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Statistical models for the effect of length on the strength of lumber Williamson, Justin Andrew 1992

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STATISTICAL MODELS FOR THE EFFECT OF LENGTH ON THESTRENGTH OF LUMBERbyJUSTIN ANDREW WILLIAMSONB.E., Auckland University, New Zealand, 1979M.E, Auckland University, New Zealand, 1983A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the reuaired standardTHE UNIVERSITY OF BRITISH COLUMBIAJanuary 1992© Justin Andrew Williamson, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegrle at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Civil EngineeringThe University of British ColumbiaVancouver, CanadaDate 30th January, 1992DE-6 (2/88)AbstractThis thesis concerns the problem of modelling the effect of length on the strength of lumber. The length effectmanifests as a shift to lower values of the strength distribution from shorter to longer boards. This effect is alsonoticeable for other materials. It is usually attributed to a statistical effect: a longer member has a greater chance ofhaving a serious defect. This effect is not currently allowed for in design codes, which results in boards of differentlength having different reliability. A method is needed to find adjustment factors that will provide equal reliability.It has been modelled with the Weibull model, which significantly over-predicts the length effect.It is deduced that the over-prediction is due to the dependence between elements of the board, which is not allowedby the Weibull model. The presence of this dependence has been found experimentally. Two classes of alternalivemodel are developed. One requires input of the magnitude of the dependence and the strength disthbution. The otherrequires only the strength distribution and gives an estimate of the dependence. A simulation study of 100,000 boardsshows with considerable certainty that both classes of model work better than the currently accepted models on datawhich contains dependence.Real lumber strength data from bending and tension tests of 3000 boards from 4 sources are used to validate theproposed models. Using a wide variety of measures it is shown with a high degree of certainty that the proposedmodels are superior to the existing models. Theory is developed which shows that this model-based approach hasmuch less uncertainty than a purely data-based approach, and this is confirmed from the data. Machine graded lumbermay be better fitted by a model developed from the Gumbel distribution.It is shown that equal reliability adjustment factors can be obtained from the model parameters very easily. Theaverage factor proposed to be applied to design strength for halving the length is 1.14. It appears that this factor islower for high grade lumber.The tools used for the length effect in single members have been extended to computing the reliability of weakestlink structures. This is a useful class of structures, and this method appears to be superior to existing methods. Thiscan be used to show that the different length adjustments make a considerable difference to the calculated reliabilityof structural systems such as trusses. Two truss types were considered and found to have effective lengths of up to7.3m, for a J3 =3. These required a (downward) adjustment to the single member strength of up to 24%.Table of ContentsAbstract.iiTable of Contents 111Table of Tables XTable of Figures XiAcknowledgements XliI Introduction ... 1A. Explanation of Length Effect 1B. Project Objectives IC. Model Selection Criteria 2D. The Statistical Nature of the Length Effect 3E. Background to Lumber Strnngth1. The Production of Lumber 42. Causes for Variation of Wood Strength Between Boards 43. Causes for Variation Along the Length 44. The Grading Process 55. Comparison of Lumber with Small Clear Specimens and Glulam 66. Tension, Bending and Compression 67. Difference between Depth, Width and Length effects 78. Three Length Effects 89. Summary of Chapter I 9II Length Effect Models 10A. Introducing The Weibull Weakest Link Theory 101. Uniform stress 102. The Weibull Length Predictions IiB. Past Applications of the Weibull Model 12C. Difficulties of the Weibull Weakest Link Model 141. The Magnitude of the Length Effect 142. The Predicted Shift of Different Percentiles 15D. Need fora Model 15E. Possible Problem Solutions 171. Alterations to the Weibull model 17a. Alterations Intended to Allow for Dependence 17b. Alterations Changing the Tail Approximating Function 18c. Miscellaneous Modifications 182. Alternative Models 19a. Non-chain Theories 19b. Simulation Models 19c. Models to Deal with Dependence 20d. Miscellaneous Models 203. Discussion 214. Conclusions 22F. Summary of Chapter II 22G. Symbols and Abbreviations 23III Weibull Assumption Violations 24A. Possible Violations of Weibull Assumptions 241. Same material 242. Weakest link 25a. Knot interference 25b. Loading point effects 26i. End Effects for Loading in Tension 26ii. Point Load Effects for Loading in Bending 27c. Added Moment effect 27d. Effect of Relative Stiffness 28e. Miscellaneous Problems 283. Tail Shape Assumption 294. Many Elements 295. Homogeneity 30a. Heterogeneous Growth Processes 30b. Heterogeneous Grading Processes 306. Independence 317. Conclusion 32IIIB. Likelihood of Element Dependence.321. Similarities in Wood 322. Similarities in Lumber Processing 32C. Likelihood of Inappropriate Lower Tail Shape 33D. Summary of Chapter III 33IV Modifications To The Weibull •. .. .. 34A. An Alternative Derivation of the Weibull Model 34B. The Relationship between Sections of One Board 35C. Within-Board and Between-Board Variation 36D. The Known Dependence Case 36E. The Other Length Effects 371. The Built-Up Length Effect 382. Grading and the Length Effect 383. The Graded Length Effect 38F. Summary of Chapter iv 39V Mixed Extreme Value Models ..... • ..40A. Dependent Extreme Values 40B. Reasoning for Selection of this Model Class 42C. Decisions for Parts of the Model 43D. The Method of Working of the ModelE. The Effectiveness of the Mixed Model 45F. Advantages of the Mixed Distribution Model 460. Disadvantages of the Mixed Distribution Model 46H. Summary of the Literature on Mixed Distributions 47I. Derivation of Mixed Distributions 491. Symbols 492. Rules for Finding New Distributions 50a. Theorem I 50b. Theorem 2 50c. Corollary I 50d. Corollary 2 50e. Theorem 3 503. Derivation of the Compound Weibull Distribution (CW) 51J. Summary of Chapter V 53K. Symbols and Abbreviations 54VI Using The Proposed Models 56A. Parameter Estimation 56I. Estimation Principles 562. Maximum Likelihood Methods 573. Maximisation Algorithms 57a. Simplex 58b. Powell 58c. Fletcher 58d. FNMIN 58B. Variance of the Estimated Distribution Parameters 58I. Cramer-Rao Bounds 582. Observed Information Matrix Method 593. Application of Observed Information Method 59C. Prediction of the Correlation Coefficient, and Model Moments 60I. Correlation Coefficient of Board Elements 602. Finding the Parameters of the Element Strength Distributions 633. Calculation of Correlation Coefficient - Alternative Method 644. Numerical Integration to Evaluate Moments 65a. Fortran-based Quadrature 65b. Spreadsheet Approach to Integration 65D. The Coefficient of Concordance 66E. Summary of Chapter VI 66F. Symbols and Abbreviations 67VII Proposed Model For Bending And Compression 68A. Adaptation to Bending 681. Weibull-based Model under Non-Constant Stress 68a. Mixed Weibull-Based Models 70ivb. Methods of Use of these Models.70c. Pure Bending for Weibull-based Models 71d. Third Point Bending for Weibull-based Models 71e. Centre Point Bending for Weibull-based Models 72f. Gumbel-based Models .. 72B. Estimation of the Shape Parameter from Fracture Position 721. Beam UnderCentral Loading 732. Extension to Dependent Elements _.. - •-•—..• 743. Conclusions.........74C. The Length Effect in Compression ...... 741. The Different Regimes for Compression 752. Length Effect for Lumber in Cnishing Regime ....... 753. Length Effect for Lumber in Buckling Regime 764. Conclusions about Applicability to Compression 76D. Summary of Chapter VII 77E. Symbols and Abbreviations 77Viii Model Validation with Simulated Data —..—....-....—...——.——-—.——— 79A. Aim of Simulation Study 79B. Advantages of Simulation Study 79C. Description of Simulations 80D. Comparison of Predictions 801. Length Effect Ratio 802. Model Bias 813. Predicted Length Effect 824. Predicted Correlation Coefficient 835. Statistical significance 846. Discussion 857. Summary of Chapter Viii 85IX Model Validation With Real Data 87A. Description of Data 871. U.S.F.P.L. Data 872. Foriniek Data 903. Old U.B.C. Data - Load Configuration Data 914. New U.B.C. Data 945. Comparisons Involving Data 95B. The Fitting Process 961. Scaling the data 962. Initial Estimates of the Parameters 963. Comparison of Different Maximisation Algorithms 96C. The Fit of the Models 971. OverallFit 992. Correlations Inferred from the Model Fits 104a. Comparisons Based on the Pearson Correlation Coefficient 104b. Comparisons Based on the Coefficient of Concordance 1063. Bias in Estimating Quantiles 1074. Choice of Mixing Distribution 107D. Evaluating the Length Effect 1081. The Length Effect as a Doubling Ratio .. 1082. Individual Doubling Ratios .._. 1093. Comparison of Models at a Particular Quantile 1154. Weibull-Based Models versus Gumbel-Based Models 1165. Testing the Fit of the Entire Distribution 118E. Comparison Between Other Models and Data 1201. The Graded Length Effect 120a. Method of Comparison 121b. DataforComparison 1212. Results of the Fracture Position Approach .. 123a. Fracture Position Model Conclusions 123F. Discussion 124G. SummaryofChapterlX 124H. Symbols and Abbreviations 125X Reliability of Single Members.126A. The Equal Reliability Length Adjustment Factor 126VB. Model Validation - 127C. The Formulation of the Reliability Problem 1271. Parameters and Variables in the Reliability Pocess 129a. The Lumber Strength 129b. The Dead Load Ratio 129c. The Live Load Ratio 1301. SnowLoads 130ii. Occupancy Loads 1302. Computation of Reliabilities 130D. Model Predictions of Reliability 1311. Weibull..based Model-based Adjustment Factors 1312. Gumbel-based Model-based Adjustment Factors 132E. Results of the Reliability Computations 1321. Model Comparisons 132a. The Simple Weibull Model ... 132b. The Censored Simple Weibull - Fiued to the Lower 30% of the Data .... 132c. The Adapted Weibull Model - Modified Using Leicester’s Adaptation 133d. The CW Model 1332. Methods of Comparison 1333. The Basic Case 1334. Results of the Basic Case 1335. Comparison of Model Length Effect Predictions 1346. Sensitivity Analysis 136a. Strength Distribution 136b. Load Distribution Assumptions 137c. Target Value of 13 137d. The Characteristic Strength Value 1377. Magnitude of the Length Adjustment Factor 137F. Discussion 1396. SummaryofChapterX 139H. Symbols and Abbreviations 140XI Reliability of Structural Assemblies .._.. ._ _....... 141A. Computation of System Reliability 141B. Series Systems 142C. Trusses 142D. Dependence 143E. Approximate Methods for Structures with Dependent Failure Modes 143F. New Method for Calculating Reliability 144G. TheLength Function 145H. Comparison Between Proposed Method and Bounds 1451. Method 1452. Parallel Chord Truss Analysis 1463. Howe Truss Analysis 1474. Reliability Analysis 1475. Comparison of Methods for an Initial Truss 1486. Summary of Results for a Variety of Trusses 150I. Assessment of Length Effect Models 153J. Discussion of the Length Function Method 155K. Summary of Chapter XI 156L. Symbols and Abbreviations 157XII Conclusions and Summary _.... 158A. Conclusions 158B. Summary 1611. The Nature of the Length Effect 1612. Approach 161a. Weakest Link 161b. Problems 1623. Dependency 162a. Current Model-based Approaches 162b. Model-based Compared to Experimentally-based Approaches 1624. Method for Dependent Extreme Models 162a. Simulation 1635. Results 163vi6. Reliability and the Application of theMethod. 164a. Equal Reliability Adjustment Factor 164b. Application to Weakest Link Structures 164C. Contribution 164XIII Appendices .. ——--..---......---.- _..__.____..._____.__..._____ 165A. Appendix I Model Prediction vs. Data Smoothing 1661. The Variance from Plotting Means 1672. The Variance of the Model-based Prediction 1693. The Variance from Plotting 5%iles 1694. Approximate Analysis for CW Model-based Prediction 1705. Comparison of the Two Approaches 1716. Variance of Doubling Ratios 1717. Symbols and Abbreviations 172B. Appendix 2 Conditional Independence ..__ 173C. Appendix 3 Moments of Conditionally Independent Elements _... __ 1751. Assumptions 1752. Notation 1753. Identity I 1754. Identity 2 1765. Results for the Correlation 176D. Appendix 4 Methods for Finding Mixed Distributions .... 1771. General Notation 1772. Theoreml 1773. Theorem2 1784. Corollary I 1795. Corollary 2 179a. Proof 1796. Theorem3 180E. Appendix 5 Derivation of other Statistical Distributions .._....... _._ 1821. Weibull Mixed with Uniform Distribution - CUW 1822. Gumbel Mixed with 3-parameter Gamma - CG 1833. Weibull Mixed with Log Gamma Distribution - CLL 185a. Effect of Size on CLL Distribution 1864. Weibull Mixed with Bessel Function Distribution - CBW 186a. Effect of Size on CBW Distribution 1875. Weibull Mixed with Four Parameter Beta Distribution - CWB 188a. Effect of Size on the CWB Distribution 1896. Symbols 189F. Appendix 6 Derivation of Correlation Coefficients ..... 1901. General Notation 1902. Calculation of Correlation Coefficients from the Fitted Model - CW 190a. E(xy) 190b. E(x 191c. E(x) 191d. Correlation Coefficient 191e. Variance 1913. Calculation of Correlation Coefficient - CG2 Distribution 191a. E(xy) 191b. E(x) and Variance 1924. Calculation of Correlation Coefficient - CG Distribution 193a. E(xy) 193b. E(x) and Variance 194c. Symbols 194G. Appendix 7 Gumbel-based Models Under Non-Constant Stress 1951. Method of Use of Non-Constant Load Gumbel-based Models 1952. Pure Bending for Gumbel-based Models 1953. Third Point Bending for Gumbel-based Models 196H. Appendix 8 Model Validation with Simulated Data . 1971. Description of Simulations 1972. Model Validation 197a. Fitting the models 197b. Preliminary Explanation of the Simulation Results 198c. Model Bias 198d. Predicted Length Effect .. 199e. Predicted Coire1ation 203VIIf. Statistical Significance of Gumbel-based Model Predictions. 2043. Discussion of the Simulated Results.204a.Bias.204b. The Effect ofDependence. 204c. Predicted and Simulated Correlation 2054. Details of Simulation Process 205a. Simulating Uniform Random Numbers 205b. Simulating Standard Normal Numbers .. 206c. Simulating Multivariate Normal Numbers 206d. Simulating Univariate gamma variates ...... 206e. Simulating Multivariaie gamma variates ...... ..... 2075. Symbols and Abbreviations ..... 208I. Appendix 9 Summaries of Simulation Results 2091. Gamma-based datWWeibull-based models 2092. Normal-based datajOumbel-based models 211J. Appendix 10 Summaries of Experimental Results 213K. Appendix 11 Summaries of Reliability Results 221L. Appendix 12 Hypothesis Testing ....... _ 2221. Method I 2222. Method 2 223a. Comparison of Models 223b. Comparison Between One Model and the Data 224c. Alternative Comparison between Model and Data 2243. Method 3 2254. Variance Used 225M. Appendix 13 Defect Elements versus Spatial Elements ._.—...... 226N. Appendix 14 The Leicester Adaptation .._....... 2291. Basis of the Proposed Adaptation 2292. The AdapLed Weibull Model 2313. The Adapted Gumbel Model 2324. Discussion of the Models 2335. Symbols and Abbreviations 2340. Appendix 15 A Model for the Graded Length Effect .. .............. 236P. Appendix 16 Discussion of References on Mixed Distributions ........... 2381. Table of Distributions 2382. Univariate Model Types 240a. Frailty Models 2403. Multivariate Model Types 240a. Dependence Structures 2434. Particular Distributional Forms 243a. Choice of Distributions. Kernel 243b. Choice of Mixing Distributions 244i. Deduction of the Mixing Distribution 244ii. The Gamma Mixing Distribution 245iii. The Compound Weibull Distribution 245iv. Multivariate Burr and Clayton’s Distributions 245c. Finding the Mixed Distribution given the Kernel and Mixing Distributions 24d. Finding the Mixing Distribution given the Kernel and Mixed Distributions 2465. The Identification Problem 246a. Relationship to the Proposed Models 2486. Estimation 248a. Moment Estimators 248b. Maximum Likelihood Estimators 248c. Dependence Function Estimators 249d. Non-Parametric Methods 250e. Effect of Model Specification 2507. Independence 250a. Simplifying Models and Testing for Independence 2518. Extremes 252Q. Appendix 17 Strength Distributions and the Length Function 2541. The Independent Case 2542. The Dependent Case 2553. Length Effects for Structures 256a. Independent Members 256b. Dependent Members 256c. Abbreviations 257VIIIR. Appendix 18 Summaries of Truss Reliability Results 258XIV References ... 261IxTable of Tables11:1 Experimental and Weibull predicted length effects .... 14IV.1 Examples of different models.V1ll:1 Signifance tests on simulated data 85IX:1 U.SJ.PL. data sets 88IX:2 Forintek strength data sets - 90IX:3 Old U.B.C. strength data sets 92IX:4 New U.B.C strength data sets .--.- 94IX:5 Negative log likelihood for different models 103TX:6 Significance of fit 104IX:7 Fitted correlation coefficient compared with experimental 1051X8 Inter-element correlations — 106IX:9 Inter-element Tau 106IX:10 Ratio of fitted 5%ile to actual 5%ile 1071x:11 Length effects as doubling ratios 108TX: 12 Model quanule prediction differences 115IX:13 Significance of Table 12 116IX: 14 Comparison with Showalter model 116IX:15 Criterion for choosing Weibull,3umbel based models 118IX:16 Old U.B.C. data used to check the graded length effect 122IX:17 New U.B.C. data used to check the graded length effect 122IX:18 Estimation of c from graded length effect data 123IX: 19 Estimation of length effect parameter from fracture position 123X:1 Summed squared deviations of adjustment factors 134X:2 Signficicance of Table 1 135X:3 Summed squared deviations for different assumptions 136X:4 Comparison of doubling ratios 138XI: I Results for one parallel chord truss 149XIII:Q Truss results for beta=3 155XII.A: I Variance of length effect predictions 171XIILH: I Results of hypotheses 204XIII.I: 1 Results from Weibull-based simulations 209X1JJJ:2 Results from Gumbel-based simulations 211XIIIJ: I Parameters of the Weibull distribution 213XfflJ:2 Parameters of the Gumbel Distribution 214XIIIJ:3 Parameters of the CW distribution 215XIIIJ:4 Parameters of the TG distribution 216XIIIJ:5 Variance of parameters of CW and CG distributions 217XIIIJ:6 Log likelihood of different distributions 218XIIIJ:7 Fifth percentiles 219XIIIJ:8 Fiftieth percentiles 220XJI.K. 1: Summary of reliability results 221XJH.O: I Proportion of variation accounted by grading 237XIII.P.1 Mixed distributions discussed in this work 239XllI:Q Parallel chord truss results 258XIII:Q Parallel chord truss results 259XIII:Q Howe truss results 260xTable of Figures11:1 Log strength / log length plot.1211:2 Two types of length effect .. 151111 Relationship between links 31IV: 1 Two types of correlation - -V:1 The mixing process 41V:2 The method of working of the mixed modelVIII:1 Length effect ratios for Weibull-based models 81VTII:2 Weibull-based model biases at 5%ile 82VIII:3 Length effect parameter alpha 83VIII:4 Estimated versus simulated correlationIX: I U.S.F.P1. Strength Data - High Grade 4” deep 88IX:2 U.S.F.P.L. Strength Data - High Grde 10” deep ........_ 89IX:3 U.S.F.P.L. Strength Data- Low Grade 4” deep 89IX:4 U.S.F.P.L. Strength Data - Low Grade 10” deep 90IX:5 Forintek Strength Data 91IX:601d U.B.C. strength data L8/L17 92IX:7 Old U.B.C strength data L14/L33 93IX:8 Old U.B.C. strength data L14/L33 93IX:9 Old U.B.C. strength data L35/L36 94IX:10 New U.B.C. strength data 95IX: 11 Model predictions on log strength vs log length plot 98IX:12 Typical CW P.D.F 100IX: 13 Typical CW P.D.F 100IX:14 Typical CO C.D.F 101IX:15 Typical Gumbel C.D.F 101IX:16 Typical Weibull P.D.F 102IX: 17 Typical Weibull C.D.F 102IX: 18 Fitted versus experimental dependence 105IX: 19 Doubling Ratios for M4 data-sets 110IX:20 Doubling Ratios for M10 data-sets 110IX:21 Doubling Ratios for N4 data-sets 111IX:22 Doubling Ratios for N10 data-sets 111IX:23 Doubling Ratios for Foriniek data-sets 112IX:24 Doubling Ratio for New UBC data-sets 112IX:25 Doubling Ratios for data-set L14/L33 113IX:26 Doubling Ratios for data-sets L9.,L12,L16 113IX:27 Doubling Ratios for data-sets L8/LI 7 114IX:28 Doubling Ratios for data-sets L351L36 114IX:29 Predicted C.D.F. and K.S. confidence intervals 119IX:30 Predicted C.D.F. and K.S. confidence intervals 120X: 1 The relation between performance factor and reliability 134X:2 The length adjustment factor 138Xl: I Parallel chord russ geometry 146XI:2 Howe truss geometry 147XI:3 Parallel chord truss reliability 151XI:4 Parallel chord truss reliability 151XI:5 Howe truss reliability 152X1:5 Howe truss reliability 152Xll.B: I Conditionally independent choice 173XIII.H.1 Weibull-based 50%ile bias 199XIII.H:2 Gumbel-based 5%ile bias 199XIII.H:3 Gumbel-based 50%ile bias 199XJII.H.4 Length effect at 50%ile 200XIII.H:5 Effect of n on Length Effect 200XIII.H:6 Gumbel length effect parameters 201XIII.H.7 LER at 50%ile .. 202X1II.H:8 Effect of n on LER at 5%ile 202X11I.H:9 Effect of n on LER at 50%ile 202XIII.H:10 LER at 5%ile 202XIII.H: 11 LER at 50%ile 203XllI.H:l2 Fitted vs Simulated Correlation 203XIII.M: 1 Spatial vs defect elements 226XIAcknowledgementsThe author would like to acknowledge the following people:Dr Karl Bury, for his help with the writing of this thesis, which is much appreciated.Dr Tony Bryant, for his useful suggestions and advice, also greatly appreciated.Dr Ricardo Foschi, for his useful suggestions.Borg Madsen, for advice and financial support from NSERC origin during the majority of my term at U.B.C., andfor use of some data.Dr Don Anderson, and Dr David Barrett for being on my commmittee.Forintek and U.S.F.P.L. for use of their data.Dr Greg Lawrence, for encouragement and tangible help. Hank Bier for some editorial suggestions. U.B.C. for asummer graduate fellowship. Keith Knight and Peter Schumacher for help with Appendix 3. Paul, who did the testingat U.B.C. Bob Leicester who tried to make useful comments about Chapter XL Encouragement from John, Alison,Dave, Dan, Andre, Bob, Jim, Jeremy, Wendy, Ann, Jacqui, Fraser, Phil, Rhys, Jennifer, Marion, Dennis, Ron, Elena,Jane, Blair, Cathy, Kathy, Mike, Con, Rod, Steph, Elf, Suzanne, Paul, Janine, and many others.xl’I IntroductionThe first chapter of this thesis introduces the problem which will be the subject of this thesis. This includes the causeof the problem, and discusses the uses to which a solution might be put. It delineates the boundaries of the problem,and the expected domain over which the solution is applicable.A. Explanation of Length EffectThis thesis considers the length effect in lumber. This phrase refers to the phenomenon of strength of lumber appearingto depend on lumber length.Because lumber is highly variable, a particular type of lumber cannot be said to have a single strength value. Evennominally identical lumber has a range of strengths, so that a statistical distribution is needed to describe lumberstrength. This distribution is found by taking a sample from a particular type of lumber, which is tested to obtain thestrengths of all the specimens. Sample strength values estimate a particular statistical distribution model. The term‘length effect’ is used to refer to the way that the distribution for a sample of shorter lumber is shifted to higher strengthvalues when compared to the distribution for a sample of longer lumber. For example, the fifth percentile will (usually)be higher in a sample of shorter lumber.This length effect has been found in many different materials of widely different types. Moreover, width and deptheffects have also been found in lumber and other materials.B. Project ObjectivesThe motivation for this topic of research comes from practical considerations. At the time of writing this thesis thereare many timber design codes which do not contain a length adjustment for strength. If the strength distribution islower for a longer board, its reliability will be less unless some adjustment is included. Salinas (1986) has shown thatat least one common type of lumber structure shows a significant decrease in reliability for longer length.I Introduction Model Selection CriteriaCost prevents structural engineers from increasing the safety of structures ad infirtuum. The goal must be to providean equal and adequate level of reliability across a range of structures. This is a unifying concept that is now the basisof many design codes.In this study it is not possible to solve the general problem of preserving equal reliability of all timber structures ofdifferent sizes. The central aim is the finding of a method that engineers can use in design for ensuring that longermembers have the same reliability as shorter members.This study will not attempt to consider the effect of longer member length on stresses. It does not deal with differencesin connectors, that may occur in larger structures. No attempt is made to solve the problem for predicting the strengthof boards of one depth from another depth, or one width from another width, for reasons that will be discussed.The effort will concentrate on establishing a method for finding a correction. Many different types of lumber (i.e.grade, size, species) are used in construction in Canada alone, and many more world-wide. There is no reason tobelieve that the length effect will be the same for these different types of lumber. Nor is there a reason to believe thatit will be the same for bending, tension, and compression loading. Given the limited scope of this study, lengthadjustment factors cannot be found for all these different types; rather, a general method will be proposed in this studywhich can be used for all lumber types. This will be checked for a limited number of lumber and loading types.The objectives of the thesis will be to answer the following questions:1. What is the best form for the length adjustment in the design code, so that boards of different lengths will havethe same reliability, when subjected to permissible design stresses?2. What is the best way of obtaining the necessary empirical information?3. What is the best way of computing the adjustment factor?For reasons that will be explained, the length effect is greater for lumber in tension and bending, than in compression.Therefore the emphasis on deriving, applying, and testing the theories will be on tension and bending.C. Model Selection CriteriaWhen comparing different options for this length effect adjustment factor it is necessary to judge them on someobjective basis. This is difficult because statistical variation in the quantities under consideration, causes samplingvariation in the estimates of the length effect (often called sampling error). There are two qualities which are desiredin the predictive model:1. The estimate should give close to the correct value when averaged over a large number of samples. This istermed low bias.2. The estimates should have a small variance about the correct value. This is termed low estimate variance.2I Introduction The Statistical Nature of the Length EffectA major problem anses in comparing the proposed model against alternative models: the correct length effect isunknown. The best information available is found in experimental strength distributions fornominally identical lumberof different lengths. The distribution estimated from a sample will almost always differ from die true distribution.This sampling variation is smaller for a larger sample. It will be shown that the variance of the experimentalvalueabout the correct value, is higher than the variance of the estimate from the proposed model. This means that iftheproposed model is good and has low bias, the predicted values from the model are likely to be closer to the truevaluethan the experimental ones, because they have a lower variance. This causes serious difficulties when the model ischecked, or when different models are compared.A model is taken to be better on two grounds:• if it gives adjustment factors which deviate less from those derived from data, or• if it gives adjustment factors which are more unlikely to differ from data.D. The Statistical Nature of the Length EffectThe cause of this length effect is not known precisely, but a substantial part of it is due to a statistical phenomenon.This type of length effect has been observed in hundreds of materials (Harter, 1978). It arises because there is strengthvariation within any particular specimen. It will be shown that this gives rise to a length effect because a longer boardhas a greater chance of having a serious defect. This will be referred to as the statistical length effect. The mainproposition of this thesis is that an adequate treatment of this statistical length effect will result in a nearly equalreliability for lumber of different length.The key concept for dealing with the statistical length effect is the weakest link proposition, which slates that aspecimen will fail completely when any small part of it becomes over-stressed. The name comes from analogyto achain, the strength of which is decided by the strength of the weakest link. This makes the problem tractable, as itfollows that the strength is the extreme of the statistical distribution of strength ofparts of the specimen. An asymptoticsolution can be found from this proposition which allows simple application to data. This is called the Weibull weakestlink theory. It requires certain assumptions which include: independence of strength ofparts of the board,a requirementthat the specimen is reasonably large (compared to the size of the defects), and is homogeneous, and certain limitsabout the shape of the strength distribution of the defects. These will be detailed in Chapter II.The way that trees grow, and that lumber is cut from logs, and then processed affects the strength variation withinand between specimens. It therefore affects the validity of these assumptions.3I Introduction Background to Lumber StrengthE. Background to Lumber Strength1. The Production of LumberMills normally receive logs cut from their locality. Depending on the size of the log, and the current demand for thedifferent sizes and grades, the mill operator will decide how to saw them. The sawn lumber is then graded into twogroups: lumber suitable for structural applications and other lumber. For special purposes it is graded into finerdivisions, but often this is not done, because the process itself, and the increased stock-holding is expensive. Thisgrading may be done visually which produces three grades: Select Structural, #1 and #2. Alternatively, machinegrading based on machine measurement of the stiffness of the lumber (which is correlated to the strength) can be used.The lumber will also have been grouped according to ‘species group’. These species groups may include more thanone species, and were formulated to include similar species that are inconvenient to separate. Samples are then takenfrom these species/grade groups, and they are tested for strength. The fifth percentile is then obtained from the estimatedstrength distribution, and various factors are then applied to this fifth percentile to give the design strength.As far as setting the design strength of the lumber tested is concerned, the lumber could have come from a blackbox process. For the purposes of modeling, it is helpful to examine the details of that black box.2. Causes for Variation of Wood Strength Between BoardsThere are two major groups of sources of variation of board strength. The first group includes the causes for woodof a particular board being stronger or weaker than the average.Every tree represents a source of wood of different average strength. This is because the tree has its own geneticcharacter, and its own growing site conditions. Some trees, for example, have closer growth rings, and this is likelyto result in stronger lumber, although they may produce less of iL On the other hand, close growth rings may also bethe result ofdry growing conditions. Density, ring width, moisture content and knottiness are all examples of propertiesof lumber which vary from one log to another, and which affect board strength (see Williamson 1982, Kollman Cole196).There are also consistent differences in these variables from log centre to periphery, and from tree top to tree bottom(Cown 1980). This means that the wood from almost every board will have a different average strength to that fromany other board, and this explains partly why there is a distribution of board strengths for nominally similar boardsfrom the same species/grade group.3. Causes for Variation Along the LengthThe second group of causes gives rise to variation in wood strength within the length of a single board. Their randomnature also introduces random variation in the strength of boards.4I Introduction Background to Lumber StrengthMost boards do not appear uniform, because they have defects like knots, checks, needle holes, and pitch pockets.Knots are the remains of branches which grew out of the tree at that point, and checks are cracks in the wood usuallydue to shrinkage stresses. The relationship between the dimensions, position and orientation of knots and other defectshas been extensively studied experimentally: see e.g. Williamson (1982), Koilman CoLe (1968), Johnson and Kunesh(1975), Schniewind and Lyon (1971). It has also been modelled in various ways: see e.g. Williamson (1982), Dabholkar(1980), Cramer (1984). Observation of lumber testing reveals that failure in the vast majority of lumber originates atknots. Moreover, these studies have confirmed that the strength of a board has a reasonable correlation with the sizeof the knot where failure occurs. Since the knot size varies considerably, some boards will have relatively small largestknots, while others will have relatively large largest knots. In this way the variation of strength within the board givesrise to a variation in board strength.Knots cause weakness in boards in several ways. The first way is the loss of stressed area at the cross section includingthe knot. The second way is that the knot causes the grain to slope as it goes around the knot. In a whole trunk thisactually reinforces the ‘hole’ caused by the branch; this does not happen in a board because the sawing process leavessloping grain at the edge of the board. Since wood has of the order of one thirtieth of the tension strength across thegrain compared to along the grain, cross-grain appearing in the zone of maximum tension stress severely lowers thestrength of the board.4. The Grading ProcessThe grading rules were set early in this century using guess-work and some empirical principles, rather than extensivetesting or analysis (see Williamson 1982 and Orosz 1969). Their purpose goes beyond engineering; they are alsoexpected to exclude unsightly lumber. Since the grading is done by eye, there can be substantial discrepancies betweendifferent graders, or the same grader at different times. For these reasons, the grading process is not highly effectivein sorting lumber into different strength classes. The result is that there is still a great deal of strength variation betweenboards. Consider the extreme case where the grading process is perfect, and very many grades are used. All the boardstrengths would be perfectly identified, and there would be no length effect because all the boards in the same gradewould have the same strength, regardless of their length. Thus length effect is entangled with grading, which will bediscussed further in section IV.D.2.For the most part, the grading rules apply to the worst defect in a board. Although the second worst defect may besimilar, it is not possible to say with any certainty what grade the rest of the board would be when the worst defect iscut from the board. Therefore, one of the results of cutting up boards is that some parts of the boards may move todifferent grades from that which contained the original board, in a way that is very difficult to predict.5I Introduction Background to Lumber Strength5. Comparison of Lumber with Small Clear Specimens and GlulamLumber testing philosophy was traditionally based on testing small clear specimens of wood. For example,experimental results on the effect of moisture content on the strength of small clear specimens were assumed to holdfor all other lumber. Since knots are specifically excluded, defects in the specimens that cause failure are very smalland probably widely dispersed. In bending, it is usual for failure to start with the compression side of the specimenbecoming plastic, thus increasing stresses on the tension side of the specimen which then fails. This is not brittlefailure and may not be associated with any sort of defect. It is therefore unlikely that the Weibull weakest link theoryis suitable.Glulam (glued-laminated timber) is made of lumber glued together to make large sections. Large defects in thelumber are often removed and the resulting short boards joined together. The result is a large number of small defectsfrom different boards scattered randomly throughout the member. The defects in glulam are smaller relative to themember than they are in standard lumber. They are also more random, since in glulam their position in the memberoccurs by chance, and they originate from many different boards. Therefore the Weibull weakest Jink theory is likelyto work well for Glulam specimens.6. Tension, Bending and CompressionIn section 1 .D, it was noted that the Weibull weakest link theory requires that first failure must proceed to completefailure of the member. This seems quite likely for lumber in tension. Failure of a small part of the board means thatit carries no stress, which means that the rest of the board section must carry more load. If the defect is eccentric thiseffect is magnified. The increased load makes it very likely that the rest of the section fails.In compression, a failed area usually still carries load, only this is less than that of the un-failed areas. Thus theremaining areas only need to carry a slightly increased load and this may not precipitate complete failure. This typeof failure may preclude use of the Weibull weakest link model (see section VIll.C). If failure is actually due to buckling,then the failure might be linked with the worst defect. However the position of the defect would be of importance, sothe length effect model discussed in section D can not be expected to be suitable.The situation for bending can be expected to be somewhere between the two. Poor lumber with many defects failsin tension and its strength may be represented by the Weibull weakest link theory. Clear lumber often fails in compression and should therefore not be represented by the Weibull theory. It is probably permissible to use a weakestlink theory for a particular grade that includes some clear lumber at the highest percentiles if the only percentiles ofinterest are in the left hand tail of the strength distribution.6I Introduction Background to Lumber Strength7. Difference between Depth, Width and Length effectsIf the length effect is due to there being a higher chance of a bad defect then it is clear that there should be depthand width effects. For example, greater depths would give a greater chance of containing a bad defect. If the Weibullmodel fails to model these two effects for no good reason then this should adversely affect confidence in the Weibullmodel’s ability to represent the length effect. The depth and width effects will therefore be briefly discussed. Anotherimportant reason to discuss the depth effect is that in many studies the depth and length effects have been confused.The reader may wish to consult Williamson (1982) for a more extensive summary of the literature.It has been found experimentally that when testing small clear specimens, depth affects the strength (see e.g. KollmanCote, 1968). The non-linear stress distribution (discussed in section I.E.5) may well be affected by depth, and sometheories attribute the depth effect to this cause.When the testing philosophy changed to full size tests, it was thought that it was important to keep the length todepth ratio constant, so deep boards were tested at a correspondingly long span. When these deep long boards werefound to be weaker than shallow short boards, it was natural that it was explained as a depth effect. This is one reasonwhy design codes often incorporate an adjustment for beam depth, but not length adjustment factors. In order toexplain the experimental findings, similar reasons to those considered for the small clear situation were examined,and therefore the difference between the two sizes was described as a depth effect.In reality, lumber usually has enough knots to lower the tension strength below the compression strength. It followsthatplastic compression behaviour will be less likely, and will be limited. Therefore, a depth effect caused by non-linearbehaviour is less likely in lumber than in small clear specimens.Strength variation within a beam is much more likely in real lumber than small clear wood, because the former hasknots. Thus the statistical length effects will be much more likely for real full-size lumber than small clear woodspecimens. There is still a possibility of confusing the statistical length and statistical depth effects, because manysets ofdata have both the length and depth changing simultaneously (e.g. Fewell and Curry 1983). This does not allowseparation of the two effects. The need is for data when either the length or depth is change singly.There is no reason for the depth effect to be equal to the length effect, or even closely related. Consider tensionstrength as a simple case. The only difference between a longer board and a shorter board is that the longer has moreof the same material stressed in exactly the same way. One assumption of the Weibull analysis is that any changes ofthe constituent material of a longer length specimen are so small they can be ignored. For a wider (rather than longer)board, the material is essentially different for two reasons. Different width boards are often sawn from different partsof a log. Since the knot populations are completely different in the different parts of the log, the material cannot bethe same. The second reason is that the grading process is going to impact on the lumber produced differently fordifferent widths. The grading rules are given in terms of ratio of knot size to board depth; however if the strength does7I Introduction Background to Lumber Strengthnot depend on this ratio, or if the frequency of different sizes of knots does not depend on this ratio, then the gradingprocess impacts upon the make-up of the material. For these reasons the assumption that the material will stayapproximately the same for deeper boards is untenable.Similar problems are found with analysing the width effect. The problem of knot scale with respect to the specimenbecomes even more serious. In the width direction the typical size of the defect is considerably larger than the relevantspecimen dimension. Consider the case of a typical knot, of the type where the branch’s longitudinal axis would havebeen extending in the thickness direction of the board. Making the board wider has no effect on increasing the numberof knots likely to appear in the board. A wider board will probably be affected by these knots in a similar fashion toa thinner board. Now consider the case of spike knots which are knots from branches running in the depth directionof the board. A large knot of this type may have little effect on a wide board, because it is buried in the reinforcinggrain. In a narrow board iL will probably have a serious effect. That is the opposite of the size effect expected. Thedifficulty lies in failure of the Weibull assumption that the size of the specimen relative to the defect must be large.For these reasons, it can be concluded that the size effects in the depth and width directions require different models,and constitute different problems. Some recent work has concentrated on separating the depth and length effects(Madsen and Buchanan 1985, Foschj and Yao 1988). Since the width and thickness effects are separate, failure tomodel them does not disqualify a model for use on the length effect, which is the topic of this thesis.8. Three Length EffectsBroadly speaking, there are three routes for obtaining lumber of different length. These three different methods havedifferent impacts on the defect populations, and therefore will give rise to three different length effects.The basic way of obtaining boards of different length leads to the ‘Cut-down’ length effect. This results when a userbuys graded boards of a certain length, and for parts of the application requires shorter boards. These are obtained byrandomly cutting down the original boards. The material of the original and the shorter boards is identical. This caseis recognised as corresponding closely to the classical Weibull weakest link theory.Another way of obtaining shorter boards is to actually buy shorter boards. In this case the material of the originallength and that of the shorter boards is not necessarily the same, for two reasons. Lumber of different lengths is oftencut from different parts of the tree, and, for reasons discussed in section I.E.2, this means that the defect populationwill be different. Second, the grading process takes place after the length change has occurred, which could changethe populations of each grade by moving all the boards from one grade to another. For a perfect strength gradingprocess, there would be no length effect because all the boards in the same grade would have the same strength,regardless of their length. This length effect does not correspond to the Weibull weakest link theory, and will be calledthe ‘Graded’ length effect. A way of quantifying this effect is derived in section IV.D.2.8I Introduction Background to Lumber StrengthThe third way of obtaining lumber members of a different length is by building longer members from shorterindividual boards. Some type of connector is obviously needed, and discussion of this detail is beyond the scope ofthis work. Nevertheless, this could be obviated by the assumption that the connection is sufficiently strong and thatthere will always be a weaker defect in the board itself. Providing the connected lumber was all graded at the samelength, it is easily seen that this case is very similar to the first, and corresponds to the Weibull weakest link theory.However it differs in a subtle way, which increases the potential success of that theory. This ‘Built-up’ length effectwill be discussed in section IV.D.2.The Cut-down length effect will be taken as being the basic subject of this thesis.9. Summary of Chapter I• The ‘length effect’ is the apparent shift in the strength distribution to lower values as specimens become longer.• This effect is generally attributed to a statistical phenomenon, and is usually modelled by the Weibull weakest linkapproach. This approach is based on the board strength being the extreme lowest value of the element strength.• The main objective of this thesis is to obtain an adjustment for the length effect such that equal reliability is preservedfor lumber structural elements of different length, and to find a method for estimating this adjustment.• An estimate of this length adjustment from a sample of limited size will have sampling error. The adjustment shouldaverage close to the correct value in the long run, and have as small a variance as possible.• Width and thickness effects do not meet the requirements of the Weibull model, and are essentially linked to thegrading process used. They are not considered further.• There are actually three different statistical length effects on lumber strength the basic Cut-down length effect, theGraded length effect, and the Built-up length effect.9II Length Effect Models Introducing The Weibull Weakest Link TheoryII Length Effect ModelsA. Introducing The Weibull Weakest Link TheoryThe Weibull weakest link theory has been widely used, and is easily applied. It is not a universal approach, but it isso widespread that any discussion of size effect is difficult without an appreciation of this theory. The followingderivation should clarify the basis of the theory and some of the requirements for its use, and will enable subsequentdiscussion.Let a board be conceptually divided into M elements of unit length. Failure of the board occurs when any of theelements fails. This assumption implies that fracture of a real board is identified with the unstable propagation of acrack from the most severe defect, through other parts of the board, regardless of the local strength of the other elementsin the board. This is called the weakest link assumption, by analogy to the links of a chain.Let the probability of failure of one element at (or below) stress a (i.e. the probability that the strength is a) be1F(a),and let the probability of failure of the board (with M elements) at (orbelow) stress abe MF(o). In statistical terminologythese are called cumulative distribution functions (c.d.f.). Since failure and survival are mutually exclusive andexhaustive events, the probability of survival of one element at stress a will be 1 —‘ F(a).Because failure occurs when any of the elements fails, the probability of survival of the entire board is the probabilitythat all the elements survive. If it is assumed that the elements are all independent, then the probability that they allsurvive is the product of the probabilities of each them surviving. This givesP{survival of board) fl P{survival of each element). (1)1. Uniform stressFor identical and independent elements subjected to the same stress, eqn (I) becomesP{survival of board) = [P{survival of element)]TM. (2)Substituting from above,MF(a) = 1 —[1 —F(o)]M. (3)If M is large then[l_x]M=eMx. (4)10II Length Effect Models Introducing The Weibull Weakest Link TheoryThis givesMF(a) = I — exp[—M1F(o)]. (5)With M large, the weakest defect will likely fall on the lower tail of the defect strength distribution. A reasonableand simple approximating function for the shape of a lower tail is a power law function,(6)where a is commonly known as the shape parameter.Substituting this into equation (5) for board strength distribution givesMF(a) = I _exp[_Mroal. (7)No mention has been made of the absolute length of the board, or the elements. The parameter changes with theelement length. For a base element length of 1, with M elements, and Mt = t, equation (7) becomes:F(o) = I — exp[—to”j. (8)Equation (8) is recognised as a Weibull c.d.f. (see e.g. Bury 1975), with shape parameter a, and t being a combinationof shape and scale parameters.2. The Weibull Length PredictionsEquation (8) can be used to predict a strength distribution for a different length. For a specimen of unit length withM elements, the strength distribution is given by equation (8). For a specimen of length N with NM elements, thestrength distribution is given by:NMF(G) = I —exp[—Ntãi. (9)The distribution is a function of the length of the board. As the board gets longer the distribution is squeezed up onthe strength axis, and moves away from the unit length distribution. The length effect is this movement. A simple wayof characterising the movement of the distribution is to obtain an expression for the movement of each quantile. Thequantile for the distribution for the longer length is found as a function of the distribution for the unit length board.For the same quantile (e.g. the 5th percentile) of the two distributions:NMF(0)—F(0). (10)Substituting equations (7) and (9) in (10) gives a relationship between the parameters of the two board disthbutions,which is:(11)where ñ is the quantile for the longer board. This shows that the size effect of this model is only a function of onedistribution parameter, normally the shape parameter a.11II Length Effect Models Past Applications of the Weibull ModelTaking logarithms givesiog=iogo’2. (12)Equation (12) implies that a plot of the logarithmof the strength against the logarithm of the lengthshould give a straight line, with the slope equal to1/a. This gives a simple method which can be used engthto interpolate between experimental results. Forexample point Pin Figure 1 might represent the fifthLog Lengthpercentile from some tests of specimens of I mlength, and point Q the. fifth percentile of somespecimens of 4m length. Then the fifth percentile ofother lengths can be cstima ted from other points on Fig. I The Log Strength / Log Length Plotthe straight line.The slope of the line gives the magnitude of the size effect. The parameter a can be estimated from a set of strengthdata from specimens of one length. It follows that the size effect can be predicted using this model from tests of justone length. A simple explanation is: the length effect is due to a longer board having a greater chance of a seriousdefect - it therefore depends on strength variation - the strength variation can be estimated from the specimen strengthdistribution - therefore the strength distribution can be used to estimate the length effect.This length effect could, in principle, be applied to any empirical statistical model used for the strength distribution.It is almost universal to apply it to the Weibull distribution used as a statistical model, which makes the modellingprocess internally consistent.B. Past Applications of the Weibull ModelSumiya and Sugihara (1957) were probably the first to use a weakest link model to represent the effect of size onwood strength. Their model used the normal distribution rather than the Weibull. The first application of the Weibullmodel (8) was by Bohannan (1966) who used it to relate a range of data from clear specimens to 5”x12” clear glulambeams. He found reasonable fit to the theory with a shape factor a of 9. He left the specimen thickness out of thetheory to obtain a better fit, arguing that thickness elements are actually in parallel, rather than in series. This workprovides the factors which are still often used to adjust small clear specimen strength to full-size strength (FPL, 1974).12II Length Effect Models Past Apphcations of the Weibull ModelIf the data plots close to a straight line on a log-log plot (Fig. I), then the model can be said to fit the data. One shouldnot put too much significance on this finding, because log-log plots are notorious for fitting any data. Other similarstudies include that of Leicester (1973), who applied the Weibull weakest link model to data from clear bendingspecimens, and found the data also plotted close to a straight line. Buchanan (1984) tried plotting clear specimen datafrom a variety of sources on the same plot, and failed to get a good fit to a straight line, but each individual set of dataproduced nearly a straight line. Schneewei6 (1969) also took historical data from both clear and knotty specimensand put them in a Weibull framework. He concluded that there was not only a depth effect but also a length and widtheffect.The Weibull weakest link theory has been successfully applied to tension perpendicular-to-grain strength and shearstrength of clear wood and glulam (Barrett 1974, Barrett and Foschi 1979). They used finite element techniques tofind the equivalent stressed volumes for a variety of specimen shapes, and were able to show that the experimentalvolume and shape effects could be described by the Weibull theory. Note that this success was found for glulam whichhas small widely distributed flaws as was pointed out in section LE.5.Liu (1982) and CoIling (1 986a,1986b) provided analyses of specimens of different shapes, specifically used in woodtesting or application, so that equivalent volumes can be found. This allows comparison between specimens ofdifferentshape.Madsen and Buchanan (1985) applied the Weibull weakest link theory to in-grade strength data in which both boardlength and depth changed, both together and singly. They pointed out that there was no reason to believe that the sizeeffect was the same in the length, depth and width directions. Therefore they evaluated the size effect parameter forlength and depth by plotting the data on the log strength vs log size plot and estimating a slope. They concluded thatthere was little or no depth effect in bending, but that there was a signficant depth effect in tension. A beam configurationof 3m with third point loading was suggested as a reference standard. Foschi and Yao (1988a) also applied the Weibullweakest link theory to lumber strength data with both length and depth changing. They used a least-squares fittingmethod to find the length and depth effect that best fit the data. The depth effect they found was significant. This isimponant, because for data with both length and depth changing, it makes the predicted length effect correspondinglysmaller.Madsen (1989) interpreted an extensive set of length effect data within the Weibull weakest link theory framework.He showed that the length effect is important and should be incorporated within design codes. This data is used inChapter X.Lam and Varoglu (1990) also used the Weibull weakest link framework to analyse length effect data. This data isused in Chapter X.13II Length Effect Models Difficulties of the Weibull Weakest Link ModelHarter (1978) and Weibull (1952) have provided surveys of the hundreds of studies of size effects in differentmaterials, using the Weibull weakest link theory. This provides some assurance of the robusmess of its assumptions.However, a straight line plot on a log-log plot could be due to many relationships other than the Weibull weakest linktheory.Another reassuring piece of evidence is that the Weibull statistical distribution usually fits real lumber strength databetter than other strength distributions with the same number of parameters. In addition, Pellicane (1985) shows the3 parameter Weibull to be superior to the normal and 2 parameter log-normal but inferior to the 4 parameter JohnsonSb (this analysis did not includeallowance for the number ofparameters, and the significance of it should be consideredin the light of section IX.C).C. Difficulties of the Weibull Weakest Link ModelAs noted above, the Weibull theory seems fairly successful at fitting data, both as a statistical distribution and as acurve-fitting technique for the length effect. However, there are two significant problems when it comes to applicationof the model to prediction of the length effect.1. The Magnitude of the Length Effect.It is well established that the length effect predicted from fitting a 2 parameter Weibull distribution to one data-setis consistently too large (see, for example, Madsen 1990).Source Grade Experimental a from Predicted a from Weibullplotting method Model fitted to individual(Fig I): eqn (12) data sets: eqn (8)Madsen and Buchanan Select Structural 6.5 3.2(1985)No.2 5.9 2.3Barrett (1974) Clear 7.7 6.4Commercial 4.7 4.0Table 1: Comparison of experimental and Weibull predicted length effect - aA common sample size for testing of lumber is approximately 100 specimens. There will be sampling error in boththe experimental length effect and the length effect predicted from the Weibull model. This sampling error makescomparison between the two more difficult; however the over-prediction is very obvious in some data-sets e.g. Madsenand Buchanan (1985), Leicester (1985b), Showalter (1986). As an illustration, consider Table 1, which is based ondata from Madsen and Buchanan (1985), and Barrett (1974), where ii. can be seen that the experimental a is higherthan the predicted a. The two length effect estimates are compared in a more rigourous way in Chapter IX.14II Length Effect Models Need for a ModelIt is significant that this over-prediction has been noted for many other materials under a variety of different loadconfigurations. See for example Moreton (1969) who used carbon fibres, and Wagner et al (1984).2. The Predicted Shift of Different Percentiles.The Weibull weakest link model predicts a particular type ofmovement of the strength distribution for a different length. The___________________Longer /‘2rterzero percentile does not move at all, and the one hundredth board boardC.D.F. /1percentile moves most. This corresponds to the upper diagram in,1Fig.2. ///Comparison with experimental findings shows that although this Strengthis sometimes found experimentally, other data-sets show all theLonger / /Shorterpercentiles moving approximately the same amount. This board boardC.D.F. / /corresponds to the lower diagram in Fig.2 . These two types of //movement can be seen in the Showalter (1986) lumber tension. /Z:: /data. These data show about an even split between the two typesof movement. For another example see Bullock and Kaae (1979)who used glassy carbon specimens. Fig. 2: Two Types of Length EffectMadsen (pers. comm.) has pointed out that a third type of shift is sometimes apparent: where the lower percentilesmove most and upper percentiles move least. One explanation is that the lower percentiles correspond to tensionfailures (which will show a large length effect) and upper percentiles will correspond to compission (which willshow a relatively small length effect). These differences could be dealt with by making the length effect a functionof the strength, but this would have the drawback of requiring a great deal of data to identify this function. Variationsin the type of length effect could be due to sampling variation in the strength distribution.D. Need for a ModelIt appears clear that a framework is needed to at least interpolate and extrapolate between the limited amount ofexperimental data available.Despite the fact that the length effect magnitude prediction is poor from the Weibull, other authors have proposedacceptance of the prediction of the theory that the results should plot as a straight line on a log-log plot (Madsen andBuchanan, 1985, Foschi and Yao, 1988a, Madsen, 1990). The model is used as a framework, and it is this feature thatallows interpolation and extrapolation.15II Length Effect Models Need for a ModelThere are several reasons why this approach is not an adequate one, and why a more complete model should bedeveloped:First, the above approach would require testing of all species/grade/depth combinations at two lengths (at least), inorder to obtain predicted length effects for each. This would be a formidable test series, since it is approximately twicewhat is needed to establish the 5th percentile for design strength purposes, at a single lumber length.A second reason is that the model makes predictions which have proven incorrect, and therefore should be rejected.Another reason is that the length effect found in this manner carries considerable uncertainty. L.am and Vanglu(1990) have given a method for considering the uncertainty. They found the slope by regression, using three differentlengths, and their analysis concentrates on the uncertainty from a regression approach. They attempt to lower theuncertainty by matching the samples used for the different lengths. Sampling matching is a difficult process.The uncertainty is calculated in Appendix I and compared to the uncertainty associated with a strictly model-basedapproach (with the length effect predicted from a fitted parameter). This uncertainty is calculated for the Weibullmodel.In order to calculate this uncertainty a few assumptions and simplifications have been made. Firstly, the estimatedmodel-based uncertainty is based on the minimum-variance bound. This is an often used statistical measure, but it isknown to be optimistic.Secondly, the quantile plotting uncertainty is quite large if the model is based on fitting a length effect to 5thpercentiles, and is difficult to calculate. This was calculated approximately. If the Weibull model is valid, then theaverages of each distribution can be used to estimate a distribution-wide length effect. The uncertainty from fittingthe model to averages is much smaller and easier to calculate. Therefore this uncertainty was also calculated.For a greater ratio of the two test lengths the uncertainty becomes less, because the measured length effect becomesgreater, compared to the sampling variance. If the ratio of the two tests is less than 27, ii is found that the model-basedprocedure is superior (on the grounds of lower variance). Ratios of that order would be very difficult to achieve, andthe short length strengths would be unlikely to be accurate because end effects (see section III.A.2.b for a discussionofend effects) would be significant. Tests with low ratios (1-2) give results with variance so high that they are virtuallyunusable.Barrett (1974) made a similar study on the two strategies for finding the size effect on tension perpendicular to grainstrength. He used simulation to show that the model based estimate had a lower variance. However he warned againstusing this method because different fitting methods, and different types of material gave different estimates of thesize effect. Good, consistent fitting methods are available, and if they are used the first problem should notbe significant.The second problem can be avoided by fitting and predicting for one type of material only.It is concluded that a model-based approach must be used.16II Length Effect Models Possible Problem SolutionsE. Possible Problem SolutionsThere are a number of modifications to the Weibull model and also alternatives to the Weibull model. These couldprovide alternatives to the Weibull weakest link model. At the least they could give an insight into correction of theproblems associated with the Weibull weakest link model.It should be pointed out that some ofthese modifications and alternatives in the literature have already been formulatedfor particular purposes. In particular, several of them are designed to correct the first problem mentioned above, i.e.correcting the magnitude of the prediction of the length effect.1. Alterations to the Weibull modelThese alterations fall into three groups:a. Alterations Intended to Allow for DependenceKennerley, Newton and Stanley (1982) studied the problem of strength of pharmaceutical tablets. They examinedthe problem where the specimens are not strictly identical, but have different values of an independent variable (e.g.density or thickness) which affects the strength. It follows that two elements within one specimen are more closelyrelated than two elements within different specimens. This is equivalent to some dependence between the strength ofelements from one specimen (this is discussed further in Chapter III).The result of this is that the fracture stress data will have two sources of variability: the inherent strength variabilityof the material itself (i.e. with the independent variables held constant), and the variability of the independent variables.They used an analysis which conditioned on this independent variable, but this relies on individual values of thisindependent factor being known for each individual specimen. In practice, it is infeasible to obtain this sort of detaileddata, for lumber tests.Leadbetter (1982), considering the general problem of extremes of ‘weakly correlated’ elements, suggested‘measuring the size of a material not as its physical dimension but in terms of an integral of a positive local sizefunction’. The result gives standard extremal models, but this idea is not extensively developed. ‘Weakly correlated’means that the correlation dies down sufficiently quickly, that, at a sufficient distance, it has little or no effect. In theproblem considered in this thesis, it is likely that the elements at two ends of a board are still significantly correlated.Therefore these weakly correlated models are unsuitable.L.eicester (1985b) examined the problem oflength and load configuration effects for lumber in bending. He consideredthe problem of bought and cut lengths, the effect of keeping the worst defect in the span and putting the worst defecton the tension side. He adapted the Weibull model so that the predicted length effect is adjusted for an amount ofcorrelation that is known a priori. This adaptation is discussed further in Chapter IV.17II Length Effect Models Possible Problem SolutionsLeicester experimentally verified this modification using sample sizes of fifty, but he did not compare his fmdingsto the Weibull weakest link model predictions. He judged the model to be successful at describing the broad featuresof the experimental evidence, but it was ‘not accurate enough to predict the magnitudes of the low percentiles withan accuracy sufficient for practical applications’. He attributes this to a particular approximation in his analysis.b. Alterations Changing the Tail Approximating FunctionThe next group of modifications includes several that change the approximating tail function that is contained in theextreme value functions (in the Weibull model as a power function, equation 6). The authors give a variety of reasonsfor doing this, but none of them correspond to the length effect problem described in section ll.C.The most comprehensive treatment of this group of modifications is by Trustrum and Jayatilaka (1983). Theyconsidered a wide variety of statistical distributions for the defect size, and used basic fracture mechanics relationshipsto derive the strength distribution. They give the size effect law that follows from this. Their work emphasizes thatthere is a multitude of different possible assumptions, each of which gives a different possible size effect law. Theirresults are not directly applicable to the lumber strength problem, because the flaws in lumber are mainly knots, ratherthan sharp cracks as in the fracture mechanics idealisation.Sumiya and Sugihara (1957) use the normal distribution for the elemental distribution, which gives a size effectlaw different from the Weibull. The reason for this choice is not clear. Petrovic and Hoover (1987) used their ownsimple function for the. elemental distribution which is able to account for the presence of a plateau in the strength/lengthrelationship for ceramic fibres. Schneeweif3 (1966) used both the Weibull and the Gumbel as basic distributions in aweakest link model for the strength of steel chains of different length. Phani (1988) found a different distribution withupper and lower bounds that gave a better fit to the data. This should have been expected, because extra parametersallow a better fit to data.c. Miscellaneous ModificationsFinally, there are a number of miscellaneous modifications which are supposed to improve the original model.Goda and Fukunaga (1986), working with ceramic fibres, found that two modes of failure gave them two lines ona Weibull probability plot. They found good agreement by using a mixture of two separate Weibull distributions asthe strength distribution. They make the assumption that the mode of failure can be identified from the failed specimen.This is an extreme of the dependence case, where each specimen falls into only one of two groups. In the lumberstrength problem there is a continuum of different types of boards; moreover, the different types cannot be easilyidentified, even after failure.Knoff (1987) found poor agreement of the Weibull model for strength of short Keviar fibres. He modified theWeibull analysis for the case when specimen length is not restricted to an integer multiple of the element length. Hisanalysis adjusts for the chances of having n + 1 or n + 2 links in a specimen that is between n and n +1 links long.This extra sophistication has an insignificant effect for large n, but predicts a lower strength for smaller specimens.18II Length Effect Models Possible Problem SolutionsThis idealisation is not particularly useful for the analysis of lumber strength, as the link length (the scale of lengthover which the strength can be assumed to be nearly constant) is very small in lumber - approximately the width ofa knot. The length effect in lumber at this scale is very seldom of interest.Kiu.l and GUnther (1981) found a smaller dependence on beam height than expected for compacted cement pastespecimens. They re-derived the Weibull equations ‘based on the idea that increasing the volume does not implyincreasing the size of flaws’. However, it is not clear that the original Weibull model assumes that it does increase.Madsen (1990) suggested using a three parameter Weibull model estimated from the lower 25% of the data. Thiscould be expected to give a better fit in the tail, because the shape does not have to fit the upper 75% of the data. Hefound that the predicted length effect agreed with that estimated from three sets of data. One explanation is that theupper percentiles are dominated by failures from a different mechanism to those at lower percentiles. Using somelower fraction of the data would remove the problem.2. Alternative ModelsThere are a number of other approaches which differ substantially from the Weibull weakest link model.a. Non-chain TheoriesThe Weibull weakest link theory has the elements arranged like links in a chain. When one link fails the chain fails.There are other possible arrangements for the links.Freudenthal (1967) describes some of these in detail. The bundle model is the main alternative. The links are inparallel, and when one link fails it sheds its load onto the other links. If the load is shared evenly, it can be shownthat, for a large number of links, the load per link has a normal distribution. For a larger number of fibres the meanstrength remains the same, but the variance decreases. Neither the arrangement of fibres, nor the length effect seemappropriate for the lumber in tension.Phoenix and Smith (1983) review models for fibrous materials based on chain-of-bundles approaches with differentload-sharing rules. The result is still almost exactly a Weibull distribution. This model may be applicable to the problemof the effect of width and depth as well as length on the strength of lumber. The transverse dimensions would beexpected to have a smaller effect on the strength of the boards.b. Simulation ModelsBy dropping the requirement for an analytical solution to the problem, the problem is made easier to solve.Showalter (1986) used a simulation-based model which could allow for dependence between the elements of boards.The model used the relationship between strength and stiffness. Serially correlated stiffnesses are simulated from asecond-order Markov model and combined with serially correlated simulated residuals to give simulated strengths.The strength of the generated piece of lumber was determined by using the weakest-link concept. This model requiresmeasurements of the stiffness at points along the board; such measurements are usually not available.19II Length Effect Models Possible Problem SolutionsFosehi and Barrett (1980) formulated a simulation-based model for the strength of laminated timber. This could beused to find size effects. Their model uses the relationship between knot size and strength, and requires data aboutthe occurrence of knots, which is also usually not available.Taylor and Bender (1989) developed a simulation model which allowed for saptial correlation of strength and stiffnesswithin a member. This was based on a transformed normal distribution. The purpose of this was to simulated thematerial for glue laminated beams.c. Models to Deal with DependenceDitlevsen (1981) studied the problem of defect generated fracture, giving as one example knots in timber boards.He used second moment approximations to give upper and lower bounds on the probability of failure of correlatedsystems. These are expressed in terms of: the correlation between strengths of individual defects, occurrence rate ofdefects, and the strength distribution of individual defects. He makes the assumption of a bivariate normal distributionto check how far apart the bounds are. For a reliability index [3 (an approximate measure of probability of structuralsafety corresponding to z from the Gaussian distribution) of 5, he found that a correlation of 0.65 resulted in errorsof only 5% when assuming independence.In practice these bounds would be difficult to use for the purpose of predicting a length effect. They give the boundson a single probability, not a strength distribution. They could only be used to find the probability for a specimen ofa length of a whole number multiple of the reference length.Ditlevsen also briefly considered the case where dependence is due to each specimen having its own value of anindependent variable (similar to the problem considered by Kennerley etal mentioned above). He suggests conditioningthe probabilities using the Rackwitz-Fiessler algorithm. This is impractical for lumber strength applications, becausethe value of the independent variable is not known.Garson (1980) considers the reliability of weaRest link systems with correlation between failure-modes. He usesconditional probabilities, distributional assumptions and various approximations to find the effect of correlation. Heconcludes that for low risk level (<0.001) and correlations less than 0.7 the effect of correlation can be neglected.Bechtel (1986) formulated a model based on conditional probabilities specifically to deal with the length effect inlumber in tension. This is based on conditional distributions and Weibull theory, with many approximations. It wasnot applied to any real data.d. Miscellaneous ModelsRiberholt and Madsen (1979) proposed a model for the longitudinal variation in strength of a lumber board. It isassumed that the strength is constant between defects, but varies from board to board. The occurrence of a defect isdescribed by a Poisson process and the strength of a cross section with a defect is controlled by a stochastic process.This model does not require an asymptotically large number of defects in each board. It does not accommodatedependence in the analysis, but Riberholt and Madsen investigated the likely seriousness of this problem. They20U Length Effect Models Possible Problem Solutionsestimated serial correlation of strength from serial correlation of both stiffness and defect size. They found no evidenceto reject the assumption ofequal correlation along the board (rather than some decaying level of correlation), and theyused Ditlevsen’s (1981) findings (see above) to justify ignoring dependence.Neville (1987) has a model that is equivalent to the Weibull model, but it uses fracture toughness instead of strength.He states the model can be expected to work for all materials. The derivation is based on a measure of the highlystressed volume around a crack tip. However this sharp crack idealisation is unlikely to correspond to the failure atdefects in lumber.Torrent and Brooks (1985) have an empirically based model which relates strength to the volume of material whichis stressed to 95% or more of the maximum tensile stress in the material. They find that this gives identical results tothe Weibull analysis in some cases, and that it explains the size effect in geometrically similar specimens. It does notcontribute anything significant beyond the Weibull model.3. DiscussionThe extreme value approach typified by the Weibull weakest link model has been widely accepted as the basic toolfor statistical size effects in material strength. It has been successfully applied to several different situations in woodstrength, where the defects are small, well-dispersed and truly random.Moreover, the Weibull weakest link model does morc than give a theoretical basis with which to interpolate andextrapolate length effect data. It gives a method which allows the length effect to be predicted from strength data fromspecimens of one length. This can be extremely useful as usually a strength distribution at one length is available, butdistributions at other lengths usually are not. However, two major problems exist in its application to the length effecton lumber strength: over-prediction of the length effect, and inconsistent prediction of the movement of the higherpercentiles.The first problem of over-prediction of the length effect has been recognised in other materials as well. It has beensuggested that this is due to dependence between elements. Two studies (Ditlevsen, 1981 and Garson, 1980) haveshown that for the multivariate normal assumption the correlation between elements would have to be greater than0.7 to have a significant effect on the reliability. There are several reasons why the first problem may be due tosignificant dependence, despite these studies which suggest dependence is unimportant.First, the correlation may actually be this high. Leicester (1985) found for low grades a correlation coefficient of0.1, and for high grades a correlation of 0.5. Riberholt and Madsen (1979) conclude that the correlation coefficientis ‘small’ for poor quality lumber and 0.5 - 0.66 for high quality lumber. However, some of their measurements givevalues over 0.8. Taylor and Bender have computed the correlation between adjoining elements to be approximately0.81.21II Length Effect Models Summary of Chapter IISecond, the conclusions apply to multivariate normal disthbutions, and may notbe applicable to the real distributionswhich are far from normal.Third, though there is an insignificant effect on reliability, there may be a noticeable movement in the strengthdistributions. The reliability may be less sensitive to the correlation coefficient than would be expected from movementof the strength distributions.A number of modifications and alternatives to the Weibull model to deal with dependence have been suggested.Some of these rely on the dependence being evaluated directly or indirectly from experimental evidence. Such dataare rarely available. For lumber the correlation coefficient may be difficult to find and it may change from species,grade and cross-section size, necessitating a large test series.The second problem of a lack of fit of the predicted lower percentiles to data has not been specifically mentionedin the literature. However, it is clear that there are very many different tail approximating functions that can be used,and that this choice alters the way the predictions move away from the original tail. It may be that the Weibull predictionis valid, but the sampling variation often makes it look incorrect.4. ConclusionsIt appears that the Weibull weakest link model is a good basis for constructing a model of the length effect on lumberstrength. However a modified version that deals with the two major problems needs to be found. At this stage it seemslikely that the first problem is due to dependence. If this is the case, then the new model should accommodatedependence, but it should not increase the data requirement. The second problem is likely to be with the approximatingtail function, so a new model may need to provide alternatives. Chapter V describes a new model for the length effect;this model satisfies the above requirements.F. Summary of Chapter II• The Weibull model has been used to deal with the length effect in many materials, including timber, with moderatesuccess.• There are two common problems with the application of the Weibull model to lumber: over-prediction of the lengtheffect, and inconsistent prediction of the upper percentiles.• A model is needed to deal with the length effect, because experimental estimates of the length effect usually havean unacceptably high variance.• Several researchers have concluded that the over-prediction of the length effect is due to the presence of dependence.• The problem with the upper percentiles may be due to a poor choice for the approximating function chosen for theleft-hand tail.22II Length Effect Models Symbols and AbbreviationsG. Symbols and Abbreviationsa Length effect parameter (i.e. shape parameter of Weibull)Reliability index0 StrengthMOdified scale parameter of the Weibull disiributionc.d.f. Cumulative distribution functionF(x) Cumulative distribution function of x1F(o) C.d.f. of element strength,4F(x) C.d.f. of strength of board of M elementsN Length of longer boardP( ) Probability of event23Ill Weibull Assumption Violations Possible Violations of Weibufl AssumptionsIll Weibull Assumption ViolationsA. Possible Violations of Weibull AssumptionsThe practical application of a length effect model usually involves conversion of strength measures of specimens ofone length, to make them suitable for application to specimens of another length.Given that the Weibull model is deficient, there are two possible causes: one of the assumptions of the Weibullmodel is incorrect, or that there is some other additional length effect operating. If this other length effect is in theopposite direction to the Weibull length effect (i.e. causing longer specimens to be stronger), then the predicted lengtheffect would be greater than the length effect found.It is useful to draw up a single list of requirements including both the Weibulj assumptions and which will alsoexclude the possibility of other length effects operating:1. The material must be the same for both lengths.2. The member is as strong as its weakest element.3. The lower tail of the element strength distribution must have the assumed shape.4. There must be enough elements, so that the asymptotic distribution is appropriate.5. The material must be homogeneous.6. The elements must be independent of each other.These requirements will be examined one by one, searching for a cause that would make the predicted length effectgreater than the real length effect. Length effects that cause the opposite effect are not of interest. Length effectsthat would only operate in bending, or would only operate in tension or some other class of tests which form a partof the total category of tests for which a length effect is observed are also unlikely to be the cause which is beingpursued. This would require multiple causes, one for each category of tests. Intuitively, this is less likely.It will not be possible to prove the cause of the discrepancy, because this would require the proof that there are nomore possible causes of length effects. It can be shown that one cause is more likely than the other obvious causes.1. Same materialThe material must be the samefor both lengths.24Ill Weibull Assumption Violations Possible Violations of Weibull AssumptionsThis requirement is so basic that it may easily be forgotten. It implies that the defect strength distributions and thedefect spatial distributions must be the same for both lengths. Neglecting the remote possibility that simultaneouschanges cancel each other out, this means that:a) the defect size distributions are the same for both lengths,b) the distributions of distances between knots are the same for both lengths,c) the defect size I defect strength relationships are the same for both lengths.The first two assumptions are clearly satisfied if the shorter lengths are randomly cut from of the longer lengths.The third requirement c) is discussed below.2. Weakest linkThe member is as strong as its weakest element.This is the basic assumption upon which the model is based. In the situation of lumber in tension, failure at any crosssection causes failure of the entire specimen. The strength at that cross section dictates the strength of the specimen.This assumption will not be satisfied if the strength of a particular cross section depends upon the length of thespecimen it is in. Consider the situation where there are two identical worst knots: one in a board of a certain length,and the other in a board of different length. The weakest link assumption requires that these two knots have the samestrength. There are several mechanisms by which this could be prevented from being so.a. Knot interferenceThe strength of a knot may depend on other defects in the board. It has been observed experimentally that the crackcausing failure of a board tested in tension often extends from one knot to another. This seems to indicate some degreeof interference between the two knots. Since it must be assumed that the final failure mechanism is the weakest one,it appears to follow that the other knot has weakened the knot where the failure originated.A shorter board will have a smaller chance of having knots to interfere and lower the strength. Therefore on averagethe same knot will have a higher strength in a shorter board than a longer board. This implies that the longer boardwill be weaker than expected from short board results, or equivalently that the length effect will be bigger thanexpected from the Weibull theory.There are several reasons which suggest that this mechanism is likely to have a small effect.There are two possible scenarios for the way that one knot can affect another:1. the secondary knot can affect the other knot before failure starts by changing the stress spatial distribution, or2. the secondary knot can affect the failure sequence after failure starts.The former situation can happen because knots have a low elastic modulus compared to the surrounding wood, sothat they act much like holes. Consider the case where the original knot is on the bottom side of the board, and thesecondary knot is on the top, and the board is in a state of constant strain. The secondary knot will have diverted stress25Ill Weibull Assumption Volattons Possible Violations of Weibull Assumptionsonto the bottom of the board in its immediate vicinity. If the original knot is very close, the stress distribution willnot have equalised and the original knot will have a higher stress on it than otherwise. Its strength will appear lowerthan it would otherwise be. St. Venant’s Principle states that the stress distribution will revert back to approximatelyuniform within about one depth (for an isotropic material). The grain deviation around knots is likely to divert thestress back to an approximately even distribution relatively quickly. Wang and Bodig (1991) concluded that fewerthan 2% of knots would interact by as much as 10%, in their study of transmission poles, indicating grain aroundknots quickly redistributes stress to an even distribution.What is the likelihood that this will affect the length effect? As noted above there will be some length effect if thelength influences the chance of this secondary knot interference. Since the secondary knot must be less than aboutone depth away from the primary knot, this would appear to be impossible unless board lengths are of the order ofthe depth of the board.The second situation where the secondary knot affects the primary knot after failure initiation is also unlikely. Oncea board starts to fail at a knot, this causes considerable eccentricity which raises the stresses in the remainder of thecross section of that board. The cross section fails very rapidly as a result and since the stiffness of the board is muchlowered, the load on the board drops. This is found in the vast majority of tests of lumber. It follows that the interferenceis unlikely to affect the maximum load, and can be ignored.The strength of wood when loaded parallel to the grain is about 20-30 times that when it is loaded perpendicular tograin, so a crack selectively follows the direction of the grain. Therefore it is not surprising that it is common for thefailure crack to move longitudinally along a board as it follows the grain. It often does so until it reaches the nextknot, where it can follow the disturbed grain to the outside of the board. If the load is not rising, knot interference willhave no effect on the strength of the board. Therefore, even if a secondary knot is involved in failure, this interferenceis unlikely to effect the strength distribution.A small number of boards do show a rising load path, and these will be affected. Since this will only have an indirectinfluence on the length effect, and since only a very small number of boards are likely to be affected, this knotinterference is considered unlikely to be important.Finally, if these mechanisms were important they would be manifested as making the length effect bigger than thatpredicted by the Weibull theory.b. Loading point effectsi. End Effects for Loading in TensionThe stresses at the end of the board, where is held by grips, are not exactly as idealised. The axial tension does notinstantaneously decrease to zero, because the axial force is transferred by shear over the length of the grips. The woodin the grips is thus under some longitudinal tension, and in lateral compression.26Ill Weibull Assumption Vio’ations Possible Viotations of Weibull AssumptionsThus failure can occur in this area, but this depends on the rare absence of serious defects in the main part of theboard, and the presence of a serious defect in the grip area. This will clearly depend on the length of the main part ofthe board. If failure in the grips is treated as a censored data point there should be no effect on the apparent strengthdistribution.When the failure defect is near the grips, it is quite possible that the failure crack would prefer to move into the gripzone, but it is unable to do so because of the restraining action of the grips. The grips must therefore have a reinfoicingaction in this area. This will happen more often in shorter boards, giving rise to a length effect. The length effectwould then be greater than expected from the Weibull theory.Another major end effect is due to the misalignment of the grips. If the grips do not allow rotation of the specimen,then a lack of collinearity of the longitudinal axis of the specimen, and the lines of actions of the grip tension forceswill produce substantial moments in the specimen. If the average magnitude of this discrepancy does not vary, thenthe moments imposed in short specimens is substantially higher than that in long specimens. This will cause a lengtheffect that is additional to the statistical length effect.ii. Point Load Effects for Loading in BendingThe stress distribution around the point loads is substantially different to that given by linear elastic bending theory.This perturbation is to be expected in isotropic materials, but the orthotropy of wood causes the perturbation to bemore severe and extend further (Hooley and Hibbert 1967). These zones of high stress around the load points substantially increase the chance of failure at these points. This failure event is not affected by the length of the board.Thus the experimental length effect would be smaller than otherwise. This could explain the discrepancy in the Weibullpredictions. However, there are two reasons to discount this possibility:1. Failure directly under the load points for boards with a reasonable number of defects is not noticeably morecommon than it should be.2. The sante over-prediction is found in the tension case, where the point load effect is absent.Thus point load effects are ignored in searching for a new length effect model.c. Added Moment effectIf a knot is positioned at the edge of a board, it causes eccentricity. For tension loading this will cause a moment tobe added to the tension force on the cross section. The maximum tension stress will be significantly increased overwhat it would be for a symmetric knot, so these knots usually affect strength most seriously.These bending stresses depend on the distance between the knot and the loading points. Orosz (1975) used experimental evidence to find the magnitude of the lateral deflection and second order theory to find the resulting stresses.His results indicate that this effect may be neglected for lengths greater than 20D approximately. His analysis showed27Ill Weibull Assumption Violations Possible Violations of Weibull Assumptionsthat the strength of a fixed-ended specimen is increased for longer boards, while the opposite is true for pin-endedconditions. In the former case there would be an extra length effect, and in the latter case there would be less of alength effect.The Weibull over-prediction has been found in experimental results from both types of testing machine. It has alsobeen noted in bending. The added moment effect should not be expected in bending, because there is no net load onthe section and the eccentricity will not cause a moment. It is unlikely that this is a major cause of the main problem.d. Effect of Relative StiffnessThe relative stiffness of specimen and testing machine is usually accepted to influence in a major way the successthat the load/displacement path of a specimen can be followed after the peak load has been reached. If the problemis examined within a fracture mechanics framework it can be seen that the relative stiffness may also affect themaximum load and thus the apparent strength.In a tension test of a specimen with a centre crack and constant fracture toughness there is so much energy availablethat unstable crack growth is certain, once the critical load is reached. Because of the twisting grain and other varyingproperties amund a knot, the crack may move into regions of higher fracture toughness, which will halt unstable crackgrowth. Some stable crack growth may follow. Under stable crack growth, the stiffness of the system can affect theload at which the crack growth becomes unstable and complete. failure occurs. For the case where the only substantialchange in the test set-up is the longer specimen, the test machine will become relatively stiffer, allowing some stablecrack growth before failure and a slightly higher load. However, many lumber tension testing machines use extensionsfor boards of different lengths, so the relative stiffness between machine and specimen is relatively unchanged. Inbending, a longer specimen also requires a longer support beam, which implies that the relative stiffness will also beapproximately unchanged.Since this effect depends on relative stiffness, and this does not change for longer specimens it appears unlikely tocause an important length effect.e. Miscellaneous ProblemsBaratta (1984) examines in detail the many requirements for fiexure testing. He examines the requirements for theuse of simple beam theory such as small deflection, equal elastic modulus in compression and tension, lack of initialcurvature and absence of buckling. He also details a second set of possible causes of errors which he calls externalerrors: load mis-location, beam twisting, friction, local stresses, contact point tangency and surface preparation.It is reasonable to assume that reseaichers have carried out tests according to established testing standards whichare designed to reduce or obviate these problems. It is also reasonable to assume that it is very unlikely that they havenot all coincidentally misplaced the loads. It follows that these sources are unlikely to be a major cause of the lengtheffect problem. This is reinforced by the finding that there is a discrepancy in the length effect for tension as well asbending.28UI Weibuu Assumption Violations Possible Violations of Weibull Assumptions3. Tail Shape AssumptionThe lower tail of the element strength distribution must have the assumed shape.It has been shown that there are three possible asymptotic extreme value distributions (Bury 1975). Rigourousrequirements for convergence to each type have been found, and these can be translated to depend on the shape ofthe lower tail of the original strength distributions.If the tail is very long, then the result would be a type II distribution. This is very difficult to imagine for lumbertension strength, as the tail appears to be at least bounded by zero.If the tail is relatively short it can be approximated as a power function. This possibility corresponds to the type ifior Weibull asymptote. This possibility allows for a lower bound.The third possibility is intermediate to the other two and is called the type I or Gumbel distribution. The lower tailcan be approximated by an exponential function, which is unbounded.Conventional treatment of the problem has always used the Weibull distribution, because strength distributions arealways bounded from below. With this assumption the length adjustment entails scaling the strength distribution (seesection II.C.2). The Gumbel assumption adjustment entails shifting the location of the distribution. It appears that thismodification is capable of explaining some of the observed length effects which are not explainable by the Weibullmodel. The incorrectness of this assumption would explain why higher percentiles were predicted to move too much,for data-sets which show this location shifting length effect. It does not explain the over-prediction for those data setswhich show a scaling type length effect.4. Many ElementsThere must be enough elements so that the asymptotic distribution is appropriate.If there are enough elements then the weakest element (and therefore, the strength of the specimen) will have oneof the asymptotic distributions mentioned in the above section. If the element distribution is already one of theasymptotic distributions, then the need for many elements is relaxed. The further the element distribution is from oneof these asymptotic distributions, the greater the number of elements needed to obtain an asymptotic distribution forthe strength.Ifthereare notenough elements this is likely toaffect the shape of the distribution. his much less likely toconsistentlyaffect the magnitude of the length effect, and is therefore unlikely to be causing the over-prediction by the Weibullmodel. In addition, it will be shown in section IV.A that the requirement for many elements can be modified, makingthis requirement slightly less restrictive.29Ill Weibull Assumption Violations Possible Violations of Weibull Assumptions5. HomogeneityThe material must be homogeneous.The board strength is assumed to be uniformly random throughout. There are two reasons why this can be onlyapproximately true.a. Heterogeneous Growth ProcessesOne source of heterogeneity is that trees, and therefore logs and boards, have more and smaller branches at thetree-top end. Thus it should be expected that knots in a board have a small non-random size change. Within the lengthof a board this effect is likely to be small, because boards are short compared to trees.This has two effects. The first is to introduce a small amount of dependence, which will be discussed in the nextsection. It also means that very long boards have a higher chance of having larger defects at one end. This wouldcause an extra length effect above that of the Weibull theory, and thus can be ignored in this investigation.Another source of heterogeneity is that for some species branches grow in groups, spaced of the order of I m. apart.This will cause the knots to appear at reasonably regular intervals. CoIling and Dinort (1987) and Riberholt and Madsen(1979) give some data on this problem, which does show some regularity in the occurrence of knots. This means thatthe number of knots in a board of a given length will be less random than would otherwise be the case. The numberof knots will still be related to the length of the board. It is suggested in section IV.A that regularity in knot spacingwill not make it less likely that the Weibull theory is suitable.b. Heterogeneous Grading ProcessesThe grading process is based on the worst defect in the board. In order to simplify the discussion, assume that thestrength discrimination part of the grading process is perfect.The grading process has the effect of restricting the worst defects in each board to a smaller range than they wereoriginally. The other defects in each board can be in that range, or stronger than that range. In contrast, imagine a setof boards with the same strength distribution but which came from an ungraded process. The worst defects must alsocome from the same range, but in this case the other defects come from that range as well.In the latter case all the defects in the board can be thought of as coming from the same statistical process, which istherefore homogeneous. In the former case the worst defects come from a modified process, and the defects cannotbe said to be homogeneous. For the graded boards, the range of the non-worst defects is much wider than it appearsfrom the range of the worst defects. Since the range of defects corresponds to the strength distribution, the Weibullweakest link theory will predict a length effect for graded boards that is smaller than the one corresponding to theoverall range of defect strengths.30Ill Weibull Assumption Violations Possible Violations of Weibufl AssumptionsThis should, however, be a small effect, since the actual grading process is very inefficient and does not restrictstrength to a narrow band. Also, this effect is partially nullified by the presence ofdependence which will be discussedin the next section. Finally, the predicted effect is additional to the Weibull length effect.6. IndependenceThe element strengths must be independent ofeach other.This assumption is about whether the defects in one board have any similarities with each other or whether they areas randomly distributed as they are between boards.____Independent—cxx=axx:c links___Dependent__in SFigure 1: Relationship between links for a chain under tensionThis is illustrated in Figure 1. Consider the chain analogy introduced in section I.D. This idealisation has the materialdivided into imaginary elements each of which corresponds to a link.If the defects in a board are independent then the ‘links’ are chosen at random, and the chains would look like theexamples at the top of the diagram. If some chains contain many strong links, and others many weak links then thedefects are dependent. This is shown in the bottom of the diagram.Finding the length effect corresponds to finding the effect of adding on extra elements. In the independent case, theprobability of having an element of a particular strength does not depend on the strength of the other elements in theboard. In the dependent case, the probability does depend on the other elements. This is much simpler in the independentcase.Thedependence between a number of random variables can be ofa general form. It iscommon as a firstapproximationto quantify the dependence by measuring the strength of the linear relationship. The usual measure of this is thecorrelation coefficient. A correlation coefficient of I corresponds to complete dependence, and a value of0 correspondsto an absence of any linear relationship.31Ill Weibull Assumption Violations Likelihood of Element DependenceCon-elation between defect strengths in the same board makes a significant difference to the length effect. If thecorrelation coefficient is zero the Weibull assumption of independence is fulfilled approximately, and the length effectis nearly equal to the Weibull predicted effect. If the correlation coefficient is 1, then all the defect strengths areidentical, and the length has no effect (provided there are enough defects so that all boards have defects). The realcorrelation must lie somewhere between the two. Therefore, the Weibull predicted length effect is an upper boundon the actual length effect.7. ConclusionThe problems with the Weibull model appear to be explainable by some of the suggested possible problems withthe assumptions. The presence of dependence seems to be the most plausible explanation for the over-prediction ofthe length effect. The second problem, concerning the relative movement of upper and lower percentiles, may beexplained by an inappropriate assumption about the lower tail shape.The next step is to look at these two explanations, and examine their chance of occurrence.B. Likelihood of Element DependenceAs noted above, the statistical concept of dependence of element strengths within a board is equivalent to strengthsbeing closer than expected otherwise. It is useful to examine reasons why elements of one board would be related.1. Similarities in WoodThe first group of reasons explain that the wood and defects within one board are related, because they have beengrown in one environment.Wood elements in one board have clearly grown in the same forest, and therefore the same environmental conditionssuch as temperature and rainfall. They have grown in the same tree, and therefore will have the same genetic characteristics. They come from approximately the same height in the tree, and the same distance out from the centre ofthe log.All these factors imply a similarity in the density, ring width and other qualities of the wood. These factors imply asimilarity in the size, frequency or orientation of the defects. All these factors have a strong influence on the strengthofthe wood (see section I.E.3), so the elements of the same board must berelated, and the elementcorrelation coefficientmust be greater than zero.2. Similarities in Lumber ProcessingElements of the same board have been through the manufacturing process in a similar way. They are likely to havebeen through the same drying process, and have a similar moisture content, and been subject to similar damage from32Ill Weibull Assumption Violations Likelihood of Inappropriate Lower Tail Shapehandling. Since boards are sawn to a certain tolerance and are not identical in size, there is also a similarity in dimension.Two elements within one board are more likely to have the same depth, than two elements from different boards. Thevariance of board dimensions, and its effect on strength is discussed in Gerhards (1983). These factors all affect theapparent strength of an element, so the elements within the same board must be related. -C. Likelihood of Inappropriate Lower Tail ShapeMany assumptions could be made for the element distribution because there is no a priori reason for accepting oneover another. Each of these would give a different length effect. In order to keep the number of choices reasonableonly asymptotic distributions are considered. As noted above, this effectively restricts the choice to either the Weibullor the Gumbel models. The choice between the two hinges mostly on whether a distribution with a lower bound ismost appropriate.Because specimens are (usually) in one piece at the beginning of a test it is reasonable to assume that zero is a lowerbound for strength distributions. Therefore if the quantiles of interest are very low and the grade of lumber is low theassumption of a lower bound of zero is probably a good one. In this case the shape of the left hand tail must be closeto appropriate for a Weibull. For higher quantiles and grades this argument may not be a good one, because 0 is sofar away it may have no effect on the shape of the left hand tail in the area of interest. It may be that a three parameterWeibull is appropriate. However, recent large-scale testing programmes have indicated that the left hand tail is veryspread out, because there are a few very low strengths even for higher grades. This may be because each grade isallowed to contain a small number of mis-graded boards. This type of very long tail may correspond better to a Gumbel.Therefore, it is a possibility that the Gumbel is superior for predicting the length effect at the level of the median, forhigh grades.D. Summary of Chapter III• It is not possible to prove the identity of the cause of over-prediction of the length effect.• The most likely cause of over-prediction is the presence of dependence between elements in a board.• There are other length effects, which are likely to cause an under-prediction of the length effect by the Weibullmodel.• Other causes are shown to be likely to be small and/or would occur for only one of the cases where the over-predictionis observed.• It can be argued from the nature of lumber that substantial dependence is presenL• The tail shape may deviate from that assumed in the Weibull model.33IV Modifications To The Weibull An Alternative Derivation of the Weibull ModelIV Modifications To The Weibull VThis chapter deals with modifications made to the Weibull model, which will be used in this thesis. There ait twopurposes for these modifications. First, to allow the model to deal with dependence, and second to allow some of themodel assumptions to more closely reflect the situation for lumber strength.The chapter starts with a discussion of a modified derivation of the Weibull theory partially relaxing two of theassumptions. This will also clarify the abstract concept of an ‘element’. The suitability of different dependenceassumptions is then discussed. Leicester’s(1985b) adaptation of the Weibull model is shown to have a suitable formofdependence. This adaptation forms the basis of a modification suitable for use when the magnitude of this dependence(of assumed form) is known a priori. The chapter finishes with discussion of the alternative length effects mentionedin Chapter I, within the framework developed for discussion of Leicester’s adaptation.A. An Alternative Derivation of the Weibull ModelIn Appendix 13, the Weibull model is derived on a slightly different basis. The major difference is in the assumptionregarding the element which makes up the specimen. In the derivation in section II.A, the elements were assumed tobe small lengths of the specimen. In the alternative derivation they are assumed to be the defects which may causefailure of the specimen.The alternative derivation requires the added complication of a variable number of elements in the specimen. Thebenefit is that the elements are no longer spatially connected, and it thus easier to uphold that the elements areindependent from the next element along the specimen. It also makes it clear that the power law approximation in thederivation needs only to approximate the tail of the strength distribution of defects, and that the strength of clear woodcan be ignored.34IV Modifications To The Weibull The Relationship between Sections of One BoardB. The Relationship between Sections of One BoardHaving decided that dependence between elements is present, the form of this dependence must be decided upon.For example, axe proximal defects in one board more closely related than distant defects? This is a difficult questionto answer. Looking through the sources ofdependence listed in section III.A.5, it is seen that they indicate a relationshipmostly due to the two defects being in the same board, rather than being close together. On the basis oftheir experimentalwork Riberholt and Madsen (1979) conclude ‘that it may be assumed that the coefficient of correlation is constantalong the beam’. Therefore it will be assumed that the dependence between two defects in the same board is the samefor any pair. This question is discussed in the tight of new experimental evidence in section IX.C.2.This dependence isrepresented by the lowerpart of Figure 1, andlabelled Uniform Correlation. Each element isrelated to other elements inthe board equally. Analternative option is thatelements close togetherare more closely related.I. 1 1. ‘ I,indirtdepepencedirect dependenceOne model for dependence of this type is shown in the upper part of Fig. I entitled Markov Correlation. This is basedon a given level of dependence between one element and the neighbouring element. The element next along will havea lower level of dependence with the first, by virtue of also being dependent on the neighbouring element. Betweenany pair of elements there will be dependence, but this becomes smaller as they become further apart.1.eadbetter (1982) has shown that the Weibull model is still valid if there are enough elements and the dependenceis of a ‘weak’ type. This means that the dependence dies out quickly enough that an element at one end of the boardis unrelated to one at the other.Unfortunately this does not extend to a case where there is a finite dependence throughout the board. Showalter(1986) developed a Markov correlation model with a significant level of dependence even between two ends of aboard. However, this approach required simulation to obtain results.Markov Correlationdirect dependence(Th iTh iTh iTh (Th Th (Th iTh11.1 i•I1 ‘ ‘II 111I IIUniform CorrelationFigure 1: Two Types of Correlation35IV Modifications To The Weibull Within-Board and Between-Board VariationSo far the exact nature of the dependence has not been specified. The strengths might have a linear relalionship withan error term. This appears to be a simple type of dependence, and this would be preferred. However, many othertypes of dependence may yield reasonable results. It is necessary that the dependence can be expressed in somemathematically tractable way so that the equations fall into place, and a useful model results.C. Within-Board and Between-Board VariationThere is an equivalent way of looking at element dependence. The variation in the strength of elements can be brokeninto two sources: within-board variation and between-board variation. The relationship between the dependenceconcept and the between/within board variation concept is best explained by some examples. There are two extremetypes of dependence. Completely dependent boards will have all elements in a particular board identical; this corresponds to zero within-board variation, and a specified level of between-board variation. Completely independentboards will have elements within a board unrelated; this coffesponds to zero between-board variation and a specifiedlevel of within-board variation. Real boards are somewhere between these two limits. For example, highly dependentboards have little within-board variation, and substantial between-board variation.The length effect is due to longer boards having a greater chance of having a serious defect. The chance of havinga serious defect depends on the within-board variation, and not the between-board variation. Therefore only within-board variation leads to a length effect. For example, completely dependent boards will have no length effect, becauseas noted above they have no within-board variation. Since the distribution of the strength of boards derives from thestrength of elements, this distribution also has two sources of variation.D. The Known Dependence CaseIt may be that the dependence (between elements in one board) is known. This would be the case if the boards werecut and the pieces tested. This is an impractical requirement for all species, grade and size groups. It may be that thedependence is similar for boards from similar groups, so that dependence can be inferred from a relatively smallamount of data. For example, both Leicesler (1985) and Riberholt and Madsen (1979) found that the correlation forhigh grades is significantly higher than that for low grades (see section II.E.3).Leicester (1985) adapted the Weibull weakest link model to allow for known dependence. He proposed a sizeadjustment factor which is a function of the correlation coefficient between elements in the same board. Dependenceis modelled by letting the element strength having a scale factor which stays the same for each board.This same approach will be used as the basis of a proposed model to be used for data where the dependence is known.This model will be called the ‘Adapted model’, because the user fits the Weibull distribution to the data and thenadapts the shape parameter to give an estimate of the length effect parameter.36IV Modifications To The Weibull The Other Length EffectsThe model is derived in Appendix 14. The scale factor form of the dependence is used to derive a relationshipbetween the within-board variance of strength and the total variance of strength. The correlation coefficient is usedto quantify the dependence. The corrected length adjustment parameter & to be used in equation 11(12) is given as- (V(cz)”&= V9 I. (1)(1 p)2)where V(x) is a function relating the coefficient of variation of the Weibull distribution to its shape parameter, andis defined in (12) of Appendix 14. Leicester (1985) gave an equation similar to (1), but it involved additionalapproximation.This length adjustment factor is used in a similar way to that from the simple Weibull weakest link model. Thestrength distribution predicted for another length is given by equation (14) in Appendix 17.It was pointed out in section III.A.3, that a plausible explanation for inconsistent prediction of quantiles is due tothe fact that both Weibull- and Gumbel-based models are needed in order to explain different types of movement ofthe strength distribution. The corrected Gumbel length adjustment parameter is given in Appendix 14.E. The Other Length EffectsThe three different types of length effect were discussed in section I.E.8. There is a fourth type of length effect thatresults when the specimens are the same length, but one set of specimens were graded at a different length. This isreally a combination of the Graded length effect, and the Cut-down length effect. Examples are given in Table 1.Predicting Specimen Predicted Specimen Model to useGraded length Test length Graded length Test length4m 4m 4m 2m Cut-down4m 2m 4m 4m Cut-down4m 4m 2m 2m Graded2m 2m 4m 4m Graded4m im 4m 4m (4xlm joined) Built-up4m 4m (4xlm joined) 4m im Built-up4m 2m 2m 2m Graded combined with Cut-down2m 2m 4m 2m Graded combined with Cut-downTable 1: Illustrative example of suitable use for different models37IV Modificanons To The Wejbufl The Other Length Effects1. The Built-Up Length EffectThe Built-up length effect refers to the length effect that results when random boards are joined together to make alonger member (see section l.E.8). Section III.A.6 may make it clear how this differs from the Cut-down length effect.In the Cut-down case the elements in one member all have the same value of the scale factor. In the Built-up case theelements can be considered to be the board parts, and have different values of the scale factor. Since the elements areindependent for the Built-up length effect, the pure Weibull weakest link theory can be used.One application where this might be needed is the length effect in the bottom chord ofa truss. Consider the exampleof a bottom chord made of 3 pieces of lumber each of 2m length, fmger-jointed together. Finger joints are usuallystronger than knots of a reasonable size, so for this example their presence is ignored. Let us also assume that the 2mlumber was obtained by cutting 2.5m lumber, for which test results are available. Then the 5th percentile might bepredicted using the following method:1. The Weibull model is fitted to the data, giving an estimate of a..2. The 5th percentile is adjusted using the Adapted length adjustment parameter given by (1), in equation 11(12)with a length ratio N of 2/2.5.3. The 5th percentile is adjusted using a. and 11(12) with a length ratio N of 3/1Overall, this will move the distribution to the left of the test results, and thus lower the design load.2. Grading and the Length EffectGrading may have an effect on the length effect. If the specimens have been graded at their respective lengths it isnecessary to use the Graded length effect discussed in the next section. If the specimens are currently at the samelength but were graded at different lengths this could be modelled by a combination of the Cut-down and Gradedlength effects.Grading could be expected to have some indirect effects on the Cut-down length effect. Effective grading shouldhave the effect ofplacing all the strengths in a clearly delineated band. In this case it might be necessary to use truncatedWeibull distributions. In reality, grading processes are never this accurate. It may be necessary to use a Weibulldistribution with a location parameter (see section 13.d.6). Usually a few boards escape through the grading process.This produces a long lower tail on the distribution which might be better modelled by a Gumbel-based model.3. The Graded Length EffectThis refers to the difference between distributions of strength of boards bought at different lengths. In section I.E.8,it was pointed out that this is due partly to the boards of different lengths tending to come from different sources, andpartly due to the impact of the grading process.38IV Modifications To The Weibull Summary of Chapter IVThe first effect is very difficult to model. It may actually be impossible to model the difference in sources. Forexample, decisions to cutup logs are not made according to a strict set of rules, and they are affected by many externalfactors like the market price of lumber. This subject is well away from the area of the rest of this thesis, so it will notbe considered further. The second effect requires quantification of the effect of grading.If the grading process is completely ineffective the Graded length effect would be the same as the Cut-down lengtheffect. If the grading process is perfect, then there can be no length effect. Intermediate to these extremes, there willbe component of the strength variation that the grading process takes out, and an unidentified or unseparated componentof strength variation. This latter will give rise to a length effect, because a longer board will have a greater chance ofa weak defect.In Appendix 14, this situation is modeled approximately within the framework used for the Adapted models. It isassumed that the grading process accounts for a constant proportion of strength variation.The result is a simple modification of the length effect parameter,(2)By looking at some typical lumber strength data, it is possible to work out that visual grading schemes allow forabout 20% of the strength variation of lumber. This gives c 0.8.F. Summary of Chapter IV• By altering the assumption of the Weibull model concerning the nature of the elements making up a member, it isshown that the Weibull model is suitable for more situations than initially expected.• The dependence structure with equal relationship between any two elements in a board is more likely to be suitablefor lumber than one with the relationship dying away with distance.• Leicester’s adaptation of the simple Weibull model has this equal type of dependence. It assumes that the magnitudeof the dependence is known a priori, and that the strength distribution will be an asymptotic distribution of thesame type as would be given by independent elements. This is internally inconsistent.• An adaptation of the Gumbel model allowing for dependence is found.• The Built-up length effect should be modelled with an application of the unchanged Weibull weakest link theory.• A method is found for relating the Graded-length effect to the Cut-down length effect.• The 3 basic length effects can be combined to give length effects for other situations.39V Mixed Extreme Value Models Dependent Extreme ValuesV Mixed Extreme Value ModelsThis chapter covers the development of a class of models derived from a basis that there is dependence in thespecimens. The purpose of developing these is to remove some of the problems associated with the models whichinvolve a posterior adjustment for dependence.The first section is a simple explanation of logic of derivation of these models. The non-statistical reader may wishto read this section and then move onto the next chapter.The following section deals with the reasoning why this is the optimal class of models, and how they relate to othermodels. This is followed by a discussion of how the dependence implied by the model relates to the data, and theadvantages and disadvantages of the model. The relevant literature is introduced, and some useful results laid ouLThese give methods for deriving the class of models. The derivation of the distributions used in this thesis are given.A. Dependent Extreme ValuesConsider a group of imaginary specimens which are from the same part of the same tree. The parts of all of themwould be generated from the same statistical process, and they could be said to have independent elements. In thiscase the assumptions of the Weibull model are fulfilled and the statistical distribution of strength could be expectedto be Weibull. The Weibull model prediction of the length effect would be appropriate.Conditional on these requirements, the parts of a board are independent. The parts of a board could be said to beconditionally independent.When lumber is sold, it comprises a mixture from many different parts of many different trees. In, this case thestrength distribution could be expected to be a mixture of Weibull distributions. If follows that a mixture of Weibulldistributions should better fit lumber strength data, than a simple Weibull distribution. The strength distribution forboards of different lengths will also be a mixture of Weibull distributions.40V Mixed Extreme Value Models Dependent Extreme ValuesA mixture of Weibull distributions gives a distribution usually termed a mixed Weibull distribution. This is foundby weighting the individual distributions by their relative frequency. If the relative frequency of different Weibulldistributions is given by a continuous mathematical function, then this becomes an integral:F(x;i)=$G(x;O,).p(O;)49 (1)where F(x;!1)represents the mixed distribution of x with a parameter set E., G(x;O, E) represents the Weibulldistribution of x with a parameter set including 0 and , and p(0;) represents the density function of 0 with aparameter set !3.The Weibull acts as a type of kernel distribution, the weighting distribution is called the mixing distribution, and theresulting distribution is called the mixed distribution. This process is shown in Figure 1.Kernel DistributionsWeighted Kernel DistributionsG(x)xPWeighting/Corn iningMixed Distribution H(x)Figure 1 The mixing process41V Mixed Extreme Value Models Reasoning for Selection of this Model ClassIf the mixing distribution degenerated to a spike (i.e. a Dirac function), then there would be no mixing. This wouldcorrespond to a zero correlation between the elements of the board. If the kernel distribution degenerated to a spikethe distribution would be a transformed version of the mixing distribution. This corresponds to complete dependenceof the elements in a board.There are only a few functions for the mixing frequency for which convenient closed form solutions to the integral(1) are available. This function also needs to be able to accurately reflect the distribution in the data. Given theserequirements several models were selected to be tried with the data, but only one was found practical. The results ofthis application are given in Chapter IX.B. Reasoning for Selection of this Model ClassIt may be possible to deduce the best model for the extremes drawn from a dependent random process.In order to model dependence between the elements of a board it appears that it is necessary to use amuhivariate statistical distribution, with a variable for each part. This is in contrast to the Weibull model whichis a univariate statistical distribution. In principle the following approach could be used:1. Some assumption could be made about the distributional form for this multivariate distribution. This meansboth the marginal distributions, and the dependence form.2. Experimental data could be used to find the parameters of this distribution.3. The distribution of the extremes of this model would have to be found.4. The effect of the number of elements on the distribution of extremes could then be found. This would bethe length effect.In practice there are a number of difficulties. It is very difficult to obtain data for the strength of more than a fewelements of a board. This is because a substantial length is necessary for grips for tension testing, or for the span inbending. This type of data is not commonly available from past testing programs, so special testing would have to bedone.Consider reversing steps 2. and 3. above. The extreme distribution is inferred from the postulated model. This willcorrespond to the strength of the whole specimen, so it can be fitted with specimen strength data. If only wholespecimen strength data is available, then there is little choice but to use this univariate technique.For reasons of mathematical convenience, one of the best choices for the form ofdependence is the mixture approachdescribed in the previous section. This will be confirmed in the literature review. A combination of mixture typedependence, and the univariate (steps 1/3/2/4) technique gives the model described in section 1.The assumptions of the model have implicitly allowed for the presence of dependence; however only strength dataare required to fit the model.42V Mixed Extreme Value Models Decisions for Parts of the ModelC. Decisions for Parts of the ModelThe kernel distribution is chosen to be an extreme value distribution because:• This is the simplest assumption, given that we wish to have an analytical solution for the minimum chosenfrom it. Intuition suggests simple solutions are often best..• The mixed distribution is known to be close to an extreme value distribution, and the two can be expected tobe similar.• Each element of the specimen can be thought of as being made of sub-elements. The strength of the elementwill be the minimum of the strength of the sub-elements. Therefore it is likely to be extreme value, itself (seesection XIII.P.4.a for a discussion of the literature).The way that the kernel and mixing distributions interact is not self-evident. Some measure (called here the mixedparameter) of the kernel distribution must change from one specimen to the other. For this application, it is necessarythat the model reflects the fact that there are some strong boards and some weak boards in an ordinary grade of timber.However, it may be that some boards also have a greater spread of strengths than other boards. The details of theseheterogeneities are unknown. However a decision must be made on which measure will be allowed to change fromboard to board.Similarly, the mixing distribution must reflect the distribution of this measure of the kernel distribution in the realpopulation, but this distribution is unknown.Some guidance on these problems are available from the literature. This is discussed in section XIIl.P.4.b. Howeverthe preliminary choices are made principally on the grounds of mathematical convenience, using the principle thatsimplest is best. If the choices are good and reflect the situation in real grades of lumber, then the mixed distributionsselected will fit real strength data well. The best fitting model can be adopted.Simplicity dictates that the measure which is allowed to vary should be one of the two parameters of the Weibulldistribution. One of them is a shape parameter, and varying this would not allow modelling of the mixture of strongand weak boards very well. The other parameter, which reflects a change in scale, is chosen as the mixed parameter.D. The Method of Working of the ModelA comparison between the simple Weibull model, and the proposed mixed Weibull is shown in Figure 2.Curve A represents the strength distribution for a set of short boards, and curveD represents the strength distributionfor a set of long boards. It is desired to predict curve D, given curve A.The simple Weibull is based on fitting a curve A to the data and predicting curve D directly. It attributes all thevariation in strength data to a process leading to a length effect.43V Mixed Extreme V&ue Models The Method of Woiking of the ModelC.D.F.0.I0.70.6050.40.3020., 2 The method of working of the mixed modelThe mixed Weibull model is also based on fitting a curve A to the data. However, it acknowledges that the curvereally represents a mixture of curves B. For each one of these curves, the Weibull weakest link theorytransformation0.944V Mixed Extreme Value Models The Effectiveness of the Mixed Modelis applicable, and gives a curve in the set C. These combine to give curve D, the predicted strength distribution. Thesteps of splitting up curve A to give curves B, transformation to curves C, and combination to curve D are not doneexplicitly by the modeller, but by the model implicitly.The user fits a curve to the data, and transforms parameters according to derived rules to predict the new curve. Thusit appears to the user that the curve A directly gives curve D.E. The Effectiveness of the Mixed ModelThe mixed model does not require input of some quantified measure of the dependence in the process which generatedthe data, unlike the Showalter and Leicester models. Thus the mixed model is operating with less information, andcould be expected to work less well than either of these models. In this section, some deduction will be done aboutthe effectiveness of the mixed model.In the discussion of Leicester’s model the dependence was placed in a framework which considered within-boardand between-board variation. The within-board variation corresponds to the kernel distribution of the mixed model,and the between-board variation corresponds to the mixing distribution. The dependence corresponds to the relativeamount of variation in the kernel and the mixing distributions.It is possible to deduce from the process which generated the data, the expected shape of the distribution of the data.For low dependence, the mixing distribution would need to have low variance, i.e. have a very narrow distribution.The resulting mixed distribution shape would correspond closely to thatof the kernel distribution. Forhigh dependence,the kernel distribution would need to have low variance. The mixed distribution shape would then correspond moreclosely to a modified version of the mixing distribution.Thereare problems involved in modelling these extreme situations. If the kernel distribution is a Weibull distribution,then it can only degenerate to a singularity at zero, so there would be problems modelling complete dependence.If the logic is reversed, then the shape of the distribution of the data is used to indicate the logic of the process whichgenerated the data. This is common practice when, for example, normal (Gaussian) distribution-shaped data is preswiied to be from a normal process.This reasoning indicates that if the distribution of the data corresponds closely to the kernel distribution, the mixedmodel fitted will correspond closely to independence. If the data are not well represented by the kernel distribution,the mixed model will interpret this as due to dependence. The set of parameters found from fitting the distribution tothe data, can be used to give an estimate of some quantifiable estimate of this dependence.It is not possible to predict how successful the model can be at inferring this dependence for several reasons:it is difficult to quantify difference in shape between the data and the kernel distribution,45V Mixed Extreme Value Models Advantages of the Mixed DisffibUt)On Model• sampling eor will change true Weibull distributions from exactly Weibull shape, which will be interpretedas due to dependence,• the true mixing distribution and mixing parameter would have to be known,• a Weibull distribution is also a mixture of other Weibull distributions, which throws doubt on the way themixed model will work. This problem and some of the relevant literature is discussed in section XflI.P.5.ft may be that the model adjusts well to the amount of dependence in data, or it may be that it has a level ofdependencewhich barely changes for different levels of dependence in the data. This will be checked using simulations, andapplication to data.F. Advantages of the Mixed Distribution ModelThe proposed mixed model has the advantage that it gives an analytical solution, and yet it allows for the presenceof dependence. It should fit a set of data generated from an extreme value process with some dependence, better thana simple extreme value model. There are two reasons for this.At least one type of dependent extreme value processes should be better modelled by these distributions than by asimple extreme value distribution. This would be the process implied by the derivation of the model. It seems intuitivelylikely that other dependent extreme value processes would be better approximated by this model than by the simple(independent) extreme value process.Secondly, the extra parameters make these distributions more flexible in shape. This gives a distribution whichshould fit a data sample better.This model only needs a set of strength data. It does not require additional information on the dependence structure,which would necessitate additional experimental work. The proposed approach relies on a statistical distribution, sousing the model on a set of data merely requires the fitting of the statistical distribution to data.G. Disadvantages of the Mixed Distribution ModelSince the mixed model infers a degree of dependence among elements in the boards from the data, this inferencemust be subject to sampling error. Thus it should be expected that the length effect predicted from this model will besubject to more error than the length effect error from a model in which the dependence is an input, and consequentlythere is no sampling error. Naturally if the dependence is input with error then the two length effect predictions mayhave similar errors.46V Mixed Extreme Value Models Summary of the Literature on Mixed DistributionsThe power of this model to infer the level of dependence from the data is unknown at this stage and will depend onthe dependence in the data. It may be that the ‘real’ mixing distribution (i.e. corresponding to the actual dependencein the data) is of a shape that would not alter the extreme value distribution significantly. In this case it is difficult orimpossible for the model to infer the correct level of dependence in the data.This model forces the user to make choices between different options,which will make a difference to the predictions,and for which little guidance can be found. For example, the choice of the mixing distribution in the model makes adifference to the success of the model. If the choice were poor, then it can be expected that too much of the strengthvariation would be left in the within-board variation component and too little would go to the between-board cornponen The inferred level of dependence would be too low, resulting in an excessive predicted length effect.It is vital for the above reasons that each board is long enough to ensure that the asymptotic extreme value distributionis reached. Otherwise the discrepancy between the actual strength distribution and the asymptotic extreme valuedistribution may be interpreted by the model as dependence and poor prediction would result. The number ofelementsthat make up a board is difficult to define. As a guide-line, low grade boards of 3m often contain 20-50 individualdefects of a size significant with respect to the size of the board (flaws described by Madsen 1991 as macroscopic).High grade boards usually contain a similar number of defects that are less obvious but which still affect strength(including flaws described by Madsen 1991 as microscopic). Asymptotic extreme value models are reasonableapproximations to the exact model for 20 or more elements (see e.g. Bury 1974), so the model has a significant chanceof working.This is possibly a pessimistic view of the mixed model. It shows that there are several ways in which it could beprevented from working. These ways provide insight into how the model should be tested.H. Summary of the Literature on Mixed DistributionsThis section covers the diverse literature covering mixed extreme value distributions as described in the first sectionsof this chapter. The term mired appears to be used in the more recent literature (e.g Marshall and 01km, 1988), whereasin earlier publications the term compound was used (e.g. Dubey, 1968).The term mixed is also used for the result of mixing variates from two or more completely separate distributions(e.g. Weibull). These are also calledfimte mixture distributions (Hill, Saunders and Laud, 1980). They are remotelyrelated to the distributions used in this work.Mixed Weibull distributions were first investigated by Dubey in 1965. who was interested in using them as generalpurpose flexible distributions. Relatively little appeared in the literature for some period.47V Mixed Extreme Value Pioc&s Summary of the Literature on Mixed DistributionsVaupel, Manton and Stallard (1979) introduced the idea of frailty distributions for use in the area of(human) lifetimedistributions. They used mixing distributions to allow for differing human frailties. Other authors proposed usingmixed Weibull distributions, because the Weibull distribution is flexible.In 1978 Clayton used multivariate Weibull distributions with a dependence tjn.jcure corresponding to mixing. Thisappeared to start a considerable amount of work in two areas: analysis of lifetimes in the epidemiologicalarea, andduration of unemployment periods in economics.One publication has mentioned the use of mixed Weibull distributions as a model for dependent univariate extremes(Leadbetter, 1982). No workable model was developed, or applied.Recent reviews have brought the frailty and muhivariate distribution areas together (Oakes 1989, Marshall and 01km1988, Tawn 1988, Hougaard 1987).The general applications of the univariate and multivariate models is discussed in Appendix 16. Individual models,and reasons for choosing them, the problems involved in using these models, and their application to extreme dataare also discussed. The important conclusions from this Appendix are:• A number of studies have used mixing to allow for heterogeneity in data (heterogeneity between boards anddependence between pieces ofa board describe the same phenomenon, but from a different viewpoint).• Most of these studies used the Weibull as a kernel function because it has a flexible shape. One author hassuggested the use of a mixed Weibull as a model for dependent extreme values. His suggested modelwouldnot provide an analytical solution, and was not applied to any practical problems.• Mixed Weibull models have two forms: a univariate form and a multivariate form. A number ofauthors havesuggested mixing models as a general form of dependence in multivariate models.• There are a number of ways that the dependence can be expressed as a functional form. These dependencefunctions can yield measures of the dependence such as the correlation coefficient between the variables.• A number of different mixing distributions have been employed. Certain limits can be put on themixingdistribution, for it to be identifiable (i.e. possible to find all the parameters uniquely). The gamma distributionprovides a mathematically tractable solution, and has certain other properties which may make it desirable.The multivariate form has been used in a number of studies.• The problem of whether the Weibull distribution mixed with the gamma is identifiable has been subject tosome argument. It appears that it is identifiable.• Methods are available for identifying the functional forms of the mixed distribution, from themixing andkernel distributions, and also the mixing distribution from the kernel and mixed distributions.• It appears that it is necessary to specify both the functional form of the kernel and mixing distributions. It istherefore difficult to find the true form of these distributions from a set of data.48V Mixed Extreme Value Models Derivation of Mixed Distributions• There is some dispute over whether incorrect specification of the kernel and mixing distributions will seriouslyaffect the success of the models.• Most workers have used maximum likelihood estimators for the parameters ofthe mixed distributions. Measuresof the dependence can also be incorporated in the estimating process.• There are considerable problems which arise when testing the model for simplifications, such as dependence.Some authors have suggested using a standard goodness-of-fit Lest.• There are indications that considerable amounts of data are necessary to accurately identify the model.I. Derivation of Mixed Distributions1. SymbolsThe following notation is used:Upper case functions: Cumulative distribution functionLower case functions: Probability density functionG: Kernel disthbutionP: Mixing distributionF: Mixed distributionPre-subscript of N Function for N elementsThus the symbol NF(x) refers to the mixed cumulative distribution function for N elements.Sometimes a function will be written NF(X ;‘l’). This is the same as NF(x), but emphasizes that the function is indexedby a set of parameters P.Elandt-Johnson (1976) found a method for finding closed-form solutions for the cumulative distribution functionfor a class of mixed distributions. The kernel distribution needs to be of the following form:G(x;tP)= 0 forxx3= I — exp(—r u(x;P)) for x x0 (2)where t is one parameter of the distribution, P is the remaining set ofparameters, and u(x;’P) is an increasing functionof x, with u -+0 as x —, x0, and u —, cc as x —, . Elandt-Johnson called this type of distribution function a ‘simpleexponential type’.49V Mixed Extreme Value Models Derivation of Mixed DistributionsThe parameter that will be allowed to vary is t. So it will now be considered as a random variable. The distributionoft (i.e. the mixing distribution) is called P,,(y;!), where ! is the set of parameters of the distribution. The momentgenerating function of the mixing distribution is M(s). If the moment generating function does not exist, this methodcannot be used.While t is not a true scale parameter according to the definition of Bury (1974)11 does allow scale changes in thekernel distribution.2. Rules for Finding New DistributionsThere are some useful results that make the derivation of mixed distributions quite straightforward. These arepresented here, but arc proven in Appendix 4. Theorem I is from Elandt-Johnson (1976). Theorem 2 is well knownand found in many textbooks such as Bury (1975). It applies for both simple and mixed distributions, and is centralto classical weakest link theory.a. Theorem 1F(x) is equal to I—Mr(s) with s replaced by —u(x,P)where —u(x, ‘I’) is part of the kernel distribution as shown in equation (12), F(x) is the mixed distribution and M1(s)is the moment generating function of the mixing distribution.b. Theorem 2If the strength distribution for unit length is F(x), the strength distribution of length N is I — [I F(x)1”, if the lengthis assumed to be made up of N independent elements.c. Corollary 1If the parental strength distribution is Weibull G (x) = I — exp(_txa) and t has some mixing distribution P(y) to givea mixed strength distribution F(x) for a unit length, then the strength distribution for length N is NF(x) = F(xN’).d. Corollary 2if the kernel strength distribution is Gumbel with G(x) = I —exp(—qexp(x/O)) and q has some mixing distributionP,,(y) to give a mixed strength distribution F(x) for a unit length, then the strength distribution for length N is,.,F(x)=F(x+OlnN).e. Theorem 3If the kernel distribution is Weibull with location parameterx0(often called the 3 parameter Weibull) so that thec.d.f. is O(x) = I —exp(—t(x —x1), then let G(x) = I —exp(--cx’) ( which is the same distribution without a locationparameter). If r is given a mixing distribution P.(y) with moment generating function Me(s) then the mixed distribution50V Mixed Extreme Value Models Derivation of Mixed DistributionsE(x) iS F(x—k). This is the same distribution as in Theorem 1, but with a location parameter. This means that all thefollowing distributions obtained by using a Weibull as a kernel distribution can be modified easily to include a locationparameter.All that needs to be done is to write (x —xe) wherever x appears in the original equation. In this thesis mixed Weibullmodels with location parameters axe not used. This is because there is no convincing evidence that there should be athreshold strength above zero. However, in an application where it is known that a location parameter is suitable thesemodels are easily found.3. Derivation of the Compound Weibull Distribution (CW)The distribution that forms the basis for much of the work in this thesis is given in the next section. Other distributionsthat proved less useful are contained in Appendix 5. These include Weibull distributions mixed with the Uniform,Log gamma, Bessel function and Beta distributions, and the Gumbel distribution mixed with the Gamma distribution.The kernel distribution is Weibull soG(x)= I —exp(--tx°) (3)which has the simple exponential form (see section 1)u(x)=xa. (4)The mixing distribution is the three parameter gamma distribution( )ex—Y)(Y’& (5)—roc)where f(K) is the gamma function and yis the location parameter, 6 is the scale parameter and c is the shape parameter.The moment generating function (see section XIll.E.l for an example of finding the moment generating function ofa distribution) is:(6)(6—Using Theorem 1 from section 2,s is replaced by —u(x) =XU in I —M(s) so the mixed distribution is:F(x;a, y, ic, 6) = I— 8cexp(_x9 (7)(8 + x°)The p.d.f. f(x) of the mixed distribution is found by differentiation:51rV Mixed Extreme Value Models Derivation of Mixed DistributionsaF(x)f(x;a,’)ciçö)= a_czexp(_)x8K + 1 (8)—(8+xx [‘ (8+x)jBy definition the Jog likelihood is:L .L. = h{flf(xi) x 11 1— F(x,)] (9)j=m—r +1where n is the total number of data points, and r. is the number of censored data points.Substituting forf(x) and F(x) from (6) and (7) gives:LL.= :iI1()+ lcln(8)+(cL— 1)ln(x— xa icln(8+x)++ [xln6—p--wln(6+x,°)]. (10)1=”—’ —1The maxim isation of this function with respect to the parameters , y, ic, 6 yields maximum likelihood estimates ofthe parameters.Assume that equations (7) and (8) are for a specimen of unit length. Then corollary I in section 2 is used to find thedistribution of x for a length N asNF(x,a,Y&)_8Kkexp(_yVxa). (11)(8 + NXa)By comparison with equation (7) it is seen that the distribution is unchanged although two of the parameters aremodified to account for the extra length:NYNT (12)N6= (13)Since the distribution form is unchanged it could be said that it is ‘closed under the length transformation’. The c.d.f.of the longer boards is related to the c.d.f. of the shorter boards byNF(x;a,y,1c8)=F(x;a,Ny,N,&N). (14)For the Built-up length effect, the elements will be independent. Under this assumption, the result of increasing thelength by a factor ii is a different set of parameters. By using Theorem 2, it is seen that the resulting c.d.f. F(x) hasparameters:52V Mixed Extreme Value Models Summary of Chapter VIt is also possible to determine the length effect suitable for the Built-up length effeci discussed in section LE.8.Assume that this model has been fitted to a set of board strength data, and it is desired to use the model to predict thestrength of longer boards made by joining n boards together end to end. The elements in a board come from a simpleextreme value model with a single value of the mixing parameter, but elements from another board come from asimple extreme value model with a different value of the mixing parameter. Therefore the length effect found above(equations (1 1)-(14)) is not suitable.The extra elements can be assumed to come from the overall distribution, notjust the simpleextreme value distributionthat the other elements came from. The effect of adding elements from the entire distribution can be found from thestraightforward application of Theorem 2. This independent-elements length effect is an upper bound of the dependentelements length effect. The independent length is equivalent to fitting a simple extreme value distribution, and usingits associated length effect. The result is:(15)w=Nw (16)with the other parameters unchanged. Since the distribution form is unchanged it could be said that it is ‘closed underthe independent length transformation’:F(x;cx.,y,ç8)=F(x;a,Ny,Nic8). (17)By setting the parameter y equal to zero, the mixing distribution becomes the two parameter gamma distribution.The resulting three parameter mixed distribution is termed ‘CW2’. The distribution, and likelihood can be found bythe same simple substitution. This distribution is considerably simpler than the full version, but less flexible.J. Summary of Chapter V• The mixed Weibull model arises naturally from the assumption that a sample of lumber is made up from a mixtureof boards from different sources.• A number of studies have used mixing models to allow for problems equivalent to the presence of dependence.• One study suggested using mixed extreme value models for extreme values from dependent processes, however thesuggested model was unworkable. The models presented in this thesis appear to be the first application of practicalsolutions.• The Weibufi distribution is the first choice for the kernel (i.e. the basic) distribution. The gamma distribution is anobvious choice for the mixing distribution, but this is more on the basis of convenience than for any other reason.This distribution has been shown to be identifiable. It was originally called the Compound Weibull distribution,and will be abbreviated CW.53V Mixed Extreme Value tLAOCteIS Symbols and Abbreviations• It is necessary to use parametric models for the kernel and mixed distributions. There are problems with comparingcompeting models, but one simple method that has been suggested is to use goodness-of-fit tests.• Virtually all studies have used the method of maximum likelihood to esümate the parameters of mixed models.• The level of dependence is not input, and the model must infer the level of dependence from the data. The power ofthe model to do this is unknown at this point. The literature indicates that a considerable amount of data may benecessary.K. Symbols and AbbreviationsCUW Weibull mixed with uniform distributionCW Weibull mixed with 3 parameter gamma distributionCW2 Weibull mixed with 2 parameter gamma distributionF(x) mixed c.d.f.NF’(X) mixed c.d.f. for N elementsF(x) mixed c.d.f. for N elements, under independent assumptionP(x) mixed c.d.f. with location parameterf(x) mixed p.d.f.G(x) kernel c.d.f.O(x) kernel c.di. with location parameterg (x) kernel p.d.f.L.L. log likelihoodMT(s) moment generating function of Tfl number of data pointsN number of unit length elements in longer boardP(x) mixing c.d.f.p(x) mixing p.d.f.number of censored data pointsu(x) part of ‘simple exponential distribution’location parametera shape parameter of Weibull, and CW distributionsparameter of gamma distributionparameter of gamma, and CW distributions8 parameter of gamma, and CW distributions54V Mixed Extreme Value Models Symbols and Abbreviationsic parameter of gamma, and CW distributionsset of parametersparameter of Weibull distribution‘I’ set of parameters55VI Using The Proposed Models Parameter EstimationVI Using The Proposed ModelsThis chapter covers the application of the proposed mixed models. It starts with a discussion about various methodsof fitting the models to data. A method for calculating the variance of the length effect predicted from the parametersfitted to a particular set of data is given. The last section shows how to derive a predicted level of dependence froma set of strength data.A. Parameter Estimation1. Estimation PrinciplesThere are at least three methods for fitting the proposed distributions to sets of data.1. The method of matching moments. This requires that a sample moment is equated to the correspondingmoment of the fitted distribution for each unknown parameter. Since the moments of the distribution areexpressions containing the parameters, a system of equations is generated which is solved to obtain estimatesof the parameters.2. The method of matching quantiles. This is very similar to method 1., but instead of matching moments thequantiles of the data and quantiles of the distribution are matched.3. The method of maximum likelihood (ML). The maximum likelihood estimates are the parameter valueswhich are most likely to yield the sample data.56VI Using The Proposed Modets Parameter EstimationThere are advantages for each method. For example, the two matching methods require relatively simple manipulationof the data, producing statistics which are effectively summaries of the data. However, because of this, they loseinformation that is available in the entire data-set. The method of matching quantiles is superior to the method ofmatching moments in one respect it is less sensitive to the effect of outliers. Maximum likelihood estimators havesome desirable properties, this is especially so for large samples. Each method, when applied in its usual form to thedistributions found in the last chapter, would yield a set of non-linear equations of equal size.Since a typical data-set size is one hundred, the maximum likelihood estimators should perform well. Cunently, themethod ofmaximum likelihood is usually used to fit the Weibull distribution to lumber strength data. For these reasons,and because the method of maximum likelihood uses all the information in the sample, this method was adopted tofit the distributions. Subsequently, a number of publications have become available which also used this method forrelated distributions (see section XIll.P.6.b).2. Maximum Likelihood MethodsThe usual implementation of this method involves the following steps (Bury 1975):1. Expressing the likelihood in terms of the parameters and the data.2. Finding the derivatives with respect to the parameters of the likelihood. This yields as many equations asparameters.3. Manipulation of the equations to yield analytical solutions or solution of the equations by numerical methods.This method was applied to some of the distributions derived in this thesis. However, the derivatives are complicatedand difficult to find. Numerical methods had ià be used for the solution of the equations and they were found to hedifficult to solve.An alternative method is to directly maximise the likelihood function, without attempting to fmd the non-linearequations. There are several alternative algorithms which can maximise functions without supplied derivatives. Since,at the beginning of the research, it was not known which distributions would be the best, these methods were consideredpreferable because they would work with any of the distribution functions when only supplied with the function itself.The derivatives of some of these distributions proved difficult or impossible to find analytically.3. Maximisation AlgorithmsSeveral different algorithms were used. The source code for each maximisation subroutine was obtained from theUniversity of British Columbia Computing Centre and each subroutine is described in Vaessen (1984). Experiencewith the different algorithms is described in section IX.B.57VI Using The Proposed Models Variance of the Estimated Distribution Parametersa. SimplexThe simplex method is based on the algorithm of Nelder and Mead (1965). Ii is based on evaluation of the functionat the vertices of a geometrical figure called a simplex’, which is moved, expanded or contracted in the search for themaximum. This is not the same algorithm as the one with the same name used in linear programming.b. PowellPowell’s method ofconjugate directions is based on searching through linearly independenidirections (Powell 1964).The name comes from the choice of directions, which alter at each iteration but are chosen to be mutually conjugate.c. FletcherThis is a variation of the variable metric method, which is a robust implementation of the well known quasi-Newtonapproach. In this version the derivatives are found using finite differences, with special attention given to the choiceof finite difference formulae and the linear search subproblem (Fletcher 1972).d. FNMINThis is very similar to the Fletcher subroutine, but it is based on supplied derivatives, instead ofnumerical derivatives.B. Variance of the Estimated Distribution ParametersIt is straightforward to find some bounds on the estimates by using simulation. This would need to be repeated foreach data-set for which bounds were needed and is a considerable amount of numerical computation. It is preferableto use analytical methods if at all possible. Because the method of maximum likelihood was used to find the estimatesof the parameters it is possible to find a lower bound on the variance of the estimates. This method unfortunately doesnot provide estimates of small sample bias which the simulation method would give.1. Cramer-Rao BoundsThe asymptotic variance-covariance matrix of the Ml. estimators for the parameters is obtained by inverting theFisher information matrix (Bury 1975). The latter is a matrix with elements which are the negative expected valuesof the second order derivatives of the log likelihood function. The elements of the variance-covariance matrix arecalled the Cramer-Rao minimum variance bounds. These variances are appropriate if the estimators are unbiased.Maximum likelihood estimators are asymptotically unbiased, so this is a reasonable approximation.The likelihood functions in the case of the proposed models are rather complicated and the process of differentiatingtwice yields a large number of terms. Since it is not even feasible to find the mean of this distribution analytically, itis not surprising that it is infeasible to find analytically the expected values of these complicated terms. Moreover,since there are so many terms it would be a daunting task to find them numerically.58VI Using The Proposed Models Variance of the Estimated Distribution Parameters2. Observed Information Matrix MethodAn alternative method, which solves of the above problems, is based on inverting the observed information matrix,rather than the Fisher information matrix. The terms ofthe observed information matrix are the second order derivativesof the log likelihood function for a particular sample.Hinkley (1978) showed that for location-scale models, that the inverse of the observed information matrix isa betterapproximation to the variance-covariance matrix than the inverse of the Fisher information matrix. Thus, using theobserved information in confidence statements, is superior to using Fisher information. Cohen (1965) and Lemon(1975) suggested using the same method for finding the variances and covariances of the maximum likelihood estimators of Weibull parameters for censored samples.Prescott and Walden (1983) used the same method for finding the variance-covariance matrix of the three parametergeneralised extreme-value distribution, a related distribution to those used in the proposed mixed model. They carriedout a simulation study in order to check the validity of the method. The procedure was to specify the distributionparameters, simulate from the distribution and then fit maximum likelihood parameters to each sample. This processyields four distinct variance-covariance matrices:1) The ‘true’ matrix directly from the simulations.2) The Fisher matrix found by averaging the results of inverting the Fisher information matrix.3) The theoretical asymptotic matrix found by using the specified parameters.4) The observed matrix found by averaging the results of inverting the observed information matrix.Prescott and Walden found that the Fisher matrix closely approximated the theoretical mathx, and, as expectedtended to underestimate the true values. The variances and covariances obtained from the observed information wereclosely distributed about the true values. They conclude that it is preferable to use the observed information to estimatethe variance-covariance matrix of the maximum likelihood estimators.3. Application of Observed Information MethodSince the observed information method has been shown theoretically to be superior for a very general distributionfamily, and shown by simulation to be superior for a closely related distribution, it was deemed to be satisfactory forapplication of the proposed models. It is also a practicable method to use for these models.The second order derivatives of the log likelihood function are found from the second derivatives of ln(f(x)) andln(1 — F(x)), using equation . These derivatives were found using symbolic computation with the program REDUCE.These were output as Fortran expressions, which were incorporated directly into a computer programme whichevaluates the variances. The derivatives were checked by using numerical derivatives.59VI Using The Proposed Models Prediction of the Correlation Coeffic4ent, and Model MomentsThe observed information matrix elements include both the maximum likelihood parameter estimates and the datavalues; this is in contrast to the Fisher information matrix elements, which include only the maximum likelihoodparameter estimates. Thus each observed information element has many terms and makes analytical inversionimpractical. When each element of the matrix had been evaluated it was inverted using a standard pivoting routine.This process has to be repeated for every data set, although the derivatives can be re-used.C. Prediction of the Correlation Coefficient, and Model MomentsThe more a model predicts correctly, the better it is. The proposed model is fitted to a set of data and then can beused to predict the strength distribution at another length. This is the main prediction of the model. As will be shownin the following, it is also possible to predict something concerning the magnitude of the dependence among boardelements from fitting the model to a set of data. The dependence is reduced to a single measure, so it can be dealt within a practical matter.One test of the proposed model is to compare the experimental correlation coefficient between the strengths of partsof a board with that predicted by the model. This test is not completely disconnected from the main prediction. Theability of the model to predict dependence is actually a measure of its sensitivity to the dependence in the data. Asthe functioning of the model depends on its sensitivity to dependence, comparison of the experimental and fittedcorrelation coefficients provides some insight into the internal workings of the model.In order to obtain this fitted correlation coefficient, it is necessary to find the mean and variance of the proposeddistributions. Since this problem was not considered elsewhere in this thesis, the methods for finding moments of themixed distributions are considered in this section. For the CW2 distribution (2 parameter Weibull mixed with a 2parameter Gamma) the moments, and the correlation coefficient can be found analytically; therefore this derivationwill be given as an example. For the other distributions, see Appendix 6.1. Correlation Coefficient of Board ElementsAs noted above the dependence may be measured by the correlation coefficient p, which is defined as— E(xy) — E(x)E(y)[var(x)var(y)]1’2cov(xy)—112(1)[var(x)var(y)]where E(x) is the expected value of x and var(x) is the variance of x.60VI Using The Proposed Models Pre&tion of the Correlation Coefficient, and Model MomentsIn this application x and y are the strength of two elements in the same board. Since the two strengths are from thesame board they must come from the same Weibull distribution (this follows from the assumptions of the theory).Elements from other boards would come from Weibull distributions with the same shape parameter cz but differentvalues of parameter t. These varying parameters have their own distribution - in this case a two parameter gammadistribution.Therefore-E(x)=E(y) (2)andvar(x) = E(x2)— (E(x))2=var(y) (3)andE(xy)=f f xyg(x,y)dxdy (4)where g(x,y) is the joint p.d.f. of x andy.For the same board, t is constant and x and y are independent. Thereforeg(x,y)=g(x)g(y). (5)SoE(xylt)=j xg(xlt)dxj yg(ylt)dy. (6)As t varies from board to board, it is necessary to integrate over all values of t with the p.d.f. of t as weightingfunction to getE(xy)= f [f xg(x t)dx j yg(y I t)dY]p(t)dt (7)where p(r) is the p.d.f. oft.ButJ xg(x I t)dx = f yg(y I=E(xlt). (8)This gives:E(xy) = f (E(x t))2p(t)dt. (9)Now, g(x) is a standard Weibull distribution so E(x I r) is straightforward (e.g. Bury 1975).61VI Using The Proposed Models Prec5ction of the Correlation Coefficient, and Model MomentsE(x lt)=tF(1/a+1) (10)and p (r) is a standard gamma p.cif. - eqn V(5) with =O.Substituting into (7)t -i Ji “18’t’exp(—&)E(xy)=J r 1]+1) F(w) dr=fHexp&-otitr+ i)riic_!)= 2(11)roc)8The terms in the bottom line of equation (1) are found next.Using equation (3)xf(x)dx. (12)Substituting forf(x) from equation V(8) with O,E(xc)= I dx-bO (6+x”=a6’xI dx..o (8+xar1(13)where B() is the beta function.So, substituting equation (13) into (12) givesvar(x)= r (1c_&)r(&+ iJ — rc—)F;+ (14)The different terms can now be substituted into equation (1), and after some manipulation this gives:1-(15)ç1j________Each term should be computed using logs so that precision problems are limited, e.g.F)=exp[2lnr()c_J_lnr(x)]. (16)62VVI Using The Proposed Models Prediction of the Corretation Coefficient and Model MomentsSome values of p cannot be computed, even with Fortran double precision.2. Finding the Parameters of the Element Strength DistributionsThe analysis that gives (15) corresponds to finding the correlation coefficient between Lest specimens cut from thesame board. It is also of interest to know the correlation coefficient between elements of a single specimen.It can be shown that the correlation coefficient is the same for specimens, and for elements from the same board.For the mixed Weibull distributions, the disthbution of an element is just the distribution of the specimen scaled bya factor depending on the relative length of specimen and element. This can be seen from the size effect law (V.1Corollary I). For the mxed Gumbel the distribution of an element is the distribution of the specimen shifted by aconstant depending on the relative size of the specimen and element (V.1 Corollary 2). Since adding a constant tovariables, or multiplying them by a factor, does not change the correlation coefficient, the correlation coefficientcalculated from (15) must be the same specimens and element.This can be shown for the Weibull as follows. As before the symbols for the board (which is a collection of Nelements ) are distinguished by the prefix N• These parameter values can be found from fitting the model to lumberstrength data. From equation (8)NE(x I t)= f xg (x r)dx=$g([xN I ‘r)dx (17)when Corollary I of section is used. Transforming the variable of integration gives:,%E(x It)=(J E(x It). (18)Similarly, from equation (9),NE(XY)=)E(xY). (19)From equation (12),(20)and, also from (12),= ()E(X2). (21)Substituting these into equation (1), gives63VI Usrng The Proposed Models Prediction of the Corr&ation Coefficient, and Model Moments(22)Thus it is concluded that the parameter values fined from the strength distributions, can be used directly to give thefitted correlation coefficient.3. Calculation of Correlation Coefficient - Alternative MethodUnfortunately, for models other than CW2, there are numerical problems with finding some of the numerical integralsin the equations given in Appendix 6 for the correlation coefficients, for models other than CW2. A slightly differentmethod is needed.The method of finding the correlation coefficient, as described in section VI.C.l above, was based on a derivationfrom the muhivariate distribution which resulted from mixing extreme value distributions. This process essentiallymixes a scale-type parameter. This can be compared to the L.eicester assumption equation (XIILN(l)), which is alsoa scale mixture. The similarity suggests that a simple expression for the correlation which can be found from theLeicester assumption may be approximately appropriate for the mixed models proposed here.From equations XJII.N(7) and XIII.N(l 1)1var(T) (23)var(S)and similarly,1[cv(T)]2 (24)[cv (S)]2The terms var(S) and cv(S) represent total variation, so they can be calculated from the variance of the fitted mixeddistribution, using equations (8) and (28) of Appendix 6.Finding var(T) and cv(T) is more complicated. The within-board variation is the variance contained in the kerneldistributions. One would expect that this variance would be difficult to find because one of the parameters for thesekernel distributions is allowed to vary according to its own distribution. In the general case, the variance would haveto be conditioned over the varying parameter. However for both of the formulations used, either the variance or thecovariance can be found as a function of the other non-varying parameter only, which obviates the need for theconditioning. Therefore a set of flued parameters yields an estimate of the within-board variation.For the Weibull the cv term can be found from XuI.N(12). For the Gumbel the variance is given by XIII.N(21).For die CW distribution the result is—(2]2 25— CL6 c,?62E1For the CG distribution the result is-64VI Using The Proposed Models Prediction of the Correlation Coefficient, and Model Moments1.6449392p=1—, ,()where 1 13,15,16 are numerical integrals defined in Appendix 6.This alternative method of caiculating p avoids the necessity for calculating E(xy). using numerical integration. Itwas found that this integration presents practical difficulties, in finding upper and lower limits withoutunderfiow oroverflow. The disadvantage of this method is that it makes the additional approximation that the mixing process isclose to the Leicester assumption. However, it was found to give correlation coefficients that were within 5% of thoseof the more rigorous equation, when they could both be calculated. Therefore, this method was used togive thecorrelation coefficients.4. Numerical Integration to Evaluate MomentsTwo methods of numerical integration were used to evaluate the integrals ‘2’ J3 15,16 which were infeasible to findanalytically.a. Fortran-based QuadratureAlthough it is known that the function dies down to near zero at finite values, the main problem is that theintegrationsextend to infinity. With problems like this it is often possible to use a transformation tochange theproblem to integrationover fixed limits. This was tried unsuccessfully. This lack of success was likely due to a singularity created at thelower integration limit.Provided that the function approaches zero at finite x, the problem reduces to finding a workable upperlimit. Thelower limit was selected as zero or the location parameter. The upper limit was selected using an iterative scheme.The starting value was the median of the relevant p.d.f., and this was multiplied by 1.5, until the p.d.f. atthe pointwas less than lxi 1O These values were subsequently checked and found to be reasonable. This relatively complicatedapproach had to be used because the functions include the gamma function. This function increases quickly andbecomes difficult to evaluate, so the upper limit must be kept workably small.The algorithm chosen uses Simpson’s rule with refined error control (it was adapted from U.B.C. MTS mainframeroutine DQUANK). This routine keeps halving the individual spacing between function evaluations until the error isbelow specified limits. Thus it will refine the intervals in the area where refinement is needed. It also provides estimatesof computation errors, which were found to be insignificant.b. Spreadsheet Approach to IntegrationThis method was tried because it was felt that the high-powered Fortran-based method used abovehas severaldisadvantages, namely:65VI Using The Proposed Models The Coefficient of Concordance1. It requires an alteration for each different integral being found because a slightly different way must be usedto find an initial guess for the upper limit.2. The user gets no feedback as to whether the procedure is working as expected.A spreadsheet-based method is a useful alternative with several potential advantages.1. Although the spreadsheet requires alterations for the different functions being integrated these are more visibleand corrections are easier to implement.2. An experienced spreadsheet user would be more comfortable with this method.3. The user can interactively check that the terminal points are satisfactory and that there are enoughintervals.4. A spreadsheet can contain explanations and be ‘self-documenting’.This method was checked against the sophisticated method described above and found to give very close answerswith about one hundred intervals. Most results described in Chapters VIII and IX were found using the Fortranmethod.However, for calculating the mean and variance of these models, the spreadsheet method has much to recommend it.D. The Coefficient of ConcordanceThe coefficient of concordance is a non-parametric measure of the dependence between two variables. It is oftenreferred to as Kendall’s r (Oakes, 1989). This is a function of the order of the data, and thus it is not altered by monotonetransformations of the data.Oakes gives an equation which can be used to find t from any Archimedean distribution as a single integral:t=4,fuØ(uW’(u)du —1. (27)The important advantage of this measure is that it does not depend on the marginal or alternatively kernel distributions.For the CW2 distribution(28)This is a very simple method for finding the mixing distribution parameter K.E. Summary of Chapter VI• The parameters of the proposed distribution can be estimated by maximising the likelihood, whichis a well-knownstatistical method.• The variance of these parameter estimates can be estimated by a relatively new method, based onobserved information.66VI Using The Proposed Models Symbols and Abbreviations• The magnitude of the dependence between elements in a board, measured with the correlation coefficient, can beestimated from the parameters fitted to a set of strength data. This dependence does not vary with the length of theelement.• The dependence between elements can also be measured with coefficient of concordance, which provides an easyestimate of a mixing distribution parameter.F. Symbols and Abbreviationscv(x) coefficient of variation of xcov(x,y) covarianceofxandyE(x) expected (mean) value of xexpected mean of x for board with N elementsf(x) p.d.f.ofxf(x I T) p.d.f. of x for a given value ofTnumerical integral results defined in Appendix 6S strength variableT variable representing between board strength variationvar(x) variance of xV (a) cv of Weibull as function of shape parameterV1(x) shape parameter of Weibull as function of cva parameter of Weibull and CW2 distributionsr(x) Gamma function of x8 parameter of gamma and CW2 distributionsparameter of gamma and CW2 distributionsp correlation coefficientNP correlation coefficient between boards of length Nparameter of Weibull distribution67VII Proposed Model For Bending And Compression Adaptation toBendingVII Proposed Model For Bending AndCompressionThis chapter covers the problems involved in extending the proposed models to loading configurations other thantension. Suitable equations are derived for bending, where the principal problem is that the stress is no longer uniformthroughout the board. The next part of the chapter covers the possibility of estimation of the length effect parameterfrom data on the position of fracture. Finally, the possible application of these models to compression is discussed.A. Adaptation to BendingIn order to adapt the proposed models to the bending case, it is necessary to extend them to non-constant stress andarbitrary size. The modification will be needed for both the Weibull-based models and for the Gumbel-basedmodels.It is shown that it is possible to fit the model to strength data from one testing configuration and predict the strengthdistribution for another configuration.In section IV.E.1 it was noted that the Built-up length effect can be modelled by the Weibull model. Size effects ina glulam beam should correspond to the Built up length effect because the elements are independent, but this is subjectto the cautions in section I.E.7. These cautions warn that the size of the defects must be small compared tothe sizeof the member. This is often the case for the depth dimension in glulam, and could be the case for the width dimension.Therefore these size effects will be incorporated in the following discussion. However, the reader should note thatthe depth and width size effects predicted by the models described beJow are not appropriate for typical sawn lumber.1. Weibull-based Model under Non-Constant StressThis model is derived from the results for the Weibull model under constant stress, given by 11(1) and 11(8). Anotherway of writing 11(8) is68OVII Proposed Model For Bendáng And Compressionptabonto BendingP{survival of unit length element) = exp(Txf) (1)where a is the strength.Even when the stress changes throughout the body, it may be assumed that it is constant for the differential volumeliv. Since N represents the volume in equation 11(1), substitute N = liv in (1) (0 get,P{survival of differential volume) = exp(livt&). (2)For a whole specimen with varying stress(i,ltwP{survival of whole body) = exp Z —livta . (3)v,1 IIn the limit, as the elements become infinitesimalP{survival of specimen) = exp[—tJ(o(x y, z)Ydv]r (L (D (H 1=ex —tj J (a(x,y,z))adzdydr (4)L zO ,=0J2=O Jwhere x, y, z are the coordinates of a point in the board measured from the front, bottom, left-hand corner in thelength (L), depth (D), and width (H) directions respectively.Lettingr=(5)(6)(7)givesa(x,y,z)=a)(r,r,,r7) (8)where a is now some characteristic stress, and c1 is the term which describes how the stress is distributed around thebody. ThereforeP { survival } = exi{—IXfLBH L=0 r,, (9)So the c.d.f for the specimen is given by,,F(a) = I — eXI{_tGLLBH L= r,,2))adrdr,dr]. (10)69VII Proposed Mod& For Bending And Compression MPt5tOflBendingIf the stress is constant then hi drdr,dr =1 and the equation reduces essentially to 11(8). Comparing these twoequations, it is found that o in 11(8) is replaced by OLBH hJJcIñLr,dr,dr in (10). Thus:(OIJLLBH $$f.afr&) (11)where a’ is the strength in the constant stress model, and a is the strength in the non-constant stress model.In the constant stress case the length effect appears in the distribution as an extra scale strengthfactor, and is afunction of the length (see section II.A). In the general case of non-constant stress and arbitrary size there is also ascale factor but it is a function of length, depth, width and the spatial stress distribution. The integral term in (11) canbe thought of as a factor relating the volume under non-constant to the equivalent volume under constant stress.a. Mixed Weibull-Based ModelsThere is no reason why the mixing process in Chapter V cannot be used with the distribution given in (10). Sincethe mixing process only affects the mixing parameter and not if, it is possible to simply take the results from mixingthe Weibull disthbution and make the replacement given above.For example, making the replacement in the CW distribution V(7) givesexrJ-’LBH $Jfcu&&,&2]F(a)=l— . (12)(i H$JJøa(j,&drx)KIt is useful that the. equation is of the same basic form as the constant stress case equation V(7), so that the otherresults from this distribution can also be used. It is convenient to use equation V(l0) for estimation instead of (12),and to distinguish the parameters of the constant stress case by the prefix N. Parameters a and Kare unchanged andthe other two parameters are given by:866 (13)BLH $fJadr&dr,3’=LHfff-t,-,, (14)where 8 and are the equivalent parameters which belong to the pure tension and unit volume case,6 and aT arethe parameters for the non-constant case.b. Methods of Use of these ModelsIt may be desired to use these models to predict the strength distributions of boards under constant stress from testsof boards under non-constant stress. The first step then is to fit the basic mixed model V(7) to the data. This providesvalues for the set of parameters for the non-constant stress case (with subscript B). The relationships given by (13)70VII Proposed Model For Bending And Compress4on Apt0uito Bendingand (14) allow the calculation of the set ofparameters for the constant stress case. The substitution of these parametersinto equation V(7) yield the prediction strength distribution. In addition, two different non-constant stress cases canbe related by using the constant stress case parameters as an intermediate stage.This process requires evaluation of the integral in equations (13) and (14). This is evaluated for two common loadingsituations in the following two sections.Other loading situations will require evaluation of the integral. However, it is still possible to find the length effectfor other configurations without evaluating the integral provided that the spatial stress distribution is identical for eachlength. Comparison of (13) and (14) with V(12) and V(13) reveals that the length effect is the same as that predictedfor the uniform tension case. Therefore, the procedure for prediction of the strength distribution at a different lengthis exactly the same as that used for the constant stress case.Another useful finding is that the parameter a which governs the length effect is the same, whatever the type of test.In principle, it should therefore be possible to estimate the length effect in tension from bending tests if the bendingspecimens fail in tension mode.c. Pure Bending for Weibull-based ModelsWhen lumber is tested in bending it is usually observed that it fails on the tension side of the board. This is nearlyuniversal for low grade lumber and is even more likely to be so for lumber in the lower tail of the strength distribution.This is the lumber which is most significant when considering the reliability of lumber because lumber in the lowertail is much more likely to fail. This justifies ignoring the probability of failure due to compression stresses.In this case the spatial stress distribution will be triangular in shape with the maximum stress at the lower edge ofthe beam. If the characteristic stress is given by the maximum stress, the stress distribution is given byI —2r, (15)and the integral term isf •f f Lø(r1r,rji”drdrdr = $ —2(a+1) (16)This is used in (13) and (14), as noted in the preceding section.d. Third Point Bending for Weibull-based ModelsA common loading method for bending specimens is third-point bending, which involves supporting the specimenat its ends and placing two equal loads at points one third of the length from the ends. For this case simple bendingtheory gives two functions for the spatial stress distribution, and symmetry means that the integral only need to becomputed for one half of the beam.71VII Proposed Model For Bending And Compression Estimation of the Shape Parameter from Fracture Position4(r,r,rj= (1 —2r,) for 1/3 <r 1/2=(1—2r,)(3r) forr 1/3. (17)Sof f f (P(r, r,, r)Ydrdr,dr= f0j(1 — 2r(3rjadrdr + — 2r,)adrxdr,]I r i ii= +—I (18)(cz+1)L3(x+1) 6fe. Centre Point Bending for Weibull-based ModelsIn a similar way the integral can be found for centre point loading.CI)(r,r,r) = (19)The integrals for the Weibull-based models are clearly very simple, and can be found for other dispositions of stressquite easily.f. Gumbel-based ModelsIn a similar way the mixed Gumbel models can be extended to different loading configurations, where there isnon-constant stress. The derivations and results are given in Appendix 7. The application of these models is morecomplicated than Weibull-based models, because each configuration has a unique distribution to fit to the data.B. Estimation of the Shape Parameter from Fracture PositionInstead of working with the Weibull strength distribution, it is possible to work with a fracture location distributionthat is compatible with the Weibull model. This distribution is available in the literature, but has not been applied inthe area of lumber strength. To allow for dependence, the distribution can be mixed.In a tension test the stress throughout the specimen is (at least nominally) uniform. Since defects are equally likelyto occur at all points along the board, the probability of fracture at all points along the board is equal. As there isnothing to differentiate one end of a board from the other, the point of fracture can be conveniently measured fromthe closest end. The p.d.f. for the fracture position is given by:for0x. (20)This p.d.f. will also be applicable where the stress varies in the width and depth directions, but not the length direction.This is true for the pure bending case. In the case where the test configuration causes stress variation in the length72VII Proposed Model For Bending And Compression Estimation of the Shape Parameter from Fracture Positiondirection this p.d.f. will no longer be applicable. If the stress is higher in the centre part of the board, then it shouldbe expected that more failures will occur in this region. The exact position of the failure still varies because elementstrength varies. This element strength variation is also the cause of the length effect. Information about the distributionof fracture position will give information about this element strength distribution, which in turn gives informationabout the length effect. In general, the parameters ofthe fracture position distribution will be functions of the parametersof the element strength distribution.1. Beam Under Central LoadingFor the fracture position approach to work best, the stresses need to change along the board as much as possible. Incentre point loading the stress in the beam varies along the entire length of the board. The other common experimentalloading set-up is third-point loading. In this method the centre third of the board has the same stress, and thus fractureis equally likely in this area. Therefore this is not providing information. Thus the most useful common testingconfiguration is centre-point loading.Trusu-um (1987), and Schultrich and Fahrmann (1984) have found the distribution for fracture position, for the centrepoint loading configuration. The steps for the derivation of this distribution are:1. Finding stress as a function of position and the maximum stress in a specimen.2. Finding the joint probability density of failure as a function of maximum stress and position.3. Integration over the y and z directions yields the joint probability density of failure as a function of stress andfracture distance along the board.4. Integration of this function with respect to the maximum stress gives the desired probability density of failurelocation.The result ish(x)=21+1.J forOx. (21)Integration with respect to x gives the c.d.f.:H(x)=jJ forOx-. (22)It is significant that this equation has only one parameter, and that this parameter is the one which dictates the lengtheffect. This is a very convenient result. Trustrum (1987) found the unique minimum variance unbiased estimator of73VII Proposed Model For Bending And Compression The Length Effect in Compression(23)The variance of this estimator is:var&=2. (24)The existence of a minimum variance unbiased estimator is also fortunate because it means that all the informationin the sample is utilised. It might be expected that information about the position of failure might be a weak sourcefor finding the magnitude of the length effect. The mathematical tools are available for utilising the information fully.Unfortunately the same cannot be said for applying this method to the Gumbel distribution. In this case, step 2. canbe achieved with increased difficulty, but the complication would appear to present extreme difficulties in proceedingto step 3.2. Extension to Dependent ElementsThe method of fracture position is easily adapted to mixed Weibull models as follows. The argument is still validfor a single value of the mixing parameter t, even if the population is a mixture. Just as it has its own strengthdistribution, each board will have its own fracture position distribution. The mixing process can be carried out on thefinal result for the p.d.f. of the fracture position x, in the same way that the strength distribution was mixed in sectionV.1.3. However, it is shown in equation (21) that the fracture position distribution is not a function of the t parameter.Therefore, the mixing process has no effect and the final equation is still equation (21). It can be concluded thatequation (21) holds regardless of the assumed mixing dependence of element strength within boards.3. ConclusionsThe fracture position information adds extra information about the magnitude of a and is easy to collect. Moreover,the estimate of a is unaffected by dependence.C. The Length Effect in CompressionIt is known that the variance of compression strength is substantially less than that of either tension or bendingstrength (Leicest.er 1985). It follows from any version of the Weibull model that the length effect will be smaller forcompression loading. As a result less emphasis has been placed on finding the length effect experimentally, or infinding suitable models. The reality is that strength in compression is more complicated because of the complicatingpresence of instability failure.74VH Proposed Model For Benng And Compression The Length Effect in CompressionSince there is not a comprehensive set of data with which to compare theoretical prediction, it is not possible toproperly evaluate any model or set of models for compression. Nevertheless, his relevant to discuss the compressionloading case and compare it to bending and tension, in order to deduce the likelihood of the proposed models alsobeing suitable for compression. Discussion of the length effect for instability problems is also relevant to bendingstrength.1. The Different Regimes for CompressionThe first regime includes compression in lengths of lumber sufficiently short and/or sufficiently well supported thatinstability failures do not occur. In this case failure is usually ductile in nature, and is described as ‘crushing failure’.Failure usually starts at knots, with fibres being crushed in a characteristic pattern often called compression creases.In a deflection controlled experiment, considerable deflection occurs as these develop, without increasing load. Theexact behaviour depends strongly on the lumber moisture content and loading rate. Final failure usually occurs becausethe defect causes wedging stresses perpendicular to the grain.The unstable regime includes compression in length of lumber sufficiently long and/or lacking in support that thefailure occurs by instability. The member bows out, producing high bending stresses throughout the member. Thiscan give brittle failure as tension failure is usually brittle. The third regime is intermediate to these two, where somefailures will be due to buckling and others due to crushing.2. Length Effect for Lumber in Crushing RegimeIt is difficult to define failure in compression because it is almost plastic in nature. Weakest link failure is usuallyassociated with brittle failure rather than ductile failure. Nevertheless, it is possible that the Weibull weakest linkmodel is appropriate, under the following assumptions.Assume that failure is defined as that point where the load/deflection curve becomes flat. It is reasonable to assumeelasto-plastic behaviour for lumber in compression (Buchanan 1984). The member load/deflection curve will becomeflat when any complete cross-section becomes plastic. Therefore the key question is whether failure ofa cross-sectiontends to occur all at once. Since compression creases are usually seen to extend all the way across the member(Williamson 1982), this is probably true. Therefore failure of the entire member is associated with failure of any ofits (length-wise) elements - as required by the Weibull weakest link theory. It appears that there may be a Weibullweakest link statistical length effect for crushing failure in compression loading.75Vii Proposed Model For Bending And Oompress4on The Length Eff in Compressofl3. Length Effect for Lumber in Buckling RegimeThe classical analysis of this situation is based on uniform (i.e. defect-free) material. Either by energy considerations,or by solving a partial differential equation the Euler buckling load can be obtained. This has the buckling loadproportional to the Young’s modulus and inversely proportional to the length squared of themember. Variousmodifications can be made to account for eccentricity of load, and other deviations from ideality.Would there be a statistical length effect, as well as the Euler length effect? For a uniform material thebuckling loadcould be expected to depend on weighted average of the stiffness of the specimen. The average stiffness will notchange with length, so it appears that there will be no statistical length effect on average. The varianceof the weightedaverage stiffness will decrease for a longer member. Therefore the variance of the buckling load will decrease forlonger members. This suggests that the lower percentiles may be slightly higher than otherwiseexpected from theEuler buckling load analysis. This is a statistical length effect opposite to that expected from the Weibull weakest linktheory.For material with substantial defects this model appears inappropriate. A failure model should reflect the fact thatfailure takes place at one point. Williamson (1991) developed a model of this type, with the member failing at a‘hinge’. A simple deterministic model would have the failure point in the middle of the specimen,because this willlead to the lowest buckling load. The buckling load will be proportional to the length (not the length2).Allowing forthe effect of changing the position of the failure gives rise to an additional ‘statistical’ length effect. It is found thatthe statistical length effect is very similar to that of the Weibull weakest link effect.4. Conclusions about Applicability to CompressionIt appears that if failure is associated with a single point, then a statistical length effect of the typeassociated withthe Weibull weakest link model should be expected. If failure occurs due to unstable bucklingof the classical typewhere the specimen bows in a smooth curve a different length effect would be expected.The mode of failure of real structures is clearly significant. Compression failures in real structuresare not commonlyseen. The most common wooden compression member is probably the wall stud. This would probablyfail in bucklingin composite action with the wall panel, which is a uniform material. Performance of the fasteners is critical andconfounds the situation.Unrestrained lumber compression members may well fail according to the hinge model described above. It is moreimportant to consider the changed buckling length effect than the statistical length effecL76VU Proposed Mode’ For Bending And Compression Summary of Chapter VIID. Summary of Chapter VII• The proposed mixed models can be applied to non-uniform stress conditions. In the Weibull-based case thisapplication is straightforward and the length effect can be analysed in the same way as in tension. For Gumbel-basedmodels this is more difficult than the tension case.• The proposed models can be applied to bending by neglecting the possibility of failure in compression.• The length effect parameter a for the Weibull model can be estimated from data on the position of fracture, if thestress changes along the board. This Fracture Position model estimate is not affected by dependence of the typeassumed.• It is expected that the proposed models work for lumber in compression where failure starts at a single point. In thecase of instability, many failures occur starting with bending of the entire specimen. A different statistical lengtheffect would be expected.E. Symbols and AbbreviationsFor Chapter VII and Appendices 5,6 and 7CG Gumbel mixed with 3 parameter gamma distributionCG2 Gumbel mixed with 2 parameter gamma distributionCUW Weibull mixed with uniform distributionCW Weibull mixed with 3 parameter gamma distributionCW2 Weibull mixed with 2 parameter gamma distributionF(x) mixed c.d.f.NF(x) mixed c.d.f. for N elementsF(x) mixed c.d.f. for N elements, under independent assumptionE(x) mixed c.d.f. with location parameterf(x) mixed p.d.f.G(x) kernel c.d.f.O(x) kernel c.d.f. with location parameterg(x) kernel p.d.f.h(x) integraliH(x) fracture position p.d.f.fracture position c.d.f.L.L. log likelihoodM,.(s) moment generating function of T77VII Proposed Model For Bending And Compression Symbols and Abbreviationsnumber of data pointsN number of unit length elements in longer boardP{) probabilityof()P(x) mixing c.d.f.p(x) mixing p.d.f.number of censored data pointsnon-dimensional measure of position in x directionF, non-dimensional measure of position in y direction/TX non-dimensional measure of position in z directionu(x) part of ‘simple exponential distribution’location parameterx distance in x directionY distance in y directionZ distance in z directiona shape parameter of Weibull, and CW distributions& estimator of a from fracture position dataparameter of gamma distribution7 parameter of gamma, and CW distributions8 parameter of gamma, and CW distributionssmall change£ parameter of gamma distributionC parameter of gamma, and CO distributionsTI parameter of Gumbel0 scale parameter of GumbelK parameter of gamma, and CW distributionsstrengtha set of parametersparameter of Weibull distributionspatial stress distribution function‘P set of parametersw parameter of CO distributionparameter of CO distribution78VUI Model Validation with Simulated Data Aim of Simulation StudyVIII Model Validation with SimulatedDataThis chapter summarises the testing of the proposed model against simulated data. It starts with describingthe purpose of using simulated data and continues with an account of validating the proposed model. Theresults are presented and conclusions are drawn. The mixed Weibull model CW is emphasized because resultsin the next chapter indicate this model is most useful. Other results are contained in Appendix 8.A. Aim of Simulation StudyThe aim of the simulation study is to show that the mixed extreme value models deal with dependence in the databetter than do the simple extreme value models. In particular, it is desired to show that the prediction of strength dataofone length from strength data obtained by testing another length, is better with the mixed model CW than the simpleWeibull model.B. Advantages of Simulation StudySimulations have a number of advantages over using real strength data.• The characteristics of the data can be known exactly, therefore it is possible to discover whether the model can dealwith data known to have dependence. This information is needed to show that the mixed model can be used inother non-timber strength applications that have dependence.• Since the level of dependence in the data is known, and the level of dependence can be predicted from the fittedmodel the ability of the model to relate to the data can be examined.• The number of simulations can be made very large, so that differences between models can be made statisticallysignificant and clear conclusions drawn.79VIII Model Vajidation with Simulated Data DeSCflPtiOfl of SimulationsThe simulated data can be used to examine different mixed models, before a fmal check with real data.C. Description of SimulationsTwo series of data-sets were simulated: one that was expected to be fitted by the Weibull-based models, and onethat was expected to be fated by the Gumbel-based models. The results from the latter are described in Appendix 8.A gamma distribution was used for the distribution ofelements. There are many disuibutions whose minima shouldalso follow a Weibull distribution (Bury 1975). This was an arbitrary decision based on die ease with which gammavariates can be generated with a given level of dependence. The algorithm used is described in Appendix 8.The simulated numbers represent the strength of elements of a board. To simulate the strength of a board of lengthI with ,i elements, n numbers are simulated and the lowest number is taken to be the strength. By doing this 100 times,a sample of 100 boards, each with n elements, is simulated. The models could then be applied to this set of data, anda prediction of the strength distribution for another length can then be made. The other length was arbitrarily set at21. The strength distribution at 21 could then be simulated by the same process as the original length, by simulating2n elements. This sample size and ratio of lengths is typical of lumber testing, and for which a prediction might beuseful. The number of elements n was varied from 20 to 500, and the correlation coefficient between elements inthesame board is varied from 0.3 to 0.7. Each set of simulations is repeated 50 times. Discussion about the reasoning forthese results is contained in Appendix 8.The fitting of the models and other details are also described in Appendix 8.D. Comparison of Predictions1. Length Effect RatioResults can come from three sources. Some came directly from the data. Some from ‘fitted’ results, arising fromfitting a model to data. ‘Predicted’ results were made by fitting the model to the data simulated with the basic numberof elements (10, 50 or 500), and predicting for boards with twice as many elements (20,100 or 1000 respectively).Three models are compared: the simple (Weibull) model, the Adapted (Leicester’s) model, and the CW (mixed) model.The models can be compared on the basis of their ability to predict the length effect. A non-dimensionalised measureis used: it is the ratio of the predicted length effect to the simulated length effect. The length effect ratio is defined asthe ratio of the average quantile difference predicted from the models, to the average simulated quantile difference.This is defined in XI1l.H(4). A perfect estimator of the length effect would give an average length effect ratio of 1.80VIII Model Validation with Simulated Data Comparison of PredictionsThe length effect ratio was calculated for the fifth and fiftieth percentiles. The fifth percentile is commonly used intimber engineering tocharacterise the strength distribution of lumber, because the lower tail is of much greater practicalimportance.Length effect ratio for 5%ile4——-- ._.G_..2: — -21.5 —1- - —-------—-0.6 I I0.2 0.3 0.4 0.5 0.6 0.7 0.8Correlation coefficient of simulated dataACTUAL (DESIRED) MIXED WEIBULL ADAPTED WEIBULL WEIBULL---b--- ----0 —-E—Figure 1: Length effect ratios for Weibull-based estimatorsIt is seen that the mixed Weibull is superior to the other models. It is important to note that a perfect model wouldnot give a length effect ratio of I from every set of simulations. Because the simulations are only repeated 50 times,they are subject to significant sampling error. The statistical significance of the findings is discussed below in sectionVIII.D.5.The relative success of the models is composed of several components. A successful model will fit the distributionof the model well, and predict the size effect well. These two aspects are examined individually below.2. Model BiasIf the model fits the simulated data well, the fitted fifth percentile will be close to the fifth percentile from the data.If it does not, then it can be the model can be said to be biased. A non-dimensional measure of bias is used, being theratio of the discrepancy between the fitted percentile and the percentile from the data, to the percentile between thedata (equation XIII.H(l) ). An ideal model would have a bias of zero, but sampling error will cause some bias inlimited size samples.81VIII Model Validation with Simulated Data Comparison ot PredictionsUnbiasing factors are available for often-used distributions such as the Weibull (Bury 1975). These were not usedfor two reasons. Since they are available for the Weibufi and not the mixed Weibull, the use of these for the Weibullwould cause comparison to be uneven. The second reason is that for sample sizes of 100 the bias is of the order of1%, which is insignificant with respect to the levels of bias found. -5%iIe bias (%)40—---30 -2010 —-1,.0 I0.2 0.3 0.4 0.5 0.6 0.7 0.8Corre’ation coefficient of simuJated dataMixed Weibull Weibull------E--Figure 2: The model biases at the 5% levelIt is seen that the mixed Weibull has very much lower bias than the Weibull distribution. Itappears that the dependencecauses considerable bias in the Weibull distribution. The mixed Weibull distribution appears highly resistant to thisbias.3. Predicted Length EffectThe length effect in the Weibull-based models depends on the length effect parameter a. For the models, parametervalues result from the estimation process. They can also be extracted from the data by taking the actual quantile fromtests of the basic length, and the quanule from tests of double length, and finding the length effect parameter thatwould have predicted this length effect. This is defined in equation XIIJ.H.(2).82VIII Model Validation with Simulated Data Companson of PredctionsLength effect parameter alpha87654321 0.8Figure 3: The magnitude of the length effect at the 5%ileThe mixed Weibull model gives a length effect of similar magnitude to that from the data. The other length effectsare too large (i.e. the parameter is too small).4. Predicted Correlation CoefficientThe parameter estimates from fitting the models give an estimate of the level of correlation, using methods describedin section VIC. The real level of correlation is known from the parameters used in the simulation process. These canbe compared.0.2 0.3 0.4 0.5 0.6 0.7Correlation coefficient of simulated data5%ile of data Mixed Weibull Adapted Weibull Weibull—4E----83VIII Model Validation with Simulated Data Comparison of PredictionsPredicted correlation coefficientI0. 4: Comparison of estimated correlation and simulated correlationThe estimated correlation in the Weibull model does not appear to be sensitive to the correlation in the data. Theestimated correlation in the Gumbel model appears to depend on the correlation in the data, but only to a small extent.5. Statistical significanceIn the sections above the superiority of the mixed models over the simple models is graphically demonstrated. Sincethere is sampling error present it is necessary to show that the differences between the size of the predicted lengtheffects are statistically significant. The variances of the estimates from the different models are relatively close whencompared to the variance of estimates from data (see Appendix I). Therefore it is most important to show that theaverage magnitude of the length effect predicted from the mixed model is significantly closer to the real length effectthan the length effect predicted from the simple model.The length effect parameter a is a highly non-linear measure of the magnitude of the length effect. Therefore a wasconvened to the predicted doubling ratio, i.e. the predicted ratio between quantiles for a doubling of length. Thevariance of this ratio was computed from the simulated sample of 50. Method 2.c of Appendix 12 was used to computethe statistical significance. The null hypothesis was that the model doubling ratio was the same as the data-baseddoubling ratio, with the alternative that the doubling ratios are different.0.2 0.4 0.6 0.8 1Correlation coefficient of simulated dataWeibull Gumbel DesiredD —--&-- ----O--•84VIII Model Validation with Simulated Data Comparison of PredictionsModel p=0.3 p=O.5 p=O.5 pO.5 p=O.7n=50/100 n=50/100 n=10/20 n=500/1000 n=50/100Mixed Weibull Accept Accept Accept Accept AcceptWeibull Reject Reject Reject Reject RejectAdapted Weibull Reject Reject Reject Reject AcceptTable 1: The results of significance tests. H0:Model parameters are equal to those from data (5% significancelevel)The CW mixed Weibull model is superior to the other models with a high level of significance. This is consistent forthe different cases.6. DiscussionThere are a number of interesting findings in the more detailed examination in Appendix 8. Some of them are onlyobserved for one of the sets of data. For example, it is shown that for gamma-based data when the length effect isinterpreted within the Weibull framework, the length effect at the 5%ile appears to differ from that at the 50%ile,when the dependence is high.Most important findings are observed for both sets of data. For increased dependence in the data, the models predictgreater length effects. In reality the length effects should decrease. This is only seen in some of the data, probablybecause the error terms are reasonably large. If the length effect is expressed as the apparent Weibull length effectparameter, the standard deviation would be approximately 1.It appears that the Gumbel model adapts to the dependence in the data to a small extent, and the Weibull does not.This suggests that the models are likely to work better for data which has a correlation coefficient in the 0.4-0.6 range.The main conclusion is that the mixed models perform better than the simple, or Adapted extreme valuemodels. There are two reasons for this. They are less likely to be affected by bias caused by dependence in thedata. They also predict the length effect more accurately, and this can be shown with an extremely high levelof confidence.The Adapted models perform better than the simple models, but suffer from the same serious bias. Theirperformance is slightly inferior to the mixed models at solely predicting the length effect, even though theyrequire the level of correlation to be known.7. Summary of Chapter VIII• Approximately 100,000 boards were simulated and the different models fitted and compared.• The simple extreme value models over-predict the length effect for the simulated data.85VIII Model Validation with Simulated Data Comparison of Pre&tions• The proposed mixed models fit simulated data better than the simple extreme value models. The fit of all modelsdeteriorates with increasing dependence.• The proposed mixed models predict the length effect belier than the simple extreme value models, or the extremevalue models with the Leicester-type adaptation.• There are anomalous effects. For example, all the models predict a higher length effect for higher dependence,whereas the real length effect should become smaller.• The probability that the proposed mixed models are superior to the simple extreme value models is extremely high.86IX Model Validation With Real DataDescription of DataIX Model Validation With Real DataThe work in this chapter is designed to evaluate how well the proposed modelswork with real data. The data andits sources is described, and the usefulness of individual data-sets is discussed. New data for experimentally measureddependence is presented. The experimental evidence for the need for the Gumbel-based models is then examined.Some of the more practical details of the fitting procedure, as applied to the real data, are then discussed. The criteriafor comparison between the different length effect models are laid out. In particularthe mixed extreme value modelsare closely compared to the simple extreme value models. Their merit is judged by comparison ofmeasures of thefollowing: closeness between the predicted and actual entire strength distributions,closeness between predicted andactual fifth and fiftieth percentiles, and closeness between predicted and model length effects. The statistical significance of the difference between the alternative models is easily calculated.A. Description of DataThe following data sets were used to check the proposed models.1. U.S.F.P.L. DataThis data is described in detail in Showalter (1986). It was obtained by special agreement from the UnitedStates Forest Products Laboratory. It comprises twelve individual data-sets: four different types of lumberand three different lengths for each different type. The cumulative frequency histograms for these data setsare shown in Figures 1-4. Data sets which differ only in specimen length areplotted together, clearly showingthe length effect.87IX Model Validation With Real Data Description ofDataData Set Grade Depth Length NumberAbbreviationM430 2250F-1.9E 4”(lOOmm) 30”(0.762m) 100M490 2250F1.9E 4” 90”(2.286m) 99M4120 2250F-1.9E 4” 120”(3.048m) 100M1030 2250F-1.9E 10”(250mm) 30” 100M1090 2250F-1.9E 10” 90” 98M10120 2250F-1.9E 10” 120” 99N430 No.2 KDI5 4” 30” 98N490 No.2 KDI5 4” 90’ 98N4120 No.2KD15 4” 120” 98N1030 No.2 KD15 10” 30” 104N1090 No.2KD15 10” 90” 104N10120 No.2KD15 10” 120” 104Table 1: U.S.F.P.L. Data sets used to check the model0.4030.20.I+ 16430FPL Machine Graded, 4 Nom.Fig.1: U.S.F.P.L. Strength Data - High Grade 4” deep c.d.f... UP.0 16490 A 16412088IX Model Validation With ReaJ Data Description of Data0. M1030FPL Machine Graded, 10 Norn.Fig.2: U.S.F.P.L. Strength Data - High Grade 10” deep c.d.f..FPL Visual Grade #2, 4 Nom. N430Fig.3: U.S.F.P.L. Strength Data - Low Grade 4” deep c.d.f..60S8.ngt UP.0 h41090 U1012000Sflnget UP.0 N400 A N412089IX Model Validation With Real Data Descnphon of Data0.90.80.70.SOsOA0.30.20.1+ N1030FPL Visual Grade #2,10 Nom.0 N1090Fig.4: LLS.F.P.L Strength Data - Low Grade 10” deep c.d.f..2. Forintek DataThis data came from Forintek Canada Corporation by special arrangement with the Canadian Wood Council, forwhom the tests were done. The tests are further described in Lam and Vamglu (1990).Only the tests carried out on select structural grade were used. The other data sets (obtained by testing grade No.2)were unsuitable, because the grade controlling defects were always included in the length inside the tension grips.Since a major proportion of the basic length effect is due to the possibility that this defect is not in the test length, theexperimental length effect measured in these data sets does not correspond to that of the other experimental data setsused in this thesis. In fact, this data represents a situation close to the graded length effect described in Chapter I. Thecorrespondence is not exact, because the selection method described above does not actually correspond to the methodsused for real production of different lengths. The cumulative frequency histograms for these data sets are shown inFigure 5.Data Set Grade Depth Length NumberAbbreviationF104 S.S. 89mm 104”(2.642m) 133F145 S.S. 89mm 145(3.683m) 133F192 S.S. 89mm 192”(4.877m) 134Table 2 : Forintek strength data sets used to check the models20 40 60 90 100 120S0.nØ UPaN1012090IX Model Validation With Real Data Descriptionof Data0.S0.50.7060.5040.3020.10Fig. 5: Forintek Strength Data c.d.f..Forintek Data, Flo F15 F193. Old U.B.C. Data- Load Configuration DataThese data sets are described in Madsen and Buchanan (1986), and come from a test series titled ‘Load Configuration’.The data sets used here to evaluate the length effect are those whose change in length corresponds to the basic lengtheffect. There are other data sets measuring a quasi-length effect due to change in the load configuration, which couldalso be used to check the models. Because this type of data is more awkward to use, it was not used to check theproposed models. L35 and L36 were tension tests with a fixed grip tension testing machine, and the rest are frombending tests. The bending tests with fixed ended conditions were not used, since the stress disthbution isnot wellestablished. The cumulative frequency histograms for these data sets used are shown in Figures 6-9. For data-sets L35and L36 a large number of boards pulled out of the grips before failure. These were taken to be censoredin this study.The method of hazard plotting (see Bury, 1975) was used to plot Fig.9.60SV.ngttl a• FlO * FIS FIS91IX Model Validation With Real Data Description of DataData Set Grade Depth Length NumberL8 S.S. 140mm 3.08m(1/3) 100L17 S.S. 140mm 4.20m(1/3) 100L14 No.2 140mm 1.54m*(1f3) 100L33 No.2 140mm 3.08m*(1/3) 10()L9 No.2 140mm 1.54m(1/3) 100L12 No.2 140mm 2.38m(1/3) 100L16 No.2 140mm 3.08m(1/3) 100L35 No.2 140mm 0.914m 100L36 No.2 140mm 3.048m 100•:Ionger graded lengthTable 3: Old U.B.C. strength data sets used to check the models0.90807090504030.20.1Old UBC Data, LB L17Figure 6: Old IJ.B.C. strength data - L8/L1760SI.ngth hFa+ 1.8 0 1.1792IX Model Validation With Real DataDescription of Data0.e010706050.40.3020.10Old UBC Data, L14 L33Figure 7: Old U.B.C. strength data - L14/L33Old UBC Data, L9 L12 L16090.80.70.605040.30.20.1+ 19 0 1.12Figure 8: Old U.B.C. strength data - L91L12/L16ha4. 1.14 0 1.3360Sl.h bFaLIS93IX Model Validation With Real Data Description of DataOQ050.70.60.5040.302010Old USC Data, L35 L36Figure 9: Old U.B.C. strength data - L351L364. New U.B.C. DataThese data sets comes from a test series carried out at the University of British Columbia in 1988. The testsarefurther described in Madsen (1989).The first four sets are from tension tests, and the last three from bending tests. Only the first three data sets areusedto evaluate the length effect in this thesis. The latter four became available too late for inclusion in the workonvalidating the models. Therefore they were only used to find the correlation coefficient between strengths of sectionsof boards (see section IX.C.2). The cumulative frequency histograms for the three data sets used in validation of theproposed models are shown in Figure 10.Data Set Grade Depth Length (load pt) NumberP5 No.2 & bir 4” O.67m 119P8 No.2 & Ni 4” 1.53m 120P16 No.2&btr 4” 3.962m 116C2G4T No.2 & Ni 4” 0A57m 134C8B3 No.2 & l,tr 4” 2.32m (mid) 120C8B4 No.2 & Ni 4” 2.25m (1/3) 80C2G2B No.2 & Ni 4” 0.78m (1/3) 135Table 4: New U.B.C. strength data sets0 20 40 60 50 100 120Se.ng0i hca* L35 13094IX Model Validation With Real Data Desenption of DataOQoe0. UBC Data, P5 P8 P16Figure 10: New U.B.C. strength data c.d.f..5. Comparisons Involving DataThe primary goal of this Chapter is to show the superiority of the models that allow for dependence over the simpleextreme models. A secondary goal is to show that the predictions from the mixed models are not significantly differentfrom the data.The desirable properties of potential length effect models were listed in section LD. The average predicted lengtheffect should be correct, and the variance of the predicted length effect about the true length effect should be assmallas possible. The models will be compared on this basis.In order to maximise the statistical power and increase the opportunity to reject the mixed models, all data-sets willbe used. This includes those which are known not to be fitted by the simple Weibull well, and those which were usedin developing the mixed models.The reader should be aware that the sampling variation of the length effects taken from the data ishigh. This isdiscussed further in section IX.D.2, but for all of the data sets considered it is higher than the samplingvariation ofthe length effect estimate from both the simple Weibull and CW models.There is an additional source of variation in the length effect estimates. The lumber comes from different sources,and therefore the (true) length effect varies from data-set to data-set. This variance is significantly larger than thesampling variance of the length effects. There are two approaches.S%ngIh.+ PS 0 P8 A P1695IX Model Validation With Real DataThe Fitting Processi. If only the sampling variance is considered in a test of statistical significance, then the conclusion applies only tothe data-sets being examined.ii. If the total variance is considered then the data-sets being examined represent a sample from all lumber. Theconclusions are being drawn about the entire population of lumber, provided that the data-sets used are arepresentativesample.B. The Fitting ProcessThere are several practical problems with fitting the models to data.1. Scaling the dataThe p.d.f.’s and c.d.f.’s of these models contain many exponentials, power functions, andlogarithms in awkwardconfigurations. Scaling of data is necessary so that under-flow and over-flow does notoccur even when doubleprecision Fortran is used. Most data sets had a maximum no more than ten times greater than the minimum. Thegreatest number which can be dealt with in evaluating the p.d.f. is approximately 50. For convenience, each data setwas scaled by the same amount, which had the result that all the data values were less than approximately 20. Theparameters presented in Appendix 11 are for data scaled to units of 10 MPa.2. Initial Estimates of the ParametersThe most difficult part of using the model is finding a set of initial parameter estimates of the parameters which willconverge to the maxim urn likelihood set. Unlike the simulation exercise, very little is known about the set ofparameterswhich is likely to be correct. It appears that the likelihood function has valleys and ridgeswhich lead the algorithmaway from the maximum value unless the starting values arc close to the solution. Most directions away from thecorrect value lead quickly to under-flow or over-flow during program execution, but some leadto local maxima whichcan be seriously in error. The best method for fitting a new set of data is to set up a batch filewhich calls the fittingprogramme many times, each time with a different set of starting parameters. These sets of starting parameters shouldhave been obtained by fitting other sets of similar data. This procedure avoids interruption in program execution.3. Comparison of Different Maximisation AlgorithmsThe algorithms described in Chapter VI were used.In general, the two algorithms which axe not based on derivatives are slower, but can be made to converge moreoften than those that are based on derivatives. The Simplex algorithm is probably the mostlikely to converge when96IX Model Validation With Real Data The Fit of the Modelsthe user has little or no information about parameters. However, it is prone to false convergence. The Powell algorithmwas almost as unlikely to fail, and was not prone to false convergence. It was also significantly faster than the Simplexalgorithm.The two algorithms that use derivatives were faster by a factor of up to 10, compared to the Simplex and Powell.However, they are more prone to the overflow/underflow problems discussed in the previous section. This wasespecially true for the CW model, although the reasons for this are not clear. As expected, the Fletcher algorithm,which uses numerical derivatives, is a little slower than DFMIN.The suggested approach is as follows. If good initial estimates are unavailable the Simplex algorithm is a goodalgorithm to use. The resulting parameter values can be checked by using the Powell algorithm, with starting valuesobtained as final answers from the Simplex algorithm. With good initial estimates, it is not necessary to use the Simplexalgorithm step. If possible, symbolic manipulation programmes should be used to find derivatives, in which caseDFMIN can be used instead of the Powell algorithm.C. The Fit of the ModelsThe relative success of the CW model compared to the Weibull model is illustrated in Figure 11, for three typicaldata sets. The log-log plot was introduced in section II.A.97r1.58lsngth mFigure 11: Model predictions on log strength vs log length plotUBC Data 0.67f1 .53/3.96m. 50%lIes a Cant. Intervals . WelbullThe Fit of the ModelsIX Model Validation With ReaJ Data6.315.013.983.162.512.001.581.266.315.013.983.164)2.51Cl,2.001.581.26I.englh m0.40 0.63 1.00 2.51 3.9698IX Model Validation With Reel DataThe Fit of the ModelsEach graph shows three lines (one line for the model prediction for each of three data sets). Each line passes throughthe fitted 5th percentile for its data set, and has a slope corresponding to the inverse of thefitted parameter a. Alsoincluded are 65% non-parametric confidence intervals for the 5th percentiles. An ideal model would have the linescollinear. Because of the sampling error some discrepancy should be expected. The following qualitative points azenoticeable in Figure 11:• The Weibull model lines are too steep.• The Weibull model lines pass close to the confidence limits at the length at which thetests were carried out.• The Weibull lines are far apart.• The CW model lines are approximately the same slope as a line through the points, areapproximately collinearand the quantiles pass closer to the middle of the confidence limits at the length at which the tests were carriedout.However, in order to show that the proposed model is superior, it is necessary to find quantified differences,and then obtain the statistical significance of theimportant differences. The following aspects of model performance will be measured:• How well the statistical distribution fits an individual data set.• The closeness of the fitted and experimental correlation coefficients.• The closeness of the predicted to actual quantiles.• The closeness of the fitted and actual quan tiles.• The closeness of the predicted and actual length effects.• The closeness of the predicted and actual strength distributions.1. Overall FitThe fit of the models can be judged subjectively by visualcomparison of the fitted distributions tothe sampledistributions. Examples of typical fitted cumulative distribution functions (c.d.f.’s) and probability distributionfunctions (p.d.f.’s) are given in figures 12 to 17 (the filled and predicted cumulative distributionfunctions in Figure13,14,15 and 17 can be compared to Figure 10, which shows the corresponding cumulative frequency histograms ofthe data).Both the mixed models allow 2 extra inflexion points more than the simple models and can be bi-modaL The complexshape of the upper tail of the CW model in Figure 12 cannot be duplicated by the Weibullmodel. Figures 14 and 15show that the Gumbel-based models allow small finite probabilities of negative strengths.This is undesirable, sincethe strength cannot be negative.99IX Model Validation With Real Data0. Fit of the Models0.4PEW of CW Distribution0.350.30.25IL0 MPaFigure 12 : Typical CW p.d.f. (New U.B.C. data)CDF of CW DistributionUBC DMa I 53m Pr.öcng 0.67m 83 96mIL00s* Mp.Figure 13 : Typical CW c.d.f.. (New U.B.C. data)100‘FIX Model Validation With Real Data0.Q0.80.70.$0.5040.,70.60.5040.3020.10The Fit of the ModelsCDF of CG Distribution50Sv.ng6MPa00Figure 14 : Typical CG c.d.t.. (New U.B.C. data)CDF Gumbel DistributionU.C06098.ngIl biPaFigure 15: Typical Gumbel c.d.f.. (New U.B.C. data)101IX Model Validation With Real Data The Fit of the Models00.45040.3503025020.150.10.05Figure 16 : Typical Weibull p.d.f. Weibull DistributionSVan UPs(New U.B.C. data)COF Welbull DistributionUSC D.Ia 1 53m Pr.cbng 0 67m 83 96mLI.CC.)Sfln UP.Figure 17 : Typical Weibull c.d.f.. (New U.B.C. data)The goodness of fit can be judged objectively by comparing the log likelihoods given in Table 5. Since these arenegative, a smaller value means a model fits more closely to data.102IX Model Validation With Real Data The Fit of the Models_______________Average Negative Log LikelihoodGumbel CO Weibull CWAverage 192.507 170.163 175.796 169.419Std Dev 26.462 20.122 19.260 19.987S.D. of 5.190 3.946 3777 3.920AverageTable 5: Negative log likelihood for different modelsThe Gunibel fits less well than the other models. For most data sets the Weibull model is not as good as the twomixed distributions CC and CW. Akaike (1974) developed a criterion for model fit (called AIC). His work impliesthat the difference in log likelihood between a 2 parameter and a 4 parameter distribution should be 2, because of thedifference in flexibility provided by the extra 2 parameters. The mixed distributions average more than 2 more thanthe relevant simple distributions. For most data sets the CW model is superior to the CG model, but by comparingthe difference between the averages to the size of the standard deviation, it is seen that this difference is not statisticallysignificant, if the likelihood is adjusted for the number of parameters.Linhart (1988) gives a way of finding the significance of the differences for each data-set. His method has theadvantage that it can be used on non-nested models. The results of applying his method to the problem of comparingthe CW model against the Weibull model is given in Table 6 (z is a standard normal test statistic). Some data-setsinclude censored data and are not included because Linhart’s method does not appear to allow for this case.In one case the Weibull model is significantly better, and in seven cases the CW model is significantly better. Theother 16 cases do not show a significant difference.It follows that the CW model is better than the Weibull model, by a degree which is greater than that expected solelydue to the increased flexibility, due to the greater number of parameters. When the results from all data-sets areconsidered together, there is a very high probability that the CW distribution is superior. This strongly suggests thatthe physical basis of the CW model is superior to the Weibull model (for this application).103IX Model Validation With Real Data The Fit of the ModelsData-set z Significance Data-set z SignificanceM430 1.56 n L8 -1.54 nM490 -0.14 a L9 0.60 flM4120 -0.048 a L12 -11.47 3’M1030 -0.18 n L14 0.80 nM1090 -0.23 a L16 0.46 flM10120 -0.87 n L17 -0.81 nN430 0.60 n L33 1.57 nN490 2.03 y P8 0.21 nN4120 2.41 y P16 1.09 nN1030 1.81 y FlO 1.48 nN1090 1.70 y F15 2.18 yN10120 3.18 y F19 2.35 yTable 6: Significance of fit for Weibull and CW models, based on likelihoods2. Correlations Inferred from the Model Fitsa. Comparisons Based on the Pearson Correlation CoefficientThere is some experimental information on the level of dependence between elements in these data-sets. For someof the data sets belonging to the new U.B.C. series, and the U.S.F.P.L. data, the test length was small enough that twoor more specimens were obtained from the same board. This means that it is possible to experimentally evaluate thecorrelation coefficient between sections of the same board.The test length of these different data tests is different, but the material is similar (data sets 2G4T and C2G2B weregraded and supplied at half the original length). As shown in section VI.C.2, the correlation coefficient betweenelements does not depend on the length of the elements. It follows that the different correlation coefficients shouldbe comparable. The results are shown in Table 7.Standard tables can be used to obtain confidence intervals for the correlation coefficient, assuming that it is fromsamples of a multi-normal population. Obviously, this assumption is not satisfied, and it is known that the distributionof the correlation coefficient is quite sensitive to non-normality (Kowalski 1971). However, derivation of a moresuitable distribution is beyond the scope of this thesis, so the normal assumption will be used as an approximation.104IX Model Validation With Real Data The Fit of the ModelsMore data is available from the New U.B.C. data. The average value is 0.63. Assuming a sample size of 80 gives95% lower and upper confidence limits of 0.47 and 0.75. It is seen in Table 7 that the data support the hypothesis thatthe correlation coefficient is the same for the different data sets.In section VI.C.3 it was shown how to determine the correlation coefficient p implied by the model. These are shownin Table 7.Data Set CW Model CG Model Experimental pp pShowalter, MG, 4” 0.310 0.742 0.444Showalter, MG, 10” 0.283 0.65 0.402Showalter, N2, 4” 0.50 1 0.938 0.831Showalter, N2, 10” 0.605 0.99 1 0.589P5 0.276 0.809 0.6 12P8 0.525 0.898 0.723P16 0.353 0.734C2G4T 0.612C8B3 0.695C8B4 0.483C2G2B 0.657avg 0.630Table 7: Fitted correlation coefficient compared with experimental correlation coefficientIt is seen that the implied correlation coefficients follow the experimental ones. However, those for the Gumbel-basedmodel appear too high and those for the Weibull-based model too low. Nevertheless, Figure 18 shows that the mixedmodels are adapting to the level of correlation in the data.090, 0.4 0.6Exp.rwnraI Corml.tian Co.8ci.ntD CWP,,&trnns + CGPr.dn.Figure 18: Relationship between fitted and experimental dependence0.2 0.8105IX Model Validation With Real Data The Fft of the ModelsThe strength of this relationship can be found by calculating the correlation coefficient between the fitted andexperimental correlation coefficients. For the CW model this is 0.50 and for the CG model this is 0.56. This showsthat these models can adjust to the level of dependence in the data.The mixed models are based on the assumption of uniform dependence between the elements of a specimen. Thisassumption can be examined with the aid of data which includes the strength of 3 elements from each specimen. Thisis available for 2 data-sets from the new U.B.C. data. The boards were cut into 3 sections. and each section was tested.There were approximately 40 specimens for each element. The results are shown in Table 8.PS Data-set C2G2Belement 1 2 3 element 1 2 31 1 0.685 0.521 1 1 0.7 17 0.6022 0.685 1 0.630 2 0.717 1 0.6533 0.521 0.630 1 3 0.602 0.653 1Table 8: Inter-element correlationsIt is seen that the correlation is inside the confidence limits mentioned above. It is not possible to reject the hypothesisthat the dependence is uniform. However, the element 1 to element 3 correlation is lower in each case, so it appearsthat the correlation may be lower at significant distances.b. Comparisons Based on the Coefficient of ConcordanceIn section IV.D, an alternative method was described for relating experimental information about the dependencebetween elements, and the fit of mixed models. CW2 distributions were fitted to the U.S.F.P.L. high grade data. Avery simple estimate of the coefficient ofconcordance t is provided by ic. The coefficientofconcordance was computedat the same time as p. The comparison is provided in Table 9.Data Set CW2 Model Experimental CW2 Model ExperimentalKendall’s t Kendall’s r p pM0430 0.285 0.337 0.527 0.444MG490 0.139 - 0.241 -MG4120 0.156 - 0.273 -MC 1030 0 0.265 0 0.402MGIO9O 0 - 0 -MGIOI2O 0.108 - 0.181 -Table 9: Inter-element coefficient of concordance106IX Model Validation With Real Data The Fit of the ModelsA 2 parameter gamma distribution is not flexible enough to model the mixing distribution and express the relativediversity of boards. The result is that estimates oft and p are too low. It is necessary to use the CW distribution, whichuses a 3 parameter gamma mixing distribution. For the CW model, the function relating the estimated distributionparameters to t is complicated.3. Bias in Estimating QuantilesThe bias can be evaluated by taking the ratio of the fitted fifth percentile to the actual fifth percentile (eqn XllI.8(l)). If the bias is zero, then for a large number of data sets the average ratio should be 1.CW Weibull Censored 3 Parm. CG GumbelWeibull WeibullAverage 5% Bias 1.030 0.859 1.022 0.994 1.023 .0.187Expected 0.00896 0.0262 0.00729 0.0133 0.0134 0.207Std Dev. of Avg BiasTable 10: Ratio of fitted fifth percentile to actual fifth percentile.It is seen in Table 10 that the mixed distributions have a much lower bias than the simple models. The CG modelappears to have a lower bias than the CW model, but since the difference is about one half of the standard deviationof the two biases, this finding is not significant. The 3 parameter Weibull distribution has the lowest bias. The negativefigure for the Gumbel model stems from the fact that the 5th percentile from the distribution is very often negative.4. Choice of Mixing DistributionMost of this chapter is devoted to the simple distributions and distributions mixed with the 3-parameter gammadistribution. In section XI1I.P.4, there is discussion indicating that the gamma distribution is the most likely candidate.A number of distributions were tried on a small sub-set of the data described in this chapter. The main criterion usedat this stage was the fit of the distribution model to the data. This indicated that the 3-parameter (rather than the2-parameter) version of the gamma distribution was needed. It also indicated that the other models described inAppendix 5 either gave a very similar quality of fit to the data and gave more difficulty in estimating parameters, orgave an inferior quality of fit. The most promising alternative model appeared to be the uniform miing distribution.Problems with estimation of the uniform-based distribution were overcome after the 3-parameter gamma had beenchosen. The uniform-based model has the advantage that it has one fewer parameter. This will cause a significantlowering of the variance of the length effect parameter, but will produce a less flexible distribution.107IX Model Validation With Real Data Evaluating the Length EffectD. Evaluating the Length Effect1. The Length Effect as a Doubling RatioThe experimental length effect and the predicted length effect can be compared in isolation. One possibility is toexpress the length effect as an equivalent Weibull shape parameter. This is not a good choice as it is a highly non-linearmeasure of the length effect. Instead, the length effect is expressed as the ratio of actual quantile for basic length topredicted quantile for twice the length. This is called the doubling ratio (1)). Equation ll:(8) givesD=2lk (1)For the data-based measures (labelled ‘Actual’ in Table 11), equation IL(8) is used first to get a corresponding tothe actual length effect. The average ratio is converted back into an equivalent shape parameter, using (1).It is also possible to fit a model to only part of the data, as long as the absence of the rest of the data is properlyallowed for. For example, fitting the Weibull to the lowest 30% of the data increases the sensitivity of the Weibullmodel to the lowest strengths of the data. Also included are the ratios obtained from the parameters found by fittingthe Weibull, and CW models to the lowest 30% of the data. These are marked ‘Cens’ in Table 11. In the CW censoredcase only a representative sample of the data sets was chosen, and the equivalent experimental length effect from thatsample is shown to its right.Doubling Ratio 5% Actual 50% W Model CW Model Ad.W W Model 3 Par.Actual Model Cens WeibullAverage 1.130 1.154 1.277 1.140 1.158 1.180 1.396Est. S.D.of 0.051 0.052 0.102 0.0076 0.0105 0.010 0.031AvgEquivalent 5.674 4.831 2.838 5.299 4.737 4.189 2.080Shape Par. aDoubling Ratio CW Cens 5%Model* Actual*Average 1.106 1.084Est.. S.D.of Avg 0.016 0.054Equivalent 6.868 8.639Shape Par. a* representative sample of data setsbTable 11: Length effects as doubling ratiosThe results ofTable 11 show that the length effect predicted by the CW model is close to that obtained experimentally.The CW model is superior to any of the other models at predicting the length effect at the fifth percentile level. Itgives both the closest average length effect and the least variation.108IX Model Validation With Real Data Evaluating the Length EffectThe significances of the differences between the various models can be obtained by using the approach used inMethod 2.b of Appendix 12. This approach rejects the null hypothesis that the Weibull and 3 parameter Weibullpredicted doubling ratios are the same as that from the data. The other models can not be rejected, mostly because thevariance of the estimate from the data is so high.The 3 parameter Weibull gives a very poor result. This may be because large samples of lumber often include veryweak boards, so a location parameter may be inappropriate. It is not considered further.It is interesting to note that both the Adapted Weibull and the censored Weibull give length effects which aresignificantly closer to the actual length effect than the simple Weibull model. If using just one part of the data givesa different result, then it follows that the data comprises a number of parts which have different characteristics. Thisassumption corresponds loosely to the assumption which formed the basis of the mixed models, i.e. that the datacomes from a mixture of sources. Thus, it is not altogether surprising that the results from fitting to censored datashould bear some relation to the results from the CW model. Censoring the CW distribution has a poor effect. Thisis probably because information is being lost as data is left out of the fitting process, and there is little to gain fromcensoring.It is also interesting to note that the standard deviation of the length effect prediction from the CW model is verymuch smaller than the standard deviation of the experimentally estimated length effect. There is no way of tellingfrom experimental evidence whether this apparent length effect variation is due to the true length effect varying, orwhether it is due to sampling variation around the true length effect. Appendix One shows that., theoretically, thestandard deviation of the model-based length effect prediction should be smaller than the standard deviation of theexperimental-based length effect for many practical situations. It appears that this has been experimentally verified.2. Individual Doubling RatiosThe Doubling Ratio is a useful way of plotting the individual results. Appendix I shows how an estimate of thevariance of the length effect parameter a can be turned into estimates of the variance of the equivalent Doubling RatioD. Groups of data-sets of the same material, but different lengths, provide estimates of D which may be compared.These are plotted in Figures 19-28, including 80% confidence intervals.These show that the estimates of the length effect found directly from the data have an excessively wide confidenceinterval. The interval from the CW model is similar to that of the Weibull model. The CW a has a much greatervariance than the Weibull a, but the higher values of CW a counteract this. The CW D values tend to fall inside theconfidence intervals from the experimental work.log— t3 J3 q53— “ “ c. ‘ “‘ c’Source of estimateFigure 19: Doubling Ratios for U.S.F.P.L. high grade 4” dataDoubling ratio D1.4 - of estimateFigure 20: Doubling Ratios for U.S.F.P.L. high grade 10” dataIX Model Validation With Real Data Evaluating the Length EffectUpper C.L.—ft Best EstimateLower C.L.UpperC.L.Best EstimateLower C.L.:f: ZIEEZZZZEExperinenta!CWModelI I I I I I110 rEvaluating the Length EffectUpper C.L.Best EstimateLower C.L.UpperC.L.—i— Best EstimateLower C.L.IX Model Validation With Real DataDoubling ratio 01.4-”±±Experir ental CW Model Weibull0.6-\O çb° (% 2.0Source of estimateFigure 21: Doubling Ratios for U.S.F.P.L. low grade 4” dataDoubling ratio D1.’’ental CW Model WeibullI I I I___,_,‘ ç’ ‘Source of estimateFigure 22: Doubling Ratios for U.S.F.P.L. low grade 10” data.01110.80.6c’ çSource of estimateFigure 24: Doubling Ratios for New U.B.C. dataH-.------- H--H---IX Model Validation With Real DataDoubling_ratio D1.41.2•10.8 -Experir ental CW Model Weibull0.6 —I I I I I I IEvaluating the Length EffectUpper C.L.Best EstimateLower C.LUpperC.L.Best EstimateLower C.L.4% çS O ,,Source of estimateFigure 23: Doubling Ratios for Forintek dataDoubling ratio D1.41.21--IExperimental / CW Model I Weibull Model--I i I - - I —-112IX Model Validation With Real Data Evaluating the Length EffectDoubling ratio D1.41.21I IUpper C.L.—f— Best Estimate -Lower C.L.0.8 —ExpenmentalCW Mode WciP!L.I I I114/133 114 133 114 133Source of estimateFigure 25: Doubling Ratios for JJ.B.C. L14/L33 dataDoubling ratio D1.41.2 —+ ±I UpperC.L.H— Best EstimateI Lower C.L.0.8 —Figure 26: Doubling Ratios for U.B.C. L9/L 121L16 data0.60.6Experim ntal CW ModelI — — I I I I I I I.Weibu II19/112 112/116 116/19 19 112 116 19 112 116Source of estimate113lX Model Vabdation With Real Data Evaluating the Length EffectDoubling ratio DUpperC.L.1.4 ——i--- Best EstimateLower C.L.1.2I__I__10.8Experimental CW Mod& Weibull0.6—I I I117/18 18 117 18 117Source of estimateFigure 27: Doubling Ratios for U.B.C. L8/L17 dataDoubling ratio DUpper C.L.1.4—f-- Best Estimate_______Lower C.L.1.2--- -----1—0.80.6 Experimenta QW ModeI I I I135/136 135 136 135 136Source of estimateFigure 28: Doubling Ratios for U.B.C. L351L36 data114IX Model Validation With ReaJ Data Evaluating the Length Effect3. Comparison of Models at a Particular QuantileAs noted in the introduction to this chapter, it is desired to measure the ability of the models to predict the strengthdistribution of lumber of a different length. One way of measuTing this ability with a single measure, is to compareat one quainile. In this thesis, this approach will be used for: the fifth percentile, which is a reasonable measure ofthe left hand tail without having too great a variance, and the median, which is useful since it has the lowest varianceand thus provides a more rigourous but less relevant test.A suitable measure of success of a particular model is the deviation between the actual quantile(i.e. from the data).and the prediction of that quantile. Squaring the deviations penalises poor predictions, and solves the problem ofopposite-signed errors cancelling. The overall success of the models is appraised by totalling the sum of these squareddeviations. The actual quartiles are non-parametric estimates, found using the methods described in section XIll.H.2.d.Predictions can be made from one data set to any other data set which only varies in the length of the specimen.These data sets are grouped in series in the description Tables I to 4. The total number of comparisons possible withthe data is 48. These comparisons are summarised in Table 12.%ile CW Weibull Adapted Censored CO Gumbel AdaptedWeibull Weibull GumbelSum of Squared 5% 2.635 6.828 4.688 4.986 3.944 290.42 227.72DeviationsS.D.of Sum 0.445 1.36 0.782 0.833 0.969 68.30 45.3Sum of Squared 50% 6.107 40.96 8.671 11.77 5.054 59.77 20.397DeviationsS.D. of Sum 1.764 9.989 1.994 2.621 1.522 11.99 4.654Table 12: Sum of squared deviations of model quantile predictions from actual quantilesIt is seen that the mixed models are better than the Adapted (Leicester) models, which are in turn better than thesimple models. The CW model is significantly better than the CO model for prediction of fifth percentiles, arid slightlyinferior for prediction of medians. The simple and Adapted Weibull models are clearly superior to the equivalentGumbel models. The order of these models is not affected by considering the sum of the absolute deviations, ratherthan the squared deviations.The significance of these deviations is examined using a method based on the sum of squared deviations havingapproximately a normal distribution (method 1 of Appendix 12). The significances of the model comparisons aregiven in Table 13.115IX Model Vahdahon With Real Data Evaluating the Length EffectModel CW Weibull Adapted Censored CG Oumbel AdaptedWeibull Weibull GumbelCV? reject reject reject reject reject rejectWeibull accept accept accept accept reject rejectAdapted Weibull accept reject reject reject reject rejectCensored Weibull accept reject reject reject reject rejectCG accept reject reject reject reject rejectGumbel accept accept accept accept accept rejectAdapted Gumbel accept accept accept accept accept rejectTable 13: Significance of comparisons in Table 12, with hypothesis H: column model is superior to row modelAll the differences in the totals are statistically significant. The CW model is significantly better than the othermodels according to this criterion.In order to make direct comparison with the Showalter model discussed in section ll.E.2.b, it is necessary to makea subset of Table 12, including only those predictions which are made in Showalter (1986). These are predictions ofthe longer lengths from the 30” sets of data in the data series described above and called U.S.F.P.L..Showalter CW WSum of Squared 0.80 0.44 3.16Deviations: 5%Standard Deviation 0.36 0.20 0.99of SumTable 14: Comparison with Showalter model, sum of squared quantile deviationsTable 14 shows that the CW model appears to be superior to the Showalter model. The statistical significance ofthese results can be found by using the same approach as above (Method I of Appendix 12). The null hypothesis thatthe Weibull model is better than the Showalter model is rejected. The probability that the Showaker model is superiorto the CW model is 0.189. The reader should remember that the CW is also superior to the Showaker model in therespect that it does not need input of the correlation coefficient.These deviations are due to three causes: random variation, model bias in estimating the quantiles, and model biasin estimating the length effect.ft can be expected that the bias found in the estimated quantiles in section IX.C.3, will also be found in the predictedquantiles. If this is the case, then the bias is the cause of a significant part of the deviations in quantiles.4. WeibuH-Based Models versus Gumbel-Based ModelsThe magnitude of the length effect can only be discussed in the context of the measure to be used. This depends onwhether a Weibull or Gumbel based models is used. It is necessary to discuss the appropriateness of these models.hRIX Model Validation With Real Data Evaluating the Length EffectThe non-experimental evidence is briefly recapitulated:1. If there is a finite left hand tail then asymptotic theory indicates that the Weibull assumption is most appropriate.2. The Gumbel model was introduced in order to explain the length effect at higher percentiles. These percentilesare obviously well away from the extreme left hand tail where the asymptotic theory is most appropriate.3. The simulations showed that dependence can also cause a difference in the length effect at different percentiles(see section VIII.C.6). Thus an alternative ‘Gumbel’ type of length effect may be due to the form of dependenceand not because the basic extreme value model is a Gumbel.The experimental evidence is as follows.1. Table 12 shows that the CG model predictions for the median are closer than those for the CW model. Thusthis alternative model does work as required.2. Some data sets seem to have a length effect corresponding to the shift expected for a Gumbel model. This mayalso be due to sampling variance being reflected in the shape of the distribution. However, in the case of theU.S .F.P.L. data from testing of machine-graded lumber (Figures 1 & 2) this type of shift is consistent. It isreasonable to suppose that the different grading process could impose a different shape on the distribution, andthat this might make the Gumbel model more appropriate.It would be interesting to know whether it is possible to find an empirical measure of the shape of the distributionthat indicates that the Gumbel model is most appropriate. It is expected that a distribution with a flatter left hand tail,which is also far from zero would be more likely to require a Gumbel-based model. Six different criteria were checked.The best performance was given by the following criterion:R— (Xc, —X5)X 2( .5X15)(Values of R are listed in Table 15.The lowest value of R for the apparent Weibull models is 0.063 ; for the apparent Gumbel models the highest is0.051. Therefore the criterion has separated the two types of data sets. Since R is solely based on measures of thestrength distribution, the conclusion is that it is possible to separate the data sets which show a Guxnbel model typelength effect, merely by considering the shape of an individual strength distribution. However, it may be that thisaltered length effect is due to a coincident change in the pattern of dependence.The group of Gumbel-type data sets includes all the machine graded data sets. It does not include the other highgrade data from the Fonntek and Old U.B.C. data sources. It appears that the tail of the distribution which resultsfrom machine grading may be of a shape which is best modelled by the Gumbel-type models.The conclusion is that it is possible that the different length effect may be due to the altered left hand tail shape,which would require a different model. In order to incorporate this into an overall model, both the CW and CG modelswould be required, along with the criterion found above. There are several arguments against this:117IX Model Validation With Real Data Evaluabng the Length EffectApparent Weibull R Apparent Gumbel RData Sets Data SetsN1030 0.2908 M4120 0.0514N1090 0.2735 M430 0.0480L17 0.2646 M490 0.0441L8 02315 M1030 0.0328N10120 0.2105 M1090 0.0210Fl5 0.2075 M10120 0.0203N430 0.1764N490 0.1764FlO 0.1680F19 0.1465L9 0.1464P5 0.1332P8 0.1253L35 0.1249N4120 0.1172L16 0.1093P16 0.08782L14 0.08568L33 0.08416L12 0.07178L36 0.06291Table 15: Criterion for choosing between Weibull and Gumbel based models1. It would approximately double the complication, because there would be 2 ‘sub’-models.2. The uncertainty associated with the criterion adds to the general uncertainty of the overall model.3. The left hand tail is the most important part of the distribution for practical purposes because it is this part ofthe distribution that dictates reliability. In this area Table 12 shows that the CW model is superior.For these reasons the Gumbel model will not be considered further.5. Testing the Fit of the Entire DistributionA final, more comprehensive test of the models, is to test the entire predicted distribution against the actual distributionusing a ‘goodness-of-fit’ criterion. Since this criterion uses all the quantiles, it utilises more information and is morepowerful than a test based on a single quantile. The test used is the Kolmogorov-Smirnov, which is based on themaximum discrepancy between the predicted c.d.f. and the sample disiribuuon of the data. This test gives a significantmaximum distance, which yields a confidence interval. The interval can be plotted as in Figures 29 and 30, which areexamples of a relatively successful prediction and unsuccessful prediction. The interpretation of these confidenceintervals is this: if the model is perfect, then all of the cumulative frequency histogram will fall inside the confidenceintervals 95 times out of 100.118IX Model Validation With Real Data Evaluating the Length EffectI0. Predicted by P540 60 80 100Strength MPaFigure 29 : Example of predicted c.d.f. and K.S. confidence intervalsApplying the Kolmogorov-Smirnov test to all the data sets, the following results are found.1. The maximum discrepancy for the CW model is less than the maximum discrepancy for the simple Weibullmodel 24 times out of 25, when the filled distribution is compared to the sample distribution of the same data.This indicates that the CW model fits the data considerably better than the Weibull model.2. The maximum discrepancy for the CW model is less than that for the simple Weibull model 42 times Out of46, when predicted distributions are compared to corresponding sample distributions. This shows that the prediction for the CW model is considerably better than for the simple model.3. The Weibull model falls outside the 90% confidence limits 38 times out of 48. The CW model falls outsidethe 90% confidence limits 14 times Out of 48. On average, a model that could predict the length effect perfectlywould fall outside the 90% limits approximately 5 times out of 48.The results can be used in hypothesis tests as indicated in Method 3 of Appendix 12. The null hypothesis that therate at which the Weibull and CW model predictions fall outside the confidence limits is the same as the expectedrate due to sampling error is rejected.0 20 120119IX Model Validation With Real Data Comparison Between Other Models and Data0. Predicted by M1012020 40 60 80 100 120Strength MPaFigure 30: Example of predicted c.d.f. and K.S. confidence intervalsThe probability that a set of length effect data is modelled by the Weibull model better than the CW model can betaken to be a binomial random variable. If the parameter p is greater than 0.5, then the Weibull could be said to bebetter than the CW model. If p is less than 0.5 then the CW model is superior. The normal approximation to thebinomial used in Method 3 of Appendix 12 can be used to derive a standard normal test statistic, for the probabilitythat p is less than 0.5. The value of Z=-6.95 implies that there is an extremely high probability that the CW model isbetter than the Weibull.E. Comparison Between Other Models and Data1. The Graded Length EffectAs defined in Chapter IV, the graded length effect is the result of changing the graded length of the board, with thetest gauge length a constant proportion of the gauged length (and in the case of non-uniform stresses, identical stressspatial distribution). The graded length effect needs to be evaluated if there is a need to fmd a relationship betweenstrength distributions from different groups of lumbers which were graded at different lengths. However, in mostcases it is also necessary to know the magnitude of the cut-down length effect, because in most cases the ratio of thegauge length to the graded length is not constant.0120IX Model Validation With Real Data Comparison Between Other Models andDataa. Method of ComparisonThe proposed model for the graded length effect in section IV.D.2 gives a relationship between the graded lengtheffect, and the cut-down length effect. The experimental data will be used to check that this relationship correspondsto that predicted by eqn IV:(33). One convenient comparison is to compare strength distributions from board groupswith identical gauge length, but which came originally from boards graded at different lengths. The difference betweenthe two distributions should be solely due to the grading at different lengths. This gives a fourth type of length effect,which can be found in terms of the cut-down length effect and the graded length effect.Let a quanWe from the strength distribution of the first group be q1, and a quantile from the second group be q2. Thegraded length of the first group is I, and that of the second is ni. If it is assumed that the gauge lengths of the twogroups were the same fraction of the original length, then the strengths of the two groups could be related using thelength effect law given by II:(l 2), with the length effect parameter equal to the graded length effect parameter a,, andN=n.q2=—j;;-• (3)flNow assume that a third group is obtained by cutting down boards from the second group so that the lengthis 1.The relationship between the strength of the third group and the second group is obtained by using the length effectlaw II:(1 1) with ci., and N = 1/n.q3 = q2,z I/a (4)Substituting IV:(33) and (3) into (4) givesq1q3= (Ii2,’•(5)This equation relates the fourth length effect (from just changing the graded length) to that from the cut-down lengtheffect a. The former length effect has the same length effect law IV:(33), with parameter(6)where c is the parameter which relates the graded length effect to the cut-down length effect. Data from the fourthlength effect can be used to estimate c, which in turn allows estimation of the graded length effect.b. Data for ComparisonIn the Old U.B.C. data there are two suitable groups of data, each with data sets of two graded lengths but otherwiseidentical and in the New U.B.C. data there are two suitable groups of data. These are listed in Table 16 and 17.121IX Model Validation With Real Data Comparison Between Other Models and DataData Set Grade Depth Length Config. Graded NumberLengthL15 S.S. 140mm 3.08m 113 Pt 4.27m 100L17 S.S. 140mm 3.08m 1/3 Pt 3.05m 100L14 No.2 140mm 3.08m 113 Pt 427m 100L16 No.2 140mm 3.08m 113 Pt 3.05m 100Table 16: Old U.B.C. data used to check the graded length effectData Set Grade Depth Length Config. Graded NumberLengthP5 No.2 & btr 89mm(4”) 0.67m Tension 4.877m 120C2G5T No.2 & tar 89mm(4”) 0.67m Tension 2.438mP8 No.2 & bir 89mm(4”) 1.53m Tension 4.877m 120C2G8T No.2 & btr 89mm(4”) 1.53m Tension 2.438mTable 17: New U.B.C. data used to check the graded length effectFor each of these pairs of data sets, the same quantile can be compared and an implied length effect parameter canbe found. Alternatively, the ‘doubling ratio’ D used in section IX.D.1 can be found. The results axe given in Table18. These quantiles can be found from the cumulative frequency histogram (i.e. the data) and also from the cumulativefrequency distribution (i.e. the fitted distribution). The doubling ratio was found for the 5th and 50th percentilesobtained by both methods. These were averaged for each data source, and then converted to an equivalent lengtheffect parameter a4.It is interesting to note that the old U.B.C. data has a doubling ratio of less than 1, which is not predictable bystatistical models. The reason is very likely to be sampling error. The difference in graded length is only 40%, andthe graded length effect is relatively small compared to the sampling error in the distributions. The new U.B.C. datacomes from experiments with the graded length different by a factor of 2, which means that the graded length effectis of a reasonable size compared to the sampling error.With an estimate of the cut-down length effect parameter, an estimate of the parameter c can be found, using (5).The cut-down length effect parameter a was taken to be 5.4 (see Table 11). The length effect parameter cz comesfrom the data in Tables 16 and 17. It is found by using equation 1I:(1 1).Because the Old U.B.C. data has a length effect opposite to that predicted, it cannot provide a sensible answer. Thisis probably because the length effect has been covered by sample variance. The new U.B.C. data gives an estimateof c which is not close to 0.8, the value estimated by the method in Appendix 0. This length effect is smaller thanthe Cut-down length effect, therefore sampling errors in samples of equal size will be have a relatively greater impacton the resulting estimates.122IX Model Validation With Real Data Comparison Between Other Models and DataData-Set Group Fit Quantile Doubling a4 ImPlied cRatioNew U.B.C. 1.040 -17.67 0.48Old U.B.C. 0.917Table 18 : Estimation of c from graded length effect data2. Results of the Fracture Position ApproachThe results of tests of 0.038x0. 140m boards in centre-point bending were reported by Madsen (1989). They are partof the testing programme which also included the data described as Old U.B.C. data in section IX.A. The position offracture of the board was recorded. Estimates of the length effect from fitting the fracture position model VII:(23) aregiven in Table 19.Data grade l/d n a var(cx)L2 s.s. 11:1 100 5.573 0.4409L3 #2 11:1 100 4.005 0.2556L5 s.s. 22:1 100 10.93 1.451LA #2 22:1 100 4.756 0.3381Table 19 : Estimation of length effect parameter from fracture position dataWhen compared to the results in Table 11, these estimates of a are seen to be of the right order of magnitude . Thevalue of 10.93 may be explained as follows. A very uniform strength material will tend to have most of the fracturesat the position of maximum stress, which is at the centre point of the board. A highly variable material will tend tohave a lower value of a (implied by eqn IV( 17)), and will be associated with more breaks away from the centre. Selectstructural material has fewer knots, and has more nearly equal tension and compression strength. Because wood ismuch less strong in the transverse direction, compression failure is encouraged by the presence of transverse loadsfrom the load head. This takes the form of highly localised softening, which strongly increases the tension stresseson the tension side of the board. Thus the transverse compression forces will tend to increase the number of boardsfailing in the centre, above that which would otherwise be expected. This will raise the apparent value of a..a. Fracture Position Model ConclusionsThe Fracture Position model is easy to fit to the data, and the position of fracture is easy to measure. The varianceof the result seems to be acceptably low. The theory indicates that the prediction is resistant to the presence ofdependence, and this is not contradicted by the results. The disadvantage is that the prediction may be poor for selectstructural lumber. More comparison with data is needed before this method can be considered validated.123IX Model Validation With Real Data DiSCUSSIOnF. DiscussionThe most important conclusion of this chapter is that in almost every respect the mixed models perform in a superiormanner to the simple models. This has been shown with a very high degree of confidence.In this chapter comparisons were made of particular arbitrary quantiles. In the next chapter comparisons will bemade of special weighted averages of the quantiles. Differences between the length effect for different grades willalso be considered in the next chapter.The conclusions reached from the simulation data can be extended to cover the length effect in real lumber. Themodels are compared on the basis of predictability of particular quantiles, entire distributions and length effects. Theproposed mixed models prove to be superior to the simple models in each of these respects, except that they are harderto fit to data.The results also indicate that a model-based approach to determine the length effect is superior to an experimentalapproach. The variance of the CW model length effect estimate is less than that found from the experimental approach.This confirms the results of the theoretical analysis.It is judged that the CG model does not provide sufficient advantages to merit the extra complication. The CW modelappears to be a good compromise for the lumber strength application. It appears to have some power to adjust to thedependence in the data. It was shown that the improvement in fit was more that could be expected due to the increasedflexibility due to the greater number of parameters. This is evidence that the CW model is superior to the Weibullmodel, and that other predictions of the CW model (such as of the length effect) are also likely Lobe superior to thoseof the Weibull model.The graded length and fracture position models show some promise, but the evidence used was insufficient to beable to test these models in any significant manner.G. Summary of Chapter IX• Data from approximately 3000 boards was obtained from 4 different test programmes, the models were fitted,predictions made and the models compared.• The mixed models are superior to the simple models in almost every respect. This can be shown with a very highdegree of confidence.• The mixed models fit a set of strength data better than the simple models. This can be shown with high probability,even when adjustment is made for the extra parameters.• The mixed models predict the length effect better than either the simple models, the simple models with the Leicesteradaptation, or the censored models. These differences are statistically significant.124IX Model Validation With Real Data Symbols and Abbreviations• The mixed Gumbel model has a weaker theoretical basis than the mixed Weibull model. It does not provide sufficientadvantages over the latter model to merit the extra complication involved in its use.• The variance of the Doubling Ratio length effect estimates from the mixed models do not appear to be higher thanthose from the simple models.• The mixed models are more difficult to fit than the simple models.• A model-based approach toestimating the length effect has a much smaller variance than that ofan experimental-basedapproach.• The model for the graded length effect and the model for estimating the length effect from fracture position showpromise, but more testing of these models is needed.H. Symbols and Abbreviationsc parameter relating a (cut-down) tosc.d.f. cumulative distribution functionCG Gumbel mixed with 3 parameter gamma mixing distributionCW Weibull mixed with 3 parameter gamma mixing distributiond depthD doubling ratioE(x) expected value of xlengthn number of data pointsp.d.f. probability distribution functionquantile from group iR criterion variable for dividing data into basic model groupss.d. standard deviationvar(x) variance of xXa ith quantile of Xa length effect parameter for Weibull-based modelsa1 a for graded length effecta4 a for constant length, but changed graded lengthp correlation coefficientsum of squared deviations of Weibull predictions from actualsum of squared deviations of CW predictions from actual125X Reliability of Single Members The Equal Reliability Length Adjustment FactorX Reliability of Single MembersThis chapter is concerned with finding the model which can best ensure equal reliability of single lumber membersof different length. The question of whether reliability of all lumber, or any particular type of lumber is adequate isnot examined.First, the equal reliability adjustment factor is defined, to give a basis upon which different models can be compared.Then the parameters and random variables for the general problem of a single lumber board under typical loads aredefined so that the reliability can be calculated. The reliabilities calculated for boards of different length yield theadjustment factors from the data.It is shown that the adjustment factors from the models can be calculated directly from the model parameters. Thesecan be compared to those from the data to find the best model.A. The Equal Reliability Length Adjustment FactorThe former chapters have considered how the strength distributions of lumber grades are affected by the length ofspecimen used. For example, in Chapter IX, the emphasis was on the effect of length on the fifth percentiles and thefiftieth percentiles. This choice of comparison at a particular percentile is based on convention. Actually the statedgoal of this thesis is to ascertain the effect of length on reliability, which depends on more than the effect of lengthon just 2 percentiles.Since lumber at lower percentiles is much more likely to cause failure, the length effect at lower percentiles is moreimportant. Thus the average length effect should be weighted towards lower percentiles. The optimal weighting schemewould be one which weighted percentiles according to their chance of causing failure.There will be an added benefit of looking at the length effect weighted over many percentiles. Cumulative frequencyhistograms from a limited sample naturally deviate from the true cumulative distribution function because of samplingvariance. The result is that the apparent length effect at a particular percentile from a particular pair of samples (of126X Reliability of Single Members MOdel Validationdifferent lengths) may have a large apparent variation from what is the true length effect. The apparent length effectaveraged over the entire strength distribution is more likely to be close to the correct value than the apparent lengtheffect at just one percentile. It follows that the weighted length effect will have a smaller variance.As noted above, one of the major goals of this thesis is to produce a factor which would ensure that the reliabilityof timber structures with members of different length stays constant. This factor is denoted E: the equal reliabilitylength adjustment factor. This would be an adjustment in the design code taking the form of a factor by which thedesign strength is divided. With E of this form, some value ofE greater than I for boards longer than a standard lengthwill ensure reliability equal to that of the standard length.The quantity E is clearly a measure of the magnitude of the length effect, since a small length effect would requirea small value of E. Moreover, E will be properly weighted if the reliability is calculated over the entire tail of thestrength distribution. Thus E will be a suitable measure of the size effect.B. Model ValidationIf the strength distributions for two different lengths are available, E might be used to validate the models as follows.The first step is to fit distributions to the two strength distributions. The second step is to assume a typical loaddistribution. Then, the reliability with each of the two load distributions is found and E calculated. This will be referredto as the data-based equal reliability length adjustment factor (ED), since the measure depends principally on the data.However, the value found will depend also on the distribution assumed for the strength distributions.The factor (ED) can be compared to the model-based equal reliability length adjustment factor EM. In this case thestrength distribution for the second length is replaced by the distribution predicted by the model from the first length.The reliabilities are calculated and EM found.A good model will have EM and ED closer than a poor model. However, even a perfect model is unlikely to have thetwo measures coincident if the data-sets are of limited size, because a sample distribution will always deviate fromthe model c.d.f. due to sampling error.C. The Formulation of the Reliability ProblemIf the reliability analyses are set up in such a way that the conclusions differed for different designs, then manydifferent designs would have to be analysed in order to be able to draw meaningful conclusions. In addition, the loaddistributions for each case would have to be derived. These problems are avoided by using the normalised approachused by Foschi and Yao (1988).127X Reliability of Single Members The Formulation of the Reliability ProblemThe stress in a particular member is related to the loads in a linear structure by an equation of the following form:T=(Df+Lf) (1)where T is the stress in the member, k is some constant depending on the structure form,A is some constant dependingon the dimensions of the member, and iL), L1 are the dead and live loads respectively. Since the dead and live loadsare random variables, the stress in the member is also a random variable.Matters are simplified by referring to the dead and live load effects, D and L respectively, which are the stresses dueto the appropriate loads, and are defined as(2)L=Lr (3)Failure occurs when the stress exceeds the strength. It is convenient to define the failure set by means of the ‘failurefunction’ G of the random variables. When a particular set of outcomes of the variables gives a value of the G whichis less than zero, then failure occurs. For the simple case which this approach deals with:G=R—(D+L), (4)where R is the (random) strength variable.The Limit State Design equation in the Canadian Timber design code CAN3-086. I -M84 has the following formatI .25D + 1 .5OL = óR0, (5)where Dais the design dead load, L, is the design live load, 6 is the performance factor, and R0 is the characteristicstrength of the lumber. The design loads are usually taken to be a high percentile from the load distribution, and thecharacteristic strength is taken to be the 5th percentile from the load distribution. The performance factor is chosenso that a target reliability is achieved.Combining (4) and (5) allows writing the failure function G in terms of modified variables which are normalised,and applicable to more than one design case.ØR05 (6)where(7)d=-, (8)128X Reliability of Single Members The Formulation of the Reliability Pmblem(9)The failure function (6) was used in this study to define the failure set for the reliability calculations, see section C.2following.1. Parameters and Variables in the Reliability ProcessThe problem has three deterministic parameters:performance factor, chosen to give the required reliability,y: dead to live load ratio, fixed at 0.25 which is a representative value (Foschi and Yao 1988).R05: characteristic strength, obtained from data. Foschi and Yao (1988) used a non-parametric 5th percentile.In this study a non-parametric 5th percentile described in section X1I1.H.1.c was used.There are three random variables:R : the lumber strength,1: the live load ratio,d : the dead load ratio.a. The Lumber StrengthAlthough the lumber strength distribution is based on actual data, the calculation of reliability is eased by fitting adistribution to the sample. For reasons of economy, lumber testing programmes are usually limited to sample sizesin the range of 100-200. Reliability computations span the lower tail of the strength distribution, and this sample sizeprovides sparse information on this tail area. Fitting a distribution to the entire data set implies that information fromthe higher percentiles affects conclusions on the lower tail; this effect depend on the strength distribution.It follows that the reliability may depend on the distribution assumed for the strength variable. For the purposes ofthis study, two alternatives were followed, in order to check that the conclusions reached did not depend on thedistribution assumption. The first alternative was the two parameter Weibull model, which is most likely to be thetwo parameter distribution best fitting lumber strength data. Two fitting options were used: fitting to all the data, andfitting to the lower 30%. The second alternative was the CW distribution which has four parameters and is clearlymore flexible than the two parameter distribution.b. The Dead Load RatioIn general, the design dead load is calculated using average sizes and densities of members and building materials.The design dead load should be the mean value of the dead load distribution, and it follows that the dead load ratio dshould have a mean of 1. Following Foschi and Yao (1988) it was assumed that d has a normal distribution with amean of I and a standard deviation of 0.1.129X Rehabihty of Single Members The Formulation of the Reliability Problemc. The Live Load RatioLive loads can arise from a number of sources, but the design of most lumber structures in Canada is governed byeither snow or occupancy loads.i. Snow LoadsSnow loads are the result of the weight of snow combined with rain that accumulates in a snow pack. The snow andrain load statistics are usually measured on the ground, and they must be adjusted to reflect the snow pack that wouldaccumulate on a roof.According to Foschi and Yao (1988) this latter adjustment is poorly documented. They follow the recommendationsgiven by Taylor and Allen (1987) of the National Research Council of Canada, and they use the proposed design snowload from the 1990 issue of the National Building Code of Canada. The result is the random variable 1 with1=rg, (10)where r is a random variable representing the variability between actual ground-to-roof conversion factor and theground-to-roof conversion factor in the code, and g is a random variable representing the variability of the total groundsnow and rain load.The distribution used in this work for r is a log-normal, with mean=0.6 and coefficient of variaiion=0.45. Thedistribution for g is Gumbel, with parameters given by Foschi and Yao (1988) suitable for Ottawa, A = 5.644 andB’ = 1.003.ii. Occupancy LoadsThe distributions obtained by Foschi and Yao (1988) are used. In that study, occupancy loads are assumed to be dueto the super-position of two live load processes: sustained and extraordinary. The sustained load is assumed to remainconstant between changes in occupancy. The extraordinary load is due to typically unforeseen causes such as crowdingduring emergencies. Changes in occupancy and extraordinary loads are assumed to occur according to a Poissonprocesses. All loads are modelled as equivalent uniformly distributed loads, although they result from a combinationof uniform loads and point loads. Both load processes are assumed to have Gamma distributions of magnitude withparameters taken from surveys.Foschi and Yao (1988) used these models for load processes to generate loads which were combined to simulate thetotal maximum load on a structure over a 30 year period. The upper 10% of the data were fitted with a Gumbeldistribution, and this is the distribution that is used in this thesis.2. Computation of ReliabilitiesComputation of the reliability implies finding the probability that a particular set of outcomes of the three (or four,for snow loads) random variables falls into the ‘safe’ set defined by equation (6). The straightforward approach to130FX Reliability of Single Members Model Prections of Reliabilitythis is to find the 3-variate p.d.f. which is then integrated over the safe set to yield the probability of failure. In thisparticular case, the three random variables are independent and the p.d.f. can be easily written down. However,inspection of the 0 function indicates that direct integration would be impractical.Various methods have been developed that fmd the reliability more easily and quickly, at the expense of requiringsome approximation. A common method used in the area of structural reliability is the Rkwitz-Fiessler algorithmwhich yields , the reliability index. The reliability index gives the approximate probability of failure, with the degreeof approximation depending on the distributions of the variables and the form of the 0 function.For this study the approximation is likely to be good. The method was used by Foschi and Yao (1988). Their computerprogram RELAN was modified to include some of the distributions used in this work.D. Model Predictions of Reliability1. Weibull-based Model-based Adjustment FactorsIn the case of the proposed Weibull-based models, finding EM is much simpler than it would at first appear. A simplemethod for finding this adjustment is obtained by comparing the predicted length effect with the length effect impliedby an adjustment factor.The model predicts that for a longer board each quantile will be reduced by the same scale factor. This factor isrelated to the shape parameter a of the Weibull-based models, and comes from equation II:(l I).C=N1’°, (11)where N is the ratio of lengths.The resulting G function is:C =R/C—+1). (12)An equivalent 0 function (i.e. defining the same failure set) is:C =R1C+1). (13)In section A, it was decided that the adjustment E would be a factor by which the characteristic strength must bedivided. The value predicted from the Weibull model will be disgnated E. The resulting 0 function is:G =RØC(ROSEW)(1) (14)1.5+ l.25TThe reliability of an adjusted longer board, and the original board is required to be the same. If the adjustment Eis equal to the predicted length effect C this will be satisfied. Thus131PX Reliability of Single Members Results of the Reliability Computations(15)2. Gumbel-based Model-based Adjustment FactorsIt was shown in section III.A.3 that the Gumbel distribuuon might fit the length effect at higher percentiles for somelumber grades, but that at lower percentiles the Weibull is more likely to be appropriate.It is illustrative to consider what difference the Gumbel model assumption would make. In this case, the modelprediction involves the entire disiribution by a constant additive term. The most appropriate adjustment in this casewould be a constant subtracted from the characteristic strength. In this case, like the Weibull case, one of the fitteddistribution parameters can be used to directly give the equal reliability length adjustment constant.EG=OlnN (16)E. Results of the Reliability ComputationsThe actual results of reliability calculations are not of direct interest. The ( parameters calculated are in Table I ofAppendix 11.1. Model ComparisonsThe objective of these computations is to compare the different models in terms of the closeness of the relevantfactors EM and E. The models to be compared are:a. The Simple Weibull Model.This is the unmodified weakest link model.b. The Censored Simple Weibull- Fitted to the Lower 30% of the Data.Foschi and Yao (1988) found that substantially different levels of reliability were obtained depending on the distribution fitted to data and used in the computations. If only the lower part of the data is fitted then these differencesdisappear. However, this occurs at some loss of information contained in the higher percentiles. This may be anadvantage, as the fit at lower percentiles may improve, and these are more directly important. For data-sets of size100, retaining the lowest 30% was taken as a reasonable compromise.The relevance of the higher percentiles is in some doubt. For example, lumber in bending can fail in compressionor tension. Most often high percentile lumber fails in compression, whereas low percentile lumber fails in tension.Since the latter are most important for purposes of computing reliability, it can be argued that the higher percentilescan be ignored.132X Reliability of Single Members Results of the Reliability Computationsc. The Adapted Weibull Model - Modified Using Leicester’s Adaptation.This model utilises the simple adaptation due to Leicester (see section IV.C.3). The model requires input of thecorrelation coefficient between parts of a board. A value of 0.6 was used for the correlation coefficient. This value isbased on the experimental evidence reported in section TX.C.2, which has values mostly varying in the 0.6-0.7 range.Other values reported in the literature are lower (see section ILE.3), therefore the lower end of the range was used.d. The CW Model.This is the model proposed in Chapter V.2. Methods of ComparisonThe model-based and data-based adjustment factors can be compared. However, it has to be emphasized that evenif a model is perfect its adjustment factor would not coincide with the data-based adjustment factors because of thesampling error in a finite data set. However, if there is enough data, then the data-based adjustment factors will averageout to the true value. Therefore, the best way ofjudging the model adjustment factors is to compare them to the averagedata-based adjustment factors.3. The Basic CaseThere axe several deterministic and several random variables in the basic G function (6). Depending on differentassumptions, different conclusions might be drawn. Therefore the approach will be to adopt a set of assumptionswhich appears to be basic, or most likely. The assumptions will be changed one at a time, in order to detect sensitivityto the corresponding assumption.Although this approach is not an exhaustive search through the possible conclusions that might be drawn, it comprisesa practical number of cases to check. If this sensitivity analysis indicates that the conclusions are not sensitive tochanges in the assumptions, then useful conclusions have been found.The basic case is taken to be1. Snow load suitable for Ottawa, plus dead load.2. Strength distribution fitted by a CW distribution.3. Target value of = 3.4. Results of the Basic CaseThe Rackwitz-Fiessler algorithm (Madsen et al,1986) with G function (6), gives a value of for a given data set,load distribution and assumed value for 9. This process is repeated fora number ofdifferent values of, and the resultis a curve as in Figure 1.133X Reliability of Single Members Resul of the Reliability ComputabonsA value of ci can be interpolated from Fig.1,for any desired value of [3. In practice asimple Newton-Raphson scheme was usedto find the desired value of.The values offor the specified value of [3 for each dataset are given inTable 1 of Appendix 11.3.7353.5343.32.23.13to the model based estimates of the length adjustment factor, EM. These axe obtained using (15). The value of a in(15) corresponds to the length effect parameter predicted by the different models described in Chpater IX. In thissection the models are compared solely on the basis of length effect prediction (and not as statistical distributionmodels - this is examined in section 6A. In Table 1 these factors are compared by summing the squared deviationsbetween the model based estimates and the data based results.Simple Model CW Adapted Model Censored Model!(E —ED)2 4.6278 2.6592 3.5389 2.7982Std Dev. Of Dev2 0.1955 0.0956 0.1420 0.1067Std Dev. Of Sum 1.3547 0.6626 0.9840 0.7393Sum Abs. Dev’s 10.417 7.8727 9.1669 8.0721S. D. Of Abs Dev 0.2244 0.1706 0.1951 0.1751Std.Dev.Of Sum 1.5548 1.1820 1.3514 1.2130Table 1: Summed squared deviations of model adjustment factors (E etc.) from data-based adj. factors Ea..Figure 1: The relation between cp and J35. Comparison of Model Length Effect PredictionsIt is seen from (14) that an estimate of ED can be found from dividing the value of p for the shorter length by ç fora longer length. This value would keep the reliability constant for the two data sets.ED=:.(17)For each pair of data sets which only differ by length, it is possible to estimate ED. These estimates can be comparedPh134X Reliabihty of Single Members Results of the Reliability ComputationsSimple CW Adapted Cens.Weibull Weibull WeibullSimple Weibull Significant N.S. SignificantCW Significant - Significant N.S.Adapted Weibull N.S. Significant - N.S.Cens. Weibull Significant N.S. N.S. -Table 2: Statistical significance of Table 1Table 1 shows that the mixed and censored model options appear to be superior to the simple and adapted models.The statistical significances are computed using Method I of Appendix 12 and given in Table 2. There are severalinteresting observations that can be made from this table:1. The margin by which the CW model is superior to the other models is much smaller than that which wasapparent from the comparison of quantiles (Table 8 of Chapter IX).2. The censored model performs better than the simple, or adapted models.3. The same conclusions are drawn when comparing the models on the basis of the sum of the absolute deviationsrather than the sum of the squared deviations.This relative lack of statistical significance may be due to the fact that sampling variance in the measure being usedis higher than the sampling variance of the measures used in the last chapter. The far left hand tail of the strengthdistribution dictates the reliability of the system. It is possible that the left hand tail of the estimated distribution isstrongly affected by the bottom two or three data points, and thus has a greater random element than, for example thefifth percentile. This effect would be greater for a very flexible distribution like the CW, than it would be for theWeibull.Another factor to consider in appraising the importance of this lack of significance is the inclusion of the variabilitybetween data-sets in the overall variance used to assess significance. This is discussed further in Appendix 12, section4.In order to get a better idea of the closeness of the relationship between the CW and the data-based adjustmentfactors, the correlation between the two was calculated. The result was 0.58, which indicates a significant relationship.135X Reliability of Single Members Results of the Reliability Computations6. Sensitivity AnalysisThe conclusions from the above analysis are checked for sensitivity to the assumptions made for the basic case insection E.3. The same procedures as above are carried out, but with the alternative assumptions altered one at a time.The results are shown in Table 3.a. Strength DistributionThe strength distribution affects the overall reliability and the adjustment factors, but does not alter the conclusions.Note that this is the distribution model assumed for the strength, not the source of the length adjustment. The adjustmentfactors appear to be altered in a consistent manner, so that if the data-sets were ranked according to the magnitude ofthe summed square deviations, this ranking would not be altered by the load distribution assumed.Simple Weibull CW Adapted Weibull Censored WeibullLive Load, CW fitSum of Dev.2 5.713 4.296 5.067 4.306Std. Dev. of Dev.2 0.2355 0.1469 0.1801 0.1486Std. Dcv. of Sum 1.632 1.018 1.248 1.030Live Load, W. fitSum of Dev.2 6.097 3.889 4.395 3.848Std. Dcv. of Dev.2 0.3035 0.1639 0.2340 0.1618Std.Dev.ofSum 2.103 1.135 1.621 1.121Snow Load, Cens.W.fitSum of Dev.2 9.650 3.344 6.586 4.758Std. Dcv. of Dev.2 0.307 1 0.09460 0.2282 0.1362Std. Dcv. of Sum 2.128 0.6554 1.581 0.9438Snow Load, W.fitSum of Dev.2 5.974 3.719 4.243 3.698Std. Dcv. of Dcv? 0.2985 0.1577 0.2282 0.1557Std. Dcv. of Sum 2.068 1.092 1.581 1.079Snow Load,CW fit1B=2.5Sum of Dcv.’ 3.702 1.287 2.294 1.535Std. Dcv. of Dcv? 0.1532 0.04963 0.1020 0.06784Std. Dcv. of Sum 1.062 0.3438 0.7066 0.4700Snow Load,CW fitB=3.5Sum of Dev. 7.083 5.823 6.480 5.774Std. Dcv. of Dev.2 0.2908 0.2268 0.2437 0.2178Std. Dcv. of Sum 2.015 1.572 1.689 1.509Table 3: Summed squared deviations for different assumptionsThere is a trend for the models to appear of relatively equal merit, under the assumption of the Weibull distribution.On the other hand, under the assumption of the CW distnbution the good models appear even better, and the poormodels appear even worse. This is consistent with the hypothesis that the reliability is highly affected by a smallnumber of low strengths. Since the Weibull model is less flexible than the CW model, its left hand tail shape is affectedmore by the higher quantiles. This lessens the ‘randomness’ in the left hand tail, and this draws the reliability-basedresults closer to those from the quantile-based results in Chapter IX.136X Rehability of Single Members Results of the Reliability Computationsb. Load Distribution AssumptionsThe load conditions which are most commonly critical for design purposes are occupancy and snow loads. Thesegive very similar results, and exactly the same conclusions can be drawn for both load conditions. It appears that theload distribution has very little effect on the length effect relationship that is found. Because of this, other loaddistributions were not examined.c Target Value of 13This value substantially alters the length adjustment factor. In general, the higher the target value 13 the greater willbe the length adjustment factor (for a given ratio of lengths). However the order of merit of the models does notchange.d. The Characteristic Strength ValueThe choice of value for the characteristic strength is open. Foschi and Yao (1988) used a non-parametric estimateof the 5%ile. For the study reported here several data-sets with different 5%ile estimates have to be given a commoncharacteristic strength. The data-set of length closest to 3m was chosen as the basic data-set, and the non-parametric5%ile was used as the characteristic strength. However, this estimate could deviate substantially from the population5%ile.The effect of altering the characteristic value is identical to changing the value. This in turn is equivalent to alteringthe target f3 value. Since this has been shown to have little effect on the conclusions, it follows that altering tuecharacteristic strength value will not change the conclusions.7. Magnitude of the Length Adjustment FactorFor these samples of lumber, it is possible to find an estimated length effect. The length adjustment factor E fromeach data set is converted to an equivalent E for doubling the length, so that results from different data sets can becompared. Comparing IX:(l) with (15), it is seen that E for doubling the length is equal to D, the doubling ratio fromChapter IX. The results are averaged across the data sets which used the same lumber and testing method, and aregiven in Table 3.It is seen that a value of 1.14 is a good estimate for the magnitude of the length effect adjustment factor, for doublingof the length.137X Rehability of Single Members Resutts of the Reliability ComputationsData Sets - Basic Case - Weibull Model - CW ModelU.S.F.P.LM4 1.16 121 1.13M10 1.19 1.17 1.11N4 0.86 1.36 1.15N1O 1.08 1A9 1.16Forintek 1.06 1.21 1.10NewU.B.C. 122 1.26 1.14Old U.B.C. bending 1.31 1.24 1.16Old u.B.c. tension 1.40 1.46 1.13Average 1.15 1.28 1.14Equivalent a 4.87 2.81 5.32Prob (High Grade Length 0.94 Very high Very highEffect is_SmalIerlAverage High Grade 1.14 1.19 1.11Average Low Grade 1.18 1.32 1.16Table 4: Comparison of Doubling Ratios DThis is a substantial adjustment as is illustrated in Figure 2. For example, tests carriedout on specimens of length im, would have tobe adjusted by 30% to obtain the strength forboards of length 4m.ISOLength MultipleFigure 2: The length adjustment factorUsing the approach of Method 2 of Appendix 12, it is also possible to calculate the probability that high grade lumberhas a lower length adjustment factor, and this is given in Table 4. This is seen to be high for the model-based results,which have a lower variance, and is seen to be significant from the data-based results. It appears that the grade isimponant, rather than whether the lumber has been machine or visually graded.It may also be possible to show that there are other factors which influence the magnitude of the adjustment factor,but the available data is inadequate for this to be done. For example, although the bending results are significantlydifferent from the tension results, all the bending results came from one data source. It follows that many details for1.4L35I .21.2$1.2E1.154138 FX Reliability of Single Members DiSCUSSIonthe bending tests may not be representative, and the differences may not represent those due to the difference betweenbending and tension alone. It is not possible, therefore, to conclude that the length effect for bending is significantlydifferent.Another point of interest is that the adjustment factor from the proposed model shows comparatively low variance.The most divergent value comes from the Forintek data. This had a very small factor between the longest and shortestspecimens, and thus the variance of the apparent length effect is very high. It may be that professional designers wouldjudge it acceptable to use a single value for the length adjustment parameter upon which to base the length adjustmentfactor.F. DiscussionThere are three important conclusions from this chapter. The proposed model and the data give an overall averagelength adjustment factor, for doubling of the length, of approximately 1.14 . Secondly, for the Weibull-based models,it is very easy to obtain an equal reliability adjustment factor from the estimated length effect parameter. Thirdly, asin the comparisons in chapters VIII and IX, the CW model gives predictions which yield adjustment factors closer tothe experimentally derived adjustment factors, than does the simple Weibuil model.The latter conclusion is unaffected by major changes in the assumptions. Fitting a censored Weibull model givesresults that are nearly as good as the CW model, and the adapted Weibull model gives results that are intermediatebetween the simple and proposed models. These findings are consistent with those found from comparing the performance of the models at predicting the distributions directly (i.e. Chapter IX).However it is harder to show that the difference between the models is statistically significant. This can be explainedby the dependence of reliability on the behaviour of quantiles in the extreme left hand tail of the strength distribution.These extreme percentiles have a higher variance and thus conclusions will have a lower statistical significance.It appears likely that the adjustment factor should be less for higher grade lumber, requiring an adjustmentapproximately 30% smaller.G. Summary of Chapter X• The CW model and the data give an overall average length adjustment factor, for doubling of the length, ofapproximately 1.14.• For the Weibull-based models, an equal reliability adjustment factor can be simply obtained from the estimatedlength effect parameter.• The CW model yields predictions of the adjustment factors closer to the experimentally derived adjustment factors,than the simple Weibull model. This conclusion is unchanged by major changes in the assumptions.139X Reliability of Single Members Symbols and Abbieviations• The Weibull model with Leicester’s adaptation, and the censored Weibull model both give better results than thesimple Weibull model. The CW models is statistically significantly better than the adapted model, and better thanthe censored Weibull model. The lack of significance of the last comparison is due to various factors raising thevariance of the measure used, because the comparison was significant in Chapter 1X.• The length effect adjustment factor may be smaller for high grade lumber than for low grade lumber, but there is noclear evidence that the type of grading (machine or visual) is important.H. Symbols and AbbreviationsA measure of dimension of boardC length adjustment factor for Weibull-based modelsCO Compound GumbelCW Compound Weibull modeld ratio of dead load to design dead loadD dead loadDr dead load effectE length adjustment factor for equal reliabilityE from Gumbel-based modelsDD doubling ratio from datadoubling ratio from modeldoubling ratio from Weibull-based modelsg ground snow load variableG failure function defining failure setk constant relating load to stressI ratio of live load to design live loadL live loadlive load effectr snow load conversion factorR strengthcharacteristic strengthreliability indexperformance factordead to live load ratio140Xl Reliability of Structural Assemblies Computation of System ReliabilityXl Reliability of StructuralAssembliesWeakest link structures form an important class of civil engineering structures. The best existing method for computing the reliability of weakest link structures uses Dit.levsen’s bounds. The proposed Length Function method isbased on extending the weakest link principle from a single member to an entire structure. The extreme value modelprovides a length function which relates member load to an effective length. This method is exact, fast and deals withboth types of dependence that may be present. This method is conveniently used to assess the differences of theestimates from the different length effect models, in the context of multi-member structures.A. Computation of System ReliabilityReal structures generally consist of many elements, and for each element there may be a number of failure modes.For real structures the reliability of all modes of each of the single members must be calculated, and these values mustbe combined to give an overall reliability.For many systems this is extremely difficult. However, systematic methods have been developed (Madsen, Krenkand Lind, 1986). First, a relation between the state of the components and the state of the structure is developed. Thisis usually done by representing the structure as an arrangement of parallel and series components, because methodsare available for these two types ofarrangemenL This relationship allows the calculation of the reliability of the systemfrom the reliability of the components.For statically indeterminate structures some components may fail before the structure fails. Relatively straightforwardcomputational procedures are available for the case where the reliabilities of the components are independent, but theamount of computation may be considerable for reasonably simple structures (Madsen et al, 1986).141XI Reliability of Structural Assemblies Series SystemsThey also note that for virtually all civil engineering structures independence of member reliabilities cannot beassumed. Fortunately, a great many real structures can be idealised as series systems, for which there are practicalsolutions, with the independence assumption relaxed.B. Series SystemsThe class of series systems includes the following structures (Madsen et al 1986):• all statically determinate structures,• statically indeterminate structures when perfect plasticity of the elements is assumed and failure is defined as thefirst failure of any element.This is a useful subset of engineering structures. Solutions for series systems are valuable tools for the calculationof reliabilities for developing structural design codes, and possibly have direct uses for design.C. TrussesTrusses are a commonly used assembly of structural elements in wooden buildings. Theyhave previously been usedas an example for the development of techniques of computing structure reliability (Foschi et al 1989). Foschi’s workwas centred on work required for a Canadian reliability-based timber design code. The work described here will focuson improving methodology, within the framework adopted by Foschi.Trusses have a variety of complex geometrical shapes and many modes of failure. A comprehensive analysis of atruss would require consideration of the semi-rigidity of the joints (including the non-linear behaviour of the connectors), gaps between adjoining members of the truss, interactions with sheathing or minor structural elements appliedto the truss, interactions with the truss supports, and many other complicating factors.Most analyses of wooden trusses use the assumption of pin joints. With common geometries this means that thestructure will be statically determinate, and therefore fall in the class of series systems.Foschi’s study focuses only on the performance of tension members comprising the truss. This means that thebehaviour of compression members and connectors is ignored.The performance of compression members can be ignored because for lumber of commonly available grades thetension strength is almost always less than the compression strength, and therefore compression failures are less likely.In addition, compression failure is much less often catastrophic than tension failure.Connectors can be increased in load capacity at relatively low cost therefore failure at connectors can be held torelatively low levels by suitable design. Failures in real trusses designed in the past may have been due to connectors,which were poorly designed, or of old technology.142Xl Reliabhity of Structural Assemblies D)fldflRecent results have confirmed these approximations. Wolfe and McCarthy (1989) found that in a sample of 24trusses, 16% failed at connections, and 16% failed in combined bearing and axial compression, the rest failing atdefects in the length of the members, presumably as a result of tension stresses.D. DependenceDependence between failure modes has at least two sources. Consider the case of an unusually high snow load ona truss roof structure. This increases the chance of breakage of an individual member, because of the higher load.However, all the members of the truss are affected, because they are simultaneously under the high load. The individualmember failures all depend on the same loads, so the reliabilities cannot be independent. Garson (1980) called thistype of dependencefuncrional correlation.The second source of dependence is related to the dependence which was discussed in Chapter III. The strength ofindividual members of a truss may be more closely related to each other than members of other trusses. Indeed, it iscommon to have the bottom chord of a truss made of one board, even though the analysis idealises the bottom chordas being made of a individual members between panels. Pieces of one board are more similar to each other, than theyare to pieces of other boards. Garson calls this dependence statistical correlation.Statistical correlauon is ignored in Foschi’s work. Initially, only functional correlation is considered in this work.E. Approximate Methods for Structures with Dependent Failure ModesExisting methods are focussed on finding the reliabilities of each component (using the load and strength distribulions), and combining them by some approximate method to give the overall reliability of the system. The approximation is necessary because functional correlation ensures that the reliabilities will be dependent. In principle it wouldbe possible to use n-dimensional numerical integration where n is the number of variables in the problem. This isusually impractical.Simple bounds are available for the obvious extreme cases ofcomplete independence (of the reliability of the differentfailure modes), and complete dependence. These bounds are generally too far apart to be of practical use.Ditlevsen (1979) obtained much closer bounds and these were used by Foschi (1989). These bounds are obtainedby considering joint probabilities of each two failure modes. Ways of finding these joint probabilities from thereliabilities and sensitivity coefficients output from the commonly used Rackwitz Fiessler algorithm were developedby Madsen et al (1986).Garson (1980) studied the effect of failure mode correlation in weakest link systems. He developed a first orderapproximation, similar to Ditlevsen’s bounds. His method only uses the bivariate joint probability of each mode withthe lowest reliability mode, rather than all bivariate joint probabilities. Consequently enors can be of the order of143Xl Retiabhty of Structural Assembties New Method for Calculating Reliability100% if the least reliable modes have approximately equal reliability. His method involves finding functionalcorrelation from moment approximations. He showed that his correction for dependence is nearly independent ofsystem size for high mode reliability, so that the adjustment from bivariate failure modes can be used for multiplefailure modes. He gives charts from which the adjustment can be read.F. New Method for Calculating ReliabilityInstead of combining the reliabilities of individual components, the proposed method combines the strength distributions of components. Since the load distribution has not entered the consideration at this stage,the functionalcorrelation has no effect. A way of combining strength distributions of components made from the samematerial canbe derived from methods developed for studies of the effect of length on material strength.Length effect studies principally concern methods of converting the strength distribution for one specimen length,to the strength distribution for another length. This means finding the quantile for the second length as afunction (thelengihfunction) of two quantities: the relevant quantile of the first length, and the ratio of the lengths. It turns out thatthis function has a single parameter and it does not change the basic shape of the distribution, although it causes ascale change (Madsen et al, 1986).When considering the strength distribution of single members it is usual to express the probability of survival as afunction of the load on the individual member. When considering the strength distribution of an entire Structure, it isconvenient to relate probabilities of survival for each member to the structure loads. This will not cause a change inthe shape of the distribution, but there will be a scale change. Members that are relatively lightly loaded (e.g. tensionchord members away from the centre of the span of a parallel chord truss) will have a distribution which is movedmore to the right than heavily loaded members.Since the effect of relative loading and the effect of length are both scale changes it is possible to find a way toequate the effect of different member lengths and relative loading. The length function can be usedto do this.The truss is a series system. The fact that it has a two (or three) dimensional shape is irrelevant, except insofar asthis means that the member loads have a different relationship to the structure loads.The proposed method is based on adjusting member lengths toaccount for the member and structure load relationshipsand then adding these effective lengths to obtain the effective length of the entire truss. This is thenconvened to givethe strength distribution for the entire truss. Once this is found, this strength distribution and the load distributionscan be used to give the reliability of the truss.144Xl Reliability of Structural Assemblies The Length FunctonG. The Length FunctionAs discussed in Chapter I the Weibull distribution is commonly used as a statistical model for strength data and thiswill be used for the examples. This is given byF(a)=1_exp(rL,aa) (1)where a is the shape parameter and t is a modified scale parameter. These are obtained by testing specimens oflength LrThe length function is given byL.1=[)xLi (2)where L. is the real length of a member, o is the stress in member i for a unit structure load,L1is the effective lengthof the member, a, is the stress to which the member is being standardised. It is Convenient to make o the stress inthe most highly loaded member for a unit structure load.The total effective length L1 of the structure is the sum of the effective lengths of each member. ThusL=ZLi. (3)The Weibull strength distribution that is used to represent the entire structure is:F(o)=1_exl{_t&UJ. (4)If the strength distribution is fitted with a Gumbel distribution (see Appendix 5) then the length function is given byL.,j=exl{0e0JxLj. (5)and a function based on the Gumbel, but an equation analagous to (4) would be used to represent the structure.H. Comparison Between Proposed Method and BoundsA Fortran program was written to fmd Ditlevsen’s bounds, as used by Foschi, in order to provide comparative results.1. MethodTwo trusses were analysed. The combined uniformly distributed dead load I) and live load Q acting on the truss,was replaced by equivalent point loads applied at the joints of the top chords. Only trusses with an even number ofpanels were analysed, resulting in symmetry which saved considerable computation.The tensile stress T1 in the members in the i-th panel (i.e. i panels from the support) is given by145XI Reliability of Structural Assemblies Comparison Between Proposed Method and BounFor the bottom chord member of the jth panel:sK.T,=.(D+Q)L( 1 2jK1=-1+-.K1 = 0 — 1)]. (8)(6)where s is the spacing between the trusses, B and H are the breadth and depth of the tension member, D and Q arethe actual dead and live loads per metre, and K1 is a geometric parameter, being the tension force in a member causedby a unit load per metre on the truss. This is equivalent to equation X(1) in the analysis of single member reliability.2. Parallel Chord Truss AnalysisThe geometry of the type of parallel chord truss that was studied is shown in Figure 1:1 2 3__N /LFigure 1: Parallel chord truss geometryFor = 45, a unit uniformly distributed load applied to the truss, K1 for the diagonal member in panel j can beexpressed as:(7)146Xl Reliability of Structural Assemblies Comparison Between Proposed Method and BoundsL—HT(j—l)forj22tanGL—=-‘ forj=l.2 tan 0— HT(i—2The formulation of structural reliability problems was described in Chapter X. In this problem, the capacity is a teststrength modified for the effect of length of the individual members. The demand is a stress, derived from the crosssection size and member load, which (in turn) comes from the structure load and geometry. The following G functionwas used3. Howe Truss AnalysisThe geometry of the type of Howe truss that was studied is shown in Figure 2:L HFigure 2: Howe truss geometryThe snow load is taken to be proportional to the projection of the roof on a horizontal plane. The influence of theangle of the roof on the snow accumulation, is ignored.For a unit uniformly distributed load applied to the truss, K1 for the bottom chord member in panelj can be expressedas:(9)For the vertical member of the ith panel:(10)4. Reliability Analysis147XI Reliabdity of Structural Assemblies Comparison Between Proposed Method and Bounds(L”?’ SK.QNGi=RrJ— HB •(yd+l). (11)where R, is the strength random variable, L’ is the reference or test length, L, is the member length, a is the lengthadjustment parameter, S is the spacing between the misses, Q is the nominal live load, H is the depth of the member,B is the width of the member, yis the dead to live load ratio, d is the ratio of dead load todesign dead load, and I isthe ratio of live load to design live load. Further explanation of this formulation is contained in Foschi et al 1989.The statistical distribution used for strength was ofWeibull type given by equation (3). This was used so that resultscould be compared to those of Foschi et al (1989). The reader should note that choice of statistical distribution isseparate from choice fo length effect model.There are a number of ways of finding the reliability of a member given the G function. A siraighforwanl approachis to integrate the muhivariate density function over the safe seL Usually this is impossible analytically, andcomput.ationally-intensive numerically. A convenient method is the Rackwitz-Fiessler alogorithm which yields , thestructural reliability index. This index gives the approximate probability of failure, with the degree of approximationdepending on the distributions of the variables and the form of the G function.For this study the approximation is likely to be good. The method was used by Foschi et al (1989), to which thereader is referred for a detailed discussion. Their computer program RELAN was slightly modified and used for thiswork.Garson’s method requires calculation of the correlation of failure modes. This was found approximately, usingmoment approximations as suggested by Garson (1980). RELAN provides numerical derivative estimates of thesensitivity of the reliability with respect to the random variables. These were used in performance moment approximations from Bury (1975).5. Comparison of Methods for an Initial TrussThe Length Function method was compared to the results of Foschi et al (1989) from the Ditlevsen bounds methodfor a single miss.Following Foschi et al (1989), the strength distribution for each element was adjusted for length using the simpleWeibull weakest link method. The distribution was adjusted from the test length of 2.642m using the length effectparameter a= 5.6.The reliabilities of the members correspond very closely to those found by Foschi et al, as seen in Table 1. Thisconfirmed that the computer codes were likely to be correct.Only a few members have significant probability of failure. The others can virtually be ignored in the computations.This depends on the geometry of the truss considered.148Xl ReIiablity of Beuctural Assemblies Comparison Between Proposed Method and BounTruss detailsThe length of the truss: 6.00 mThe depth of the truss: 0.50 mThe spacing between the misses: 2.00 mThe snow load corresponds to: ArvidaThe maximum 30-year load value: 2.240 kNj’m1The material of the truss: DFir, SS, ‘2x4’Single member results (Foschi) P (cz=5.6)Diagonals 4.330 4.3294.614 4.6134.950 4.9495.37 1 5.3705.959 5.9577.069 7.067Bottom Chords 4.893 4.8924.047 4.0463.564 3.5643.270 3.2703.107 3.107Overall reliability resultsSimple lower bound 2.707Simple upper bound 3.108Dit]evsen’s lower bound 2.725Ditlevsen’s upper bound 2.749Garson’s estimate 2.709Length function method 2.758Table I Results for one parallel chord truss149Xl Reliability of Structural Assemblies Comparison Between Proposed Method and BounAs noted above, the correlation of individual failure modes with the least reliable mode was calculated for use withGarson’s method. This varies from p=O.29 to 0.56. where p is Pearson’s moment correlation coefficienL The highestcorrelation is found with the member of equal reliability which is in the corresponding position on the opposite sideof the truss, and the least with the lightly loaded, highly reliable members.The simple bounds are not usefully close. The bound corresponding to independence is much closer to the correctanswer than the bound corresponding to complete dependence. This accords with other findings. Garson (1980) statedthat small errors result by approximating levels of correlation less than 0.7 as 0, but that even with correlationcoefficients as high as 0.99, significant errors result from using 1.0 as an approximation.Garson’s estimate is barely different from the estimate based on independence of failure modes. This is probablybecause there are two equally dominant failure modes. The error in the predicted probability of failure is well withinGarson’s maximum anticipated error for this method, which is 100%.6. Summary of Results for a Variety of TrussesA number of different trusses were analysed to examine the effectiveness of the length function method. Thesefollowed the spirit of the analysis of Foschi et al (1989), in that the same member cross section size was used for agroup of trusses of different length. Therefore a graph such as Figure 3 shows the influence of truss length on increasingmember loads, as well as the statistical length effect of decreasing strength.Data set P8 was used to provide some illustrative results, showing how the truss reliability changes with geometry.These are shown in Figures 3-6 (the significance of the three lines is described in the next section).15034333. Between Proposed Method and Bounds21.91.81.75 7I IXI Reliability of Structural AssembliesI2Truss sp m+ WeuI.adj. CWI.a. V S4ngIeIowchordFigure 3 The influence of truss length on the reliability of parallel chord trusses - O.5m deep, PSdata343. I I I3Truss .9an m+ Wb41 I.j. CW Iaq. V Iae dlordFigure 4 The influence of truss length on the reliability of parallel chord trusses - im deep, PS data5____ 7151Xl Rehabibty of Structural Assemblies Comparison Between Proposed Method and BoundsTruss span m+ W.M i.sq A cw sq. v Sne $ower dard3 5 7343.3323.13.2.9282.7a 26IFigureS The influence of truss length on the reliability of Howe trusses .25 degree slope, P8 data34333. 2.6I II I ITop wd anØ. dsg..s+ w.ci t.aq cw .sq. v sr. • ardFigure 6 The influence of top chord angle on the reliability of Howe trusses - 6m span, PS dataThe following points can be noticed from Figures 3-6, and Table I in Appendix 18.A35 37 39152Xl Reliability of Structural Assemblies Assessment of Length Effect Models• The length function method gives results that are very close to that from Ditlevsen’s bounds. The differences are small enough that design would not be significantly altered (e.g. of the order of 0.2% change inmember depth).• As expected, the reliability of the trusses decreases with span, increases with depth, and increases withroof slope.• For highly reliable Irusses, Ditlevsen’s bounds are very close, and although the length function methodgives answers which are relatively close to the Ditlevsen bounds, the length function results fall above theupper bound, i.e. predict a reliability higher than Ditlevsen’s bounds.• For less reliable trusses, the Ditlevsen bounds are further apart and the length function answer falls insidethe bounds.It was not expected that the new method would predict reliabilities outside Ditievsen’s bounds. All numericalprocedures in the program were checked, and tolerances and error limits were lowered, and the Gaussian quadraturerefined. This made no difference.This leaves three possibilities:• There is a precision problem, despite the fact that double precision variables was used in the (Fortran) program.• One of the two algorithms is incorrect in some detail. Ditlevsen’s method is much the more complicatedmethod.• One of the two methods interacts with other approximations. The use of 3 is a significant approximation tothe reliability. It may be that this approximation is amplified in one of the algorithms.These are not worth investigating for two reasons:• The discrepancy makes virtually no difference to the design application.• Madsen et al (1986), compute the reliability of a truss in their example 5.10, using Ditlevsen’s bounds.They also use a sophisticated simulation method to yield results which they consider exact, which liesoutside their bounds. Their discrepancy is a difference in the second decimal place of I, which is of thesame order as the discrepancy found in this work. Therefore Ditlevsen’s bounds have errors that are of theorder of these errors, and they cannot be taken as correct to this order of accuracy.I. Assessment of Length Effect ModelsIt is useful to know whether the length effect models which have been developed in this thesis imply differences inreliability of trusses which are of practical importance. In order to test this, each tension data set listed in Chapter lxwas used in two trusses, to compare the differences between three different length effects.153Xl Reliability of Structural Assemblies Assessment of Length Effect ModelsThe first length effect corresponded to the Weibull weakest link model, using length effect parameters from Table1 ofAppendix 10. The second corresponded to the CW model, using length effect parameters from Table 3 of Appendix10, allowing for a dependent length effect. This was assumed to be the best model for a truss made from pieces oftimber, with no continuous chord. The third corresponded to the CW model, modified to allow for a single continuoustension chord.There are two ways in which the CW model could be used to allow for a dependent length effect. The first way isrelated to the two types of length transformation in Chapter V section I. The second way is based on the length functionmethod and is equally applicable to the known-dependence models. This method and the version for single membertension chords is described in Appendix 17. This gives a function for finding the effective length, which yields thestrength distribution to be used in a reliability analysis.For each of these length effect models the reliability was calculated at several spans for both the parallel chord andHowe trusses. These allowable spans are set by constraints - e.g. there must be an integer number of bays. The parallelchord trusses were 0.5m deep, with 0.5m bays. The Howe trusses were 2.517m deep (this makes a 40 degree slopefor the 6m span), with lm bays. The Arvida snow load was used. The span, and effective length for f =3 wereinterpolated from these results by linear interpolation. This follows the methods used by Foschi et al (1989). Theresults are given in Tables 2 and 3 of Appendix 18.These are summarised in Table 2. The following points can be made:• There is a considerable difference between the spans allowable under the Weibull model, CW model, and singletension chord assumption - of the order of 5%.• The average effective size for the single tension chord assumption is less than 3m. It may be possible to ignore thisadjustment for most trusses, since the standard length is usually 3m.• For a truss without a single piece tension chord the effective length may be as large as 7.3m. For the Howe truss,the effective length is usually less than the span, but for the parallel chord truss they are of similar magnitude.• It has been proposed that design strengths should be adjusted to a standard length of 3m. If a truss was designedsolely on the most highly loaded member then it will be under-designed. The factor which should be appplied tostrength is given by: (effective 1ength/3)’This ‘truss’ factor was calculated for each truss, and had a minimum value of 0.76(24% under-design).• The ‘truss’ factor was markedly different for the different length effect models. For the parallel chord truss, theWeibull model gave an average 11% increase in strength compared to a 6% decrease from the CW model. For theHowe truss, the Weibull model gave an average 1% decrease compared to an 8% decrease from the CW model.154Xl Reliability of Structural Assemblies Discussion of the Length Function MethodWeibull CW One piecespan,m effective span,m effective span,m effectivelength,m length,m length,mParallelchordmax 6.44 2.50 6.11 6.61 7.37 4.16mm 1.27 1.48 1.03 1.50 1.04 1.25avg 3.82 2.14 3.54 3.54 3.75 2.62std 1.77 0.28 1.75 1.19 1.94 0.88Howemax 9.96 4.18 9.66 7.28 10.3 3.73mm 3.44 2 2.54 1.64 3.32 1.09avg 6.17 3.00 5.88 3.83 6.52 2.39std 2.37 0.84 2.41 1.55 2.46 0.85Table 2 Truss results, for 3 length effect models. Interpolated results for f = 3J. Discussion of the Length Function MethodFor this method to work there must be a length function available. Length functions are available for the Weibulland Gumbel distributions.A better model for truss reliability would allow for statistical correlation between members. This is present becauseit is common to make the bottom and top chords from single boards, or only a small number of boards. In both theDitlevsen bound, and length function methods described above, the bottom chord is modelled as being composed ofmany members: one for each panel.If the strength of parts of the board were independent, then this would make no difference. However, it was shownin earlier chapters that it is likely that dependence is present.. This will make the reliability of the truss greater than isindicated by both the Ditlevsen bound and length function methods. In Chapters IV and V there a number of suggestionsfor modified methods for finding suitable values of a to be used in the length function method so that suitable valuesof the reliability are predicted.It is difficult, but possible, to deal with statistical correlation using Ditlevsen bounds.The length function method has the following advantages:155XI Rehabihty of Structural Assemblies Summary of Chapter XI• The method is much easier to program, and to use than alternative methods.• The method is much faster to compute than alternative methods, usually taking of the order of one tenth ofthe time for the bound method.• It can relatively easily allow for dependence between strengths of members, such as occurs if they aremade from a single board.The disadvantages are:• It can only be used if the strength is modelled with one of the extreme value models mentioned in thischapter.• If the structure is made from different materials, then these must have the same length effect parameter. Ifthe Weibull model is being used, this is equivalent to the materials having the same coefficient of variation.Currently, the best method is to use Ditlevsen’s bounds. Garson’s method appears to give little or no advantage overthe assumption of independence. However Ditlevsen’s bounds require a specialised program, and are only approximate.For many structures functional dependence can be expected to be small, and it is not expected that it would alwaysbe needed.However, the proposed method is superior to existing methods which ignore functional dependence, because theyrequire a reliabilit calculation for each member. Thus the length function method is both more accurate, and requiresless computation.K. Summary of Chapter XI• The length effect methods developed in this thesis can be used as the basis for a new method for calculating thereliability of weakest link structures. These are an important class of civil engineering structures.• This new method is based on a length function which converts unequal loads to equivalent lengths which may besummed. It can be used to accomodate both types of dependence that may be presenL• This was used to show that there is a considerable difference between current models, the proposed model and theproposed model allowing for the bottom chord being in one piece.• Assuming that all the strengths were adapted to a standard 3m length, it is possible to find an adjustment for thesingle member design strength which would allow equal reliability for a truss. In one case this was 0.76, a decreaseof 24%.156Xl Reliability of Structural Assemblies Symbols and AbbreviationsL. Symbols and AbbreviationsB width of memberd ratio of dead load to design dead loadD dead loadDf dead load effectG failure function defining failure setH depth of memberHT truss dimensionK1 constant relating stress in member ito unit loadi panel numberI ratio of live load to design live loadL effective length of structureL1 effective length of member iL. real length of member ilength of members at testQ live loadr strengthR characteristic strengths spacing between trussesa Weibull distribution parameterreliability indexa1 stress in member i for unit loada, stress to which member is being standardisedperformance factory dead to live load ratiop Pearsons moment correlation coefficientt Weibull distribution parametero Gumbel distribution parameterO truss geometric angle157XII Condusons and Summary COnClUSIOnSXII Conclusions and SummaryA. ConclusionsThe conclusions of this thesis can be summarised as follows:There is a length effect ofprobabilitistic origin. It appears that the length effect as found in a variety of sources ofdata can be explained in terms ofa statistical model. This follows from the fact that the proposed model (an asymptoticweakest link model including allowance for dependence) predicts the average length effect of the data closely.Reinforcing this is the fact that statistical size effects have been found in many materials including timber.It does not follow from this conclusion that it is impossible to have a length effect from other causes as well. It appliesto good experimental configurations and practice, as exemplified by the data sets used in this study.It is necessary to find a modelfor this statistical length effect. The most basic reason for needing a model is that itis necessary to interpolate and extrapolate from whatever experimental evidence is available. For this purpose, a modelis more efficient than an empirical approach like regression. Another reason is that it is been shown that the samplingvariance of experimental estimates of the length effect can be unacceptably high, arid that model-based estimates havea much lower sampling variance. The third reason is that it appears from experimental evidence that the length effectfor different types of lumber is different, and that it is unreasonable to expect that experimental data will be availablefor all types of lumber. It follows that a model-based approach which only requires strength data at one board length,is needed. Such data are generally available.The problem withfinding a model reduces to finding an asymptotic solution for extreme values ofdependent data.It is difficult to find fault with the assumption that the strength distribution should be some type of extreme valuedistribuuon. This flows from the weakest link assumption.An asymptotic solution is suitable if there are many elements in a board. By giving an expanded interpretation to theelements in the Weibull model, it is shown that it is likely that an asymptotic solution is suitable.The most likely asymptotic extreme value model for independent data (the Weibull model) gives an over-predictionof the length effect. It is deduced that the only likely cause for this over-prediction is the presence of dependence.This observation is reinforced by experimental fmdings that dependence is present. It is also reinforced by the findingthat simulations indicate that allowance for dependence gives better estimates of the length effect.158 FXII Conclusions and Summary ConclusionsThe length effect found in a structural application is actually a combination of I or more of3 fundamental lengtheffects. It can be deduced that there will be three different length effects: the Cut-down length effect which occupiesmost of the discussion in this thesis, the Graded length effect which can be obtained approximately from the Cut-downlength effect, and the Built-up length effect which is straightforward it fmd from the Weibull weakest link theory.The Cut-down length effect model has been extensively validated. The Graded length effect model requires morevalidation. The Built-up length effect was not compared to data.It is not possible to say with certainty that the suggested model is the best at explaining the Cut-down length effect.It is argued that some types of high quality lumber may require a different model, using the same logic but differentbasic distribution, to best model the length effect on the entire distribution. There is some evidence that this is so formachine-graded lumber. However, a single model was adopted and extensively tested because of the uncertainty andcomplication involved in 2 models.Some data-sets may require a location parameter. Moreover, there may be models with other dependence structureswhich may be more suitable for the lumber strength application.The proposed CW model is superior to existing models. The CW model allows for the presence of dependence. Itcan be shown with great certainty that the proposed model is superior at predicting the length effect to other models,using both simulated and real data. On a number ofbases (fitting to strength data, predicting length effect, and predictingstrength distributions) the proposed model average prediction is closer than the Weibull weakest link and other models.This difference is statistically significant when the comparisons are directly on the size of the length effect. However,the sampling variance of the length prediction from the proposed model is a little larger than that of the Weibull model,and the proposed model is more difficult to fit to data.The proposed CW model is workable and practical. The proposed model only requires the strength distribution forthe lumber to which it is being applied. This is in contrast to other models put forward to replace the Weibull model.Algorithms are suggested, and equations and starting values are provided which allow the use of the proposed modelfor any other type of lumber, for which strength data are available.The proposed models are suitable for tension and bending, and may be suitable for compression. The models arebased on the weakest link assumption which is reasonable for tension and bending. It should be suitable for compression,but this has not been validated. The length effect can also be estimated from fracture position data from single pointbending tests, but this has not been comprehensively checked.The average length effect adjustment factorfor equal reliabilityfor doubling the length is approximately 1.14. Thebasic goal of this thesis was to obtain a method for finding an adjustment factor which preserved equal reliability forlumber of different lengths. The average factors obtained directly from data, and the proposed model are close. The159XII Conclusions and Summary Conclusionsdesign strength of lumber is increased by a factor of 1.14 when the length is halved, and decreased by the same factorwhen it is doubled. It appears that the factor should be lower for higher grade lumber: an average of 1.11 for highgrade lumber, and 1.16 for low grade lumber.The Length Function method is a superior methodfor computing the reliability of weakest link structures. Theapproaches used for dealing with the length effect in single members can be extended to these structures, and theresult is a method for computing reliability. Many civil engineering structures can be modelled as weakest linkstructures. The proposed method is simpler and has fewer approximations than commonly used methods. It is possibleto allow for members made from the same board, which is common practice and is difficult to deal with by othermethods.The different length effect model predictions have a major effect on the predicted reliability of trusses. The othermembers effectively increase the length of the most highly loaded member, requiring a lower design strength - thehighest adjustment was 24% (for a 3m standard member). The longest effective size of a truss was 7.3m, but a singlemember for the tension chord considerably lowers this. It is shown that the acceptable span varies by approximately5% on average for two of the models.160XII Conclusions and Summary SummaryB. Summary1. The Nature of the Length EffectThe strength property of a material is variable, and is thus represented by a statistical distribution. Any sample ofspecimens will thus have a range of strengths, and any two samples will have a different disthbution of strengths. Thestrength properties of lumber are more variable than most civil engineering materials.For tension loadings the strength distribution for a longer lumber member is usually shifted downward (i.e. loweroverall strengths), when compared to that for a shorter member. This is referred to as the length effect, and is alsonoticeable for a variety of materials, under a variety of loading conditions. Elementary mechanics of materials theoriesdo not predict this finding. It can be attributed to a statistical effect, because a longer board has a greater chance ofhaving a serious defect.The aim of this work was to find an adjustment that may be used in design to ensure that structures with membersof different length are uniformly reliable.2. Approacha. Weakest LinkIn the literature, a chain has been used as an analogy for specimen strength. The specimen (chain) is as strong as itsweakest part (link). This has been used extensively to find a suitable function for the distribution of strengths. Thisis often called a statistical model. For the case when the specimen (chain) has many parts (links) one of three extremevalue distribuiion.s is appropriate. These solutions are asymptotically correct as the number of links becomes large.They predict simple functions for the shape of the distribution curve. It can be argued that the Weibull is the mostappropriate for material strength distributions.This model provides an estimate of the statistical length effect and it predicts a straight line on a log length versuslog strength percentile graph. The slope of this line depends on the magnitude of the variation in material strength. Aset of tests with specimens of one length provides an estimate of this variation, and thus also the statistical lengtheffect.The process of grading lumber interacts with the length effect, so it is necessary to define a basic length effect (calledherein the ‘Cut-down’ length effect) as resulting when shorter specimens are cut from longer ones, with the length atwhich grading took place as being unchanged. Other length effects can be defined and modelled with additionaldifficulty.161XII Conclusions and Summary Summaryb. ProblemsUnfortunately, it has been found experimentally with timber and other materials that the magnitude of the ‘Cut-down’length effect predicted by the Weibull model is too high, on average. It appears that since the model predicts somethingthat is not observed experimentally it should be rejected.It may be that poor experimental technique is to blame, but it is deduced that this possibility is unlikely to be thecause of this problem.3. DependencyOne of the assumptions of the weakest link model was found to be suspect. Good evidence is available from woodanatomy, and small scale wood testing that wood from the same tree is more similar in strength to wood elsewherein the tree, compared with strength of wood from other trees. This implies that the strength of elements of a board arenot independent, contrary to one of the assumptions of the theory.a. Current Model-based ApproachesSome researchers have suggested that the problem with the Weibull is due to other causes. However, it appears thatagreement is developing that the problem is due to dependence.Leicester developed a method that retains the asymptotic weakest link solution but corrects the magnitude of thestrength variation for a pre-known level of dependence. This model was further developed in this thesis.Other methods use a simulation-based model, which allows greater freedom in choice of model form. Most workersusing these have chosen to use a type of dependence that dies away with distance between the dependent points.b. Model-based Compared to Experimentally-based ApproachesSome researchers have suggested that it is necessary to carry Out tests of specimens of different lengths in order toquantify the length effect. This would have the disadvantage that even an abbreviated test programme would beunmanageably large.There is a second disadvantage related to the sampling error of the estimate. It was shown that a model-based estimateof the length effect will have a much smaller sampling error than the experimental approach. The only way to obtainexperimental estimates with even moderate errors is to have the ratio of lengths of the specimens of the order of threeor more.4. Method for Dependent Extreme ModelsThe author has developed an asymptotic solution for dependent extreme values - the CW model. Other workers haveused similar methods to allow for dependence in other statistical models.162XII Conclusions and Summary SummaryThe basic idea stems from the fact that the parts of a single member are actually conditionally independent, i.e.independent conditional upon the knowledge that they are from the same board. Therefore a Weibull distribution isappropriate fora single board. However, the whole population is a mixtureof individual boards; therefore the populationmust be a mixture of Weibull distributions.There is a whole class of mixed Weibull distributions, because there are many choices for the distribution functiondescribing the mixture. Only a few are mathematically convenient.It has been shown by others that under the circumstances used in this thesis that the resulting distribution isidentUiable.This means that the original Weibull, and the level of dependence (given by the mixing distribution) can be foundwhen the function is fitted to a set of strength data. A crude measure of the dependence is the correlation coefficientp between the strength of two equal sized pieces from the same board. It was shown here that the CW model yieldsa modified estimate of the length estimate (which allows for the presence of dependence), and also an estimate of thedependence.a. SimulationIn order to verify that mixed extreme value distributions can be used to model extremes from dependent processes,a simulation study of 100,000 boards was performed. This had the dual advantage of a high data set size, and that thegenerating process was known. This allowed virtually conclusive confirmation that the proposed models would workbetter than the Weibull model. This was mostly due to a belier fit of the statistical distribution to the data. It was alsoshown that the size effect predicted corresponded more closely to the observed one and that the dependence predictedfrom the models was weakly related to the dependence built into the data.5. ResultsData from four different experimental programmes was obtained. This had the advantage that experimental errorsfrom a single testing programme would be unlikely to have a serious influence on the conclusions.The different models were applied to the data, and various comparisons were made.It was shown that the mixed model predicts the fifth percentile of tests on different lengths of wood much betterthan any other model. The basic aim of the model has thus been met.The principal improvement is that the mixed distributions (including the CW model) fit the strength distributionmuch better than the Weibull distribution.It was also shown that the length effect predicted by the CW model is more accurate than the other models. Theimproved Leicester model is a good model if nothing is known about the dependence in the material. There areindications that it may be fairly consistent for wood. It is confirmed experimentally that the variance ofthe model-basedestimates is much smaller than that of the experimental estimates.163XII Conclusions and Summary ContibutoBoth the Weibuli and the proposed CW models make predictions about the distribution offracture location for beams.Experimental results indicate agreement with these predictions.Experimental results show that the relationship between two elements of a board is approximately independent ofthe length between them.6. Reliability and the Application of the MethodThe model could be used within a Working Stress Design code fonnat. However modem design codes are based ona Limit States Design format, with equal structural reliability as the fundamental principle.a. Equal Reliability Adjustment FactorIt is shown that the length adjustment parameter identified from the mixed Weibull model gives a straightforwardadjustment which provides equal reliability.b. Application to Weakest Link StructuresThe truss is probably the structure with most need for a length adjustment. This is an example of a weakest linkstructure. Methods for calculating the reliability of whole structures have been developed as research tools, but arenot generally used in design.Based on the length effect tools described above, a method was developed which computes the reliability of weakestlink structures and allows for length effects. This method is potentially exact, and requires an order of magnitude lesstime than the current methods of computing reliabilities which are approximate.This method is applied to two trusses: parallel chord trusses and Howe trusses. It was shown that the method givesanswers which are practically the same as the current methods.It was used to show that the various length adjustments make a considerable difference in the calculated reliabilityof multi-member structures, such as trusses. The Weibull, and CW models were considered, and the CW was alsomodified to allow for dependent members, so that it could be used for a one-piece tension chord truss. The effectivelength of the trusses examined was at the most 7.3m. The largest effective strength reduction (to a standard 3m member)caused by the length effect of the miss was 24%.C. ContributionThe following major contributions were made:a useful, convenient model for length effects in material strength was developed.• models for dependent extreme value processes were devised.• a superior method for computation of reliability of weakest link structures was constructed.164XIII AppendicesXIII Appendices165XIII Appendices Appendix 1 Model Prediction vs. Data SmoothingA. Appendix 1Model Prediction vs. Data SmoothingIt may appear to the reader that the discrepancy in the Weibull prediction of the length effect is not very important,because it may be avoided by emphasis on experimentation. Within the Weibull weakest link framework there aretwo basic strategies which may be used to find the length effect for a particular grade, species and depth of lumber.I. The model-based approach uses the strength distribution from one length of lumber. Fining the Weibullmodel to the data yields an estimate of the slope of the line on the log strength versus log length line.2. The experimental approach which uses the strength distributions from at least two lengths. The same quantilefrom each is plotted to directly give the log strength versus log length line.The latter approach seems intuitively to be giving a fair estimate of the length effect, with only a small modelingassumption. Close scrutiny reveals that in most cases it is actually a very poor method. The following analysis seeksto compare the two methods based on the quality of estimates of the two methods, when using the same amount ofdata.The merit of the alternative methods can be judged on the relative closeness of the prediction of the strength atanother arbitrary length, to the real strength at that length. Since the prediction will be made from estimating a line,there are clearly two sources of variation in the final estimate: the height of the line, and the slope of the line. Theheight of the line corresponds to the original data point or points from which the prediction is being made. For thisdiscussion, it is assumed that the same method will be used to obtain this data point.The slope of the line represents the magnitude of the length effect. It has two error components: bias and variance.The bias is the tendency of the model to give an incorrect answer when averaged over the long-run. The experimentalapproach must have no bias, since the experimental estimate averaged over the long-run is the definition of the correctanswer.The bias in the model-based approach depends on the estimation approach. Proper development of the estimationmethod should remove most bias. For example, un-biasing factors are available for estimation of the Weibull distribution using the method of maximum likelihood. Therefore bias in the estimate of the shape parameter will be putaside.166XIII Appendices Appendix Mod Prediction vs. Data SmoothingA properly developed, appropriate model should have little bias. Considerable space has already been given over toensuring that the models are appropriate. Some bias may result from the estimation process. This can often be removedby the use of appropriate factors. Bwy (1975) gives un-biasing factors for the estimation of the Weibull distributionestimated by the method of maximum likelihood.It is concluded that the two approaches should be compared on the basis of variance of the estimate of the lengtheffect, i.e. the variance of the slope of the length effect prediction tine.1. The Variance from Plotting MeansIt will be assumed that data from N12 tests of unit length and N/2 tests of length M are available.In principle it is permissible to plot any quantile on the log strength / log length plot (Fig. 11:1) in order to find theslope of the line. However it will be shown below that the method is also valid for plotting the average. The meanhas a smaller variance than the more extreme quantiles. The result is that even if the quantile to be predicted is thefifth percentile, the variance will be smaller if the slope of the line is estimated from plotting the mean.The mean of a (simple) Weibull distribution (equation 11:8) is given by:(1)where is the mean strength.Equation 11(9) shows that for a longer board t is scaled by M for a longer board. Therefore the mean for a longerboard is:rti+!)U (2)U tMwhere a pre-subscript of M refers to the longer board of length M.ThusMj (3)Thus a similar relationship to 11(11) holds, and plotting the means and on a log strength/log length plot willalso give the slope for the log-log plot for quantiles.The variance of the means of each set of N12 tests isvar(s) . 2var(a) (4)var() 2167XIII Appendices Appendix 1 Model Prediction vs. Data SmoothingSince the experimental relationship, equation (3), is plotted as a log strength/log length plot, the plotting points areln and ln. The variance of these points can be found as a function of the variance of the mean strengths using themethod ofperformance moments (see Bury 1975). This method is based on MacLaurin expansions The two equationswhich will be used are:E{G}=G(i1,x,...,x,)+Z: [yar(xi). (6)var(G)=E[L)2vaa)+E It3(x1). (7)For a variable which has an approximately normal distribution, the second summation in (7) can be ignored, becauseIL3 =0. Equation (7) gives—var(a)var(lna)= —[E(a)]2var(a) (8)Similarly,—2var(a)var(lnMo)N(9)This is the variance of the plotting points in terms of the variance of the original data. The slope of the line in termsof the plotting points islflMO — In 01/a= (10)in M — In 1From application of (7) the variance of the slope is approximately:(lnM (in’var(1/cx)—var — (11)These variances are found from another application of (7),varI!. = var(ln) (12)%1nM) (mM)2Substituting from equation (8) and (12) into (11) gives4 var(o)var(1/cz) =N(lnM)2(E(o )= [cv(a)] (13)N(ln M)2168XIII Appends Appendix I Model Prediction vs. Data SmoothingEquation (13) gives the variance of the estimate in terms of a measure of the variance of the data, the amount of dataand the ratio of the two test lengths.There has been no explicit mention of the comparative benefit if the tests are carried out at more than two lengths.Subjectively it seems that this will give a better estimate of the slope of the line. However, with a fixed total numberof tests this will lower the number of tests carried out for the two basic lengths. It appears that it is optimal to havetests at lengths as far apart as possible (as the analysis shows). These tests will give a maximum amount of information.Therefore the result found should serve as an upper bound for the case where more than two lengths are tested, andN is the total number of tests. One benefit of doing tests of boards of more than two lengths is that the theory can betested by examining the collinearity of the three points.2. The Variance of the Model-based PredictionIn this case, all N tests are carried out at the same length. The variance of a is approximated by the minimum variancebound (Bury 1975),0.60793 2var(a)= N a.(14)Equation (7) is used to find var(1/a) in terms of var(a). Since a is a maximum likelihood estimator it is asymptoticallynormal, and 113 is zero.var(a)var(l/a) [E(a)]40.60793 (15)Ncs?For a Weibull model, there is a useful approximation relating the cv(x) to a ( see Bury 1975).(16)6 ‘2aSubstituting (16) into (15),var(1/a) Z[cv(o)]2 (17)3. The Variance from Plotting 5%ilesft is necessary to find a relationship like (4), relating the variance of a quantile to the variance of the strength.If N/2 =20 then the 5%ile would be the minimum from the sample. The distribution of the minimum has the samecoefficient of variation, but is scaled by a factor 1/20110. Thereforecv(a)=cv(o), (18)169XIII Appendices Appendix 1 Model Prediction vs. Data Smoothing(19)This givesvar(a) ()2var(TOLQS) == var(a) (20)2021aForN not equal to 40, the mean will be the same, but the variance will be approximately scaled by a factor of 40/N.var(o) 40var(a0)’= 2021a •__(21)The method used for finding the variance from plotting the averages can be followed. In this case it3 0 becausequantiles are asymptotically normal (Bury 1975). Instead of (8) we have4Ovar(a)var(ln aQ(5) (22)That is, the variance of the plotting point is 20 times larger than that of the plotting points when using the averages.The result is80 var(a)var(1/cz) =N(lnM)2(E(o )= 80 [cv (a)]2. (23)N(ln M)2For reasons of economy, it is often a good strategy to use censored data to estimate a quantile. This can also be usedto estimate the shape parameter a. If the total number of tests was used as N, then the variance would be increasedfrom that calculated in (23).4. Approximate Analysis for CW Model-based PredictionIt is not possible to obtain a simple expression like (14) for the CW distribution. An empirical approach was used.For the data sets used in this thesis the variance ofa from the CW distribution was found using the approach discussedin Chapter VI. The variance of 1/a was found using performance moments as in (15). For the Weibull model thevariance of 1/a was found using (15). The ratio of these was taken, and this was found to average 1.92 (with a standarddeviation of 0.98). Thus the following equation was usedvar(lIa) 0.71 [cv(a)]2. (24)170XIII Appendes Appendix 1 Model Prec5ction vs. Data Smoothing5. Comparison of the Two ApproachesThe variance of the length effect has been found in terms of non-dimensional measures of the original strengthdistribution, for both approaches for a fixed data-base. They are compared for different ratios of the two test lçngthsin Table 1.Ratio - M Model based I Model based Experimentally ExperimentallyWeibull CW (plotting means) (plotting 5%iles)x[cv(a)]1/n1.3 0.37 0.71 58.1 11621.5 0.37 0.71 24.3 4862.0 0.37 0.71 8.3 1663.0 0.37 0.71 3.3 665.0 0.37 0.71 1.5 3010.0 0.37 0.71 0.75 1520.0 0.37 0.71 0.45 927.0 0.37 0.71 0.37 7.450.0 0.37 0.71 0.26 5.2100.0 0.37 0.71 0.19 3.8Table 1: Variance of length effect slopes 1/aIt is intuitively obvious that for the experimental approach a greater ratio of lengths gives an estimate with lowervariance, and this is seen to be true. However a surprisingly large ratio is needed before it becomes better than amodel-based approach.6. Variance of Doubling RatiosThe sections immediately preceding give the variance of the slope of the log-log plot, 1/a. Alternatively the varianceof the doubling ratio D may be needed. This can be found from the above as follows.The doubling ratio is given byD=2. (25)Using equation (7)var(D) = 2(1n2)var(1/a). (26)The expressions for var(1/a) (i.e. (13), (17), (23) and (24) ) can be substituted into (26).Alternatively, the 5%ile and variance of the 5%ile may be computed for a particular data set using exact expressions,appropriate only for the relevant statistical model. The variance of 1/a could be found as follows.(27)In MUsing (7)i r varo vara01var(1/a)= + I (28)(lnM)2[ (o)2 (a)2 j171XIII Appendices Appendix 1 Model Prediction vs. Data Smoothing7. Symbols and Abbreviationscv(x) coefficient of variation of xE(x) mean of xM ratio of length of longer board to shorter boardN total number of boardsvar(x) variance of xa shape parameter of Weibull distributiona strengthstrength of longer board5%ile of strengtha mean strength172XIII Appendis Appendix 2 Conditional IndependenceB. Appendix 2Conditional IndependenceA suitable assumption for the relationship between elements within a board is that they are conditionally independent;they are only independent when considered as coming from one board.A simpler example may help to explain thisconcept. Consider the example ofa man pickingmarbles from 4 bags, each of which containswhite and black marbles. The marbles are________________replaced after being chosen, and the bags arebehind a curtain, so the man cannot see the bagor marble which he is choosing. Is the sequenceof marble colours independent?It will be shown that this depends on how he ischoosing from these bags.Consider first the case that he is choosing from among the bags randomly. There are an even number of black andwhite marbles overall, and this will not change during the sequence of choices, since the marbles are being replaced.Clearly, the last choice cannot affect the next choice, and the colour of marbles in a sequence is independent.Secondly, consider the case where he randomly chooses the bag, and from there on must keep choosing from thesame bag. If he chose a black marble initially it is much more likely that he happened to choose one of the bags onthe left. Given that is so, it is much more likely he will choose another black ball, because there are so many moreblack balls in these bags. The second and subsequent colours are almost always the same as the first. This means thereis strong dependence in the colour of balls.Thirdly, consider the case where the man is directed to the left-hand lower bag, and must keep choosing from thatbag. He will have a 8, chance of picking a black ball, and a 1,9 chance of picking a white ball. This chance does notchange (provided the balls are replaed), so the first choice does not affect the latter choices. Therefore the coloursare independent.•o• °oI•.•IIFigure 1 Conditionally independent choice173rXIII Appen&es Appendix 2 Conditional IndependenceOn the surface, these last two cases appear identical, yet in one case the colours are strongly dependent, and in theother case they are independent. The apparently irreconcilable difference lies in the context of the problem. Specificallyit depends on the assumed population.In the former case the odds of getting a black ball are nominally 112 because the population is all the marbles in thefour bags. In the latter case the odds are nominally 89 because the population is just the marbles in the one bag. Thekey concept is that the population is defined by the context of the problem.The colours are generated from a process that must be independent. Provided they are judged within that context,they are independent. They may appear dependent when considered in a wider context, so they are said to be conditionally independent. In this particular problem, they are independent conditional upon knowledge of the bag fromwhich they are drawn.The models allowing for dependence within a board are based on assumption that the elements within a board areconditionally independent. When the population is considered to be elements in one board, the elements are independent. When the population is all the elements in all the boards in a grade, then they are dependent.174XIII Appendices Appendix 3 Moments of Conditionally Independent ElementsC. Appendix 3Moments of Conditionally Independent ElementsThe author would like to acknowledge that most of the following work was carried out by Keith Knight and PeterSchumacher of the Statistics Department U.B.C..1. AssumptionsThe following assumptions are made:I. All elements have the same distribution.2. Elements within the same board are independent, except in so far as they have the same value of a singlerandom parameter of their distribution.Therefore, the. elements can be said to be ‘conditionally independent’ (see Appendix 2).2. NotationX is the strength of one element of a boardY is the strength of another element of the same boardA is the parameter which remains constant over one board.var(X) is the variance of XCOV(X, Y) is the variance of X3. Identity 1It will be shown thatvar(X)=E[var(X IA)]+var[E(X IA)]. (1)The first term on the r.h.s. of the equation isE[var(X IA)J=E[E(X2)—( ( IA))2]. (2)The second term on the R.H.S. of the equation isvar[E(X IA)] =E[(E(X )A))2]—[E(E(X IA))]2. (3)Substituting (2) and (3) into (1) gives175XIII Appendices Appendix 3 Moments of Conditionally Independent ElementsE[var(X IA)1+var[E(X IA)i=E[E(X2—[E( IA))]2=E(X)—[E( )]=var(X). (4)4. Identity 2It will be shown thatCOV(X,Y)=E[COV(X,YIA)]+COV[E(XIA),E(YIA)J. (5)The first term on the R.H.S. is given by:E[COV(X,Y IA)]=E[E(X,Y IA)—E(X IA)E(YIA)]=E[E(X•YIA)]—E[E(XIA)•E(YIA)]. (6)The second term isCOV[E(XIA),E(YIA)]=E[E(XIA).E(yIA)]—E[E(xIA)].E[E(YIA)]. (7)Substituting (6) and (7) in the R.H.S. of (5) givesE[var(X IA)1+var[E(X IA)]=E[E(X Y IA)]—E[E(X A)].E[E(Y IA)]=E(XY)—E(X)E(Y)= COV(X, Y). (8)5. Results for the CorrelationAssumption 1 givesE(XIA)—E(YIA) (9)var(X IA)=var(YIA) (10)var(X) = var(Y). (11)It follows from the definition of p and (11) thatCOV(X,Y) 12var(X) (The conditional independence assumption gives:COV(X, V IA) = 0. (13)Using Identity 2COV(X,Y)=COV[E(X IA),E(Y IA)]. (14)Since E(X IA) =E(Y IA), rewriting the right hand side gives,COV(X,Y)=var[E(X IA)]. (15)176XIII Appendkes Appendix 4 Methods for Finchng Mixed DistiibutionsD. Appendix 4Methods for Finding Mixed Distributions1. General NotationThe notation matches that of Chapter V:Upper case functions: Cumulative distribution functionLower case functions: Probability density functionG: Kernel distributionP: Mixing distributionF: Mixed distributionPre-subscript of N Function for N elements2. Theorem IThis is the method presented by Elandt-Johnson (1976) for finding closed-form solutions for the cumulative distribution function for a class of mixed distributions. The kernel distribution needs to be of the following form:G(x;t, ‘F) = 0 for x x0= I — exp(—tu(x;’P)) for x x0. (1)where r is one parameter of the distribution and ‘F is the remaining set of parameters, and u(x;’+’) is an increasing—ooThe parameter that is allowed to vary is t. So it can now be considered as a random variable. The distribution of thisparameter (i.e. the mixing distribution) is calledP1(y;), where is the set of parameters of the distribution. Themoment generating function of the mixed distribution is Mi(s), if it exists.TheoremF(x) is equal to 1— M1(s) with s replaced by —u(x,’Y).ProofPmbability of survival is 1 —F(x), thereforeP{survival} = I —F(x;’F,!). (2)The probability of survival of x is the probability of survival at a given t integrated over all values oft.177XIII Appendices Appendix 4 Methods for Finding Mixed DistributionsP{survival} = f(1 — G(x;t, P))dP(t)= Jexp(—tu(x;’P))dP(t). (3)Equating (2) and (3) gives1 —F(x,’P,S) =Jexp(—tu(x;’P))dP(t). (4)Lets=—u(x,”I’) (5)then1 —F(x,1’,) = J’exp(st)dP(t)=M(s), (6)by definition (see Bury 1975). It follows thatF(x;’l’, ) = I — Ms(s) (7)as required.3. Theorem 2If the strength distribution for unit length is F(x), the strength distribution of length N is I — [1 — F(x)]”, if the lengthis assumed to be made up of independent elements.This theorem is central to classical weakest link theory, and applies whether the distribution is mixed or not.ProofThe probability of failure is F(x), so the probability of survival is given byP{ survival } = I —F(x). (8)Survival of a length N follows if each link survives. Since these are independent, this is the product of survival ofeach one.Therefore the probability of survival of length N is given byP{ survival of length N } = (I F(x))”. (9)Therefore1vF(x) = 1 (1 _F(x))N (10)as required.178XIII Appendices Appendix 4 Methoc for Finding Mixed Dist,ibutions4. Corollary 1If the kernel strength distribution is Weibull G(x) =1 exp(tx and t has some mixing distribution P(y) to givea mixed strength distribution F(x) for a unit length, then the strength distribution for length N NF(x) = F(xN).ProofFor a single board the probability of failure aix is G(x).The elements within one board are independent. Using Theorem 1:,1G(x) = 1 —[1 _G(x)]lv. (11)Substituting for G(x)NG(x) = I — [exp(t?’)f=1 _exp(_Ntxa)=G(Nlax,t,a), (12)so= u(N”’x). (13)Using Theorem I it is seen that= F(N1x) (14)as required.5. Corollary 2If the kernel strength distribution is Gumbel with G(x) = I —exp(—rexp(x/O)) and i has some mixing distributionP(y) to give a mixed strength distribution F(x) for a unit length, then the strength distribution for length N isNF(x)=F(x+OlnN).a. ProofFor one board the probability of failure aix is G(x). Therefore the probability of survival is given byP{survival} = 1— P{failure}=exp(_iiexp(--)J. (15)Since these are independent within one board179XIII Appendices Appendix 4 Methoc for Finding Mixed DistributionsP{survival of N lengths) = [exP(_1l exp(-j-))]N=exi{_mlexP(j-)]=e_iiex+0m_)], (16)soNu(x)=u(x +Oln(N)). (17)Using Theorem 1 it is seen thatNFCr) = F(x + Oln(N)) (18)as required.6. Theorem 3If the kernel distribution is Weibull with location parameter xg so that the c.d.f. is O(x) = 1— exp(—’r(x _X0)a). thenletG(x) = 1 — exp(_txa) (which is the same distribution without a location parameter). Iftis given a mixing distributionP(y) with m.g.f. Mr(s) then the mixed distribution E(x) is F(x —x0). This is the same distribution without a locationparameter.This means that all the following disthbut.ions obtained by using a Weibull as a kernel distribution can be modifiedeasily to include a location parameter.All that needs to be done is to write (x —x0) wherever x appears in the original equation. Therefore these will notbe written ouLProofThis follows from Theorem 1:Theuterm forO(x),is(xxo)Q.TheutermforG(x),jsxa.Therefore(19)using Theorem 1F(x) = 1— M(—u(x)), (20)so180XIII Appendices Appendix 4 Methods for Finding Mixed DistributionsE(x) = I —M(—ü(x))=1—M(—u(x—x)=F(x—x1), (21)as required.181XIII Appendices Appendix 5 Derivation of other Statist DistributionsE. Appendix 5Derivation of other Statistical Distributions1. Weibull Mixed with Uniform Distribution - CUWThe kernel distribution is two parameter Weibull, soG(x)=1_exp(_t?) (1)which is of simple exponential form (see section 13.D.2) withU(X)=Xa. (2)The mixing distribution is the uniform or rectangular distribution which has two parameters:forayb=0 fory<a,y>b. (3)The moment generating function of p(y;E) can be found in many standard texts, but as an example it is found asfollows:M(s) = fexp(sx)f(x)dr. (4)Substituting for f(x) givesM(s)= jb(b—a) dx— [exp(bs)—exp(as)) (5)—(b—a)sUsing Theorem I from Appendix 4,s is replaced by —u(x) =Xa in 1 —M(s) soF(x) = I — [exp(.bxa) — exp(—ax] (6)(a—b)xAfter differentiation the p.d.f. is given by:—a [[exp(_bxa) — exp(—ax°)J + Lb exp(bx°) — a exp(axa)JX (a—b)[ x’ x(7)and by definition the log likelihood isL.L.=ln[flf(x1)x 11 1_F(xj)] (8)=1 j=—r +1182XIII Appendices Appendix 5 Derivation of other Statistical Distributionswhere n is the totai number of data points, and r is the number of censored data points.Substituting forf(x) and F(x) from (6) and (7) into (8) gives:L L. = ln(c) — ln(a — b) + 1Jexp(—bx,$) — exp(—axfl + b exp(—bx) — a exp(—ax$i=i[+ [ln[exp(—bx) — exp(—ax] — In(a — b) — a lnx1. (9)j=k—r +1The maximisation of this function with respect to the parameters a and b yields maximum likelihood estimates.Assume the distribution for unit length is given by (6) and (7). Corollary 1 of Theorem 2 of Appendix 4 gives:NF(x)=1(exp(_bNxa)_exp(_aNx()) (10)(a _b)Nxafor the distribution of a specimen of length n, and by differentiation:- r exp(-bNx6)- exp(aNx’) bN exp(_bNxz) - aN exp(_aNxu) 11x°1 + xThe p.d.f. of a longer board can also be expressed as (7) but with two of the parameters changed. They becomefunctions of the old parameters and the ratio of sizes, N withNa=aN (12)(13)AlternativelyNF(x;a,b) = F(x;ri,aN,bN). (14)Since the distribution for length N is of the same form as for a unit length, the disthbution is termed ‘closed underthe length transformation’.The Built-up length effect is found from the application of Theorem 2 of Appendix 4. This gives the distribution forlength N as[exp(_bxz)_exaa)JF(x)=1— . (15)(a—bfx’’This distribution is not the same type as the original distribution. Thus the distribution is not closed under theindependent lengths transformation.2. Gumbel Mixed with 3-parameter Gamma - CGThe kernel distribution is a type 1 extreme value (or Gumbel) distribution of minima, modified so that it is of simpleexponential form:183XIII Appendices Appendix 5 Derivation of other Statistical DistributionsG(x;q,O)= i_exP[_T1exP()] (16)soIx ‘1u(x)-exPl-p (17)The mixing distribution is the three parameter gamma distribution:exp(—3(y —E))(y —E)1 (18)p’(y)=This is written with notation different from the CW case, so that the mixing distribution will have differentparameters.The m.g.f. is13Cexp(es) (19)(l3_s)CSubstituting s = —u and using Theorem 1, Appendix 4, givesF(x;C,,E,O)= exl{—Eexp()] (20)Substituting w = — 1n(), and =exp[—exp(+c)] (21)F(x;C,o,2,e)=[1+ex+o)]This can be differentiated to give:—exp[ + — exp(+ )] + exp[ + o — exp( + (22)f(x;C,0,f2,O)—o[i +exp+oJ]C i +exp(÷co)]The L.L. is given by:-LL.=i=l[ch{1 +exp(+coJJ+xex+-ex+JJ+i+ex+w))r+ E (23)j = II — r + 1LTo find the length effect, using Comllary 2 of Appendix 4, x is replaced by x + 0 ln(N), soN’Q’) (24)184XIII Appendices Appendix 5 Derivation of other Statistical DistributionsN—th(N) (25)orexp(_exp(÷ InN))NFCz) (26)1 +exp(+to+lnN)Therefore the CG distribution is closed under the size transformation.Using Theorem 3 of Appendix 4, one fmds that it is also closed under the independent links size transformation with(27)(28)When the variable E is set equal to zero, the mixing distribution becomes the two parameter gamma distribution.The resulting three parameter mixed distribution is termed ‘CG2’. The distribution is considerably simpler, but willhas less flexibility.3. Weibull Mixed with Log Gamma Distribution - CLLThe kernel distribution is Weibull,G(x)= 1exp(tx’). (29)So that(30)The mixing distribution is(31)This is the log gamma distribution given by Johnson and Kotz (1970) . It is a three parameter distribution similar tothe three parameter gamma distribution, however it can have negative skewness. Johnson and Kotz gave the m.g.f.(32)for a standardised log gamma.Hoel (1984) gave some transformations so that the m.g.f. for a three parameter log gamma distribution can be foundM= M.(cs), (33)M÷(s)= M(s)exp(sc). (34)So185XIII Appendices Appendix 5 Derivation of other Statistical DistributionsMe(s) = F(A±sX),) (35)Substituting u=-s (see Theorem 2 of Appendix 4)F(x)= i_r’(A_x°x)exi,,x, (36)Differentiating the c.d.f. gives the p.d.f.:f(x) = rXA— x) a)al[,(Xa 1)X+], (37)where iv(x) = r(x)Ir(x) and is usually called the Psi or Digamma function.So for an uncensored sample:L.L. =!h1[—4x+ lncz+(c— 1)lnx — lnf(A)+ 1)+)J.(38)a. Effect of Size on CLL DistributionReplacing x by xN (Corollary 1 of Theorem 2, Appendix 4)NF(x) = 1—r(A Nx”X) exp(4Nxa) (39)orNX=NX (40)(41)NF(X ;(x, , A, C1) = F(x;cc, NZ, A, Nc1), (42)however for independent links Appendix 4 Theorem 2 givesxF(x) = 1—[ exp(_4xa)] (43)and this cannot be reduced to the same distribution.4. Weibull Mixed with Bessel Function Distribution - CBWThe kernel distribution is WeibullG(x) = 1 exp(txa). (44)So that(45)The mixing distribution is186XIII Appendices Appendix 5 Derivation of other Statistical Distilbutions1_t2=I I Y I exp(—--K(I y/q I). (45)qThis is the Bessel function distribution, given by Johnson and Kotz (1971). It was hoped that this would give adifferent set of shapes for the mixing distribution from the Gamma distribution, although since it does not have moreparameters it cannot be expected to have much more flexibility.Johnson and Kotz give the m.g.f.1_12Mt(S)=[l.._(c_qs)1] (47)Substituting u=-s (Appendix 4, Theorem 1)r i—F(x)=1—I 21 (48)1_(c_qxa) JDifferentiating the c.d.f gives the p.d.f.:f(x)= 2aq(+)(_qx+ tx(u_)(1 _t2t5 (49)(1—Q—qx)So for an uncensored sample:LL. =[ln(2)+ln()+ln(q)+ln()+ +ln(qx”+tx +Q+O.5)ln(1—12) Q.+O.5)ln(1 —t2) (A+ L5)ln(1 —(1 _qxa)2)]. (50)a. Effect of Size on CBW DistributionReplacing .x by xN1 (Corollary 1 of Theorem 2, Appendix 4)rNF(x)_hI 21 (51)[1_(c_qNxa)Jor= Nq (52)which shows that the distribution is closed under the length transformation.However for independent links Theorem 2 Appendix 4 givesl_12NF(1)1_[ 1_-(c_qs)2] (53)which can also be reduced to the same distribution, but with the following parameter modification,187XIII Appendices Appendix 5 Derivation of other Statistical Distributions(54)5. Weibull Mixed with Four Parameter Beta Distribution - CWBThe kernel distribution is WeibullG(x)= I _exp(_txa)so=The mixing distribution is(55)where B(a,b) is the beta functionB(a,b)=’’. (56)This distribution gives a lot of flexibility in choice of skewness and kurtosis. However, it cannot provide infinitetails. Johnson and Kotz (1971) gave an expression forthe m.g.f. fora standardised (i.e. twoparameter) beta distribution:Mi(s) = M(;4+v;s), (57)where M(x;y;z) is a confluent hypergeometric function.In order to find the m.g.f. for a four parameter beta it is necessary to transform (57) using equations (33) and (34).If y has the above distribution and(58)thenz=(b—a)y+a (59)and the m.g.f. of(b —a)y isM(b_4)Y = M(;Ø+v;s(b —a)). (60)Similarlyb-)y+ =M(O;Ø-i-v;s(b—a))exp(as) (61)soMr(s) = M(Ø;Ø + v;s(b — a)) exp(as). (62)Substituting s = —u188XIII Appenckes Appendix 5 Derivation of other Statistical DistributionsF(x;Ø,v, a, b) =1— M(Ø,4i + v;(b — a)xexp(—ax (63)Abramowit.z and Stegun (1972) gaveM ‘(abx) M(a + 1b + 1x) (64)Differentiating F(x)f(x;Ø,v,a,b) = cx ax)ax’[a M(Ø;(Ø+v);(a _b)xa)_(a —b)-M(Ø+ 1;Ø+v+ 1;(a(65)So the L.L. for an uncensored sample is:L.L. Zh1[—ax,”+ln(a)+(cz— 1)ln(x)+ln(a M(;Ø+v;(a b)x)(a —b)jM(0+ 1,Ø+v+ 1,(a _b)S].(66)a. Effect of Size on the CWB DistributionFor correlated links substitution ofxN11’for x using Appendix 4 Theorem 2 gives:NF(XA),’, a, b) = I — M(t,, (0 + v), (b — a)Nxa) exp(_aNxa). (67)This is closed under the length transformation withNa=aN, (68)(69)For independent links:,,,F(x;Ø,v,a,b) = 1— [M(Ø;(ô4-v;(b a)x”)exp(—ax’)]” (70)which is not closed under the length transformation.6. SymbolsPlease see section V.K.189XIII Appendices Appendix 6 Derivation of Correlation CoefficientsF. Appendix 6Derivation of Correlation Coefficients1. General NotationThe notation matches that of Chapter V:Upper case functions: Cumulative distribution functionLower case functions: Probability density functionG: Kernel distributionP: Mixing distributionF: Mixed distributionMoment generating functionPart of c.d.f. see Chapter VPrefix of N Function or parameter for N elements2. Calculation of Correlation Coefficients from the Fitted Model - CWThe CW distribution is the Weibull mixed with the 3 parameter Gamma. As in the calculation for the CW2 distribution(section VI.C.l), this is based on equation VI:(l). Each of the terms are found as follows:a. E(xy)This is based on the use of equation VI(9) for E(xy), and equation V(8) for f(t).E(xy) = j44-+ l)6K(t_*1exP(_6(t_)d=t “(t — 1 exp(-6(t — ))dt.(1)The integral is too difficult to find analytically, so it has to be found it numerically. Unfortunately, this integral isimpossible to find numerically as it stands because of the gamma p.d.f. component of the function. It is known (forexample see Bury,1975) that this function approaches infinity at zero, if ic <0. The solution is to use a substitution of(t — = exp(—x). This gives an alternative formulationE(xy)= f exp[—ln(y+ej+ 2lnrQ-+ 1)+ icln8— lnT(ic)—(icx + 8e]dx. (2)190XIII Appendices Appendix 6 Derivation of Correlation CoefficientsThis transformation can be checked for cases where ic> 1, because the original integral can be obtained. It appearssatisfactory. Let:ra(!+ i)E(xy)=r’Oc)(3)b. E(x)As before E(x) = E(y)xf(x)dx_______ic•‘ldz. (4)o (8+x)c (6+x°)jThis integral is also too difficult to find analytically so it will be found numerically. LetE(x)=cz672. (5)c. E(x2)In a similar wayE(x2)= Iy+——;ltx0 (&+xa)c 8+X )= CZSJ3. (6)d. Correlation CoefficientSubstituting these results in V1(l) gives— r2(+ i )o’i, —a26Ir’(ic)— CLl3roc) — 8Ir(c)e. VarianceUsing equation VI(3), and substituting (5) and (6) gives:var(x) = cz6’(13— cz61d1). (8)3. Calculation of Correlation Coefficient - CG2 DistributionThe CG2 distribution is the Gumbel mixed with the 2 parameter gamma distribution. As before, p is derived fromequation VI:(l). Expressions for each of the terms are given by:a. E(xy)Eqn VI(9) is used for E(xy)Bury (1975) gives an expression for E(x)191XIII Appendices Appendix 6 Derivation of Correlation CoefficientsE(x) =—O1nTi—0.57722 (9)SoE(xy) = f(_OIn — 0.57722O)2f(T)di (10)and f(rI) is a two parameter gamma distribution. Because the parameterisation is modified (see VI(32) ) the form ofthis p.d.f. is changed slightly:Jo = exP(1exP(—o)))qexP(—(oC) (11)This givesE(xy) = InTl — 0577220)2exP( Ti ex,(—o)) TIC-1exp(—o))d1. (12)0 T()This integral is too difficult to find analytically. Unfortunately as it stands it is impossible to find numerically becausethe function becomes asymptotically infinity at zero, if < 1.The solution is to use a substitution Ti = exp(—x). This givesE (xy)= J exp[ln((Ox — 0.577221)2) — exp(—(x + w)) — C(co + x) — ln T(C)Jdx. (13)This transformation was checked by performing the numerical integration for one set of parameters with 1> 1. Inthis case the numerical integration can be found for both the transformed (12) and untransformed (13) formulations,and these were found to agree.b. E(x) and VarianceIt will be useful to find the moment generating function for this distribution. It is easiest to fmd the m.g.f. for astandardised distribution f(z ;C) first. This is given byf(z;C)= Cexpz (14)(1+expz)Then the m.g.f. will be given by= 1Ce2(1)2. (15)J— (l+expz)Tables of integrals givefexp(z(s-i-1))dz=B(s+1,C—s). (16)- (1+expz)Using the transformation equations XIILE(33) and (34), and eqn XII.E(56) forB(s +1, — s), the m.g.f. of x is found.192XIII Appendices Appendix 6 Derivation of Correlation CoefficientsM2(s) = CFOS+1fC_OSexp(s() (17)The mean of x is -aM2(s)E(x) as 12=0. (18)This givesE(x)— F(C÷1)(er(1)r(C)—9r(1)r(O— (OOr(1)r(Q). (19)To compute the variance E(x2) is also needed. This is given by‘M ‘u(s)E(X2)=as= r(i+ ) (r’(l)r(C) - 2r(1 )V(C) - r(1)() + r’(C)T(l)— 0r(1)T(C) + wr(1)r(ç) +02r(1)F(C)). (20)Then the variance is given byvar(x) = E(x2)— (E(x))2. (21)Thus all the terms in VI:(l) are found.4. Calculation of Correlation Coefficient - CG DistributionThe CG is the Gumbel distribution mixed by the 3 parameter gamma distribution. As before, equation VI:(l) givesp. Each of the are terms as follows.a. E(xy)Equation Vl(9) is used for E(xy)The equation for E(x),is the same as for the CG2 disthbution - (9).and f(ri) is a three parameter gamma distribution. Because the parameterisation hadLo be modified (see V(21)) theform of this p.d.f. is—exp(—(t — expc) exp(—o)) (i — exp)1exp(—COC)This givesE(xy)= f (—0 lnr — 0577220)2ex1*-(11 — exp fl) exp(—o)) (i — exp £2) ‘exp(—w)dr. (23)0(23)193XIII Appendices Appendix 6 Derivation of Correlation CoefficientsThis integral is too difficult to find analytically. For the same reason (a singularity) as was noted in section 2.a, thisintegral cannot be integrated numerically as it stands.The solution is to use a substitution (q — = exp(—x). This givesE(xy)= f exp[ln((—O ln(exp(—x + £) — 0.577229)2) — exp(—(x + co)) — C(w + x) — in r(O]dr. (24)It is seen that this equation is very similar to eqn (12).b. E(x) and VarianceAlthough it would be useful to find the moment generating function for this distribution, the integrals are too difficultto make this possible. Therefore E(x) and E(x2)are found numerically.‘.. Fex +_exp(+)] Cexp+o)_exp(+)]E( J) ii ° + e d (25)J_, [ +exj(+O))]CLet the results of these numerical integrals beE(x)=15 (26)E(x2)=16. (27)Then the variance is given by substituting these into eqn (21)var(x) 6(5) (5)The correlation coefficient is found by substituting these quantities into VI:(l)(29)c. SymbolsSee section VI.D194XIII Appendices Appendix 7 Gumbet-based Modete Under Non-Constant StressG. Appendix 7Gumbel-based Models Under Non-Constant StressThe mixed Gumbel models for non-constant stress can be found using a similar argument to that used for fmdingthose for the mixed Weibull models. The equation for the Gumbel distribution XJfl.E(16), takes the place of VI(3),which changes VII(l). This gives the following c.d.f. for the specimen:NF’(O) = 1 — exr{ —TjLBH s:.0f:=0f exP(ø(r r,, r)drzcLr,&x]. (1)When this is compared with the Gumbel distribution, it is found that exp(a/6) is replaced byLBH Jexp((o/O)c)dr,dr,dr. So the non-constant analogue to XllI.E(35) is:exi{—(expf)LBH iffexp(}ir1sr,&]—— [i +(expo)LBHfjfexp())ir1dr,&]The parameters for the non-constant stress case to be used in XIII.E(35) are:N0= £2 + h{ LBH •f•JjxP)}irzdrydrz] (4)with 6 and 6 unchanged.1. Method of Use of Non-Constant Load Gumbel-based ModelsThese models are used in a very similar way to that discussed for the Weibull-based models in section VII.A. 1. Theparameter which dictates the length effect is also unchanged for different load configurations and allows the possibilityof estimating the length effect for one load configuration from tests of another configuration.There is one important difference from the Weibull-based models. For the Gumbel-based models it is not possibleto fmd the effect of length without evaluating the integrals which must be evaluated in order to obtain the p.d.f. to fitto the data. Once the integrals have been obtained, the effect of length is very similar to that obtained for the puretension case and is straightforward.2. Pure Bending for Gumbel-based ModelsThe derivation used for Weibull-based models is followed. The stress distribution is given by VU(15). The tripleintegral term in equations (l)-(4), is given by195XIII Appendices Appendix 7 Gumbet-based Models Under Non-Constant Stress.fffexI{4(r/,ri]drfr,dr = fex[(1 — 2r,)]dr,or (o’ 1=j[eL)_lJ.This term is more complicated than the equivalent case for the Weibull-based models. It depends on the ratio of thecharacteristic stress to one of the fitted distribution parameters. The resulting c.d.f. is:F — 1exj{—(expQ)[exp()_ i]] (6)(°[1+(expco)[exl()—1]r3. Third Point Bending for Gumbel-based ModelsThis derivation also follows that used for the Weibull-based models. The stress distribution is given by V(17) andV(18). The triple integral term is given by:J fj exJ{cl}irdr,dr= 2L0i::exp( (3r,) (1 — 2r)drdr, +2 i:0ex (1 — 2ri}irdr,.(7)The first integral can be integrated over r to give:‘i =_J’ -[exp(r,J_ 1]th.,1 O_____=—+—-----+—-—+... . (8)3 120 5492 013ii!where the exponential is expressed as a series.The second integral can be integrated over r to give:or (°“ 1 (9)Substituting these into (2) gives:F(a) =([e) i] W13.i!)] (10)[i 4-(expo)LBH(L{exP°)_1J+Zç.,)]196XIII Appen&es Appendix 8 Model Validaflon with Simulated DataH. Appendix 8Model Validation with Simulated DataThis appendix fills in some of the less important details excluded from Chapter VIII. In particular it gives some ofthe details of the simulated normal data which were fitted with the Gumbel-based models. It includes some figuresand discussion which were not of central importance.1. Description of SimulationsTwo basic sets of data were simulated. The gamma-based data, which is fitted with Weibull-based models is describedin section VIIl.C.The multi-normal distribution is commonly used, and fulfils the requirements for giving data to be fitted by theGumbel-based model. This distribution can be simulated using the principle that n dependent random variables areconstructed from n+1 independent random variables. Devroye (1986) called such methods tn-variate reductionmethods, when referring to the bivariate case.The details of the analogy between boards, and simulations is covered in section VIII.C.The number of elements n that corresponds to a board is not known. This was discussed in section VllI.c, where itwas pointed out that it bears some relationship to the number of defects in a board. The number of defects dependson the grade and can be over 100 in poor quality lumber. Select structural lumber may have no apparent defects, butit will have many very small, possibly invisible defects. There are two possible scenarios:1. There are effectively as few as 10 elements, in which case the board strength distribution will onlyapproximately be an asymptotic modification of the element disthbution.2. There axe effectively at least 50 elements, in which case the modification is almost truly asymptotic.Most simulations were carried out with 50 elements in the basic length. Some simulations were carried out with 10and 500 elements for comparison purposes.2. Model Validationa. Fitting the modelsThe models were fit using the maximum likelihood method described in section VI.A . The maximising algorithmsused were the Simplex algorithm alone for fitting the Gumbel-based models, and the Simplex and Powell algorithmsin tandem for the Weibull-based models.197XIII Appendices Appendix 8 Model Validation with Simulated Datab. Preliminary Explanation of the Simulation ResultsThere are two types of data, fit by three models with two variables, and results in terms of two quarniles. Obviouslythere is a great deal of information contained in the results, but only the most significant observations will be noted.As described in Chapter VIII, there are three sources of results: direct from the data, fitted from the data. and resultspredicted from models. Three models are compared.1. The basic extreme value models are labelled Weibull or Gumbel. The basic Weibull model and length effectare given in XIILD.4. The basic Gumbel model and length effect are given in XIILD.5.2. The proposed mixed models are labelled Mixed. If unstated, the base model corresponds to the basic modelin the comparison. The Mixed Weibull model is given by VI(7), and the length effect is given by VI(12) andVI(13). The Mixed Gumbel model is given by XIII.E(21) and the length effect is given by XIII.E(24) andXIILE(25).3. The models using the Leicester type adjustment are labelled Adapted Weibull or Adapted Gumbel asappropriate. The fitting equations, and the length effects are given by the same equations as the basic models.However, for the Weibull the parameter governing the length effect is modified using equation XIII.N(15).Equation XIII.N( 16) is not used, because the approximation inherent in obtaining (15) from (16) is considerable.The quantiles are essentially non-parametric. They are found from fitting a curve to the three data points above andthree points below the quantile of interest. The closest points are weighted with a factor of 3 and the intermediatepoint with a factor of 2. The curve used was a Weibull, fitted by least squares, but since a very short length is fittedthe distributional assumption is insignificant.c. Model BiasFigures 1 and 2 show the bias of the models, and how it is affected by the correlation in the data. This is found bycomparing the fitted quantile to the actual quantile of the data. it is measured by taking the discrepancy between thetwo as a percentage of the actual quantile:(1)where q is the actual quantile, and q is the fitted quantile.198XIII Appendices Appendix 8 Model Validation with Simulated DataFigure XIII.H.1: Figure XIII.H.2:Weibull-based model bias at 50%ile (%) Gumbel-based mode) bias at 5%ile (%)50%ila bias (%) 5%fle bias (%)12 - 10ID ...$a -6 ;;—•4 WQ3Q4 0.6 OJ .. :Correlation coefficient of simulated data Correlation ceefficient of simulated dataMzed We4,jI Weed MUed aiee GLereel--s-.Figure XIH.H.3 Gumbel-based model bias at5O%ile (%)50%Je bias (%)15zzErE:zzz::zzzzzzzz::z::0 ...-.-L_—_0 0.2 04 0.6 0.8 1Correlation coefficient of simulated data0eedGLrn,beI Gjveei--6-- —4(--The following observations can be made, for both types of data.• Bias is larger for higher correlation.• Bias is smaller for mixed models.• Bias is larger at the 5th percentile level than the 50th percentilelevel, for all models except the mixed Gumbel.d. Predicted Length EffectLength Effect Parameter199XIII Appendices Appendix 8 Model Validation with Simulated DataThe length effect in the Weibull-based models depends on the one parameter CL. In the simple Weibull model this isoften called the shape parameter. The length effect in the Gumbel-based models also depends on one parameter: 0.These length effect parameters can be obtained from the estimation process. They can also be extracted from the dataas follows. For the Weibull-based models eqn 11(12) gives:inN (2)In q—where ae is the value of a from the data, q is the quantile from the basic length, N(= 2) is the ratio of the lengthsand q6 is the quantile from data from the longer length. For the Gumbel-based models XII1.D(18) gives:_q4—Nq 3inNwhere 0 is the value of 6 from the data.Figure XIII.H.4: Figure XI1I.H.5:Effectofponcxat5O%jle(%) Effectofnon aLength effect parameter alpha Length effect parameter alpha:::::z::rz:::E:E::::E:::::::::::: E22 ——————-——._...__—:5 I II Q I II I0.2 0.3 04 0.5 0.6 0.7 0.610 20 50 100 200 500 1.000Correlation coefficient of simulated data Basic number ofelements50%de øt data MIaed WeOI AdapSO Wedai WeCA 50%ie oldata 5%Ie of data MIxed Wedaxi Adaped WedaA Weaul—9--- - -s-- - -- . -*-- —8-- --a-- --G--- —W—200XIII Appen&es Appendix 8 Model Validaton with Simulated DataFigure XIII ii.6:Effect of p on 0Length effect parameter theta0.l•1--0-...::1Correlation efficient of simulated dataSO%ie 01 data 5%k otdaat Utoed G,n*el *dapd G,ratel Gta’*eI- -----&-- --O-- —4-—Figures 4-6 show the length effect parameter and how it is affected by the correlation in the data, and the numberof elements assumed in each board. For the Weibull-based models a large value of that parameter implies a smalllength effect, but the opposite is true for the Gumbel-based models. The following observations are noted, for bothtypes of data.• At a low level of dependence the length effect predicted from the mixed models and that from the data areapproximately equal. That from the simple models is slightly too high.• As correlation increases, for the normal-based data the length effect decreases. This is expected. For thegamma-based data it decreases at the 5th percentile level, and increases at the 50th percentile level. This canbe compared with the following:• Except for the adapted Gumbel model, the models show a length effect that increases with greater correlation.This is not expected.Length Effect RatioIn a typical practical situation, the model is fitted to a data-set from tests of one length, and this is used to predictthe quantiles of another length. The effects of model bias, prediction error and sampling error are combined. Theeffectiveness of the different models must be compared on this basis with a non-dimensionalised measure, such asthe ‘Length Effect Ratio’ (LER), which is the ratio of the predicted quantile difference to the actual quantile difference.The equation used is:Length Effect Ratio=q (4)qd 2q201Figure XIll.H.9:Effect of n on LER at 50 %iIeLength effect raSo for SOth%ile2.51.50.55 10 20 50 100 200Basic number of elementsUiedWh8 MadWibA WaiFigure XJII.H.8:Effect of n on LER at 5%ileLength effect ratio 5th%ileFigure XII1.H.1O:Effect of p on LER at 5%ileLength effect ratio for 5th%IleXIN Appences Appendix B Model Validation with Simulated DataThe data is represented as having values of 1. Even a perfect model would not always have a value of 1, because ofthe sampling enor. The length measurement is the number of elements n in the basic length. The results are plottedin figures 6-1 1.Figure XIU.H.7:Effect of p on LER at 50%ile (%)Length effect ratio for 50%ile252————•60Is___a.--‘E3-‘00.5————0.2 0.3 0.4 0.5 0.6 07 0.8Correlation coefficient of simulatad dataACTUAL DESIRED) MED WEIBULL. ADAPTED WELBUL.L WEIBULL-.-----s-- -ow-20Basic number of elementsU.dWebjI AdapedWeti Wetd—o,-7’----pI I500 1,000 0 0.2 0.4 0.8 0.6asrrelation coeffloent of simulated dataU4GA,e Ad.pGrad OiiIel202XIII Appencoes Appendix B Model Validation with SimulatedDataFigure Xffl.H.11: Figure Xm.II.12:Effect of p on LER at 5O%ile z Fitted p versus Simulated pLength effect ralto for SOth%JIe Predied correlationIt is seen that:• The CW model gives a length effect that is approximately correct, and the simple and Adapted Weibullmodels give length effects that are too large compared to the data at 5%, and too small at 50%.• In the Gumbel case the model predictions are almost all too large. At the 5% level, the CG is closer than theAdapted Gumbel, which is closer than the simple Gumbel. At the 50% level, the adapted Gumbel is closerthan the CG, which is closer than the simple Gumbel.• As a general rule, the models predict a greater length effect for a higher correlation. It does appear that thislevels off or actually decreases for the highest levels of correlation, for the mixed and adapted models.• A greater number of elemerns does not appear to change the predicted length effect at 5%, but increases thelength effect predicted at the 50% level.e. Predicted CorrelationThe fit of the models implies a level of correlation which can be calculated according to section VLC. 1. A model ismore likely to be good if the implied level of correlation is equal to that which was used to simulate thedata. Theimplied level of correlation is plotted against the actual correlation in Figure 12, and the following observations arenoted.• The implied correlation is clearly related to the actual correlation. The relationship for the Guxnbel set of dataappears to be much stronger.•The range ofvariation in the implied correlation appears tobe much smaller than the actual rangeofcorrelation.The implied correlation is in the region of 0.4- 01 0.1 0 D.2 040.0correlation coeffiaent of simulated data Correlation coefficient of simulated data@e.d) Ub,eC Gun* øda G.rC Gia,tel WetJ 50 elemerts Webi 100 eIem.nG.rtel SO &smen Oreel 100 elererW—00---- - -a-- ---e -- •—0(— --00- —-6------e-203XIII Appendices Appendix 8 Model Validation with Simulated Dataf. Statistical Significance of Gumbel-based Model PredictionsThe need for the calculation of statistical significances is discussed in section Vu1.D.5.In this case, method 2.b from Appendix 12 was used to find examine the null hypothesis that the model length effectparameter 8 is the same as the length effect parameter 0 directly from the data. These are shown in Table 12, whereit is seen that the models were virtually all rejected. The main goal of this exercise as stated is to show that the CGmodel is superior to the Gumbel model. The null hypothesis that the Gumbel model is superior to the CG model wasexamined using method 2.a from Appendix 12. The result shows that the CG model is almost certainly superior.Model p=O.l p=O.5 p=O.7CG Accept Reject Reject RejectGumbel Reject Reject Reject RejectAdapted Gumbel Reject Reject Reject RejectH0:Gumbel is better Reject Reject Reject Rejectthan CGTable XIII.H:l Results of hypotheses H0: Gumbel-based model parameters are equal to those from data3. Discussion of the Simulated ResultsSome insight into how the proposed model works can be gained by studying the general trends in the figures, whichyields observations as listed above. Some of the trends are clearly not completely smooth, and some of the resultsappear anomalous. This is probably due to the number of simulations being limited, so that the error term is appreciable.Each point on the figures represents 50 simulations of 100 boards, with up to 1000 elements. The effect of this limitedsample size is, for example, to produce a typical standard deviation of the Weibull length effect parameter ofapproximately I. Notwithstanding the above, it is remarkable that so many similar observations can be made aboutthe two sets of results, considering that each set is based on its own data, and its own models.a. BiasFor example, the presence of dependence increases the bias in all the models. However, the mixed models are muchless affected. This is a significant result for more purposes than modeling and predicting the length effect. It meansthat dependence in the lumber would cause bias in Weibull-based estimates of the characteristic strength used to obtainthe design strengths for structuraJ purposes. The proposed mixed models give distributions which should be superiorin modeling lumber strength.b. The Effect of DependenceThe way that the level of correlation in the data affects the length effect parameters is particularly revealing. Thereader should remember that the variance of the element strength distributions is fixed, so increasing the correlation204XIII Appenoes Appendix 8 Model Validabon with Simulated Datacoefficient has the effect of making the elements in a single board vary more in unison, lessening the length effect.However, the simple models actually predict a larger length effect. The mixed model estimates also predict a largerlength effect, but not as large. This discrepancy is difficult to explain, except possibly by dependence chang ng theshape of the distribution thus causing large bias in the parameter estimates. The adapted model estimates followapproximately the data trend, because they have the real correlation as a model input, and this forces the predictionsto decrease as they should.c. Predicted and Simulated CorrelationIf the fitted correlation bears a close relationship to the real correlation, then it is more likely that the model willwork well with a variety of data. It is seen that for the Gumbel-based models there is a weak relationship betweenreal (i.e. simulated) correlation and the fitted correlation, so these models do weakly adapt to the correlation in thedata. However it is unlikely that these models are powerful at inferring correlation from a set ofdata. Therefore successfor data with a given level of dependence, is not good evidence that the model will work for data with a different levelof dependence. The fitted correlation for the mixed models is in the range of 0.4-0.6, and it is likely that they willperform best for data with correlation in this range.It has been shown (see section Xlll.P.8) that the minima of a multi-normal distribution axe asymptotically normal.This means that as the number of elements rises, the predicted correlation should fall. This is not observed in Figure12. However, the effect is likely to be smaller than the sampling error.The success of the different models can be judged on a practical basis by comparing the length effect ratio results.These clearly show the superiority of the mixed models over both the adapted and simple extreme value models. Itcan be inferred from the above discussion that a substantial part of this success is due to the bias of the mixed modelsbeing less affected by dependence. This is because the mixed model distributions continue to fit the data well, evenwhen there is a substantial level of dependence. The simple model distributions (and consequently the adapted modeldistributions) fit data with dependence poorly. The mixed models also predict the length effect more closely than thesimple and adapted models.4. Details of Simulation Processa. Simulating Uniform Random NumbersVirtually all algorithms for generating random numbers produce uniform random numbers. These generators producelong cycles of random numbers. These algorithms should produce random numbers that have all the properties whichare usually considered desirable, such as very low serial dependence. The algorithm used was adapted from Wichmanand Hill (1987) which has an exceptionally long cycle length of 6.95E12. It achieves this by using a combination ofshorter cycle length generators.205XIII Appendices Appendix 8 Model Validation with Simulated DataThe seed values for the generator were derived from the seconds, minutes and hours of the clock built into the IBMAT computer used.b. Simulating Standard Normal NumbersThe algorithm used to generate multivariate normal numbers uses univariate normal numbers. These are generatedusing the standard method of Box and Muller (1958). This method uses two standard uniform numbers to generatetwo standard normal numbers.c. Simulating Multivariate Normal NumbersThere are very many multivariate disthbut.ions which have normal marginal distributions. Devroye (1986) calls thesemultivariate normal distributions. The multinormal (as termed by Devroye) has a p.d.f. of:I I—— ——(xflx)f([x])=(2it) 2e 2 (2)This is the distribution which was simulated. For the case where the variables are equally correlated the covariancematrix contains a set of equal values except for a leading diagonal of l’s. Gupta, Nagel and Panchapakesan (1973)give an equation for the i correlated variates X1:X= —p”24+(1 — p)”2Z1 (6)whereZ are standard normal variates. This equation was used to simulate multinormal variates. The resulting numbershave a mean of 0, variance of 1 and the specified correlation p. Since the proposed models used are designed only tobe used on numbers greater than 0 (i.e. strength data), an arbitrary constant of 10 was added to the simulated data.This changed the data mean to 10.d. Simulating Univariate gamma variatesThe algorithm for generating multivariate gamma variates requires univariate gamma variates. Because the c.d.f. ofthe gamma disthbution is not of closed form, there is no simple way of simulating gamma variates. Devroye (1986)p.410 gives an algorithm attributed to Best (1978) for generating gamma variates. This algorithm is based on usingthe rejection method. This method rejects many of the simulated random numbers, hence the need for a very longcycle random number generator.This generator is only valid for values of the gamma shape parameter greater than 1. However, some variates withshape parameters less than one are needed in the multivariate algorithm. For these variates, the algorithm is based ona transformation of the case where the shape parameter is greater than 1, as follows:Ga=Ga+iUlla (7)where Ga is a gamma variate with shape parameter a, Ga+i is a gamma variate with shape parameter a + 1, and U isstandard uniform variate.206XIII Appences Appendix 8 Model Validation with Simulated DataThese algorithms give a gamma variate with a unit value of the scale parameter. Therefore the generated numbershave to be multiplied by the scale parameter to get the desired variates.e. Simulating Multivariate gamma variatesDevroye (1986) gives the following method for simulating from the bivariate gamma distribution:Consider independent gamma random variables G1,G2,G3 with shape parameters a1,a2,a3. Then the random vector(X1,2)= (G14-G3,G2+G3) (8)is a bivariate gamma. The marginal gamma distributions have parameters a1 +a3 anda2+respectively. The ccwtelation is:a3 (9)[(a1 + a3)(a1+This can easily be extended to the multivariate case with equal correlations and identicaJ marginal distributions. Thesimulated numbers are given by:x1=G0+ , (10)where X is the ith desired multivariate gamma variate, G0 is a gamma variate with shape parameter a0, and G is theith gamma variate with shape parameter a.The shape parameters of the marginal distribution of X, is given by:(11)The correlation is given by:(1)a0 + aIt was considered preferable to have the normal-based and gamma-based data sets similar, so that the shape of theleft-hand tail is the major difference between the two. The gamma distribution is more like a normal distribution fora higher value of the shape parameter. Therefore the shape parameter 4. was set at 10, which should give a fairlynormal shaped distribution. The difference is that the left hand tail is limited at zero. Since the normal variates had avariance of 1, the variance of the gamma variates was also set equal to 1. This effectively sets the scale parameter.The variance of a gamma distribution is given byvar(x) = C1b2 (13)where b is the scale parameter. Since var(x) = 1, this givesb=(lIa). (14)For a value of 4 =10, combining (11) and (12) gives:207XIII Appendices Appendix 8 Model Validation with Simulated Dataa=1O(1—p). (15)5. Symbols and Abbreviationsquantile from dataquantile from data of tests of double the basic lengthquantile predicteda length effect parameter for Weibull-based modelsa from length effect dataa from Weibull modela from CW modela meanao length effect parameter for Gumbel-based models0 0 from length effect data208XIII Appendices Appendix 9 Summaries of Simulation ResultsI. Appendix 9Summaries of Simulation Results1. Gamma-based data/Weibull-based modelsp 0.5number of elements 10 20 50Measure/Source mean s.d. mean s.d. mean s.d.5% CW fit 0.6135 0.0392 0.5430 0.0355 0.4535 0.03565% Actual 0.5933 0.0487 0.5265 0.0513 0.4379 0.04295% Weib.fit 0.4661 0.0408 0.4022 0.0346 0.3280 0.03265% CW pred. 0.5345 0.0440 0.4733 0.0384 0.3938 0.03935% Ad.W.pred. 0.4386 0.0426 0.3762 0.0360 0.3036 0.03425% Weib.pred. 0.4262 0.0434 0.3645 0.0365 0.2928 0.034950% CW fit 1.1621 0.0718 1.0322 0.0567 0.9052 0.052250% Actual 1.1284 0.0465 1.0085 0.0491 0.8770 0.040950% Weib.fit 1.1878 0.0356 1.0735 0.0472 0.9431 0.042950% CW pred. 1.0110 0.0604 0.8985 0.0467 0.7844 0.04 1050% Ad.W.pred. 1.0849 0.0381 0.9716 0.0437 0.8398 0.037750% Weib.pred. 1.1168 0.0366 1.0031 0.0440 0.8716 0.0382aCW 5.1344 0.9318 5.1488 0.8981 5.0369 1.0422yCW 0.0120 0.0176 0.0171 0.0137 0.0387 0.0359icCW 0.9448 0.9526 0.7124 0.3193 0.6105 0.38248 CW 1.7438 2.2221 0.7086 0.5575 0.3025 0.3462tWeib. 0.4309 0.0421 0.5802 0.0687 0.8109 0.0906aWeib. 2.7931 0.2381 2.6624 0.2146 2.4792 0.2387aAd.W. 4.1054 0.3462 3.9152 0.3128 3.6476 0.3489pCW 0.3540 0.1125 0.3838 0.0951 0.4109 0.0991Table X1II.I:1 Percentiles and parameters from Weibull-based models209XIII Appendices Appendix 9 Summaies of Simulation Resultsp 0.5number of elements 100 500 1000Measure/source mean s.d. mean s.d. mean s.d.5% CW fit 0.4059 0.0323 0.3079 0.0235 0.2744 0.02 175% Actual 0.3975 0.0400 0.3040 0.0275 0.2657 0.02725% Weib.fit 02857 0.0321 0.2054 0.0193 0.1829 0.02085% CW pred. 0.35 16 0.0345 0.2622 0.0265 0.2305 0.023 15% Ad.W.pred. 0.2621 0.0331 0.1836 0.0199 0.1617 0.02125% Weib.pred. 0.2516 0.0335 0.1742 0.0201 0.1526 0.021350% CW fit 0.8214 0.0470 0.6864 0.0542 0.6547 0.052050% Actual 0.8055 0.0440 0.6688 0.0461 0.637 1 0.042450% Weib.fit 0.8730 0.0430 0.7335 0.0352 0.6930 0.037750% CW pred. 0.7104 0.04 12 0.5827 0.0369 0.5484 0.037750% Ad.W.pred. 0.7671 0.0420 0.6205 0.0302 0.5767 0.034450% Weib.pred. 0.7995 0.04 16 0.6546 0.0309 0.6116 0.0346aCW 4.9163 0.8798 4.4357 0.9380 4.0727 0.8332yCW 0.0429 0.0312 0.0826 0.0702 0.1243 0.0969icCW 0.5332 0.1806 0.5319 0.2325 0.5368 0.22396 CW 0.1546 0.1163 0.0977 0.1324 0.0859 0.0916‘r Weib. 0.9614 0.1131 1.3242 0.1632 1.4413 0.2109a Weib. 2.3392 0.2127 2.0516 0.1507 1.9594 0.1570a Ad.W. 3.4428 0.3116 3.0199 0.2224 2.8836 0.2323p CW 0.4404 0.0762 0.4592 0.0988 0.4387 0.1000Table XIII.I:1 Continuedp 0.350 100Measure mean s.d. mean s.d.5% CW fit 0.7398 0.0406 0.6580 0.03895% Actual 0.7258 0.0517 0.6437 0.04355% Weib.fit 0.5645 0.0436 0.483 1 0.05005% CW pred. 0.6635 0.0462 0.5886 0.04295% Ad.W.pred. 0.5332 0.0459 0.45 18 0.05285% Weib.pred. 0.5266 0.0464 0.4453 0.053350% CW fit 1.2423 0.0648 1.1201 0.043250% Actual 1.2197 0.0479 1.1024 0.04 1450% Weib.fit 1.2871 0.0451 1.1798 0.039350%CWpred. 1.1188 0.0808 1.0010 0.039150% Ad.W.pred. 1.1997 0.0432 1.0857 0.043450% Weib.pred. 1.2149 0.0431 1.1020 0.0421aCW 6.5186 1.1624 6.3152 0.9736yCW 0.0054 0.0062 0.0054 0.0055w CW 0.637 1 0.2797 0.5932 0.23096CW 1.8496 1.1688 0.8862 0.5315tWeib. 0.3144 0.0385 0.4299 0.0496a Weib. 3.1748 0.2905 2.93 16 0.3010a Ad.W. 3.8701 0.35 16 3.575 1 0.3654p CW 0.4034 0.1097 0.4384 0.0860Table Xliii: 1 Continued210XIII Appences Appendix 9 Summaries of Simutalon ResuIp 0.7number of elements 50 100Measure mean s.d. mean s.d.5% CW fit 0.1708 0.0182 0.1391 0.01455% Actual 0.1641 0.0230 0.1354 0.01665% Weib.flt 0.1104 0.0168 0.0868 0.01415%CWpred. 0.1384 0.0177 0.1106 0.01425% Ad.W.pred. 0.0981 0.0164 0.0751 0.01395% Weib.pred. 0.0877 0.0160 0.0655 0.013550% CW fit 0.4754 0.0526 0.4429 0.066150% Actual 04502 0.0426 0.4 174 0.036050% Weib.fit 0.4973 0.0403 0.4594 0.029350% CW pred. 0.3840 0.04 10 0.3503 0.044850% Ad.W.pred. 0.3933 0.037 1 0.3445 0.029550% Weib.pred. 0.4409 0.0380 0.3962 0.0286a CW 3.3464 0.6072 3.0816 0.5275y CW 0.3425 0.2069 0.3884 02222w CW 0.5968 0.2184 0.5436 0.2947ö CW 0.0466 0.0409 0.0463 0.0773‘t Weib. 2.3595 0.3697 2.3627 0.3137a Weib. 1.7304 0.1343 1.5631 0.1275a Ad.W. 3.3814 0.2653 3.0493 0.2543pCW 0.4128 0.0867 0.4289 0.1019Table X1I.I:1 Continued2. Normal-based data/Gumbel-based modelsp 0.1 0.1 0.5number of elements 100 50 100measure/source mean s.d. mean s.d. mean s.d.5% TO fit 6.7313 0.0990 6.9179 0.1205 6.9877 0.15255% Actual 6.6875 0.1394 6.8641 0.1564 6.9223 0.16505% Gumbel 6.4555 0.1479 6.6219 0.1325 6.4184 0.21805% Ad.G.Pred 6.1445 0.1759 6.2879 0.1626 6.0545 0.24955% Gumb.Pred. 6.1277 0.1774 6.2698 0.1642 5.9037 0.26285% TO Pred. 6.5300 0.1272 6.6952 0.1573 6.7103 0.187050% TO fit 7.6292 0.0656 7.8855 0.0477 8.2403 0.108950% Actual 7.6306 0.0724 7.8819 0.0556 82396 0.113150% Gumbel 7.687 1 0.0533 7.9448 0.0494 8.35 16 0.092550% Ad.G.Pred 7.3761 0.0716 7.6107 0.0570 7.9877 0.104250% Gumb.Pred. 7.3593 0.0727 7.5926 0.0579 7.8370 0.111750% TO Pred. 7.4278 0.0634 7.6628 0.0597 7.9630 0.0998C TG 0.8660 1.1159 0.8955 0.5706 0.7717 0.4667e TG 0.2905 0.0558 0.32 13 0.0590 0.4002 0.0743c TO -25.8302 52180 -24.0028 4.4466 -19.8395 3.8244co TO -294298 6.7525 -29.0529 7.4755 -25.8996 6.5722C Gumb. 7.8605 0.0482 8.1310 0.0536 8.6238 0.09150 Gumb. 0.4730 0.0445 0.5081 0.0485 0.7425 0.0698OAd.Gumb. 0.4487 0.0422 04820 0.0460 0.5250 0.0493pTG 0.4170 0.1759 0.3902 0.1615 0.5092 0.1408Table XTI.1:2 Results from Gumbel-based simulations211XIII Appendices Appendix 9 Summanes of Simulation Resultsp 0.5 0.7 0.7number of elements 100 100 50mean s.d. mean s.d. mean s.d.5% TG fit 7.1172 0.1354 7.2286 0.1500 7.3165 0.16845% Actual 7.0584 0.1727 7.1850 0.2198 7.2704 0.22565% Gumbel 6.5716 0.1955 6.5692 0.2093 6.6483 0.20965% Ad.G.Pred 62031 0.2277 6.2492 0.2346 6.3237 0.23795% Gumb.Pred. 6.0505 0.2412 5.9849 02557 6.0557 0.26175% TG Pred. 6.8261 0.1743 6.9246 0.1919 7.0074 0.202050% TG fit 8.4194 0.0931 8.6276 0.0875 8.7395 0.108350% Actual 8.4163 0.0933 8.6308 0.0928 8.7454 0.125050% Gumbel 8.5291 0.0776 8.7637 0.0710 8.8746 0.097250% Ad.G.Pred 8.1606 0.0833 8.4437 0.0808 8.5500 0.093850% Gumb.Pred. 8.0079 0.0896 8.1795 0.0940 8.2820 0.098650% TO Pred. 8.1283 0.0801 8.3235 0.0826 8.4304 0.0974C TO 0.8032 0.5552 0.6665 0.3065 0.7046 0.44406 TG 0.4200 0.0749 0.4386 0.0782 04459 0.0774TG -19.2003 2.6691 -18.7135 3.2485 -18.5670 2.7516oTG -25.5496 6.3041 -24.4765 5.2605 -24.7910 5.0876C Gumb. 8.8046 0.0830 9.0727 0.0704 9.1880 0.10926 Gumb. 0.75 18 0.0705 0.8429 0.0702 0.8551 0.0820OAd.Gumb. 0.5316 0.0498 0.4617 0.0385 0.4683 0.0449pTG 0.4859 0.1502 0.5386 0.1274 0.5363 0.1351Table XIII.I:2 Continuedp 0.9 0.9number of elements 100 50mean s.d. mean s.d.5%TGfit 7.6415 0.1546 7.7361 0.16115% Actual 7.5669 0.2066 7.7097 0.21205% Gumbel 6.8613 0.24 17 7.0192 0.28065% Ad.G.Pred 6.6532 0.2598 6.8169 0.29975% Gumb.Pred. 6.2034 0.2994 6.3795 0.34 165% TO Pred. 7.3095 0.2002 7.4033 0.205 150%TGfit 9.1756 0.1138 9.2766 0.118050% Actual 9.1753 0.1237 9.2953 0.128750% Gumbel 9.3326 0.0991 9.4220 0.100950% Ad.G.Pred 9.1245 0.0987 9.2197 0.107050% Gumb.Pred. 8.6747 0.1104 8.7823 0.130450% TG Pred. 8.8437 0.0923 8.9438 0.1129C TO 0.6656 0.3471 0.6599 0.34280 TO 0.4789 0.0904 0.4801 0.08332TG -18.9818 3.0452 -19.0459 2.8437w TO -24.6243 4.3390 -24.5683 4.1993Gumb. 9.6805 0.108 1 9.7602 0.09970 Gumb. 0.9492 0.0894 0.9228 0.0929OAd.Gumb. 0.3002 0.0283 0.2918 0.0294pTG 0.5568 0.1383 0.5382 0.1314Table XII.I:2 Continued212XIII Appendices Appendix 10 Summaries of Experimental ResultsJ. Appendix 10Summaries of Experimental ResultsNote: all results are for strengths scaled to units of 1OMPa.Data Set var(t) a var(a)M430 O.9754E-03 0.3225E-06 0.3505E+01 0.7467E-0lM490 0.9726E-03 0.3274E-06 0.3808E+01 0.8995E-0lM4120 0.1159E-02 0.4349E-06 0.3714E÷01 0.8387E-0lM1030 0.1384E-03 0.1027E-07 0.4564E+0l 0.1266E+00MI 090 0.2763E-03 0.3594E-07 0.4509E+01 0.126 1E+00M10120 0.5330E-03 0.1 134E-06 0.4191E+01 0.1079E÷00N430 0.3272E-01 0.1090E-03 0.2277E+01 0.3217E-01N490 0.5 128E-01 0.2146E-03 0.2222E+0l 0.3062E-01N4120 0.4831E-01 0.1964E-03 0.2286E+01 0.3243E-0lNI 030 0.751 1E-01 0.3533E-03 0.1 849E+01 0. 1999E-01N1090 0.1410E+00 0.8501E-03 0.1 707E+01 0.1703E-01N10120 0.1726E+00 0.1114E-02 0.1632E+0I 0.1558E-01L8 0.7393E-03 0.1990E-06 0.44 IOE+01 0.1 182E+00L9 0.7480E-02 0.1018E-04 0.3326E+01 0.6723E-01LI2 0.1562E-01 0.3358E-04 0.3033E+01 0.5592E-01L14 0.2823E-0I 0.85 1 IE-04 0.2672E+01 0.4341E-01L16 0.1878E-01 0A499E-04 0.2992E÷01 0.5441E-01L17 0.7876E-03 0.2223E-06 0.4730E+01 0.1360E+00L33 0.9143E-01 0.4863E-03 0.2628E+01 0.4198E-01L35 0.6046E-01 0.2682E-03 0.2379E+01 0.3440E-0IL36 0.1188E+00 0.7006E-03 0.2266E+01 0.3121E-01P5 0.1535E-01 0.2722E-04 0.3043E+01 0.4692E-01P8 0.1 749E-01 0.3378E-04 0.3085E+0l 0.4863E-01P16 0.3609E-01 0.1069E-03 0.2842E+01 0.4234E-01Fl 0 0.6488E-02 0.6053E-05 0.3774E+01 0.65 12E-01F15 0.5819E-02 0.5055E-05 O.4025E÷0l 0.7406E-0lF19 0.1 336E-Ol 0. 1952E-04 0.3360E+0l 0.51 23E-01Table 1: Parameters of Weibull distribution213XIII Appendices Appendix 10 Summaries of Experimental ResultsC var(C) 9 var(9)M430 0.7543E-i-01 0.5056E-01 0.2136E+01 0.2773E-01M490 0.6405E+0I 0.3096E-0i 0.1654E+01 0.1698E-0lM4120 0.6412E+01 0.3306E-01 0.1727E+01 0.1813E-01M1030 0.7181E-i-01 0.2366E-0l 0.1461E+01 0.1297E-01M1090 0.6312E+01 0.1957E-01 0.1315E+Ol 0.1073E-01M10120 0.6220E+01 0.1921E-0i 0.1460E+01 0.1124E-0lN430 0.4956E+01 0.5019E-0i 0.2106E+01 O.2752E-01N490 0.4230E--0l 0.3919E-Ol 0.1861E+O1 O.2149E-0lN4120 0.4166E+01 0.3823E-01 0.1838E÷O1 0.2096E-O1N1030 0.4699E+0i 0.6019E-0i 0.2376E+01 0.3300E-0lN1090 0.3765E+01 0.5251E-0l 0.2219E+0i 0.2879E-01N 10120 0.3553E+01 0.4498E-0l 0.2054E+01 0.2466E-01L8 0.5266E+0i 0.1355E-01 0.1 106E-i-01 0.7432E-02L9 0.4558E+0i 0.1690E-01 0.1235E+01 0.9266E-02L12 0.4 166E+01 0.1 833E-0i 0.1286E+01 0. 1005E-.01L14 0.4079E+01 0.2273E-01 0.1432E-i-01 0.1247E-01L16 0.4000E+0i 0.1928E-01 0.1319E+01 0.1057E-0iLi 7 0.4635E+Oi 0.9041 E-02 0.903 iE+O0 0.4958E-02L33 0.2679E+Oi 0.1 130E-01 0.lOiOE-i-01 0.6199E-02L35 0.3404E.,-01 0.1449E-01 0.1143E+01 0.7947E-02L36 0.2772E+01 0.1254E-01 0.1064E+0i 0.6877E-02P5 0.4 160E÷01 0. 1638E-01 0.133 1E+01 0.8979E-02P8 0.3916E+01 0.1373E-01 0.1214E+01 0.7527E-02P16 0.34 36E+01 0. 1425E-0 1 0.1 221E+01 0.781 3E-02FlO 0.3940E+01 0.9418E-02 0.i063E-i-Oi 0.5164E-02F15 0.3711E+Oi 0.7499E-02 0.9485E+00 0.41 12E-02Fl 9 0.3787E+01 0.1 227E-0 1 0.121 8E+0i 0.6726E-02Table XIH.,.J:2 Parameters of the Gumbel Distribution214XIII Appendices Appendix 10 Summaries of Experimental ResultsDataset a y 8M430 0.7620E+01 0.1962E-07 0.3494E+00 0.17 18E.i-06M490 0.5496E+01 0.1029E-04 0.6182E+00 0.7392E+04M4120 0.4850E+01 0.1744E-04 0.1606E+01 0.8135E+04M1030 0.7632E+01 0.1 122E-06 0.2007E÷00 0.1 104E÷06M1090 0.7380E+01 0.4803E-06 0.2132E+00 0.3079E+05MiOl 20 0.5001E+01 0.245 1E-04 0. 1989E+01 0.1477E+05N430 0.389 1E+01 0.471 3E-03 0.4327E-t.00 0.451 3E-i-02N490 0.5223E+01 0.4932E-04 0.3302E-i-00 0.4002E+02N4120 0.6046E+01 0.7932E-05 0.3420E+00 0.9528E+02N 1030 0.4696E-,-O1 0.8062E-04 0.2408E+00 0.973 IE÷01N1090 0.4061E÷O1 0.4275E-03 0.3319E+00 0.3725E+01N10120 0.5226E+01 0.6420E-04 0.2270E-i-00 0.1745E+01L8 0.5151E+01 0.1787E-03 0.3338E-01 0.6038E+02L9 0.5801E-i-01 0.7165E-04 0.1 120E+00 0.4477E÷02L12 0.3100E+01 0.9338E-02 0.3074E+01 0.5773E+03L14 0.5261 E+01 0.1 396E-03 0.2655E÷00 0.4161 E÷02L16 0.5038E+01 0.2109E-03 0.4300E+00 0.1071E+03L17 0.6740E+01 0.2014E-04 0.1 139E+00 0.7453E+03L33 0.5920E+01 0.2989E-03 0.3120E+00 0.7451E÷01L35 0.5758E+01 0.1411E-03 0.1784E÷00 0.7101E+01L36 0.3543E+01 0.11 12E-01 0.2790E+00 0.2392E+01P5 0.7254E÷01 0.1948E-05 0.3619E+00 0.9327E+03P8 0.501 IE+01 0.3527E-03 0.3185E+00 0.6556E+02P16 0.4248E+01 0.6210E-03 0.1068E+01 0.8585E÷02Fl 0 0.7404E+01 0.4640E-05 0.3826E+00 0.1 386E+04F15 0.8028E+01 0.1520E-05 0.5613E+00 0.4143E+04F19 0.7172E+01 0.1970E-05 0.5792E÷00 0.1355E+04Table XIII.J:3 Parameters of the CW distribution215XIII Appendices Appendix 10 Summaries of Experimental ResultsTable XIII.J:4 Parameters of the TG distributionM430M490M41 20M1030M1090M10120N430N490N4 120N1030N1090N10120L8L9L12L14L16L17L33L35L36P5pgP16FlOFl 5F190.3001E-i-O00.4691E+000.5548E-i-000.2774E-i-000.3186E+000.9440E+000.2289E+000.1 748E+O00.1 585E+000.5882E-010.771 IE-Ol0.6012E-010.6566E+000.1987E-i-010.55 19E+000.2247E+000.3413E+000.33 19E+000.9607E-010.5261E-010.1 537E+000.2301E÷000.5259E+000.3558E+000. 1728E+000.3459E+000.2653E+000.62 12E+000.6861E+00O.7109E+O00.6626E+O00.6062E+000.7910E+000.4575E+000.2961E+000.2532E+000.1475E+000.1463E+000.1 107E+000.7346E-i-000.9386E+000.5805E+000.3594E+000.4126E+000.5616E+000. 108 IE+000.1 109E÷000.21 36E+O00.2993E÷O00.5098E+000.361 3E+000. 198 IE+000.2804E+000.2562E+00-0.7653E+01-0.6720E+01-O.6779E+01-0.7294E+0l.-O.7188E+01-0.6850E-i-01-0.4717E+0l-0.5869E+01-0.7166E+01-0.8061E-i-01-0.633 1E+01-0.7237E+01-0.6028E+0l-0.5126E+01•0.5053E+01-0.5472E+01-0.5750E÷01-0.5945E÷01-0.1041E+02-0. 1005E+02-0.4783E+01-0.7924E-i-01-0.521 3E÷01-0.5736E+0l-0.1 188E+02-0.9299E+0l-0. 1874E+02-0.1367E+02-0.1417E+02-0.1 321E+02-0. 1295E+02-0.8160E+03-0.2036E+02-0.7760E+02-0.3 163E+02-0.6069E+02-0.6373E+02-0.7042E-i-02-0.8901E-i-01-0.2323E÷04-0.11 13E+02-0.18 19E+02-0.1637E+02-0.9648E+01-0.433 1E+02-0.4295E+02-0.2140E+02-0.21 88E+02-0.1 884E+04-0.1 579E+02-0.3244E+02-0.201 1E-i-02-0.2801E+02216XVI Appendices Appendix 10 Summaries of Experimental ResultsDataset CW var(a) CO var(C)M430 0.1967113+01 0.2078413-01M490 0.12907D+01 0.16550D-01M4120 0.6508813+00 0.16391D-01M1030 0.2509213+01 0.1279113-01M1090 0.2183413÷01 0.12868D-01M10120 0.6609013+00 0.1254113-01N430 0.56576D+00 0.1573013-01N490 0.8073213+00 0.62407D-02N4120 0.1044013+01 0.3983413-02N1030 0.69177D+00 0.29844D-02N1090 0.4697613÷00 0.26785D-02N10120 0.78594D-i-00 0.21888D-02L8 0.76371D÷00 0.20692D-01L9 0.8676713+00 0.1134013-01L12 0.2122013+00 0.1094513-01L14 0.9087613+00 0.8443813-02L16 0.97 18913+00 0.9880813-02L17 0.1884613+01 0.13296D-01L33 0.1261613+01 0.1472413-02L35 0.1944413+01 0.2406713-02L36 0.6842613+00 0.71537D-02P5 0.1559113+01 0.3641313-02P8 0.86034D+00 0.24405D-01P16 0.4037713+00 0.4140713-02FlO 0.1348813+01 0.4169113-02F15 0.1541713+01 0.24279D-02F19 0.1161013+01 0.3106813-02Table XIIIJ:5 Variance of length effect parameters of mixed distributions217XIII Appendices Appendix 10 Summaries of Experimental Results- Log LikelihoodsData Set Gumbel CO Weibull CWM430 0.2244E-i-03 0.2054E+03 0.2102E-i-03 0.2035E+03M490 0.1957E+03 0.1837E+03 0.1848E+03 0.1830E+03M4120. 0.2025E+03 0.1874E+03 0.1906E+03 0.1887E+03M1030. 0.1920E.+-03 0.1858E+03 0.1880E+03 0.1864E÷03M1090. 0.1771E+03 0.1711E+03 0.1721E+03 O.1704E+03MiOl 20 0.1 854E+03 0.1 763E+03 0.1 770E+03 0.1761E+03N430 0.2172E+03 0.1924E+03 0.1943E+03 0.191 1E+03N490 0.2049E+03 0.1 739E÷03 0.1791E+03 0.1 708E+03N4 120. 0.2028E+03 0. 1635E+03 0.175 1E+03 0.163 1E+03N1030. 0.2430E+03 0.1979E+03 0.21 13E+03 0.2027E+03NI 090. 0.2326E+03 0. 1807E+03 0. 1903E+03 0.18 12E+03N10120 0.2270E+03 0.1726E+03 0.1860E÷03 0.1708E+03L8 0.1638E+03 0.1613E+03 0.1609E+03 0.1604E÷03L9 0.1742E÷03 0.1722E÷03 0.1681E+03 0.1646E+03L12 0.1748E+03 0.1638E+03 0.1651E+03 0.1651E+03L14 0.1855E+03 0.1677E+03 0.1698E+03 0.1658E÷03L16 0.1756E+03 0.1582E+03 0.1592E+03 0.1563E+03L17 0.1446E+03 0.1426E+03 0.1426E+03 0.1417E+03L33 0.1482E+03 0.1 192E+03 0.1272E+03 0.1204E÷03L35 0.1516E+03 0.1305E+03 0.1375E+03 0.1326E+03L36 0.1549E+03 0.1374E+03 0.1373E+03 0.1359E+03P5 0.21 37E+03 0.1 824E+03 0.1 925E+03 0.181 5E+03P8 0.201 IE+03 0.1885E+03 0.1854E+03 0.1831E+03P16. 0. 1932E+03 0. 1647E+03 0.1701 E+03 0. 1657E+03FlO 0.2042E+03 0.1800E+03 0.1852E+03 0.1784E+03F15 0.1884E+03 0.1585E+03 0.1689E+03 0.1584E+03F19 0.2193E+03 0.1767E+03 0.1905E+03 0.1766E+03Table XII.J:6 Log likelihood of different distributions218XIII Appendices Appendix 10 Summaries of Experimental ResultsData set 5%Actual 5%CG 5%Gumbel 5%CW 5%WeibullM430 0.3654E+01 0.37 1OE-t-01 0. 1200E+O1 0.3813E+01 0.3098E+01M490 0.2893E+01 0.3128E+01 0.1492E-i-01 0.3167E+01 0.2833E+01M4120 0.3147E-i-O1 0.3158E+01 0.1283E+01 0.3101E+01 0.2774E+01M1030 0.3806E+O1 0.3769E+01 0.2842E-t-01 0.3856E+01 0.3655E+01M1090 0.31 34E+01 0.3293E+01 0.2405E+01 0.3363E+01 0.31 85E+O1M10120 0.3100E+01 0.3140E+01 0.1890E+01 0.3180+01 02970E+01N430 0.1490E+01 0.1526E+01 -0.1301E÷01 0.1541E+01 0.1218E+01N490 0.1361E+01 0.1419E+01 -0.1298E-t-01 0.1439E+01 0.1000E+01N4120 0.1477E+O1 0.1571E+01 -0.1294E+01 0.1571E+01 0.1027E+01N1030 0.1209E÷O1 0.1238E+01 -0.2358E-i-01 0.1194E+01 0.8136E+00N1090 0.8699E+OO 0.9182E÷00 -0.2827E+01 0.8886E-t-00 0.5531E+00N10120 O.8192E+OO 0.8339E-t-00 -0.2548E+01 0.8554E+00 0.4754E+00L8 0.2358E÷O1 0.2519E+01 0.1982E÷01 0.2455E+01 0.2615E+01L9 0.1716E+01 0.1391E+O1 0.8910E÷00 0.1740E+01 0.1784E+01L12 0.1333E-i-01 0.1579E+01 0.3476E+00 0.1498E+01 0.1480E+01L14 0.1453E+01 0.1477E÷01 -0.1739E+00 0.1507E+01 0.1250E+01L16 0.1680E+01 0.1622E+01 0.8375E-01 0.1659E+01 0.1399E÷01L17 0.2451E+01 0.2288E+01 0.1953E+01 0.2388E+01 0.2418E+01L33 0.1080E+O1 0.1087E+01 -0.3198E÷00 0.1048E÷01 0.8025E-i-00L35 0.1 128E+01 0.1 170E+01 0.7690E-02 0.1 160E+01 0.9332E+00L36 0.7856E-i-00 0.8237E+00 -0.3872E÷00 0.7898E+00 0.6902E+00P5 0.1904E÷01 0.1956E+01 0.2056E+00 0.1979E+01 0.1486E÷01P8 0.1597E+01 0.1496E+01 0.3104E+00 0.1601E+01 0.1417E+01P16 0.1193E+01 0.1399E+01 -0.1907E-i-00 0.1387E÷01 0.1132E+01FlO 0.2152E+O1 0.2143E+01 0.7829E+00 0.2039E÷01 0.1729E+O1F15 0.2065E+01 0.2094E÷01 0.8942E+00 0.2104E-t-01 0.1717E+01F19 0.1931E+01 0.1990E+01 0.1707E+00 0.1960E+01 0.1492E÷01Table XIIIJ:7 Fifth percentiles219XIII Appendices Appendix 10 Summaries of Experimental ResultsData set 50%Actual 50%CG 50%Gumbel 50%CW 50%WeibullM430 0.6233E+01 0.6123E+01 0.6761E+01 0.6136E+01 0.6511E+01.M490 0.5423E+01 0.5441E+01 0.5799E÷01 0.5470E+01 0.5613E+01M4120 0.5467E+01 0.5464E+01 0.5779E+01 0.5494E+01 0.5592E+01M1030 0.6391E+01 0.6362E-i-01 0.6645E+01 0.6413E+01 0.6466E+01M1090 0.5567E+01 0.5555E+OI 0.5830E+01 0.5616E+01 O.5675E+01Ml 0120 0.54800+01 O.5480E+O1 0.5690E+01 0.5470E+01 O.5530E+01N430 0.3550E÷O1 0.3521E+01 0.4184E.i-01 0.3599E+01 0.3822E+01N490 0.2839E+01 0.2906E÷01 0.3548E+01 0.2928E+01 0.3228E+01N4120 0.2845E+01 0.2918E+01 0.3492E+01 02895E÷01 O.3206E+01N1030 O.2954E.i-(J1 O.2927E+O1 0.3828E+01 0.2925E÷01 0.3326E+01N1090 0.2 165E+O1 0.2242E÷01 0.2952E+01 0.221 8E+01 0.2542E+O1N10120 0.1934E+01 0.2077E+01 0.2800E+01 0.1973E+01 0.2343E+01L8 0.4769E+01 0.4736E+01 0.4860E+01