Statistical Modeling Of Fracture Toughness Data by Guru Prakash A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in Civil and Environmental Engineering Waterloo, Ontario, Canada, 2007 © Guru Prakash, 2007 AUTHOR'S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract The fracture toughness of the zirconium alloy (Zr-2.5Nb) is an important parameter in determining the flaw tolerance for operation of pressure tubes in reactor. Fracture toughness data have been generated by performing rising pressure burst tests on sections of pressure tubes removed from operating reactors. The test data were used to generate a lower-bound fracture toughness curve, which is used in defining the operational limits of pressure tubes. The thesis presents a comprehensive statistical analysis of burst test data and develops a multivariate statistical model to relate toughness with material chemistry, mechanical properties, and operational history. The proposed model can be useful in predicting fracture toughness of specific in-service pressure tubes, thereby minimizing conservatism associated with a generic lower-bound approach. iii Acknowledgements I would like to thank my supervisor Professor Mahesh D. Pandey for his support, encouragement, guidance, and patience during my study at the University of Waterloo. I would like to thank Professor Galena Morozova for providing me Teaching Assistantship for the CIVE153 course. I would like to thank Mike Cocker for supporting me in fixing my computer. I would also like to thank the University of Waterloo for providing financial support in the form of the International Master Student Award. Finally, I would like offer my thanks to my friends for their financial, emotional, and social support. iv Table of Contents AUTHOR'S DECLARATION ............................................................................................................... ii Abstract ................................................................................................................................................. iii Acknowledgements ............................................................................................................................... iv Table of Contents ................................................................................................................................... v List of Figures ..................................................................................................................................... viii List of Tables ......................................................................................................................................... ix Chapter 1 Introduction ............................................................................................................................ 1 1.1 Introduction .................................................................................................................................. 1 1.2 Objectives ..................................................................................................................................... 1 1.3 Thesis Organization ...................................................................................................................... 2 Chapter 2 Preliminary Analysis .............................................................................................................. 3 2.1 CSA Method to determine lower bound fracture toughness ........................................................ 3 2.1.1 Statistically based fracture toughness .................................................................................... 3 2.2 Data .............................................................................................................................................. 4 2.2.1 Material ................................................................................................................................. 4 2.2.2 Burst Test Procedure and Analysis ....................................................................................... 5 2.3 Preliminary Analysis .................................................................................................................... 6 2.4 Concluding Remark ...................................................................................................................... 8 Chapter 3 Linear Regression Model ..................................................................................................... 10 3.1 Introduction: ............................................................................................................................... 10 3.2 The Linear model and Assumptions ........................................................................................... 11 3.3 Least Square Estimation ............................................................................................................. 12 3.4 Accuracy of Estimates ................................................................................................................ 13 Λ 3.4.1 Variance of β 1 .................................................................................................................... 13 ∧ 3.4.2 Variance of β 0 ................................................................................................................... 14 3.5 Predicted Values and Residuals .................................................................................................. 15 3.6 Analysis of Variation in the Dependent Variable ....................................................................... 15 3.7 Tests of Significance and Confidence Intervals ......................................................................... 17 v 3.8 Effect of Texture Parameter ....................................................................................................... 19 3.9 Case study: Fracture Toughness Vs Temperature ...................................................................... 20 3.9.1 Temperature less than 1500C .............................................................................................. 20 3.9.2 Temperature greater than 1500C ......................................................................................... 22 3.10 Concluding Remark ................................................................................................................. 23 Chapter 4 Linear Regression Model .................................................................................................... 24 4.1 The model .................................................................................................................................. 24 4.2 Normal Equations and Solutions ................................................................................................ 27 4.3 Correlation Analysis .................................................................................................................. 28 4.4 Correlation among Covariates ................................................................................................... 29 4.4.1 Pair-wise Correlation Analysis ........................................................................................... 29 4.4.2 Stepwise Estimation: Selecting the first Variable ............................................................... 30 4.4.3 Selecting the second Variable ............................................................................................. 35 4.4.4 Selecting the third Variable ................................................................................................ 35 4.4.5 Predictive Model for samples with Low Chlorine Content ................................................ 41 4.5 Concluding Remark ................................................................................................................... 44 Chapter 5 Stepwise Regression ............................................................................................................ 46 5.1 Stepwise Regression Methods ................................................................................................... 46 5.1.1 Forward Selection ............................................................................................................... 46 5.1.2 Backward Elimination ........................................................................................................ 46 5.1.3 Stepwise Selection .............................................................................................................. 47 5.2 Stopping Rules ........................................................................................................................... 47 5.3 Model development fracture toughness data .............................................................................. 48 5.4 Results ........................................................................................................................................ 50 5.5 Discussion .................................................................................................................................. 53 5.6 Model development for low Chlorine content data .................................................................... 55 5.7 Results