Characterisation of Model Uncertainty for Reliability-Based Design of Pile Foundations by Mahongo Dithinde Dissertation presented for the degree of Doctor of Philosophy in the Department of Civil Engineering, Faculty of Engineering at the University of Stellenbosch Promoters: Dr. M. de Wet; Prof. J. V. Retief; Prof. K. K. Phoon (National University of Singapore) December 2007 i Declaration I, the undersigned, hereby declare that the work contained in this dissertation is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree. Signed: ????????????? Date: ?????????? Copyright ? 2007 Stellenbosch University All rights reserved Stellenbosch University http://scholar.sun.ac.za ii Abstract To keep pace with international trends, the introduction of geotechnical limit state design in South Africa is inevitable. To pave the way for implementation of limit state pile design in the country, the study quantifies model uncertainty in the classic static pile design formula under the Southern African geologic environment. The generated model uncertainty statistics are used to calibrate partial resistance factors in a reliability-based design framework. A series of pile performance predictions by the static formula are compared with measured performances. To capture the distinct soil types for the geologic region of Southern Africa as well as the local pile design and construction experience base, pile load tests and associated geotechnical data from the Southern African geologic environment are used. The methodology of collecting, compiling, and analyzing the pile load tests to derive the measured ultimate pile capacities is described. To facilitate the computation of the theoretical capacities, the site specific geotechnical data in the database are transformed to the desired engineering soil properties through well established empirical correlations. For a given pile test case, model uncertainty is presented in terms of a model factor computed as the ratio of the measured to the theoretical capacity, leading to n realisations of the model factor. To facilitate further interpretation and generalisation of the model factor realisation data, statistical analysis is carried out. The statistical analysis comprises of graphical representation by histograms, outliers detection and correction of erroneous values, and using the corrected data to compute the sample moments (mean, standard deviations, skewness and kurtosis) needed in reliability analysis. The analyses demonstrate that driven piles depict higher variability compared to bored piles irrespective of materials type. Furthermore, for a given pile installation method (driven or bored) the variability in non-cohesive materials is higher than that in cohesive materials. In addition to the above statistics, reliability analysis requires the theoretical probability distribution for the random variable under consideration. Accordingly it is demonstrated that the lognormal distribution is the most appropriate theoretical model for the model factor. Another key basis for reliability theory is the notion of randomness of the basic variables. To verify that the variation in the model factor is not explainable by deterministic variations in the database, an investigation of correlation of the model factor with underlying pile design parameters is carried out. It is shown that such correlation is generally weak. Stellenbosch University http://scholar.sun.ac.za iii Correlation can have a significant impact on the calculated reliability index if not accounted for. Accordingly, the effects of the exhibited correlation is investigated through an approach based on regression theory in which systematic effects of design parameters are taken into account (generalised model factor). The model factor statistics from the conventional approach and those from the generalised model factor approach are used to determine reliability indexes implied by the current design practice. It is demonstrated that no significant improvement in values of the reliability indexes is gained by taking into account the effects of the weak correlation. The model factor statistics derived on the basis of the standard model factor approach are used to calibrate resistance factors. Four first order reliability methods are employed for the calibration of resistance factors. These include; the Mean Value First-Order Second Moment approach, an Approximate Mean Value First-Order Second Moment approach, the Advanced First-Order Second Moment approach using Excel spreadsheet, and the Advanced First-Order Second Moment approach (design point method). The resistance factors from the various calibration methods are presented for the target reliability index values of 2.0, 2.5, and 3.0. The analyses of the results demonstrate that for a given target reliability index, the resistance factors from the different methods are comparable. Furthermore, it is shown that for a given material type, the resistance factors are quite close irrespective of the pile installation method, suggesting differentiation of partial factors in terms of materials types only. Finally, resistance factors for use in probabilistic limit state pile design in South Africa are recommended. Stellenbosch University http://scholar.sun.ac.za iv Samevatting Ten einde in pas te bly met internasionale neigings, is dit onafwendbaar dat geotegniese limietstaat-ontwerp in Suid Afrika ingevoer word. Ter voorbereiding vir die plaaslike toepassing van limietstaatontwerp op heipale, kwantifiseer hierdie ondersoek onsekerheid rondom die model vir klassieke statiese heipaalontwerpformules in die Suid Afrikaanse geologiese omgewing. Die statistiek van modelonsekerheid wat gegenereer is, word gebruik om parsi?le weerstandsfaktore in ?n betoubaarheid-gebasseerde ontwerpraamwerk te kalibreer. ?n Reeks voorspellings van die gedrag van heipale volgens die statiese formules word vergelyk met die gemete gedrag. Om die kenmerkende grond-tipes in die geologiese gebied van Suidelike Afrika sowel as die plaaslike ondervinding met heipaalontwerp en - konstruksie vas te l?, word heipaaltoetse en die gassosieerde geotegniese data vanuit hierdie geologiese omgewing gebruik. Die metodiek vir die versameling, saamstelling en analise van heipaaltoetse om uiterste kapasiteite daarvan te bepaal, word beskryf. Terreinspesifieke geotegniese data in die databasis word getransformeer na die vereisde ingenieurseienskappe volgens gevestigde empiriese korrelasies. Vir ?n gegewe heipaaltoets word modelonsekerheid weergegee in terme van ?n modelfaktor wat bereken word as die verhouding van die gemete tot die teoretiese kapasiteit waaruit n uitkomstes van die modelfaktor dus gegenereer word. Om verdere interpretasie en veralgemening van die modelfaktordata te vergemaklik, word ?n statistiese analise daarop uitgevoer. Die statistiese analise bestaan uit grafiese voorstellings deur middel van histogramme, uitkenning van uitskieters en verbetering van foutiewe waardes, waarna die statistiese momente (gemiddeld, standaardafwyking, skeefheid en kurtose) vir gebruik in betroubaarheidsanalise bereken word. Volgens die analises toon ingedrewe heipale ?n groter veranderlikheid as geboorde pale, ongeag die grondtipe. Verder is die veranderlikheid van heipale in kohesielose materiale ho?r as in kohesiewe materiale, ongeag die installasiemetode (ingedrewe of geboor). Bykomend tot bogemelde statistiek, vereis betroubaarheidsanalise die teoretiese waarskynlikheidsdistribusie van die ewekansige veranderlike onder beskouing. Ooreenkomstig word illustreer dat die log-normale verspreiding die mees toepaslike verspreiding vir die modelfaktor is. ?n Verdere sleutelvereiste vir betroubaarheidsteorie is Stellenbosch University http://scholar.sun.ac.za v die mate van ewekansigheid van die basiese veranderlikes. Om te bepaal of die variasie in die modelfaktor nie deur deterministiese veranderlikes in die databasis verduidelik kan word nie, word ?n ondersoek na die korrelasie van die modelfaktor met onderliggende heipaal- ontwerpfaktore uitgevoer. Sodanige korrelasie is in die algemeen as laag bevind. Korrelasie kan ?n belangrike invloed op die berekende betroubaarheidsindeks h? indien dit nie in ag geneem word nie. Dienooreenkomstig word die effek van die getoonde korrelasie ondersoek met behulp van die metode van regressie-analise waarin sistematiese effekte van ontwerpparameters in berekening gebring word (veralgemeende modelfaktor). Die modelfaktorstatistiek wat volg uit die konvensionele benadering en di? van die veralgemeende benadering word gebruik om betroubaarheidsindekse te bepaal wat deur die bestaande ontwerppraktyk ge?mpliseer word. Die bevinding is dat daar nie ?n noemenswaardige verbetering in die waardes van die betroubaarheidsindekse is wanneer die effek van die swak korrelasie in berekening gebring word nie. Die statistiek van die modelfaktor wat afgelei is volgens die standaardbenadering word gebruik om die weerstandsfaktore te kalibreer. Vier eerste-orde betroubaarheidsmetodes word gebruik om die weerstandsfaktore te kalibreer, naamlik die Gemiddelde Waarde Eerste-Orde Tweede Moment benadering, die Benaderde Gemiddelde Waarde Eerste-Orde Tweede Moment benadering, die Gevorderde Eerste-Orde Tweede Moment benadering waarin ?n Excel sigblad gebruik word en die Gevorderde Eerste-Orde Tweede Moment benadering (die ontwerppuntmetode). Die weerstandsfaktore vanaf die verskillende kalibrasiemetodes word weergegee vir waardes van 2.0, 2.5 en 3.0 van die teiken- betroubaarheidsindeks. ?n Ontleding van die resultate toon dat vir ?n gegewe teiken betroubaarheidsindeks die weerstandsfaktore vanaf die verskillende metodes vergelykbaar is. Verder word getoon dat vir ?n gegewe grondsoort, die weerstandsfaktore vir verskillende metodes van installasie van die heipaal nie veel verskil nie. Dit wil dus voorkom asof parsi?le faktore in terme van die grondsoort uitgedruk kan word. Ten slotte word weerstandsfaktore vir gebruik in plastiese limietstaatontwerp van heipale in Suid Afrika aanbeveel. Stellenbosch University http://scholar.sun.ac.za vi Acknowledgements I would like to express my sincere gratitude to my promoters Dr. M. De Wet, Prof. J. V. Retief and Prof. K. K. Phoon (external promoter from the National University of Singapore) for their time, assistance, guidance, and continual support throughout this study. Their comments have been of great value to me and this dissertation. In addition I would like to acknowledge the financial assistance from Dr. De Wet which enabled me to travel to Johannesburg twice to collect data and to travel to Taipei, Taiwan for conference attendance. I am grateful to Franki Africa for permission to search their archives for pile load test data. In this regard, I acknowledge gratefully the valuable assistance and trouble taken by Alan Berry in making my two visists to Franki Africa a success. A also wish to record my thanks to the following companies for providing additional pile load test data and associated geotechnical data used in the study: Dura Piling Jones and Wagener Gauteng Piling Stefanutti and Bressan Stellenbosch University http://scholar.sun.ac.za vii Table of Contents Declaration ............................................................................................................................. I Abstract .................................................................................................................................II Samevatting ......................................................................................................................... IV Acknowledgements ............................................................................................................. VI Table of contents ................................................................................................................VII CHAPTER 1. INTRODUCTION........................................................................................1 1.1 BACKGROUND ..........................................................................................................1 1.1.1 Uncertainties in geotechnical design ...................................................................1 1.1.2 Status of probabilistic geotechnical limit state design.......................................3 1.1.3 State of limit state design in SA ...........................................................................3 1.2 RESEARCH STATEMENT .......................................................................................5 1.3 AIMS AND SCOPE OF THE STUDY........................................................................7 1.4 LAYOUT OF THE DISSERTATION ........................................................................8 CHAPTER 2. GENERAL INTRODUCTION TO GEOTECHNICAL LIMIT STATE DESIGN................................................................................................................................10 2.1 INTRODUCTION ............................................................................................................10 2.2 EVOLUTION OF GEOTECHNICAL DESIGN ....................................................................11 2.2.1 Experience based design approach ...................................................................11 2.2.2 Working stress design.........................................................................................12 2.2.3 Limit state design philosophy ............................................................................14 2.2.3.1 Verification of non-exceedance of limit states...........................................18 2.2.3.2 Deterministic limit state design ..................................................................18 2.2.3.3 Probabilistic limit state design....................................................................21 2.3 BASIS OF GEOTECHNICAL LIMIT STATE DESIGN TO SANS 10160 ..............................23 2.3.1 Fundamental requirements................................................................................23 2.3.2 Design situations .................................................................................................24 2.3.3 Limit states ..........................................................................................................26 2.3.4 Design methods ...................................................................................................26 2.3.5 Design by calculations ........................................................................................27 Stellenbosch University http://scholar.sun.ac.za viii 2.3.5.1 Load types, load combinations and load factors.......................................27 2.3.5.1.1 Load types..............................................................................................28 2.3.5.1.2 External actions ....................................................................................29 2.3.5.1.3 Characteristic and design values of external actions.........................30 2.3.5.1.4 Geotechnical actions .............................................................................31 2.3.5.1.5 Characteristic values of geotechnical actions.....................................36 2.3.5.1.6 Load statistics........................................................................................37 2.3.5.1.7 Fundamental load combinations and load factors.............................37 2.3.5.2 Geotechnical materials properties..............................................................39 2.3.5.2.1 Determination of characteristic values of geotechnical parameters 40 2.3.5.2.2 Statistical approach to characteristic geotechnical properties .........42 2.3.5.2.3 Characteristic value as mean value at 95 % confidence level ..........44 2.3.5.2.4 Characteristic value as a 5% fractile ..................................................45 2.3.5.3 Geometric data.............................................................................................47 2.3.6 Design approaches for STR/GEO limit state ...................................................48 2.3.7 Selection of design approach for the study.......................................................50 CHAPTER 3. RELIABILITY BASIS FOR GEOTECHNICAL LIMIT STATE DESIGN................................................................................................................................57 3.1 INTRODUCTION ............................................................................................................57 3.2 RELIABILITY ANALYSIS ..............................................................................................60 3.2.1 Probability of failure ..........................................................................................60 3.2.2 Reliability index ..................................................................................................62 3.2.2.1 Mean value first order second moment (MVFOSM) analysis.................63 3.2.2.2 Advanced first order second moment (AFOSM) analysis .......................67 3.2.2.2.1 Extension to non-linear performance functions and non-normal random variables .................................................................................................70 3.2.2.2.2 Extension to correlated random variables .........................................72 3.2.2.3 Computation of ? using Excel.....................................................................73 3.2.3 Relationship between Pf and ? and interpretation ..........................................75 3.2.4 Choice between probability of failure and reliability index ...........................77 3.3 SIMPLIFIED RELIABILITY BASED DESIGN ....................................................................78 3.3.1 Calibration of resistance factors........................................................................78 3.3.2 Reliability calibration .........................................................................................83 3.3.2.1 Target reliability index................................................................................83 Stellenbosch University http://scholar.sun.ac.za ix 3.3.2.2 Reliability Calibration methods .................................................................87 3.3.2.2.1 Advanced first-order second moment method ...................................88 3.3.2.2.2 Mean value first-order second moment method (MVFOSM) ..........89 3.3.2.2.3 Approximation to the MVFOSM approach .......................................91 CHAPTER 4. PILE LOAD TEST DATA COLLECTION AND PROCESSING........94 4.1 INTRODUCTION......................................................................................................94 4.2 DATA COLLECTION......................................................................................................94 4.3 COMPILATION OF THE DATA .......................................................................................95 4.3.1 Test pile characteristics and projects description............................................95 4.3.2 Associated geotechnical data ...........................................................................100 4.3.3 Pile load test database summary .....................................................................106 4.4 EVALUATION OF ULTIMATE PILE CAPACITY FROM AVAILABLE LOAD TESTS ..........106 4.4.1 The South African pile load test procedure ...................................................106 4.4.2 Analysis and interpretation of load test results .............................................108 4.4.2.1 Fitting a hyperbolic curve using Chin?s method.....................................111 4.4.2.2 Davisson?s failure criterion .......................................................................115 4.4.2.3 Interpreted capacities and hyperbolic parameters obtained.................116 4.4.2.3 Accuracy of the Chin/Davisson approach ...............................................120 CHAPTER 5. EVALUATION OF GEOTECHNICAL DESIGN PARAMETERS AND COMPUTATTIONS OF PREDICTED CAPACITIES.................................................127 5.1 INTRODUCTION ..........................................................................................................127 5.2 METHODS FOR PREDICTING ULTIMATE PILE CAPACITY IN SOUTH AFRICA............127 5.3 THE STATIC FORMULA ..............................................................................................128 5.4 DERIVATION OF SOIL PROPERTIES FROM AVAILABLE MEASUREMENTS ..................131 5.4.1 Background and interpretation of SPT results..............................................132 5.4.1.1 Historical development..............................................................................132 5.4.1.2 Test procedure ...........................................................................................132 5.4.1.3 Factors Affecting SPT N -values ..............................................................133 5.4.1.4 Correlation between SPT N-vales and soil properties............................136 5.4.1.5 Correlations for properties of non-cohesive materials...........................137 5.4.1.6 Correlations for cohesive materials .........................................................143 5.5 EVALUATION OF DESIGN PARAMETERS....................................................................147 5.5.1 Selection based on commonly assumed values in practice............................148 5.5.1.1 Driven piles in non-cohesive materials ....................................................148 Stellenbosch University http://scholar.sun.ac.za x 5.5.1.2 Bored piles in non-cohesive materials......................................................150 5.5.1.3 Driven piles in cohesive materials ............................................................150 5.5.1.4 Bored piles in cohesive materials..............................................................151 5.5.2 Selection based on results of a parametric study ...........................................152 5.5.2.1 Parameters for driven piles in non-cohesive materials ..........................153 5.5.2.2 Bored piles in non-cohesive materials......................................................154 5.5.2.3 Driven piles in cohesive materials ............................................................155 5.5.2.4 Bored piles in cohesive materials..............................................................157 5.5.3 Comparison of results from the two approaches ...........................................158 5.5.4 Selected design parameters ..............................................................................159 5.6 PREDICTED CAPACITIES ............................................................................................159 CHAPTER 6. STATISTICAL ANALYSIS OF THE MODEL FACTOR REALISATIONS ...............................................................................................................161 6.1 INTRODUCTION ..........................................................................................................161 6.2 BIAS FACTORS GENERATED .......................................................................................161 6.3 THE BIAS FACTOR AS A MEASURE OF VARIOUS SOURCES OF UNCERTAINTIES.........161 6.4 Statistical analysis of the model factor data ......................................................163 6.4.1 Histogram of the model factors .......................................................................164 6.4.2 Detection of data outliers .................................................................................166 6.4.2.1 Univariate approach ..................................................................................167 6.4.2.1.1 Sample z-score method.......................................................................167 6.4.2.1.2 Box plot method ..................................................................................170 6.4.2.2 Multivariate approach...............................................................................173 6.4.2.3 Correction to the identified outliers .........................................................176 6.4.3 Summary statistics............................................................................................177 6.4.4 Probabilistic model for the model factor ........................................................181 6.4.4.1 Verification of the assumption of lognormal distribution for the data.183 6.4.4.2 Goodness of-fit statistical tests..................................................................184 6.4.5 Investigation of correlation with underlying factors .....................................186 6.4.5.1 Correlation with pile length ......................................................................191 6.4.5.2 Correlation with pile diameter .................................................................193 6.4.5.2.1 Correlation with shaft diameter ........................................................193 6.4.5.2.2 Correlation with base diameter .........................................................195 6.4.5.3 Correlation with soil properties ...............................................................196 Stellenbosch University http://scholar.sun.ac.za xi 6.4.5.3.1 Correlation with friction angle (?) ....................................................196 6.4.5.3.2 Correlation with undrained shear strength .....................................198 6.4.5.4 Correlation with predicted and interpreted pile capacities ...................201 6.4.5.5 Summary and discussion of correlation investigation results ...............204 CHAPTER 7. REMOVAL OF STATISTICAL DEPENDENCIES BETWEEN THE MODEL FACTOR AND THE PREDICTED CAPACITY...........................................206 7.1 INTRODUCTION ..........................................................................................................206 7.2 GENERALISED MODEL FACTOR APPROACH ..............................................................206 7.2.1 Normalisation schemes .....................................................................................208 7.2.2 Regression results after normalisation ...........................................................208 7.2.2.1 Results for DNC .........................................................................................208 7.2.2.2 Results for BNC .........................................................................................210 7.2.2.3 Results for DC ............................................................................................211 7.2.2.4 Results for BC ............................................................................................212 7.2.2 Generalised model factor statistics and discussion of the regression results .....................................................................................................................................213 7.2.3 Verification of removal of systematic dependency ........................................215 7.3 CONDITIONED M .......................................................................................................217 7.3.1 Results for Regression of M on Qp ..................................................................217 7.3.2 Derivation of model factor statistics ...............................................................219 7.4 COMPARISON IN TERMS OF RELIABILITY INDEXES...................................................220 7.4.1 Design equation and performance function for the working stress design approach .....................................................................................................................221 7.4.2 Beta values based on the standard model factor statistics ............................223 7.4.3 Beta values based on the generalised model factor statistics ........................225 7.4.4 Comparison of magnitudes of beta values......................................................228 CHAPTER 8. RELIABILITY CALIBRATION OF RESISTANCE FACTORS ......230 8.1 INTRODUCTION ..........................................................................................................230 8.2 TARGET BETA ............................................................................................................230 8.3 CALIBRATION METHODS ..........................................................................................233 8.3.1 Advanced first-order second moment approach ...........................................233 8.3.1.1 Resistance factors for a target beta of 2.5................................................235 8.2.1.2 Resistance factors for a target beta values 3.0 ........................................244 8.2.1.3 Resistance factors for a target beta values 2.0 ........................................246 Stellenbosch University http://scholar.sun.ac.za xii 8.3.2 Mean value first-order second moment approach (MVFOSM)...................247 8.3.3 Approximate to first-order second moment approach..................................250 8.2.4 Design value approach .....................................................................................251 8.2.5 Calibration by fitting with WSD approach ....................................................253 8.2.6 Summary of resistance factors from various calibration approaches .........255 8.4 COMPARISON WITH PARTIAL FACTORS GIVEN IN EUROCODE 7 ..............................256 8.5 COMPARISON WITH OTHER PUBLISHED RESISTANCE FACTOR VALUES ...................259 8.6 RECOMMENDED RESISTANCE FACTORS ...................................................................260 CHAPTER 9. SUMMARY AND CONCLUSIONS ......................................................262 9.1 SUMMARY ..............................................................................................................262 9.1.1 Geotechnical limit state design ........................................................................262 9.1.2 Reliability background for the geotechnical limit state design ....................263 9.1.3 Data collection, processing and evaluation of interpreted capacities ..........265 9.1.4 Evaluation of engineering soil properties and design parameters ...............266 9.1.5 Model factor statistics.......................................................................................269 9.1.6 Treatment of correlation ..................................................................................271 9.1.7 Calibration of resistance factors......................................................................273 9.2 CONCLUSIONS ......................................................................................................275 9.3 RECOMMENDATIONS FOR FURTHER RESEARCH ....................................276 REFERENCES ..................................................................................................................278 LIST OF FIGURES...........................................................................................................293 LIST OF TABLES.............................................................................................................295 APPENDIX A. AVAILABLE STATISTICS ON GEOTECHNICAL PARAMETER UNCERTAINTIES ............................................................................................................298 APPENDIX B. TYPICAL PILE TEST RECORD AND GEOTECHNICAL DATA 306 APPENDIX C. RELIABILITY INDEX CALCULATION SPREADSHEET AND THE ASSOCIATED EXCEL FUNCTIONS DESCRIPTION......................................310 APPENDIX D. CALIBRATION SPREADSHEET AND THE ASSOCIATED EXCEL FUNCTIONS DESCRIPTION.........................................................................................313 Stellenbosch University http://scholar.sun.ac.za 1 Chapter 1 INTRODUCTION 1.1 BACKGROUND The presence of uncertainties and their significance in relation to performance of the designed systems has long been acknowledged by the engineering profession. Traditionally, a single global factor of safety is adopted to account for all the associated uncertainties. However, in today?s increasingly safety conscious civil engineering industry, it is becoming not only necessary to determine if the proposed design is safe, but also to determine how safe or reliable it actually is (Cameron, 2002). The need to quantiy the level of safety explicitely calls for identification, explicit quantification and systematic incorporation of the key uncertainties in the design process. Currently only the reliability-based design approach is capable of treating uncertainties in a rational and explicit manner. Accordingly reliability- based design is on the forefront of design philosophies in civil engineering worldwide (Harr, 1977). 1.1.1 Uncertainties in geotechnical design In common with other engineering designs, geotechnical design is performed under a considerable degree of inherent uncertainty. Sources of uncertainties in civil engineering designs have been identified by a number of researchers (e.g. Phoon, 1995; Whitman, 1984; Jaksa, 1982; Griffiths et al 2002) and may incorporate one or more of the following: ? Variability of material properties; ? Uncertainties associated with the measurement and conversion of design parameters; ? Inaccuracies that arise from the models which are used to predict the performance of the design; ? Inconsistencies associated with the magnitude and distribution of design loads; ? Anomalies that occur as a result of construction variability; ? Human gross errors. Stellenbosch University http://scholar.sun.ac.za 2 For most geotechnical designs, two predominant sources of uncertainties can be distinguished: (i) uncertainties associated with the evaluation of design soil properties and (ii) calculation model uncertainties. Soil parameter uncertainty arises from the variability exhibited by properties of geotechnical materials from one location to the other, even within seemingly homogeneous profiles. Tang et al (1984), Whitman (2000), Baecher (1986), Tabba et al (1981), Christian et al (1994) Phoon and Kulhawy (1999a) and many other researchers have identified the following as the key sources of geotechnical parameter prediction uncertainties: inherent spatial variability; measurement noise/random errors, systematic measurement errors, and statistical uncertainties. Conversely, model uncertainty emanates from imperfections of analytical models for predicting engineering behaviour. Mathematical modelling of any physical process generally requires simplifications to create a useable model. Inevitably, the resulting models are simplifications of complex real world phenomena. Consequently there is uncertainty in the model prediction even if the model inputs are known with certainty. For a rational design, both soil parameter and model uncertainty need to be quantified. With regard to geotechnical properties uncertainties, significant research work has been carried out to generate statistics on individual components of soil parameter uncertainty. Accordingly first-order estimates of inherent variability, measurement errors and correlation uncertainties have been reported (e.g. Kulhawy et al, 1991; Phoon, 1995; Phoon and Kulhawy, 1996; Kulhawy and Trautmann, 1996; Lacasse and Nadim, 1996; DeGroot, 1996; Phoon and Kulhawy, 1999a; Phoon and Kulhawy, 1999b; Jones et al, 2002). Relevant tables from some of the published works are presented in Appendix A. Now statistics on soil parameter uncertainties are available for more rigorous calibration of geotechnical reliability based design equation. Conversely model uncertainty statistics are generally scarce. In fact, the lack of model statistics is a key impediment to the development of geotechnical reliability based design (Phoon, 2005). Therefore there is a need for concerted research effort to develop model uncertainty statistics worldwide. In civil engineering, limit state design represents the-state-of-the-art design philosophy in which ultimate failure and serviceability conditions can be evaluated considering the uncertainties in load and materials resistances. In structural design, limit state design has developed to a level where it has been merged with reliability theory leading to incorporation of probabilistic concepts in structural codes. Prompted by developments in Stellenbosch University http://scholar.sun.ac.za 3 structural design, in recent years much work has also been devoted to developing limit state methodologies for geotechnical engineering based on probabilistic techniques (Christian, 2003). Accordingly probabilistic limit state design (level 1 reliability methods) has been internationally accepted as the standard basis on which the new generation of geotechnical codes are being developed today. 1.1.2 Status of probabilistic geotechnical limit state design As already alluded to, initiatives to bring geotechnical design within the reliability based design framework as is the case in structural engineering are in progress in most parts of the world. Already there are some geotechnical codes with the same design format as structural codes. Examples include Eurocode 7, the Canadian Foundation Engineering Manual, the AASHTO Standard Specification for Highway Bridges, Geoguide 1, etc. Although the Load and Resistance Factor Design (LFRD) or its equivalent in Canada and Europe have the same design format as that for structural codes, several researchers (e.g. Phoon, 2004a; DiMagio et al, 1999; Paikowsky and Stenersen, 2000) have expressed concern that in general there is a lack of analytical calibration and verification. Mostly the partial factors have been developed on the basis of judgement by rearranging the existing global safety factors (Phoon et al 2003, Phoon, 2004a). Therefore, in essence geotechnical limit state design is being implemented in a deterministic framework. Phoon et al (2003), assert that implementation of limit state design within a deterministic framework does not address the drawbacks associated with the traditional working stress design approach. For a probabilistic limit state design, the partial factors need to be derived using reliability theory. In this regard, the key long-term objective of the probabilistic limit state initiative is to determine all the load and resistance factors using reliability theory. Accordingly advancement of reliability based methodologies on determination of partial factors is among the terms of references for the Technical Committee (TC 23) of the International Society of Soil Mechanics and Geotechnical Engineering on Limit State Design in Geotechnical Engineering. 1.1.3 State of limit state design in SA Currently in South Africa, foundation design codes of practice (SABS 088- Pile foundations and SABS 0161: 1980 ? Design of foundations for buildings) are still based on the working Stellenbosch University http://scholar.sun.ac.za 4 stress design philosophy. However, in order to keep pace with international developments, the Geotechnical Division of South African Institute of Civil Engineers embarked on some initiatives to convert to limit state design as far back as 1993. A chronological account of these initiatives is as follows: ? A few years before 1993, the SABS canvassed the opinion of the industry as to whether the existing piling code was to be updated or re-written. Most respondents were in favour of re-writing the code. ? In 1993, a committee to undertake the task of re-writing the new piling code was formed. ? The committee advised that, in order to keep pace with recent developments overseas, the new code should preferably be written in terms of limit state principles. ? The committee also observed that it would not be possible to introduce limit state design into geotechnical engineering on an ad hoc basis and recommended that the profession should look into the merits of adopting the Eurocodes approach for geotechnical design as a whole. ? As an initial step, the Geotechnical Division of SAICE invited Dr Ovesen and Dr. Simpson, chairmen of Eurocode 7 drafting committee to address a seminar held in Pretoria in October 1995. At the seminar it was agreed that: - a number of geotechnical designers would apply the latest version of Eurocode 7 (ENV-1997-1) in parallel with design methods currently in use for a trial period of two or three years. - At the end of the trial period, another seminar will be held and then a final decision on adopting Eurocode 7 in South Africa would be made. ? Following the seminar, a meeting between the Geotechnical division and the SABS Building Codes section resolved that should the former recommend the adoption of Eurocode 7, it would be acceptable to the latter. It was also agreed that with permission of the European Committee for Standardisation, the code would then be published by the SABS as joint Eurocode and SABS code. Despite the above resolutions, the implementation of Eurocode 7 was not entirely successful (Day, P. 1997). The main stumbling block to the adoption of the Eurocodes in general is the difference between the load factors in the Eurocodes in those contained in the South African Loading Code (Day, P. 1997, Day et al, 2000). In view of this problem, the initiatives were held back pending the revision of the South African Loading Code (Day et al, 2000). Stellenbosch University http://scholar.sun.ac.za 5 The revision of the South African loading code is at an advanced stage. The code is being revised along the lines of the Eurocodes with a strong reliability basis. A provision for geotechnical design has been made. The code forms the foundation for the materials codes including geotechnical design codes. In a way, the revised South African loading code indirectly forces materials codes including geotechnical design to be based on reliability principles. It is evident from these latest developments that future geotechnical design codes in South Africa will be a probabilistic limit state design code. The impending transition from the current design practice to reliability based geotechnical design requires some reference materials in the form of detailed studies into various aspects of limit state design under the local conditions. Such studies will form the basis on which code writing committees will base their decisions. 1.2 RESEARCH STATEMENT The South African piling industry expressed the desire to rewrite the current South African Code of Practice for Pile Foundation design (SABS 088-1972: Pile foundations) on the basis of limit state design principles in 1993. However, geotechnical design as a whole in South Africa and most African countries is still based on the traditional working stress design approach. Recent international and local developments have now added impetus to the introduction of probabilistic limit state design in South Africa. These include; (i) the international acceptance of probabilistic limit state as the standard basis on which the new generation of geotechnical codes are being developed today (ii) the formulation of the draft South African Loading Code (SANS 10160) on the basis of reliability framework, which implies that the subsequent materials codes including geotechnical design code will be based on the same framework (iii) increased interest in harmonisation of technical rules for design of building and civil engineering works across disciplines and national borders. To keep pace with all these developments, geotechnical design in South Africa has no choice but change to limit state design as a matter of urgency. Given that uncertainties are the hallmark of geotechnical engineering, it is only logical to adopt limit state design in a probabilistic framework. Probabilistic limit state is currently implemented in the form of level 1 reliability design approach in which a prescribed level of reliability is imparted to the designed element through the use of predetermined partial factors rather than performing Stellenbosch University http://scholar.sun.ac.za 6 reliability analysis. Central to this approach is the determination of load and resistance factors. Calibration of partial factors is dependent on local design practice, experience and environment such as local geology, soil type and conditions, site investigation practices (extend, methods, standards, equipment advances). Since these factors generally differ from one country to another, values developed for a specific country can not be simply adopted by another country. The need to calibrate geotechnical resistance factors for different applications utilising local databases was further emphasised in a study to review developments of limit state design or LRFD methods in Canada, Germany, France, Denmark, Norway and Sweden (DiMaggio et al, 1999). For the same reason, Eurocode 7 leaves the calibration of partial factors to individual countries. Therefore to introduce probabilistic geotechnical limit state design to the design of pile foundations in South Africa, calibration studies in which the South African design experience is captured are required. A prerequisite to such reliability calibration of partial factors is the identification and quantification of uncertainties associated with pile design models. In common with other geotechnical design, the major uncertainties can be broadly classified as those relating to determination of soil design parameters and those pertaining to calculation models. As already alluded to, statistics on parameter uncertainties as well as first order second moment models for combining such uncertainties are available. Conversely statistics on model uncertainties are scarce worldwide. In spite of scarcity of model uncertainty statistics, prediction symposiums and research studies have indicated that model uncertainty is normally the predominant source of uncertainty for pile design. A typical example is the results of an international competition to test the ability of experienced practising engineers to predict the behaviour of plain driven and jet grout enhanced piles under static and cyclic loading (Jardine et.al, 2001). The results were such that even for the conventional driven piles, the ultimate capacity predictions were not within 50% of the measured capacities while the settlement was overestimated by about 100%. Further more, Ronold and Bjerager (1992) carried out a first order reliability analysis which incorporated model uncertainty on tension piles. Through sensitivity factors of variables for the performance function, the study showed that the model uncertainty exclusively is the most important source of uncertainty by a contribution to the total Stellenbosch University http://scholar.sun.ac.za 7 uncertainty by close to 100%. It concluded that other sources of uncertainties such as evaluation of soil parameters could just as well be neglected. The high contribution of model uncertainty to the total uncertainty was attributed to the limited physical understanding of the pile-soil interaction problem. In recognition of the significance of model uncertainty, EN 1997-1 recommends that if the ultimate pile capacity is determined on the basis of a theoretical model, a model factor may be introduced to ensure that the predicted compressive resistance is sufficiently safe. However, EN 1997-1 does not provide reference values of model factors and instead recommends that such values be set by national annexes. This is a further demonstration of the scarcity of model uncertainty or model factor statistics. Motivated by the need for the introduction of probabilistic limit state pile design in South Africa, this study quantifies model uncertainty in the classic static pile design formula. The static formula is the main theoretical pile design method; hence model uncertainty associated with this approach is of international interest. In fact, due to lack of model uncertainty statistics, characterisation of model uncertainty in any geotechnical design model is a major contribution to the development of probabilistic geotechnical limit state design internationally. As a contribution to the South African geotechnical practice, particularly the piling industry, the developed model uncertainty statistics are used to calibrate resistance partial factors for pile foundations. 1.3 AIMS AND SCOPE OF THE STUDY The main aim of the study is to characterise model uncertainty in analytical pile design methods and to use the generated model factor statistics to calibrate partial resistance factors in a reliability based framework. Specific research objectives are: ? To quantify model uncertainty in the classic static pile design formula in terms of model factor statistics. ? To provide insight into the degree of conservatism in the static formula. ? To compare model factor statistics developed on the basis of the conventional approach (standard model factor) to that based on regression theory where systematic effects of design parameters are taken into account (generalised model factor). Stellenbosch University http://scholar.sun.ac.za 8 ? To use the developed model factor statistics as input in reliability calibration of partial resistance factors. ? To recommend a suitable target reliability index for design of pile foundations in South Africa. ? To recommend suitable resistance factors for design of piles in South Africa. ? Since the Eurocodes are the reference codes in South Africa, the study will make recommendations as to which of the three EN 1997-1 design approaches will be more appropriate for South Africa. ? To contribute to the general introduction of probabilistic geotechnical limit state design in South Africa. The study is limited to pile foundations for structures within the scope of the South Afican Loading Code (SANS 10160-Draft). These include buildings and industrial structures. Both driven and bored piles in cohesive and non-cohesive soil are considered. Even though pile foundations can resist various types of loading such as lateral loads, tension loads and compression loads, the study focuses on piles subjected to axial load only. Although limit state design requires the structure to satisfy two principal criteria of ultimate limit state and serviceability limit state, this study focuses on ultimate limit state. 1.4 LAYOUT OF THE DISSERTATION The dissertation details the current research undertaken to quantify model uncertainty in the static formula for computing the ultimate capacity of pile foundations. The generated model factor statistics were used to calibrate partial resistance factors in reliability based design framework. In Chapter 2 the principles and basis of geotechnical limit state design are set. Given that the future South African geotechnical design code will be based on the Eurocodes format, the basis of design will be in accordance with Eurocodes approach. Among the issues considered include selection of characteristic geotechnical design parameters, design approaches, actions and action combinations, verification of ultimate and serviceability limit states. Stellenbosch University http://scholar.sun.ac.za 9 Chapter 3 presents the fundamental reliability background required for reliability-based geotechnical design. This entails an overview of reliability theory and reliability calibration principles. Chapter 4 describes the compilation of a database of local static pile load tests along with the associated geotechnical data (soil profiles, field and laboratory test results). The Chapter also describes the processing of the pile tests records to evaluate the interpreted capacities (measured capacities). In Chapter 5, the specific site measurements presented in Chapter 4 are transformed to the desired engineering soil properties by means of empirical correlations. Furthermore, evaluated soil properties are used to derive other required pile design parameters (e.g. Nq, Ks, Nc, ?, and ?). Finally the soil parameters and the associated pile design parameters are used to calculate the theoretical or predicted capacities. In Chapter 6, the results obtained from Chapter 4 and Chapter 5 are used to determine the model factors. The generated model factors are analysed statistically to facilitate their further interpretation and generalisation beyond the database. In Chapter 7, the statistical dependencies between the model factor and the predicted capacity observed in Chapter 6 are either removed or taken into account. The effects of further treatment of the correlation on the model factor statistics and the calculated reliability indexes are investigated. Chapter 8 applies the results of model uncertainty quantification in reliability calibration of partial resistance factors. A summary and conclusions of the study, as well as areas for further research are then presented in Chapter 9. Stellenbosch University http://scholar.sun.ac.za 10 Chapter 2 GENERAL INTRODUCTION TO GEOTECHNICAL LIMIT STATE DESIGN 2.1 Introduction As mentioned in Chapter 1, uncertainties are an inherent part of all civil engineering works. In spite of the existence of uncertainties, society expects the designed structures to be safe for the people who use them or who are in their vicinity (Schneider, 1997). Accordingly, the civil engineering profession has risen to the challenge and has been continuously refining design methodologies to ensure that structures are designed to fulfil certain performance criteria. One such design approach is limit state design. The basis of the limit state method is the acknowledgement that the structure may fail to meet its design requirements through a number of possible shortcomings (Day, R. 1997). In this context, limit state design is a formal and methodical way of ensuring each of the design criteria will be properly considered. The performance of a structure or structural element is described with reference to a set of limit states. Limit states are states beyond which the structure no longer satisfies the design performance requirements (EN 1990). Each such limit state is considered separately, and its occurrence is shown to satisfy the design criteria. The impetus in the development of limit state design comes from the structural engineering profession. However in recent years much work has been devoted to development of geotechnical limit state design. A noticeable initiative is the establishment of a technical committee (TC 23) on Limit State Design in Geotechnical Engineering under the auspices of the International Society of Soil Mechanics and Geotechnical Engineering in 1990. The committee was mandated with promoting and enhancing professional activities in the limit state design in geotechnical engineering practice. Accordingly the committee has been organising international symposiums on limit state design in geotechnical engineering practice. Examples of such include; Copenhagen Symposium on Limit State Design in Geotechnical Engineering (1993), seminar in London entitled Eurocode 7 ? Towards Implementation (1996), LSD 2000 in Melbourne Australia, LSD 2002 in Kamakura Japan, Stellenbosch University http://scholar.sun.ac.za 11 LSD 2003 in Boston USA and an International Symposium on New Generation Design Codes for Geotechnical Engineering Practice (Taipei, Taiwan, 2006) and many others. Historically the limit state design philosophy has developed in parallel with the application of statistics, probability theory and partial factors to the design of structures (Borden, 1981; Day, R. 1997). The three concepts (limit state design, partial factors, and probabilistic considerations) are often wrongly thought of as being inseparable. In essence, there is no fundamental connection between them (Simpson, 2001).The original concept of limit state design does not specify a particular way in which non-ascendance of the relevant limit states are ensured (Borden, 1981; Simpson et al, 1981; Becker, 1996a). The level of safety may be provided through partial safety factors, global factor of safety or any other means. Therefore limit state design should not only be associated with partial factor method and a probabilistic design as it is commonly the case. Two forms of geotechnical limit state design have emerged. One can be termed deterministic limit state design and the other termed probabilistic limit state design. The fundamental difference is in the derivation of the associated partial factors. With the deterministic limit state design, partial factors are determined by engineering judgement and by fitting to the existing design practice. Conversely partial factors for the probabilistic limit state are based on reliability calibration. Given that coping with uncertainty is the hallmark of geotechnical practice, it is only logical to develop geotechnical limit state in a probabilistic framework. This Chapter reviews the evolution of geotechnical design methodologies and fundamental concepts of the modern limit state design. 2.2 Evolution of geotechnical design Over the years, geotechnical design procedures have evolved to ensure that designs meet performance requirements in the face of uncertainties. Generally, civil engineering design philosophies evolved from experience based design, working stress design, limit state design and the current probabilistic limit states design (level 1 reliability based design). 2.2.1 Experience based design approach In the old days, the experience of the ancient builders guaranteed the safety of structures. The Gothic cathedral of Amiens in France built during the period 1220-1280 is generally Stellenbosch University http://scholar.sun.ac.za 12 viewed as the best example of the knowledge of the medieval builders (Van Straalen, 1999). These experience based design approaches were used for many centuries. A turning point in the guarantee of safety of designs occurred during the industrial revolution in the early 19th century (Becker, 1996a). The industrial revolution brought with it new materials and proliferation of new technology which forced modern builders to develop more rational design procedures. This new design approach came to be known as working stress design or allowable stress design. 2.2.2 Working stress design Since its introduction in the early 1800?s, the working stress design has been the traditional design basis in geotechnical engineering world wide. The working stress design philosophy attempts to ensure that the applied service load or the stresses induced in the soil mass do not exceed some allowable limit, often taken to be the limit of elastic resistance. In other words, safety is achieved by restricting the applied loads to values less than the ultimate geotechnical resistance divided by a factor of safety using the mathematical relationship of the general form: ?? QFSR [2.1] in which, R = ultimate geotechnical resistance; FS = factor of safety; ?Q = summation of loads effects (dead and live loads). In this context, the factor of safety provides reserve strength in the event that an unusually high load occurs or in the event the resistance is less than expected. Therefore it is an empirical, but arbitrary number greater than unity used to reduce the potential for adverse performance. However in practice different engineers use different approaches of selecting the ultimate strength and load. Some may use mean values while others use nominal or characteristic values. Consequently two alternative definitions of the factor of safety emerge and are defined as follows: a) Mean factor of safety = Q R [2.2] b) Nominal factor of safety = n n Q R [2.3] Stellenbosch University http://scholar.sun.ac.za 13 The numerical values of FS for the two cases are not equal. The mean FS is higher than the nominal FS. Therefore for the same numerical value of safety factor, the margin of safety can be very different. It is evident from the forgoing that all uncertainty in the variation of the applied load and the ultimate capacity of the soil are lumped in a single factor. In geotechnical design, the factor of safety is generally applied to the resistance side on account that uncertainties in soil parameters are the largest uncertainties affecting geotechnical design. The values of the global factors of safety for various geotechnical structures have been developed from previous experience with similar structures under similar conditions (Becker, 1996a). They reflect past experience and the consequences of failure. In geotechnical design the concept of a factor of safety in stability estimation was introduced in the 18th century by Belidor and Coulomb who both suggested placing a value of 1.23 on the width of retaining walls determined from earth pressure theory and later Kery introduced a factor of about 1.5 for stability of slopes and retaining walls and recommended a range of 2-3 on ultimate bearing capacity of foundations (Meyerhof, 1995). Similar global factors of safety became customary for geotechnical design in Europe, North America and other parts of the world. Ranges of global factor of safety commonly used in geotechnical design throughout the world were compiled by Terzaghi and Peck (1967) and are presented in table 2.1. Table 2-1: Ranges of global factor of safety for geotechnical design (After Terzaghi and Peck, 1948, 1967) Failure type Item Factor of safety Shearing Earthworks Earth retaining structures Foundations 1.3-1.5 1.5-2 2 ? 3 Seepage Uplift heave Exit gradient, piping 1.5-2 2-3 Ultimate pile loads Load tests Dynamic formulae 1.5-2 3 Stellenbosch University http://scholar.sun.ac.za 14 As illustrated by table 2.1, similar values of factor of safety have become customary for geotechnical design through out the world regardless of the different soil conditions encountered. Obviously the use of the same numerical value of safety factor for conditions that involve widely varying degrees of uncertainty, results in different levels of safety. Working stress design is a simple and straight forward approach that is still in use in some countries including South Africa. However, when viewed from the perspective of current advances in engineering design, the working stress design approach has several limitations. These limitations are discussed below: a) As already alluded to, with this approach, all uncertainties are lumped under a single factor of safety and therefore there is no distinction between model and soil parameter uncertainty. This makes it difficult to justify any reduction in safety level if there is additional information or advances in the state-of-the-art (Phoon 1995); b) The fact that values of safety factor for various geotechnical applications have been developed subjectively on the basis of experience suggests that the approach can not be extrapolated rationally and consistently to accommodate new design situations; c) The level of safety associated with the value of FS depends on its definition (mean or nominal model factor). Therefore the use of a global safety factor does not lead to consistent level of safety; d) Earth structures have failed even though the computed FS was greater than one (Becker, 1996a). This indicates that a value of FS greater than one does not necessarily ensure safety. Similarly failure does not necessarily occur when the computed FS is less than 1. Therefore the actual level of safety implied by the FS is not known to the engineer; e) In foundation design, the safety factor applied to the ultimate bearing capacity is deemed to limit settlement to acceptable limit without computing the actual settlement. Even though this is a common practice, Becker (1996a) argues that the specific values of FS were not derived for the separate consideration of soil rupture or collapse under bearing capacity considerations. 2.2.3 Limit state design philosophy Given the drawbacks of the working stress design approach, the quest for a better design methodology continued, leading to the introduction of the limit state design approach. As already pointed out, with limit state design, the performance of a structure or part of a structure is described with reference to a set of limit states beyond which the structures is Stellenbosch University http://scholar.sun.ac.za 15 deemed to have failed to satisfy the fundamental requirements. Therefore essentially limit state design entails: - Identification of all potential limit states and related failure modes; - Checking of the occurrence of each limit state; - Demonstration that the occurrence of the limit states is improbable or within acceptable risk. In general, two main categories of limit states are normally considered. These include the ultimate limit state and serviceability limit state. Ultimate limit states are associated with the total or partial collapse of the structure (e.g., strength, ultimate bearing capacity, overturning, sliding, etc). It concerns the safety of the structure and its contents including people. Typical examples include (i) loss of equilibrium of a part or all of a structure as a rigid body leading to overturning, sliding, etc (ii) rupture of critical components, causing partial or complete collapse. Conversely, serviceability limit states correspond to those conditions beyond which specific requirements of the structure or structural element are no longer met. Serviceability limit states concern the functionality of the structure, the comfort of people and aesthetic appearance of the structure. Examples include deformations, settlement, vibrations, cracks, and local damage of the structure in normal use under working loads such that it ceases to function as intended. The distinction between ultimate and serviceability limit states is also applicable to geotechnical practice. In this regard, Geotechnical ultimate limit states have been defined as failure or excessive deformation of the ground where the strengths of the soil or rock are significant in providing resistance (EN 1997-1). With respect to foundation structures, ultimate limit state emanates from soil bearing capacity failure. Bearing capacity failure usually occurs as a shear failure of the soil. The three principal modes of shear failures (general shear failure, punching shear and local shear failure) lead to an occurrence of an ultimate limit state. However, the development of the ultimate limit state associated with the three modes of shear failure differs. General shear failure is charecterised by the development of a well defined failure pattern, consisting of a continuous slip surface form the edge of the footing up to the ground level. Stellenbosch University http://scholar.sun.ac.za 16 Ground failure is sudden and catastrophic accompanied by substantial rotation of the foundation. For structures for which by virtue of their configuration, do not prevent the rotation of the foundation, general shear failure lead to total collapse of the structure. Examples of such structures include silos, tanks, and towers. In conventional structures, rotation of the entire foundation system is prevented by the configuration of the structure. Although rotation may be prevented, the distress caused by the ground movement may lead to rupture of critical elements of the structure, resulting in partial or total collapse. In contrast to the general shear failure, the punching shear failure mode is charecterised by a failure pattern that is not easily observed. Slip lines do not develop and little or no bulging occurs at the ground surface. However, there is a large vertical movement of the foundation due to the soil compressibility. The soil outside the loaded area remains uninvolved and there are no movements of the soil on the sides of the foundation (Vesic, 1973). Similarly local shear failure is characterised by a failure pattern that is not well defined except immediately beneath the foundation. The slip surfaces end somewhere in the soil mass. Failure is not catastrophic and tilting is insignificant. However, the vertical movement of the foundation is significant. From the description of the punching and local shear failure, it can be concluded that the two modes of failure do not lead to catastrophic collapse of the ground which may result in the total collapse of the structure and hence the attainment of ultimate limit state condition. However, ultimate limit states in the structure can be reached as a result of excessive deformation of the ground associated with both the punching and local shear failure. From the geotechnical engineering perspective, serviceability limit state is settlement. Due to the presence of voids, soils always settle under loading. Soil settlement when related to the structure can be classified as uniform or differential settlement. Uniform ground settlement is not really detrimental to the structure. However, it leads to distress in service pipes and cables connected from external mains to the structure. Such distress may result in the fracturing of the service pipes. It is the differential settlement that may affect the overall efficiency of the structure. Serviceability limit states are reached when the angular distortion is 1/300 (Burland et al, 1978). The most common symptom is the development of cracks in elements such as walls, floors, beams, etc. Cracks lead to water penetration, resulting in corrosion of reinforcement and gradual distortion of the structure. Cracks also result in reduced weather- tightness, dampness; heat loss and reduced sound insulation. Another Stellenbosch University http://scholar.sun.ac.za 17 symptom of violation of serviceability limit state is deflection of elements. Deflection affects operation of lifts and other precision machinery. Both cracks and deflection of elements render a structure visually unacceptable. Due to this inter-relationship between the ground and the structure, limit states are generally defined in terms of damage to the structure as damage to the ground is rarely of significance in itself (Simpson, 1981). Actions from the superstructure lead to accidence of the above limit states in the ground. The violation of the ground limit states in turn lead to distress in the structure or its elements which may lead to failure or serviceability problems. It is evident from the foregoing discussion that limit state approach to design requires that the designer should check the adequacy of the structure against collapse and serviceability. Many geotechnical engineers (e.g Boden, 1981; Simpson, 1981; Becker, 1996a) are of the opinion that this approach has always been used in one form or the other in geotechnical design and therefore limit state design is not a radically new method compared to earlier design practice. First limit states concept in geotechnical engineering include work by Coulomb and Rankine (Meyerhof, 1995). Coulomb derived the critical height of a vertical embankment in cohesive soil based on limit states considerations while Rankine established limit states of active and passive earth pressures. However, the classical geotechnical limit state approach became well established when Terzaghi introduced the modern approach to soil mechanics. In 1934, Terzaghi classified geotechnical problems into two categories, namely stability problems and elasticity problems. The stability problems deals with conditions immediately before ultimate failure by plastic flow without consideration of strain effects while elasticity problems deal with soil deformation either under self-weight or external forces without consideration of stress condition for failure. The above two classes of geotechnical problems coincide with ultimate limit state and serviceability limit state respectively in the current limit state design philosophy. The first limit state code of practice was the 1956 Danish Standard for foundations. This resulted from work by Taylor and Brinch Hansen. Taylor (1948) introduced separate factors of safety on the cohesive and frictional components of the shear strength parameters in the analysis of stability of slopes. The approach was generalised by Brinch Hansen (1965) when he proposed partial factors on different type of loads, shear strength parameters and pile capacities for ultimate limit state design of earth retaining structures and foundations. From the forgoing, it appears that in earlier days, geotechnical engineering was ahead of structural engineering in the knowledge and application of limit state design philosophy. Stellenbosch University http://scholar.sun.ac.za 18 The current format of limit state design represents a clearer formulation of some widely accepted principles. From this premise, limit state design is regarded as a calculation tool in the design process where satisfactory performance is the primary objective. In fact the design philosophy is similar to that for working stress design approach with the emphasis shifted from elastic theory and materials strength to failure of the structure to perform its intended function (Becker 1996a). 2.2.3.1 Verification of non-exceedance of limit states In principle any method (e.g. global factor, partial factors) can be used to give confidence that the limit states are satisfied. However, during the development of the approach the partial factors format became the routine method for ensuring non-exceedance of the limit states. Originally the partial factors were determined by subjective means such as engineering judgement and fitting to the working stress design approach. However the advent of the application of probability theory to design provided a direct link between the partial factors and probability of failure. Accordingly a second method of deriving partial factors based on probabilistic considerations emerged. Both approaches are currently in use, leading to two distinct types of limit state design (i.e. deterministic and probabilistic). 2.2.3.2 Deterministic limit state design Limit state design in the partial factors format became the general design approach in structural practice in the 1970s. Limit state design in this format was also permeating into geotechnical design. Christian (2003) asserts that the interest was driven by the desire to apply the same mathematical insights to geotechnical practice that have proved successful for structural engineering. The move was also motivated by the desire to achieve compatibility between geotechnical and structural engineering. However in geotechnical engineering, the use of the partial factors has generated some controversy. The proponents of the approach argue that if the use of the approach is felt to represent an improvement in structural design, why should its use not be equally valid in geotechnical design? Some of the reservations expressed by the geotechnical profession (e.g. Boden, 1981; Semple, 1981; Simpson et al, 1981; Semple, 1981; Ovesen , 1981; Fleming, 1989) include: Stellenbosch University http://scholar.sun.ac.za 19 ? Not too many foundations and substructures are failing at the moment or appear to have excessively conservative designs, so why the fuss? ? The method is cumbersome to use, with a multiplicity of coefficients and increases the chances for computational errors; ? Results of analysis using partial factors must fit experience and therefore do not produce substantial differences in overall safety factors; ? Splitting the safety margin into components associated with loads and resistances introduces uncertainty as to whether any function of the original safety factor has been omitted; ? When failures occur, it is due to serious errors in unforeseen conditions. Failures due to excessive variation of recognised parameters are rare and should not be given undue emphasis by focusing attention on partial factors; ? Prescribed factors applied to soil properties might define soils which could not possibly exist. Under such circumstances the design could hardly be considered to be realistic; ? The method of applying a safety factor in the working stress design approach have been developed to enable the best use to be made of decades of full-scale and model evidence. The factors are applied to the ground properties or loads or to some derived quantity depending upon which approach has shown to be the most appropriate. Therefore there is considerable reluctance to depart from this practice unless there are convincing reasons of overriding importance; ? The majority of geotechnical design procedures used throughout the world are rooted in engineering judgement and empiricism. However when used in the traditional way which draws on many years of experience, satisfactory designs have been produced. Under these circumstances, the replacement of the traditional method of factoring by a range of new prescribed partial factors has obvious dangers; ? The partial factor approach integrates safety factors into analysis, tending to distort perception of parameter and behaviour. Attention is likely to focus on satisfying the code requirements than on the truly important aspect of understanding soil distributions, properties and behaviour. The conventional separation of behavioural assessment from safety considerations enhances geotechnical design practice; ? Partial factors can lead to probability theory and encourage statistical assessment of measured data. In geotechnical engineering, these provide little insight and divert attention from reality. Statistical analysis may cause major errors in selection of soil parameters, there being no comparable problem with structural design in manufactured materials; Stellenbosch University http://scholar.sun.ac.za 20 ? The true measure of safety is how easily lost is the excess of resistance over load. This can be studied by sensitivity study analyses which are compatible with the global safety factor approach. Partial factors specified in codes and incorporated into analyses may discourage proper in-depth study in complicated problems just as they tend to complicate routine design without apparent benefit. Phoon (1995) adds the following reservations: ? Partial factors for soil parameters are not dependent on influential factors such as the design equation and the procedure for determining the soil strength; ? Definition of nominal values to be factored is not clearly given, resulting in different engineers adopting different values. For an example, one might use average soil parameters while another may use conservative soil parameters as the nominal value; ? Partial factors do not account for the variation in uncertainty of soil parameters from site to site; ? It is not clear how partial factors can assist in extrapolating the experience of safe practice to new conditions or can permit full advantage to be taken of improvements in the knowledge base. Closely associated with the partial factors is the use of statistics and probability. The following reservations against the application of statistics and probability in geotechnical engineering have been expressed: ? Man-made steels and concretes are fundamentally different from naturally occurring geotechnical materials. The former are ideally suited to definitions by statistics of variations; it is the essence of their quality control during construction. No such control was applied during the formation of soil or rock and the subsequent modification of the earth crust. In view of this situation statistics have no place in geotechnics; ? There is danger in the use of statistics as a less experienced engineer might place too great an emphasis on the role of statistics to the detriment of the geotechnics; ? The properties and the three-dimensional geometry of the ground are often typified by extreme values (e.g. a soft layer under one corner of the structure) rather than by randomly distributed values having means and standard deviations which are valid for design purposes. Under such conditions, the use of statistics might not help much. The use of statistics might encourage an unjustified feel of security in the designer; Stellenbosch University http://scholar.sun.ac.za 21 ? The probability theories are even more complex than the geotechnical theories themselves. In these circumstances there would be a danger that the attention of the designer could be diverted from understanding the main problem (i.e. the geotechnics) to attempting to understand the probability theory which is just an aid to the design process. Despite all the controversy, geotechnical limit state design continued to develop to its current status. The motivation to continue applying limit state design to geotechnical practice stems from three main factors: ? The need to achieve compatibility between structural and geotechnical design. Structural design codes of practice have progressed to the new limit state approach while geotechnical codes have not. Therefore different design procedures are being used on either side of the interface between the structure and the ground. Although the methods and analysis used on both sides produce reasonable designs within their own current- state-of-the-art, it is desirable for the two design processes to be compatible (Borden, 1981); ? There has been an intensive limit state design awareness campaign by the Technical committee (TC23) of the International Society of Soil Mechanics and Geotechnical Engineering on Limit State Design in Geotechnical Engineering. As already mentioned the committee has organised many conferences on limit state design in geotechnical engineering practice; ? The application of statistics, probability and reliability theory to analysis and design of structures made it possible for probabilistic assessment of level of safety in terms of probability of failure. The load effects and resistance can now be expressed in terms of their distributions and the overlap of the two distribution curves signifies a condition where the resistance is less than the load effect and hence a probability of failure. This lead to a transparent and rational treatment of uncertainties. Even the partial factors could now be derived on the basis of reliability theory. 2.2.3.3 Probabilistic limit state design The controversy and severe criticism on the application of limit state design with partial factors to geotechnical practice appear to stem from the lack of a theoretical basis on which the partial factors were developed. Just like the global factors, the partial factors were developed purely on basis of intuition and judgement. Further more the new approach was Stellenbosch University http://scholar.sun.ac.za 22 required to produce designs similar to the existing working stress design approach. If the new design approach yields the same results as the old approach then what is the need for the new approach? Clearly there is no noticeable benefit from the transition from the working stress design to the limit state design in a non-probabilistic framework. Without reasons of overriding importance, it is understandable that there was a considerable reluctance to depart from the familiar factoring method. Therefore an important step in justifying the use of limit state design is to ensure that the partial factors are derived on the basis of reliability theory and not through simply scaling to achieve the same design as the methods it would replace. The non-probabilistic limit state design just like the working stress design does not explicitly quantify the level of safety achieved. Therefore in recent years, probability and reliability theory have been added to the original limit state design leading to the semi- probabilistic limit state design. A further motivation is the explicit and rational treatment of uncertainties. Essentially, semi-probabilistic limit state is a level 1 reliability based design method. Level 1 reliability methods are design methods in which appropriate levels of reliability are provided on the structural component by the use of prescribed partial factors. The derivation of the partial factors explicitly and systematically incorporates the major sources of uncertainties. Ideally the partial factors are calibrated on the basis of reliability theory. The use of such partial factors ensures consistent level of reliability over a range of structures. Although the approach is set within a probabilistic framework, it does not require explicit use of the probabilistic description of the variables. This is an advantage in that even engineers with no knowledge of probability and reliability theory can produce designs at a prescribed level of reliability. Due to their convenience and simplicity, level 1 reliability methods form the basis of current reliability based design codes. Accordingly new generation design codes for geotechnical engineering are now developed on the basis of level 1 reliability methods. The many advantages and benefits of using reliability based design methods include the following (Ayyub et al, 2000): - They provide the means for the management of uncertainty in loading, strength, and degradation mechanism; - They provide consistency in reliability; - They result in efficient and possibly economical use of materials; Stellenbosch University http://scholar.sun.ac.za 23 - They allow for future changes as a result of gained information in prediction models and material and load characterisation; - They allow for performing system reliability analysis. In addition to the above advantages, reliability analysis provides a unifying framework for risk assessment across disciplines (especially structural and geotechnical design) and national boundaries. Traditionally geotechnical engineering has been a localised practice based on the excuse that geotechnical materials differ from one region to the other. However, globalisation dictates harmonisation of technical rules for design of buildings and civil engineering works. The harmonisation of design rules for all structures dictates that geotechnical design being based on the same limit state design as for structural design involving other materials. This need has led to what is termed as basis of design in the Eurocodes (EN 1990, 2002) and the revised South African Loading code (SANS 10160- Draft). 2.3 Basis of geotechnical limit state design to SANS 10160 Basis of design provides a common basis and general principle for the design of building and civil engineering works within the limit state design framework. In general, common rules are required for performance requirements, specification of the limit states, design situations to be checked, reliability requirements, and treatment of basic variables (actions, materials properties, and geometric data). This section presents the basis of design pertaining to geotechnical limit state design in the context of EN 1990, EN 1997-1, JCSS model code, ISO 2394 and SANS 10160-Draft. 2.3.1 Fundamental requirements The fundamental requirements for all structures are outlined in EN 1990, JCSS model code, ISO 2394 and SANS 10160-Draft. In accordance with these references, a structure is required to fulfil the following requirements: ? Remain fit for the use for which they are required (serviceability limit state requirements); ? Withstand actions and influences occurring during their construction and anticipated use (ultimate limit state requirements); Stellenbosch University http://scholar.sun.ac.za 24 ? Shall not be damaged by accidental events like fire, explosions, impact, or consequences of errors to an extent disproportionate to the original cause (robustness requirements). These requirements can be achieved through the choice of suitable geotechnical parameters, use of appropriate calculation models and by specifying control procedures for design, production, execution, and use relevant to the particular project. 2.3.2 Design situations Variations of material properties, environmental influences and actions with time can be accommodated by the selection of an appropriate design situation. The design situations are generally classified as (JCSS model code, 2001): ? Persistent situations, which refer to conditions of normal use of the structure and are generally related to the working life of the structure; ? Transient situations, which refer to temporally conditions of the structure, in terms of its use or exposure; ? Accidental situations, which refer to exceptional conditions of the structure or its exposure. In establishing the appropriate design situation for geotechnical design, EN 1997-1 gives a comprehensive list of factors to be taken into account as follows: - the actions, their combinations and load cases; - the general suitability of the ground on which the structure is located with respect to overall stability and ground movements; - the disposition and classification of the various zones of soil, rock and elements of construction, which are involved in any calculation model; - dipping bedding planes; - mine workings, caves or other underground structures; In the case of structures resting on or near rock: - interbedded hard and soft strata; - faults, joints and fissures; - possible instability of rock blocks; Stellenbosch University http://scholar.sun.ac.za 25 - solution cavities, such as swallow holes or fissures filled with soft material, and continuing solution processes; The environment within which the design is set, including the following; - effects of scour, erosion and excavation, leading to changes in the geometry of the ground surface; - effects of chemical corrosion; - effects of weathering; - effects of freezing; - effects of long duration droughts; - variations in ground-water levels, including, e.g. the effects of dewatering, possible flooding, failure of drainage systems, water exploitation; - the presence of gases emerging from the ground; - other effects of time and environment on the strength and other properties of materials: e.g. the effect of holes created by animal activities; - earthquakes; - ground movements caused by subsidence due to mining or other activities; - the sensitivity of the structure to deformations; - the effect of the new structure on existing structures, services and the local environment. To establish the above factors, detailed soil exploration is required. In order to establish minimum requirements for the extent and content of geotechnical investigations, calculations and construction control checks, the complexity of each geotechnical design need to be identified together with the associated risks. In this regard EN 1997 specifies three categories of complexity as follows: ? Geotechnical Category 1, which includes small and relatively simple structures for which it is possible to ensure that the fundamental requirements will be satisfied on the basis of experience and qualitative geotechnical investigations. Generally for this category, risk is considered negligible; ? Geotechnical Category 2, which include conventional types of structure and foundation with no exceptional risk or difficult soil or loading conditions. Design of Stellenbosch University http://scholar.sun.ac.za 26 structures within this category requires quantitative geotechnical data and analysis to ensure that the fundamental requirements are satisfied; ? Geotechnical Category 3, which includes structures or parts of structures, which fall outside the limits of Geotechnical Categories 1 and 2. Geotechnical Category 3 structures require alternative provisions and rules. 2.3.3 Limit states Generally limit states are classified into ultimate limit states and serviceability limit states. Depending on the nature of the geotechnical structure at hand five ultimate limit states are recognised in SANS 10160-Draft and EN 1997-1. These include: a) Loss of equilibrium of the structure or the ground, considered as a rigid body, in which the strengths of structural materials and the ground are insignificant in providing the resistance (EQU); b) Internal failure or excessive deformation of the structure or structural element including footings, piles and basement walls in which the strength of the structural materials is significant in providing resistance (STR); c) Failure or excessive deformation of the ground, in which the strength of the soil is significant in providing resistance (GEO); d) Loss of equilibrium of the structure or the ground due to uplift by water pressure or other vertical actions (UPL); e) Hydraulic heave, internal erosion and piping in the ground caused by hydraulic gradients (HYD). In practice, experience will often show which type of limit state will govern the design and the avoidance of other limit states may be verified by a control check. 2.3.4 Design methods Consistent with the practice world wide, four fundamental ways of carrying out geotechnical design are recognised in EN 1997. ? Using calculations based on analytical , semi-empirical or numerical models; Stellenbosch University http://scholar.sun.ac.za 27 ? Adopting prescriptive measures involving conventional and generally conservative procedures; ? Using experimental models and load tests carried out on a sample of construction; ? Using the observational methods, in which the design is continuously reviewed during construction. For category 2 structures (i.e. commonly encountered structures) general designs are carried out by the calculations approach. Further more, design by calculation is amenable to harmonisation of basic rules with that for design of structures involving other materials, hence it is further explored in the subsequent subsections. 2.3.5 Design by calculations Consistent with structural design, geotechnical limit state design calculation involves: ? Considerations of actions (i.e. load types, load combinations and load factors); ? Geotechnical materials properties; ? Geometric data; ? Calculation models. These components are generally referred to as the basic variables. Uncertainties in the basic variables propagate through the rest of the calculation and eventually affect the performance. In the spirit of harmonisation of design rules, a standardised approach to characterisation of the basic variables in terms of characteristic values and design values have been adopted in ISO 2394, Eurocodes and SANS 10160-Draft. 2.3.5.1 Load types, load combinations and load factors The design of any structure requires the understanding of the types and magnitude of the loads that are expected to act on the structure during its life span. Although a major input in the design, loads (except geotechnical loads) required for geotechnical design are usually provided by the structural engineer. Although it is not within the scope of geotechnical design to develop loads and load combinations required, it is useful to understand the principles underlying the development of load combination schemes and the associated load Stellenbosch University http://scholar.sun.ac.za 28 factors. It is for this reason that this section is dedicated to description of load types, load factors and load combinations. 2.3.5.1.1 Load types Eurocode 7 provides a comprehensive list of sources of loads or actions to be considered for geotechnical design. The list is as follows: ? the weight of soil, rock and water; ? stresses in the ground; ? earth pressures and ground-water pressure; ? free water pressures, including wave pressures; ? ground-water pressures; ? seepage forces; ? dead and imposed loads from structures; ? surcharges; ? mooring forces; ? removal of load or excavation of ground; ? traffic loads; ? movements caused by mining or other caving or tunneling activities; ? swelling and shrinkage caused by vegetation, climate or moisture changes; ? movements due to creeping or sliding or settling ground masses; ? movements due to degradation, dispersion, decomposition, self-compaction and solution; ? movements and accelerations caused by earthquakes, explosions, vibrations and dynamic loads; ? temperature effects, including frost action; ? ice loading; ? imposed pre-stress in ground anchors or struts; A close scrutiny of the above list of possible actions reveals that they can be classified into two broad classes of: Stellenbosch University http://scholar.sun.ac.za 29 a) External load applied to the soil or to the geotechnical structure. This comprises of loads from the superstructure such as column load on a foundation; b) Loads originating from the soil itself. These are actions generated by the soil itself such as earth and water pressure and are generally known as geotechnical loads. 2.3.5.1.2 External actions In recognition that external actions are random variables in time, they are normally classified by their variation in time (EN 1990, SANS 10160) as follows: (i) Permanent actions: These are gravitational loads caused by the weight of structural materials such as concrete, steel, etc. In general, the magnitude of this class of actions usually does not change during the life of the geotechnical system. Also it can be estimated relatively accurately. Tang (1981) identified the following as possible sources of uncertainties associated with permanent loads: - Variability in materials density due to inhomogeneity and tolerance of manufacturers; - Discrepancy in dimensions from design values; - Uncertainty in the final choice of building materials; - Variability in the non-structural components such as decorative architectural forms. To account for these uncertainties, a coefficient of variation (COV) of 10% is usually adopted for the variation of permanent actions in design of superstructures. In foundation design, the total dead load is composed of the sum of contributions from many components of dead loads and therefore in accordance with the law of large numbers, the variability in the total dead load will be less than the 10% values suggested for the superstructure (Tang, 1981). (ii) Variable actions: These are loads that vary with time. They can be further classified into sustained live loads and transient life load. Sustained life loads represents weight of people and their possessions, furniture, movable partitions and other portable fixtures and equipment (Nowak and Collins, 2000). These are basically imposed loads on building floors, beams and roofs. The magnitude depends on the Stellenbosch University http://scholar.sun.ac.za 30 type of occupancy. The occupancy classes and their respective load magnitudes are given in most national loading codes including SANS 10160. Surveys to obtain statistical data on the sustained live load have been carried out by many researchers (e.g. Corotis and Doshi, 1977; Ellingwood et.al 1980). The results of the surveys indicate that sustained live load can be modelled as a gamma distributed random variable. Transient live loads or extraordinary loads on the other hand include forces due to wind, snow, storm waves, and anticipated rare events, such as concentrations of persons or of furniture, or the moving or stacking of objects which may occur during reorganisation or redecoration (Tang 1981). The major characteristics of this type of load are: - For most of the time they do not occur - When they occur, the magnitude and the duration of the load are uncertain. For design purposes both the sustained and the transient loads need to be taken into account. Ellingwood et.al (1980) suggests that the total variable load is modeled by an Extreme value Type I distribution. (iii) Accidental actions: These haven been defined in EN 1990 as actions that are usually of short duration but of significant magnitude and are unlikely to occur on a given structure during the design working life. Typical examples include fire, explosions or impact loads. Further more, EN 1990 and SANS 10160 classify actions due to earthquakes as accidental actions. Although other classification schemes are available, the classification based on variation in time is more appropriate for the establishment of action combination schemes. 2.3.5.1.3 Characteristic and design values of external actions In accordance with SANS 10160 and the Eurocodes approach, all actions are introduced in design calculations through their characteristic values. Depending on the available data and experience, characteristic values may be specified as a mean, an upper or lower value of a Stellenbosch University http://scholar.sun.ac.za 31 nominal value. Table 2.7 presents the description and statistical meaning of actions in accordance with SANS 10160. Further more, for variable actions, several independent variable actions may act on the structure. However it is unlikely that all the actions will reach their maximum values in a given reference period simultaneously. To take account of this, a combination factor is applied to the accompanying variable actions. In SANS 10160, a single combination value is specified for each variable action, as opposed to the Eurocode scheme of specifying the three values of combination, frequent and quasi-permanent values. For accidental actions, SANS 10160 and EN 1990 state that the design value Ad should be specified for individual projects. Table 2-2: Definitions of characteristic values TYPE OF LOAD CHARACTERISTIC VALUE DESCRIPTION STATISTICAL DEFINITION PERMANENT Small variability (cov < 0.1) High variability (cov > 0.1) Mean value Two values Gk = 5% fractile Gk = 95% fractile VARIABLE Upper value with intended probability not to be exceeded Nominal 98% fractile based on a reference one year period (i.e. expected maximum over 50 years design life) No reference to statistical distribution 2.3.5.1.4 Geotechnical actions In addition to the permanent and variable actions described above, substructures such as foundations and retaining walls are subjected to geotechnical loads. Depending of the nature of the geotechnical structure, geotechnical loads may consist of some of the following action types: - Vertical earth loading - Earth pressure - Water pressure (groundwater and free water) - Actions due to ground displacement - Earth surcharge loading Stellenbosch University http://scholar.sun.ac.za 32 Vertical earth loading: Vertical earth loading comprises the weight of backfilling material (i.e. overburden pressure). Overburden pressure is classified as permanent load. It is calculated on the basis of nominal dimensions shown on the drawing and the mean unit weight. Lateral earth pressure: The force effects of lateral earth pressure due to partial or full embedment into soil must be considered in the design of substructures. The magnitude of the lateral earth pressure on the structure is a function of: - Type of structure (e.g. gravity, cantilever anchored or mechanically stabilized earth wall or flexible or rigid buried structure); - Type of retained materials (unit weight and strength properties); - Magnitude and direction of lateral movement of the structure relative to the retained materials; - Compaction effort used during placement of soil fill; - Location of ground water table and the seepage forces within the retained materials; - Presence of surcharge loads on the retained soil mass. Nonetheless, the stiffness of the structure and the type of the retained materials are the most significant factors governing the distribution of the developed earth pressure. Structures that can move laterally away from the retained earth mobilise an active state stress in the retained soil mass. Consequently such structures should be designed using an active earth pressure distribution. Conversely structures which for one reason or the other are forced to move laterally towards the retained soil mass mobilise passive earth pressure and should therefore designed to resist the passive earth pressure. When no movement of the structure relative to retained soil mass is anticipated, the pressure is calculated from the at-rest state of stress. Surcharge load: A surcharge load is any load imposed upon the surface of a retained soil mass. Consequently it causes a lateral pressure to act on the retaining structure in addition to the basic earth pressure. Examples of surcharge loads include materials stockpiles or construction machinery, traffic loads on adjacent streets, spoil embankments adjacent to the structure, loads from adjacent structures, and rail traffic. Given the wide range of sources of surcharge loads, some will fall under permanent while others under variable action. Stellenbosch University http://scholar.sun.ac.za 33 Generally the loads are transmitted through the soil to the substructure. The magnitude of the additional lateral earth pressure due to surcharge load is governed by the type (i.e. point or uniformly distributed), the shear strength of the retained soil mass, the stiffness of the retaining structure relative to lateral displacement, and the magnitude and proximity of the surcharge load to the structure. The distribution of lateral pressure due to surcharge loads is usually estimated using Boussinesq?s theory. Water pressure: Forces on structures due to water action include static pressure, buoyancy and stream pressure. Static water pressure needs to be considered when differential water loads develop on a structure. Consideration for buoyancy is critical when the structure is constructed below the water table (e.g. a spread footing or pile cap located below the water table). Effects of stream pressure including floating debris, waves and stream currents and scour. The main parameters in the calculation of the water pressure are the unit weight of water and the level of the water (water table or free water level). While the unit weight is well known, the water level poses the greatest uncertainties in determining the characteristic or design water pressure. This is due to the fact that the water level is dictated by: - Seasonal changes - Rainstorms - Dewatering activities in the neighbourhoods If piezometric measurements are available, long-term predictions could be made from the recorded seasonal cycles and short-term fluctuations caused by rainstorms. Eurocode 7 concurs with this approach and states that the geometric data which determine the regime of free water or ground water shall be made on the basis of locally available data for the hydraulic and hydrogeological conditions at the specific site. Bauduin (1998) further elaborates this clause by suggesting the following approach to the selection of the design water level when reliable measurements of water levels are available: - select the characteristic water level on the base of a high water level or low water level fractile based on a reference period of one year, - establish the design value by adding or subtracting a certain value on the characteristic levels. Stellenbosch University http://scholar.sun.ac.za 34 If piezometric data are not available, Tang (1981) suggests the use of hydrological models considering precipitations and infiltration rates to predict water levels. Bauduin (1998) on the other hand suggests that, when only a few measurements are available, the design value may be found by adding or subtracting a certain value to the most unfavourable known or measured water level. Actions due to ground displacement: The ground may be subjected to displacements as a result of a number of natural processes such as consolidation, swelling, subsidence of adjacent deep excavations or mines, creeping of the soil, land slides and earthquakes. Structures which were initially in equilibrium with the supporting strata are required to react to the imposition of additional actions due to ground movement. In the process of reacting to the additional forces, internal stresses develop in structural components leading to tilting and rotation of the entire structure. Burland et al (1978) identified four possible consequences resulting from ground movement. These are: (i) Impair visual appearance of the structure as a whole: This is usually in the form of tilting of walls, floors and the whole building. A structure in such a condition could be unpleasant or even alarming for the occupants and visitors; (ii) Impair visual appearance of architectural materials: Visible damage such as cracks and deflection of cladding is unsightly; (iii) Affect serviceability or function of the structure: Structural damage from ground displacement can affect the overall efficiency of the building such as reduced weather-tightness, rain penetration, dampness, draughts, heat loss, reduced sound installation, windows and doors sticking. Ground movement can also affect the basic function of the structure such as the operation of lifts or precision machinery, access ramps and fracturing of service pipes; (iv) Stability problems: In addition to the violation of serviceability limit states presented above, large ground displacement can lead to violation of ultimate limit state. This manifest itself in severe damage to the cladding and fittings and eventually to the collapse of the structure. Depending on the nature of the structure and the soil type, principal actions imposed on the structure by ground movements include: down-drag, heave, and transverse loading. These Stellenbosch University http://scholar.sun.ac.za 35 forces exclude forces and associated damages stemming from landslides, earthquakes and mining subsidence as they are generally regarded as accidental actions. Down-drag acts as a permanent additional axial load on a pile and if the force is of sufficient magnitude, bearing capacity failure at the base is possible. This is a downward drag force on the pile by the soil surrounding (Das, 2003). Down-drag generally occurs when a hard layer is overlain by a consolidating stratum. Typical conditions include: - When a fill of clay soil is placed over granular soil layer into which a pile is driven. Under this condition the consolidation process of the fill will produce a downward drag force on the pile; - When a fill of granular material is placed over a layer of soft clay. In this case the fill will induce the process of consolidation in the clay layer, thereby producing a down-drag force on the pile; - Lowering of ground water table. This increases vertical effective stress on the soil and thereby causing consolidation in the soil with the resultant downward drag forces being developed on the pile. In contrast to down-drag, heave is an upward force which can be generated beneath shallow foundations and along the pile shaft. It is essentially caused by the expansion of the ground. Typical conditions that may result in the expansion of the ground include: - When clay is able to absorb more water than it had hitherto. Under these circumstances, the clay expands and thereby producing an upward force on affected structures. The increase in water content may result from the removal of trees, cessation of abstraction from aquifers, prevention of evaporation and from accidents (EN 1997); - Reduced pressure on over-consolidated clay. As a result of reduced pressure the clay tries to expand vertically to be restored back to what it was prior to being compressed in the past. Reduced pressure is associated with excavation and unloading activities; - Heave may also be caused by frost action and driving of adjacent piles. Both down-drag and heave are a result of vertical ground movement. Horizontal ground movements are also possible in practice. Structures that are likely to be subjected to horizontal loads due to ground displacement are those constructed in areas of mining subsidence, above tunnels, adjacent to deep excavation and in hillsides. With respect to pile Stellenbosch University http://scholar.sun.ac.za 36 foundations, Eurocode 7 lists the following design situations under which horizontal loads need to be considered: - Different amounts of surcharges on either side of a pile foundation; - Different levels of excavation on either side of the pile foundation; - The pile foundation is located at the edge of an embankment; - The pile foundation is constructed in a creeping slope; - Inclined or battered piles in settling ground; - Piles in seismic regions. 2.3.5.1.5 Characteristic values of geotechnical actions No specific information on geotechnical actions is provided in EN 1990 and reference is made to EN 1997 for guidelines. However, EN 1997 does not provide general guidelines for treatment of geotechnical actions. In fact EN 1990 states the characteristic values of actions (all actions including geotechnical actions) should be derived in accordance with EN 1990. In SANS 10160-Draft, there is provision for a separate treatment of geotechnical actions based on EN 1997. Table 2.3 summarises the treatment of characteristic values of geotechnical actions. Table 2-3: Characteristic values of geotechnical actions Action Type Characteristic value description Vertical earth loading Permanent Mean value calculated on the basis of nominal dimensions and mean unit masses Lateral earth pressure Permanent Resultant (mean) Water pressure - Permanent (ground water pressure) Resultant/mean - Variable (free water above ground) Accidental (floods) N/A Surchage load -Permanent Resultant force -Variable Ground displacement - Down-drag Permanent Mean -Heave Permanent Mean - Transverse loading Permanent Mean - Subsidence Accidental N/A - Earthquake Accidental N/A Stellenbosch University http://scholar.sun.ac.za 37 2.3.5.1.6 Load statistics The calibration of load and resistance factors requires statistical data (bias factors and coefficient of variations) for load and resistance. The collection of this data is a significant effort in the calibration process. In principle, the bias factors and associated coefficients of variation for load can be established from physical measurements or data provided by professional publications and engineering reports. In this study, work by Kemp et al (1987) serves as the main reference on load statistics assumed in the South African loading code Table 2.4 presents the load statistics compiled by Kemp et al (1987). It is evident from table 2.4 that the load statistics assumed by the South African loading code (SABS 0160) are comparable to load statistics in ANSI A58 and Australian loading codes. Accordingly the load statistics suggested by SABS 0160 were adopted for this study. Table 2-4: Load statistics (Kemp et al, 1987) Type of load Code Mean load / Nominal load Coefficient of variation Type of distribution Dead (permanent) load ANSI A58 Australian SABS 0160 1.05 1.05 1.05 0.10 0.10 0.10 Normal Lognormal lognormal Live (office): Lifetime max. ANSI A58 Australian SABS 0160 1.0 0.7 0.96 0.25 0.26 0.25 Type 1 Type 1 Type 1 Live (office): Point in time ANSI A58 Australian SABS 0160 0.25 0.19 0.25 0.71 0.79 0.60 Gamma Weibull Gamma Wind: Lifetime max. ANSI A58 Australian SABS 0160 0.78 0.30 0.41 0.37 0.43 0.52 Type 1 lognormal Type 1 Wind: Point in time ANSI A58 Australian SABS 0160 0.0097 0.022 0.05 6.95 0.94 1.08 Type 1 Weibull Weibull 2.3.5.1.7 Fundamental load combinations and load factors In accordance with EN 1990, the fundamental combinations for ultimate limit states verification are as follows: ? Persistent and transient design situations Stellenbosch University http://scholar.sun.ac.za 38 ?? ++ ? 1 ,0,1,1, 1 ,, """" fi ikiiQkQ j jkjG QQG ???? [2.4] ?? ++ ? 1 ,,0,1,1,01, 1 ,, """" fi ikiiQkQ j jkjG QQG ????? [2.5] ?? ++ ? 1 ,,0,1,1, 1 ,, """" fi ikiiQkQ j jkjGj QQG ????? [2.6] Where: ?+? implies ?to be combined with? ? implies ?the combined effect of? jG ,? is the partial factor for the permanent action j; jkG , is the characteristic value of permanent action j; 1,Q? is the partial factor for the leading variable action; 1,kQ is the characteristic value of the leading variable action; iQ,? is the partial factor for the accompanying variable action i; ikQ , is the characteristic value of the accompanying variable action i; i? is the action combination factor corresponding to the accompanying variable action i; ? is a reduction factor for unfavourable permanent actions G. In addition, EN 1990 allows for the modification of Eq. 2.5 to cater for permanent actions only, also applied in the present South African loading code (SABS 0160). The resulting expression is given by: kGd GE ?= [2.6] Eq. 2.4 is the single expression combination scheme presented in EN 1990 as Expression 6.10 while Eq. 2.5 and Eq. 2.6 constitute the alternative dual expression combination scheme (EN 1990 Expressions 6.10 a & b). For geotechnical design, expression 6.10 (Eq. 2.4) has been adopted. Accordingly EN 1997-1 only provides recommended values of partial factors corresponding to equation 6.10 (table A.3 of Annex A). This derives from the fact that the recommended geotechnical values come from a few calibration studies performed using values of expression 6.10, while on the other hand there is no experience Stellenbosch University http://scholar.sun.ac.za 39 on the use of expressions 6.10a and 6.10b in geotechnical engineering (Frank, 2005). Likewise, SANS 10160 has adopted EN 1990?s expression 6.10 for geotechnical design. Partial factors to be used with expression 6.10 vary with the specific ultimate limit state. Generally three set of partial factors to be applied to the characteristic values of actions are introduced in SANS 10160 (Set A, Set B and Set C): ? Set A is used to verify EQU limit state ? Set B is for verification of STR limit state ? Set C is for verification of GEO limit state Table 2.5 presents the recommended values of partial factors for Sets A, B and C extracted from SANS 10160. Table 2-5: Values of action partial factors for Set A, B and C Action type Value of partial factors Set A (EQU) Set B (STR) Set C (GEO) Permanent - unfavourable 1.2 1.2 1.0 - favourable 0.9 0.9 1.0 Variable - unfavourable 1.6 1.6 1.3 - favourable 0.0 0.0 0.0 For the GEO limit state, the values of the partial factors are the same as those prescribed in EN 1997-1. 2.3.5.2 Geotechnical materials properties Properties of materials including soils are described by measurable physical quantities corresponding to the properties considered in the calculation model (ISO 2394). Further more, EN 1990 requires that properties of materials including geotechnical materials should be represented by characteristic values. Unlike the properties of other structural materials, soil properties are not specified but determined by testing on a site specific basis. With all the uncertainties discussed earlier, determination of characteristic values of geotechnical parameters from available measurements is extremely difficult. In fact, the assessment of subsoil properties and conditions from available laboratory and field measurements is an old Stellenbosch University http://scholar.sun.ac.za 40 problem that has continuously plagued the geotechnical profession (Tabba and Young, 1981). In acknowledging this problem Kulhawy (2004) states: The geotechnical parallel to the mythological quest for the Holy Grail is the search for a means by which geotechnical properties of a soil or rock may be determined straightforwardly and reliably from relatively simple in-situ tests. 2.3.5.2.1 Determination of characteristic values of geotechnical parameters Sources of geotechnical parameter prediction uncertainties have already been mentioned as inherent spatial variability, measurement noise/random errors, measurements/model bias, and statistical uncertainties. In recognition of these uncertainties, the concept of characteristic/nominal values has been introduced into the geotechnical design process. Uncertainties in the prediction of in-situ soil parameters are accounted for implicitly at the characteristic value selection stage. Within the current format of limit state design approach, partial safety factors are applied to the characteristic values of the design variables (loads and material properties) to obtain the respective design values. With this approach, the safety level achieved depends not only on the partial safety factor values specified by the code but also on the way the characteristic values are obtained (Orr, 2002). Therefore a clear definition and rational methodology for its selection are essential. However, geotechnical design codes give little guidance as to how such values should be determined (Borden, 1981; Hicks and Samy, 2002). In the absence of well defined guidelines, the procedure for the determination of geotechnical characteristic values has become an empirical undertaking left to the designer?s discretion. Cardoso and Fernandes (2001) argue that it is not logical to apply partial factors of safety to poorly defined characteristic values. But what exactly is this characteristic value? The characteristic value has been defined as the value assigned to a basic variable associated with a prescribed probability of not being violated by unfavourable values during some reference period (EN 1990). In essence, this definition implies a fractile value of the measured data, usually a 5% fractile. This is applicable to structural materials such as concrete and steel as in these materials failure is governed by weaker localised portions. However in geotechnical design, failure is governed by average soil strength along the failure plain and therefore defining the characteristic value as a 5% fractile is generally not acceptable in geotechnical engineering. Stellenbosch University http://scholar.sun.ac.za 41 A definition of the characteristic value that seems to be more representative of the geotechnical design environment is provided in Eurocode 7. Eurocode 7 defines the characteristic value is a ?cautious estimate of the value of the parameter governing the limit state?. Depending on the extent of the zone of ground governing the behaviour of the geotechnical structure at the limit state being considered, the governing parameter can either be (i) the mean value over the affected surface or volume or (ii) the lower bound value. A value close to the true mean governs the limit state when: ? A large volume within the homogenous layers is involved, allowing for compensation of weaker areas by stronger areas, ? The structure carried by the soil allows transfer of forces from weaker foundation points to stronger points. Under these circumstances, the characteristic value of a geotechnical parameter is to be regarded as a cautious estimate of the mean value. A lower value close to the lower bound values of the soil parameter governs the limit state when: ? A small soil volume is involved, which does not allow for compensation, ? The structure carried by the soil can not resist local failure. Under these circumstances, the characteristic value of a geotechnical parameter is to be regarded as an estimate of the lower bound value. In most geotechnical design situations, the zone of ground governing the behaviour of a geotechnical structure is usually much larger and therefore geotechnical performance is often governed by the spatial average of soil properties, such as average compressibility of a volume of soil beneath the footing, or the average shear strength along a potential failure surface. Hence the characteristic value of geotechnical parameters should be the cautious or conservative estimate of the mean value. The use of adjectives ?cautious or conservative? is to reflect the fact that the true mean is not known and therefore it can only be estimated on the basis of the available test data with unavoidable uncertainties. Stellenbosch University http://scholar.sun.ac.za 42 Even though the definition seems to be suitable, detailed procedure on how to determine the characteristic value has not been provided. Eurocode 7 mentions in general terms various methods for deriving characteristic values. The methods suggested by Eurocode 7 are shown in figure 2.1. Even though the sources of the uncertainties are known, they are accounted for in an implicit manner. Since each of the various methods employed to derive the characteristic value accounts for uncertainties differently, it is not surprising that each method yields different results. A number of research works have been reported in the geotechnical literature where experienced geotechnical engineers were given the same raw data to evaluate the characteristic value (e.g. Fellin, 2004; Ovesen, 1995). In all the reported cases, totally different answers were given to the same problem. For example, Ovesen (1995) reports of an experiment where twenty five Eurocode 7 committee members were asked to determine the characteristic value for the ultimate state for the following ten test results of a certain soil; 138,140,170,171,179,179,182,232,258,272 kN/m2. The determined characteristic values ranged from 145 to 200 kN/m2. This indicates that given the same tests results, different engineers with varying background and experience would come to completely different results. Derived values of geotechnical parameters Figure 2.1: Various methods for determining the characteristic value 2.3.5.2.2 Statistical approach to characteristic geotechnical properties Despite the initial criticism to the application of statistics to the geotechnical practice, in recent years its use has gained popularity. A recent survey conducted by Orr et al (2002) regarding the usefulness of statistical approach, it was found out that 64% of the Derived values of geotechnical parameters Experience Statistical methods Cautious methods Standard tables Characteristic value Stellenbosch University http://scholar.sun.ac.za 43 respondents accept the effectiveness of statistical approach in determining characteristic values. The outcome of a similar questionnaire administered in Japan by Shiroto et al (2002) also indicated that over 50% of the respondents were in favour of the use of statistical approach. Those in favour of the use of statistical approach mentioned objectiveness and transparency as the key advantages. Further comments in favour of statistical approach were as follows; ? Personal differences in determining the parameter values can be eliminated; ? It is possible to avoid biased judgement; ? It is an excellent tool in terms of accountability and objectivity to explain to a third party the basis for selecting a particular value. In any case, using statistical methods as the basis for determining characteristic values structures the decision making process and makes it clearer and more exchangeable between different engineers (Fellin, 2004). Lump (1974) adds that: ?Statistical methods can be of great value to the designer since it is possible to express many of the decision uncertainties in terms of numerical probabilities, thus allowing quantification of judgement to some extent and clarification of the problems.? As shown in figure 2.1, statistical approach is one of the methods for determining characteristic values permitted by Eurocode 7. In this regard Eurocode 7 further states that if statistical methods are used, the characteristic value should be derived such that the calculated probability of a worse value governing the occurrence of a limit state is not greater than 5%. The statement is further interpreted to mean that: ? In the case that the characteristic value is a cautious estimate of the mean value, then the calculated characteristic value is an estimate of the true mean such that there is a probability of 95% that the true mean value is higher than the calculated value. In this respect the characteristic value is the lower bound of the 95% confidence interval of the mean value; ? In the case that the characteristic value is a cautious estimate of the lower bound value, it should be taken as the 5% fractile of the distribution of the parameters i.e. the value for which there is only 5% chance that a lower value may be found. Stellenbosch University http://scholar.sun.ac.za 44 2.3.5.2.3 Characteristic value as mean value at 95 % confidence level In statistical terms, a mean value at a given confidence level is an interval estimate for the mean. Interval estimate for the mean can be defined as the process of utilising the available observations to calculate two numbers (the lower and upper limits) that define an interval that will enclose the true mean with a high degree of confidence. The resulting interval is termed a confidence interval and the probability that it contains the true mean is the confidence level or confidence coefficient. The confidence level is expressed by; (1-?) 100%, where ? is level of significance. For the 95% confidence level, (1-?) 100% = 95%. Although the choice of confidence level is arbitrary, Eurocode 7 specifies a 95 % confidence interval. The theoretical interpretation of the 95% confidence level is that, if a sample size n were to be repeatedly collected from the population and a 95% confidence interval computed for each sample, then 95 % of the intervals will enclose the true parameter value. In practice, only one sample of size n is collected from the population and therefore a 95 % mean confidence interval is assumed to imply that the calculated interval will enclose the true mean with a probability of 95%. Eurocode 7 applies this practical interpretation of 95% confidence interval to the statistical determination of characteristic values of geotechnical properties. Accordingly the characteristic value is the lower limit of the 95% confidence interval. This lower limit value has a 95 % probability that the true mean is equal or higher than it. The basic equation for the characteristic value as mean value at 95% confidence level is given by (Bauduin, 1998): ??? ? ??? ? ?= n VZXX k 1 1 2/? [2.7] Where; Xk is the characteristic value, X is the arithmetic mean of the test results, V is the coefficient of variation of the desired property, n is the number of test results, and Z?/2 is the standard normal variate that locates an area of ?/2 to its left. Stellenbosch University http://scholar.sun.ac.za 45 Equation 2.7 is based on the central limit theorem, which states that for a large sample size (n > 30), the mean is approximately normal irrespective of the distribution assumed. However, since the central limit theorem applies to large samples only, for small sample size the assumption that the sampling distribution of the mean is approximately normal no longer holds. Therefore the sample standard deviation may not be a satisfactory approximation to population standard deviation if the sample size is small. For small sample size as is the case in the geotechnical practice, the student distribution is recommended. The general expression for the lower limit (characteristic value) of a confidence interval for the mean of a small sample, based on the student?s t-distribution is given by: ??? ? ??? ? ?= ? nVtXX nk 1 1 95.0 1 [2.8] Where; Xk is the characteristic value, X is the arithmetic mean of the test results, V is the coefficient of variation of the desired property, n is the number of test results, and t is the value of the student distribution corresponding to a confidence level of 95% and a degree of freedom of n-1. Equation 2.8 is the basic equation for determining characteristic values for geotechnical properties. When prior information about the coefficient of variation of the desired property is available, V is regarded as known and a limiting value of t = 1.645 is used. 2.3.5.2.4 Characteristic value as a 5% fractile In the case that the characteristic value is a cautious estimate of the lower bound value, a 5% fractile is considered as the characteristic value as it is the case with structural materials. In general, the p fractile xp is the value of the variable X for which the probability that the variable X is less than or equal to xp is equal to p. Mathematically, p-fractile is expressed as; ( ) ( ) pxPxXP pp ==? [2.9] For characteristic properties of materials, EN 1990, set the probability p to 5%. This means there is only 5% chance that a lower value may be found. Several methods have been Stellenbosch University http://scholar.sun.ac.za 46 developed to estimate the p-fractile. The prediction method is amongst the common methods and it has been specifically mentioned in the geotechnical literature. With this method, the lower p-fractile is assed by the prediction limit xp, pred, determined in such a way that an additional value xn+1 randomly taken from the population would be expected to occur below xp, pred with the probability p. Thus, ( ) pxxP predpn =?+ ,1 [2.10] For the case of small sample size the prediction methods lead to the following expression for determining characteristic value as a 5% fractile. ??? ? ??? ? ?? ??? ? +?= ? 111 95.0 1 nVtXX nk [2.11] Where; X is the characteristic value, X is the arithmetic mean of the test results, V is the coefficient of variation of the desired property, n is the number of test results, and t is the value of the student distribution corresponding to a confidence level of 95% and a degree of freedom of n-1. From equations 2.8 and 2.11, it can be concluded that the characteristic value is a scaled mean value. For geotechnical properties, the value of the scaling factor depends on whether the characteristic values is considered as a 5% fractile or an estimate of the mean value at 95% confidence level. For the purposes of this study, the scaling factors are designated ?1 and ?2 for the 95% confidence level and 5% fractile respectively. These scaling factors are given by the following expressions; n Vtn 1 1 95.0 11 ??=? [2.12] ?? ??? ? +?= ? 111 95.0 12 nVtn? [2.13] From the expressions for the scaling factors, it is apparent that for a given value of V and n, ?2 gives a smaller factor compared to ?1. Therefore ?1 lead to a characteristic value that is Stellenbosch University http://scholar.sun.ac.za 47 closer to the mean compared as to ?2. To further illustrate this point, figure 2.2 shows the relative positions of the characteristic value as a 95% confidence level and the characteristic value as a 5% fractile on a probability density function. It is evident from figure 2.2 that the characteristic value as a 95% confidence level yields values that are close to the mean compared to the characteristic value as a 5% fractile. Despite the theoretical basis underlying the determination of characteristic values statistically, there is a danger of averaging data from different populations. This can be avoided by rationally dividing the profile into sub-regions, each of which is treated as an independent population prior to the application of statistical techniques. This is consistent with the general feeling in the geotechnical practice that statistics should only be used to help in the assessment of the soil or rock properties after the geological and geotechnical aspects of the problem have been examined. In other words statistics can not replace the importance of understanding the true underlying geology. Figure 2.2: Difference between characteristic values as a 5% fractile and mean estimate at 95% confidence level (After Bauduin, 1998) 2.3.5.3 Geometric data Stellenbosch University http://scholar.sun.ac.za 48 Geometric quantities describe the shape, size and overall arrangement of structures, structural elements and cross-sections (ISO 2394). In geotechnical engineering this translates to the level and slope of the ground surface, water levels, levels of interfaces between strata, excavation levels and the dimensions of the geotechnical structure. The variability of geometric quantities is generally small or negligible compared to that of materials properties and actions, hence characteristic values of geometric quantities are generally taken as specified on drawings or measured on site. 2.3.6 Design approaches for STR/GEO limit state There has not been a consensus worldwide regarding where in the calculation process partial factors should be applied (Driscoll and Simpson, 2001, Simpson 2000). The main alternatives include: ? Factors be applied to the primary variables (actions and material properties); ? Factors be applied to action effects and resistances; ? Factors be applied to action and resistance models. To accommodate the divergent opinions and practices, Eurocode 7 provides three design approaches for design of substructures or structural members involving geotechnical actions. These are: (a) Design Approach 1, (b) Design Approach 2, and (c) Design Approach 3. The attributes of the three design approaches are detailed in Eurocode 7 and have been discussed by many commentators (e.g. Driscoll and Simpson, 2001; Orr, 2002, 2006; Frank, 2002, Schuppener and Frank, 2006). These three design approaches differ in the way they distribute the partial factors on actions and ground resistances/properties. With regard to actions, there are two sets of partial factors: ? Set A1 in which the factors are greater than one for both unfavourable permanent and variable actions. ? Set A2 in which the factors are equal to unity except for unfavourable variable actions. Material factors also comprises of two sets; M1 and M2. Stellenbosch University http://scholar.sun.ac.za 49 ? Set M1 in which geotechnical properties are unfactored (i.e. partial factors are equal to one). ? Set M2 in which the partial factors on geotechnical properties are greater than one except for unit weight. There are also 4 sets of resistance factors as follows: R1 applicable to Design Approach 1, R2 applicable to Design Approach 2, R3 applicable to Design Approach 3 and R4 applicable to Design Approach 1 for design of piles and anchors. The numerical values of the various sets of partial factors are given in table 2.6: The main features of the three design approaches are outlined below. Design Approach 1: Except for the design of piles and anchors, two combinations of partial factors need to be investigated. Combination 1 aims to provide safe designs against unfavourable deviation of the actions from their characteristic values (Schuppener and Frank, 2006). Hence, partial factors greater than one (Set A1) are applied to the permanent and variable actions from the structure and the ground. With regard to ground resistance, calculations are performed with characteristic values of soil properties (i.e. unfactored parameters). Accordingly partial materials factors, ?? , c? , and cu? are all set to unity (Set M1). Further more, the calculated resistance is also not factored (i.e. Set R1). Conversely, Combination 2 aims to provide safe design against unfavourable deviations of the ground strength properties from their characteristic values and against uncertainties in the calculation model (Schuppener and Frank, 2006). Hence, geotechnical parameters are factored for the calculation of geotechnical actions and for the calculation of resistances. Regarding action from the structure, permanent actions are assumed to be at their characteristic values while the variable actions are assumed to deviate only slightly from their characteristic values and hence a relatively smaller factor is applied. Partial factors Sets A2, M2 and R1 are applicable for this combination. When it is obvious that one of the combinations governs the design, it is not necessary to carry out calculations for the other combination. Accordingly it has become customary that: Combination 1 governs structural design of the elements while Combination 2 governs the sizing of the elements. Stellenbosch University http://scholar.sun.ac.za 50 For the design of axially loaded piles and anchors, again two combinations need to be considered: Combination 1: A1 + M1 + R1 Combination 2: A2 + (M1* or M2**) + R4 in which * applies when calculating the resistance of piles and anchors, ** applies when calculating unfavourable actions on piles owing to negative skin friction or lateral loading. Design Approach 2: In this design approach, partial factors of greater than unity are applied to actions and resistances while the partial factors on soil parameters are set to unity. Thus design approach provides essentially a resistance factor approach. The partial factors applied to the geotechnical actions and effects of actions are the same as those applied to the action on or from the structure. In this regard, partial factors for Sets A1, M1 and R2 are used with this design approach. Design Approach 3: Applies partial factors for Set A2 to geotechnical actions and simultaneously applying partial factors for Set A1 to the other actions on/from the structure. In calculating the ground resistance partial factors are applied to the ground strength parameters and not to the obtained resistance (i.e. Sets M2 and R3). 2.3.7 Selection of design approach for the study The selection of a particular design approach is a matter for national determination. Accordingly South Africa will have to select one of the three design approaches. As part of the current study, an investigation was carried out as to which of the three design approaches was suitable for the South African environment. The basis for deciding which of the three design approaches will be more suited to the South African design environment were as follows: (a) Design approach that can accommodate a wide spectrum of geotechnical design; situations (e.g. spread foundations, pile foundations, retaining walls, slope stability, etc); (b) Design procedure that is close to the current design; (c) Design approach that leads to safe but economic designs. Stellenbosch University http://scholar.sun.ac.za 51 Table 2-6: Numerical values of various sets of partial factors Design Approach 1 Design Approach 2 Design Approach 3 1. Partial factors for actions Any other structures Piles and Anchors Set Set Set Set A1 A2 A1 A2 A1 A1 A2 Permanent (unfavourable) 1.35 1.00 1.35 1.00 1.35 1.35 1.00 Variable (unfavourable) 1.50 1.30 1.50 1.30 1.50 1.50 1.30 Pamanent (favourable) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Variable (fafourable) 0 0 0 0 0 0 0 2. Partial material factors Set Set Set Set M1 M2 M1 M1* or M2** M1 M2 Shearing resistence 1.0 1.25 1.0 1.0 1.25 Effective cohesion 1.0 1.25 1.0 1.0 1.25 Undrained strength 1.0 1.40 1.0 1.0 1.40 Unconfined strength 1.0 1.40 1.0 1.0 1.40 Weight density 1.0 1.00 1.0 1.0 1.00 3. Partial resistance factors Set Set Set Set R 1 R 1 R 1 R 4 R 2 R 3 Spread foundations Bearing resistance 1.00 1.00 1.40 1.00 Sliding resistance 1.00 1.00 1.10 1.00 Driven piles Base 1.00 1.30 1.10 1.00 Shaft (compression) 1.00 1.30 1.10 1.00 Total/combined (compression) 1.00 1.30 1.10 1.00 Shaft (tension) 1.25 1.60 1.15 1.10 Bored piles Base 1.25 1.60 1.10 1.00 Shaft (compression) 1.00 1.30 1.10 1.00 Total/combined (compression) 1.15 1.50 1.10 1.00 Shaft (tension) 1.25 1.60 1.15 1.10 CFA piles Base 1.25 1.45 1.10 1.00 Shaft (compression) 1.00 1.30 1.10 1.00 Total/combined (compression) 1.15 1.40 1.10 1.00 Shaft (tension) 1.25 1.60 1.15 1.10 Retaining structures Bearing capacity 1.00 1.00 1.40 1.00 Sliding resistance 1.00 1.00 1.10 1.00 Earth resistance 1.00 1.00 1.40 1.00 Stellenbosch University http://scholar.sun.ac.za 52 The three design approaches were compared on the basis of the above criteria. (a) Accommodation of a wide spectrum of geotechnical design situations Uncertainties are accounted for by the use of partial factors. During the process of drafting the Eurocode 7, it emerged that a partial factor should be applied to basic variables (material properties and actions), or to quantities derived later in the calculation (resistances and action effects) according to where its effect will be most severe. This leads to two broad geotechnical design situations. The first situation is where the greatest uncertainties are with the geotechnical parameters. Examples include slope stability problem, spread foundations, earth retaining structures, etc. Under this condition, it is logical to apply partials factors to the geotechnical properties. The second design situation is where the greatest uncertainty is with the calculation model. Specific examples for this category are pile foundations and anchors. When the predominant uncertainty is associated with the model, it is logical to apply the partial factor to the resistance. Only Design Approach 1 accommodates both material factor and resistance factor approaches depending on the nature of the geotechnical structure being designed. Design Approach 2 is a resistance factor approach and therefore does not represent uncertainties in material properties adequately. For example, bearing capacity increases more, in proportion, than the angle of shearing resistance from which it is calculated, so it is appropriate to apply the factor to the material property rather than the bearing resistance. From this perspective, Design Approach 2 is suited to design of pile foundations and anchors only. Design Approach 3 is a material property factor approach and therefore does not allow for uncertainty in the calculation model. Based on the above considerations, Design Approach 1 ranks higher than the other two design approaches. (b) Design procedure that is similar to the current design The choice between factored materials and factored strength has become a matter of convenience and familiarity. In concept, the resistance factor approach is similar to the global factor of safety used in the working stress design and therefore the method is simple and familiar to the South African geotechnical engineers. On this basis, Design Approach 1 and 2 are preferable to Design Approach 3. For pile foundations, the main difference Stellenbosch University http://scholar.sun.ac.za 53 between design approach 1 and Design Approach 2 is on the action side. For the geotechnical ultimate limit state, Design Approach 1 factor only variable actions while permanent actions are at their characteristic values. On the other hand, both variable and permanent actions are factored in Design Approach 2. Therefore Design Approach 1 is more representative of the geotechnical design tradition whereby unfactored load are used in the design. On this account, design approach 1 ranks above design approach 2. (c) Design approach that lead to safe but economic designs A preliminary assessment was carried on the basis of a few published examples and design examples carried out by the author. The design examples reviewed are for geotechnical structures within the scope of SANS 10160-Draft (foundations and earth retaining structures). (i) Spread foundations Orr and Farrell (2000), present an example of a pad foundation with a vertical load on cohesive soil. For this loading and ground conditions, the results were as follows; - With design approach 1, combination 2 controls the design. - Of the three design approaches, design approach 1 gives the smallest foundation size while design approach 3 gives the largest foundation size. In a presentation to the South African Loading committee, Holicky (2005) presented two examples of a footing on cohesive soil and another footing on non-cohesive soil. The results were as follows: - For the footing on cohesive soil, the trend is similar to that reported by Orr and Farrell above, - For the footing on non-cohesive soil, design approach 2 gives the smallest footing size while design approach 3 still gives the largest size. For design approach 1, still combination 2 controls the design. Stellenbosch University http://scholar.sun.ac.za 54 As part of an in-house report, the author carried out a design example of an axially load pad foundation on cohesionless soil. The results showed a similar trend to that reported by Holicky. Based on the above design examples, it can be concluded that, in cohesionless soil, Design Approach 2 gives the smallest footing size while Design Approach 3 leads to the largest size. However in cohesive soil, Design Approach 1 produces the smallest foundation and Design Approach 3 still produces the largest foundation size. In terms of safety, Design Approach 3 is the safest but the also the most uneconomic. Therefore Design Approaches 1 and 2 rank higher than Design Approach 3. (ii) Pile foundations Driscoll (2005) presented an example of a bored pile in clay. The design was carried out in accordance with the semi-empirical alternative for designing pile foundations on the basis of ground test results. For the design situation considered, it was observed that; - Combination 1 and 2 of Design Approach 1, gave very similar results for pile length. - Similar pile lengths were obtained in all the design approaches. As part of the present study, the design of a bored pile in cohesionless soil was carried out. The final results are presented in table 2.7. Table 2-7: Final results of a pile design example Example Parameter Design approach DA1.1 DA1.2 DA.2 DA.3 Pile foundation from soil parameter values L 24.0 24.7 22.5 26.7 A study of table 2.7 lead to the following observations: - As expected, for Design Approach 1, combination 2 controls the design length. However the two combinations produce close results as reported by Driscoll (2005); - For this design situation, design approach 2 gives the shortest pile length while design approach 3 gives the longest pile length; Stellenbosch University http://scholar.sun.ac.za 55 - The pile length for Design Approach 1 is equal to the average length for Design Approaches 2 and 3. In other words, Design Approach 1 falls midway between Design Approaches 2 and 3. Therefore in terms of safety and economy, it is the optimum design. The results seem to follow the trend depicted by shallow foundations (i.e. Design Approach 2 produces the smallest foundation size, Design Approach 3 yields the largest foundation size while Design Approach 1 produces an optimum design. Therefore Design Approach 1 ranks higher than the other two Design Approaches. (iii) Retaining walls Examples provided by Orr and Farrell (2000) indicated that the three Design Approaches produced approximately the same design (wall size). Therefore from safety and economic perspective, the three Design Approaches are comparable. From the comparison of the three Design Approaches under various criteria, Design Approach 1 ranked higher than the other two Design Approaches in two out of the three selection criteria. On the third criteria (safety and economic considerations), there was no strong evidence to suggest the one Design Approach was superior to others. However, a design example on pile foundation in cohesionless soil indicated that Design Approach 1 produces an optimum design. The results of the recent international workshop on the evaluation of Eurocode 7 confirm the foregoing observations. Prior to the workshop, ten geotechnical design examples involving five different areas were distributed. Orr (2005) provided model solutions to the various design examples. A summary of the results of the model solutions are presented in table 2.8. On the basis of the model solution, it was concluded that Design Approach 3 gave the most conservative designs, Design Approach 2 the least conservatism designs, and Design Approach 1 generally gave designs between the other two approaches. These results agree with the findings of the preceding preliminary investigation. Therefore for this study, Design Approach 1 was selected as the basis for calibrating the partial factors. Based on the same line of reasoning, SANS 10160 has also opted for Design Approach 1. Stellenbosch University http://scholar.sun.ac.za 56 Table 2-8: Summary of model solutions (After Orr, 2005) Example Parameter Design approach DA1.1 DA1.2 DA.2 DA.3 1. Spread foundation vertical central load B 1.62 2.08 1.87 2.29 2. Spread foundation inclined eccentric load B 3.46 3.98 3.77 4.23 3. Pile foundation from soil parameter values L 14.9 14.6 14 16.7 4. Pile foundation from load test results N 9 9 10 - 5. Gravity retaining walls B 3.85 5.03 4.21 5.03 6. Embedded retaining wall D 3.14 4.73 4.68 4.73 7. Anchored retaining wall D 2.6 3.64 3.67 3.64 8. Uplift T 0.60 9. Heave H 6.84 10. Embankment H 2.90 2.40 2.15 2.40 Stellenbosch University http://scholar.sun.ac.za 57 Chapter 3 RELIABILITY BASIS FOR GEOTECHNICAL LIMIT STATE DESIGN 3.1 Introduction Uncertainties associated with the geotechnical design process were pointed out in Chapter 1. To deal with such uncertainties, the geotechnical fraternity has developed several strategies. Traditionally a global factor of safety is applied to the resistance side of the equation to cater for all uncertainties. As pointed out earlier, this approach has several drawbacks including the fact that the level of safety achieved is not known. Because it is not known as to what uncertainties are accounted for, Seidel (2002) rightly referred to it as the factor of ignorance. Christian (2004) in the thirty-ninth Terzaghi lecture outlined the following additional approaches for dealing with geotechnical uncertainties: ? Ignoring it ? Being conservative ? Using the observational method ? Quantifying it Ignoring the uncertainties lead to baseless decisions with catastrophic consequences while being conservative, although guarantees safety, is usually uneconomical. The two design extremes are illustrated in figure 3.1 for the case of rock-bolting. Clearly these two approaches do not meet the fundamental design requirements of simultaneously achieving safety and economy. The observational method or ?learn-as you go? is suitable for large projects with complex ground conditions necessitating contract documents tailored for the specific project. The approach is closely related to the Bayesian updating technique. However, under the normal design setting where a complete design is required to facilitate tendering of the construction stage of the project by various contractors, the approach is not feasible. Concerning quantifying the uncertainties, there is consensus that the approach is the most rational and transparent. Christian (2004) asserts that quantifying the uncertainties is consistent with the philosophy of the observational method and should therefore be considered as a logical extension of the approach that accommodates modern developments in probabilistic methods. Stellenbosch University http://scholar.sun.ac.za 58 Communication of risk within a transparent and rational framework is further motivated by the increasing pressure in code harmonisation as results of greater economic cooperation and integration brought by the advent of the World Trade Organisation, public involvement in defining acceptable risk levels, and risk-sharing among client, consultant, insurer and financier. The need for a framework that can treat uncertainties in transparent and rational manner can not be over emphasised. The critical question now is what framework is capable of dealing with uncertainties in this desired manner? Historically, probability theory has been the primary tool for modelling uncertainties. Therefore the framework should ideally be based on probability theory. If the framework is not reliability analysis, then what alternative is available (Phoon, et al, 2003; Phoon, 2004)? Certainly with the current state of knowledge, only reliability analysis and design can provide a consistent method for propagating uncertainties throughout the design process. In addition to dealing with uncertainties, the reliability based design framework provides a unifying framework for risk assessment across disciplines and national boundaries. This is important for achieving compatibility between structural design and geotechnical design so as to avoid the current scenario whereby different approaches are applied to two sides of the soil-structure interface. Consistent with the robustness of reliability theory in dealing with uncertainties, geotechnical limit state design is now generally based on level 1 reliability methods. Therefore this Chapter provides the reliability background for geotechnical limit states in general. This entails an overview of reliability theory and reliability calibration principles. Although probability theory is the basis for reliability analysis, it is of peripheral nature to this study and therefore it will not be reviewed. Besides, several text books have been devoted to probability theory. Text books on basic probability concepts specifically written for engineers are also available (e.g. Benjamin and Cornel, 1970; Ang and Tang, 1975; Harr, 1987; Smith, 1986; Baecher and Christian, 2003; etc). Stellenbosch University http://scholar.sun.ac.za 59 (a) Ignoring uncertainties (b) Being conservative Figure 3.1: Rock-bolting alternatives (After a carton in a brochure on rock falls published by the Department of Mines of Western Australia) Stellenbosch University http://scholar.sun.ac.za 60 3.2 Reliability Analysis Reliability analysis falls within the broad subject of probabilistic methods. Accordingly the U.S. Army Corps of Engineers defines probabilistic methods as techniques that may be called or include reliability analysis, risk analysis, risk based analysis, life-data analysis and other similar terms (ETL 1110-2-556). Based on the dependency of reliability on probability theory, Phoon (1995) defines reliability analysis as simply the consistent evaluation of design risk using probability theory. But what does the term reliability mean? The reliability of an engineering system is defined as its ability to fulfil its design purpose for a specific time period (ISO 2394, 1998; Harr, 1987; Ayyub and Popescu, 1998). Probability theory provides the basis for measuring this ability. In a probabilistic setting, the reliability of an engineered system is the probability of its satisfactory performance for a specific period under specific service conditions (i.e. the probability of survival). This definition recognises the fact that design variables (loads, materials properties, etc) are uncertain and therefore success of failure of a designed engineering system can only be assessed in probabilistic terms. It is also consistent with the earlier observation that probability and reliability analysis are interrelated. The complement of probability of survival or reliability is the probability of failure, hence reliability and probability of failure sum to unity. Mathematically the relationship between probability of failure (Pf) and reliability (R) is expressed as follows: 1=+ fPR [3.1a] fPR ?= 1 [3.1b] RPf ?= 1 [3.1c] The reliability of a structure is generally expressed in terms of either the probability of failure or reliability index. 3.2.1 Probability of failure The basic measure of reliability is the probability of failure. In evaluating the probability of failure, the behaviour of the system is described by a set of basic variables ( )nxxxX ,....., 21= characterising actions, material properties, geometric data and model Stellenbosch University http://scholar.sun.ac.za 61 uncertainty. Moreover, the limit state (ULS or SLS) is defined by the performance function given by: ( ) 0=Xg [3.2] in which g(X) is the performance function, the vector X is a set of n random variables. The performance function defines three different regions: ? ( ) 0=Xg , limit state ? ( ) 0>Xg , safe region ? ( ) 0 100) Stellenbosch University http://scholar.sun.ac.za 105 Table 4-8: Geotechnical data for BC Case No. Soil type SPT N-value CPT base shaft base shaft base shaft 122 V. Soft rock Residual material Ref.* 20 123 Very soft rock Residual material Ref. 80 124 Very soft rock Residual material Ref. 80 125 Residual material Residual standstone Ref. 60 126 Residual material sandy clay alluvium 100 15 127 Very soft rock sandy clay alluvium Ref. 15 128 Residual material sandy clay alluvium 60 15 129 decomposed shale decomposed shale (socket) 100 20 130 Weathered shale Weathered shale (socket) 100 20 131 Weathered rock Residual material 100 20 132 Weak rock, Shale Weak rock, Shale Ref. Ref. 133 Very soft rock Very soft rock (socket) Ref. 20 134 Very soft rock Very soft rock (socket) Ref. 20 135 Weathered diabase Clay alluvium 20 10 136 weathered granite Residual granite 100 20 137 V.soft rock Residual materials Ref. 20 138 V.soft rock silty clay Ref. 20 139 Residual granite silty clay 100 20 140 Residual granite silty clay 60 20 141 Residual material sand alluvium 100 20 142 Residual diabase Clay 32 28 143 Residual diabase Clay 32 28 144 Residual shale Residual shale 90 70 145 Residual shale Residual shale 90 70 146 Residual granite silty sand Ref. 17 147 Residual Andesite Clay alluvium 38 25 148 weathered granite Clay alluvium 40 13 149 Residual granite Clayey sand Ref. 35 150 Residual dorelite Clayey sand 100 17 151 Residual dorelite Clayey sand 100 17 152 Residual sandstone Silty sand Ref. 12 153 Residual sandstone Silty sand Ref. 12 154 Residual sandstone Silty sand Ref. 12 155 Residual sandstone Silty sand Ref. 12 156 Residual sandstone Silty sand Ref. 12 157 Soft rock sandstone Clayey sand 100 19 158 Soft rock granite Clayey granite Ref. 20 159 Very soft rock, mudrock Clay silty sand 90 15 160 Residual Andesite Clayey sand Ref. 15 161 Soft rock lava Clayey silt 100 28 162 Soft rock lava Clayey silt 100 28 163 Soft rock lava Clayey silt Ref. 31 164 Soft rock lava Clayey silt 100 31 165 Gneiss soft rock silty sand Ref. 12 166 Soft rock, mudstone silty clay Ref. 20 167 Soft rock, mudstone silty clay Ref. 20 168 Soft rock, mudstone silty clay Ref. 20 169 Soft rock, mudstone silty clay Ref. 20 170 Soft rock, mudstone silty clay Ref. 20 171 Soft rock sandstone calcrete Ref. 70 172 Soft rock sandstone calcrete Ref. 77 173 Very soft rock, siltstone calcareous clay 100 60 174 Very soft rock, siltstone calcareous clay 100 60 * SPT Refusal (N > 100) Stellenbosch University http://scholar.sun.ac.za 106 4.3.3 Pile load test database summary After discarding the load test data with insufficient information, 172 cases were selected for the study. Out of this, 60 cases comprise of piles in cohesionless materials while 112 cases consist of piles in cohesive materials. The 60 cases of piles in cohesionless materials are further subdivided as follows: ? 29 cases of driven piles ? 31 cases of bored piles Conversely the 112 cases of piles in cohesive materials are further subdivided into: ? 53 cases of driven piles ? 59 cases of bored piles 4.4 Evaluation of ultimate pile capacity from available load tests Once the pile load test data and the associated geotechnical data have been processed as described in the previous sections, the next step was to evaluate the ultimate capacities of all the selected test piles. In accordance with worldwide practice, the ultimate capacities were interpreted from the respective load-settlement curves. 4.4.1 The South African pile load test procedure The most common form of load test in South Africa is the static compression test in which a load is gradually applied to the pile head while the deflection is monitored. Although in principle piles can also be tested using dynamic or semi-dynamic testing procedures, such procedures are not currently in use in Southern Africa (Byrne et al, 1995). Static load testing of piles can be performed on working piles or trial piles specifically installed for the purpose. With working piles, the maximum test load is limited to avoid damaging the pile. In South Africa, the maximum test load is normally limited to one and half the design load. The main reason for performing load test on a working pile is to check the actual performance of the pile against that specified in the contract documents. Secondary benefits include confirmation of pile design parameters and a check on the structural integrity of the piles. Conversely, trial piles are loaded to failure. Testing piles to failure provides more Stellenbosch University http://scholar.sun.ac.za 107 accurate and meaningful design data. A trial pile programme enables the installation of more than one pile type whose performance for a given site are desired. This assists in deciding which pile type provides the most economic solution. More over, the trial piles can be installed to different depths so as to establish the optimum founding level. However, the prohibitive cost of installing additional piles just for testing purposes outweighs their advantages. Therefore, except for very large contracts, most pile load testing is carried out on working piles. On very large contracts the substantial cost of trial piles testing can often be compensated for many times over through achieving economic designs (Byrne et al, 1995). SABS 1200 F-1983: Standardised Specification for Civil Engineering Construction Part F- Piling constitutes the pile testing standard in South Africa. SABS 1200F outlines two compressive load test procedures. Theses are termed the British and the Danish procedure. Both procedures entail a series of test load cycles in which the pile is loaded in gradual increments and then unloaded in a similar manner. Nonetheless, Byrne et al, 1995, suggest that in South Africa most specifications call for the use of the British method or a variation thereof as the Danish procedure is very time consuming. The common procedure reflected in the pile test records is as follows: ? A first cycle of loading is started and it entails increasing the load in 25 percent increments up to the design load. ? The first cycle is then unloaded in 25 percent increments back to zero. ? The load kept at zero for a period of one hour after which the second cycle commences. ? The second cycle is also loaded in 25 percent increments up to a maximum load of one and half times the design load. ? The second cycle is then unloaded in 25 percent increments back to zero. The intermediate load increments are maintained until two successive readings 30 minutes apart show that the head deflection has not changed by more 0.1 mm. Appendix B.2 shows a typical load-deflection curve pile in which the procedure described above is captured. Stellenbosch University http://scholar.sun.ac.za 108 4.4.2 Analysis and interpretation of load test results For each test pile there was a load test results in the form of Appendix B 1. The load test results were further analysed by plotting the load versus the head deflection to produce load- deflection curves. Having plotted the load deflection curve, the major task was to determine the ultimate capacity. However, establishing the ultimate pile capacity from a load test is not a straight forward matter. The problem arises as to what constitutes failure or ultimate capacity of the test pile. Ideally, ultimate capacity of a pile would be defined as the point along the load-deflection curve where the test experiences continuous deflection at no increase in load (i.e plunging in case of an axial compression test). However, large movements are required for a pile to reach a plunging state. In most cases, a distinct plunging failure is not obtained in the test and therefore the ultimate capacity has to be estimated by some other methods. Fellenius (2000) warns that without a proper definition, ultimate pile capacity interpretation becomes meaningless for cases where obvious plunging has not occurred. Ideally an ultimate capacity definition should be based on some mathematical rules and should result in repeatable values that are independent of scale effects and individuals? personal opinion (Fellenius, 1990; Prakash & Sharma, 1990). Research to develop proper failure criteria has been ongoing for decades. Consequently a considerable number of different failure criteria have been proposed in the geotechnical literature. Some of the methods developed are actually used in specifications and codes of practice around the world. Some of the failure criteria reported in the geotechnical literature are listed below: ? Terzaghi?s 10% criterion (Terzaghi, 1942) ? Brinch-Hansen?s 80% criterion (Hansen, 1963) ? Brinch-Hansen?s 90% criterion (Hansen, 1963) ? Brinch-Hansen?s parabolic construction (Hansen, 1963) ? Vander Veen (1953) ? DeBeer (1970) ? Fuller and Holly (1970) ? Buttler and Holly (1977) ? O?Neill and Reese (1999) ? Mazurkiewicz (1972) ? Davison?s offset limit criterion (Davison, 1972) Stellenbosch University http://scholar.sun.ac.za 109 ? Kyfor et al (1992) ? Texas A & M Method for drilled shafts (Barker et al. 1993) ? Chin ?s extrapolation method (Chin, 1970, 1971) ? Fleming?s method (1992) ? Decourt extrapolation (Decourt, 1999) Most of these failure criteria are achieved after a substantial pile head movement. Therefore such failure criteria are applicable to cases where the test pile is loaded near to failure. As noted earlier, testing piles to failure is only done on very big piling contracts and such contracts are generally very few. For the common medium to large contracts, only working piles are tested. Consequently the majority of the test piles in the database are working piles tested to a maximum load of one and half times the design load. This limits the movement to which the pile head is subjected and requires an extension to the load-settlement curve to determine the ultimate capacity. Currently the estimation of ultimate capacity from proof tests (test on working piles) is only achieved through extrapolation procedures. Only a few extrapolation techniques have been reported in the literature. These include: ? Brinch-Hansen?s 80% criterion (1963) ? Brinch-Hansen?s polynomial construction (1963) ? Chin?s extrapolation method (1970, 1971) ? Decourt extrapolation (1999) ? Fleming?s method (1992) Out of these extrapolation techniques, Chin?s method is most popular worldwide. Paikowsky and Tolosko (1999) investigated extrapolation techniques for obtaining the ultimate capacity from proof tests. The extrapolation techniques investigated are the Chin?s method and Brinch-Hansen?s 80% criterion. These two approaches allow for fitting of a theoretical curve to the test data according to some mathematical relations. The techniques were examined through a database of 63 driven piles loaded to failure. The load-settlement data were truncated to 25%, 33%, 50%, 75% and 100 % of the actual failure load. The truncated data were extrapolated using the different techniques and the obtained ultimate capacity compared to the actual measured capacities. However, it was observed that the use of mathematical equations alone to extrapolate proof tests results in higher values of Stellenbosch University http://scholar.sun.ac.za 110 ultimate capacity than the measured values. Other researchers including Prakash and Sharma (1990) and Fellenius 1990, and Fleming 1992 have drawn attention to the fact that the Chin method appears to over-predict the ultimate capacity. Chin and Vail (1973) attributes the over-prediction to the fact that in the pile load test, it is the settlement at the pile head and not the displacement of the supporting soil that is measured. Therefore the measured displacement not only reflects the soil settlement but also the elastic shortening of pile. Further more, the measured displacement may include deformation of the pile due to any eccentricity. To obtain more conservative estimates Paikowsky and Tolosko combined the use of Davisson?s criterion and the extrapolation methods. Therefore the pile capacity estimation methods studied include: Chin?s method, Brinch-Hansen?s 80% criterion, Chin/Davisson?s method, and Brinch-Hansen?s 80% criterion/Davisson?s method. That study showed that the ultimate capacity from Chin/Davisson method was the most conservative. Building on the work by Paikowsky and Tolosko, Ooi et.al (2005) conducted a similar study on a database of bored shafts. In addition to the extrapolation methods used by Paikowsky and Tolosko, two more methods were investigated. The additional methods were: Brinch- Hansen?s polynomial construction and Brinch-Hansen?s polynomial construction/Davisson method. The study concluded that among the three extrapolation/ Davison methods, the Chin/Davisson procedure appears to be the most reliable with least scatter, confirming Paikowsky and Tolosko?s findings Based on the results of the above studies, it can be concluded that the Chin/Davisson procedure is superior to other methods in terms of consistency and conservatism. Therefore the Chin/Davisson procedure was adopted for the current study to determine ultimate capacity of proof tests results. The key steps of the procedure are as follows: ? Use Chin?s method to fit a hyperbolic curve to the available load-settlement points of a proof test and extrapolate. ? Use Davisson?s offset criterion to determine the pile capacity at the intersection of the extrapolated load-settlement curve and Davisson?s offset line. Stellenbosch University http://scholar.sun.ac.za 111 4.4.2.1 Fitting a hyperbolic curve using Chin?s method The method evolved from work by Kondner (1963) on hyperbolic stress-strain response of cohesive soils in a compressive triaxial test. Kondner demonstrated that the load- deformation behaviour of soils follow hyperbolic laws and therefore the nonlinear stress- strain curves for soils may be approximated by hyperbolae with a high degree of accuracy. The hyperbolic equation proposed by Kondner was of the form: ( ) ? ??? ba +=? 31 [4.1] Where ?1 and ?3 = the major and minor principal stresses respectively; (?1 ? ?3) = the deviator stress; ? = axial strain; a and b = constants (hyperbolic parameters) whose values may be determined experimentally. The resulting hyperbola is shown in figure 4.3. From figure 4.3, it can be seen that the constants a and b have physical meanings as follows: a is the reciprocal of the initial tangent modulus EI and b is the reciprocal of the asymptotic value of the stress, i.e. ultimate value of (?1 ? ?3). Figure 4.3: Hyperbolic representation of stress-strain relationship Stress-strain curve 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 Strain Stress Asymptote = ?ult = 1/b EI =1/a Stellenbosch University http://scholar.sun.ac.za 112 Kondner also showed that the values of the constants a and b may be determined most readily if Eq. 4.1 is transformed to represent a linear relationship as follows: ( ) ??? ? ba +=? 31 [4.2] Figure 4.4 shows the transformed plot of 21 ?? ? ? as a function of ?. The hyperbolic parameters a and b now represent the intercept on ordinate ? / (?1-?3) and the slope of the straight line respectively. The inverse slope of the straight line is a measure of the position of the horizontal asymptote and corresponds to the ultimate stress. ? / (?1-?3) b 1 a ? Figure 4.4: Transformed linear plot Following the work by Kondner, Chin (1970, 1971) extended the hyperbolic model to load- settlement behaviour of pile foundations. In accordance with Chin?s formulation, equations 4.1 and 4.2 become equations 4.3 and 4.4 respectively. bSa S Q += [4.3] bSa Q S += (4. 4] Where Q is the applied load, S is the pile head movement, a and b are constant parameters as described in Kondner?s formulation. Stellenbosch University http://scholar.sun.ac.za 113 The validity of equation 4.4 with regard to pile load-settlement data was investigated through full scale pile tests and published load test results. It was found that the plot of S/Q versus S is linear and therefore the load ?settlement behaviour of piles is hyperbolic and the inverse slope of this linear relationship is then the ultimate value of Q (i.e. the ultimate capacity of the test pile). This method of determining the ultimate capacity of a test pile is termed Chin?s method. For a given test pile data the step by step extrapolation procedure was as follows: ? First the hyperbolic parameters a and b were obtained as follows: - The transformed linear plot of S/Q versus S was plotted using the load-deflection data. - Best fit line was fitted to the plotted data. - From the equation of the best fit line, the hyperbolic parameters a and b were obtained (i.e. a = y-intercept and b= the slope of the line). Typical transformed plots are shown in figure 4.5. These are the two possible forms of the transformed plots and are interpreted in accordance with Chin and Vail (1973). The first case is a single straight line, denoting a situation in which the resistance is wholly provided by either the shaft materials only (friction piles) or the base materials only. The second case comprises a bilinear plot, denoting a situation in which the resistance is provided by both the shaft and base materials. In this case the first line represents shaft resistance while the second represent the ultimate capacity. In this case the hyperbolic parameters are obtained from the equation of the second line. ? The load-deflection curve (Q Vs S) was plotted. In the load-defection curve, the unload curves were left out. ? Using Eq. 4.4 in collaboration with the determined hyperbolic parameters, a theoretical load-deflection curve was generated and superimposed on the load deflection curve generated from the actual test data. The theoretical curve can be extrapolated to the desired magnitude of settlement to reach a plunging state. A typical theoretical load- deflection curve superimposed on the curve for actual test data is shown in figure 4.6. Stellenbosch University http://scholar.sun.ac.za 114 y = 0.000294x + 0.001406 R2 = 0.993266 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 1 2 3 4 S S /Q y = 0.000114x + 0.002022 R2 = 0.991064 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0 5 10 15 20 S S /Q (a) (b) Figure 4.5: Transformed plots (a) shaft or base resistance and (b) Both shaft and base resistances. Figure 4.6: Extrapolated curve 0 1000 2000 3000 4000 5000 6000 0 10 20 30 40 S (mm) Q (KN) Test data Ideal curve Stellenbosch University http://scholar.sun.ac.za 115 4.4.2.2 Davisson?s failure criterion With the Davisson failure criterion, the ultimate load is defined as the load corresponding to the intersection between the elastic deformation curve of the pile, shifted along the settlement axis by a value equal to 4mm plus the diameter of the pile (in millimetres) divided by 120, and the load-deflection curve from the load test data (Robert, 1997). The general equation for the Davisson?s offset line is given by: 4 120 ++= D AE QL S [4.5] Where S is the pile head deflection in mm, A is the cross-sectional area of the pile shaft, E is the Young?s modulus of the pile material, L is the length of the pile, Q is the maximum applied load and D is the pile diameter or width in mm. The step by step procedure is as follows: a) Calculate the elastic compression of the pile (? = QL/AE); b) Draw the elastic line on the extrapolated load-settlement curve (the initial straight line portion of the curve); c) Draw the Davisson?s offset limit line parallel to the elastic line at a distance of : 4 120 ++ D AE QL d) The ultimate capacity is then at the intersection of the Davisson?s offset limit line with the extrapolated load- deflection curve (fig. 4.7). Stellenbosch University http://scholar.sun.ac.za 116 Figure 4.7: Davisson?s failure criterion 4.4.2.3 Interpreted capacities and hyperbolic parameters obtained The interpreted capacities and the associated hyperpolic parameters are presented in table 4.9 through table 4.12. For trial piles (total of 8 cases) only the Davisson failure criterion was applied while for working piles (total of 164 cases) the combied Chin-Davison approach was used to obtain the interpreted capacities. The hyperbolic parameters can be used to re-draw the respective load-deflection curves if required. The Qi values range from 1310 ? 6000 kN for DNC, 440 ? 6000 kN for BNC, 490 ? 7000 kN for DC and 510 ? 13000 kN for DC. The Qi values are influenced by both the pile size and the soil properties. The average values for Qi are 2785 kN for DNC, 1992 kN for BNC, 2613 kN for DC and 2600 kN for BC. The average Qi values for piles in cohesive materials are quite close. Conversely average Qi values for piles in non-cohesive materials are appreciably different. The difference is attributed to the coincidental absence of smaller pile capacities (Qi <1000 kN) from the database for BNC. 0 1000 2000 3000 4000 5000 6000 0 10 20 30 40 S (mm) Q (kN) Test data Ideal curve Qi Davisson's offset limit line Elastic deformation Stellenbosch University http://scholar.sun.ac.za 117 Table 4-9: Interpreted capacities and hyperbolic parameters for DNC Case No. Qi (kN) a (mm/kN) b 1 2850 3.81E-03 2.32E-04 2 2550 1.51E-03 3.00E-04 3 2700 1.06E-03 3.02E-04 4 1900 2.60E-03 3.60E-04 5 3650 2.43E-03 1.68E-04 6 2080 1.45E-03 3.86E-04 7 1350 1.91E-03 5.84E-04 8 2280 2.00E-03 3.00E-04 9 2200 2.10E-03 3.00E-04 10 4100 2.10E-03 1.40E-04 11 3880 1.54E-03 1.58E-04 12 1725 3.87E-04 5.47E-04 13 3100 1.50E-03 2.00E-04 14 3400 1.70E-03 2.00E-04 15 3075 2.28E-03 2.29E-04 16 1800 3.81E-03 4.16E-04 17 2400 4.62E-03 2.91E-04 18 2200 9.56E-04 3.86E-04 19 2175 8.85E-04 3.84E-04 20 2600 9.08E-04 3.03E-04 21 1480 3.00E-03 4.50E-04 22 2200 2.52E-03 3.22E-04 23 1310 3.20E-03 5.00E-04 24 3400 3.78E-03 1.47E-04 25 2800 2.59E-03 2.41E-04 26 1550 5.20E-03 4.00E-04 27 6000 4.66E-03 6.00E-05 28 4800 6.77E-04 1.66E-04 29 5220 1.26E-03 1.23E-04 Table 4-10: Interpreted capacities and hyperbolic parameters for BNC Case No. Qi (kN) a (mm/kN) b 30 1375 4.00E-05 7.20E-04 31 4600 1.60E-04 2.04E-04 32 3000 1.35E-04 3.24E-04 33 1050 7.60E-03 5.78E-04 34 1357 1.03E-03 6.54E-04 35 1380 1.05E-03 6.69E-04 36 1050 7.64E-04 8.79E-04 37 1225 1.61E-03 6.83E-04 38 840 4.41E-03 8.63E-04 39 5600 6.99E-04 1.54E-04 40 1250 5.97E-04 7.52E-04 41 1500 1.39E-03 5.36E-04 42 970 4.00E-03 7.58E-04 43 540 5.35E-03 1.35E-03 44 600 1.82E-02 7.60E-04 45 1175 4.50E-03 5.00E-04 46 800 7.40E-03 9.00E-04 47 480 1.64E-02 1.20E-03 48 625 1.30E-02 7.00E-04 49 1025 1.02E-02 5.00E-04 50 1160 3.70E-03 6.00E-04 51 1500 2.60E-03 5.00E-04 52 2010 4.20E-03 2.00E-04 53 6000 6.18E-04 1.27E-04 54 1200 4.00E-03 5.00E-04 55 4650 2.02E-03 1.14E-04 56 1690 4.63E-04 1.22E-04 57 1620 7.00E-04 6.00E-04 58 1650 1.80E-03 5.00E-04 59 2100 1.60E-03 4.00E-04 60 2680 2.00E-03 3.00E-04 61 3650 1.26E-03 1.92E-04 62 5600 2.07E-03 1.40E-04 Stellenbosch University http://scholar.sun.ac.za 118 Table 4-11: Interpreted capacities and hyperbolic parameters for DC Case No. Qi (kN) a (mm/Kn) b 63 3290 2.41E-03 1.04E-04 64 680 2.03E-03 1.28E-03 65 1160 1.66E-03 7.39E-04 66 1800 1.50E-03 4.90E-04 67 2800 1.20E-03 3.10E-04 68 3400 1.00E-03 2.40E-04 69 3600 1.15E-03 2.13E-04 70 2050 2.90E-03 3.00E-04 71 2330 1.59E-03 3.09E-04 72 3150 2.10E-03 2.00E-04 73 2490 1.27E-03 3.19E-04 74 2300 1.20E-03 3.50E-04 75 2140 1.73E-03 3.56E-04 76 6080 9.00E-04 1.00E-04 77 2750 9.00E-04 3.00E-04 78 2180 9.00E-04 4.00E-04 79 4050 8.00E-04 2.00E-04 80 5200 8.02E-04 1.45E-04 81 5600 1.24E-03 1.19E-04 82 2780 1.40E-03 3.00E-04 83 4600 1.40E-03 1.50E-04 84 2380 2.06E-03 3.09E-04 85 7000 1.34E-03 7.51E-05 86 3300 1.60E-03 2.00E-04 87 1750 1.91E-03 4.34E-04 88 2300 4.05E-03 2.42E-04 89 1860 3.20E-03 3.80E-04 90 1500 1.75E-03 5.29E-04 91 1650 3.43E-04 5.73E-04 92 980 2.40E-03 8.00E-04 93 1320 3.40E-03 5.30E-04 94 3140 1.90E-03 2.00E-04 95 960 1.40E-03 9.00E-04 96 850 1.90E-03 1.00E-03 97 550 2.90E-03 1.50E-03 98 1280 2.80E-03 6.00E-04 99 1050 1.20E-03 8.44E-04 100 1800 9.00E-04 4.80E-04 101 1180 2.48E-03 6.30E-04 102 5100 3.60E-04 1.72E-04 103 1340 5.00E-04 7.00E-04 104 2880 2.41E-03 2.15E-04 105 2600 2.24E-03 2.67E-04 106 3700 1.31E-03 2.06E-04 107 3150 1.70E-03 2.19E-04 108 1140 1.37E-03 7.67E-04 109 3600 1.54E-03 1.88E-04 110 2550 1.41E-03 3.23E-04 111 4400 5.12E-04 1.92E-04 112 3600 4.47E-04 2.42E-04 113 2500 8.95E-04 3.39E-04 114 3090 7.26E-04 2.70E-04 115 1560 1.57E-03 5.25E-04 116 490 4.96E-03 1.56E-03 117 3470 1.54E-03 1.91E-04 118 2700 7.96E-04 3.09E-04 119 2200 8.06E-04 3.86E-04 120 2470 9.72E-04 3.31E-04 121 2120 2.72E-04 4.46E-04 Stellenbosch University http://scholar.sun.ac.za 119 Table 4-12: Interpreted capacities and hyperbolic parameters for BC Case No. Qi (kN) a (mm/kN) b 122 4800 1.00E-04 2.00E-04 123 4100 7.00E-04 2.00E-04 124 7100 5.00E-04 1.00E-04 125 760 7.00E-05 1.30E-03 126 2300 1.48E-03 3.47E-04 127 3100 1.60E-03 2.70E-04 128 3360 3.70E-03 1.90E-04 129 2650 2.00E-03 2.70E-04 130 1800 6.00E-04 5.00E-04 131 3700 8.25E-04 2.17E-04 132 2930 6.00E-04 3.00E-04 133 1700 1.50E-03 4.50E-04 134 1900 1.50E-03 4.00E-04 135 520 5.10E-03 1.20E-03 136 1175 7.00E-04 8.00E-04 137 1080 1.10E-03 8.00E-04 138 3500 8.35E-04 2.28E-04 139 1240 6.57E-04 7.50E-04 140 825 2.13E-03 1.02E-03 141 1450 1.00E-03 6.00E-04 142 3000 1.61E-03 2.50E-04 143 2450 1.60E-03 3.00E-04 144 1400 1.70E-03 5.00E-04 145 510 1.97E-03 1.82E-03 146 3600 7.42E-04 2.32E-04 147 1150 2.92E-03 6.42E-04 148 2310 1.29E-03 3.55E-04 149 8500 4.64E-04 9.30E-05 150 1230 6.83E-03 4.03E-04 151 1820 9.10E-04 4.65E-04 152 2580 5.88E-04 3.46E-04 153 2670 8.01E-04 3.21E-04 154 2790 8.57E-04 3.00E-04 155 2900 7.58E-04 2.92E-04 156 4200 9.19E-04 1.80E-04 157 2800 6.21E-04 3.11E-04 158 9600 4.40E-04 7.90E-05 159 1660 1.01E-03 5.15E-04 160 4800 3.04E-04 1.87E-04 161 7050 1.52E-04 1.31E-04 162 5900 1.67E-04 1.58E-04 163 9700 4.72E-04 7.39E-05 164 7000 2.54E-04 1.23E-04 165 1880 1.93E-03 4.01E-04 166 5430 6.40E-04 1.46E-04 167 3000 4.35E-04 3.06E-04 168 6250 3.08E-04 1.42E-04 169 4450 4.37E-04 1.96E-04 170 13000 4.13E-04 5.80E-05 171 3300 4.98E-04 2.62E-04 172 4810 1.83E-04 1.93E-04 173 2200 1.27E-03 3.70E-04 174 5300 7.61E-04 1.46E-04 Stellenbosch University http://scholar.sun.ac.za 120 4.4.2.3 Accuracy of the Chin/Davisson approach The Davison failure criterion is well known and widely used to estimate ultimate pile capacity for trials piles. However, besides the few research studies mentioned earlier, there is not much information in the literature on the use Chin/Davisson approach. Therefore the applicability of the trends observed by the few studies to different databases is questionable. To double check the reported results, the load-settlement data for the cases that were tested to failure were truncated to a maximum load of one and half times the design load. The truncated data were extrapolated using Eq. 4.4 and the ultimate capacities obtained on the basis of Davisson?s failure criterion. The capacities were then compared to the ones obtained from full test data load-settlement curve using Davisson?s failure criterion. Pile load test data from two trial piles at different sites were used in this exercise. These are pile cases 63 and 56. The particulars of these test piles and the associated geotechnical data are given in the relevant tables. Pile case 63 The load-settlement data are presented in table 4.13. The last column of table 4.13 contains the results of S/Q required for drawing the transformed linear plot. The data was truncated at Q = 1500 KN. This is the load corresponding to the maximum load of one and half times the design load. The transformed plot for the truncated data is shown in figure 4.8 and the hyperbolic parameters a and b were obtained from the equation of the best fit line. Using the hyperbolic parameters in conjunction with Eq. 4.4, a theoretical curve was fitted to the truncated test data. Figure 4.9 shows the resulting theoretical curve superimposed on the truncated test data curve. The value of the ultimate capacity interpreted from the extrapolated curve is 3290 kN. Using the full test data, the actual load-deflection curve was plotted (figure 4.10). From figure 4.10, the interpreted capacity is 3490 KN. The two values of the interpreted capacities are quite close. Further more, the extrapolated curve has been superimposed on the full test data curve in figure 4.11. It is evident from figure 4.11 that the extrapolated curve reasonably fits the actual full test data curve. Stellenbosch University http://scholar.sun.ac.za 121 Table 4-13: Load deflection data for case 63 y = 0.000241x + 0.001037 R2 = 0.911519 0.000000 0.000200 0.000400 0.000600 0.000800 0.001000 0.001200 0.001400 0.001600 0.001800 0 0.5 1 1.5 2 2.5 3 S S /Q Figure 4.8: Transformed plot for case 63 Q S S/Q 0 0.0 250 0.2 0.000800 500 0.5 0.001000 750 1.0 0.001333 1000 1.3 0.001300 1250 1.8 0.001440 1500 2.5 0.001667 1750 3.5 0.002000 2000 4.8 0.002400 2250 5.8 0.002578 2500 7.2 0.002880 2750 9.0 0.003273 3000 12.0 0.004000 3250 15.5 0.004769 3500 21.8 0.006229 3750 31.0 0.008267 Stellenbosch University http://scholar.sun.ac.za 122 0 500 1000 1500 2000 2500 3000 3500 4000 0 10 20 30 40 Truncated test data Theoretical curve S = 11.9 mm Qi = 3290 kN Figure 4.9: Extrapolated curve for case 63 0 500 1000 1500 2000 2500 3000 3500 4000 0 10 20 30 40 S (mm) Q (k N ) Full test data S=17.2 mm Qi = 3490 kN Figure 4.10: Full test data curve for case 63 Stellenbosch University http://scholar.sun.ac.za 123 0 500 1000 1500 2000 2500 3000 3500 4000 0 10 20 30 40 Full test data Extrapolated curve Figure 4.11: Extrapolated curve superimposed on full test data curve for case 63 Stellenbosch University http://scholar.sun.ac.za 124 Pile case 56 To double check that the results obtained for case 63 have not occurred by chance, load test results from a trial pile at another site were analysed in a similar way. For this case the load- deflection data are presented in table 4.14. The load of 1.5 times the design load corresponds to 600 kN and therefore the data were truncated at this value. Table 4-14: Load-settlement data for pile case 56 Q (kN) S (mm) S/Q 0 0 100 0.09 0.0009 200 0.23 0.00115 300 0.42 0.0014 400 0.6 0.0015 500 0.8 0.0016 600 1 0.001667 700 1.1 0.001571 800 1.3 0.001625 900 1.6 0.001778 1000 1.8 0.0018 1100 2 0.001818 1200 2.4 0.002 1300 2.8 0.002154 1400 3.5 0.0025 1500 5 0.003333 1625 8 0.004923 1700 10 0.005882 1850 15 0.008108 1900 20 0.010526 1950 30 0.015385 1950 40 0.020513 1950 50 0.025641 The ensuing transformed plot, extrapolated curve and full test data curve are presented in figures 4.12, 4.13 and 4.14 respectively. The ultimate pile capacity from extrapolated curve is 1800 kN while that from full test data is also 1800 kN. Figure 4.15 shows the extrapolated curve superimposed on the full test data curve and it can be seen that the fit between the two cases is good. The results of the two cases confirm that the extrapolated curves using Eq. 4.3 reasonably simulate load-deflection curves for piles tested to failure. Also the interpreted capacities from extrapolated curves are quite close to those from full test data. Therefore the interpreted capacities derived in this study can be used with confidence. Stellenbosch University http://scholar.sun.ac.za 125 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0 0.2 0.4 0.6 0.8 1 1.2 S S /Q Figure 4.12: Transformed plot for pile case 56 0 500 1000 1500 2000 2500 0 10 20 30 40 50 60 S (mm) Q (k N ) Truncated test data Theoretical curve S=9.7mm Qi = 1800 kN Figure 4.13: Extrapolated curve for case 56 Stellenbosch University http://scholar.sun.ac.za 126 Full test data 0 500 1000 1500 2000 2500 0 10 20 30 40 50 60 S (mm) Q (k N ) Full test data S=12.5mm Qi=1800 kN Figure 4.14: Full test data curve for case 56 0 500 1000 1500 2000 2500 0 20 40 60 S (mm) Q (k N ) Full test data Extrapolated curve Figure 4.15: Extrapolated curve superimposed on full test data curve for case 56 Stellenbosch University http://scholar.sun.ac.za 127 Chapter 5 EVALUATION OF GEOTECHNICAL DESIGN PARAMETERS AND COMPUTATIONS OF PREDICTED CAPACITIES 5.1 Introduction The site specific geotechnical data presented in Chapter 4 were in the form of soil profile descriptions and penetration tests results. However, the theoretical computations of the bearing capacity of geotechnical materials require the engineering properties of the materials as input in the calculation model. Therefore the available measurements (in-situ or laboratory measurements) need to be transformed to the desired soil engineering properties through empirical correlation. In general the process of mapping the available measurements to engineering properties is not well defined. The procedure is still arbitrary in the sense that upon repetition it does not yields the same results whether by the same engineer of by several engineers (Tabba and Young, 1981). Accordingly this Chapter presents the basis for the selection of the various geotechnical parameters used in this study. The selected parameters are then used to compute the theoretical pile capacities. 5.2 Methods for predicting ultimate pile capacity in South Africa In South Africa there are two codes of practice which specifically relate to geotechnical aspects of pile foundations. These are: (1) SABS 1200 F-1983: Standardised Specification for Civil Engineering Construction, Part F- piling and (2) SABS 088-1972: South African Standard Code of Practice for Pile Foundations. SABS 1200 F-1983 covers the construction of piles and does not cover any design criteria while SABS 088-1972 provides a general description of the types of piling at present in use, together with precautions recommended for observance in their design and application. In addition to the two above codes, a Guide to Practical Geotechnical Engineering in Southern Africa was published by Frankipile South Africa in 1976. The main purpose of the guide was to create a practical reference on all aspects of soil investigation and piling as carried out by the company in Southern Africa. The guide is now in its third edition and has become a standard text for all those in the piling industry in Southern Africa (Knight, 1995). Stellenbosch University http://scholar.sun.ac.za 128 Both SABS 088-1972 and the Guide to Practical Geotechnical Engineering in Southern Africa state that the load that a pile can support, in relation to the strength of the soil, can be determined by: (i) Static analysis using engineering properties of the soil as determined from laboratory or in-situ field testing. (ii) Empirical analysis by directly using standard field tests results - CPT - SPT (iii) Dynamic driving resistance - Pile driving formulae - Wave equation (iv) Full scale pile load tests These four methods constitute the main categories of pile capacity prediction world wide. However for each category, many variations of the basic approach have evolved. SABS 088-1972 recognises the variability of preference for formulae for static, empirical and dynamic analysis by refraining from prescribing specific pile capacity prediction equations. The code recommends the use of standard text books on foundation and soil mechanics and data from specialised piling firms familiar with the pile type and soil conditions as reference for the detailed design of any given pile foundation. Therefore, there is no distinctively South African method for determining pile capacity. However, from published local case studies and the Guide to Practical Geotechnical Engineering in Southern Africa it is evident that the static analysis using engineering properties of the soil is the predominant pile design approach in South Africa. For this reason, the static analysis approach was selected for quantification of model uncertainty in this study. The analytical model based on static analysis is generally referred to as the ?Static Formula?. 5.3 The static formula The static analysis for the ultimate axial compressive capacity is based on the assumption that the capacity comes from two distinct physical mechanisms: (i) Shaft resistance due to the development of shear stresses along the pile shaft as a result of sliding between the ground and the pile material. Mathematically the side shear is modelled using Mohr-Coulomb failure criterion. With this failure criterion, Stellenbosch University http://scholar.sun.ac.za 129 shearing comprises of an adhesion component as well as a stress-dependent friction component. The summation of the shear stress along the pile shaft produces a resultant force that provides resistance to the applied load. (ii) Bearing capacity at the base of the pile foundation that is conceptually identical to that of shallow foundations. Based on the above mechanisms, the basic formula as given by SABS 088-1972, the guide to piling and foundation systems, and several foundation engineering text books is expressed as; WQQQ sbult ?+= [5.1] Where Qult is the ultimate pile capacity, Qb is the end-bearing capacity and Qs is the frictional capacity along the pile perimeter and W is the weight of the pile. The terms in equation 5.1 are generally further expanded as follows; For base capacity: ( )qfcbb NDBNcNAQ ?? ? ++= 5.0 [5.2] The term ( ??BN5.0 ) is very small compared to the other two and is therefore generally ignored in practical designs. When this term is left out, equation 5.2 reduces to: ( )cubb NcAQ = for piles in cohesive materials [5.3] ( )qvbb NAQ ? ?= for pile in non-cohesive materials [5.4] For shaft capacity: LfpQ LL L ss ?= ?= =0 [5.5] Also the specific equation for shaft capacity is dependent on the soil type as follows: Stellenbosch University http://scholar.sun.ac.za 130 Cohesive materials: sus AcQ ?= [5.6] Non-cohesive materials: svss AKQ )tan( ?? ?= [5.7] The complete theoretical equations for axial compression capacity of a pile in non-cohesive and cohesive materials respectively are as follows: ( ) ( )??? tanvssvqbult KANAQ ?+?= [5.8] ( ) sucubult ACNCAQ ?+= [5.9] Where: bA = base area, sA = shaft area qc NandNN ;; ? = bearing capacity factors uc = undrained shear strength ? = unit weight, B = width or diameter of the pile fD = depth to the pile tip v? ? =effective vertical stress P = pile perimeter sf = unit shaft friction L = length of the pile ? = soil-pile interface friction ? = adhesion factor The pile weight is routinely neglected in practice. The basis for this is that the weight of pile is approximately equal to weight of soil removed or displaced to make way for the pile. Since in the calculation of the net bearing capacity of the soil, the overburden pressure at the base of the pile is not taken into account, it follows that the weight of the pile should not be included. Stellenbosch University http://scholar.sun.ac.za 131 For piles in non-cohesive materials, the effective vertical stress in Eq. 5.4 and Eq. 5.7 suggests that the pile resistance increases with depth. However field pile tests have shown that both the base capacity and the shaft capacity do not increase continually with depth. In fact they increase with depth up to a maximum limit at a depth of about 15-20 times the pile diameter and remain constant thereafter (Das, 2003). The depth at which the resistance stops to increase with depth is termed the critical depth. Although the concept of the critical depth is a contentious issue, it was adopted in this study. Conceptually equations 5.7 and 5.8 are straight forward to use. The validity of the equations has been confirmed repeatedly throughout the years by research (Horvath, 2002). The difficulty in applying the equations in practice has always been to correctly quantify the various variables that appear in the equations. The determination of the variables is further complicated by the fact that some of the variables are dependent on both the geostatic stress state in the ground prior to pile installation and the specifics of the installation process (i.e. bored or driven). Consequently various approaches for the determination of the soil input parameters have been suggested by different researchers. Generally the determination of the soil design parameters is a two-step process that encompasses (i) derivation of soil properties from the available laboratory or field measurements and (ii) correlating design parameters such as bearing capacity factors, cohesion factor and earth pressure coefficient from the derived soil properties. 5.4 Derivation of soil properties from available measurements As already pointed out, the available geotechnical data was limited to soil profile descriptions and penetration tests results. The penetration tests results included Standard Penetration Test (SPT) blow-counts, Dynamic Probe Super Heavy (DPSH) results and Cone Penetration Test (CPT) results. However, measurements from SPT were the most common for most of the sites. This suggests that SPT is the most popular subsurface investigation method in Southern Africa. For derivation of soil properties, the DPSH and CPT measurements were converted to equivalent SPT N-values. The SPT N-values were then used to estimate the desired geotechnical properties. Stellenbosch University http://scholar.sun.ac.za 132 5.4.1 Background and interpretation of SPT results 5.4.1.1 Historical development The standard penetration test owes its origin from the extensive work geared toward developing an effective pipe sampler in the 1920s. The samplers evolved from the simple Gow sampler in 1902 followed by the split sampler in 1920s. The split sampler was further modified to the Raymond sampler. During the same period, investigation towards measuring the resistance of the material in terms of the number of blows required to penetrate a specific depth got under way. In 1954, the conventional procedure whereby blows are recorded for each of three 6 inch (15.25 cm) increments was introduced. The procedure was such that the value for the first round of advance is discarded because of fall-in and contamination in the borehole. The second pair of numbers are then combined and reported as a single value for the last 12 in. (30.5 cm). Terzaghi furthered the research on the Raymond sampler and realised that the penetration resistance of the split spoon sampler could provide useful in-situ test data which might be correlated to the consistency and density of the soil encountered. During the writing of the text on Soil Mechanics in Engineering Practice, Terzaghi in collaboration with Mohr developed the correlations between the number of blow-counts and various salient soil properties including: relative density of sands, consistency and unconfined strength of clays, and allowable bearing pressure on both sands and clays. In 1947, Terzaghi named the Raymond sample procedure as the ?Standard Penetration Test? in a presentation titled ?Recent Trends in Subsoil Exploration?. The presentation was derived at 7th Conference on Soil Mechanics and Foundation Engineering. 5.4.1.2 Test procedure National standards outlining the procedure for carrying out the test are available in many countries. Some of the most commonly followed standards are the British Standard (BS 1375: Part 9, 1990), the American Standard (ASTM D1586, 1984), and the Japanese Standard (JIS-A219, 1976). Further more, CIRIA report CP/7 (Clayton, 1995) gives the procedures adopted around the world as well as describing its strength and weakness and its use for geotechnical design. The test procedure is illustrated in plate 5.1 and briefly described below. Stellenbosch University http://scholar.sun.ac.za 133 ? The boring is advanced to the desired sampling depth and the split sampling apparatus is lowered to the bottom of the borehole. ? The sampler is than driven into the soil by a 63.5 kg hammer raised 0.76m above the upper face of the drivehead assembly. Common methods of raising and dropping the hammer include trip hammer, semi-automatic and automatic drop systems (plate 5.2). ? The numbers of blows for each of the three 150 mm increments of penetration are recorded. ? The penetration resistance (N-value) is taken as the sum of the blows required for the second and third 150 mm increments of penetration. The first 150 mm penetration is considered to be a seating drive. 5.4.1.3 Factors Affecting SPT N -values Several research studies on factors influencing SPT results have been carried out (e.g. Schmertmann, 1979; Decourt, 1989; Skempton, 1986; Kulhawy and Tautmann, 1986). Table 5.1 presents a summary of the various factors that have been found to influence the SPT results. A closer look at table 5.1 reveals that most influences are either attributed to equipment effects or procedural/operator effects. Therefore with good equipment care and site supervision, most of the factors can be minimised. Given that SPT tests are carried out by professional and experienced drilling companies, it can be assumed that quality control is good and therefore the SPT measurements gathered for this study are reliable. Stellenbosch University http://scholar.sun.ac.za 134 Figure 5.1: SPT test procedure (After FHWA-HI-97-021) Figure 5.2: SPT hammer drop systems Stellenbosch University http://scholar.sun.ac.za 135 Table 5-1: Factors affecting SPT N-vales Factors Comments Inadequate cleaning of the bore hole SPT is only partially made in original soil. Sludge may be trapped in the sampler and compressed as the sampler is driven, increasing the blow count. Not seating the sampler spoon on undisturbed material Incorrect N-values obtained Driving the sample spoon above the bottom of the casing N-values are increased in sands and reduced in cohesive materials. Failure to maintain sufficient hydraulic head in boring The water table in the borehole must be at least equal to the piezometric level in the sand, otherwise the sand at the bottom may be transformed into loose state thereby decreasing the blow count. Attitude of operators Blow counts for the same soil using the same rig can vary, depending on who is operating the rig, and perhaps the mood of operator and time of drilling Overdrive sampler Higher blow counts usually results from an overdriven sampler. Sampler plugged by gravel Higher blow counts results when gravel plugs the sampler, resistance of loose sand could be highly overestimated. Plugged casing High N-values may be recorded for loose sand when sampling below groundwater table. Hydrostatic pressure can cause the sand to rise within the casing. Overwashing ahead of casing Low blow count may result for dense sand since overwashing loosen sand. Drilling method Drilling technique (e.g. cased holes Vs mud stabilised holes) may result in different N-values for the same soil. Free fall of the drive weight is not attained Using more than 1-1/2 turns of rope around the drum and or using wire cable will restrict the fall of the drive weight. Not using correct weight Drillers frequently supply drive hammers with weights varying from the standard. Weight does not strike the drive cap concentrically Impact energy is reduced, increasing N-values. Not using a guide rod Incorrect N-values obtained Not using a good tip on the sampling spoon If the tip is damaged and reduces the opening or increases the end area the N-values can be increased. Use of drill rods heavier than standard With heavier rods more energy is absorbed by the rods causing an increase in the blow count. Stellenbosch University http://scholar.sun.ac.za 136 5.4.1.4 Correlation between SPT N-vales and soil properties The results of SPT have been used extensively to obtain soil properties for input into routine geotechnical design calculations world wide. A wide range of properties for almost all soils and weak rocks can be obtained with ease and convenience and at modest cost (Clayton, 1995). The SPT N-value is an index or measure of the in-situ firmness or denseness of the materials being penetrated. Consequently many correlations relating SPT blow count and soil properties have been developed. The first published SPT- soil properties correlation appeared in Terzaghi and Peck (1948). This was followed by correlation relating blow counts to consistency for silts and clay and relative density for sands in Peck, Hanson and Thornburn (1953). To date, many other correlation schemes have been developed. In the 1980s, a rather new thinking to the development of SPT correlations emerged. The series of new correlations seek to correct the SPT data for a number of site specific factors (table 5.1) to improve its repeatability. Examples of published papers on the subject of correction to SPT data include: Riggs (1986), Skempton (1986), Liao and Whitman (1986), Clayton (1995), etc. Following the new line of thinking, the corrected SPT blow count is given by (Skempton, 1986): CBFASRBENm CCCCCCCCNN =)60(1 [5.9] Where: )60(1N = measured blow count corrected to 60% of the theoretical free-fall hammer energy, 100kPa effective overburden pressure and other factors. mN = measured SPT blow count NC = overburden correction factor EC = energy correction factor BC = borehole correction factor RC = rod length correction factor SC = sampling method correction factor AC = anvil correction factor BFC = blow counts frequency correction factor Stellenbosch University http://scholar.sun.ac.za 137 CC = hammer cushion correction factor Certainly the above correction factors are just too many to be applied in practical design situations. Nevertheless, only the overburden and hammer energy correction factors are usually considered. Both the new and old correlations have one thing in common; they are developed from a large database of results based on past experience. Therefore when using any specific correlation, the following critical points should be borne in mind. ? The selected correlation is only as good as the data used to develop it. ? Many correlations for sands were developed for clean, uncemented, and uniform sand. ? A correlation provides an approximate answer and will undoubtedly exhibit scatter among the data points. ? The selected correlation will be most accurate if calibrated to local soil conditions. The correlation schemes can be broadly classified into schemes for non-cohesive materials and schemes for cohesive materials. 5.4.1.5 Correlations for properties of non-cohesive materials For design of piles in cohesionless soils, the key parameters to be obtained from SPT test data are the in-situ density and the friction angle (?). The correlation between soil consistency, unit weight and SPT N- values is presented in table 5.2. The table is based on the correlation developed by Terzaghi and Peck (1948). The correlation has been adopted by most geotechnical codes of practice worldwide including the Guide to Practical Geotechnical Engineering in Southern Africa. The consistency field identification descriptions and the approximate unit weight values presented in table 5.2 are in accordance with the Revised Guide to Soil Profiling for Civil Engineering Purposes in Southern Africa (Jennings et.al, 1973). The unit weights required for the calculation of the ultimate pile capacity in this study were based on the correlation presented in table 5.2. The resulting unit weights for various case histories are presented in tables 5.3 and 5.4. Stellenbosch University http://scholar.sun.ac.za 138 Table 5-2: Correlation between SPT, consistency, and unit weight Consistency Filed identification SPT (N) ? (kN/m3) Terzaghi and Peck (1948) Jennings et.al (1973) Terzaghi and Peck (1948) Jennings et.al (1973) V.Loose No resistance to shoveling 0 ? 5 16 Loose Easily penetration by 12mm bar driven by hand. Small resistance to shoveling. 5 ? 10 16.5 Med. Dense Easily penetration by 12mm bar driven with 2 kg hammer. Considerable resistance to shovel. 10 ? 30 17.3 Dense Hard penetration with 12 mm bar driven with hammer. Hand pick needed for excavation. 30 ? 50 18 V.Dense Penetration only to 75mm with 12mm bar driven with hammer. Power tools for excavation >50 19.5 Table 5-3: Unit weight (bored piles cases) Table 5-4: Unit weight (driven pile) Case No. ? 1 16.5 2 16.5 3 16.5 4 16.5 5 16.5 6 16.5 7 16 8 16 9 18 10 18 11 18 12 16 13 16 14 16 15 18 16 18 17 19 18 16 19 18 20 18 21 17 22 17 23 16 24 16 25 16 26 16 27 18 28 18 29 18 Case No. ? 30 19 31 19 32 19 33 19 34 18 35 18 36 18 37 18 38 18 39 20 40 17 41 16 42 17 43 17 44 17 45 17 46 17 47 16 48 16 49 17 50 16 51 16 52 18 53 18 54 18 55 18 56 18 57 18 58 16 59 16 60 17 61 18 62 18 Stellenbosch University http://scholar.sun.ac.za 139 Concerning the friction angle, several correlations that have been suggested by various researchers are available. Some of the correlations are based on corrected N-values (N1 (60)) while others are based on uncorrected values (N). Currently there is no consensus on the use of corrected versus uncorrected SPT blow count. In this study correlations based on uncorrected N-values were used. The reasons for this selection are as follows: ? Uncorrected values are used in practical designs worldwide; ? Most reliable correlations are based on uncorrected blow count; ? The required correction factors are too many; ? The models for deriving the correction factors are not perfect and therefore introduce additional uncertainties to the interpretation of SPT results. For example, several formulae and charts have been published for overburden correction with completely different results; ? State-of-the- art equipment are in use these days and therefore most equipment-related effects are eliminated; ? SPT is carried out by specialist drilling companies with sufficient experience and therefore they are conversant with the test. Hence operator/procedural effects are minimal. With regard to the essential corrections of overburden pressure and hammer energy loss, the position taken in this study is that: ? The reference overburden stress of 100 kPa is just a hypothetical value and therefore does not reflect the operational stress around a pile. The increase in SPT N value with depth is a reflection of the increase in strength and stiffness caused by in-situ overburden stress and therefore correction for overburden stress may not be necessary. ? The correction for energy loss makes sense if the exact loss has been measured on site for each case considered. But usually a generic correction factor is applied. Surely in some sites, advanced SPT equipment might have been used with no energy loss at all while in some sites older equipment might have been used with higher energy losses. Since the exact SPT equipment used to measure the N-values as well as their site measured energy losses are not known, it is not worth while to apply the correction factor. Stellenbosch University http://scholar.sun.ac.za 140 Furthermore preliminary investigations on the effects of correcting the N-values on various pile design parameters were made. Using the two sets of SPT data (i.e. N and N1 (60)) various design parameters were derived. The parameters were used as input into the static formula to calculate the predicted capacity for each case leading to two sets of predicted capacities. The two sets of predicted capacities were then compared with the respective measured capacities evaluated in chapter 4. The comparison was based on the ratio of measured capacity to predicted capacity (i.e the bias). The bias statistics were then used to evaluate the fit of the two data sets. The bias factor statistics in this case provides an indication of the accuracy and precision of the design parameters from the two data sets. A better approach will yield a mean bias factor of close to 1 and small variability. For the 26 pile cases investigated, the bias factor statistics are presented in table 5.5. Table 5-5: Bias factor statistics Data set Mean Standard deviation Coefficient of variation Uncorrected N-values 1.11 0.29 0.26 Corrected N-values 1.19 0.46 0.39 The conclusions drawn from this preliminary investigation are as follows: ? In comparing the design parameters derived on the basis of corrected and uncorrected N- values, no significant improvement in terms of reduced variability was gained from correcting the N-values. ? The data set for uncorrected N-values provides the best fit to the measured capacities as shown in table 5.5 (i.e. a mean value of closer to 1 and a smaller variability). It is therefore concluded that for reliability based design of pile foundations, correction of N-values for overburden and hammer efficiency do not add any value. Some of the published correlation between the angle of internal friction and the uncorrected SPT N-values are presented in figure 5.3. From figure 5.3, it can be seen that curves corresponding to Peck et.al (1974) and Teng (1962) coincide which is an indication that the correlations might have been based on the same database. In terms of conservatism, ?- values by Nixon (1982) and Byrne (1995) are the most conservative while those by Meyerhof (1965) and Das (1984) are less conservative. Values by Peck et.al and Meyerhof (1974) are moderate. However, at N-values of less than 46, Nixon?s ?-values are on a less Stellenbosch University http://scholar.sun.ac.za 141 conservative side. The correlation by Peck et.al (1974) seems to lead to reasonably conservative ?-values and was therefore selected for this study. This correlation is further shown in figure 5.4. Many researchers including Meyerhof (1965) have drawn attention to the fact that values of angle of internal friction given by the correlation schemes in figure 5.3 are conservative estimates based on experimental data for relatively clean sands. Therefore, these values need to be adjusted to account for the presence of fines and coarser materials. The correction that has been generally adopted is to reduce the values by 3 to 5? for clay or silty sands and increase them by the same range for gravely sands. These adjustments were followed in this study. The procedure followed to obtain ? values from SPT N values is as follows: a) Obtain average N values for both shaft and base materials from the borehole logs. For base material, the average N value was taken over a depth of four pile base diameters above and one base diameter below pile toe. b) Obtain ? value corresponding to the determined average N value using Peck et.al (1974) correlation. c) Adjust the ? value for presence of fines or gravel as follows: - Reduce by 3? for clay and silty sands; - Increase by 3? for gravely sands. 0 10 20 30 40 50 60 70 80 25 30 35 40 45 50 55 PHI S P T N Peck et.al (1974) Meyerhof (1974) Nixon (1982) Teng (1962) Byrne et.al (1995) Meyerhof (1965) Das (1984) Figure 5.3: N ? ? correlation by different Authors Stellenbosch University http://scholar.sun.ac.za 142 0 10 20 30 40 50 60 70 80 28 30 32 34 36 38 40 42 44 46 Friction angle S P T N v al u e Figure 5.4: N ? ? correlation (After Peck et.al (1974) The values of the angle of friction obtained are presented in tables 5.6 and 5.7. Further examination of the ? values lead to the following observations: ? In general, for a given case ?-base values are higher than ?-shaft. This is in accordance with expectations as piles are normally founded on denser materials. ? On average, the ? values for materials in which bored piles were constructed are relatively higher than for materials in which driven piles were installed. This is attributed to the fact that under normal circumstances, bored piles are preferred in very dense materials (higher ? values) while driven piles are preferable when the granular material is not very dense. ? The magnitudes of the ? values are within the expected range of ? values for non- cohesive materials. However, some ?-shaft values are on the lower side, indicating that the pile shaft passes through some silty soils. Stellenbosch University http://scholar.sun.ac.za 143 Table 5-6: ?-values (driven piles) Table 5-7: ?-values (bored piles) Case No. ?-base ?-shaft 30 35.9 33.1 31 35.9 33.1 32 35.9 33.1 33 40.3 31.2 34 38.9 30.8 35 38.9 30.8 36 38.9 30.8 37 38.9 30.8 38 38.9 30.8 39 40.1 36.4 40 34.5 32.2 41 34.5 31.6 42 33.7 32.2 43 32.8 31.9 44 32.8 31.9 45 33.7 32.2 46 34.5 31.9 47 34.3 31.0 48 34.5 31.3 49 34.8 32.2 50 31.0 31.6 51 31.0 32.2 52 40.3 40.3 53 47.2 47.2 54 40.3 40.3 55 47.2 47.2 56 30.3 32.5 57 30.7 31.9 58 30.7 29.7 59 32.2 29.7 60 32.2 31.3 61 37.3 36.4 62 35.9 33.1 5.4.1.6 Correlations for cohesive materials In this study, cohesive materials do not entirely refer to clay soils. Cohesive materials predominately refer to residual soils. The area between latitudes 35? North and 35? South is characterised by the extensive occurrence of residual soils (Blight and Brummer, 1980). Southern Africa falls within these bounds and therefore the prime soil materials are of residual origin. The residual soils are overlain by transported materials and underlain by the fresh parent rock. A typical Southern African soil profile consists of (1) top horizon of transported materials, (2) middle horizon of residual materials and (3) bottom horizon of Case No. ?-base ?-shaft 1 33.4 32.8 2 30.7 30.0 3 33.6 34.0 4 32.9 31.9 5 33.6 31.9 6 33.6 31.9 7 32.1 26.4 8 32.1 33.4 9 38.2 34.5 10 38.2 34.5 11 38.2 34.5 12 30.5 31.6 13 31.5 31.6 14 34.5 33.7 15 39.4 38.9 16 39.4 38.9 17 39.4 38.9 18 34.2 31.9 19 33.0 34.3 20 33.2 34.3 21 33.2 31.9 22 35.0 31.9 23 33.2 31.0 24 33.2 31.3 25 33.3 31.3 26 30.1 31.0 27 43.3 40.1 28 43.3 40.1 29 43.3 40.1 Stellenbosch University http://scholar.sun.ac.za 144 parent rock. The depth of the transported and residual soil horizons is influenced by the local geology, climate and geomorphology that produced the prevailing landform. In general pile foundations are founded on the residual soils rather than on the transported materials. Residual soils exhibit special engineering properties and characteristics that distinguish it from sedimentary soils (Blight, 1991; Fookes, 1991). The definition of residual soil varies from country to country. In South Africa residual soil is defined as a soil-like material derived from the in-situ weathering (both physical and/or chemical weathering) and decomposition of rock which has not been transported from its original location (Blight, 1991). The materials are described in terms of the rock from which they have been derived. The fundamental geotechnical property required for design of piles in cohesive materials is the undrained strength (Cu). Correlation between SPT N-values and the consistency as well as the undrained shear strength of cohesive materials have been developed. Table 5.8 presents the correlation between rock consistency and the undrained shear strength while table 5.9 presents the correlation between in-situ tests and the required design parameters. Table 5-8: Correlation between rock consistency and undrained shear strength (Byrne et al, 1995) Consistency Field Identification Cu kN/m2 V. Soft Material crumbles under firm blows with geologist's pick. Can be peeled off with knife. 350 -1500 Soft Indentation 1 mm to 3 mm with firm blows with geologist's pick. Can just be scraped with knife. 1500 -5000 Hard Hand held specimen breaks with hammer end of geologist's pick with single firm blow. Cannot be scraped with knife. 5000 ? 10000 V.Hard Hand held specimen breaks with hammer end of geologist's pick with more than one blow. 10000 ? 35000 V.V Hard Hand held specimen requires many blows with geologist's pick to break through intact material. >35000 Stellenbosch University http://scholar.sun.ac.za 145 Table 5-9: Correlation between in-situ tests and pile design parameters for cohesive materials (Byrne et al, 1995) Consistency Filed identification SPT CPT Cu ? N qc (MPa) kN/m 2 Very Soft Easily moulded with fingers. <2 <0.15 <20 1.00 Geologist's pick can be easily pushed in up to its handle Soft Easily penetrated by thumb. 2 ? 4 0.15 ? 0.3 20 - 30 1.00 Moulded with strong pressure. Geologist's pick pushed in up to 30 t0 40 mm (sharp end) Firm Intent by thumb with effort. Very 4 ? 6 0.3 ? 45 30 -40 0.90 difficult to mould with fingers. 6 ? 8 0.45 -600 40 - 50 0.80 Geologist's pick pushed in up to 10mm (sharp end) Stiff Penetration by thumb nail. Can not 8 ? 10 0.6 ? 0.75 50 - 60 0.70 be moulded with fingers. 10 - 12 0.75 ? 0.9 60 - 70 0.60 Geologist's pick makes slight 12 -14 0.9 ? 1.05 70 -80 0.55 indentation (sharp end). 14 - 16 1.05 ? 1.2 80 -90 0.50 Very Stiff Indentation by thumb nail difficult. 16 - 18 1.2 ? 1.35 90 -100 0.45 Slight indentation with blow of 18 - 20 1.35 ? 1.5 100 -110 0.40 geologist's pick. Power tools 20 -22 1.5 ? 1.65 110 -120 0.38 required for excavation. 22 -24 1.65 ? 1.8 120 130 0.36 24 -26 1.8 ? 1.95 130 -140 0.34 26 -28 1.95 ? 2.1 140 -150 0.32 28 - 30 2.1 ? 2.25 150 -160 0.31 Hard >30 30 - 31 2.25 ? 2.75 160 - 170 0.30 31 - 32 2.75 ? 3.25 170 - 190 0.29 32 - 35 3.25 ? 3.75 190 - 210 0.28 35 - 38 3.75 ? 4.35 210 - 230 0.27 38 - 42 4.35 ? 5.0 230 - 250 0.26 42 - 50 5.0 ? 6.3 250 - 300 0.25 Very Hard 50 - 65 6.3 ? 8.8 300 - 400 0.22 > 65 8.8 ? 12 400 - 500 0.20 Another widely used correlation for obtaining the undrained shear strength from SPT results is that developed by Stroud (1989). In accordance with this correlation, the undrained strength of a cohesive material including weak rocks is given by: NfCu 1= [5.10] Where: uC = undrained shear strength of the material N = number of SPT blow count 1f = a factor depending on the plasticity of the material. Generally 1f is taken as 5 (Clayton, 1995). Stellenbosch University http://scholar.sun.ac.za 146 Due to its simplicity, Stroud?s correlation was selected for estimation of the undrained shear strength of cohesive materials as well as weak rocks. However, for cases where only SPT refusal has been recorded, the undrained shear strengths were inferred from the consinstency description (Tables 5.8 & 5.9). The undrained shear strength values obtained are presented in tables 5.8. Table 5-10: Undrained shear strength values Driven piles Bored piles Case No. Cu-base Cu-shaft Case No. Cu-base Cu-shaft 63 175 85 122 1000 100 64 150 75 123 1000 400 65 150 75 124 1000 400 66 250 0 125 500 300 67 300 0 126 500 75 68 300 0 127 1000 75 69 300 75 128 300 75 70 300 75 129 500 100 71 200 0 130 500 100 72 250 60 131 500 100 73 250 60 132 1000 1000 74 250 60 133 1000 100 75 250 60 134 1000 100 76 1000 100 135 100 50 77 150 75 136 500 100 78 250 50 137 1000 100 79 350 100 138 1000 100 80 500 50 139 500 100 81 750 175 140 300 100 82 1000 175 141 500 100 83 500 175 142 160 140 84 500 175 143 0 140 85 1000 175 144 450 0 86 1000 175 145 0 350 87 300 100 146 1000 85 88 350 100 147 190 125 89 300 100 148 200 65 90 300 100 149 1000 175 91 300 100 150 500 85 92 250 75 151 500 85 93 250 75 152 1000 75 94 250 75 153 1000 75 95 300 200 154 1000 75 96 300 105 155 1000 75 97 300 105 156 1000 75 98 500 125 157 500 95 99 500 125 158 2000 150 100 300 200 159 450 75 101 300 75 160 1000 125 102 500 75 161 500 140 103 300 75 162 500 140 104 500 100 163 1000 155 105 175 70 164 500 155 106 160 140 165 1000 75 107 450 175 166 1000 100 108 200 60 167 1000 100 109 400 75 168 1000 100 110 400 75 169 1000 100 111 400 75 170 2000 100 112 400 75 171 1000 300 113 500 85 172 1000 300 114 500 85 173 500 200 115 500 75 174 500 200 116 200 75 117 500 115 118 500 115 119 500 135 120 500 135 121 175 105 Stellenbosch University http://scholar.sun.ac.za 147 Close inspection of the undrained shear strength values show that: ? Values for Cu-base are much higher than those for Cu-shaft. This is consistent with the variation of the SPT measurements with depth and is attributed to the fact that the degree of weathering in a typical residual soil tends to decrease with depth. The weathered materials gradually merge into the unweathered rock. Accordingly the consistency varies from soft for the transported soil overlying the residual soil to very stiff for weathered rock. ? The Cu values for materials in which bored piles were constructed are generally higher than those for driven piles. These higher Cu values which are associated with higher consistencies have necessitated the construction of bored piles as it is difficult to drive piles in such materials. In fact in South Africa, the most common type of pile in residual soils is the bored cast in situ pile (Blight, 1991). ? The magnitudes of the Cu values are in accordance with what is expected of a typical profile in a residual soil. The lower values (Cu <100 kPa) denote the predominance of transported materials while middle values (i.e. Cu values of up to 500 kPa) indicate the prevalence of the residual soil. There are also a number of cases depicting rock consistency (i.e. Cu of 1000 kPa or greater). The higher values are mostly associated with Cu-base, thus confirming that piles in residual soils are usually founded on a rock consistency stratum. 5.5 Evaluation of design parameters In the computation of pile capacity using the static formula, design parameters other than the geotechnical properties presented in table 5.4 through table 5.9 are required These design parameters can be classified as those required for piles in non-cohesive materials and those for piles in cohesive materials. For non-cohesive materials the required parameters are the bearing capacity factor (Nq) and the earth pressure coefficient (Ks) while the parameters for piles in cohesive materials are the bearing capacity factor (Nc) and the adhesion factor (?). These parameters are estimated from empirical correlations with soil properties. In any case, currently the process of determining these parameters is not standardised. Consequently different engineers follow different procedures thereby producing different design parameters for even the same site. Stellenbosch University http://scholar.sun.ac.za 148 To select the appropriate design parameters for the South African geological setup, two approaches were followed. The first approach entails using values commonly assumed in practice while the other approach entails conducting a parametric study of the possible values for a given parameter. The second approach is tantamount to assessment of design parameters by back analysis of load test results. 5.5.1 Selection based on commonly assumed values in practice This approach captures the practical design situation where site specific data is limited. The selection of the key pile design parameters such as Nq, Nc, Ks and ? is based on experience or adoption of published empirical correlations. 5.5.1.1 Driven piles in non-cohesive materials Equation 5.7 constitutes the traditional theoretical equation for axial compression capacity of a pile in non-cohesive materials. To use the equation, the parameters Nq, Ks and ? need to be determined on the basis of empirical correlations with the angle of friction. Various authors have proposed various empirical correlations between the angle of friction and Nq. Figure 5.5 illustrates some of the proposed correlations. Vesic (1967) has pointed out that there is a great variation in the theoretical values of Nq, but the values by Berezantzev (1961) appear to fit the pile full-scale test data. For this reason, Nq values by Berezantzev are used by many practicing engineers and researchers as well as some geotechnical codes (Cheng 2004). Accordingly, in this study, Berezantzev Nq values have been adopted. Figure 5.6 shows an enlarged version of the ? - Nq correlation proposed by Berezantzev et.al (1961). Also there is a wide variation of Ks values as obtained by different investigators using different theories as well as back analysis of pile tests data. In general Ks is considered to be greater than the active earth pressure coefficient but less than the passive earth pressure coefficient. It has also been found the value of Ks varies with depth such that it is approximately equal to the passive earth pressure at the top of the pile and less than the at rest earth pressure coefficient at the pile tip (Das, 2003). Further more values of Ks increases with the volume of displacement leading to smaller values for small displacement piles such as H piles and bored piles and higher values for large displacement piles. Generally the value of Ks is dependent on friction angle of the material, pile installation method and the pile characteristics (i.e. size and materials). Although the values of Ks are dependent on Stellenbosch University http://scholar.sun.ac.za 149 these factors, in practice generic values are used depending on whether the pile is driven or bored. For driven piles values ranging from 1 to 2 have been used. Based on this range of values a value of 1.5 was chosen. 1 10 100 1000 10000 25 30 35 40 45 Phi N q De Beer (1945) Meyerhof (193) Canquot - Kerisel (1956) Brinch Hansen (1961) Skempton et.al (1953) Brinch Hansen (1951) Berezantzev (1961) Vesic (1963) Vesic (1972): Ir =200 Vesic (1972): Ir = 60 Terzaghi (1943): General shear Terzaghi (1943): Local shear Figure 5.5: Bearing capacity factors for piles in cohesionless soils (Based on table by Coyle and Castello, 1981) Stellenbosch University http://scholar.sun.ac.za 150 10 100 1000 25 30 35 40 Phi N q Figure 5.6: ? ? N correlation by Berezantzev et.al (1961) 5.5.1.2 Bored piles in non-cohesive materials Equation 5.7 is also used for computation of the predicted capacity of bored piles in non- cohesive materials. However, different values of Nq, Ks, and ? are used to reflect the loosening effects of the boring operation. Again, ? - Nq correlation proposed by Berezantzev et.al (1961) was selected due to its popularity. However ? values were reduced by 3? to account for the loosening effects of the boring operation as proposed by Paulos and Davis (1980). For bored piles, generic values of Ks range from 0.5 to 0.9. Ks value of 0.7 is commonly assumed and accordingly the same value was assumed in this analysis. 5.5.1.3 Driven piles in cohesive materials For cohesive materials equation 5.8 is the governing theoretical equation for determination of axial compressive pile capacity. It is apparent from equation 5.8 that in addition to the undrained shear strength, the parameters Nc and ? are required for the computation. The dimensionless bearing capacity factor Nc for piles has been studied by many researchers. Outcomes of some of the studies are as follows: Stellenbosch University http://scholar.sun.ac.za 151 ? Meyerhof (1951) reported values of Nc = 9.3 and 9.8 for smooth and rough bases respectively. ? After examining various analytical and experimental values, Skempton (1951) concluded that Nc = 9 was accurate enough for practical designs. ? Whitaker and Cook (1966) suggested values of 8 to 8.5 for small diameter piles and Nc = 6.75 for large diameter piles (D? 600 mm). However, they suggested the adoption of Nc = 9, with a factor of 0.75 being introduced for large diameter piles. ? For expanded base piles, the local literature quote values of Nc ranging from 12 to 40. This is attributed to the high impact energy of the hammer when building the expanded base. Based on this information, Nc = 9 and Nc = 16 were adopted for plain and expanded base piles respectively. Concerning the adhesion factor, also different values have been published. Generally the parameter ? depends on the nature and strength of the material, pile dimensions, and method of pile installation. However values derived by Tomlinson (1970) are widely used. This is because the derivation of these values takes into account factors that influence ?. Therefore Tomlinson?s ?. values were adopted. 5.5.1.4 Bored piles in cohesive materials Generally Nc values adopted for bored piles are the same as for driven piles. Therefore again Nc = 9 and Nc = 16 were adopted for plain and expanded base piles respectively. However adhesion factors for bored piles differ from those for driven piles. Again various correlation schemes between the undrained shear strength and adhesion factors have been published (e.g. Byrne et al 1995 and Coduto, 1994). However, the use of these correlation schemes did not yield good results in term of the variability of resulting model factor statistics. Consequently a new scheme based on a combination of local and international practices was devised for bored piles. Based on this new scheme, the adhesion factors and the corresponding Cu values are presented in table 5.11. Table 5-11: Adhesion factors for bored piles in cohesive materials Undrained shear strength (Cu) ?-value >75 0.8 76-150 0.6 < 150 0.45 Stellenbosch University http://scholar.sun.ac.za 152 5.5.2 Selection based on results of a parametric study In the previous section, it was established that for a given design parameter (e.g. Nc) a wide range of possible values have been proposed in the literature. Most of the design parameters have been derived from empirical correlations based on databases specific to certain geological setup. Extrapolation of the correlations to the Southern African distinct geologic environment may not give satisfactory results. Therefore in the absence of local correlations, it is important to establish which of the reported parameters suit the local conditions. This was accomplished by performing a parametric study on all design parameters that are selected subjectively. For a given design parameter, the parametric study entailed using the possible values that have been reported in the literature to calculate the predicted capacity. The predicted capacities were then compared with their respective measured capacities. Within a given range, the value of a given parameter that yields the best fit to the measured capacity was then considered to be the most appropriate under the Southern African soil conditions. The criterion for determining ?best fit? was based on the equation of the best fit line of predicted versus measured capacity with the corresponding coefficient of determination (R2). On the basis of regression analysis, the general equation of the best fit line is given by: pfit bQQ = [5.11] in which fitQ is the least squares average measured capacity corresponding to a given predicted capacity values; b is a regression constant denoting the slope of the line; and pQ is the predicted capacity. Associated with each regression equation is the coefficient of determination (R2). This is a statistical measure of goodness of fit between the predicted and measured values. More specifically, R2 measures the proportion of the total variance in the dependent variable explained by the independent variable. For the purposes of this section, was taken as a measure of the degree of agreement between the measured capacity and predicted capacity. For a perfect fit, b = 1 and R2 = 1. These perfect fit values were used as the datum for evaluating the degree of fit achieved by specific values of the design parameter of interest. Stellenbosch University http://scholar.sun.ac.za 153 Accordingly for a range of values of a given design parameter, the value that yields a relatively high R2 and a value of b that is relatively closer to 1 gives the best fit between predicted capacity and measured capacity. Design values selected on the basis of best fit principle described above at least avoids the use of conservative parameters as within the framework of reliability based design, conservatism and safety are best introduced into the design process through partial factors. Given the transparency of the procedure for their derivation, such design parameters provide guidance on what constitutes appropriate design values in South Africa. Further more, such values can be tabled at a code committee for further discussion and perhaps adopted as normative values by the impending South African geotechnical code. This avoids the current chaotic situation where the selection of the design values is left to the discretion of the designer. 5.5.2.1 Parameters for driven piles in non-cohesive materials As already noted, the required parameters are Nq, Ks, and ?. However, the use of Nq values by Berezantzev et.al (1961) and ? = 0.75? have become very popular and therefore they were maintained in this analysis. Conversely, there is no common preference for Ks values. To establish which value out of the range of Ks values proposed in the literature (i.e. Ks of 1 ? 2) is more appropriate for the South African conditions, a parametric study of the values was performed. The regression parameters (measures of fit) for the various Ks values are presented in table 5.12. Further inspection of table 5.12 lead to the following observations: ? The parameter b decreases with the increase in Ks value. ? The coefficient of determination slightly increases with increase in Ks value within a range of Ks of 1.0 ? 1.3, and then starts to decrease with the increase in Ks values. On the basis of R2, it is apparent from table 5.12 that the best fit is produced by Ks value of 1.3. Therefore Ks = 1.3 was considered to be more appropriate. The Qi versus Qp plot for the selected case of Ks = 1.3 is shown in figure 5.7. Stellenbosch University http://scholar.sun.ac.za 154 Table 5-12: Variation of the regression parameters b and R2 with Ks Ks b R2 1.0 1.090 0.432 1.1 1.070 0.439 1.2 1.050 0.444 1.3 1.030 0.446 1.4 1.010 0.445 1.5 0.990 0.443 1.6 0.970 0.439 1.7 0.950 0.434 1.8 0.940 0.427 1.9 0.920 0.419 2.0 0.900 0.410 Figure 5.7: Scatter plot of Qi Vs Qp for Ks = 1.3 5.5.2.2 Bored piles in non-cohesive materials Also for this design situation, Nq and ? values were fixed while the Ks values were varied from 0.5 ? 0.9 in accordance with the generic values reported in the literature. The variation of the measure of fit parameters with Ks value is presented in table 5.14. Examination of table 5.13 lead to the following observations: ? As was the case for driven piles, the parameter b decreases with the increase in Ks values. On the conservative side, the b value closest to 1 corresponds to Ks value of 0.8. ? The R2 values are almost the same for entire range. The range of values for bored piles is very narrow and accordingly a small increment of 0.1 from one point to the next can not be expected to bring drastic changes in the degree of fit. y = 1.031x R2 = 0.445 0 1000 2000 3000 4000 5000 6000 7000 0 2000 4000 6000 Qp (kN) Q i ( kN ) Ks=1.3 Linear (Ks=1.3) Stellenbosch University http://scholar.sun.ac.za 155 On the basis of R2 it can be concluded the best fit is achieved with a Ks range of 0.6 ? 1.0. To select the most appropriate value from within this range, use was made of the parameter b. The Ks value corresponding to the parameter b of relatively more close to 1 was taken as the most appropriate value. Inspection of table 5.13 shows that the above criterion is satisfied by Ks = 0.8 with b = 1.03 and Ks = 0.9 with b = 0.99. Comparison of these b values indicates that the b value for Ks of 0.8 is on the conservative side while that for Ks of 0.9 is on the unconservative side. Since it is better to err on the conservative side, Ks value of 0.8 was selected as the most appropriate. The plot of Qi versus Qp for the selected case of Ks = 0.8 is shown in figure 5.8. Table 5-13: Variation of M statistics and resistance factors with Ks Ks b R2 0.5 1.150 0.73 0.6 1.110 0.74 0.7 1.070 0.74 0.8 1.030 0.74 0.9 0.990 0.74 1.0 0.960 0.74 Figure 5.8: Scatter plot of Qi Vs Qp for Ks = 0.8 5.5.2.3 Driven piles in cohesive materials For piles in cohesive materials, input geotechnical parameters that are not well defined include the bearing capacity factor Nc and the adhesion factor ?. Due to their popularity, ? values by Tomlinson have been maintained while the Nc values were varied from 9 to 30. y = 1.0272x R2 = 0.7397 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 Qp (kN) Q i ( kN ) Ks=0.8 Linear (Ks=0.8) Stellenbosch University http://scholar.sun.ac.za 156 Varying of Nc values was only applicable to expanded base piles (Franki piles). For the plain piles the value of Nc = 9 was maintained on the grounds that this is the value that is commonly used worldwide. The resultant measures of fit are presented in table 5.14. The following conclusions were drawn from the inspection of table 5.14. ? There is a rapid decrease in the parameter b with the increase in Nc values, suggesting that lower Nc values (Nc of 9 ? 14) tend to under-predict the predicted capacity while higher Nc values (Nc > 14) tend to over-predict the predicted capacity. ? There is also a rapid decrease in R2 values with the increase in Nc values, implying a better fit for lower Nc values. ? In general Nc values ranging from 9 to 14 seem to yield better overall prediction (i.e. under predicts theoretical capacity and relatively higher R2 values). The results support the lower range of Nc values suggested by the local literature. Based on the above observations, the most appropriate Nc value for the design of expanded base piles under the Southern African soil conditions falls within a range 9 ? 14. Within this narrow range of Nc values, R2 values corresponding to Nc of 9 and 10 are the highest. However the corresponding b values are appreciably greater than 1 leading to a significant degree of conservatism (33% for Nc = 9 and 24% for Nc =10). Conversely the Nc value of 14 produces the best b value but the lowest R2 value. The fit parameters for Nc value of 12 are somewhat between the extremes. The R2 value is quite close to that shown by Nc values of 9 and 10 while the degree of conservatism is just 10 %. Therefore the Nc value of 12 yields the best combination of R2 and b and was consequently selected as the most appropriate value. The Qi versus Qp for the selected case of Nc =12 is shown in figure 5.9. Table 5-14: Variation of fit parameters b and R2 with Nc Nc b R 2 9 1.33 0.713 10 1.24 0.717 12 1.10 0.707 14 0.98 0.687 16 0.89 0.662 18 0.81 0.635 20 0.74 0.609 22 0.68 0.584 24 0.63 0.560 26 0.59 0.538 28 0.55 0.52 30 0.52 0.498 Stellenbosch University http://scholar.sun.ac.za 157 Figure 5.9: Scatter plot of Qi Vs Qp for Nc = 12 5.5.2.4 Bored piles in cohesive materials In exception of two pile cases, all the piles in the database for this category are plain piles (i.e. without expanded bases). For piles without expanded base, normally Nc is taken as 9. For the two cases with expanded bases, an Nc value of 12 was assumed as was the case for driven piles. Again adhesion factors previously assumed were adopted. The Qi versus Qp plot for bored piles in cohesive materials is shown in figure 5.10. From figure 5.10, the fit parameters are; b= 1.12 and R2 = 0.87. Relative to parameter for cases previously considered, these values were considered reasonable. y = 1.1015x R 2 = 0.7078 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 Qp (kN) Q i ( kN ) Nc=12 Linear (Nc=12) Stellenbosch University http://scholar.sun.ac.za 158 Figure 5.10: Scatter plot of Qi Vs Qp for BC 5.5.3 Comparison of results from the two approaches The range of parameters for the parametric study included values that were selected in section 5.4.1. Table 5.15 presents a summary of the measure of fit parameters. The parameters have been separated into those given by the commonly assumed design parameters (approach 1) and those selected on the basis of the best fit principle (approach 2). It is apparent from this table that for a given pile class, R2 values for design parameters selected on the basis of best fit principle are slightly higher than those for commonly assumed parameters in practice. An exception to this trend is the parameters for BC which are the same for both approaches. The results for BC are attributed to the fact that this class contains only two expanded base piles. Since the varying of Nc was applicable to expanded base piles only, only two cases were affected and this could not change the overall results. Nonetheless, on the basis of a general improvement of the R2 values, the design parameters selected on the basis of best fit principle were considered as the most appropriate for the South African conditions. Accordingly results from the second approach were adopted for further analysis. Table 5-15: Comparison of measure of fit parameters Pile class Approach 1 Approach 2 b R 2 b R 2 DNC 1.09 0.45 1.01 0.47 BNC 1.15 0.73 1.03 0.74 DC 1.02 0.69 1.14 0.71 BC 1.12 0.87 1.12 0.87 y = 1.1236x R2 = 0.8669 0 2000 4000 6000 8000 10000 12000 14000 0 2000 4000 6000 8000 10000 12000 Qp (kN) Q i ( kN ) Stellenbosch University http://scholar.sun.ac.za 159 5.5.4 Selected design parameters The selected design parameters are summarised in table 5.16. Table 5-16: Selected design parameters Parameter Value/correlation scheme 1. Driven piles in non-cohesive materials (a) ? (b) Nq (c) Ks (d) ? Peck, Hanson and Thornburn (1974) Berezantzev et.al (1961) 1.3 0.75? 2. Bored piles in non-cohesive materials (a) ? (b) Nq (c) Ks (d) ? Peck, Hansen and Thorburn (1974) Berezantzev et.al (1961), but ? reduced by 3 0.8 ? 3. Driven piles in cohesive materials (a) Cu (b) ? (c) Nc Stoud (1989) Tomlison (1970) 9 for plain piles 12 for expanded base piles 4. Bored piles in cohesive materials (a) Cu (b) ? (c) Nc Stroud (1989) Table 5.11 9 for plain piles 12 for expanded base piles 5.6 Predicted capacities The geotechnical design parameters determined as per table 5.16 were used as input into the relevant static formula to compute the predicted capacity for each case. The predicted capacities obtained are presented in table 5.17. The following observations were drawn from the detailed examination of table 5.17: ? Qp ranges from 1101 ? 5154 kN for driven piles in non-cohesive materials, 472 - 6754 kN for bored piles in non-cohesive materials, 556 ? 5908 kN for driven piles in cohesive materials and 462 ? 10172 kN for bored piles in cohesive materials. ? It is apparent from the above ranges of Qp values that for a given soil type, the upper bound values for bored piles are higher than their respective values for driven piles. This is in line with the earlier observation that bored piles are normally installed in very stiff materials, hence the higher resistances. Stellenbosch University http://scholar.sun.ac.za 160 Table 5-17: Predicted capacities Piles in non-cohesive materials Piles in cohesive materials Driven Bored Driven Bored Case Qp (kN) Case Qp (kN) Case Qp (kN) Case Qp (kN) 1 3078 30 1321 63 2361 122 4807 2 1666 31 3783 64 626 123 6447 3 3389 32 2705 65 963 124 6521 4 2001 33 1084 66 1325 125 1398 5 2564 34 1202 67 1590 126 2062 6 2564 35 1202 68 2290 127 3280 7 1951 36 1131 69 3547 128 2545 8 1628 37 1202 70 3547 129 3040 9 1572 38 1178 71 1460 130 2235 10 2380 39 6754 72 2469 131 4179 11 2380 40 1287 73 2469 132 3658 12 2888 41 1145 74 2469 133 1196 13 1745 42 940 75 2469 134 2333 14 4013 43 679 76 3699 135 707 15 2109 44 1226 77 2069 136 1225 16 1592 45 1583 78 2093 137 975 17 1802 46 605 79 3100 138 3223 18 2768 47 472 80 3398 139 1221 19 3209 48 793 81 5908 140 834 20 2718 49 1060 82 2740 141 1444 21 1978 50 916 83 3186 142 2375 22 3648 51 1132 84 1325 143 1948 23 1080 52 2028 85 5908 144 1184 24 2136 53 3051 86 2770 145 462 25 2549 54 2028 87 1340 146 2392 26 1887 55 3051 88 1549 147 907 27 4766 56 2563 89 1340 148 1818 28 4797 57 2478 90 1250 149 5869 29 4968 58 2111 91 1069 150 1292 59 2451 92 694 151 1292 60 2974 93 1090 152 2110 61 3644 94 2012 153 2110 62 3739 95 697 154 2110 96 877 155 2110 97 556 156 2110 98 800 157 1602 99 800 158 8906 100 2135 159 1156 101 1265 160 2916 102 3675 161 5088 103 1265 162 5088 104 2977 163 7648 105 2580 164 4721 106 3417 165 1955 107 2986 166 4230 108 1736 167 4196 109 2851 168 6153 110 2903 169 4207 111 2903 170 10172 112 2903 171 3294 113 3588 172 6266 114 3588 173 1901 115 2621 174 3515 116 706 117 3248 118 3248 119 3221 120 3221 121 1566 Stellenbosch University http://scholar.sun.ac.za 161 Chapter 6 STATISTICAL ANALYSIS OF THE MODEL FACTOR REALISATIONS 6.1 Introduction For reliability calibration, the statistics (mean, standard deviation and coefficient of variation) as well as the type of distribution that best fit the data need to be determined for each random variable considered in the limit state function. Normally the various design variables can be classified into resistance and load related variables. Therefore load and resistance statistic are required. However, this study focuses on the resistance statistics as the load statistics are normally provided by loading codes. The required statistics are generated from the measured value of the random variable (resistance or load) and the predicted value yielded by the theoretical design model. Therefore the statistics are generally represented in terms of the the ratio of the measured to predicted values. This ratio is generally referred to as the bias. For this study the bias factor is defined as the ratio of the measured pile capacity over the predicted pile capacity. 6.2 Bias factors generated Once the interpreted capacities were compiled from pile load tests as described in chapter 4, the predicted capacities were computed for the same pile type, soil conditions and installation methods using the geotechnical design parameters developed in chapter 5 as input in the static formula. The bias was then computed for each data case. A spread sheet was set to automate the computation of both the predicted capacity and the bias factor. The generated bias factors are presented in table 6.1. However, table 6.1 does not convey much information about the general characteristics of the generated bias factors, thereby necessitating a further statistical analysis of the presented data. The statistical characterisation of the bias factor realisations is presented in the subsequent sections. 6.3 The bias factor as a measure of various sources of uncertainties In principle, the bias factor accounts for all the sources of uncertainties pointed out in Chapters 1 (i.e. model error, systematic error, inherent spatial variability, statistical error and load tests related errors). However due to the high level of expertise and experience of companies in charge of SPT tests and the pile load tests measurements errors are minimized. Also inherent spatial variability is minimized due to averaging effects along Stellenbosch University http://scholar.sun.ac.za 162 Table 6-1: Generated bias factors Pile in non-cohesive materials Piles in cohesive materials Driven piles Bored piles Driven piles Bored piles Case M Case M Case M Case M 1 0.926 30 1.041 63 1.394 122 0.999 2 1.530 31 1.216 64 1.086 123 0.636 3 0.797 32 1.109 65 1.205 124 1.104 4 0.950 33 0.968 66 1.358 125 0.544 5 1.423 34 1.144 67 1.761 126 1.115 6 0.811 35 1.148 68 1.485 127 0.945 7 0.692 36 0.928 69 1.015 128 1.320 8 1.401 37 1.019 70 0.578 129 0.872 9 1.399 38 0.713 71 1.733 130 0.805 10 1.722 39 0.800 72 1.276 131 0.885 11 1.630 40 0.971 73 1.008 132 0.801 12 0.597 41 1.310 74 0.931 133 1.422 13 1.777 42 1.032 75 0.867 134 0.815 14 0.847 43 0.796 76 1.644 135 0.778 15 1.458 44 0.490 77 1.329 136 0.959 16 1.131 45 0.742 78 1.042 137 1.107 17 1.332 46 1.322 79 1.307 138 1.086 18 0.795 47 0.933 80 1.530 139 1.015 19 0.678 48 0.789 81 0.948 140 0.990 20 0.956 49 0.967 82 1.014 141 1.004 21 0.748 50 1.267 83 1.444 142 1.263 22 0.603 51 1.325 84 1.796 143 1.258 23 1.213 52 0.991 85 1.185 144 1.208 24 1.592 53 1.967 86 1.191 145 1.104 25 1.099 54 0.592 87 1.306 146 1.505 26 0.822 55 1.524 88 1.485 147 1.268 27 1.259 56 0.659 89 1.388 148 1.271 28 1.001 57 0.654 90 1.200 149 1.448 29 1.057 58 0.781 91 1.544 150 0.952 59 0.857 92 1.413 151 1.408 60 0.901 93 1.211 152 1.223 61 1.002 94 1.561 153 1.265 62 1.498 95 1.378 154 1.322 96 0.969 155 1.374 97 0.990 156 1.991 98 1.563 157 1.748 99 1.313 158 1.078 100 0.843 159 1.437 101 0.932 160 1.646 102 1.388 161 1.386 103 1.059 162 1.160 104 0.967 163 1.268 105 1.008 164 1.483 106 1.083 165 0.961 107 1.055 166 1.284 108 0.657 167 0.715 109 1.263 168 1.016 110 0.861 169 1.058 111 1.516 170 1.278 112 1.240 171 1.002 113 0.697 172 0.768 114 0.861 173 1.157 115 0.595 174 1.508 116 0.694 117 1.068 118 0.831 119 0.683 120 0.767 121 1.354 the pile length. This leaves out model error as the predominant uncertainty reflected by the bias factor. Stellenbosch University http://scholar.sun.ac.za 163 That the bias factor mainly reflects model uncertainty is further reiterated by the following studies: Phoon and Kulhawy (2005) compared model pile test and full scale pile test results in terms of histograms and simple statistics. The model tests were conducted under controlled laboratory conditions and hence uncertainties arising from evaluation of soil parameters, construction variability and measurement errors associated with load tests were minimal. The results were such that the histograms and the simple statistics for the two sets of data were similar. Further more, p-values for equal medians statistical test showed that the null hypothesis of equal medians cannot be rejected at the customary 5% level of significance. It was concluded from these results that the ratio of measured to predicted capacity primarily represents model uncertainty. Further more, the study by Ronold and Bjerager (1992) mentioned in Chapter 1 demonstrated that model uncertainty exclusively is the most important source of uncertainty by a contribution to the total uncertainty by close to 100%. It was further concluded that other sources of uncertainties such as evaluation of soil parameters could just as well be neglected. It is appreciated by many geotechnical engineers that in pile foundations, there is significant uncertainty about pile behaviour and calculation models available to describe this behaviour. For example, in the drafting of Eurocode 7 it was reckoned that the major uncertainty was not the strength of the in-situ ground but the way the construction would interact with it (Simpson, 2000). Therefore the partial factor required is largely a factor on the resistance model rather than on the strength of material. This led to the exception in Design Approach 1 that for piles factors should be applied to the resistance rather than on the materials as is the case for other geotechnical structures. In accordance with the above discussion, in this study, the bias factor is considered to represent model uncertainty. Therefore the term model factor instead of bias factor will be used to refer to the ratio of the measured over predicted capacity. 6.4 Statistical analysis of the model factor data The model factor realisations presented in table 6.1 are just a raw dataset representing a sample from the population of interest. Such raw data do not convey much information and therefore need to be reduced to manageable forms to facilitate its interpretation. In order to reduce the model factor realisation data to a manageable form, statistical analysis was Stellenbosch University http://scholar.sun.ac.za 164 carried out. The analysis comprised of (a) graphical representation by histograms, (b) outliers detection and correction of erroneous values, (c) using the corrected data to compute the sample moments (mean, standard deviations, skewness and kurtosis), (d) determining the appropriate distribution for the model factor and (e) investigation of correlation with underlying pile design parameters. 6.4.1 Histogram of the model factors As already stated, the compiled model factors are just a set of raw data in a form of unorganized list of numbers. It is not easy to spot patterns and other useful information from such a data set. Therefore the data need to be organised and summarised in useful ways to make them more informative. A useful first step in the presentation of observed data is the graphical display. The large information content of pictures is aptly captured by the old adage that ?a picture is worth a thousand words?. The same principle holds true for graphical displays as they can uncover hidden or at least not readily noticed features in the data. The most common method of graphical data presentation is the histogram. The histogram condenses a set of data for easy visual comprehension of its characteristics. Visual inspection of the histogram often brings out the following features that are not immediately apparent from a given set of data: - Immediate impression of the range of the data, its most frequently occurring values, and the degree to which it is scattered about the mean, - Outlying observations which somehow do not fit the overall pattern of the data, - The exhibition of two or more peaks which may imply an inhomogeneous mixture of data from different samples, - Whether the data is symmetric or asymmetric, - Indication of the underlying theoretical distribution for the data. The fact that the histogram is generated by plotting the proportional frequency of observations lying within given numerical intervals renders its appearance to be influenced by the choice of the number of intervals or cells. Histograms from the same data would look very different if the number of cells is different. The produced shape affects the visual interpretation of the data. There are no universally applicable rules for determining the number of cells. However, a practical guidance has been reported in some geotechnical Stellenbosch University http://scholar.sun.ac.za 165 literature (e.g Smith, 1986; Baecher and Christian, 2003). In accordance with this guide, the number of intervals for a given data size is given by: nk 10log3.31+= Where k is the number of cells; n is the number of data points. Nevertheless, experience and experimentation with different cell numbers usually provide the best guidance (Baecher and Christian, 2003). In this study the cell number for a given sample size was obtained by the combination of the above guide and experimentation. First trial cell number was by the guide and the histogram drawn. Next histograms with cell numbers slightly less and more than the guide cell number were drawn. The various histograms were then compared and the one with a well defined shape and a smooth variation of the observed frequencies was chosen. The resulting histograms are presented in figure 6.1. Visual examination of the histograms shows similar characteristics for the four histograms. These characteristics are: ? Most of the data points are clustered around the mean value. ? The histograms are unimodal, i.e. they have one point of concentration or a single peak indicating that the data is from the same source in terms of geological set up and piling practice. ? The data are not symmetrical about the peak frequency, suggesting that the underlying distribution for the data is not normal. ? There are no extreme values (outliers) in the data. ? The histograms are skewed to the right, i.e. they show a long right tail of relatively large values. ? The widths of the histograms are an indication of the scale of variability. Given that the histograms would look very different if cell numbers used were different, over-interpretation of results should be avoided. For this reason the most reliable insight obtained from the histograms is immediate impression of the range of the data, its most frequently occurring values, and the degree to which it is scattered about the mean. Stellenbosch University http://scholar.sun.ac.za 166 0.60 0.76 0.93 1.09 1.26 1.43 1.59 Model factor 0 1 2 3 4 5 6 7 N o of o bs 0.49 0.71 0.94 1.16 1.38 1.61 1.83 Model factor 0 2 4 6 8 10 12 14 N o of o bs (a) Driven piles in cohesionless materials (b) Bored piles in non-cohesive materials 0.58 0.75 0.93 1.10 1.27 1.45 1.62 1.80 Model factor 0 2 4 6 8 10 12 14 16 18 N o of o bs 0.54 0.75 0.96 1.16 1.37 1.58 1.78 1.99 Model factor 0 2 4 6 8 10 12 14 16 18 N o of o bs (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 6.1: Histograms of the model factors . 6.4.2 Detection of data outliers Data outliers are extreme (high or low) values that appear to deviate markedly from the main body of a data set. In general, outliers in data are attributed to human error, instrument error and natural deviations in populations. There is also a possibility that the outlier is a Stellenbosch University http://scholar.sun.ac.za 167 correct observation representing the true state of nature. In the model factors data considered in this study, the most likely causes of outliers include: ? Pile load tests related errors (inaccuracy in measuring devices calibration or data recording); ? SPT blow count related errors due to equipment or procedure; ? Data entry errors in processing pile load test results to determine the interpreted capacities; ? Data entry errors in processing the measured SPT N-values to determine design geotechnical properties. The presence of outliers may greatly influence any calculated statistics leading to biased results. For instance, they may increase the variability of a sample and decrease the sensitivity of subsequent statistical tests (McBean and Rovers, 1998). Therefore prior to further numerical treatment of samples and application of statistical techniques for assessing the parameters of the population, it is absolutely imperative to identify extreme values and correct erroneous ones. A number of procedures have been developed to detect outliers. The procedures can be divided into univariate and multivariate approaches. In a univariate approach, screening data for outliers is carried out on each variable while in the multivariate approach, variables are considered simultaneously. Since there may be some correlation between the variables, the multivariate approach is considered to be statistically correct as it accounts for the correlation (Robinson et.al, 2005). However in this study, both univariate and multivariate approaches will be employed to detect outliers in the data. 6.4.2.1 Univariate approach With this approach, the ratio of the interpreted capacity to predicted capacity (model factor) is the only variable considered. Two methods were used to detect outliers in the model factor data set. The methods include (i) sample z-score approach and (ii) box plot 6.4.2.1.1 Sample z-score method The z score is a measure of the number of standard deviations that an observation is above or below the mean. A positive z-score indicates that the observation is above the mean while Stellenbosch University http://scholar.sun.ac.za 168 a negative z-score that the observation is below the mean. The z-score of an observation in a given data set is given by the expression: s xx z ?= Where: x = original data value; x = the sample mean; s = the sample standard deviation; z = the z-score corresponding to x. According to Chebychev?s rule, in any distribution, the proportion of scores between the mean and k standard deviation contains at least 1-1/k2 scores. This rule gives at least 75% of the scores between the mean and two standard deviations (? 2s), and 89 % of the scores would lie between the mean and three standard deviations (? 3s). Another well known data distribution rule is the Empirical Rule applicable to normally distributed data. According to the empirical rule, approximately 68% of the z-scores reside between mean and ? 1s, approximately 95% of the scores resides between mean and ? 2s, and approximately 99% of the scores reside between mean and ? 3s. Both these rules have led to the general expectation that almost all the observations in a data set will have z-score less than 3 in absolute value. This implies that all the observation will fall within the interval )3,3( sxsx +? . Therefore the observation with z-score greater than ? 3 is considered an outlier. The above principles were applied to the data. First the z-score for each data point was determined. The z-scores were then plotted against the model factors (original data values). The plots of z-score versus model factor are presented in figure 6.2. Examination of figure 6.2 shows that: ? For driven piles in non-cohesive materials (fig. 6.2 (a)), there are no data point with a z- score of greater than 3. Therefore there are no outliers in this data set. ? For bored piles in non-cohesive materials (fig.6.2 (b)), there is one data point plotting above a z-score of 3. The data point is for pile case number 55 with a model factor of 2.55. ? For the driven piles in cohesive materials (fig. 6.2 (c)), there are no data points with z- scores of greater than 3. ? For the bored piles in cohesive materials (fig. 6.2 (d)), there are also no data points with z-scores of greater than 3. Stellenbosch University http://scholar.sun.ac.za 169 Based on the examination of the z-scores for the four data sets, it can be concluded that there are no outliers in the data sets for driven piles in non-cohesive materials, driven piles in cohesive materials and bored piles in cohesive materials. However for bored piles in non- cohesive materials, there is one outlier in the data set. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0.000 0.500 1.000 1.500 2.000 Model factor z- sc o re (a) Driven piles in non cohesive materials -2 -1 0 1 2 3 4 0.000 0.500 1.000 1.500 2.000 2.500 3.000 Model factor z- sc or e (b) Bored piles in non-cohesive materials Stellenbosch University http://scholar.sun.ac.za 170 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0.000 0.500 1.000 1.500 2.000 Model factor z- sc or e (c) Driven piles in cohesive materials -3 -2 -1 0 1 2 3 4 0.000 0.500 1.000 1.500 2.000 2.500 Model factor z- sc or e (d) Bored piles in cohesive materials Figure 6.2: Z-scores vs model factor plots 6.4.2.1.2 Box plot method The box plot method is a more formalised statistical procedure for detecting outliers in a data set. A box plot displays a 5-number summary in a graphical form. The 5-number summary consists of; the most extreme values in the data set (the maximum and minimum values), the lower and upper quartiles, and the median. These values are presented together and ordered from lowest to highest: minimum value, lower quartile, median value, upper quartile, and largest value. Each of these values describe a specific part of a data set: the median identifies the centre of a data set; the upper and lower quartiles span the middle half of a data set; and the highest and lowest observations provide additional information about the actual dispersion of the data. Stellenbosch University http://scholar.sun.ac.za 171 In using the box plot to identify outliers in the data set, the inter-quartile range (IQR) is required. The inter-quartile range is the difference between the upper quartile and the lower quartile. Any observation that is more than 1.5 IQR beyond the upper and lower quartiles is regarded as an outlier. In this study the programme STATISTICA was used to construct the box plots for the four data sets. The resulting box plots are presented in figure 6.3. In figure 6.3, the vertical axis represents the response variable, which in this case is the model factor. To the right of each box plot is a legend explaining the meaning of the various symbols used to represent the data summaries. Visual inspection of figure 6.3 for outliers shows the following results: ? For driven piles in non-cohesive materials (fig. 6.3 (a)), there are no outliers in the data set. ? For bored piles in non-cohesive materials (fig. 6.3 (b), there are two data point marked as outlier or extreme value. The data points correspond to pile cases number 53 and 55. ? For driven piles in cohesive materials (fig. 6.3 (c)), there are no data points marked as outliers. ? For bored piles in cohesive materials (fig. 6.3 (d)), two data points have been tagged as an outlier or extreme value. These data points are again pile cases number 156 and 158. Median = 1.0402 25%-75% = (0.8112, 1.2804) Non-Outlier Range = (0.5973, 1.5919) Outliers Extremes M-DNC 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 (a) Driven piles in non-cohesive materials Stellenbosch University http://scholar.sun.ac.za 172 Median = 0.9914 25%-75% = (0.7956, 1.1482) Non-Outlier Range = (0.4895, 1.4977) Outliers Extremes #53 #55 M-BNC 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 (b) Bored piles in non-cohesive materials Median = 1.1914 25%-75% = (0.9478, 1.3878) Non-Outlier Range = (0.578, 1.7957) Outliers Extremes M-DC 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (c) Driven piles cohesive materials Stellenbosch University http://scholar.sun.ac.za 173 Median = 1.157 25%-75% = (0.9614, 1.3597) Non-Outlier Range = (0.5437, 1.9188) Outliers Extremes #156 #158 M-BC 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 (d) Bored piles in cohesive materials Figure 6.3: Box plots of model factors 6.4.2.2 Multivariate approach The main variables in the computation of model factor realisations are the predicted (Qp) and interpreted (Qi) capacities. Outliers in either of these two variables affect the calculated model factor. Based on correlative behaviour of the variables, some results with inconsistent behaviour, such as lying a significant distance from the general trend of the data is indicative that such values may be outliers. The multivariate approach adopted was limited to scatter plots of Qi vs Qp. The scatter plots of Qi vs Qp are presented in figure 6.4. Inspection of figure 6.4 shows the following: ? For driven piles in cohesionless materials (fig. 6.4 (a)), there are no data points that deviate from the general trend and therefore there are no outliers in the data. The results are consistent with the results from the z-score and Box plot approaches. ? For bored piles in non-cohesive materials, (fig. 6.4 (b)) there appear to be two observation that do not follow the general trend and therefore regarded as potential outliers. The data points correspond to pile cases 53 and 55. The pile cases were also detected by the Box plot. Stellenbosch University http://scholar.sun.ac.za 174 ? For driven piles in cohesive materials, (fig. 6.4 (c)) there are no data points that appear to deviate markedly from the main body of a data set. The results agree with the results obtained by the Box plot. ? For bored piles in cohesive materials, (fig. 6.4 (d)), pile cases 158 and 170 appear to be markedly different from the others. Case 158 was also detected by the box plot. Case 156, although tagged as an outlier by the Box plot seems to follow the general trend of the data set. Other possible statistical approaches for detecting outliers include residuals Vs dependent variable (Qi) and Residuals Vs Independent variable (Qp). However, in this study the three methods considered were deemed to be sufficient. 500 1500 2500 3500 4500 5500 Qp 1000 2000 3000 4000 5000 6000 7000 Q i (a) Driven piles in non-cohesive materials Stellenbosch University http://scholar.sun.ac.za 175 #53 #55 0 1000 2000 3000 4000 5000 Qp 0 1000 2000 3000 4000 5000 6000 7000 Q i (b) Bored piles in non-cohesive materials 0 1000 2000 3000 4000 5000 6000 7000 Qp 0 1000 2000 3000 4000 5000 6000 7000 8000 Q i (c) Driven piles in cohesive materials Stellenbosch University http://scholar.sun.ac.za 176 #156 #158 #170 0 1000 2000 3000 4000 5000 6000 7000 8000 Qp 0 2000 4000 6000 8000 10000 12000 14000 Q i (d) Bored piles cohesive materials Figure 6.4: Scatter plots of Qi vs Qp 6.4.2.3 Correction to the identified outliers A total of five observations were detected as potential outliers (i.e cases 53, 55, 156, 158 and 170). It is incorrect to automatically delete a data point once it is identified as a statistical outlier (Robinson et.al, 2005). It is only after examining why a data point is an outlier (i.e. carefully considering all factors) that a data point can be deleted. Accordingly the three data points identified as outliers were carefully examined by double checking the processes of determination of interpreted capacities and computation of predicted capacities. This entailed going back to the original data (pile testing records and derivation of soil design parameters) and checking for recording and computational errors. Following this procedure the corrections were as follows: ? Cases 53 and 55: Examination of records for these cases showed that an uncommon pile installation practice was employed. The steel piles were installed in predrilled holes and then grouted. The strength of the grout surrounding the piles contributed to the high resistance and hence the higher interpreted capacities. Therefore these data points were valid and no corrections were applied to them. However, since the Stellenbosch University http://scholar.sun.ac.za 177 installation procedure for these piles deviates from the normal practice, they represent a different population. These data points were therefore regarded as genuine outliers and were deleted from the data set. ? Case 156: This case was detected by the univariate approaches. However, on the scatter plots of Qi vs Qp, these data points seem to fit into the general pattern of the data sets (Fig. 6.4d). Therefore no correction was necessary for this data point. ? Case 158: No mistake was found with the determination of the interpreted capacity. However some details regarding the foundation materials were missed leading to incorrect base and shaft resistances. Revisiting the data pertaining to this test pile case revealed that the pile had a 2m socket in very soft rock followed by another 2m socket and end bearing in soft rock. In the original computation of the predicted capacity, only socket friction in very soft rock was considered leading to a smaller shaft resistance. Also the end bearing was wrongly taken to be in very soft rock. In general parameters for very soft rock were used in the computation of the predicted capacity. After correcting the rock design parameters to the appropriates ones (soft rock), the predicted capacity increased from 4580 kN to 8906 kN. ? Case 170: Examination of original data revealed a pile diameter of 600 was used in the computation of the predicted capacity thereby giving a smaller capacity of 6776 kN. The correct test pile diameter was 750 mm. With the correct test pile diameter the predicted capacity increased to 10172 kN. Correcting the predicted capacity made the data point for case 170 to fall in the general pattern of the data 6.4.3 Summary statistics After the detection of outliers and correction of erroneous observations, the next step was to summarise the data numerically. Although graphical presentation condenses a set of data for easy visual comprehension of its general characteristics, numerical sample characteristics are required for calculations, statistical testing, and inferring the population parameters. These are quantities used to describe the salient features of the sample. The four main sample characteristics most commonly used in practical applications are: ? Mean (m) ? Standard deviation (s) ? Skewness ? Kurtosis Stellenbosch University http://scholar.sun.ac.za 178 These sample characteristics are also known as sample moments and summary statistics. The mean is the first moment about the origin and is correctly interpreted as the centre of gravity of the frequency distribution of the data along the x-axis. It describes the centre around which the observations in the data are distributed. The mean of a set of n data x = (xi, ?, xn) is given by: ? = = n i ixn m 1 1 [6.1] in which n is the sample size. The standard deviation is a measure of variation and describes the dispersion of the data (i.e. how far the observations are from the centre). It is computed as: ( )2 1? ?= ? n xx s ix [6.2] where x is the sample mean and n is the sample size. The skewness and the kurtosis provide hints on the shape of the underlying distribution for the data. In this regard, the skewness measures the asymmetry of the data distribution while the kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. In accordance with STATISTICA, the two parameters are computed from the following expressions: Skewness = ( ) ( )[ ]33 *2*1 * snn Mn ?? [6.3] where: ( )33 ? ?= xxM i s = the standard deviation n = the valid number of cases Stellenbosch University http://scholar.sun.ac.za 179 Kurtosis = ( ) ( )[ ] ( ) ( ) ( )[ ]4224 *3*2*1 1***3*1* snnn nMMMnn ??? ??+ [6.4] where: ( ) jij xxM ? ?= n = the valid number of cases s = the standard deviation The summary statistics for the model factors are presented in table 6.2. The key statistics are the mean (m) and the standard deviation (s). In addition to the measure of centrality and dispersion, in this study the mean and standard deviation of the bias factor were considered as indicators of the accuracy and precision of the predication method. An accurate and precise method gives mM = 1 and sM = 0 respectively, which means that for each pile case, the predicted pile capacity equals to the measured capacity (an ideal case). However, due to uncertainties mentioned earlier, the results of an ideal case can not be attained in practice. Therefore in reality, the method is better when mM is close to 1 and sM is close to 0. In general when mM > 1, the predicted capacity is less than the interpreted capacity, which is conservative and safe where as when mM < 1, the predicted capacity is greater than the interpreted capacity, which is unconservative and unsafe. Table 6-2: Summary statistics for the model factor Case N Mean Std. Dev. COV Skewness Kurtosis DNC 29 1.06 0.30 0.28 0.171 -1.131 BNC 31 0.98 0.24 0.24 0.054 -0.368 DC 59 1.17 0.30 0.26 0.022 -0.713 BC 53 1.15 0.28 0.24 0.457 0.715 It is apparent fro from table 6.2, that the mean model factor is greater than one for DNC, DC and BC while for BNC, it is less than one. This implies that on average, the static formula is conservative by 6 % in DNC, 17% in DC, 15 % in BC, and unconservative by 2% in BNC. It can be concluded from these results that the use of the static formula with geotechnical parameters recommended in this study, yields slightly conservative theoretical capacities for DNC, DC, and BC. Conversely for DNC the approach is marginally unconservative. Stellenbosch University http://scholar.sun.ac.za 180 A comparison of the standard deviations or coefficient of variations, for the four cases indicates small differences. However there seem to be a distinct trend that is influenced by the pile installation method. In this regard, driven piles depict higher variability compared to bored piles irrespective of materials type. Further more, for a given installation method (driven or bored) the variability in non-cohesive materials is higher than that in cohesive materials. This is attributed to the fact that in cohesive materials the undrained shear strength derived from the SPT measurement is directly used in the computation of pile capacity while in non-cohesive materials, the angle of friction obtained from the SPT measurement is not directly used. Instead other parameters such as Nq are obtained from the derived angle of friction on the basis of empirical correlation and thus introducing some additional uncertainties. The trend that model factors from cohesionless materials seem to be more variable than those from cohesive materials has also been reported by Briaud and Tucker (1988) and Phoon and Kulhawy (2005a). Phoon and Kulhawy attributed the high variability exhibited by model factors in cohesionless materials to the correlation of the model factors to the nominal side or tip shear. In general the mean model factors for the cases considered in this study are not considerably different from unity. Model factors of close to unity have also been reported in other studies based on other different other pile capacity calculation models (e.g. Li et al, 1993; Lacasse and Nadim, 1996; Tuomi and Roth, 1995). However, the coefficient of variations obtained in this study, although within the range of values reported by other studies seem to be on the lower side. For example Lo et al (1995) reported that 50% of the model factor data set studied yielded a coefficient of variation in excess of 0.4 while Phoon and Kulhawy (2005) reported values ranging from 0.30 to 0.40. The lower coefficients of variations are attributed to procedure used to determine the soil design parameters, especially the parametric study approach described in Chapter 5. A further characterization of the data through summary statistics includes skewness and kurtosis. In table 6.2, the skewness for all pile cases are positive, suggesting that the corresponding data is right skewed (i.e. has a long right tail of relatively large observations). That the underlying distributions for the data seem to be skewed to the right was also highlighted by the histograms. Stellenbosch University http://scholar.sun.ac.za 181 According to the explanation provided in STATISTICA, a skewness that is clearly different from zero indicates an asymmetrical distribution. However no guidance have been provided as to what constitutes ?clearly different?. McBean and Rover (1998) provide a further guideline as follows; a distribution is considered highly skewed if the absolute value of skewness is greater than one; a distribution with skewness value from 0.5 to 1 is considered moderately skewed and one with skewness value of 0 to 0.5 is essentially symmetrical. In accordance with this guideline, the data for all pile cases can be considered as symmetrical. A symmetrical distribution suggests that the data is normally distributed. In any case, formal normality tests will be carried out to determine whether the underlying distributions for the data sets are normal or not. The values of kurtosis from table 6.2 are negative for driven piles in cohesionless materials, bored piles in non-cohesive materials, and driven piles in cohesive materials. However for bored piles in cohesive materials the numerical value of the kurtosis is positive. Positive kurtosis indicates a "peaked" distribution and negative kurtosis indicates a "flat" distribution. A peaked distribution implies that most of the values of the distribution are concentrated around the mean whereas a flat distribution denotes that the values are more evenly spread throughout the range. Concerning the magnitudes of the values, generally data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have long tails. Data sets with low kurtosis on the other hand tend to have a flat top near the mean rather than a sharp peak. 6.4.4 Probabilistic model for the model factor The theory of reliability is based on a general principle that the basic variables (actions, material properties and geometric data) are considered as random variables having appropriate types of distribution. One of the key objectives of the statistical data analysis is to determine the most appropriate theoretical distribution function for the data. This is the governing probability distribution for the random process under consideration and therefore extends beyond the available sample (i.e. the distribution of the entire population). Once the probability distribution function is known, inferences based on the known statistical properties of the distribution can be made. Reaching conclusions that extend beyond the available sample data is the main point of sampling. It is the population that is of more interest than the sample itself. The process of mapping from sample statistics to population parameters starts with the selection of the suitable distribution for the data. Stellenbosch University http://scholar.sun.ac.za 182 A number of probabilistic models of random variable are available. Certain types of variables follow specific distributions. However, the probabilistic characteristics of a given random variable are sometimes difficult to discern because the appropriate model to describe these characteristics is not readily amenable to theoretical formulation (Ang and Tang, 1975). Therefore in practice, data are often assumed to come from a particular distribution. The assumption that a sample follows a particular distribution is often based on the physical considerations about the nature of the process under study. For example processes whose values are determined by an infinite number of independent random events will be distributed following the normal distribution. On the other hand, if the values of the physical process are the result of very rare events, then the variable will be distributed according to the Poisson distribution (Ang and Tang, 1975). Also there may be historical or theoretical reasons that favour a certain distribution for the phenomenon under consideration. For example past data may have consistently fit a known distribution, or theory may predict that the underlying population should be of a specific form. There are situations where among a number of contending distributions, there is no single one that is preferred on the basis of underlying characteristics of a given phenomenon. Under such circumstances, the distribution is determined empirically from the available observational data. The technique entails fitting the observed distribution to a theoretical distribution by comparing the frequencies observed in the data to the expected frequencies of the theoretical distribution. The procedure is however curtailed by the limitation of the sample size to generate a well defined observed distribution. Whether the distribution is assumed or derived empirically, it must be verified in the light of data using statistical tests known as goodness of fit tests for distribution. Goodness-of-fit tests indicate whether or not it is reasonable to assume that a random sample comes from a specific distribution. From theoretical and practical point of view, the most commonly applied distributions to describe actions, materials properties and geometric data are the normal and lognormal distributions (Holicky, 2005). Therefore the candidate distributions for the model factor are the normal and lognormal distributions. In the Joint Committee on Structural Safety (JCSS) Probabilistic Model Code, model factors are modeled by the lognormal distribution. Further more, past studies (Phoon, 2005; Briaud and Tucker, 1988; Ronold and Bjerager, 1992; Titi Stellenbosch University http://scholar.sun.ac.za 183 and Abu-Farsakh, 1999, FHWA-H1-98-032, 2001; Rahman et.al, 2002) have indicated that the most suitable theoretical model for model factor realisations data is the lognormal distribution. Other considerations that suggest the lognormal distribution seems to be the most appropriate model include: ? Theoretically, the ratio of interpreted capacity to predicted capacity ranges from zero to infinity. This results in an asymmetric distribution with a zero lower bound and an infinite upper bound. The lognormal probability density function is often the most suitable theoretical model for such data as it is a continuous distribution with a zero lower bound and an infinite upper bound. ? As the logarithm of the value is normally distributed rather than the value itself, it provides a convenient model for random variables with relatively large cov for which an assumption of normality would imply a significant probability of negative values (Jones et al, 2002). ? In accordance with the central limit theorem, the distribution of products or ratios of random variables approaches the lognormal distribution as the number of random variables increases. Based on the above considerations, a lognormal distribution was assumed as the most appropriate theoretical distribution for the model factor. This assumption was verified using goodness of fit statistical tests. 6.4.4.1 Verification of the assumption of lognormal distribution for the data There are several techniques for verifying an assumed distribution. The techniques can be broadly classified into qualitative and quantitative approaches. The most commonly used qualitative approach is to construct a histogram and superimpose the assumed distribution on it. This approach suffers from an inability to distinguish between anomalies in shapes due to sampling fluctuations and theoretical considerations (Bowker and Lieberman, 1972). An alternative qualitative technique is to plot the data on a probability paper. Probability paper is a specialized graph paper with horizontal and vertical scales designed such that the cumulative frequencies of a particular form of distribution plot as straight lines. Graph papers possessing this property are available for several distributions and it is common practice to plot empirical data on the appropriate graph paper as a check for the form of the underlying distribution. This is often done for checking the assumption of normality. Stellenbosch University http://scholar.sun.ac.za 184 In general qualitative approaches greatly suffer from the lack of objective criteria for judging whether the data fit the assumed distribution and to what degree. As a solution to the above problem, quantitative techniques have been developed. Quantitative approaches are based on goodness of fit statistical tests. Well established statistical methods are used to test whether the difference between the empirical data frequencies and the assumed distributions are statistically significant or insignificant. 6.4.4.2 Goodness of-fit statistical tests When a random variable X is lognormally distributed, its natural logarithm, ln(X) is normally distributed. Accordingly the natural logarithms of the model factors were determined and some normality tests supported by the available software were applied to the resulting data sets. The normality tests included Kolmogorov-Smirnov (K-S) test, Lilliefors test and Shapiro-Wilk?s W test. Lilliefors test is a form of Kolmogorov-Smirnov test with the mean and standard deviation computed from the data. The interpretation of the results from the statistical significant tests is as follows: Null hypothesis (H0): the empirical distribution is similar to the assumed theoretical distribution (normal distribution). Alternative hypothesis (H1): the empirical distribution is not similar to the assumed theoretical distribution. Results: A small p value (p < 0.05) means it?s unlikely that the assumed distribution is correct and therefore the null hypothesis is rejected. A big p value (p ? 0.05) means that there is no evidence that the assumed distribution is not correct and therefore the null hypothesis is accepted. Where the p value is a measure of fit and therefore the larger the p value the better the fit. The results of the various tests are presented in table 6.3. The p-values for all the three tests are greater than 0.05 and therefore there is no evidence to reject the null hypothesis of normal distribution for the logarithms of the model factors of all the cases considered. Since the distribution of the natural logarithms of the bias factors is normal, it follows that the Stellenbosch University http://scholar.sun.ac.za 185 distribution of the bias factors is lognormal. It is thus verified that the lognormal distribution is an appropriate theoretical model for the bias factor. The results of the statistical tests are in agreement with that found by Lo et.al (1995, 1996). Lo et.al (1995, 1996) carried out the Kolmogorov-Smirnov test on model factor data sets and concluded that the model factor for most data sets could be well approximated by the lognormal distribution as compared to a normal distribution. Table 6-3: Normality tests results Normality tests Variable N Kolmogorov-Smirnov test Shapiro-Wilk's W test Max D K-S p Lilliefors p W p Cohesionless materials Driven piles 29 0.108 p > 0.20 p > 0 .20 0.957 0.27 Bored piles 31 0.073 p > 0.20 p > 0 .20 0.989 0.98 Cohesive materials Driven piles 59 0.079 p > 0 .20 p > 0 .20 0.981 0.51 Bored piles 53 0.072 p > 0.20 p > 0.20 0.982 0.59 To check visually how well the assumed distribution (i.e. normal distribution) fits the data, probability-probability plots (P-P plots) were constructed. In the P-P plot, the observed cumulative distribution function is plotted against a theoretical cumulative distribution function. If the theoretical cumulative distribution function reasonably models the observed cumulative distribution function, the point pattern in this plot should be approximately linear. The P-P plots for the logarithms of the bias factors are presented in figures 6.5. Examination of these P-P plots shows a linear point pattern and therefore it can be concluded that the assumed distribution is the appropriate model for the data. Stellenbosch University http://scholar.sun.ac.za 186 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Value -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 E xp ec te d N or m al V al ue 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Value -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 E xp ec te d N or m al V al ue (a) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Value -3 -2 -1 0 1 2 3 E xp ec te d N or m al V al ue 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Value -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 E xp ec te d N or m al V al ue (c)Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 6.5: P-P plots for model factors 6.4.5 Investigation of correlation with underlying factors Although the mean and standard deviation presented in table 6.2 provide a useful data summary, they combine data in ways that mask information on trends in the data. When there is a strong correlation between the dependent and independent variables, part of the total variability of the dependent variable is explainable by its association with the independent variable. Accordingly if there are hidden correlations between the model factor and some pile design parameters, then part of the variability of the model factors presented Stellenbosch University http://scholar.sun.ac.za 187 in table 6.2 is explained by these design parameters. The presence of correlation between the model factor and deterministic variations in the database would indicate that: ? Generally a trend in the bias factor with a design parameter implies that the method does not fully take the effects of the parameter into account. ? The pile design model does not fully take into account such parameters or ? A random variable probabilistic model is the appropriate model for the parameter as the variation of a random variable is not explainable by deterministic variations in the database. Reliability based design is based on the assumption of randomness of basic variables. Therefore it was necessary to check for correlation between the bias factor and various pile design parameters (pile length, pile diameter, soil properties, and the predicted capacity). This was accomplished by checking presence or absence of correlation between the bias factor and various pile design parameters (pile length, pile diameter, and soil properties). The measure of the degree of association between variables is the correlation coefficient. The basic and most widely used type of correlation coefficient is Pearson r, also known as linear or product-moment correlations. The correlation can be negative or positive. When it is positive, the dependent variable tends to increase as the independent variable increases; when it is negative, the dependent variable tends to decrease as the independent variable increases. The numerical value of r lies between the limits -1 and +1. A high absolute value of r indicates a high degree of association whereas a small absolute value indicates a small degree of association. When the absolute value is 1, the relationship is said to be perfect and when it is zero, the variables are independent. The Pearson product moment correlation coefficient is calculated is given by the following expression: ( )( ) ( ) ( )? ? ? = = = ?? ?? = n i n i ii n i ii yyxx yyxx r 1 1 22 1 [6.5] Stellenbosch University http://scholar.sun.ac.za 188 Where ix = the i th realisation of the dependent variable; x = the mean of all the realisations of the variable x; iy = the i th realisation of independent variable y; y = the mean of all the realisations of the variable y. An alternative to the parametric Pearson product moment correlation coefficient is a variety of non-parametric correlations. Generally non-parametric statistics have been specifically developed to process data from low quality data. Low quality in this context refers to small samples and unknown parameters (mean, standard deviation and distribution) of the variable. Accordingly the non-parametric correlations are most suitable when the sample sizes are small. The most popular of the non-parametric correlations is the Spearman rank order correlation. The Spearman correlation (R) is calculated by replacing ix and iy in equation 6.1 by their ranks in the respective list of number (i.e the smallest number takes the value of 1, the next smallest takes the value of 2, and so forth). In calculating the Spearman correlation, equation 6.1 reduces to (Phoon and Kulhawy, 2005): ( ) ( )( )11 6 1 2 ?? ? ?= ? nnn ts R ii [6.6] in which is and it are the ranks of ix and iy respectively. In this study both the Pearson and Spearman correlations were considered. However, in case of discrepancies between the results of the two correlations, the results of the Spearman correlation were used to reach the final decision. The preference of the Spearman correlation is based on the account of limited size of the database and unknown population parameters of the model factors. Further more, the Spearman correlation is less affected by outliers (Phoon and Kulhawy, 2005). Whether the correlation is computed on the basis of the Pearson or Spearman correlation coefficient, the technical question that needs to be answered is ?when is the numerical value of the correlation coefficient considered high and when is it low and negligible?? Several authors in various fields have suggested guidelines for the interpretation of the correlation Stellenbosch University http://scholar.sun.ac.za 189 coefficient. One such typical interpretation uses five easy rules of thumb (Franzblau, 1958) as follows: ? r ranging from 0 to about ?0.2 may be regarded as indicating no or negligible correlation. ? r ranging from about ?0.2 to ?0.4 may be regarded as indicating a low degree of correlation ? r ranging from about ?0.4 to ?0.6 may be regarded as indicating a moderate degree of correlation. ? r ranging from about ?0.6 to ?0.8 may be regarded as indicating a marked degree of correlation. ? r ranging from about ?0.8 to ?1 may be regarded as indicating a high correlation. Other more recent scholars suggest that as a rule of thumb, a correlation coefficient of less than 0.3 indicate no or negligible correlation. However, Cohen (1988) observed that all such criteria are in some ways arbitrary and should not be observed too strictly. Generally the interpretation of the correlation coefficient depends on the context and purpose of the analysis and the field of research. Other statisticians prefer the use of r2. Technically r2 is called the coefficient of determination and represents the proportion of shared variance between the variables. However even with this approach, the criterion for deciding what r2 value is significant is still subjective. In geotechnical reliability, Smith (1986) suggested the interpretation presented in table 6.4. Due to its relevancy to the current study, this is the interpretation adopted for the study. Table 6-4: Practical significance of correlation coefficient Range of r values Interpretation ? ?0.2 Weak correlation and the variables can be assumed to be independent ?0.8 > r > ?0.2 Marked correlation between variables ? ?0.8 Strong correlation and the variables can be assumed to be completely depended Stellenbosch University http://scholar.sun.ac.za 190 Another widely used convention to determine the significance of the correlation is the concept of statistical significance tests. There is always a possibility that the computed correlation could have risen by chance and begs the question, how probable it is that a similar relation would be found if the experiment was replicated with other samples drawn from the same population? The doubt regarding the reliability of the computed correlation coefficient necessitates that the results be subjected to some statistical significance tests. The statistical significance of a result is an estimated measure of the degree to which it is "true" in the sense of "representative of the population". Most recent statistical softwares including STATISTICA give the results of the significance tests in terms of the p-value. When used to check the significance of the correlation, the p-value is interpreted as follows: Null hypothesis (H0): there is no correlation between the variables (indicative of statistical independence). Alternative hypothesis (H1): there is a correlation between the variables (statistical dependency). Results: A small p value (p < 0.05) means that the null hypothesis is not valid and should be rejected. A big p value (p ? 0.05 indicates that there is no relationship and therefore the null hypothesis is valid. It should be noted that the significance of the correlation as expressed by the p-value simply means that there is statistical evidence that it is unlikely that the computed correlation have occurred by chance. It does not necessarily mean that the correlation is large, important or significant in the usual sense of the word. Therefore through out the analysis, significance of correlation inferred on the basis of practical considerations (table 6.3) supersedes that reached on the basis of statistical significance. The existence of a correlation was therefore assessed by (i) visual inspection of the scatter plots of the model factor versus various pile design parameters, (ii) the numerical value of the correlation coefficient and (iii) the statistical significance of the correlation expressed in terms of the p-value. The results of correlation assessment between M and pile design parameters are presented in the subsequent sections. Stellenbosch University http://scholar.sun.ac.za 191 6.4.5.1 Correlation with pile length Figure 6.6 presents the scatter plots of the model factor (M) versus pile lengths while Table 6.5 presents the correlation coefficients together with their associated p-values. Visual inspection of the scatter plot shows ? A weak negative correlation for driven piles in non cohesive materials ? A weak positive correlation for bored piles in non-cohesive materials ? A significant negative correlation for driven piles in cohesive materials and ? A weak negative correlation for bored piles in cohesive materials The quantitative analysis of the degree of the correlation is presented in table 6.5. The results indicate: ? Both Pearson and Spearman correlations show a positive correlation for BNC and negative correlation for the other pile classes. ? The correlation coefficients are less than 0.2, hence the parameters are considered to be independent. ? The p-values for all the cases are greater than 0.05, suggesting that at the customary significant level of 5%, there is no evidence to reject the null hypothesis of no correlation between the model factor and pile lengths. Both the practical significance and statistical significance of the correlation coefficients suggest that the degree of correlation is negligible and the parameters can be assumed to be independent. However, on the basis of the slope of the best fit line, visual inspection of the scatter plots for driven piles seem to suggest that the correlation is appreciable. This discrepancy is an indication that judgement based on visual inspection of the scatter plots can be misleading. Therefore more credence is given to the quantitative results and accordingly, it can be concluded that for all four cases there is no correlation between the model factor and pile lengths. Stellenbosch University http://scholar.sun.ac.za 192 2 4 6 8 10 12 14 16 18 20 22 24 26 28 L (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 4 6 8 10 12 14 16 18 L (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M (a) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials 0 5 10 15 20 25 30 L (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 0 2 4 6 8 10 12 14 16 18 20 22 L (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 6.6: Scatter plots of M vs pile length Table 6-5: Correlation coefficients and associated p-values for M vs pile length Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level DNC 29 -0.09 0.009 0.63 -0.15 0.45 BNC 31 0.03 0.001 0.87 0.01 0.97 DC 59 -0.09 0.008 0.51 -0.15 0.26 BC 53 -0.05 0.003 0.71 -0.05 0.73 Stellenbosch University http://scholar.sun.ac.za 193 6.4.5.2 Correlation with pile diameter For expanded base piles, the shaft diameter differs from the base diameter. The base diameter contributes to the base capacity while the shaft diameter contributes to the shaft capacity. Therefore the correlation is investigated for both base and shaft diameters. 6.4.5.2.1 Correlation with shaft diameter The correlation analysis results are presented in figure 6.7 and table 6.6. The scatter plots of the model factor versus shaft diameters are presented in figure 6.7. Qualitative inspection of the scatter plots indicates: ? Weak negative correlation for driven piles in non-cohesive materials ? Marked positive correlation for bored pile in non-cohesive materials ? Appreciable negative correlation for driven piles in cohesive materials ? Low positive correlation for bored piles in cohesive materials The quantitative measure of the strength and significance of the correlation are presented in table 6.6. The following observations can be drawn from this table: ? The signs of the correlation coefficients agree with the qualitative observations. ? Except for bored piles in non-cohesive materials, the correlation coefficients from both Pearson and Spearman approaches are less than 0.2 and therefore negligible. However, for bored piles in non-cohesive materials, the Spearman rank correlation is -0.215 which is marginally greater than 0.2. In accordance with the interpretation followed the correlation for bored piles in non-cohesive materials is practically significant. Therefore the qualitative observations are in complete agreement with the visual inspection results. ? The p-values for all the cases are greater than 0.05, indicating that the hypothesis that the correlation between the two parameters is not significant is valid. Even for bored piles in non-cohesive materials which on the basis of visual inspection of the scatter plot as well as the practical significance of the numerical value seemed to depict a significant correlation, the p-values are still greater than 0.05 (0.172 and 0.102 for Pearson and Spearman respectively). Since more credibility is given to the practical significance of the correlation coefficient rather than to its statistical significance, it can be concluded that the correlation for bored Stellenbosch University http://scholar.sun.ac.za 194 piles in non-cohesive materials is significant while that for the other three cases is insignificant. Table 6-6: Correlation coefficients and associated p-values for M vs pile shaft diameter Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level DNC 29 -0.04 0.002 0.83 -0.09 0.63 BNC 31 0.15 0.023 0.42 0.04 0.84 DC 59 -0.18 0.033 0.17 -0.22 0.10 BC 53 0.06 0.004 0.67 0.05 0.73 250 300 350 400 450 500 550 600 650 Shaft diameter (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 200 300 400 500 600 700 800 Shaft diametr (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M a) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Shaft diameter (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Shaft diameter (m) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 6.7: Scatter plots of M vs shaft diameter Stellenbosch University http://scholar.sun.ac.za 195 6.4.5.2.2 Correlation with base diameter The scatter plots are presented in figure 6.8 while the correlation coefficients and the associated p-values are presented in table 6.7. Visual examination of the scatter plots shows: ? Significant negative correlation for driven piles and negligible correlation for bored piles. Observations based on the correlation statistics (table 6.7) indicate that: ? In agreement with the visual observations, the magnitudes of correlation coefficients for driven piles are much higher than those for bored piles. On the basis of Spearman correlation, the correlation coefficient for driven piles in non-cohesive materials is greater than 0.2 while that for driven piles in cohesive materials is on the borderline. Accordingly the correlation for driven piles can be regarded as practically significant. Conversely the correlation coefficients for bored piles are from practical point of view negligible. ? However, the p-values for all the cases are greater than 0.05, suggesting that there is no evidence to reject the null hypothesis. 200 300 400 500 600 700 800 900 Base diameter (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 200 300 400 500 600 700 800 Base diameter (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M (a) ) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials Stellenbosch University http://scholar.sun.ac.za 196 200 400 600 800 1000 1200 1400 Base diameter (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 200 300 400 500 600 700 800 900 1000 Base diameter (mm) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 6.8: Scatter plots of M vs base diameter Table 6-7: Correlation coefficients and associated p-values for M vs pile base diameter Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level DNC 29 -0.25 0.063 0.19 -0.29 0.12 BNC 31 -0.02 0.0004 0.92 -0.05 0.80 DC 59 -0.15 0.023 0.25 -0.19 0.14 BC 53 0.01 0.0001 0.94 0.05 0.71 6.4.5.3 Correlation with soil properties The soil properties considered for the correlation investigations are the angle of friction for non-cohesive materials and the undrained shear strength for cohesive materials. 6.4.5.3.1 Correlation with friction angle (?) Both the base friction angle and the shaft friction angle were considered. The outcome of the regression and correlation analysis between the model factor and the friction angle are presented in figure 6.9, figure 6.10 and table 6.8. The following observations were drawn from the results: Stellenbosch University http://scholar.sun.ac.za 197 ? For ?-shaft, visual inspection of the scatter plots shows an appreciable correlation for driven piles and a weak correlation for bored piles. Consistent with this observation, the correlation coefficients indicate appreciable correlation for driven piles and weak correlation for bored piles. However, the p-values for both cases are greater than 0.05, suggesting that the computed correlations are statistically insignificant. ? For ?-base, visual inspection of the scatter plots shows a significant positive correlation for driven piles and a weak negative correlation for bored piles. The quantitative analysis results (i.e. correlation coefficients values) also imply a noticeable correlation for driven piles and a weak correlation for bored piles. At a confidence level of 5%, the degree of correlation portrayed by both Pearson and Spearman correlation coefficients are considered to be statistically insignificant as indicated by p-values of greater than 0.05. On the basis of practical significance of the correlations, it is concluded that there exist an noticeable correlation between the model factor and the friction angle in driven piles but not in bored piles. 22 24 26 28 30 32 34 36 38 40 42 Phi -shaft 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 28 30 32 34 36 38 40 42 44 46 48 Phi-shaft 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M (a) Driven piles (b) Bored piles Figure 6.9: Scatter plots of M vs ?-shaft Stellenbosch University http://scholar.sun.ac.za 198 28 30 32 34 36 38 40 42 44 46 48 Phi-base 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 30 32 34 36 38 40 42 44 46 48 Phi-base 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M (a) Driven piles (b) Bored piles Figure 6.10: Scatter plots of M vs ?-base Table 6-8: Correlation coefficients and associated p-values for M vs ? Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level ?-shaft DNC 29 -0.143 0.020 0.46 -0.25 0.19 BNC 31 -0.048 0.002 0.80 -0.10 0.60 ?-base DNC 29 0.200 0.040 0.30 0.25 0.19 BNC 31 -0.098 0.010 0.60 -0.05 0.77 6.4.5.3.2 Correlation with undrained shear strength Results for the investigation of correlation between the model factor and undrained shear strength are presented in figure 6.11, figure 6.12, and table 6.9. Visual examination of the scatter plots indicates an appreciable negative correlation for Cu-shaft and a negligible correlation for Cu-base. Further scrutiny of figure 6.11a reveals a divergent data point to the right which might have led to the observed appreciable correlation. Accordingly this data point was excluded and the resulting scatter plot is shown in figure 6.11c. Even after the exclusion of the divergent data point, the correlation for Cu-shaft is still noticeable (Fig. 6.11c). Therefore it can be concluded that indeed the correlation for Cu-shaft is significant. The quantitative assessment of the correlation (table 6.9) indicates that: Stellenbosch University http://scholar.sun.ac.za 199 - Only the correlation coefficients (both Pearson and Spearman) for Cu-shaft in bored piles are practically significant (r ? 0.2) while the values for the other cases are practically insignificant. To investigate the effects of the divergent data point, the correlation coefficients were re-calculated with the divergent point excluded. The results are highlighted in table 6.9. It is evident from this results that the exclusion of the divergent point slightly reduces the correlation coefficients. With this reduction, the Spearman rank correlation now less than 0.2 after rounding off to one decimal point while the Pearson correlation is still greater than 0.2. In any case, the value for Spearman was upheld. - After exclusion of the divergent data point, the p-values for both Spearman and Pearson correlation coefficients are greater than 0.05, indicating a statistically insignificant correlation. 0 20 40 60 80 100 120 140 160 180 200 220 Cu-shaft (kPa) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 0 200 400 600 800 1000 1200 Cu-shaft (kPa) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (a) Driven piles (b) Bored piles Stellenbosch University http://scholar.sun.ac.za 200 0 50 100 150 200 250 300 350 400 450 Cu-shaft (kPa) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M -B C (c) Bored piles after exclusion of the divergent data point Figure 6.11: Scatter plots of M vs Cu-shaft 100 300 500 700 900 1100 Cu-base (kPa) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 0 400 800 1200 1600 2000 Cu-base (kPa) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (a) Driven piles (b) Bored piles Figure 6.12: Scatter plots of M vs Cu-base Table 6-9: Correlation coefficients and associated p-values for M vs Cu Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level Cu-shaft DC 59 -0.13 0.016 0.35 -0.07 0.61 BC 53 -0.28 0.073 0.04 -0.16 0.26 52 -0.23 0.052 0.10 -0.12 0.41 Cu-base DC 59 -0.003 0.000 0.98 -0.05 0.74 BC 53 0.03 0.001 0.86 0.01 0.97 Stellenbosch University http://scholar.sun.ac.za 201 6.4.5.4 Correlation with predicted and interpreted pile capacities The model factor was previously defined as the ratio of the measured to predicted capacity (i.e M=Qi/Qp). In this definition, the measured and the predicted capacities appear as the numerator and denominator respectively. Therefore by virtue of the way the model factor is computed, it may be correlated to these two parameters. The correlation with Qp if it exists is of more practical significance in the calculation of reliability index. This arises from the fact that in practice the model factor is applied as an independent corrective random variable to predicted capacity. To apply the model factor as an independent random variable on the predictive capacity equation, it is important to verify that it does not vary systematically with the predicted capacity. Correlation can have significant impact on the calculated reliability index if not accounted for. Further more, all the pile design parameters investigated so far, appear in the equation for calculating the ultimate bearing capacity. Therefore the sensitivity of the bias factor to the combined effects of the pile design parameters can be explored through its correlation with the predicted capacity. On the other hand the correlation with the interpreted capacity has no effects on the reliability calculation and therefore it was not considered. The results of the correlation analysis between the model factor and the predicted capacities are presented in figure 6.13 and table 6.10 For driven piles in non-cohesive materials (fig. 6.13a), the scatter plot shows a significant negative correlation between the two parameters. Based on the Spearman correlation coefficient, the statistical measures of the strength of the correlation are; R = -0.413 and p = 0.026. The magnitude of correlation coefficient supports the visual observation that the correlation is significant. The significance of the correlation is further captured by p-value of less than 0.05. A p-value of less than 0.05 indicates that the null hypothesis of zero correlation is not valid. Therefore at a confidence level of 5%, the model factor is considered to be statistically correlated to the predicted capacity for driven piles in non- cohesive materials. For the case of bored piles in non-cohesive materials, the scatter plotter (fig. 6.13b) shows a relatively weak positive correlation between the model factor and predicted capacity. The Spearman rank correlation coefficient and the associated p-value confirm that the correlation between the two parameters is not statistically significant (p > 0.05). Stellenbosch University http://scholar.sun.ac.za 202 For piles in cohesive materials, visual inspection of the scatter plots indicate that the correlation between the model factor and predicted capacity appears to be strong in driven piles and weak in bored piles. For driven piles (fig. 6.13c) there two extreme data points to the right might have influenced the observed correlation. These points were excluded and the resulting scatter plot is presented in fig. 6.13e. Inspection of fig. 6.13e still shows a noticeable correlation. Therefore with or without the divergent data points, the correlation for driven piles in cohesive materials is appreciable. Both the Spearman and Pearson correlation coefficients are greater than 0.2, implying that the correlation is practically significant. Even after the exclusion of the two divergent data points the correlation coefficients are still greater than 0.2 (highlighted results). However the associated p-values are marginally greater than 0.05, indicating that the correlation is statistically insignificant. For the case of bored piles in cohesive materials, visual examination of the scatter plot (fig. 6.13d) shows a negligible correlation. In agreement, quantitative results indicate practically and statistically insignificant correlation. 500 1500 2500 3500 4500 5500 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M 0 1000 2000 3000 4000 5000 6000 7000 8000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M a) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials Stellenbosch University http://scholar.sun.ac.za 203 0 1000 2000 3000 4000 5000 6000 7000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M 0 2000 4000 6000 8000 10000 12000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials 0 500 1000 1500 2000 2500 3000 3500 4000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M (e) Driven piles cohesive materials after removal of divergent data points Figure 6.13: Scatter plots of M vs Qp Table 6-10: Correlation coefficients and associated p-values for M vs Qp Case N Pearson correlation Spearman rank correlation r r2 p-level R p-level DNC 29 -0.32 0.099 0.09 -0.41 0.03 BNC 31 0.13 0.018 0.48 -0.05 0.81 59 -0.25 0.062 0.06 -0.26 0.05 DC 57 -0.26 0.068 0.05 -0.24 0.07 BC 53 -0.04 0.001 0.81 -0.01 0.94 Stellenbosch University http://scholar.sun.ac.za 204 6.4.5.5 Summary and discussion of correlation investigation results On the basis of statistical significance (p-value) it can be concluded that at the customary confidence level of 5%, none of the pile design parameters is significantly correlated with the model factor. From the probabilistic perspective, this implies that the variation in the model factor is not caused by the variations in the key pile design parameters. Therefore it is correct to model the model factor as a random variable. With regard to predicted capacity, it was found that the model factor is correlated to Qp for driven piles but not bored piles. These results differ from that reported by Dithinde et.al (2006) where on the basis of 5% confidence level it was found that the model factor was correlated to base diameter for driven piles in non-cohesive materials, ?-shaft in driven piles, ?-base in driven piles, Cu- base in driven piles, and Qp for driven and bored piles in non-cohesive materials as well as Qp for driven piles in cohesive materials. The difference in results is attributed to the differences in the schemes for selecting the geotechnical design parameters. The former was based on commonly assumed parameters while the later was based the outcome of a parametric study as discussed in chapter 5. The differences in the two results lead to the conclusion that the use of more realistic design parameters significantly reduces the systematic dependence of the model factor on the pile design parameters. If the interpretation of the numerical values of correlation coefficients is based on practical considerations (table 6.4), then the model factor was found to be correlated with: ? Shaft diameter for driven piles in cohesive materials ? Base diameter for driven in non-cohesive materials ? Base diameter for driven piles in cohesive materials ? ?-shaft for driven piles in non-cohesive materials ? ?-base for driven piles in non-cohesive materials ? Predicted capacity for driven piles in non-cohesive materials ? Predicted capacity for driven piles in cohesive materials The numerical values of the correlation coefficients for the above cases ranged from 0.2 to 0.4. In general this range of correlation indicates a low degree of correlation. However for the purposes of this study this range of correlation was considered appreciable and therefore warrants a further investigation. This further investigation is carried out in Chapter 7. Stellenbosch University http://scholar.sun.ac.za 205 It is evident from the above results that for the individual pile design parameters, significant correlation occurs only in driven piles and not in bored piles. This scenario suggests that the static formula is more accurate and precise in bored piles. This deduction is further supported by the results of the summary statistics (Table 6.2). As observed earlier, the variability of the model factor is slightly higher in driven piles compared to bored piles irrespective of the materials type. The non-existence of correlation in bored piles was also reported by Tuomi and Roth (1995) who investigated the sensitivity of the model factor for bored piles in clay with respect to pile dimensions, predicted capacity, and soil properties. The results indicated that the model factors were not significantly influenced by the pile design parameters. For correlation with soil properties, the results show correlation with shear strength parameters for non-cohesive materials and negligible correlation with undrained shear strength. This trend was also observed by Phoon and Kulhawy (2005). Phoon and Kulhawy plotted model factors for undrained and drained analysis against side and tip shears. The results indicated a significant correlation for drained analysis and no correlation for the undrained analysis. With regard to predicted capacity, it can be seen that correlation occurs only in driven piles and not in bored piles, reiterating that correlation is more prevalent in driven piles than in bored piles. With more weight given to the practical significance of the value of the correlation coefficient rather than its statistical significance, it can be concluded that the conventional model factor exhibits some statistical dependencies with some underlying factors. Therefore to apply the model factor as an independent variable in reliability analysis, these statistical dependencies need to be removed or taken into account. Treatment of the statistical dependencies is presented in Chapter 7 of the dissertation. Stellenbosch University http://scholar.sun.ac.za 206 Chapter 7 REMOVAL OF STATISTICAL DEPENDENCY BETWEEN THE MODEL FACTOR AND THE PREDICTED CAPACITY 7.1 Introduction To apply a model factor as an independent random variable on the predictive capacity equation as it is currently done, it is desirable to verify that the model factor does not vary systematically with some underlying factors. However, it was established in Chapter 6, that the model factor is correlated to various underlying factors. Correlation can have significant impact on the calculated reliability index if not accounted for. Even though correlation can be incorporated into the reliability calculations, it makes the calculation to be cumbersome as it involves transforming the original variables to a set of uncorrelated variables. Therefore to apply the model factor as an independent random variable in reliability analysis, the statistical dependencies need to be removed. Alternatively, the model factor should be presented as a function of Qp to reflect the its dependency upon Qp. Accordingly the two approaches to the treatment of correlation were employed. Removal of statistical dependencies is presented under the generalised model factor approach while accounting for the correlation is presented under conditioned model factor approach. 7.2 Generalised model factor approach The generalised model factor approach is based on a sound regression theory and it therefore entails performing regression using the predicted values as the predictor variable. It was established in Chapter 6 that the appropriate probabilistic model for the model factor is the lognormal probability distribution. This being the case, the generalised model factor was derived from the regression of LN(Qi) on LN(Qp). The resulting functional relationship between LN(Qi) and LN(Qp) is given by a general regression model of the form: ?++= )()( pi QLNbaQLN [7.1] in which a and b are regression constants and ? is a normal random variable with zero mean and non-zero variance. Stellenbosch University http://scholar.sun.ac.za 207 Taking antilog on both sides of equation 7.1 yields; b pi QaQ ).exp().exp( ?= [7.2] Note that in equation 7.2, the error term (i.e. ?) is now modeled as multiplicative as compared to equation 7.1 where it was additive. Therefore regression model in the form of equation 7.2 removes systematic effects and the remaining component tends to appear random (Phoon and Kulhawy 2005b). Equation 7.2 can be re-written as: b pi QaQ )exp( ?+= [7.3] Let )exp( ?+a = M [7.4] Then; b pi MQQ = [7.5] Equation 7.5 is the generalised representation of the model factor M. This equation is immediately recognised as being of the same form as that for the conventional model factor (i.e. pi MQQ = ). In fact, the conventional model factor is a special case of the generalised model factor with 1=b . In equation 7.4, ? is a random variable and therefore M will likewise be random. Theoretically the probability distribution of a function of random variables such as M can be derived from the probability distributions of the basic random variables. However, such derivations are generally difficult especially when the function is non-linear. Therefore for practical purposes the moments, in particular the mean and variance are used to describe the function. Accordingly a linear function for M can be described by its mean and variance. In accordance with the computational rules for lognormal random variables, the mean and variance for M are as follows: ( )25.0exp ?? += aM [7.6] ( )[ ]1exp 222 ?= ??? MM [7.7] The generalised model factor as presented in equation 7.5 is not dimensionless in contrast to conventional model factor equation. The force unit adopted throughout this study is Stellenbosch University http://scholar.sun.ac.za 208 kiloNewton (kN) and this unit is applicable to equation 7.5. To make the generalised model factor dimensionless, both the interpreted and predicted capacities need to be normalised. 7.2.1 Normalisation schemes The objective of normalisation is to make Qi and Qp dimensionless so that the resulting model factor is dimensionless. In principle any quantity or combination of quantities associated with pile design, can be used as a normalizing factor. Obviously different model factor statistics will be obtained for different normalisation schemes. The dependence of the generalised model factor on the normalisation scheme is the greatest disadvantage of this approach. In this study the following three normalization schemes were experimented with: ? Scheme 1: dividing LN(Qi) and LN(Qp) by area of pile base x atmospheric pressure (AbPa) ? Scheme 2: dividing LN(Qi) and LN(Qp) by volume of water displayed by the pile (i.e. volume of piles x unit weight of water (Vw)) ? Scheme 3: dividing LN(Qi) and LN(Qp) by weight of pile shaft (Ws) 7.2.2 Regression results after normalisation The data for each pile class were normalised by the three approaches. The regression results for each pile class are presented in the subsequent subsections. The normalisation schemes were evaluated on the basis of the resulting fit between Qi and Qp. In this regard, the coefficient of determination (r2) was used as the measure of fit. 7.2.2.1 Results for DNC The regression results for DNC are presented in figure 7.1. A comparison of the r2 indicates that normalisation scheme 1 gives the best fit while scheme 3 yields the poorest fit. However the standard deviations about the regression lines (standard error of estimate) are comparable. Stellenbosch University http://scholar.sun.ac.za 209 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 LN(Qp/AbPa) 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 LN (Q i/A bP a) r2 = 0.8772; y = -0.1163 + 1.0312*x Std. err. of est. = 0.293 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 LN(Qp/Vw) 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 LN (Q i/V w ) r2 = 0.7819; y = 0.1031 + 0.9847*x Std. err. of est. = 0.294 (a) Scheme 1 (a) Scheme 2 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 LN(Qp/Ws) 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 LN (Q i/W s) r2 = 0.5523; y = 0.5268 + 0.8873*x Std. err. of est. = 0.291 (c) Scheme 3 Figure 7.1: Normalised regression results for DNC Stellenbosch University http://scholar.sun.ac.za 210 7.2.2.2 Results for BNC The results for BNC are presented in figure 7.2. Inspection of figure 7.2 shows that r2 and the standard error of estimate (?) for the three schemes are quite close. 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 LN(Qp/AbPa) 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 LN (Q i/A bP a) r2 = 0.7640; y = -0.1828 + 1.0297*x Std err. od est. = 0.264 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 LN(Qp/Vw) 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 LN (Q i/V w ) r2 = 0.7687; y = -0.0187 + 0.9928*x Std. err. of. est. = 0.264 (a) Scheme 1 (b) Scheme 2 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 LN(Qp/Ws) 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 LN (Q i/W s) r2 = 0.7439; y = -0.425 + 1.094*x Std. err. of est. = 0.262 (c) Scheme 3 Figure 7.2 Figure 7.2: Normalised regression results for BNC Stellenbosch University http://scholar.sun.ac.za 211 7.2.2.3 Results for DC The regression results for DC are presented in figure 7.3. The values for coefficient of determination and standard error of estimate suggest that normalisation schemes 2 and 3 produces a better fit compared to scheme 1. 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 LN(Qp/AP) 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 LN (Q i/A P ) r2 = 0.6751; y = 0.2231 + 0.9766*x Std err. of est. = 0.273 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 LN(Qp/Wv) 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 LN (Q i/V w ) r2 = 0.7426; y = 0.6695 + 0.8897*x Std err. of est. = 0.267 (a) Scheme 1 (b) Scheme 2 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 LN(Qp/Ws) 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 LN (Q i/W s) r2 = 0.7426; y = 0.5709 + 0.8897*x Std err. of est. = 0.267 (c) Scheme 3 Figure 7.3: Normalised regression results for DC Stellenbosch University http://scholar.sun.ac.za 212 7.2.2.4 Results for BC The regression results for BC are shown in figure 7.4. Both measures of fit (r2 and ?) suggest that the three schemes produce similar results. 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 LN(Qp/AP) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 LN (Q i/A P ) r2 = 0.7927; y = 0.2981 + 0.9606*x Std err. of est. = 0.244 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 LN(Qp/Vw ) 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 LN (Q i/V w ) r2 = 0.8039; y = 0.2772 + 0.9668*x Std. err. of est. = 0.244 (a) Scheme 1 (b) Scheme 2 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 LN(Qp/Ws) 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 LN (Q i/W s) r2 = 0.8039; y = 0.2475 + 0.9668*x Std. err. of est. = 0.244 (c) Scheme 3 Figure 7.4: Normalised regression results for BC Stellenbosch University http://scholar.sun.ac.za 213 7.2.2 Generalised model factor statistics and discussion of the regression results The regression results presented in figure 7.1 through figure 7.4 were then used in conjunction with equations 7.6 and 7.7 to compute the required generalised model factor statistics. The regression parameters and the ensuing generalised model factor statistics are summarised in table 7.1. Since both the statistics and the regression parameter ?b? are used as input into the performance function, they influence the calculated value of the reliability index. Table 7-1: Generalised model factor statistics for various normalisation schemes Case Normalisation Regression parameters Generalised M statistics R2 a b ? ? ? COV DN AbPa 0.88 -0.1163 1.03 0.293 0.93 0.28 0.30 Vw 0.78 0.1031 0.985 0.294 1.16 0.35 0.30 Ws 0.55 0.5268 0.887 0.291 1.77 0.53 0.30 BN AbPa 0.76 -0.1828 1.03 0.264 0.86 0.23 0.27 Vw 0.77 -0.0187 0.993 0.264 1.02 0.27 0.27 Ws 0.74 -0.4250 1.094 0.262 0.68 0.18 0.27 DC AbPa 0.68 0.2231 0.977 0.273 1.30 0.36 0.28 Vw 0.74 0.6695 0.89 0.267 2.02 0.55 0.27 Ws 0.74 0.5709 0.89 0.267 1.83 0.50 0.27 BC AbPa 0.79 0.2981 0.961 0.244 1.39 0.34 0.25 Vw 0.80 0.2772 0.967 0.244 1.36 0.34 0.25 Ws 0.80 0.2475 0.967 0.244 1.32 0.33 0.25 Scrutiny of table 7.1 lead to the following observations: ? The numerical values of the generalised model factor statistics (mean and standard deviation) reduces with the decrease in the regression constant ?a?. Consequently when the regression constant is negative, the mean generalised model factor tends to be less than or quite close to one (i.e. becomes unconservative). The standard deviation also reduces significantly. This trend is apparent in DNC with respect to normalisation scheme 1 and BNC with respect to the three normalisation schemes. Conversely for higher positive values of ?a?, the mean and standard deviation of the generalised model factor becomes significantly high. Typical examples are DNC with respect to normalisation scheme 3 and DC with respect to normalisation schemes 2 and 3. The reduction in the mean and standard deviation with degrease of the regression constant is attributed to the calculation models (Eq. 7.6) and Eq. 7.7). Stellenbosch University http://scholar.sun.ac.za 214 ? As expected, the regression parameter ?b? increases with the decrease of the regression constant ?a? and vice-verse. For negative values of ?a?, the regression parameter ?b? becomes greater or quite close to one. Conversely for higher positive values, ?b? becomes appreciably less than one. Accordingly the parameter ?b? can be related to the generalised model factor statistics as follows: o For mean model factor of less than one (with small standard deviation), ?b? becomes greater than one o For relatively high mean generalised M (with higher standard deviation), ?b? becomes appreciably less than one ? The above relationship between the regression parameter ?b? and the generalised M statistics has a significant influence on calculated value of reliability index. A smaller standard deviation and a value of ?b? that is greater than one tend to increase the reliability index. In this regard, the lower mean value is compensated for by the smaller standard deviation and higher b value. High values of standard deviation and values of the ?b? that are less than one tend to reduce the reliability index. This combination counter balances the effects of the high mean value. Therefore for a given pile class, the different normalisation schemes will result in comparable values of reliability indices. ? Because for a given pile class, the mean and the standard deviation change in the same direction, the coefficient of variation remains the same for the three schemes. ? As already noted, for a given pile class, the r2 values for the different normalisation schemes are generally comparable. An exception is DNC with respect to scheme 3. The r2 value for this case is notably smaller. Also for a given pile class, values of the standard error of estimate are quite close. It can therefore be concluded that, in general the three normalisation schemes give comparable results. Based on the above results, the best normalisation scheme was selected. The best normalisation scheme is that which produces the best fit between Qi and Qp. The best fit was evaluated in terms of the results of a perfect fit. For the perfect fit, b = 1 and R2 = 1. Accordingly a normalisation scheme that yields a relatively high R2 and a value of b that is relatively closer to 1 gives the best fit. The two parameters have been extracted from table 7.1 and presented in Table 7.2 in a manner that facilitates easy comparison. Inspection of Table 7.2 shows that scheme 3 yield the poorest overall results (smallest r2 for DNC and values of b that are appreciably different from 1 for DNC, BNC and DC). On the basis of r2 schemes 2 and 3 give very close results. However, in terms of ?b?, scheme 2 gives a value of appreciably less than one for DC whiles scheme 1 produces values that are close to one Stellenbosch University http://scholar.sun.ac.za 215 for all the pile classes. In this respect, scheme 1 is slightly better than scheme 2. Accordingly the generalised M statistics for scheme 1 were selected for further analysis. For ease of reference the selected generalised M statistics are presented in table 7.3. A comparison of the generalised M statistics to the standard M statistics (Table 6.2) show reasonable agreement. The standard deviations and coefficients of variation are comparable. Furthermore, the two sets of model factors show similar trends such as: ? The mean model factors in non-cohesive materials are less than those in cohesive materials ? Driven piles exhibit higher variability compared to bored piles irrespective of materials type Table 7-2: Comparison of r2 and b from the three normalisation schemes Pile Class Scheme 1 Scheme 2 Scheme 3 r2 b r2 b r2 b DNC 0.88 1.031 0.78 0.985 0.55 0.887 BNC 0.76 1.030 0.77 0.993 0.74 1.094 DC 0.68 0.977 0.74 0.890 0.74 0.890 BC 0.79 0.961 0.80 0.967 0.80 0.967 Table 7-3: Selected generalised M statistics CASE ? ? COV b DNC 0.93 0.28 0.30 1.031 BNC 0.86 0.23 0.27 1.030 DC 1.30 0.36 0.28 0.977 BC 1.39 0.34 0.25 0.961 7.2.3 Verification of removal of systematic dependency As noted earlier, the correlation of practical significance is that between the model factor and the predicted capacity. To verify the removal of such correlation, the regression model error (?) is plotted against LN(Qp/AP). For a given case the model error for a given data point was determined from equation 7.1 as follows: )/()/( PAQbLNaPAQLN bpbi ??=? [7.8] Stellenbosch University http://scholar.sun.ac.za 216 The scatter plots of the model error (?) versus predicted capacity are presented in figure 7.5. Visual inspection of the scatter plots shows that for all the four cases, the best fit line is horizontal, suggesting that the correlation between the two parameters is insignificant. The quantitative measure of the correlation in terms of the correlation coefficients and the associated p-values is presented in table 7.4. Again for all the four cases, the p-values are much greater than 0.05, thus implying that the null hypothesis of zero correlation is valid. Therefore both qualitative and quantitative results are in agreement that the correlation between the parameters is negligible. It is thus verified that model factor defined as the regression model error does not suffer from statistical dependencies with underlying parameters. 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 LN(Qp/AP) -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 LN(Qp/AP) -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 (a) Driven piles in non-cohesive materials (b) Bored piles in non-cohesive materials 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 LN(Qp/AP) -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 LN(Qp/AP) -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 (c) Driven piles in cohesive materials (d) Bored piles in cohesive materials Figure 7.5: Scatter plots of ? Vs LN(Qp/AP) Stellenbosch University http://scholar.sun.ac.za 217 Table 7-4: Correlation coefficients and associated p-values for ? Vs LN(Qp/AP) Case N r p-value DNC 29 -0.003 0.987 BNC 31 -0.002 0.990 DC 59 -0.001 0.996 BC 53 -0.001 0.995 7.3 Conditioned M This approach seeks to take into account the fact that the model factor varies with the predicted capacity. Since for a given M value, there is a range of possible values of Qp, a probabilistic description of the functional relationship between M and Qp is required. A probabilistic description of the relationship between variables is generally achieved by regression analysis. Accordingly regression analyses of M on Qp were performed for each piles class. The resulting regression equation gives the functional relationship between the mean model factor and Qp. The general equation for the model factor is given by: pbQaM += [7.9] Where M= mean model factor, a = y-intercept, b = slope, Qp = interpreted capacity. Since the computation of the mean and the standard deviation of M takes into account its correlation with Qp. The resulting M statistics are referred as conditioned mean and standard deviation (i.e. the standard deviation given Qp). Because the trend is accounted for through the regression line of Eq. 7.9, the variance about this line is the measure of dispersion of interest, which is the conditional variance. For the case where the conditional variance is constant within the range of Qp values of interest, the conditional standard deviation of the model factor is given by the standard error of the estimate (?), which is the variation of the model factor values about the regression line. 7.3.1 Results for Regression of M on Qp The scatter plots of regression of M on Qp for the various cases are presented in figure 7.6. The regression equations and the associated standard error of estimate are also shown in this figure. The conditional mean and standard deviation of the model factor for a given pile class are represented by the regression equation and standard error of estimate respectively. Stellenbosch University http://scholar.sun.ac.za 218 The representation of the mean model factor by an equation captures the fact that the mean model factor is a function of Qp. Although the mean model factor varies with Qp, for a given pile class values of the standard deviation are generally assumed to be constant. 1000 2000 3000 4000 5000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M M = 1.2938 - 8.9903E-5*Qp Std err. of est. = 0.287 0 1000 2000 3000 4000 5000 6000 7000 8000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 M M = 0.9321 + 2.4371E-5*Qp Std err. of. est.= 0.241 (a) DNC (b) BNC 0 1000 2000 3000 4000 5000 6000 7000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 M M = 1.3414 - 7.749E-5*Qp Std err. od est. = 0.299 0 2000 4000 6000 8000 10000 12000 Qp (kN) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 M M = 1.1661 - 3.9715E-6*Qp Std err. of est. = 0.275 (c) DC (d) BC Figure 7.6: Regression of M on Qp Stellenbosch University http://scholar.sun.ac.za 219 7.3.2 Derivation of model factor statistics The general regression equation of M on Qp (i.e. Eq. 7.9) is taken as a function of a random variable. From the computational rules of random variables, the moments of M are given by: ( ) ( ) pQp babQaEME ?+=+= [7.10] 222 pQM b ?? = [7.11] From Eq. 71, the standard deviation is given by: ??? == PQM b [7.12] The regression parameters and the ensuing conditioned model factor statistics are presented in table 7.5. It should be noted that the expression for computing the mean model factor is not dimensionless. Therefore the conditioned mean model factors presented in table 7.5 are not dimensionless. However, comparison of the conditioned and standard (Table 6.2) model factor statistics shows very little difference. In fact, it is only for driven piles in non- cohesive materials that the standard deviation reduced from 0.30 to the conditional value of 0.29. For BNC and DC, the values of conditional standard deviations are the same as their respective absolute values. Theoretically accounting for the trend significantly reduces the variability provided there is a significant correlation between the parameters under consideration. Accordingly in this study it was shown that the model factor was only statistically significantly correlated to the predicted capacity in driven piles in non-cohesive materials, which explains the slight reduction of variability for this case. For BC, the conditional standard deviation is slightly higher than the absolute standard deviation. This suggests that in the scatter plot of M vs Qp, a horizontal line about the mean model factor best fit the data compared to the regression line. Visual inspection of figure 7.6(d) seems to support that a horizontal line drawn at M=1.2 will fit the data better. Due to the weak correlation between M and Qp, it can be concluded that accounting for the trend between the two parameters does not lead to reduced variability. Since there is no noticeable reduction of variability, it follows that no significant improvement to the calculated reliability index will be achieved by using conditional model factor statistics Stellenbosch University http://scholar.sun.ac.za 220 rather than the standard model factor statistics. Accordingly the approach was not considered any further. Table 7-5: Conditioned model factor statistics Pile class Regression parameters Mean Model factor statistics a b Qp M STD COV DNC 1.2938 -8.9903E-05 2702.59 1.05 0.29 0.28 BNC 0.9321 2.4371E-05 1787.16 0.98 0.24 0.25 DC 1.3414 -7.7490E-05 2286.97 1.16 0.30 0.26 BC 1.1661 -3.9715E-06 3100.48 1.15 0.28 0.24 7.4 Comparison in terms of reliability indexes In this study the objective of accounting for the correlation was to minimize its effects on the calculated reliability index. Accordingly this section investigates the extent of improvement to the calculated beta values imparted by the removal or incorporation of the correlation. Regarding the conditioned model factor approach, the model factor statistics are similar to those for standard model factor approach. Given that the performance functions for the two approaches will be identical, the two approaches will give similar results. Therefore there is no need to compute the beta values based on the conditioned M approach. For the generalised model factor approach, the statistics are different for that of the standard approach and on the basis of the statistics, it can be seen that no significant reduction of the variability has been realized. However, the performance function for the approach incorporates the regression parameter ?b? as an exponent to the Qp. The effects of ?b? might lead to improve the calculated reliability. This necessitated the comparison of beta values derived on the basis of the standard and generalised model factor statistics. Instead of computing reliability indexes for a few selected pile design cases, the reliability indexes implied by the current working stress design method were evaluated using the compiled database. A motivation for computing the reliability indexes inherent in the current practice is that, the results can be used as the basis for the selection of target beta for pile foundations in South Africa. For a given pile class, the reliability index implied by the current practice was evaluated by the advanced first-order second moment method using the nonlinear optimisation function in Excel as discussed in Chapter 3. Stellenbosch University http://scholar.sun.ac.za 221 7.4.1 Design equation and performance function for the working stress design approach Currently, in South Africa geotechnical design is based on the working stress design approach. With this approach, the resistance is factored while the loads are unfactored. If only dead load plus live load are considered with wind load ignored as it is the case for design of pile foundations for most common structures, the design equation is given by: nn n LD FS R += [7.10] Where: nR = Nominal resistance (predicted pile capacity), nD = nominal dead load, nL = nominal life load and FS = factor of safety. For simplicity the calculation was done in the load space which entails expressing nL in terms of nD . By so doing, it is not necessary to deal specifically with pile diameter and length. When nL is expressed in terms of nD , Eq. 7.10 becomes: ??? ? ??? ? += n n n n D L D FS R 1 [7.11] From Eq. 7.11 the expressions for nD and nL are as follows: ?? ??? ? + = n n n n D LFS R D 1 [7.12] n n n DFS R L ?= [7.13] The limit state function is given by: 0=?? LDR [7.14] Where R, D, and L are random variables defined as follows: R = measured resistance Stellenbosch University http://scholar.sun.ac.za 222 D = measured permanent load and L = measured variable load In this study the resistance and load were modelled as lognormal variable. Usually the measured load and resistance are presented in terms of their respective predicted values and mean model factors as follows: nR RMR = ; nD DMD = ; nL LML = [7.15] Where: RM = mean model factor for resistance, DM = mean model factor for dead load, LM = mean model factor for live load and the other symbols are as previously defined. Substituting Eq. 7.15 into Eq. 7.14, the performance function becomes: 0=?? nLnDnR LMDMRM [7.16] For the generalised model factor approach, the design equation is the same as that for the standard model factor approach (i.e. Eq. 7.11 through Eq. 7.13). However the performance function becomes: 0=?? nLnDbnR LMDMRM [1.17] in which the exponent b is the appropriate regression parameter given in table 7.3. The formulations were set on a spread sheet to compute beta values implied by the current design practice. The spreadsheets for computation of beta values as well as the description of the Excel functions for both standard M and generalised M approaches are presented in Appendix C. The model factor statistics for load were deduced from the South African loading code (Chapter 2). The factor of safety was taken as 2.5 in accordance with common practice in South Africa. Stellenbosch University http://scholar.sun.ac.za 223 7.4.2 Beta values based on the standard model factor statistics Beta values were sensitive to the variation in Ln/Dn ratio and not the variation in the nominal resistance, implying that the calibration points are only defined by the Ln/Dn ratio. Accordingly beta values were calculated for a range of Ln/Dn ratio. The results are presented in figure 7.7. Figure 7.7: Variation of beta values with Ln/Dn ratio for standard M approach The following observations were drawn from figure 7.7: ? The beta values vary from one pile class to the other indicating that the current global factor of safety approach does not produce a consistent level of reliability across different design situations. Further more even within the same pile class beta values vary with individual cases represented by the Ln/Dn ratios. This suggests that even within the same design situation the global factor approach fails to attain a consistent level of reliability. ? It is evident from figure 7.7 that piles in cohesive materials depict higher reliability compared to piles in non-cohesive materials. The scenario is attributed to higher inherent conservatism of the static formula in non-cohesive materials compared to non- cohesive materials. 2.50 2.70 2.90 3.10 3.30 3.50 3.70 3.90 4.10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ln/Dn ratio B et a DNC BNC DC BC Stellenbosch University http://scholar.sun.ac.za 224 ? For a given material type, the beta values for bored piles are higher than that for driven piles. This is attributed to the observation noted in chapter 6 that for a given material type the variability of the model factor in driven piles was higher than that in bored piles ? In general, the variation of beta values with the Ln/Dn ratio seems to be dependent on the nature of the structure. For structures that are predominated by dead loads (Ln/Dn < 0.6), beta values tends to increase with the increase in the Ln/Dn ratio. Conversely as live loads become significant (i.e. Ln/Dn ? 0.6), beta values decreases with the increase in the Ln/Dn ratio. ? Within the range of Ln/Dn ratios where beta values decrease with increase in the ratio, the change in the beta values are not substantial. ? For a given material type, the beta values for bored piles are higher than that for driven piles. This is attributed to the observation noted in chapter 6 that for a given material type the variability of the model factor in driven piles was higher than that in bored piles. To facilitate the comparison between beta values derived by the various model factor statistics, a single representative beta from each approach is required. Ideally such a value could be determined from a weighted average of all the calculated beta values (Melchers, 1999). In practice this approach is seldom feasible owing to the large number of calibration points and the difficulty of assigning a suitable weight to each point. A practical approach is to note the complexity of the issue and select representative beta values on a semi intuitive basis (Melchers, 1999). In this study, a critical value was used instead of the representative value. The critical value was the smallest value within a typical range of Ln/Dn ratios encountered in practice. Typical range of Ln/Dn ratios are 0.5 ? 1.5 for concrete structures and 1 ? 2 for steel structures (Melccres, 1999). Based on this information, a practical range of Ln/Dn ratio of 0.5 to 2 was adopted. Since within this practical range, the beta values decrease with the increase in Ln/Dn ratio, the lowest values correspond to Ln/Dn ratio of 2. From figure 7.7 the critical values are 2.99 for DNC, 3.07 for BNC, 3.47 for DC and 3.61 for BC. Stellenbosch University http://scholar.sun.ac.za 225 7.4.3 Beta values based on the generalised model factor statistics Beta values based on the generalised M statistics vary with Ln/Dn ratio as well as nominal resistance values (Rn). This necessitates the determination of beta values for representative Rn values. To obtain the range of beta values, only values corresponding to the lower and upper bounds Rn need to be determined. The Lower and upper bounds Rn values were in accordance with values in the pile load tests database for the respective cases. The range of Rn values for the respective cases are presented in table 7.6. Table 7-6: Range of Rn (Qp) values in the database Case Range (kN) Driven piles in non-cohesive materials (DNC) 1080-4969 Bored piles in non-cohesive materials (BNC) 472-6754 Driven piles in cohesive materials (DC) 556-5908 Bored piles in cohesive materials (BC) 462-10172 The beta values for the lower and upper bounds Rn corresponding to a range of Ln/Dn ratio, are presented in figure 7.8a through 7.8d. (a) DNC 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 0 1 2 3 Ln/Dn B et a Lower bound Rn Upper bound Rn Stellenbosch University http://scholar.sun.ac.za 226 (b) BNC (c) DC (d) BC Figure 7.8: Variation of beta values with Ln/Dn ratio for generalised M approach 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 0 1 2 3 Ln/Dn B et a Lower bound Rn Upper bound Rn 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 0 0.5 1 1.5 2 2.5 3 Ln/Dn B et a Lower bound Rn Upper bound Rn 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 0 0.5 1 1.5 2 2.5 3 Ln/Dn B et a Lower bound Rn Upper bound Rn Stellenbosch University http://scholar.sun.ac.za 227 The analysis of figure 7.8 yielded the following observations: ? For driven piles in non-cohesive materials, beta values increases with the increase in Rn values. ? For bored piles in non-cohesive materials, also beta values increase with the increase in Rn values. ? For driven piles in cohesive materials, beta values decreases with the increase in Rn values. ? Also for bored piles in cohesive materials there is a decrease in beta values with the increase in Rn. ? The variation in beta values with Rn value depends on the magnitude of the exponent b. When b is greater than 1, beta increases with the increase in Rn as was the case for DNC and BNC. The b values for these two cases were 1.031 for DNC and 1.030 for BNC. Conversely when b is less than 1 the beta values decreases with the increase in Rn as was the case for DC and BC (b=0.977 for DC and 0.961 for BC). This comes from the mathematical set up of the performance function ( MQMRg bn ?= ). ? The variation of beta values with Ln/Dn ratio follows a similar trend to the one shown by beta values based on standard M statistics (i.e. in general beta values increases with the increase in Ln/Dn ratio for Ln/Dn < 0.6 and there after, beta values decrease with the increase in Ln/Dn ratio). Again, the range in beta values implied by the working stress design approach varies widely. This wide variation of beta values reiterate the fact that the current working stress design approach does not produce uniform beta values even in the same design case (e.g. bored piles in cohesive materials). As was the case for standard M approach, within the practical range of Ln/Dn ratio (i.e. 0.4 to 2), beta values decrease with the increase in Ln/Dn ratio. Therefore the critical beta value corresponds to an Ln/Dn ratio of 2 for either lower or upper bound Rn value, depending on which gives smaller beta values. From figure 7.8, the critical values are 3.09 for DNC, 3.00 for DNC, 3.02 for DC and 2.96 for BC. Stellenbosch University http://scholar.sun.ac.za 228 7.4.4 Comparison of magnitudes of beta values The critical beta values corresponding to the practical range of Ln/Dn of 0.5 to 2 are summarised in table 7.7. Close inspection of the critical beta values for the two approaches lead to the following observations: ? The critical beta values are close to the specified target beta of 3. Therefore all the three approaches yield beta values that meet the target reliability index specified by SANS 10160. This further suggests that the three approaches produce satisfactory results; hence any of them can be adopted. ? A comparison of beta values shows that values for the standard M are significantly higher than that for generalised M in cohesive materials. However, in non-cohesive materials, the difference between the results of the two approaches is negligible. Therefore the generalised M approach does not produce improved reliability indexes. Table 7-7: Critical beta values for Ln/Dn ratio of 0.5 to 2 Case Standard M Generalised M DNC 2.99 3.09 BNC 3.07 3.00 DC 3.47 3.02 BC 3.61 2.96 From the foregoing, it can be concluded that the results of the supposedly more refined approach is not radically different from those of the conventional approach (standard M approach). This state of affairs can be traced back to the degree of the correlation established in chapter 6. It was shown that except for DNC, the correlation between the model factor and predicted capacity was statistically insignificant. Therefore the removal of the existing weak correlation can not lead to considerable improvement in the beta values. Therefore, while acknowledging the potential of the generalised M approach as an alternative method for deriving model factor statistics, in this study the resistance factors will be calibrated on the basis of the standard model factor statistics. Additional advantages of adopting the standard model factor approach include: ? The approach is simple to use as the calibration points are limited to Ln/Dn ratio only. Stellenbosch University http://scholar.sun.ac.za 229 ? The approach has been used in other calibration studies, thereby facilitating the comparison of the calibration results with those obtained in other studies. This is crucial in the case of Southern Africa as this is the first exercise of its kind. ? Various calibration equations based on standard model factor statistics have been developed. This will make it possible to derive the resistance factors using well developed and tested calibration equations. Stellenbosch University http://scholar.sun.ac.za 230 Chapter 8 RELIABILITY CALIBRATION OF RESISTANCE FACTORS 8.1 Introduction As alluded to in Chapter 2, the partial factors format has become the orthodox method for verification of non-accidence of the limit states. In the context of reliability based design, the partial factor method constitutes an important step from the complex direct probabilistic methods towards simplified design procedures (i.e. level 1 reliability methods). In conjunction with the calibration principles set in Chapter 3 and the model factor statistics developed in the study, this Chapter is primarily focused on deriving the resistance partial factors. 8.2 Target beta The primary factors influencing the selection of target beta values were identified in Chapter 3 as: ? Reliability indexes implied by the current design approach ? Target beta value set by regulatory authorities for a given limit state ? Redundancy inherent in the system Reliability indexes inherent in the current design practice captures the long successful South African design experience. The beta values implied in the current working stress design approach were developed Chapter 7. The critical beta values corresponding to the practical range of Ln/Dn of 0.5 to 2 were presented in table 7.7. The same value, but now rounded to one dismal place are presented in table 8.1. On the basis of beta values presented in table 8.1, it can be concluded that reliability indexes implied by the current working stress design practice are comparable to the prescribed target beta of 3. Therefore the two selection criteria of beta values implied by the current practice and the target beta value set by the South African loading code suggest a target beta of 3. However, the redundancy inherent in pile foundation system needs to be considered. Stellenbosch University http://scholar.sun.ac.za 231 Table 8-1: Critical beta values Case ? value DNC 3.0 BNC 3.1 DC 3.5 BC 3.6 In pile foundations, it is not common for a single pile to support a structural load on its own because of overturning effects, lateral loading and difficulty of transferring the load axially down the pile (Barnes, 1995). In most cases piles are used in groups with the structural load shared between the piles. In the pile group, failure of an individual pile does not necessary imply that the group will fail as adjacent piles that may be more lightly loaded could take some of the additional load (Zhang, et al 2001). Therefore beta for pile groups can be significantly higher than that of single piles. Further more, in a pile system of several groups (most common practical case) supporting a structure, failure of a pile group does not necessarily mean the pile system will fail. Load redistribution among the pile groups through superstructure elements may occur due to system effects resulting in an additional reliability. The flexibility of the soil contributes to the ability of the pile-soil system to share and redistribute load. Therefore redundancy is prevalent in the field operation of pile foundations. Because of presence of redundancy, pile foundation can be designed for a lower target beta value than superstructure components. In the case of pile foundations, the general target beta value prescribed by regulatory authorities should be considered as a target value for pile groups. The task now is to determine the corresponding target beta values for single pile design (?TS) to achieve the target reliability of the pile group (?TG). In general (?TS) values are smaller than the (?TG) value for the pile group. In this regard Zhang et.al (2001) showed that for a (?TG) value of 3 the corresponding (?TS) values were in the range of 2.0 ? 2.8 if no system effects are considered and further decreased to 1.7 to 2.5 when a system factor of 1.25 is considered. The (?TS) values reduced even further to 1.7 ? 2.0 when a larger system factor of 1.5 is considered. From the foregoing it can be inferred that representative beta values implied in the current practice of about 3 for piles in non-cohesive materials and about 3.5 for piles in cohesive materials will significantly increase if group effects and system effects are taken into account. This further implies that the target reliability of the pile group implied in the current design practice is higher than 3. In other words current practice is over-conservative. Stellenbosch University http://scholar.sun.ac.za 232 Therefore the representative beta values of 3.0 and 3.5 need to be reduced so that when group and system effects are considered ?TG should be a minimum of 3 but not much higher than 3. Calculation of single pile reliability on the basis of group reliability or vice-versa is possible if pile test data on pile groups are available. However in this study such data is not available and therefore mapping from single pile reliability to group reliability or vice-versa is based on intuition and values reported in the literature. Hence a target beta of 2.5 is considered sufficient for building structures and other common structures (i.e. Class RC 2). For class RC 3 a target beta of 3 is adopted while for class RC 1 structures a target beta of 2 is considered. Table 8.2 compares the target values recommended for pile foundations to values given in the revised South African loading code (SANS 10160-Draft). Table 8-2: Comparison of ?T in SANS 10160 the ?T adopted for the study Class ?T ?SANS 10160 ?T - recommended for pile foundations RC 3 3.5 3.0 RC 2 3.0 2.5 RC 1 2.5 2.0 Depending on the failure consequence class the recommended target beta values range from 2.0 ? 3.0. This range of target beta values has also been adopted in other calibration studies around the world. Baecher and Christian, (2003) asserted that in modern foundation codes, target reliability indexes ranging from 2.0 to 3.0 are common. The lower range is for cases of non-essential designs with high redundancy while the upper range is for critical designs with little redundancy. Paikowsky et.al (2004), as a revision to the driven piles and drilled shaft section of the AASHTO LRFD Bridge Design Specifications, recommended ?T = 2.33 for redundant piles (5 or more piles per pile cap), and ?T =3.0 for non-redundant piles (? 4 piles per pile cap). Becker, et.al (1991) selected ?T of 2.0 to 2.5 for calibration of resistance factors of driven piles and Withiam et.al (1998) confirmed that this range of target beta is reasonable for a single pile design considering that piles are usually used in groups. Rahman et.al (2002), asserted that ?T of 2.0 to 2.5 was within conformity with ? values implied by their current design practice and hence selected this range of ?T for the calibration of resistance factors for their study. Stellenbosch University http://scholar.sun.ac.za 233 8.3 Calibration Methods A number of expressions for reliability calibration of resistance factors proposed in the literature were presented in Chapter 3. In addition, direct calibration to the existing practice was carried out for the purposes of comparison of the results. The following five approaches were explored: d) Advanced first-order second moment approach (AFOSM) e) First-order second moment approach (FOSM) f) Approximate to first-order second moment approach g) Design point approach h) Calibration by fitting to WSD 8.3.1 Advanced first-order second moment approach As pointed out in Chapter 3 AFOSM analysis entails an iteration process. For calibration of partial factors, limit state design equations are required as opposed to the working stress design equations used for deriving beta values implied by the current practice. With the limit state design equations, both the resistance and the loads are factored. Accordingly, the design equation is given by: kQkG R n QG R ??? += [8.1] Where: nR = predicted pile capacity; R? = partial resistance factor; kG = permanent action; G? = partial factor for permanent action; kQ = variable action; and Q? = partial factor for variable action. It was pointed out in Chapter 2, that for Design Approach 1 two combinations of load and resistance partial factors need to be considered. However it has become customary that combination 1 governs the structural design of the elements while combination 2 governs the geotechnical of the elements. Accordingly for the GEO limit state, the values of the load factors G? and Q? are 1.0 and 1.3 respectively. Therefore Eq. 8.1 becomes: Stellenbosch University http://scholar.sun.ac.za 234 kk R n QG R 3.10.1 +=? [8.2] Expressing kQ in terms of kG ??? ? ??? ? += k k k R n G Q G R 3.1 1? [8.3] From Eq. 8.3, the expressions for kG and kQ are as follows: ?? ??? ? + = k k R n k G Q R G 3.11? [8.4] ??? ? ??? ? ?= k R n k G R Q ?3.1 1 [8.5] The performance function is the same as that developed for the working stress design approach and is given here as Eq. 8.6. 0=?? nLnDnR LMDMRM [8.6] In equation 8.6 nD and nL have the same meaning as kG and kG respectively. The design equation, the performance function, and the model factor statistics were then set on a spread sheet to facilitate the computation of the partial resistance factor that will yield a prescribed target beta value for a given set of model factor statistics. The spreadsheet and the description of Excel functions for the various cells are presented in Appendix D. The basic algorithm of the AFOSM for resistance factor calibration is similar to that of the AFOSM reliability analysis presented in Chapter 7. Therefore the spreadsheet in figure 8.1 is similar to that presented in chapter 7. However the process of obtaining the resistance factors is a reverse of what was used to determine the beta values for a given set of model factor statistics and factor of safety. The procedure was as follows: Stellenbosch University http://scholar.sun.ac.za 235 ? Enter a trial resistance factor value on the spread sheet; ? Evoke the solver function to compute the beta value corresponding to the entered resistance factor; ? Repeat steps 1 and 2 till the computed beta value is equal to the desired target beta value. The resistance factors were calibrated for ?T = 2.5 (for class RC 2 structures), ?T = 3.0 (for class RC 3 structures) and ?T = 2.0 (for class RC 3 structures). The model factor statistics used in the calibration are presented in table 8.3. These are the same model factor statistics derived in chapter 6. Table 8-3: Model factor statistics used in the calibration of resistance factors Case Mean Std. dev. COV DNC 1.06 0.30 0.28 BNC 0.98 0.24 0.24 DC 1.17 0.30 0.26 BC 1.15 0.28 0.24 8.3.1.1 Resistance factors for a target beta of 2.5 The resistance factors as a function of the Ln/Dn ratio are is shown in figure 8.1. Stellenbosch University http://scholar.sun.ac.za 236 Target beta = 2.5 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 0 0.5 1 1.5 2 2.5 3 Ln/Dn R es is ta nc e fa ct or DNC BNC DC BC Figure 8.1: Variation of resistance factors with Ln/Dn ratio The following observations can be drawn from figure 8.1: ? For all the cases, there is a general decrease in values of resistance factors with the increase of the Ln/Dn ratio. However, the rate of decrease reduces with the increase in Ln/Dn ratio. From an Ln/Dn ratio of 0 ? 1 the rate of decease is appreciable while at an Ln/Dn ratio of greater than 1, the rate of decrease is very small. ? The values of the resistance factors appear to be influenced by the materials types. In this regard, the partial factors for piles in cohesionless materials are quite close irrespective of the pile installation method. The same trend is exhibited by piles in cohesive materials. The variation of resistance factor with the various calibration points is an indication of the variation in reliability indexes across the calibration points or design situations within a given class of pile foundations. Despite the variation in resistance factors across the calibration points, a single value applicable to all the design situations is required for the purposes of codified design. The application of a single resistance factor to all the design situations will inevitably lead to some deviation from the target reliability index for some of Stellenbosch University http://scholar.sun.ac.za 237 the calibration points. Therefore the partial factors should be calibrated such that the beta values for the various calibration points are as close as possible to the target reliability index. In principle the partial factor which best approximate the uniform target reliability can be obtained by minimizing the deviation from the target beta using an objective or penalty function penalising the deviation from the target reliability index. A number of objective functions have been proposed in the literature. Ideally such functions should be based on economic terms (i.e. cost benefit analysis). Detailed discussion on this form of objective function can be found in Ditlevsen and Madsen (2005). However, in most current calibration exercises, the mean square deviation of the calculated reliability index from the target value has been used (e.g. Gayton et al, 2004). The most common and simplest one is the weighted least function given by: ( )2Tin i iwS ?? ?= ? [8.7] Subject to the constraint: ( ) Tin i iw ?? =? with 1=?n i iw Where S = measure of closeness to the target reliability index; ?T = target reliability index; ?i = the value of the ith reliability index computed on the basis of the partial factor under consideration; wi = a weighting factor to account for the importance of the calibration point (Ln/Dn ratio) relative to design practice. It is common to express equation 8.6 in terms of probability of failure. In this regard, logarithm of the probability of failure is used to make S relatively more sensitive to very low values of failure probabilities. The resulting expression is as follows: ( )2loglog fTfCn i i PPwS ?= ? [8.8] Stellenbosch University http://scholar.sun.ac.za 238 Where PfT = ?(-?T) is the target probability of failure; PfC = ?(-?i) is the probability of failure for a given calibration point. The main disadvantage of above penalty function is that it is symmetrical about ?T and hence calibration points with a reliability index smaller than the target are not more penalised than structures with a higher reliability index. As a solution to this problem, skewed objective functions giving relatively more weight to reliability indices smaller than target compared to those larger than the target have been proposed in the literature. A commonly used function in this category is that proposed by Lind (1977). The function is given by: )1))(exp()(( 1 ???+?= ? TiTii i kkwS ???? [8.9] in which k>0 is the curvature parameter and the other symbols are as defined earlier. When the parameter k increases the Lind?s function becomes more penalising for reliability indexes smaller than the target value. Gayton et.al (2004) observed that for k=1 the Lind?s function gives results that are close to the least square function while still penalising under- designs more than over-designs. In general, it has been found that the final result is not very sensitive to the choice of the penalty function (Ditlevsen and Madsen, 2005; Gayton et.al, 2004). This suggests that any of the three expressions discussed above (Eq. 8.6, Eq. 8.7 and Eq. 8.8) can be adopted. However, for this study the three expressions were investigated using data from one of the pile classes. The function leading to conservative results was then chosen and applied to all the other pile classes. The solution to equations 8.6, 8.7 and 8.8 becomes a problem of constrained minimization for which a number of standard techniques and computer programs are available. In the absence of a ready made computer programme for the optimization process, the following procedure was adopted in this study: ? The number of calibration points was reduced by limiting the Ln/Dn ratio to a range that is most frequently encountered in practical designs. Theoretically, Ln/Dn range from 0 - ?. However, as noted in Chapter 7, in practical designs typical ranges of Ln/Dn are: 0.5 ? 1.5 for reinforced concrete structures and 1-2 for steel structures. On the basis of the Stellenbosch University http://scholar.sun.ac.za 239 typical ranges, a range of 0.5 ? 2.0 was considered to constitute a commonly employed design range. ? Weighting factors were assigned to the chosen calibration points to reflect the relative occurrence of each point in practical design practice. Greater weight has been given to the more frequent design cases. The most common design cases are represented by Ln/Dn range of 1-1.5. This is the range applicable to both concrete and steel structures. The weights assigned to the respective design cases or calibration points are presented in table 8.4. Table 8-4: Importance weighting for the respective calibration points Ln/Dn ratio Weight 0.5 0.1 1.0 0.4 1.5 0.4 2.0 0.1 ? Trial values of partial factors were used to calculate ?i for each calibration point. The results were substituted in equation 8.6 to 8.8 to calculate the S. ? After a number of trials, a set of partial resistance factors and their respective S values were obtained. ? The partial resistance factors were plotted against their respective S values (S1= least square in terms of beta, S2= least square in terms of probability of failure, S1= Lind?s objective function) to obtain a curve from which optimal resistance factor can be read. The optimal resistance factor corresponds to the minimum S value. Table 8.5 presents the S values for the three objective functions for the DNC case. The corresponding weighted average beta values are also shown in the last column of the table. From table 8.5 it can be seen that for the objective functions based on the least squares the minimum S values correspond to a resistance factor of 1.8 while for the Lind?s objective function, the minimum S value corresponds to a partial factor of 1.85. These values are approximate. To obtain more accurate values graphs of S versus resistance factors were plotted (figure 8.2). Graphs of the trial partial factors against their respective S values for the three functions show the same trend. That is, initially the S values decrease with the increase in the partial Stellenbosch University http://scholar.sun.ac.za 240 factors, then reaches a minimum value and then increases with the increase in the partial factor values. The partial factor corresponding to this minimum S value is taken as the optimum value. For DNC, the lowest S value corresponds to a partial factor of 1.82 for the two least square functions (fig. 8.2 a) and 1.87 for Lind?s function (fig. 8.2 b). The values are slightly different from those obtained from table 8.5. It is evident from these results that the Lind?s function is on the conservative side. Since it is better to err on the conservative side, Lind?s objective function was used in the selection of the optimum partial factor for all the pile classes. Table 8-5: S values for respective trial ? values ? S1 S2 S3 w? 1.5 0.3619 0.4416 0.0734 1.9 1.6 0.1560 0.2040 0.0336 2.1 1.7 0.0419 0.0581 0.0116 2.3 1.75 0.0143 0.0202 0.0054 2.4 1.8 0.0041 0.0059 0.0017 2.5 1.85 0.0098 0.0153 0.0001 2.6 1.9 0.0303 0.0484 0.0003 2.7 1.95 0.0642 0.1051 0.0020 2.7 2 0.1106 0.1853 0.0049 2.8 2.2 0.402848 0.737793 0.0257 3.1 2.5 1.0844 2.2327 0.0719 3.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.4 1.6 1.8 2 2.2 Resistance factor S Least square (beta) Least square (Pf) (a) least square penalty functions Stellenbosch University http://scholar.sun.ac.za 241 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 1.4 1.6 1.8 2 2.2 2.4 2.6 Resistance factor S Lind (b) Lind?s penalty function Figure 8.2: Optimal partial resistance factor for DNC The graphs of S versus partial resistance factors based on Lind?s objective function for BNC, DC, and BC are presented in figure 8.3. The ensuing optimal partials factors are 1.84 for BNC, 1.62 for DC and 1.57 for BC. 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 Resistance factor S (a) Stellenbosch University http://scholar.sun.ac.za 242 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 Resistance factor S (b) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 Resistance factor S (c) Figure 8.3: Optimal partial resistance factors; (a) BNC, (b) DC, (c) BC Scrutiny of the optimal partial factors obtained reveals that these values are the same as the partial factor values for Ln/Dn of 0.5 in all the pile classes. Ln/Dn of 0.5 is the minimum value of the commonly employed design range. Within the practical range of Ln/Dn (i.e 0.5 to 2), the largest partial resistance factor correspond to Ln/Dn of 0.5 as shown in figure 8.2. In terms of reliability index, the calibration point represented by Ln/Dn of 0.5 has the lowest beta value within the range under consideration. In essence, the partial factor for Ln/Dn of 0.5 provides the minimum required reliability index (target beta). Therefore Lind?s objective function yields optimal partial factors that ensure that reliability indexes for all calibration points are equal or greater than the target value. Stellenbosch University http://scholar.sun.ac.za 243 From the foregoing, it can be concluded that the optimal partial factors could have as well been selected on the basis of values that meet the required minimum reliability index within the range of selected calibration points rather than the tedious optimisation process. For all pile classes, such partial factor values correspond to the calibration point denoted by Ln/Dn of 0.5. Comparison of the numerical values of the optimal partial factors indicates that the optimal resistance factors for the four pile classes are comparable. Consequently, it is tempting to adopt one single partial factor for all the four classes. Such a partial factor could be established from an optimisation process similar to the one described previously or by simply using the partial factor for the least reliable pile class. The least reliable pile class was identified as BNC (i.e. one with the highest ?R). Accordingly the single partial factor applicable to all the cases is 1.87. The use of a partial factor of 1.87 for all the four cases gives a minimum beta value of 2.50 for DNC, 2.58 for BNC, 3.04 for DC and 3.19 for BC and an average beta of 2.83. For DC and BC the beta values are appreciably higher than the target value of 2.5, leading to high conservatism. Even the average beta value for the four cases is significantly higher than the target beta by 13.2 %. To avoid high conservatism for piles in cohesive materials (DC and BC), it seems reasonable to apply separate partial factors for piles in cohesive materials and piles in non-cohesive materials. Differentiation in terms of material types is further motivated by the closeness of the partial factors for a given material type irrespective of the installation method (i.e. 1.87 and 1.84 for piles in cohesionless materials; 1.62 and 1.57 for piles in cohesive materials). Separation of partial factors on the basis of materials type implies further subdividing the four pile classes into two categories of two pile classes each. For each category, the partial factor belonging to the least reliable pile class was selected as the overall partial factor for piles in that specific material. The resulting partial factors are: ?R = 1.87 for piles in cohesionless materials and ?R = 1.62 for piles in cohesive materials. Adopting ?R = 1.87 for piles in non-cohesive materials yield beta values of 2.5 for BNC and 2.58 for DNC with an average value of 2.54. These values are not very different from the target beta and hence not much additional conservatism has been introduced. For piles in cohesive materials, the use of ?R = 1.62 produces beta values of 2.52 for DC and 2.64 for BC with an average value of 2.58. As it was the case for piles in non-cohesive materials, the beta values for the two pile classes in cohesive materials are not appreciably different from the target value. Therefore differentiation in terms of material types satisfies the requirements of (i) achieving a Stellenbosch University http://scholar.sun.ac.za 244 minimum target reliability index across a specific domain of interest and (ii) increasing the uniformity of reliability across the domain of interest. For ease of reference and comparison with values for other target beta, the optimal values for the target beta of 2.5 are compiled in table 8.6. Table 8-6: Optimal resistance factors for a target beta of 2.5 Pile class Optimal ?R DNC 1.87 BNC 1.84 DC 1.62 BC 1.57 8.2.1.2 Resistance factors for a target beta values 3.0 The variation of resistance factors with the calibration points (Ln/Dn ratios) are presented in figure 8.4. The analysis of figure 8.4 shows similar characteristics to those shown by resistance factors for the target beta of 2.5. However, the numerical values of resistance factors have slightly increased as expected. It has been previously observation that the optimal partial resistance factor obtained by minimization of Lind?s objective function is very close to partial factors corresponding to the calibration point represented by Ln/Dn of 0.5. Therefore instead of performing the optimisation process, the partial factors for Ln/Dn of 0.5 in figure 8.4 were taken as the optimal partial resistance factors. The resulting optimal partial resistance factors for the various pile classes are presented in table 8.7. Stellenbosch University http://scholar.sun.ac.za 245 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.5 1 1.5 2 2.5 3 Ln/Dn R es is ta n ce fa ct o r DNC BNC DC BC Figure 8.4: Variation of resistance factors with Ln/Dn ratio for ?T = 3.0 Table 8-7: Optimal resistance factors for a target beta of 3.0 Pile class ?R DNC 2.17 BNC 2.09 DC 1.85 BC 1.78 Differentiating only in terms of material types yields ?R = 2.17 for piles in non-cohesive materials and ?R = 1.85 for piles in cohesive materials. Stellenbosch University http://scholar.sun.ac.za 246 8.2.1.3 Resistance factors for a target beta values 2.0 The calibration results for the target beta of 2.0 are presented in figure 8.5. Examination of the results reveals characteristics shown by results for other target beta values. The only noticeable difference is in the values of the partial resistance factors which have now been reduced as expected. Comparison of figures 8.2, 8.4, and 8.5 indicate that for a given material type, the partial factor for driven and bored piles become quite close as the target reliability index reduces. This trend is more apparent in figure 8.5 (?T=2.0) in which curves for driven and bored piles in non-cohesive materials coincide. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.5 1 1.5 2 2.5 3 Ln/Dn ratio R es is ta n ce fa ct o r DNC BNC DC BC Figure 8.5: Variation of resistance factors with Ln/Dn ratio for ?T = 2.0 The selection of the optimal resistance factors for the different pile classes was based on values corresponding to Ln/Dn of 0.5 as was the case for target beta of 3.0. The ensuing partial resistance factors are presented in table 8.8. Further classification in terms of material Stellenbosch University http://scholar.sun.ac.za 247 types only, yields ?R = 1.62 for piles in non-cohesive materials and ?R = 1.41 for piles in cohesive materials. Table 8-8: Optimal resistance factors for a target beta of 2.0 Pile class ?R DNC 1.62 BNC 1.61 DC 1.41 BC 1.38 8.3.2 Mean value first-order second moment approach (MVFOSM) The principal equation for determining resistance factor in accordance with the MVFOSM approach was derived in Chapter 3. This principal equation is given by: [ ])1)(1ln(exp)( 1 1 222 2 22 QLQDRT D L QLQD R QLQD D L QLQDR COVCOVCOV Q Q COV COVCOV Q Q ++++ + ++ ??? ? ??? ? + = ??? ??? ? [8.10] All the statistics in Eq. 9.10 are known. Accordingly Eq. 9.10 was set on a spread sheet to automate the calculations. The results are presented in figure 9.7a through 9.7d. The figures show similar characteristics as follows: ? In all the cases, for a given Ln/Dn ratio, resistance factor increases with the value of the target beta. ? As was the case for the AFOSM approach, there is a general decrease in values of resistance factors with the increase of the Ln/Dn ratio. Again for Ln/Dn ratio of 0 ? 1 the rate of decease is rapid while for Ln/Dn ratio of greater than one the rate of decrease is gentle. Stellenbosch University http://scholar.sun.ac.za 248 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.5 1 1.5 2 2.5 3 Ln/Dn R es is ta nc e fa ct or ?=2.0 ?=2.5 ?=3.0 (a) 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0 0.5 1 1.5 2 2.5 3 Ln/Dn R es is ta nc e fa ct or ?=2.0 ?=2.5 ?=3.0 (b) Stellenbosch University http://scholar.sun.ac.za 249 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 0.5 1 1.5 2 2.5 3 Ln/Dn ratio R es is ta n ce fa ct or ?=2.0 ?=2.5 ?=3.0 (c) 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 0.5 1 1.5 2 2.5 3 Ln/Dn ratio R es is ta n ce fa ct o r ?=2.0 ?=2.5 ?=3.0 (d) Figure 8.6: Resistance factors for MVFOSM method, (a) DNC, (b) BNC, (c) DC, (d) BC Stellenbosch University http://scholar.sun.ac.za 250 For a given pile class, a single representative value of the resistance factor was taken as the highest value within the practical range of Ln/Dn ratio (i.e 0.5 to 2) as was the case for the AFOSM method. The resultant resistance partial factors are presented in table 8.9. It is evident from table 8.9 that partial factors for piles classes in the same material type are quite close, thereby reiterating the need to separate the partial factors on the basis of materials types only. In comparison with the values for the same pile class and target beta presented in tables 8.5 to 8.7, the partial factors in table 8.9 are generally higher. For the reference class of structures with a target beta of 2.5, these partial factors are high by 22 % for DNC, 24% for BNC, 22 % for DC and 24 % for BC. On average the MVFOSM approach gives resistance factors that are 23% higher. Table 8-9: Resistance factors for the MVFOSM approach Pile class ?=2.0 ?=2.5 ?=3.0 DNC 1.88 2.28 2.75 BNC 1.91 2.28 2.73 DC 1.65 1.98 2.38 BC 1.63 1.94 2.32 8.3.3 Approximate to first-order second moment approach The approximation allows for separate determination of resistance and load factors. This is an advantage given that for a particular geotechnical application, resistance and load statistics are not readily available and it would be a lengthy process to collect the necessary data. In this study, only resistance statistics were collected and therefore calibrating resistance and load factors independently seems to be the most logical approach. The expression for determining resistance factors using this approach was derived in Chapter 3 and is given by: ( )RTRR COV???? ?= exp [8.11] The use of equation 8.11 requires the selection of a fitting factor (?). Although there is consensus about the range of ? (i.e 0.7 to 1), no specific value is universally used. For example Becker, (1996) and Scott et.al (2003) used ? = 0.75 while FHWA H1-98-032 (2001) recommends a value of ? = 0.87. FHWA H1-98-032 (2001) recommended value was derived on the basis of extensive trial and error exercise involving ranges of values for Stellenbosch University http://scholar.sun.ac.za 251 loading and resistance encountered in foundation design. Because of its strong basis, the value of ? = 0.87 was adopted for this study. The resistance factors obtained from Eq. 8.11 and ? = 0.87 for the various pile classes are presented in table 8.10. The values follow the trend that for pile classes in the same soil type, the partial factors are very close. For all the pile classes the partial factors are 24% less than values for the full MVFOSM approach (Table 8.9). When compared to partial factors from the advanced first-order second moment approach, the values in table 8.12 are generally lower. Using the reference case with target beta of 2.5 as an example, the values are lower by 7% in all cases. On the basis of closeness of the values to that obtained by rigorous reliability calibration, it can be asserted that the approach gives reasonable results. However, from a practical point of view, the results of this approach are unconservative and therefore will lead to unsafe designs. From this perspective, the results of the MVFOSM method are better as they err on the conservative side. Table 8-10: Resistance factors for the approximate MVFOSM method Case Resistance factors ? =2.0 ? =2.5 ? =3.0 DNC 1.54 1.73 1.96 BNC 1.55 1.72 1.91 DC 1.34 1.50 1.68 BC 1.32 1.47 1.63 8.2.4 Design value approach In accordance with the design value format discussed in chapter 3, the partial resistance factor is given by: d k R R R=? [8.12] Where: kR is the characteristic resistance and dR is the design value of the resistance given by: )1( VR RRd ??? ?= [8.13] Stellenbosch University http://scholar.sun.ac.za 252 in which R? = sensitivity factor taken as -0.8 (EN 1990) and V = coefficient of variation. Since geotechnical performance is governed by the average resistance, the characteristic resistance ( kR ), is the lower bond of the 95% confidence interval of the mean value. Form Chapter 2, the characteristic value as a mean value at 95% confidence level is given by: ??? ? ??? ? ?= ? nVtXX nk 1 1 95.0 1 [8.14] Where: Where; Xk is the characteristic value, X is the arithmetic mean of the test results, V is the coefficient of variation of the desired property, n is the number of test results, and t is the value of the student distribution corresponding to a confidence level of 95% and a degree of freedom of n-1. From Eq. 8.13 and 8.14; )1( 1 1 95.0 1 V n Vt RR nR R ??? ? ? ? ??? ? ??? ? ? = ? [8.15] Given that R is a lognormal variable, Eq. 8.32 can be expressed as: )5.0exp( 5.0 1 exp 2 295.0 1 VV V n Vt RR nR R ? ??? ? ??? ? ? = ? ??? ? ? [8.16] Most of the terms in Eq. 8.16 counsels out and the expression reduces to: )exp( 1 exp 95.0 1 V n Vt R n R ??? ??? ? ??? ? = ? [8.17] Stellenbosch University http://scholar.sun.ac.za 253 The resistance factors obtained from Eq. 8.17 are presented in table 8.11. These values are comparable with those for the AFOSM approach using an Excel spreadsheet (Table 8.9). For a beta value of 2.5, the resistance factors from the two approaches differ by 2% (DNC), 6% (BNC), 10% (DC) and 9% (BC). Theoretically the two approaches should give similar results as they are both based on evaluating the performance function at the design point. The difference in results is attributed to the approximation of the sensitivity factor to -0.8 which might be different from the actual values. Table 8-11: Resistance factors for the design point approach Case Resistance factors ? =2.0 ? =2.5 ? =3.0 DNC 1.71 1.91 2.14 BNC 1.58 1.74 1.91 DC 1.60 1.78 1.97 BC 1.55 1.71 1.88 8.2.5 Calibration by fitting with WSD approach The selection of final resistance factors is influenced by the resistance factors obtained from calibration by fitting to working stress design. This is of paramount importance if the pile load tests data are limited or not of high enough quality to produce reliable statistics and thereby casting doubt on the reliability of the resistance factors determined based on reliability theory. Since questions surrounding the quantity and quality of the available pile load test database are always there, it is necessary to at least compare resistance factors derived on the basis of reliability theory to resistance factors obtained by fitting to the current practice. This approach considers that if the factors of safety used in the past or current practice have resulted in consistently successful designs, one will at least maintain that degree of success at the same cost as required to meet previous practice. Although in this study the quantity and quality of the data are considered to be sufficient, calibration by fitting is also considered. However, in the selection of the final resistance factors, more weight will be given to results obtained from reliability calibration. The general equation for fitting to WSD was derived in Chapter 3 and is given by: )/1( )/( nn nnLD DLFS DL + += ??? [8.18] Stellenbosch University http://scholar.sun.ac.za 254 Eq. 8.17 was used to determine the resistance factors that need to be used in limit state design equation to obtain a factor of safety equal to that of the working stress design approach. Accordingly Eq. 8.17 was set on a spread sheet to calculate the resistance factors for a range of Ln/Dn ratio and a given factor of safety. The same load factors as those used for the reliability calibration (i.e. 0.1=D? and 3.1=L? ) were adopted while the safety factors ranged from 2 to 3. The calibration results are presented in figure 8.6. 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 Ln/Dn ratio R es is ta n ce fa ct o r FS=2.0 FS=2.5 FS=3.0 Figure 8.7: Resistance factors for fitting to WSD approach Consistent with the working stress design approach, no distinction is made between resistance factors for different pile classes. Therefore the partial factors shown are applicable to all the pile classes. It is evident from figure 8.7 that the variation of the resistance factors with Ln/Dn ratio follows the general trend depicted by resistance factors derived on the basis of reliability calibration. In accordance with the procedure adopted in the preceding calibration methods, the representative partial factor value for a given pile Stellenbosch University http://scholar.sun.ac.za 255 class was taken as the highest value within the practical range of Ln/Dn ratio. The representative values are 1.82 for FS=2.0, 2.27 for FS= 2.5, and 2.73 for FS=3.0. These values mean that in a limit state design format, a resistance factor of 1.82 ie required to achieve a factor of safety of 2.0 and so forth. A comparison of these resistance factors with those obtained by reliability calibration lead to the following observations: - The resistance factor corresponding to FS=2.0 fall within the resistance factors for beta of 2.0 (table 8.8), - The resistance factor for FS=2.5 is comparable with the values for ?T =2.5 (table 8.6), - The value for FS=3.0 fits the results for ?T =3.0 (table 8.7). The results seem to suggest that the use of factors of safety of 2.0, 2.5 and 3.0 lead to reliability indexes of 2.0, 2.5 and 3.0 respectively. As alluded to earlier, calibrating limit state design to existing practice leads to design much the same as that obtained using the working stress design. Therefore the approach does not lead to improved design. 8.2.6 Summary of resistance factors from various calibration approaches Table 8.12 presents a compilation of the resistance factors obtained from the various approaches. Table 8-12: Compilation of resistance factors from various approaches ?T Pile class AFOSM Design point FOSM FOSM approx. Cal. to WSD DNC 1.62 1.71 1.88 1.54 2.0 BNC 1.61 1.58 1.91 1.55 1.82 DC 1.41 1.60 1.65 1.34 BC 1.38 1.71 1.63 1.32 DNC 1.87 1.91 2.28 1.73 2.5 BNC 1.84 1.74 2.28 1.72 2.27 DC 1.62 1.78 1.98 1.5 BC 1.57 1.71 1.94 1.47 DNC 2.17 2.14 2.75 1.96 3.0 BNC 2.09 1.91 2.73 1.91 2.73 DC 1.85 1.97 2.38 1.68 BC 1.78 1.88 2.32 1.63 Stellenbosch University http://scholar.sun.ac.za 256 Comparison of the resistance factors from various calculation schemes (table 8.12) indicates that: ? The resistance factors based on the approximate reliability methods are comparable to those derived by the rigorous reliability approach. ? Regarding the results for calibration to working stress design, it appears that the results for FS =2.0 match the resistance factors for ?T =2.0 while those for FS of 2.5 and 3.0 falls within the factors for ?T =2.5 and ?T =3.0 respectively. Within these subdivisions, the resistance factors from calibration to working stress design are more close to resistance factors from the MVFOSM method. 8.4 Comparison with partial factors given in Eurocode 7 Since Eurocodes are the reference codes in South Africa, it is important to compare the partial factors derived in this study with those given in Eurocode 7. The design of pile foundation is covered in Section 7 of Eurocode 7. The principle pile design methods in Eurocode 7 are (i) full scale pile load tests approach and (ii) semi-empirical analysis using standard field tests results directly. In keeping with the rules of the Eurocodes, the design pile capacity is obtained by dividing the characteristic resistance by a partial factor. The characteristic capacity is obtained as follows: For pile load tests characteristic value of the compressive ground resistance (R) is taken as the lesser of: 1 )( ? meanR R mk = and 2 min)( ? m k R R = Where: meanRm )( = the measured resistance, min)( mR = the minimum measured resistance, 21,?? = correlation factors given in table 8.13. Stellenbosch University http://scholar.sun.ac.za 257 Table 8-13: Correlation factors based on static load tests (After EN 1997-1) ? for n = 1 2 3 4 ?5 ?1 1.40 1.30 1.20 1.10 1.00 ?2 1.40 1.20 1.05 1.00 1.00 On the basis of in-situ ground tests the characteristic resistance is given by the lesser value of: 3 )( ? meanR R ck = and 4 min)( ? c k R R = Where: meanRc )( = the mean calculated resistance, min)( cR = the minimum calculated resistance, 43 ,?? = correlation factors given in table 8.14. Table 8-14: Correlation factors based on semi-empirical methods (After EN 1997-1) ? for n = 1 2 3 4 5 7 10 ?3 1.40 1.35 1.33 1.31 1.29 1.27 1.25 ?4 1.40 1.27 1.23 1.20 1.15 1.12 1.08 For design approach 1 which has been adopted in South Africa, the partial factors with which to divide the determined characteristic resistance are given in table 8.15. Table 8-15: Partial factors on total compressive resistance Pile type ?R Driven piles 1.3 Bored piles 1.5 CFA piles 1.4 It is evident from the foregoing discussion that two sets of partial factors are applied to calculated or measured resistance. One is from table 8.13 or table 8.14 depending on the method of pile capacity determination and the other is obtained from table 8.15. Therefore for a given pile type the total partial factor applied is given by: Stellenbosch University http://scholar.sun.ac.za 258 tR ??? = [8.19] Where; R? = total partial factor applied, ? = appropriate correlation factor ( 41 ?? to ), t? = appropriate partial factor on the total resistance from table 8.16. Using the above approach, the total applied partial factor for the full scale pile tests and semi-empirical design methods are presented in tables 8.16 and 8.17 respectively. Table 8-16: Total resistance factors for the full scale pile test design method Pile Type and ? Total partial resistance factor for test piles of n = 1 2 3 4 ?5 Driven ?1 1.82 1.69 1.56 1.43 1.30 ?2 1.82 1.56 1.37 1.30 1.30 Bored ?1 2.10 1.95 1.80 1.65 1.50 ?2 2.10 1.80 1.58 1.50 1.50 CFA ?1 1.96 1.82 1.68 1.54 1.40 ?2 1.96 1.68 1.47 1.40 1.40 Table 8-17: Total resistance factors for the semi-empirical design method Pile type and ? Total resistance factor (?R ) for profiles of tests of n = 1 2 3 4 5 7 10 Driven ?3 1.82 1.76 1.73 1.70 1.68 1.65 1.63 ?4 1.82 1.65 1.60 1.56 1.50 1.46 1.40 Bored ?3 2.10 2.03 2.00 1.97 1.94 1.91 1.88 ?4 2.10 1.91 1.85 1.80 1.73 1.68 1.62 CFA ?3 1.96 1.89 1.86 1.83 1.81 1.78 1.75 ?4 1.96 1.78 1.72 1.68 1.61 1.57 1.51 The partial factors presented in tables 8.16 and 8.17 are for the reference class of structures (?T = 3.8). Since the partial factors were determined by judgement and fitting to the existing practice the target reliability index has no influence on values of the partial factors. Direct Stellenbosch University http://scholar.sun.ac.za 259 comparison of the partial factors in tables 8.16 and 8.17 with the values obtained in this study is not possible as these values are based on pile installation method only irrespective of the soil type. However the numerical values of the partial factors presented in table 8.16 and 8.17 are comparable to values of partial factors derived in this study for the reference class of structure with ?T = 2.5 (table 8.6). Further more, Eurocode 7 state that if the calculated ultimate capacity is obtained by an analytical approach (i.e. static analysis using engineering properties of the soil as determined from laboratory or in-situ field testing), then the partial factors presented in table 8.13 may need to be corrected by a model factor larger than 1 (clause 7.6.2.3(11)). However no tentative values of the model factors have been provided. Orr (2005) recommended a model factor of 1.4 when designing a pile from ground test results. Assuming that 1.4 is the appropriate model factor for European pile design practice and soil conditions, then the partial factors in table 8.15 should be multiplied by 1.4. The resulting partial factors are presented in table 8.18. Table 8-18: Resistance factors after inclusion of a model factor of 1.4. Pile type ?R Driven piles 1.82 Bored piles 2.1 CFA piles 1.96 Although direct comparison is not possible for the reason given earlier, in general the values of the partial factors presented in table 8.18 are again comparable to the partial factors derived in this study. 8.5 Comparison with other published resistance factor values The introduction of the AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications in North America necessitated the consideration of reliability methods to the geotechnical aspect of the specifications (FHWA-NHI-05-052). Major reliability calibration studies associated with this project are presented in NCHRP Report 343 by Backer et al. (1991) NCHRP Report 507 by Paikowsky et al. (2004). Another notable resistance factor calibration study is that contacted by Kim et al. (2002). All these major resistance factor calibration studies relate to pile foundation for bridges and therefore Stellenbosch University http://scholar.sun.ac.za 260 the derived partial factors can not be compared with that for building structures due to different load factors. In general the partial factors from these studies were higher than those derived in this study. With regard to building structures the calibration work carried out by Becker (1996b) for Development of Limit State Design for Foundations in the National Building Code of Canada is notable. Resistance factors for the ultimate limit state of shallow and deep foundations were derived from calibration with working stress design and from reliability based calibration. For pile foundations, the proposed partial factors ranged from 3.3 to 1.67 depending on the method of pile design and site investigation. For the semi-empirical and full scale pile load test design methods, the proposed factors were 2.5 and 1.67 respectively. These values were for achieving a reliability index of 3.4 and 3.2 respectively. Although these partial factors are for higher reliability indexes, they are not significantly different from the partial factors developed in the current study for a target beta of 3.0 (Table 8.7). On the basis of comparison of partial factors derived in this study with those given in Eurocode 7 Annex A and those recommended by Becker (1996b), it can be concluded that the reliability calibration results are within the range of values used in other countries. 8.6 Recommended Resistance factors In this study the selection of the final resistance factors is made simple by the fact that the results of the principal calibration methods (i.e. AFOSM method) are comparable to results obtained from other reliability methods (MVFOSM, approximate MVFOSM and the design point approaches) and from calibration with working stress design approach. Further more, the resistance factors were within the range of values given Eurocode 7, which is the reference code for the future geotechnical code for South Africa. Under the circumstances, there is no compelling reason to adjust the resistance factors obtained by reliability calibration. Since the AFOSM method is more accurate than the FOSM method, the results from the of the AFOSM calibration are taken as the recommended resistance factors Table 8.12 column 3. If differentiation in terms of materials type is taken into consideration as previously discussed, the recommended resistance factors corresponding to target beta of 2.0 ? 3.0 are as given in table 8.19. Stellenbosch University http://scholar.sun.ac.za 261 Table 8-19: Recommended resistance factors based on materials type Material type ?=2.0 ?=2.5 ?=3.0 Non-cohesive 1.62 1.87 2.17 Cohesive 1.41 1.62 1.85 Stellenbosch University http://scholar.sun.ac.za 262 Chapter 9 SUMMARY AND CONCLUSIONS 9.1 SUMMARY The importance of uncertainty quantification is increasingly recognised in geotechnical engineering as reliability-based methods are assuming a prominent role in the calibration of new generation design codes for geotechnical engineering practice. Accordingly this study has quantified model uncertainty in the classic static pile design formula under the Southern African geological environment. Model uncertainty was represented in terms of model factor statistics. The generated model factor statistics were used to calibrate partial resistance factors in a reliability-based design framework. The subsequent paragraphs summarises the key results obtained. 9.1.1 Geotechnical limit state design Chapter 2 introduced the general limit state design philosophy. It was established that: ? Although the original concept of limit state design does not specify a particular way in which non-violation of the relevant limit states are ensured, verification by partial factors has become the general design approach. ? Limit state design in the partial factor format can be implemented in both a deterministic and probabilistic framework. The fundamental difference is in the derivation of the associated partial factors. With the deterministic limit state design, partial factors are determined by engineering judgement and by fitting to the existing design practice while for the probabilistic limit state partial factors are based on reliability calibration. ? Limit state design in a non-probabilistic framework suffers from the same draw-backs as the working stress design it replaces. Consequently the geotechnical profession could not see the benefits of converting to limit state design, hence the severe criticism of the approach. ? Despite the initial reluctance to the use of limit state in the partial factors format, the geotechnical profession has now embraced this design approach. Three main factors has motivated this change in hearts: (i) the need to achieve compatibility between structural Stellenbosch University http://scholar.sun.ac.za 263 and geotechnical design, (ii) the application of statistics, probability and reliability theory to analysis and design which led to a transparent and rational treatment of uncertainties as well as the derivation of partial factors, (iii) limit state design awareness campaign by the Technical Committee (TC 23) of the International Society of Soil Mechanics and Geotechnical Engineering on Limit State Design in Geotechnical Engineering. ? The limitations of the deterministic limit state design serves as a motivation to adopt the more rational and transparent probabilistic limit state design in geotechnical engineering. ? Globalisation and economic co-operation requires harmonisation of technical rules for design of buildings and civil engineering works. The harmonisation of design rules for all structures dictates that geotechnical design being based on the same limit state design as for structural design involving other materials. ? The need for common design rules has led to establishment of a basis of design in the Eurocodes (EN 1990, 2002) and the revised South African Loading Code (SANS 10160- Draft). Basis of design provides a common basis and general principle for the design of building and civil engineering works within the limit state design framework. ? The common rules relate to performance requirements, specification of the limit states, design situations to be checked, reliability requirements, and treatment of basic variables (actions, materials properties, and geometric data). ? Within the Eurocodes framework, the design of elements involving geotechnical actions (e.g. foundations and retaining walls) can be carried out by any of the three alternative Design Approaches. Furthermore, the selection of a particular design approach is a matter for national determination. ? Through comparison of worked design examples, Design Approach 3 produces the most conservative designs, Design Approach 2 the least conservatism designs, and Design Approach 1 generally yields designs between the other two approaches. Accordingly Design Approach 1 was selected for this study. 9.1.2 Reliability background for the geotechnical limit state design Having established the need for the probabilistic limit state in Chapter 2, Chapter 3 presented the fundamental reliability background required for reliability-based geotechnical design. This entails an overview of reliability theory and reliability calibration principles. Highlights of this chapter include: Stellenbosch University http://scholar.sun.ac.za 264 ? In a probabilistic setting, the reliability of an engineered system is the probability of its satisfactory performance for a specific period under specific service conditions (i.e. the probability of survival). ? The basic measure of reliability is the probability of failure. In evaluating the probability of failure, the behaviour of the system is described by a set of basic variables (i.e. actions, material properties, geometric data and model uncertainty). ? Theoretically the probability of failure can be determined from the integration of the joint probability density function of n basic variables over the failure region. However, to directly evaluate the n-fold integral using standard methods of integration is a formidable task and closed form solutions do not exist except for very simple cases. Furthermore, the joint probability density function of random variables is practically not possible to obtain, and the PDF of the individual random variables may not always be available in explicit form. Therefore for practical purposes, analytical approximation of the integral is employed to simplify the computations of the probability of failure. The most common methods of approximation are the First Order Second Moment reliability methods (FOSM). ? In the FOSM approximation, the probability of failure is expressed in terms of the reliability index (?). In the computation of the reliability index, all that is required are the statistics of the random variables (mean and coefficient of variation). ? FOSM reliability methods can be further classified into the Mean Value First-Order Second Moment method (MVFOSM) in which the limit state function linearised at the mean values of the random variables and the Advanced First-Order Second Moment method (AFOSM) in which limit state function linearised at the design point. ? There are two contrasting interpretations of the calculated probability of failure (i.e. frequentist and degree of belief). However for civil engineering designs the calculated probability should be interpreted as a degree of belief. ? Given the complexity of the computations for the failure probability or the reliability index, this approach, although rigorous, might not be suitable for routine designs, hence the need for simplified reliability methods. Such design methods are referred to as level 1 reliability design methods. ? With the level 1 design approach, the appropriate degree of reliability is provided through the use of partial factors. Such partial factors are derived using level II reliability methods and are associated with the major sources of uncertainties in the basics variables. The process of assigning partial factors to resistances or loads is generally termed calibration. Stellenbosch University http://scholar.sun.ac.za 265 ? Calibration of partial factors can be carried out by engineering judgement and experience, fitting to existing design practice, and by reliability theory. ? The main draw-back of the calibration by engineering judgement and experience and fitting to existing design practice is that they result in non-uniform level of conservatism. Accordingly this study was based on reliability calibration. 9.1.3 Data collection, processing and evaluation of interpreted capacities In general, model uncertainty quantification entails comparing predicted performance with measured performance. The approach requires a substantial amount of local pile load tests results with the associated geotechnical data. Chapter 5 detailed the collection and processing of the pile load test data. Analysis of the database led to the following observations: ? The majority of the piles were made of concrete and only a few steel piles, indicating that the principal pile material in South Africa is concrete. ? The pile lengths range from 3 to 27 m for DNC, 6 to 16.5 for BNC, 3.5 to 29.3m for DC and 4.5 to 24 m for BC. Generally the extremely long piles (e.g. 27 and 29 m) were steel piles. ? The concrete pile diameters vary from 330 to 610 mm for DNC, 360 to 520 for BNC, 250 to 750 mm for DC and 300 to 910 mm for BC. Except for one case, all the steel piles comprises of H-piles of 305 x 305 mm in size. The exception is a steel tube pile of 560 mm diameter. ? The pile types include Franki (expanded base) piles, Auger piles, Continuous Flight Auger (CFA) piles and steel piles. For driven piles the Franki pile is the most popular while for bored piles the Auger and CFA are more prevalent. For each test pile (case 1 ? 174), there was some accompanying soil data. The following observations were drawn from study of the geotechnical data: ? The materials fall into one of the following four natural categories: transported soil, residual soil, pedogenic material, and rock. ? In cohesive materials the piles are generally founded in the residual materials horizon. Stellenbosch University http://scholar.sun.ac.za 266 ? The soil test results comprises mainly of Standard Penetration Tests (SPT) measurements and a few Cone Penetration Test (CPT) results, implying that the SPT is the most popular field soil test in Southern Africa. ? In general, for a given case the SPT N-value for the base materials is higher than that for the shaft materials. This trend is marked in cohesive materials, indicating that in residual materials, the pile passes through soil strata of increasing consistency and is founded on a rock consistency stratum. The load tests data were then used to evaluate the ultimate pile capacities. This was accomplished by plotting the load versus the head deflection to produce load-deflection curves. The majority of the test piles in the database are working piles tested to a maximum load of one and half times the design load. This limits the movement to which the pile head is subjected and requires an extension to the load-settlement curve to determine the ultimate capacity. The Chin/Davisson procedure was adopted for the current study to determine ultimate capacity of proof tests results. 9.1.4 Evaluation of engineering soil properties and design parameters The theoretical computations of the bearing capacity of geotechnical materials require the engineering properties of the materials as input in the calculation model. Accordingly in Chapter 5, the specific site measurements presented in Chapter 4 were transformed to the desired engineering soil properties. The penetration test results included Standard Penetration Test (SPT) blow-counts, Dynamic Probe Super Heavy (DPSH) results and Cone Penetration Test (CPT) results. For derivation of soil properties, the DPSH and CPT measurements were converted to equivalent SPT N-values. The SPT N-values were then used to estimate the desired geotechnical properties. Many correlations relating SPT blow count and soil properties have been developed. Some of the correlations are based on corrected N-values (N1 (60)) while others are based on uncorrected values (N). It was observed that currently there is no consensus on the use of corrected versus uncorrected SPT blow count. In this study correlations based on uncorrected N-values were used. The reasons for this selection are as follows: ? Uncorrected values are used in practical designs worldwide; ? Most reliable correlations are based on uncorrected blow count; Stellenbosch University http://scholar.sun.ac.za 267 ? The required correction factors are too many; ? The models for deriving the correction factors are not perfect and therefore introduce additional uncertainties to the interpretation of SPT results. For example, several formulae and charts have been published for overburden correction with completely different results; ? State of the art equipment are in use these days and therefore most equipment-related effects are eliminated; ? SPT is carried out by specialist drilling companies with sufficient experience and therefore they are conversant with the test. Hence operator/procedural effects are minimal. ? The reference overburden stress of 100 kPa is just a hypothetical value and therefore does not reflect the operational stress around a pile. The increase in SPT N value with depth is a reflection of the increase in strength and stiffness caused by in-situ overburden stress and therefore correction for overburden stress may not be necessary. ? The correction for energy loss makes sense if the exact loss has been measured on site for each case considered. But usually a generic correction factor is applied. Surely in some sites, advanced SPT equipment might have been used with no energy loss at all while in some sites older equipment might have been used with higher energy losses. Since the exact SPT equipment used to measure the N-values as well as their site measured energy losses are not known, it is not worth while to apply the correction factor. ? Results of preliminary investigations on the effects of correcting the N-values on various pile design parameters showed that: - In comparing the design parameters derived on the basis of corrected and uncorrected N-values, no significant improvement in terms of reduced variability was gained from correcting the N-values. - The data set for uncorrected N-values provides the best fit to the measured capacities (i.e. a mean value of closer to 1 and a smaller variability). On the basis of the foregoing, it was concluded that for reliability-based design of pile foundations, correction of N-values for overburden and hammer efficiency do not add any value. Stellenbosch University http://scholar.sun.ac.za 268 The main parameters derived from the SPT measurements were the internal angle of friction (?) for non-cohesive materials and undrained cohesion for cohesive materials. Analysis of the ? values led to the following conclusion: ? In general, for a given case ?-base values are higher than ?-shaft. This is in accordance with expectations as piles are normally founded on denser materials. ? On average, the ? values for materials in which bored piles were constructed are relatively higher than for materials in which driven piles were installed. This is attributed to the fact that under normal circumstances, bored piles are preferred in very dense materials (higher ? values) while driven piles are preferable when the granular material is not very dense. ? The magnitudes of the ? values are within the expected range of ? values for non- cohesive materials. However, some ?-shaft values are on the lower side, indicating that the pile shaft passes through some silty soils. Analysis of the undrained shear strength values showed that: ? Values for Cu-base are much higher than those for Cu-shaft. This is consistent with the variation of the SPT measurements with depth and is attributed to the fact that the degree of weathering in a typical residual soil tends to decrease with depth. The weathered materials gradually merge into the unweathered rock. Accordingly the consistence varies from soft for the transported soil overlying the residual soil to very stiff for weathered rock. ? The Cu values for materials in which bored piles were constructed are generally higher than those for driven piles. These higher Cu values which are associated with higher consistencies have necessitated the construction of bored piles as it is difficult to drive piles in such materials. In fact in South Africa, the most common type of pile in residual soils is the bored cast in situ pile. ? The magnitudes of the Cu values are in accordance with what is expected of a typical profile in a residual soil. The lower values (Cu <100 kPa) denote the predominance of transported materials while middle values (i.e. Cu values of up to 500 kPa) indicate the prevalence of the residual soil. There are also a number of cases depicting rock consistency (i.e. Cu of 1000 kPa or greater). The higher values are mostly associated with Cu-base, thus confirming that piles in residual soils are usually founded on a rock consistency stratum. Stellenbosch University http://scholar.sun.ac.za 269 In the computation of pile capacity using the static formula, design parameters other than the geotechnical properties are required. These include Nq, Ks, Nc, ?, and ?. These parameters are estimated from empirical correlations with soil properties. It was observed that the process of determining these parameters is not standardised. Consequently different engineers follow different procedures thereby producing different design parameters for even the same site. To select the appropriate design parameters for the South African geological setup, two approaches were followed. The first approach entails using values commonly assumed in practice while the other approach entails conducting a parametric study of the possible values for a given parameter. The second approach is tantamount to assessment of design parameters by back analysis of load test results. The results were such that for a given pile class, r2 values for design parameters selected on the basis of best fit principle are slightly higher than those for commonly assumed parameters in practice. Therefore, the design parameters selected on the basis of best fit principle were considered as the most appropriate for the South African conditions. The selected geotechnical design parameters were used as input into the static formula to compute the predicted capacity for each case. Examination of the calculated predicted capacities led to the following conclusions: ? Qp ranges from 1100 to 5150 kN for driven piles in non-cohesive materials, 470 to 6750 kN for bored piles in non-cohesive materials, 560 to 5910 kN for driven piles in cohesive materials and 460 to 10170 kN for bored piles in cohesive materials. ? It is apparent from the above ranges of Qp values that for a given soil type, the upper bound values for bored piles are higher than their respective values for driven piles. This is in line with the earlier observation that bored piles are normally installed in very stiff materials, hence the higher resistances. 9.1.5 Model factor statistics For reliability calibration, the statistics (mean, standard deviation and coefficient of variation) as well as the type of distribution that best fit the data need to be determined for each random variable considered in the limit state function. Using the results obtained from Chapters 4 and 5, Chapter 6 detailed the computations of the model factors as well as the statistical analysis of the model factor realisations. It was noted that in principle, the bias factor accounts for all the sources of uncertainties (i.e. model error, systematic error, Stellenbosch University http://scholar.sun.ac.za 270 inherent spatial variability, statistical error and load tests related errors). However due to the high level of expertise and experience of companies in charge of SPT tests and the pile load tests, measurement errors are minimized. Also inherent spatial variability is minimized due to averaging effects along the pile length. Therefore the calculated model factor mainly represents model uncertainty. The statistical analysis comprised of graphical representation by histograms, outliers detection and correction of erroneous values, using the corrected data to compute the sample moments (mean, standard deviations, skewness and kurtosis), determining the appropriate distribution for the model factor, and investigation of correlation with underlying pile design parameters. The key results of the statistical analysis were as follows: ? Based on the mean bias factors it was observed that the static formula with geotechnical parameters recommended in this study, yields slightly conservative theoretical capacities for DNC, DC, and BC. Conversely for BNC the approach is marginally unconservative. ? Although the standard deviations of the four pile classes were not significantly different, there was a distinct trend that driven piles depicted higher variability compared to bored piles irrespective of materials type. ? For a given pile installation method (driven or bored) the variability in non-cohesive materials is higher than that in cohesive materials. ? The lognormal probability distribution was found to be the appropriate distribution for the model factor. Concerning correlation with pile design parameters, the significance of the correlation was interpreted in terms of practical and statistical significance. On the basis of statistical significance (p-value): ? None of the pile design parameters was significantly correlated with the model factor. With regard to predicted capacity, it was found that the model factor is correlated to Qp for driven piles but not for bored piles, suggesting that the static formula is more accurate in bored piles than driven piles. Stellenbosch University http://scholar.sun.ac.za 271 When the interpretation of the numerical values of correlation coefficients was based on practical considerations (i.e. r ? 0.2 considered significant), then the correlation was found to be significant as follows: model factor was found to be correlated with: ? In cohesive materials, the correlation was significant for shaft diameter, base diameter, and predicted capacity in driven piles. ? In non-cohesive materials, the correlation was significant for base diameter, ?-shaft, ?- base, and predicted capacity in driven piles. It was noted that correlation coefficients of ? 0.2 occurred only in driven piles and not in bored piles. It was observed that the numerical values of the correlation coefficients for the ranged from 0.2 to 0.4 thereby implying a low degree of correlation. With more weight given to the practical significance of the value of the correlation coefficient rather than its statistical significance, it was concluded that the conventional model factor exhibits some statistical dependencies with some underlying factors. 9.1.6 Treatment of correlation To apply the model factor as an independent variable in reliability analysis, these statistical dependencies need to be removed. In Chapter 7, the statistical dependencies between the model factor and the predicted capacity were either removed or taken into account. Removal of statistical dependencies was considered under the generalised model factor approach while accounting for the correlation was considered under conditioned model factor approach. The fact the generalised M was not dimensionless necessitated the normalisation of the interpreted and predicted capacities. Accordingly three normalization schemes were experimented with. The normalisation schemes were evaluated on the basis of the resulting fit between Qi and Qp. In this regard, the coefficient of determination (r2) was used as the measure of fit. The results were as follows: ? For a given pile class, the r2 values and values of the standard error of estimate were quite close for the different normalisation schemes. It was therefore concluded that, in general the three normalisation schemes yield comparable regression results. ? Basing the selection of the best fit on r2 and the regression constant denoting the slope of the best fit line (?b?) it was found that scheme 1 produced the best overall results. Accordingly the generalised M statistics for scheme 1 were selected for further analysis. Stellenbosch University http://scholar.sun.ac.za 272 ? A comparison of the generalised M statistics to the standard M statistics showed reasonable agreement. The standard deviations and coefficients of variation were comparable. Furthermore, the two sets of model factors showed similar trends. ? Comparison of the conditioned and standard model factor statistics showed very little difference. ? On the basis of the model factor statistic it was concluded that there was no noticeable reduction in variability gained from removing and accounting for correlation. This was attributed to the weak correlation. The objective of accounting for the correlation was to minimize its effects on the calculated reliability index. Accordingly the extent of improvement to the calculated beta values imparted by the removal or incorporation of the correlation was investigated. Since for the conditioned M approach, the model factors statistics and the performance function were similar to that of the standard M, it was obvious that no significant improvement to the calculated reliability index could be achieved. Therefore the beta values were determined on the basis of standard and generalised M. The results were as follows: For the standard M: ? Beta values were sensitive to the ratio of variable to permanent load (Ln/Dn) and not the variation in the nominal resistance, implying that the calibration points are only defined by the Ln/Dn ratio. ? The beta values varied from one pile class to the other indicating that the current global factor of safety approach does not produce a consistent level of reliability across different design situations. Further more even within the same pile class beta values varied with individual cases represented by the Ln/Dn ratios. This suggested that even within the same design situation the global factor approach fails to attain a consistent level of reliability. ? Piles in cohesive materials depicted higher reliability compared to piles in non-cohesive materials. The scenario was attributed to higher inherent conservatism of the static formula in non-cohesive materials compared to non-cohesive materials. ? For a given material type, the ? values for bored piles were higher than that for driven piles. This was attributed to the observation noted in Chapter 6 that for a given material type the variability of the model factor in driven piles were higher than that in bored piles Stellenbosch University http://scholar.sun.ac.za 273 For generalised M: ? Beta values based on the generalised M statistics varied with Ln/Dn ratio as well as nominal resistance values (Rn). This necessitated the determination of beta values for representative Rn values. ? The variation in beta values with Rn value depended on the magnitude of the exponent b. When b was greater than 1, beta increased with the increase in Rn, which was the case for DNC and BNC. The b values for these two cases were 1.031 for DNC and 1.030 for BNC. Conversely when b was less than 1 the beta values decreased with the increase in Rn, which was the case for DC and BC (b=0.977 for DC and 0.961 for BC). ? The variation of ?-values with Ln/Dn ratio followeed a similar trend to the one shown by values based on standard M statistics. For both approaches, a critical value (i.e. the smallest value within a typical range of Ln/Dn ratio) was determined for each pile class. The following conclusions were drawn for examination of the critical beta values: ? For a safety factor of 2.5 both approaches produced values of close to the specified target beta of 3. Therefore all the three approaches yield beta values that met the target reliability index specified by SANS 10160-Draft. ? A comparison of the critical beta values showed that values for the standard M were significantly higher than that for generalised M in cohesive materials. However, in non- cohesive materials, the difference between the results of the two approaches was negligible. Therefore the generalised M approach did not produce improved reliability indexes. It was concluded that due to the weak correlation between M and Qp, removal or accounting for the trend between the two parameters does not lead to improved reliability. It was therefore concluded that the calibration of the partial factors be based on the standard M. 9.1.7 Calibration of resistance factors Using the model factor statistics developed in Chapter 6 and the calibration methods discussed in Chapter 3, Chapter 8 developed the required resistance factors for the ultimate limit state. Five calibration methods were employed (i) Advanced first-order second moment approach (AFOSM), (ii) Mean value first-order second moment approach Stellenbosch University http://scholar.sun.ac.za 274 (MVFOSM), (iii) Approximate MVFOSM, (iv) Design point approach and (v) Fitting to working stress design. The following conclusions were drawn from the calibration study: ? Based on considerations of reliability indexes implied by the current design approach, target beta values set in the current and draft South African loading codes, and redundancy inherent in pile groups, target beta values of 2.0 for RC 1, 2.5 for RC 2 and 3.0 for RC 3 were recommended for pile foundations. ? In general the resistance factors decreased with the increase of the Ln/Dn ratio, indicating variation in reliability indexes across the calibration points or design situations within a given class of pile foundations. This further implied that the application of a single resistance factor to all the design situations will inevitably lead to some deviation from the target reliability index for some of the calibration points. ? To achieve consistent reliability within a range of calibration points for a given pile class, an optimum partial factor which best approximate the uniform target reliability is needed. In principle this can be obtained by minimizing the deviation from the target beta using an objective or penalty function penalising the deviation from the target reliability index. ? Investigation of the optimisation schemes showed that Lind?s function gives results that are close to the least square function while still penalising under- designs more than over-designs, hence providing conservative results. Since it is better to err on the conservative side, Lind?s objective function was used in the selection of the optimum partial factor for all the pile classes. ? Further analysis revealed that optimal partial resistance factors obtained by minimization of Lind?s objective function were very close to partial factors corresponding to the calibration point represented by Ln/Dn of 0.5 (i.e. design situation with the highest partial factor). Therefore instead of performing the optimisation process, the partial factors corresponding to Ln/Dn ratio of 0.5 were taken as the optimal partial resistance factors. ? The resistance factors appeared to be influenced by the materials types. In this regard, values of partial factors for a given calibration method and material type were quite close irrespective of the pile installation method. ? On average the resistance factors from the MVFOSM approach were 23% higher than those from the AFOSM method. ? The resistance factors for the approximation MVFOSM method were 24% less than those for the full MVFOSM method. Stellenbosch University http://scholar.sun.ac.za 275 ? Resistance factors from the AFOSM and design point methods were quite close. For a beta value of 2.5, the resistance factors from the two approaches differed by 2% (DNC), 6% (BNC), 10% (DC) and 9% (BC). ? A comparison of resistance factors obtained through calibration by fitting with those obtained by reliability calibration showed that: - The optimal resistance factor corresponding to FS=2.0 fell within the resistance factors for ?T = 2.0. - The resistance factor for FS=2.5 was comparable with the values for ?T =2.5. - The value for FS=3.0 fits the results for ?T =3.0. ? Numerical values of the partial factors derived in this study were within those recommended in EN 1997-1 and the Canadian building code. 9.2 CONCLUSIONS On the basis on results obtained from various Chapters, the following overall conclusions can be drawn from the study: ? The variability exhibited by the model factors is an indication that there are significant model uncertainties in the classical static pile design method. To explicitly account for the variability and bias reflected in this study, a design approach based reliability considerations seems to be the most reasonable alternative for pile foundation design in South Africa. ? The similarity of the model factor statistics for the static formula with those reported for other static pile analysis methods (e.g. Meyerhof method, CPT methods, Nordlund methods, Vesic method) is an indication that static analysis using engineering properties gives equally good results. Therefore, with the application of a resistance factor to cater for the model uncertainty, the method should lead to safe designs and perhaps lessen the need for the customary practice of verification through proof pile tests. ? The statistically insignificant correlation between the model factor and various pile design parameters is an indication that the variation in the model factor is not explainable by deterministic variations in the database. Therefore, the model factor is a random variable warranting a probabilistic description. In this regard the lognormal distribution was found to be the most appropriate theoretical model for the model factor Stellenbosch University http://scholar.sun.ac.za 276 ? Although correlation between variables generally affects the calculated reliability index, weak correlation (r ? 0.4) do not significantly affect the results and could be ignored. ? Values of reliability indexes implied by existing practice meet the minimum beta value of 3.0 specified in the current South African loading code. However, due to redundancy in pile groups, lower target beta values are recommended as acceptable for pile foundations. ? Results of the principal calibration methods (i.e. AFOSM method) are comparable to results obtained from other reliability methods (MVFOSM, approximate MVFOSM and the design point approaches). This implies that the approximate methods yield reasonable results, further suggesting that reliability calibration can as well be based on the simple approximation procedures. ? The resistance factors were within the range of values given in Eurocode 7, which is the reference code for the future geotechnical code for South Africa. This provides further confidence in the use of the partial factors developed in this study. ? The general trend that for a given material type, the resistance factors were quite close irrespective of the pile installation method suggests differentiation of partial factors in terms of materials types only. ? Results of calibration by fitting give resistance factors that need to be used in the limit state design equation to obtain a factor of safety equal to that of the working stress design. From this premise, calibrating limit state design to existing practice results in the same minimum permissible foundation sizes as those obtained by the working stress design. Therefore the use of limit state on non-reliability based procedures does little to improve design. 9.3 RECOMMENDATIONS FOR FURTHER RESEARCH ? Owing to the nature of the available data, resistance factors were calibrated for the total capacity only. However a derivation of separate resistance factors for total, shaft and base capacity provides more insight. Accordingly a separate calibration is considered valuable. ? The load factors of 1.0 for permanent actions and 1.3 for variable actions recommended in SANS 10160-Draft and EN 1997-1 were adopted in this study. However these load Stellenbosch University http://scholar.sun.ac.za 277 factors have not been developed on the basis of reliability calibration as is the case for load factors for structural design. Therefore it will be of interest to calibrate load factors for geotechnical design. ? Both the resistance and the load were taken as lognormal variables in this study. However, studies have shown that variable loads follow an Extreme value type 1 distribution. Since the reliability index is sensitive to the distribution assumed, it will be interesting to investigate the sensitivity of the resulting reliability to the distribution applied to the variable action, considering a lognormal distribution as was done here and an Extreme value type 1 as is generally applied. ? Although the correlation with various pile design parameters was generally weak, it appeared that the correlation in driven piles was higher than in bored piles. This suggests that the static formula applied to driven piles does not adequately account for the various pile design parameters. Therefore further study directed in the refinement of the static formula in driven piles should be a fruitful investigation. ? Refinement of the model factor statistics and the ensuing partial resistance factors can be achieved through an increment of the database developed in this study. Therefore there is need to expand the current database with more effort directed to instrumented piles and piles tested to failure. ? In order to introduce probabilistic limit state design to geotechnical design as a whole in South Africa, the procedures developed in this study need to be extended to other geotechnical applications such as shallow foundations, retaining walls and slope stability. ? For the resistance, the design point is typically located in the lower tail and therefore it is important that the lognormal parameters selected produce the best fit possible in the region of the tail. Accordignly a worthwhile further research is to determine the best fit statistical parameters of the assumed distribution and use such statistical parameters in the calibration exrcise. Stellenbosch University http://scholar.sun.ac.za 278 REFERENCES Ang, A. H-S. and Tang, W.H. (1975). Probability Concepts in Engineering Planning and Design, Volume 1: Basic Priciples. Wiley, New York Ang, A. H-S. and Tang, W. H. (1984). Probability Concepts in Engineering Planning and Design, Volume II. Decision, Risk, and Reliability, John Wiley & Sons, New York. Ayyub, B. M. and Popescu, C. (1998). Web-based structural reliability assessment (WSTAR). Technical Report for Contracts DACA31-96-D-0063, Army Corps of Engineers, Washington DC. Ayyub, B. M., and Assakkaf, I. A. (2000). 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Stellenbosch University http://scholar.sun.ac.za 293 List of Figures Figure 2.1: Various methods for determining the characteristic value------------------- 42 Figure 2.2: Difference between characteristic values as a 5% fractile and mean estimate at 95% confidence level ---------------------------------------------------------------- 47 Figure 3.1: Rock-bolting alternatives ----------------------------------------------------------- 59 Figure 3.2: Domain of integration and probability density function -------------------- 62 Figure 3.3: Geometrical definition of reliability index--------------------------------------- 65 Figure 3.4: Performance function in x-space and u-space ---------------------------------- 69 Figure 3.5: Overview of calibration methods in Eurocodes -------------------------------- 82 Figure 3.6: Empirical rates of failure of civil engineering facilities ----------------------- 85 Figure 3.7: Relationship between design value, characteristic value and mean value- 89 Figure 4.1: Possible combinations of materials in a typical profile----------------------- 100 Figure 4.2: Typical weathering profile of residual soils ------------------------------------ 101 Figure 4.3: Hyperbolic representation of stress-strain relationship --------------------- 111 Figure 4.4: Transformed linear plot------------------------------------------------------------ 112 Figure 4.5: Transformed plots ------------------------------------------------------------------ 114 Figure 4.6: Extrapolated curve ------------------------------------------------------------------ 114 Figure 4.7: Davisson?s failure criterion -------------------------------------------------------- 116 Figure 4.8: Transformed plot for case 63------------------------------------------------------ 121 Figure 4.9: Extrapolated curve for case 63---------------------------------------------------- 122 Figure 4.10: Full test data curve for case 63 -------------------------------------------------- 122 Figure 4.11: Extrapolated curve superimposed on full test data curve for case 63 --- 123 Figure 4.12: Transformed plot for pile case 56----------------------------------------------- 125 Figure 4.13: Extrapolated curve for case 56 -------------------------------------------------- 125 Figure 4.14: Full test data curve for case 56 -------------------------------------------------- 126 Figure 4.15: Extrapolated curve superimposed on full test data curve for case 56 --- 126 Figure 5.1: SPT test procedure ----------------------------------------------------------------- 134 Figure 5.2: SPT hammer drop systems -------------------------------------------------------- 134 Figure 5.3: N ? ? correlation by different Authors------------------------------------------ 141 Figure 5.4: N ? ? correlation -------------------------------------------------------------------- 142 Figure 5.5: Bearing capacity factors for piles in cohesionless soils ----------------------- 149 Figure 5.6: ? ? N correlation by Berezantzev et.al ------------------------------------------ 150 Figure 5.7: Scatter plot of Qi Vs Qp for Ks = 1.3---------------------------------------------- 154 Figure 5.8: Scatter plot of Qi Vs Qp for Ks = 0.8 --------------------------------------------- 155 Stellenbosch University http://scholar.sun.ac.za 294 Figure 5.9: Scatter plot of Qi Vs Qp for Nc = 12 ---------------------------------------------- 157 Figure 5.10: Scatter plot of Qi Vs Qp for BC-------------------------------------------------- 158 Figure 6.1: Histograms of the model factors-------------------------------------------------- 166 Figure 6.2: Z-scores vs model factor plots----------------------------------------------------- 170 Figure 6.3: Box plots of model factors --------------------------------------------------------- 173 Figure 6.4: Scatter plots of Qi vs Qp ------------------------------------------------------------ 176 Figure 6.5: P-P plots for model factors -------------------------------------------------------- 186 Figure 6.6: Scatter plots of M vs pile length -------------------------------------------------- 192 Figure 6.7: Scatter plots of M vs shaft diameter --------------------------------------------- 194 Figure 6.8: Scatter plots of M vs base diameter---------------------------------------------- 196 Figure 6.9: Scatter plots of M vs ?-shaft ------------------------------------------------------ 197 Figure 6.10: Scatter plots of M vs ?-base------------------------------------------------------ 198 Figure 6.11: Scatter plots of M vs Cu-shaft---------------------------------------------------- 200 Figure 6.12: Scatter plots of M vs Cu-base ---------------------------------------------------- 200 Figure 6.13: Scatter plots of M vs Qp----------------------------------------------------------- 203 Figure 7.1: Normalised regression results for DNC ----------------------------------------- 209 Figure 7.2: Normalised regression results for BNC ---------------------------------------- 210 Figure 7.3: Normalised regression results for DC ------------------------------------------- 211 Figure 7.4: Normalised regression results for BC ------------------------------------------- 212 Figure 7.5: Scatter plots of ? Vs LN(Qp/AP) -------------------------------------------------- 216 Figure 7.6: Regression of M on Qp-------------------------------------------------------------- 218 Figure 7.7: Variation of beta values with Ln/Dn ratio for standard M approach----- 223 Figure 7.8: Variation of beta values with Ln/Dn ratio for generalised M approach--- 226 Figure 8.1: Variation of resistance factors with Ln/Dn ratio ------------------------------ 236 Figure 8.2: Optimal partial resistance factor for DNC ------------------------------------- 241 Figure 8.3: Optimal partial resistance factors for BNC, DC, and BC ------------------- 242 Figure 8.4: Variation of resistance factors with Ln/Dn ratio for ?T = 3.0 ---------------- 245 Figure 8.5: Variation of resistance factors with Ln/Dn ratio for ?T = 2.0 ---------------- 246 Figure 8.6: Resistance factors for MVFOSM method -------------------------------------- 249 Figure 8.7: Resistance factors for fitting to WSD approach ------------------------------- 254 Stellenbosch University http://scholar.sun.ac.za 295 List of Tables Table 2-1: Ranges of global factor of safety for geotechnical design ..............................13 Table 2-2: Definitions of characteristic values ..................................................................31 Table 2-3: Characteristic values of geotechnical actions .................................................36 Table 2-4: Load statistics ...................................................................................................37 Table 2-5: Values of action partial factors for Set A, B and C........................................39 Table 2-6: Numerical values of various sets of partial factors ........................................51 Table 2-7: Final results of a pile design example ..............................................................54 Table 2-8: Summary of model solutions ...........................................................................56 Table 3-1: Relationship between probability of failure and reliability index for ..........75 Table 3-2: Reliability classification as a function of consequences of failure ...............87 Table 4-1: Test pile characteristics and projects description for DNC ..........................96 Table 4-2: Test pile characteristics and projects description for BNC...........................97 Table 4-3: Test pile characteristics and projects description for DC .............................98 Table 4-4: Test pile characteristics and projects description for BC .............................99 Table 4-5: Geotechnical data for DNC ............................................................................102 Table 4-6: Geotechnical data for BNC.............................................................................103 Table 4-7: Geotechnical data for DC ...............................................................................104 Table 4-8: Geotechnical data for BC ...............................................................................105 Table 4-9: Interpreted capacities and hyperbolic parameters for DNC ......................117 Table 4-10: Interpreted capacities and hyperbolic parameters for BNC.....................117 Table 4-11: Interpreted capacities and hyperbolic parameters for DC .......................118 Table 4-12: Interpreted capacities and hyperbolic parameters for BC........................119 Table 4-13: Load deflection data for case 63...................................................................121 Table 4-14: Load-settlement data for pile case 56 ..........................................................124 Table 5-1: Factors affecting SPT N-vales ........................................................................135 Table 5-2: Correlation between SPT, consistency, and unit weight..............................138 Table 5-3: Unit weight (bored piles cases).......................................................................138 Table 5-4: Unit weight (driven pile)????????.. . ??????????.138 Table 5-5: Bias factor statistics.........................................................................................140 Table 5-6: ?-values (driven piles) .....................................................................................143 Table 5-7: ?-values (bored piles)? ??? ????? ???????????..143 Table 5-8: Correlation between rock consistency and undrained shear strength.......144 Stellenbosch University http://scholar.sun.ac.za 296 Table 5-9: Correlation between in-situ tests and pile design parameters for cohesive materials???????????????????...???.????????..145 Table 5-10: Undrained shear strength values .................................................................146 Table 5-11: Adhesion factors for bored piles in cohesive materials..............................151 Table 5-12: Variation of the regression parameters b and R2 with Ks ..........................154 Table 5-13: Variation of M statistics and resistance factors with Ks ...........................155 Table 5-14: Variation of fit parameters b and R2 with Nc..............................................156 Table 5-15: Comparison of measure of fit parameters ..................................................158 Table 5-16: Selected design parameters ..........................................................................159 Table 5-17: Predicted capacities.......................................................................................160 Table 6-1: Generated bias factors ....................................................................................162 Table 6-2: Summary statistics for the model factor .......................................................179 Table 6-3: Normality tests results ....................................................................................185 Table 6-4: Practical significance of correlation coefficient............................................189 Table 6-5: Correlation coefficients and associated p-values for M vs pile length .......192 Table 6-6: Correlation coefficients and associated p-values for M vs pile shaft diameter .....................................................................................................................................194 Table 6-7: Correlation coefficients and associated p-values for M vs pile base diameter .....................................................................................................................................196 Table 6-8: Correlation coefficients and associated p-values for M vs ? .......................198 Table 6-9: Correlation coefficients and associated p-values for M vs Cu .....................200 Table 6-10: Correlation coefficients and associated p-values for M vs Qp...................203 Table 7-1: Generalised model factor statistics for various normalisation schemes.....213 Table 7-2: Comparison of r2 and b from the three normalisation schemes..................215 Table 7-3: Selected generalised M statistics ....................................................................215 Table 7-4: Correlation coefficients and associated p-values for ? Vs LN(Qp/AP) .......217 Table 7-5: Conditioned model factor statistics ...............................................................220 Table 7-6: Range of Rn (Qp) values in the database........................................................225 Table 7-7: Critical beta values for Ln/Dn ratio of 0.5 to 2 ..............................................228 Table 8-1: Critical beta values..........................................................................................231 Table 8-2: Comparison of ?T in SANS 10160 the ?T adopted for the study .................232 Table 8-3: Model factor statistics used in the calibration of resistance factors ...........235 Table 8-4: Importance weighting for the respective calibration points.......................239 Table 8-5: S values for respective trial ? values ..............................................................240 Table 8-6: Optimal resistance factors for a target beta of 2.5 .......................................244 Stellenbosch University http://scholar.sun.ac.za 297 Table 8-7: Optimal resistance factors for a target beta of 3.0 .......................................245 Table 8-8: Optimal resistance factors for a target beta of 2.0 .......................................247 Table 8-9: Resistance factors for the MVFOSM approach ...........................................250 Table 8-10: Resistance factors for the approximate MVFOSM method......................251 Table 8-11: Resistance factors for the design point approach.......................................253 Table 8-12: Compilation of resistance factors from various approaches.....................255 Table 8-13: Correlation factors based on static load tests ............................................257 Table 8-14: Correlation factors based on semi-empirical methods .............................257 Table 8-15: Partial factors on total compressive resistance...........................................257 Table 8-16: Total resistance factors for the full scale pile test design method.............258 Table 8-17: Total resistance factors for the semi-empirical design method.................258 Table 8-18: Resistance factors after inclusion of a model factor of 1.4. ......................259 Table 8-19: Recommended resistance factors based on materials type........................261 Stellenbosch University http://scholar.sun.ac.za 298 Appendix A AVAILABLE STATISTICS ON GEOTECHNICAL PARAMETER UNCERTAINTIES Stellenbosch University http://scholar.sun.ac.za 299 A.1: COV of inherent variability for index soil properties (Phoon, 1995) Stellenbosch University http://scholar.sun.ac.za 300 A.2: COV for inherent variability for shear strength parameters (After Jones et al, 2002) Stellenbosch University http://scholar.sun.ac.za 301 A3: COV of inherent variability for consolidation and permeability parameters (After Jones et al, 2002) Stellenbosch University http://scholar.sun.ac.za 302 A 4: Measurement error for laboratory tests (After Phoon, 1995) Stellenbosch University http://scholar.sun.ac.za 303 A 5: COV of inherent variability of field measurements Stellenbosch University http://scholar.sun.ac.za 304 A 6: COV of measurement uncertainty of field tests Stellenbosch University http://scholar.sun.ac.za 305 A 7: Scales of fluctuation for common geotechnical properties Stellenbosch University http://scholar.sun.ac.za 306 Appendix B TYPICAL PILE TEST RECORDS, LOAD-DEFLECTION CURVE, AND ASSOCIATED GEOTECHNICAL DATA Stellenbosch University http://scholar.sun.ac.za 307 B1: Typical load-deflection record Stellenbosch University http://scholar.sun.ac.za 308 B 2: Typical load-deflection curve 0 100 200 300 400 500 600 700 800 0 1 2 3 4 Settlement (mm) Lo ad ( kN ) Stellenbosch University http://scholar.sun.ac.za 309 B 3: Typical soil data Stellenbosch University http://scholar.sun.ac.za 310 Appendix C RELIABILITY INDEX CALCULATION SPREADSHEETS AND THE ASSOCIATED EXCEL FUNCTIONS DESCRIPTION Stellenbosch University http://scholar.sun.ac.za 311 C1: Spreadsheet for computation of beta (standard M approach) C2: Excel functions for the various cells of the calibration spreadsheet Cell Excel function H5 : H7 H5 = LN(C5)-0.5*I5^2; H6=LN(C6)-0.5*I6^2; H7 = LN(C7)-0.5*I7^2 I5 : I7 I5 = SQRT(LN(1+D5^2); I6 = SQRT(LN(1+D6^2); I7 = SQRT(LN(1+D7^2) F11 : F13 F11 = A11*I5+H5; F12 =A12*I6+H6; F13 = A13*17+H7 J11: J13 J11 = EXP(F11); J12 = EXP(F12); J13 = EXP(F13) C15 J11*L5-J12*L6-J13*L7 G15 {=SQRT(MMULT(TRANSPOSE(A11:A13),A11:A13)))} G17 =NORMSDIST(-G15) L6 L5/L4/(1+N5) L7 L5/(L4-L6) Solver: Set target Cell: $G$15 equal to: Min By changing Cells: $A$11:$A$13 Subject to the constraints: $C$15<=0 Stellenbosch University http://scholar.sun.ac.za 312 C.3: Spreadsheet for computation of beta (generalised M approach) Chapter One C4: Excel functions for the various cells of the calibration spreadsheet Cell Excel function H5 : H7 H5 = LN(C5)-0.5*I5^2; H6=LN(C6)-0.5*I6^2; H7 = LN(C7)-0.5*I7^2 I5 : I7 I5 = SQRT(LN(1+D5^2); I6 = SQRT(LN(1+D6^2); I7 = SQRT(LN(1+D7^2) F11 : F13 F11 = A11*I5+H5; F12 =A12*I6+H6; F13 = A13*17+H7 J11: J13 J11 = EXP(F11); J12 = EXP(F12); J13 = EXP(F13) C15 J11*L5^F5-J12*L6-J13*L7 G15 {=SQRT(MMULT(TRANSPOSE(A11:A13),A11:A13)))} G17 =NORMSDIST(-G15) L6 L5/L4/(1+N5) L7 L5/(L4-L6) Solver: Set target Cell: $G$15 equal to: Min By changing Cells: $A$11:$A$13 Subject to the constraints: $C$15<=0 Stellenbosch University http://scholar.sun.ac.za 313 Appendix D CALIBRATION SPREADSHEETS Stellenbosch University http://scholar.sun.ac.za 314 D1: Calibration Spreadsheet D2: Excel functions for the various cells of the calibration spreadsheet Cell Excel function H5 : H7 H5 = LN(C5)-0.5*I5^2; H6=LN(C6)-0.5*I6^2; H7 = LN(C7)-0.5*I7^2 I5 : I7 I5 = SQRT(LN(1+D5^2); I6 = SQRT(LN(1+D6^2); I7 = SQRT(LN(1+D7^2) F11 : F13 F11 = A11*I5+H5; F12 =A12*I6+H6; F13 = A13*17+H7 J11: J13 J11 = EXP(F11); J12 = EXP(F12); J13 = EXP(F13) C15 J11*L5-J12*L6-J13*L7 G15 {=SQRT(MMULT(TRANSPOSE(A11:A13),A11:A13)))} G17 =NORMSDIST(-G15) L6 L5/L4/(1+1.3*N5) L7 L5/(L4*1.3)-L6/1.3 Solver: Set target Cell: $G$15 equal to: Min By changing Cells: $A$11:$A$13 Subject to the constraints: $C$15<=0 Stellenbosch University http://scholar.sun.ac.za