 22.77 0.135 22.68 0.149 22.82

ϕ′ -0.289 51.84 -0.268 52.06 -0.246 51.07 -0.291 53.12ψ -0.080 5.27 -0.034 5.12 -0.066 5.23 -0.053 5.19Rinter -0.043 0.863 -0.046 0.864 -0.022 0.857 -0.075 0.872


β 3.426 3.502 3.531 3.517Pf 3.07E-04 2.31E-04 2.07E-04 2.18E-04

199


Table C.10: Reliability analysis results for working pile with poor concrete zone underSF mode, considering stress state in poor concrete zone


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λ f n 0.721 1.120 0.773 1.042 0.761 1.046 0.767 0.959Sandstone E50 0.039 1.46E5 0.011 1.48E5 0.006 1.49E5 0.039 1.42E5

γsat -0.007 19.42 -0.010 19.44 -0.011 19.50 0.087 19.24ϕ′ -0.018 19.09 0.008 18.92 -0.074 19.29 0.026 19.14c′ 0.022 40.20 -0.033 40.46 -0.040 39.82 0.051 40.42Rinter -0.176 0.85 -0.025 0.815 0.162 0.748 0.071 0.781

Clayey E50 0.021 6,835 -0.003 7,038 0.086 6,970 -0.099 7,283sand γsat 0.009 20.09 -0.002 20.26 0.003 20.30 0.035 20.05

ϕ′ 0.031 20.96 -0.017 21.08 0.037 20.59 0.036 20.57c′ 0.040 45.14 -0.002 44.82 -0.082 49.22 -0.048 44.72Rinter -0.064 0.71 -0.066 0.718 0.050 0.686 -0.043 0.715

Weathered E′ 0.242 1.17E6 0.229 1.18E6 -0.042 1.27E6 0.225 1.21E6rock γsat 0.290 22.62 0.220 22.81 0.089 23.07 0.178 23.14

ϕ′ -0.517 50.57 -0.501 52.17 -0.578 49.43 -0.516 54.93ψ -0.045 5.07 -0.095 5.185 -0.159 5.31 -0.049 5.12Rinter -0.167 0.872 -0.186 0.881 0.019 0.847 0.165 0.815


β 1.534 1.951 1.950 2.530Pf 0.063 0.026 0.026 5.70E-03

Table C.11: Reliability analysis results for working pile subjected to working load underSF mode, considering stress state in good concrete zone


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λ f n 1.000 0.098 1.000 0.097 1.000 0.097 1.000 0.097Sandstone E50 5.76E-5 1.47E5 4.07E-5 1.47E5 3.92E-5 1.47E5 4.01E-5 1.47E5

γsat -2.06E-6 19.40 -1.77E-6 19.40 -3.14E-6 19.40 -2.61E-6 19.40ϕ′ -5.99E-5 19.00 -4.37E-5 19.00 -4.14E-5 19.00 -4.28E-5 19.00c′ -6.79E-5 40.11 -4.86E-5 40.11 -5.09E-5 40.11 -5.12E-5 40.11Rinter -3.87E-5 0.800 -2.61E-5 0.800 -2.60E-5 0.800 -2.66E-5 0.800


ϕ′ -9.91E-5 21.00 -7.23E-5 21.00 -6.72E-5 21.00 -6.94E-5 21.00c′ -3.12E-5 45.02 -2.29E-5 45.02 -2.16E-5 45.02 -2.10E-5 45.02Rinter -2.36E-5 0.700 -1.69E-5 0.700 -1.41E-5 0.700 -1.49E-5 0.700

Weathered E′ 6.31E-5 1.08E6 4.51E-5 1.08E6 3.63E-5 1.08E6 3.01E-5 1.08E6rock γsat 5.38E-5 22.00 3.88E-5 22.00 3.22E-5 22.00 3.08E-5 22.00

ϕ′ -1.24E-4 43.01 -8.92E-5 43.00 -7.28E-5 43.00 -7.42E-5 43.00ψ -3.11E-6 5.00 -2.25E-6 5.00 -1.50E-6 5.00 -3.19E-6 5.00Rinter -4.19E-5 0.850 -3.01E-5 0.850 -1.89E-5 0.850 -2.67E-5 0.850


β 5.774 5.779 5.779 5.779Pf 3.87E-09 3.77E-09 3.77E-09 3.77E-09

200


Table C.12: Reliability analysis results for working pile subjected to working load underSF mode, considering stress state in poor concrete zone


Xi α∗i x∗i α∗i x∗i α∗i x∗i α∗i x∗iDead load λ f n 1.000 0.196 1.000 0.151 0.999 0.181 0.999 0.133Sandstone E50 1.56E-4 1.47E5 3.83E-3 1.48E5 4.53E-3 1.48E5 2.72E-3 1.48E5

γsat -6.08E-6 19.40 -5.82E-3 19.44 -6.85E-3 19.45 -5.32E-3 19.44ϕ′ -1.54E-4 19.00 9.67E-3 18.72 9.83E-3 18.74 7.91E-3 18.81c′ -1.80E-4 40.11 -1.63E-2 40.80 -1.96E-2 40.80 -1.40E-2 40.57Rinter -1.07E-4 0.800 7.61E-3 0.793 1.47E-2 0.786 1.01E-2 0.790


ϕ′ -2.54E-4 21.00 -4.38E-3 21.09 -8.11E-3 21.11 -9.07E-3 21.13c′ -7.72E-5 45.02 -3.23E-3 45.05 -6.93E-3 45.11 -6.85E-3 45.03Rinter -6.17E-5 0.700 -3.87E-3 0.703 1.67E-2 0.686 9.52E-3 0.692

Weathered E′ 1.65E-4 1.08E6 7.64E-3 1.09E6 1.81E-3 1.10E6 2.69E-4 1.10E6rock γsat 1.40E-4 22.00 8.04E-3 22.10 1.10E-2 22.13 6.72E-3 22.09

ϕ′ -3.21E-4 43.02 -1.86E-2 44.07 -2.63E-2 44.16 -1.78E-2 43.77ψ -8.04E-6 5.00 -4.34E-4 5.00 1.25E-4 5.00 -3.83E-3 5.02Rinter -1.09E-4 0.850 -6.42E-3 0.853 -2.55E-2 0.863 1.93E-2 0.840


β 5.893 6.130 5.979 6.224Pf 1.90E-09 4.40E-10 1.13E-09 2.44E-10

201


C.2.3 The effect of soft bottom

Table C.13: Reliability analysis results for working pile with soft bottom under GF mode

Item Variable, Xi Unit α∗i x∗iDead load λQD - 0.016 1.177Live load λQL - -0.450 1.232Sandstone E50 kN/m2 -0.069 1.49E5

γsat kN/m3 -0.022 19.30ϕ′ degree -0.048 19.95c′ kN/m2 0.057 36.07Rinter - 0.291 0.569

Clayey E50 kN/m2 0.012 6,803sand γsat kN/m3 0.036 20.06

ϕ′ degree -0.014 21.20c′ kN/m2 0.001 45.82Rinter - 0.637 0.201

Weathered E′ kN/m2 -0.225 1.06E6rock γsat kN/m3 -0.188 19.86

ϕ′ degree 0.454 10.39ψ degree 0.042 4.76Rinter - 0.038 0.832

Slim layer γsat kN/m3 -4.56E-5 13.00Number of calculations (FORM): 191

β 5.593Pf 1.12E-08

Table C.14: Reliability analysis results for working pile with soft bottom under SF mode

Item Variable, Xi Unit α∗i x∗iConcrete λ f n - 0.944 0.384Sandstone E50 kN/m2 0.015 1.41E5

γsat kN/m3 0.031 19.20ϕ′ degree 0.013 18.60c′ kN/m2 0.033 40.75Rinter - -0.135 0.904

Clayey E50 kN/m2 -0.036 6,795sand γsat kN/m3 -0.048 19.77

ϕ′ degree 0.075 20.01c′ kN/m2 0.009 45.40Rinter - -0.056 0.736

Weathered E′ kN/m2 0.100 1.07E6rock γsat kN/m3 0.030 22.27

ϕ′ degree -0.070 47.47ψ degree 0.030 4.87Rinter - -0.244 0.944

Slim layer γsat kN/m3 0.005 12.99Number of calculations (FORM): 181

β 4.511Pf 3.22E-06

202


Table C.15: Reliability analysis results for working pile subjected to working load underSF mode

Item Variable, Xi Unit α∗i x∗iConcrete λ f n - 1.000 0.097Sandstone E50 kN/m2 1.89E-4 1.47E5

γsat kN/m3 1.43E-4 19.40ϕ′ degree -2.87E-5 19.00c′ kN/m2 -1.97E-4 40.10Rinter - -6.47E-4 0.801

Clayey E50 kN/m2 -1.29E-5 6,980sand γsat kN/m3 3.06E-5 20.20

ϕ′ degree -8.62E-5 21.00c′ kN/m2 -3.73E-6 45.00Rinter - -2.62E-4 0.700

Weathered E′ kN/m2 1.58E-4 1.08E6rock γsat kN/m3 -4.21E-6 22.00

ϕ′ degree -1.66E-4 43.00ψ degree -1.19E-6 5.00Rinter - -8.01E-5 0.850

Slim layer γsat kN/m3 5.18E-8 13.00Number of calculations (FORM): 181

β 5.780Pf 3.75E-09

203


204

List of Symbols

Roman Symbols

a magnitude of defect in shape of circular segment mmA set-up factor -AB cross-sectional area at the base of pile m2

AD cross-sectional area of defect in shape of circular segment m2

Ar cross-sectional area reduction %AS shaded area used by Li et al. (2005) to determine PE m2

ASI side area of pile m2

AT area of circular segment located by a chord going through centers of twoadjacent access tubes

m2

A0 cross-sectional area of reinforcing steel cage m2

c crack size mmci cohesion of interface element kN/m2

c1 minimum detectable crack size mmc2 detectable crack size with certainty mmc′ effective cohesion kN/m2

c′f design effective cohesion kN/m2

C stress wave velocity in concrete m/sCov(Xi, Xj) covariance of variables Xi and Xj -COVf n coefficient of variation for compressive strength bias factor of concrete -COVQ coefficient of variation for load effect bias factor -COVQD coefficient of variation for dead load bias factor -COVQL coefficient of variation for live load bias factor -COVR coefficient of variation for resistance bias factor -COV

′′R coefficient of variation for resistance bias factor of posterior log-normal

distribution-

COVRB coefficient of variation for base resistance bias factor -COVRS coefficient of variation for shaft resistance bias factor -COVRSE coefficient of variation for set-up resistance bias factor -COVR0 coefficient of variation for reference resistance bias factor -d distance from tangent plane of failure surface at x∗ to origin of reduce

coordinate system-

di displacement value considered in O-cell test mmdmin minimum distance from origin of reduce coordinate system to failure

surface-

dt distance from center to center for each pair of access tubes md∗ distance from origin of reduce coordinate system to failure surface at x∗ -

205

List of Symbols List of Symbols

D pile diameter m, mmDc displacement criterion mmDmax measured maximum displacement in head-down compressive test mmDdmax measured maximum downward displacement in O-cell test mmDumax measured maximum upward displacement in O-cell test mmDN Kolmogorov-Smirnov test statistic -D0 diameter of reinforcing steel cage m, mmE elastic modulus of concrete kN/m2

E′ Young’s modulus of soil kN/m2

Ed event that a defect is detected if it is indeed encountered -Ee event that a defect is encountered -Ei initial stiffness modulus kN/m2

Eoed oedometer stiffness (or tangent stiffness modulus) kN/m2

Ere foed reference tangent stiffness modulus kN/m2

E50 secant stiffness modulus for primary loading kN/m2

Ere f50 reference secant stiffness modulus for primary loading kN/m2

Eur Young’s modulus for unloading and reloading kN/m2

Ere fur reference Young’s modulus for unloading and reloading kN/m2

f frequency of ultrasonic signal kHzfc partial safety factor for effective cohesion -fg probability density function of limit state function g -fn nominal compressive strength of concrete kN/m2

fQ probability density function of load -fXi(x∗i ) original probability density function of variable, Xi, evaluated at design

point x∗i-

fϕ partial safety factor for effective friction angle -fT side shear stress at elapsed time t kN/m2

f0 predicted side shear stress for the first day kN/m2

f1 measured side shear stress at elapsed time, t1, in 1996 kN/m2

f2 measured side shear stress at elapsed time, t2, in 2002 kN/m2

f ′(θ) prior probability density function of random parameter θ -f ′′(θ) posterior probability density function of random parameter θ -FU(U) standard uniform cumulative distribution function -FXi(x∗i ) original cumulative distribution function of variable, Xi, evaluated at

design point x∗i-

g limit state function -gG limit state function for geotechnical failure mode -gS limit state function for structural failure mode -g(X) general limit state function -g(θ) scalar prediction model -k normalizing constant -kQ ratio of mean value to nominal value for load -kR ratio of mean value to nominal value for resistance -L pile length mLD length from pile top to defect mL(θ) likelihood function of random parameter θ -m the power used in Hardening-Soil model -M number of test piles out of N test piles do not fail pilen number of random variables considered -nD number of values of ω, for which detection length greater than or equal

to detection threshold-

206


n f number of simulations having g(X) < 0 -ns total number of simulations -nt number of access tubes installed in pile tubenω total number of values of ω is taken from a range of zero to π/nt -N total number of pile cases considered pileN60 SPT index corrected for field procedure to average energy ratio of 60% blows/0.3 mp′ mean effective stress kN/m2

pre f reference pressure, pre f =100 kN/m2 kN/m2

Pa atmospheric pressure kPaPD detection probability -PE encountered probability -PI inspection probability -Pf probability of failure -Ps probability of safety -pp isotropic pre-consolidation stress kN/m2

q primary deviator loading kN/m2

qa asymptotic value of shear strength kN/m2

q f ultimate deviator stress kN/m2

Q load effects kNQa actual load kNQc failure test load kNQd design value of load kNQdi downward load corresponding to displacement, di, in O-cell test kNQui upward load corresponding to displacement, di, in O-cell test kNQD dead load kNQL live load kNQmax maximum load applied to pile kNQMD measured dead load kNQML measured live load kNQn nominal value of load kNQni ith nominal load component kNQT test load kNQ mean value of load kNrB unit base resistance kN/m2

R resistance kNR mean value of resistance kNRB base resistance kNRS shaft resistance kNRd design value of resistance kNRde radius of designed pile m, mmRE reference resistance at end of driving kNR f failure ratio -RF factored total resistance kNRinter strength reduction factor in interface element -RM measured nominal resistance kNRMB measured nominal base resistance kNRMS measured nominal shaft resistance kNRSE set-up resistance kNRMSE measured set-up resistance kNRM0 measured reference resistance kNRn nominal value of resistance kNRP predicted resistance kN

207


Rr remaining radius of pile m, mmRT resistance at elapsed time kNRu ultimate resistance kNR0 resistance at reference time kNSMN measured normalized side shear stress -SPN predicted normalized side shear stress -t elapsed time daytSE the time that set-up effect takes place dayt0 reference time dayt1 elapsed time after pile installation in 1996 dayt2 elapsed time after pile installation in 2002 dayt28 28 days after pile installation dayu value of standard uniform variable U -U standard uniform variable -v measured velocity of acoustic signal m/svb baseline velocity of acoustic signal m/sW buoyant weight of pile kNx value of variable X -xd defect diameter (according to Li et al., 2005) mmx∗ design point -x∗i ith random variable value at design point in regular coordinate system -x′∗i ith random variable value at design point in reduced coordinate system -

X vector of basic variables of system -Xi ith random variable in regular coordinate system -X′i ith random variable in reduced coordinate system -y actual system response -z depth mz1 depth of the top of soil layer considered mz2 depth of the bottom of soil layer considered mZ impedance of pile kN-s/m

Greek symbols

α∗i direction cosine or influence factor of variable Xi -β reliability index -β parameter used in β method (O’Neill and Reese, 1999) -βT target reliability index -γ load factor -γD dead load factor -γL live load factor -γi ith partial safety factor applied to mean value of ith basic variable -γi load factor for ith nominal load component -γsat saturated unit weight of soil kN/m3

γunsat unsaturated unit weight of soil kN/m3

ε observed experimental outcome -ε additive model correction factor -εv volumetric strain -εe

v volumetric strain for unloading and reloading -ε1 axial strain in drained triaxial test -

208


ζ standard deviation of ln(λ) -ζ ′ standard deviation of prior normal distribution -ζ ′′ standard deviation of posterior normal distribution -η mean of ln(λ) -η′ mean of prior normal distribution -η′′ mean of posterior normal distribution -θ vector denotes uncertain parameters -Θ random variable -κ∗ modified swelling index -λ multiplicative model correction factor or bias factor -λ mean of bias factor -λ∗ modified compression index -λ f n compressive strength bias factor of concrete -λQD dead load bias factor -λQL live load bias factor -λQD mean of dead load bias factor -λQL mean of live load bias factor -λR resistance bias factor -λRi resistance bias factor of ith pile -λRB base resistance bias factor -λRS shaft resistance bias factor -λRSE set-up resistance bias factor -λR0 reference resistance bias factor -λT bias factor value corresponds to maximum test load -λ f n mean of compressive strength bias factor of concrete -λR mean of resistance bias factor -

λR′′

mean of resistance bias factor of posterior log-normal distribution -λRB mean of base resistance bias factor -λRS mean of shaft resistance bias factor -λRSE mean of set-up resistance bias factor -λR0 mean of reference resistance bias factor -λω wavelength of ultrasonic signal mµg mean value of limit state function g -µQi mean value of ith load kNµQD mean value of dead load kNµQL mean value of live load kNµR mean value of resistance kNµRSE mean value of set-up resistance kNµR0 mean value of reference resistance kNµXi mean value of ith random variable -µN

Xi mean value of equivalent normal distribution for variable Xi -ν Poisson’s ratio -ρXi ,Xj correlation coefficient between variables Xi and Xj -σc maximum compression stress in pile concrete caused by external loads kN/m2

σg standard deviation of limit state function g -σn normal stress in interface element kN/m2

σQ standard deviation of load effects kNσR standard deviation of resistance bias factor -σ′′R standard deviation of resistance bias factor of posterior log-normal dis-

tribution-

σRSE standard deviation of set-up resistance bias factor -

209


σR0 standard deviation of reference resistance bias factor -σXi standard deviation of ith random variable -σN

Xi standard deviation of equivalent normal distribution for variable Xi -σ′z vertical effective stress in soil at depth z kN/m2

σλ standard deviation of bias factor -σ′1 effective compressive pressure in oedometer test kN/m2

σ′3 confining effective compressive pressure in triaxial test kN/m2

τ shear stress in interface element kN/m2

φ resistance factor -φ resistance factor applied to mean value of resistance -φB base resistance factor -φS shaft resistance factor -φSE set-up resistance factor -φ0 reference resistance factor -φFOSM resistance factor is calibrated by first order second moment method -φFORM resistance factor is calibrated by first order reliability method -φMCS resistance factor is calibrated by Monte Carlo simulation -φ(g) probability density function of standard normal distribution of limit

state function g-

ϕ′ effective friction angle degreeϕi friction angle of interface element degreeϕ′f design effective friction angle degreeΦ standard normal cumulative distribution function -ψ dilatancy angle of soil degreeω location angle of defect radian

Abbreviations

AASHTO American Association of State Highway and Transportation OfficialsAKS Adjusted Kolmogorov-smirnov test StatisticsASCE American Society of Civil EngineersASD Allowable Stress DesignASTM American Society for Testing and MaterialsCAPWAP CAse Pile Wave Analysis ProgramCDF Cumulative Distribution FunctionCEN Comite Europeen de Normalisation (in French)CFA Continuous Flight Auger pilesCOV Coefficient of VariationCPT Cone Penetration TestCSL Cross-hole Sonic LoggingCV Critical ValueCR Correlation RatioCUR Centre for Civil Engineering Research and Codes (the Netherlands)DARS Directional Adaptive Response Surface samplingDOT Department of Transportation (the United States)DS Directional SamplingEOD End of DrivingFAT First Arrival Time (s)FEM Finite Element Method

210


FHWA Federal Highway AdministrationFORM First Order Reliability MethodFOSM First Order Second Moment methodFS Factor of SafetyFSE Factor of Safety applied to reference resistance at end of drivingFSSE Factor of Safety applied to set-up resistanceGF Geotechnical Failure modeHS Hardening-Soil modelIGM Intermediate GeoMaterialIV Increased Variance samplingJCSS Joint Committee on Structural SafetyLRFD Load and Resistance Factor DesignLSD Limit State DesignMC Mohr-Coulomb modelMCS Monte Carlo SimulationNEN NEderlandse Norm (in Dutch)NDT Non-Destructive TestNI Numerical IntegrationPDA Pile Driving AnalyzerPDF Probability Density FunctionPSF Partial Safety FactorRBD Reliability-Based DesignSE Sonic Echo methodSF Structural Failure modeSLS Serviceability Limit StateSORM Second Order Reliability MethodSPT Standard Penetration TestSS Soft-Soil modelSWEPDI South-West Electric Power Design Institute, ChinaTNO Nederlandse organisatie voor Toegepast Natuurwetenschappelijk On-

derzoek (in Dutch)ULS Ultimate Limit StateVGI Vietnam Geotechnical InstituteVR Velocity Reduction (%)WSD Working Stress Design

211


212

List of Figures

1.1 Scheme of a bored pile subjected to loads. . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Slurry method of construction (O’Neill and Reese, 1999). . . . . . . . . . . . . . . . 3

1.3 Overview of thesis outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Types of uncertainty (van Gelder, 2000). . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Definitions of load and resistance in ASD and in reality. . . . . . . . . . . . . . . . . 15

2.3 Possible load and resistance distributions. . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Comparison of limit states design approaches for the ULS (Ovesen and Orr, 1991). 18

2.5 Load and Resistance Factor Design (LRFD). . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Definition of probability of failure (Withman, 1983). . . . . . . . . . . . . . . . . . . 22

2.7 Definition of limit state function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Definition of design point on the failure surface. . . . . . . . . . . . . . . . . . . . . 27

2.9 Relation between u and x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Some types of structural defect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Scheme of sonic echo method (adapted from Sliwinski and Fleming, 1983). . . . . . 36

3.3 Scheme of cross-hole sonic logging method (adapted from Brown et al., 2010). . . . 37

3.4 Access tubes placed inside the reinforcing steel cage (Brown et al., 2010). . . . . . . 38

3.5 A CSL result for an acoustic profile (ADCOM, 2011). . . . . . . . . . . . . . . . . . . 39

3.6 Concrete cores taken from two bored pile shafts (O’Neill and Reese, 1999). . . . . . 41

3.7 Geometrical diagram determining encountered probability (Li et al., 2005). . . . . . 44

3.8 The shapes of defect located at the periphery of pile. . . . . . . . . . . . . . . . . . . 45

3.9 Pile shaft with defect in shape of circular segment (ADCOM, 2008). . . . . . . . . . 46

3.10 Geometrical diagram determining encountered probability in this study. . . . . . . 46

3.11 Encountered probability for bored pile D=1,000 mm. . . . . . . . . . . . . . . . . . . 47

3.12 Encountered probability for bored pile D=2,500 mm. . . . . . . . . . . . . . . . . . . 48

3.13 Relationship between magnitude of defect and the number of access tubes for dif-ferent pile diameters with target encountered probability, PE=0.9. . . . . . . . . . . 48

213

List of Figures List of Figures

3.14 Comparison between the detection probabilities for a general case and a specialcase (Li et al., 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.15 Geometrical diagram determining detection probability in this study. . . . . . . . . 51

3.16 Variation of detection length with location angle. . . . . . . . . . . . . . . . . . . . . 52

3.17 Variation of detection length with location angle and number of access tubes. . . . 54

3.18 Detection probability for bored pile D=1,500 mm. . . . . . . . . . . . . . . . . . . . 55

3.19 Inspection probability for bored pile D=2,000 mm. . . . . . . . . . . . . . . . . . . . 56

3.20 Inspection probability with magnitude of defect as a percentage of cross-sectionalarea for bored pile D=2,000 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.21 Inspection probability for bored pile D=1,400 mm. . . . . . . . . . . . . . . . . . . . 57

3.22 Inspection probability for bored pile D=915 mm. . . . . . . . . . . . . . . . . . . . . 58

4.1 Determination scheme of partial safety factors. . . . . . . . . . . . . . . . . . . . . . 66

4.2 The K-S test for the case of Rock-IGM-Mixed. . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Calibrated resistance factors using FOSM . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Calibrated resistance factors using FORM . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Calibrated resistance factors using MCS . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.6 Correlation analyses between calibrated resistance factors and statistical parame-ters of the resistance bias factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7 Correlation analyses between φ and ratio COVR/λR when using different reliabil-ity methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.8 Correlation analyses between resistance factors using different reliability methods. 77

4.9 Comparison of head-down compressive test and O-cell test. . . . . . . . . . . . . . 81

4.10 Overview of the Los Angeles coliseum (www.wildnatureimages.com). . . . . . . . . 81

4.11 Load-displacement curves of the O-cell test for pile 10B. . . . . . . . . . . . . . . . . 82

4.12 Load-displacement curves of the O-cell test for pile 23B. . . . . . . . . . . . . . . . . 83

4.13 Correlation between φS and φB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.14 Sensitivity analyses associated with shaft and base resistance factors. . . . . . . . . 86

4.15 Load-displacement curves by O-cell and equivalent head-down compressive testsfor pile 4A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.16 Factored total resistance results applying a common resistance factor and separateshaft and base resistance factors for 26 bored piles with βT=3.0. . . . . . . . . . . . 87

5.1 Time-dependent development of pile resistance (adapted from Skov and Denver,1988). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Soil stratigraphy at location of bored pile 7. . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Set-up side shear stress vs. time (Ability 1). . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Measured and predicted normalized set-up side shear stresses for Ability 1. . . . . 102

214


5.5 Set-up side shear stress vs. time (Ability 2). . . . . . . . . . . . . . . . . . . . . . . . 102

5.6 Measured and predicted normalized set-up side shear stresses for Ability 2. . . . . 103

5.7 Correlation between φ0 and φSE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1 CAPWAP analysis results for pile S6 (VGI, 2008). . . . . . . . . . . . . . . . . . . . . 117

6.2 Initial prior log-normal PDF for calibration case of Rock-IGM-Mixed. . . . . . . . . 118

6.3 Initial prior normal PDF, likelihood function, and posterior normal PDF based onthe three PDA test piles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4 Initial prior log-normal PDF and posterior log-normal PDF after considering threePDA test piles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Prior normal PDF, likelihood function, and posterior normal PDF based on threestatic test piles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6 Prior log-normal PDF and posterior log-normal PDF after considering three statictest piles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.7 Initial prior log-normal PDF and posterior log-normal PDFs based on test pileswith different combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.8 Statistical parameters of resistance bias factors and updated resistance factors vs.different predicted resistances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.9 Extrapolation of load-displacement curve for pile S4. . . . . . . . . . . . . . . . . . 125

6.10 Initial prior log-normal PDF and posterior log-normal PDFs with and without con-sidering ”failure” test loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.1 Coupling scheme Prob2B-Plaxis (Schweckendiek et al., 2007). . . . . . . . . . . . . 131

7.2 Fault tree under failure modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3 A bored pile construction procedure simulated in Plaxis. . . . . . . . . . . . . . . . 134

7.4 Basic idea of a linear elastic perfectly-plastic model. . . . . . . . . . . . . . . . . . . 135

7.5 Hyperbolic stress-strain relation in primary loading for a standard drained triaxialtest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.6 Logarithmic relation between volumetric strain and mean stress. . . . . . . . . . . 138

7.7 Pier T10 layout and stratigraphy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.8 Load-displacement curves obtained from static loading test and Plaxis analysis. . . 141

7.9 Uncertainty in soil property estimates (Kulhawy, 1992). . . . . . . . . . . . . . . . . 143

7.10 Nine calculation cases considered with different pile toe levels. . . . . . . . . . . . 146

7.11 Pile resistance of nine calculation cases. . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.12 Maximum compressive stress in pile of nine calculation cases. . . . . . . . . . . . . 147

7.13 Reliability of nine calculation cases under GF mode. . . . . . . . . . . . . . . . . . . 148

7.14 Factors of safety of nine calculation cases under GF mode. . . . . . . . . . . . . . . 148

7.15 Reliability and factor of safety of working pile under SF mode. . . . . . . . . . . . . 149

7.16 Reliability of nine calculation cases under SF mode. . . . . . . . . . . . . . . . . . . 150

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7.17 Description of defects in axisymmetric Plaxis model. . . . . . . . . . . . . . . . . . . 151

7.18 Effect of necking size and its location on pile behaviour. . . . . . . . . . . . . . . . . 152

7.19 Effect of necking near the pile top on reliability of pile. . . . . . . . . . . . . . . . . 152

7.20 Effect of necking near the pile top on reliability of pile subjected to working load. . 153

7.21 Effect of poor concrete zone on pile behaviour. . . . . . . . . . . . . . . . . . . . . . 153

7.22 Effect of poor concrete zone on reliability of pile. . . . . . . . . . . . . . . . . . . . . 154

A.1 Empirical distributions of calibration cases 1 through 4. . . . . . . . . . . . . . . . . 179

A.2 Empirical distributions of calibration cases 5 through 8 (continue). . . . . . . . . . . 180

A.3 Empirical distributions of calibration cases 9 through 12 (continue). . . . . . . . . . 180

A.4 Empirical distributions of calibration cases 13 through 16 (continue). . . . . . . . . 181

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List of Tables

1.1 Advantages and limitations of bored piles . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Factors of safety on ultimate axial geotechnical resistance based on level of con-struction control (AASHTO, 1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Summary of partial factors for foundation design . . . . . . . . . . . . . . . . . . . 18

2.3 Relationship between target reliability index and probability of failure . . . . . . . 24

3.1 Rating criteria for concrete quality (adapted from Brown et al., 2010) . . . . . . . . 39

3.2 Recommended number of access tubes for different bored pile diameters (synthe-sized by Li et al., 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Recommended number of access tubes for given defect sizes as a percentage of pilecross-sectional area, p(%), with target encountered probability, PE=0.9 (extractedfrom Li et al., 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Summary of the minimum detectable defect size and the detectable defect sizewith certainty (synthesized by Li et al., 2005) . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Detection threshold of CSL test (Amir and Amir, 2008) . . . . . . . . . . . . . . . . 54

3.6 Recommended number of access tubes according to pile diameters and detectablemagnitudes of defect as a percentage of cross-sectional area of pile with targettarget inspection probability of 0.99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Detectable minimum magnitudes of defect (in millimeter) according to pile diam-eters and number of access tubes with target inspection probability of 0.99 . . . . . 60

4.1 Relationship between dead load to live load ratio and span length of bridges . . . 65

4.2 Load factors and probabilistic characteristics for the dead load and live load biasfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Sixteen calibration cases for bored pile foundations . . . . . . . . . . . . . . . . . . 70

4.4 Statistical parameters and K-S test results for resistance bias factors . . . . . . . . . 72

4.5 Calibrated resistance factors for target reliability indices, βT = 2.5 and 3.0 . . . . . 75

4.6 Statistical parameters and probability distributions of the shaft and base resistancebias factors for 26 bored piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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List of Tables List of Tables

4.7 Load factors and probabilistic characteristics for the dead load and live load biasfactors (Ellingwood and Galambos, 1982) . . . . . . . . . . . . . . . . . . . . . . . . 84

4.8 Comparison of calibrated resistance factors of calibration case in Section 4.5 withthose of this case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.1 Reference time, t0, and set-up factor A . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 Safety factors used for design and pile installation (Komurka et al., 2005) . . . . . . 95

5.3 Summary of side shear stress over time for the SR20 tested bored piles . . . . . . . 98

5.4 Set-up factor A used in this case study . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5 Five calculation cases of the reference resistance . . . . . . . . . . . . . . . . . . . . 100

5.6 Statistical parameters and probability distributions of the reference resistance biasfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.7 Three calculation cases of the set-up resistance . . . . . . . . . . . . . . . . . . . . . 101

5.8 Statistical parameters and probability distributions of the set-up resistance biasfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.9 Ratio of the predicted set-up resistance to the predicted reference resistance, RSE/R0104

5.10 Ratio CR for calibration cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.11 Calibration results for φ0 and φSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.1 Statistical parameters of measured resistances and resistance bias factors at the siteof the Los Angeles Memorial Coliseum . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Testing bored piles at the site of the Uong Bi Extension No. 2 Thermal Power Plant(VGI, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Updated resistance factors based on three PDA test piles . . . . . . . . . . . . . . . 119

6.4 Updated resistance factors based on three static test piles . . . . . . . . . . . . . . . 120

6.5 Updated resistance factors based on test piles with different combinations . . . . . 122

6.6 Updated resistance factors based on three static test piles in advance, and thenbeing the three PDA test piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.7 Updated resistance factors based on assumed prediction resistances . . . . . . . . . 124

6.8 Updated resistance factors with considering ”failure” test loads . . . . . . . . . . . 125

7.1 Concrete strength and probabilistic characteristics of λ f n . . . . . . . . . . . . . . . 132

7.2 Initial and back-calculated soil parameters . . . . . . . . . . . . . . . . . . . . . . . 142

7.3 Statistical parameters and probability distributions of soil parameters . . . . . . . . 144

7.4 Correlation matrix of stochastic soil parameters . . . . . . . . . . . . . . . . . . . . . 145

7.5 Statistical parameters and probability distributions for slim layer . . . . . . . . . . 155

A.1 Shaft and base resistance bias factors of the 26 bored piles at the L.A. Coliseum . . 182

A.2 Comparison of factored total resistances when using a common resistance factorand using shaft and base resistance factors for the 26 bored piles with βT=3.0 . . . 183

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B.1 Set-up resistance bias factors for Ability 1 . . . . . . . . . . . . . . . . . . . . . . . . 186

B.2 Set-up resistance bias factors for Ability 2 . . . . . . . . . . . . . . . . . . . . . . . . 186

C.1 Reliability analysis results for 9 calculation cases under GF mode . . . . . . . . . . 187

C.2 Reliability analysis results for 9 calculation cases under SF mode . . . . . . . . . . . 191

C.3 Reliability analysis results for 9 calculation cases under SF mode with displace-ment of 2.2%D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

C.4 Reliability analysis result for working pile subjected to working load under SF mode196

C.5 Reliability analysis results for working pile with necking under GF mode . . . . . 197

C.6 Reliability analysis results for working pile with necking under SF mode . . . . . . 197

C.7 Reliability analysis results for working pile subjected to working load under SFmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

C.8 Reliability analysis results for working pile with poor concrete zone under GF mode199

C.9 Reliability analysis results for working pile with poor concrete zone under SFmode, considering stress state in good concrete zone . . . . . . . . . . . . . . . . . . 199

C.10 Reliability analysis results for working pile with poor concrete zone under SFmode, considering stress state in poor concrete zone . . . . . . . . . . . . . . . . . . 200

C.11 Reliability analysis results for working pile subjected to working load under SFmode, considering stress state in good concrete zone . . . . . . . . . . . . . . . . . . 200

C.12 Reliability analysis results for working pile subjected to working load under SFmode, considering stress state in poor concrete zone . . . . . . . . . . . . . . . . . . 201

C.13 Reliability analysis results for working pile with soft bottom under GF mode . . . 202

C.14 Reliability analysis results for working pile with soft bottom under SF mode . . . . 202

C.15 Reliability analysis results for working pile subjected to working load under SFmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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Acknowledgements

This thesis would have not been completed without the supports and contributions ofmany people. I would like to thank all of them, including those I may have forgotten tomention herein.

First, my deepest gratitude addresses to Prof. J.K. Vrijling and Prof. P.H.A.J.M. vanGelder, my promoters, for their enthusiastic guidance on applying the reliability theoryto bored pile foundations, for providing me freedom in doing research. I highly appre-ciate their profound knowledge, their sincere encouragement, and their unconditionalsupport.

Second, I am very grateful to Dr. K.J. Bakker, my mentor, for his fruitful discussions,advices on the research contents, and constant support during my Ph.D. research.

Third, I would like to thank Mrs. Mariette van Tilburg for her patience and thoroughnessin reviewing English writing for my thesis. I have learned much from her in the writingstyle.

Many thanks go to the members of the Doctoral Committee for reviewing and comment-ing on my thesis. Especially, I would like to give my sincere appreciation to Dr. Trinh VietCuong, from Vietnam Institute for Building Science and Technology, for his commentsand constructive advices for all parts of my thesis during the final stage.

This research was funded by the Ministry of Education and Training of Vietnam (throughProject 322), the TU Delft Valorisation Centre (formerly, CICAT), and the Hydraulic En-gineering Department of Delft University of Technology. These financial supports aregratefully acknowledged.

I would like to give my appreciation to all members of Vietnam International EducationDevelopment (VIED) and the TU Delft Valorisation Centre, especially Mrs. Tran Thi Nga,Dr. Paul Althuis, and Mrs. Veronique van der Varst, for their helps and supports. Mylife in Delft would have been much more difficult without their assistance.

The case studies of the thesis could not finished without the data and guidances, whichwere provided by experts and friends. Many thanks go to them with my sincere grat-itude. I would like to mention Dr. B.H. Fellenius (Canada), Dr. M.B. Hudson (AMEC,the US), Dr. P.J. Bullock (Fugro Consultants Inc.- Loadtest, the US), Dr. G. Axelsson(Skanska Teknik AB, Sweden), Dr. R.B.J. Brinkgreve (Plaxis B.V., the Netherlands), Dr.W.M.G. Courage (TNO, the Netherlands), Mr. Le Nguyen Tinh (ADCOM, Vietnam), Mr.Le Minh Tuan (PECC1, Vietnam), Mr. Bui Xuan Hoc, and Dr. Nguyen Viet Khoa (ITST,

221

Acknowledgements Acknowledgements

Vietnam).

During my Ph.D. study, I have received supports from colleagues. My sincere thanksgo to Mr. Nguyen Thanh Chi, Dr. Pham Quang Tu, Mr. Khuong Tat Chien, Dr. TranTrong An, Dr. Le Hai Trung, Mr. Pham Quoc Cuong, Mr. Le Luong Bao Nghi, Mr. VuMinh Ngan, Dr. W. Kanning, Dr. M. Eelkema, Dr. Reza Shams, Mr. Vincent Vuik, Dr. T.Schweckendiek, Dr. A.C.M. Teixeira. Many thanks also go to Mrs. Judith Schooneveld-Oosterling, Mrs. Inge van Rooij, Mrs. Chantal van Woggelum, and Mr. Mark Voorendtfor all their help and support.

It would be difficult in communicating with everybody when living abroad, if withoutthe teaching of English teachers. Let me express my deep gratitude to them, especiallyMrs. Kim van der Linden, Mrs. Claire Taylor, and Mrs. Mariette van Tilburg.

I would like to thank my colleagues at the Port and Waterway Engineering Department,the National University of Civil Engineering (NUCE), for their support. I especially wantto thank Assoc.prof.dr. Do Van De and Dr. Bui Viet Dong for their support and givingme enough time to study. Assoc.prof.dr. Pham Van Giap is highly appreciated for hisadvice and encouragement during the time I have been doing research. And I thankMrs. Dao Tang Kiem for helping me to apply for the Ph.D. position at Delft Universityof Technology.

Last but not least, I would like to thank my parents and parents-in-law for their loveand their encouragement. I would like to thank my wife, Le Thi Hai Yen, for her love,patience, enormous support, quiet sacrifices, and finally for taking care of our sons. Ialso would like to thank my two sons, Bach Le Son and Bach Le Minh, for their love andbelief in me.

Delft, 23 September 2014

Bach Duong

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Curriculum Vitae

Name: Bach DuongDate of birth: 18 September 1972Place of birth: Hanoi, VietnamEmail: [email protected]

1989 - 1994 Bachelor in National University of Civil Engineering, Hanoi, Vietnam(Field of study: Port and Waterway Engineering)

1997 - 2000 Master of Engineering in National University of Civil Engineering, Hanoi,Vietnam(Field of study: Port and Waterway Engineering)

1994 - 2004 Researcher at Institute of Transportation Science and Technology(Field of research: Port and Waterway Engineering)

2004 - present Lecturer at the Department of Port and Waterway,National University of Civil Engineering, Hanoi, Vietnam

2010 - 2014 Ph.D. candidate at the Department of Hydraulic Engineering,Delft University of Technology, the Netherlands

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Curriculum Vitae Curriculum Vitae

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