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Acknowledgement  
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Uncertainty analysis using stochastic...  
Reliabilitybased design optim...  
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Fatigue reliabilitybased load...  
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Page i Page ii Dedication Page iii Acknowledgement Page iv Page v Table of Contents Page vi Page vii Page viii List of Tables Page ix Page x List of Figures Page xi Page xii Abstract Page xiii Page xiv Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Literature survey Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Uncertainty analysis using stochastic response surface Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Reliabilitybased design optimization Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Global sensitivity analysis for efficient RBDO Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Fatigue reliabilitybased load tolerance design Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Robust design using stochastic response surface Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Summary and recommendations Page 113 Page 114 Appendices Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 References Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Biographical sketch Page 134 

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RELIABILITYBASED DESIGN AND LOAD TOLERANCE EVALUATION USING STOCHASTIC RESPONSE SURFACE AND PROBABILISTIC SENSITIVITIES By HAOYU WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Haoyu Wang To my family ACKNOWLEDGMENTS I would like to express my appreciation to my advisor, Professor NamHo Kim, for his endless encouragement and continuous support during my Ph.D. research. Without his guidance, inspiration, experience and willingness of sharing his knowledge, this work would have never been possible. Dr. Kim made a tremendous contribution to this dissertation, as well as my professional and personal life. I must express my gratitude to the members of my supervisory committee, Professor Raphael T. Haftka, Professor Stanislav Uryasev, Professor Nagaraj K. Arakere, and Professor Ashok V. Kumar, for their willingness to review my Ph.D. research and provide constructive comments to help me complete this dissertation. Special thanks go to Professor Raphael T. Haftka for not only his guidance with several technical issues during my study, but also comments and suggestions during group meetings which were extremely helpful for improving my work. Special thanks are also given to Professor Nestor V. Queipo from the University of Zulia at Venezuela, for his interaction in my research and collaboration in publishing papers during his visiting at the University of Florida. My colleagues in the Structural and Multidisciplinary Optimization Lab at the University of Florida also deserve my gratitude. In particular, I thank Dr. Xueyong Qu, Dr. Amit A. Kale, Dr. Erdem Acar, Tushar Goel, Long Ge, Saad Mukrus for their encourage and help. My parents deserve my deepest appreciation for their constant love and support. Lastly, I would like to thank my beautiful and lovely wife, Zhilan, for her love, patience and support during my study. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ..................................... .. .......... ...................................... ix LIST OF FIGURES ........................................ .............. xi ABSTRACT ............................................ ............. ................. xiii CHAPTER 1 IN T R O D U C T IO N .................................................................. .. ... .... ............... 1 M motivation ..................................................................................................... ...............1 Obj active ............................................. ............................... 2 Scope .................................................. .......................3 O u tlin e ........................................................................... .................................... . 5 2 LITERA TURE SU RVEY ..........................................................................................7 Uncertainty and Reliability Analysis of Structural Applications ..............................7 R eliabilityB ased D esign O ptim ization........................ ...........................................11 Sensitivity in R liability A analysis ...........................................................................12 D im ension R education Strategy ................................................................................13 R obust D design ............................................................................... ...................... 13 F atigu e L ife P reduction .......................................................................... ...............14 3 UNCERTAINTY ANALYSIS USING STOCHASTIC RESPONSE SURFACE .... 19 In tro d u ctio n ............... .................................................... ..................... 1 9 Description of Uncertainty M odel ...........................................................................20 Stochastic Response Surface Method (SRSM)...................................................22 Polynomial Chaos Expansion (PCE) in Gaussian Space .................................22 Numerical Example of Stochastic Response Surface.......................................26 Improving Efficiency of SRS Using Local Sensitivity Information ..........................30 ContinuumBased Design Sensitivity Analysis ................................................31 Constructing SRS Using Local Sensitivity.......................................................34 Numerical Example Torque Arm M odel.......................................................36 Sum m ary ......................................................................................... ............. ....... 37 4 RELIABILITYBASED DESIGN OPTIMIZATION...........................................39 G general R B D O M odel .................. ........ ... .............. ............. ........................... 39 Reliability Index Approach (RIA) and Performance Measure Approach (PMA)......41 Probability Sensitivity A analysis (PSA ) ................................................. ................ 43 Probability Sensitivity Analysis in FORM ..................................... ................ 44 Probability Sensitivity Analysis Using SRSM...............................................47 ReliabilityBased Design Optimization Using SRSM...........................................48 R B D O w ith R IA .................................................................................................. 4 9 R B D O w ith Inverse M measure ......................................................... ................ 52 S u m m a ry .................................................................................................................. .. 5 3 5 GLOBAL SENSITIVITY ANALYSIS FOR EFFICIENT RBDO......................... 54 In tro d u ctio n ................................................................................................................. 5 4 Sensitivity A analysis ............................................................. .. .... ............... ........ ..... 55 VarianceBased Global Sensitivity Analysis (GSA).............................................56 Global Sensitivity Analysis Using Polynomial Chaos Expansion ..........................58 Adaptive Reduction of Random Design Space Using GSA in RBDO.................... 59 S u m m a ry ..................................................................................................................... 6 5 6 FATIGUE RELIABILITYBASED LOAD TOLERANCE DESIGN................... 66 In tro d u ctio n ................................................................................................................. 6 6 F atigu e L ife P reduction ............................................... .......................................... 67 Crack Initiation Fatigue Life Prediction ............... .............. ..................... 68 Variable Amplitude Loading and Cumulative Damage .................................70 Model Preparation for Fatigue Reliability Analysis ........................ ..................... 71 Finite Elem ent M odel .. .. ................. ............................................... 71 D ynam ic L oad H history ................................................................... ................ 73 Uncertainty in Material Properties and SN Curve Interpolation .....................74 Uncertainty Modeling of Dynamic Loadings........................................................75 Linear E stim ation of Load Tolerance.................................................... ................ 76 Variability of Dynamic Load Amplitude ............... ...................................77 V ariability of M ean of Dynam ic Load ........................................... ................ 82 Safety Envelope Concept for Load Tolerance Design ..........................................84 Numerical Path Following Algorithm................ ...................................85 Example for MultiDimensional Load Envelope ...........................................88 C conservative D distribution Type............................................................. ................ 90 S u m m a ry .................................................................................................................. ... 9 2 7 ROBUST DESIGN USING STOCHASTIC RESPONSE SURFACE................... 94 In tro d u ctio n .. ..................... ... .... ............................................... ..................... 9 4 Performance Variance Calculation Using SRS .............. .....................................96 Variance Sensitivity ................................ ......... ......................97 Robust Design Twolayer Beam...... .......... ........ ..................... 102 Dynamic Response of TwoLayer Beam ..................................... ................ 102 Robust D esign for Tw oLayer Beam ....... .......... ....................................... 103 Global Sensitivity A analysis ...... ............. ............ ..................... 106 Robust Design by Tolerance Control ....... ... .... ...................... 107 S u m m ary ................................................................................................ .... ........... 1 1 1 8 SUMMARY AND RECOMMENDATIONS .....................................................113 APPENDIX A SAMPLINGBASED PROBABILITY SENSITIVITY ANALYSIS FOR DIFFERENT DISTRIBUTION TYPE...... .... .........................115 Normal Distribution X, N (,/ o2) ................................1... 15 C ase 1: 0 = u .............. ............................................... ................. .......... 115 C ase 2 : 0 = cr .............. ................................................................ ........ 116 U uniform D distribution .............. .................. ................................................ 117 L ogN orm al D distribution ...................................... ......................... ............... 119 B NATURAL FREQUENCY OF CANTILEVER COMPOSITE BEAM............... 122 B en d in g M o m en t ..................................................................................................... 12 2 Geom etric Properties of Composite Beam ....... .......... ...................................... 122 Effective Compliance for Composite Beam ...... .... ..................................... 123 Effective Mass for Composite Beam...... .... ...... ..................... 123 LIST O F REFEREN CE S .. .................................................................... ............... 125 BIOGRAPH ICAL SKETCH .................. ............................................................. 134 LIST OF TABLES Table page 31. The type of polynomials and corresponding random variables for different Askey Chaos (N >0 denotes a finite integer)................................................... ................ 22 32. Root mean square error of PDF compared with the exact PDF of performance fu n ctio n y = ex.............. ....................................................................... 2 7 33. Comparison of probability of G>520MPa obtained from different uncertainty analysis methods (Full sampling without using local sensitivity).........................30 34. Comparison of probability of G>520MPa obtained from different uncertainty analysis methods (reduced sampling using local sensitivity) ; .....................37 35. Comparison of probability of G>520MPa obtained with/without local sensitivity (7/27 sam pling points) using 2nd order SRS........................................ ................ 37 41. Probability sensitivity with respect to random parameters (unit: centimeter)............46 42. Computational efficiency of analytical method for probability sensitivity .............47 43. Definition of random design variables and their bounds. The values of design variables at optimum design are shown in the 5th column (unit: centimeter). ........50 44. Reliability Index of active constraint at optimal design........................................52 51. Variances of the Hermite bases up to the second order.........................................58 52. Global sensitivity indices considering only main factors for the torque arm model at the initial design. Only three random variables (u2, u6, and us) are preserved w hen a threshold value of 1.0% is in place ........................................ ................ 63 53. Comparison of the number of random variables in each design cycle. The threshold of 1.0% is used. The first constraint is listed ................. ..................... 64 61. Q quality of response surface ........................................ ........................ ................ 78 62. T statistic of the coefficients ....................................... ....................... ................ 79 71. Random variables for cantilevered beam structure ............................... ................ 99 72. Variance estimation of linear performance (strength)................... .................. 100 73. Variance estimation of nonlinear performance (deflection)............................... 101 74. Sensitivity of variance for linear performance (strength)...................................102 75. Sensitivity of variance for nonlinear performance (deflection)............................. 102 76. Random parameters for the composite beam structure ................. ...................104 77. Sensitivities of objective functions at the initial design (ts = 6tm, tp = 0.2[tm, L = 1000[tm ) ........................................................................................................ 105 78. Total sensitivity indices for the composite beam structure (ts = 6tm, tp = 0.2[tm, L = 10 0 0[tm ) ............................................................................. . ... ............... 10 7 79. Sensitivity of variance for linear performance (strength)...................................109 710. Sensitivity of variance for nonlinear performance (deflection)........................... 109 711. Random variables and costtolerance functions....... .................. .................. 110 712. Random variables and costtolerance functions...................................................111 A1: Accuracy of proposed probability sensitivity method for normal distribution using 200,000 sam pling M CS ....... ........... ............ ..................... 116 A2: Accuracy of proposed probability sensitivity method for uniform distribution using 200,000 sam pling M CS ....... ........... ............ ..................... 119 A3: Accuracy of proposed probability sensitivity method for Lognormal distribution using 200,000 sam pling M CS ....... ........... ............ ..................... 121 LIST OF FIGURES Figure page 31. Limit state function divides the safe region from the failure region .......................21 32. PDF of performance function y(x) = ex ..........................................27 33. Shape design param eters for the torque arm ......................................... ................ 28 34. PDF of performance function G(x) torque arm model......................................29 35. Variation of a structural domain according to the design velocity field V(x)............32 36. PDF of performance function G(x) Torque model at initial design (SRS with se n sitiv ity ) ................................................................................................................ 3 6 41. Flow chart for reliabilitybased design optimization.............................................43 42. Optimum design and stress distribution of the torque arm model with 8 random v a ria b le s ............................................................................................................. .. 5 1 43. Optimization history of cost function (mass) for the torque arm model with 8 ran d om v ariab les. ..................................................................................................... 5 1 44. PDF of the performance function at the optimum for the torquearm problem .........52 51. Global sensitivity indices for torque arm model at initial design.............................. 59 52. Adaptive reduction of unessential random design variables using global sensitivity indices in RBDO. Loworder SRS is used for global sensitivity analysis, while a highorder SRS is used to evaluate the reliability of the system. .61 53. Optimum designs for the full SRS (solid line) and adaptively reduced SRS (dotted line). Because some variables are fixed, the interior cutout of the design from the adaptively reduced SRS is larger than that from the full SRS. ........................64 61. Flow chart for fatigue life prediction..................................................... ................ 67 62. R ainfl ow and hysteresis .......................................... .......................... ................ 70 63. Front loader frame of CAT 994D wheel loader (subject to 26 channels of dynamic lo a d in g ) .................................................................................................................. ... 7 2 64. Finite elem ent m odel for front fram e .................................................... ................ 73 65. M material SN curve w ith uncertainty...................................................... ................ 74 66. Illustration of one channel of dynam ic loads......................................... ................ 75 67. Reliability index /Twith respect to random parameter Iy ................ ............... 81 68. Probability of failure Pf with respect to random parameter [y .............................. 81 69. Reliability index / with respect to random parameter a ................. ..................... 84 610. Probability of failure Pf with respect to random parameter [a ................................84 611. Safety envelope for tw o variables ....................................................... ................ 85 612. Predictorcorrector algorithm ..................................... ...................... ................ 86 613. C construction of load envelope............................................................. ................ 89 614. Safety envelop for fatigue reliability of CAT 994D front loader frame................90 615. Reliability index / with respect to random parameter a ....................................... 91 616. Probability of failure Pf with respect to random parameter a ............................... 91 617. 2D safety envelope for different distribution type with same random parameters .92 71. Cantilever beam subject to two direction loads..................................... ................ 99 72. Piezoelectric cantilevered composite beam.......... ...................................... 103 73. Pareto optimal front for the robust design of the composite beam........................ 106 B1: Free body diagram of twolayer beam............... ..........................122 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RELIABILITYBASED DESIGN AND LOAD TOLERANCE EVALUATION USING STOCHASTIC RESPONSE SURFACE AND PROBABILISTIC SENSITIVITIES By Haoyu Wang December 2006 Chair: NamHo Kim Major: Mechanical Engineering Uncertainty is inevitable in structural design. This research presents an efficient uncertainty analysis technique based on stochastic response surfaces (SRS). The focus is on calculating uncertainty propagation using fewer number of function evaluations. Due to sensitivity analysis, the gradient information of the performance is efficiently calculated and used in constructing SRS. Based on SRS, reliabilitybased design optimization (RBDO) is studied intensively in this research. Probability sensitivity analysis using the sampling technique is also proposed. Since the computational cost of RBDO increases significantly proportional to the increasing number of random variables, global sensitivity analysis is introduced to adaptively reduce unessential random variables. It has been shown that the global sensitivity indices can be calculated analytically because the SRS employs the Hermite polynomials as bases. Traditional structural design focuses on designing a reliable structure under well characterized random factors (dimensions, shape, material properties, etc). Variations of these parameters are relatively small and well characterized. However, everyday engineering life tends to use the existing structural part in a different applications instead of designing a completely new part. In this research, a reliabilitybased safety envelope concept for load tolerance is introduced. This shows the capacity of the current design as a future reference for design upgrade, maintenance and control. The safety envelope is applied to estimate the load tolerance of a structural part with respect to the reliability of fatigue life. Stochastic response surface is also applied on robust design in this research. It is shown that the polynomial chaos expansion with appropriate bases provides an accurate and efficient tool in evaluating the performance variance. In addition, the sensitivity of the output variance, which is critical in the mathematical programming method, is calculated by consistently differentiating the polynomial chaos expansion with respect to the design variables. A reliabilitybased robust design method that can reduce the variance of the output performance as well as the deviation of the mean value is proposed using SRS and efficient sensitivity analysis. Numerical examples are shown to verify accuracy of the sensitivity information and the convergence of the robust design problem. CHAPTER 1 INTRODUCTION Motivation A typical mechanical design procedure includes two steps: first, a design space is defined and a mathematical model is established, which includes the objective function and required constraints. Second, a proper optimization algorithm is selected properly based on this mathematical model to solve the design problem. In engineering design, the deterministic optimization model has been studied intensively to reduce the objective function by pushing design to the limits of system failure boundaries. However, everything in the real world involves uncertainties, and so does the design of mechanical components. After realizing deterministic design leaves very little or no room for tolerances of the imperfections in design, manufacturing and variety of service conditions, design engineers incorporate a safety factor into the structural design to leave safety margins. Without considering uncertainties and probabilistic quantification, deterministic design with a safety factor may be either unsafe or too conservative. Motivated by overcoming the bottleneck of the deterministic design, the reliability based design optimization (RBDO) model has become popular in past two decades since uncertainties exist everywhere in every phase of the structure system, from design and manufacturing to service and maintenance. If elements in the mathematical model are considered to be probabilistic with certain types of random distribution, the design problem becomes a typical RBDO problem. The probabilistic elements can be design variables, material properties, applied loads, etc. One of the most important issues in RBDO is a good model of uncertainty propagation in mathematical models. Besides RBDO, which only considers the failure mode as a constraint in the probabilistic point of view, robust design will also be considered in this research in order to design a structure 'less sensitive' to the existing uncertainty factors. In optimization point of view, that means minimization of performance variance. For a certain design, it is also important to consider the service capability of the system subject to applied loads since engineers tend to use the same design in different applications instead of a completely new design. Another motivation of this research is the load tolerance design. A good estimation of load tolerance shows the capacity of the current design, future reference for design upgrade, maintenance and control. Since static or quasistatic loading is rarely observed in modern engineering practice, the majority of engineering design projects involves machine parts subjected to fluctuating or cyclic loads. Such loads induce fluctuating or cyclic stresses that often result in failure by fatigue. In addition, because service loads are subjective, which means the load characteristic of one operator may be completely different from that of the other, it is necessary to consider the uncertainties while estimating the load tolerance of dynamic systems. Objective Uncertainty in the design parameters makes structural optimization a computationally expensive task due to the significant number of structural analyses required by traditional methods. Critical issues for overcoming these difficulties are those related to uncertainty characterization, uncertainty propagation, ranking of design variables, and efficient optimization algorithms. Conventional approaches for these tasks often fail to meet constraints (computational resources, cost, time, etc.) typically present in industrial environments. The first objective of this research is to develop a computationally efficient method for uncertainty propagation. Local and global sensitivities can then be used to improve the efficiency of estimating uncertainty propagation. Besides efficiency, the accuracy and applicability of the methods to a wide range of applications need to be addressed. The second objective of this research is to develop a computationally efficient RBDO and robust design framework based on proposed uncertainty analysis. In the gradientbased algorithm, the sensitivity information is required during the optimization procedure. The computational cost can be significantly saved if the gradient can be obtained analytically, instead of using the finite difference method. The probabilistic sensitivity analysis is utilized to calculate the gradient of the reliability constraints. In the framework of robust design, sensitivity analysis of performance variance is also studied. Traditional structural design usually makes assumption on randomness of factors involved in modeling a structural system such as design variables, material properties, etc. However, it is also important to consider the capacity of the system subject to uncertain loads. The final objective of this research is to present a reliabilitybased load design method, which provides the safety envelope, for a structure subject to fatigue failure. Scope In the standard framework of RBDO, constraints are provided in terms of the probabilities of failure. The uncertainties involved in the system are modeled by assuming random input variables with a certain type of probabilistic distribution. RBDO achieves the design goal by meeting the requirement of structural reliability constraints. The RBDO involving a computationally demanding model has been limited by the relatively high number of required analyses for uncertainty propagation during the design process. The scope of this research is to present an efficient uncertainty propagation technique based on stochastic response surfaces (SRS) constructed using model outputs at heuristically selected collocation points. The efficiency of the uncertainty propagation approach is critical since the response surface needs to be reconstructed at each design cycle. In order to improve the efficiency, the performance gradient, calculated from local sensitivity analysis, is used. Even if the local sensitivity information can reduce the number of required simulations, the dimension of the SRS is still increased according to the number of random variables. If the contribution of a random variable is relatively small to the variance of the model output, it is possible to consider the random variable as a deterministic one. In this research, the global sensitivity index is used for that purpose. The role of the global sensitivity is to quantify the model input's contributions to the output variability, hence establishing which factors influence the model prediction the most so that i) resources can be focused to reduce or account for uncertainty where it is most appropriate, or ii) unessential variables can be fixed without significantly affecting the output variability. Reliability constraint in RBDO requires probability sensitivity analysis for gradientbased algorithms. In this research, both FORMbased and samplingbased reliability sensitivity analysis are investigated. The analytical expression for probability sensitivity based on SRS is derived and used for RBDO. Variations in dynamic loads are usually too complicated to be predicted. A simplified uncertainty modeling technique based on the mean and amplitude of the load history is proposed. Using the uncertainty in the load history, a reliabilitybased safety envelope is constructed that can provide load tolerance of the current design. In addition, the effect of different distribution types is investigated so that the design engineers can choose the conservative distribution type. This research involves uncertainty modeling and quantification, design sensitivity analysis, fatigue life prediction, reliabilitybased design optimization (RBDO) and robust design. Methodologies investigated or applied in reliability analysis include moment based methods such as first and secondorder reliability method (FORMISORM), approximation methods such as Monte Carlo Simulation (MCS) with stochastic response surface method (SRSM). Furthermore, sensitivity analysis for reliability constraints of RBDO is investigated to improve the computational cost involved in reliability analysis and design. Performance variance and sensitivity are calculated based on SRSM for robust design. Computationally affordable reliabilitybased optimization and robust design method, and safety envelope for load tolerance are presented in this work. Outline A literature survey is presented in chapter 2, which includes all aspects involved in this research such as reliability analysis, reliabilitybased design optimization, robust design, sensitivity analysis, dimension reduction strategy and fatigue analysis. Chapter 3 describes the uncertainty modeling and widely used reliability analysis methods. A stochastic response surface method (SRSM) coupled with the sensitivity analysis of performance measure is introduced. It is shown that the local sensitivity information improves computational efficiency significantly by reducing required number of samples. Convergence and accuracy of the proposed SRSM scheme are also discussed in this chapter. In Chapter 4, the mathematical model is defined for RBDO. RBDO using either direct probability measure or inverse measure is investigated and compared. The difference of numerical procedures between RBDO and deterministic optimization are also compared. As required by RBDO, probability sensitivity analysis is studied in this chapter. In Chapter 5, a dimension reduction strategy is proposed by introducing the concept of variancebased global sensitivity analysis, which saves the computational resources further by fixing the unessential design variables. Chapter 6 demonstrates a fatigue reliabilitybased load tolerance design by using reliability sensitivity information. A reliabilitybased safety envelope is constructed by path following continuation method. Chapter 7 proposes an optimization model for robust design where SRS is used to calculate the performance variance and its sensitivity. Chapter 8 concludes this research followed by recommendations for future research work. CHAPTER 2 LITERATURE SURVEY Uncertainty and Reliability Analysis of Structural Applications Reliabilitybased design optimization (RBDO) provides tools for making decision within a feasible domain of design variables with consideration of uncertainties underneath. In the past decades, tremendous amount of work has been carried out in this area and it is still moving forward. Compared to deterministic optimization, design variables included in RBDO are random and usually modeled with specific distribution types, so do the random parameters such material properties as Young's modulus and Poisson's ratio. Usually random parameters do not change during optimization, but their effects to the probability propagation must be counted due to its uncertainty. Reliability, which is defined as the probability that a system response does not exceed the limit threshold, is often considered as constraints. The system response is a function of design variables and random parameters, which is called a performance function in this research. Performance function is usually implicit and nonlinear prediction of random variables, making probabilistic description of a system response difficult. Several approximation methods for reliability analysis have been developed in the literatures. Among them, Monte Carlo Simulation (MCS) (Metropolis and Ulam 1949; Rubinstein 1981) has been widely used due to its simplicity and dependability. However, the large sample size required in MCS in order to reduce the noise and to reach a certain level of accuracy makes it practically formidable in computationally intensive engineering applications, such as Finite Element Analysis (FEA). Even improved version of MCS are developed, such as importance sampling, Latin Hypercube Sampling (Wyss and Jorgensen 1998), Stratified Sampling, etc, they are still expensive in structural reliability analysis. Momentbased methods (Breitung 1984; Haldar and Mahadevan 2000; Hasofer and Lind 1974) have been developed to provide less expansive calculation of the probability of failure compared to MCS. However, they are limited to a single failure mode. As the most widely used momentbased methods, the development of the theory of First and SecondOrder Reliability Method (FORMISORM) is claimed to be finished and only technical work left to do (Rackwitz 2000). FORM/SORM are based on the linear/quadratic approximation of the limit state function around most probable point(MPP), which is defined in standard normal space as the closest point from the origin on the response surface. For highly nonlinear problems, predictions of reliability from FORMISORM are not accurate enough because they approximate the response using a linear or quadratic function. The response surface method (Khuri and Cornell 1996; Myers and Montgomery 1995) is proposed to resolve this difficulty. This method typically employs polynomials bases to approximate the system performance and facilitate reliability analysis with little extra computational cost by combining with MCS. Since the accuracy of MCS with fixed sample size relies on the seeking level of probability of failure which sometimes is extremely low in structural design, the probability calculated by MCS near optima is too rough to represent the true value of failure probability. Reliability analysis using safety factor (Wu et al. 2001) or probability sufficiency factor (PSF) (Qu and Haftka 2004) is proposed to ameliorate this effect. With the PSF as the constraints in RBDO, the variation of magnitude of constraints is usually several orders of magnitude lower than that of the probability of failure, and so is the magnitude of the numerical noise caused by MCS. One of the significant advantages of the momentbased approach is that the sensitivity of the system reliability or probability of failure can be obtained with very little extra computation (Yu et al. 1998). However, moment based approach such as FORM/SORM still has limitations when the performance function is highly nonlinear (Ghanem and Ghiocel 1996). The evaluation of the probabilistic constraints may have large errors in this case. Mahadevan and Shi (Mahadevan and Shi 2001) presented a multipoint linearization method (MPLM) for the reliability analysis of nonlinear limit states, which determines the multiple linearization points through the secant method. It increases the complexity of the problem with limited accuracy improvement. The response surface method can approximate the system response and with little extra computation for MCS, the probability of failure can easy to be obtained. Compared to the conventional deterministic design response surface, Stochastic Response Surface (SRS) (Isukapalli et al. 1998) has the advantage that it only approximates the function around most probability region which highly improved accuracy. Another advantage of SRS is the choice of basis function. The monomial bases(Qu et al. 2000) are widely used due to its simplicity. Other polynomial bases are also being studied intensively such as radial basis function (RBF) (Krishmamurthy 2003), orthogonal polynomials (Xiu et al. 2002),etc. Since Ghanem and Spanos proposed the spectral approach of stochastic finite element method (Ghanem and Spanos 1991), the homogeneous Polynomial Chaos Expansion (PCE) has been widely utilized to represent the uncertainties due to the nature of stochastic process. To make better approximation with less model analyses, sampling methods are studied intensively. Different sampling methods were studied and brought in different applications recent years, such as Latin Hypercube Sampling (LHS) (Choi et al. 2003; Qu et al. 2000) and collocation sampling method (Webster et al. 1996). In the collocation method, Webster and Tatang derived a set of polynomials from the probability density function of each input parameter such that the roots of each polynomial are spread out over the high probability region of the parameter by deriving orthogonal polynomials. Because the uncertainty is usually evaluated by transforming all the random variables and parameters into the Gaussian space, the corresponding orthogonal polynomials are Hermite polynomials. To obtain additional accuracy of SRS, moving least square (MLS) method (Youn 2001, Dec; Youn and Choi 2004) is proposed by introducing weight functions. The number of simulations can be reduced if the sensitivity information is available. Isukapalli (Isukapalli et al. 2000) used an automatic differentiation program to obtain the sensitivity and utilized it in constructing the response surface. However, the computational cost for automatic differentiation is usually very high(Van Keulen et al. 2004), which reduces the significance of the method. Design sensitivity analysis can provide analytical sensitivity information of response with little extra computation (Kim et al. 2000). Thus, coupling the regression based stochastic response surface method (SRSM) with sensitivity can save large amount of computational cost, especially when the required number of design variables is large (Kim et al. 2004b). Several methods(Lauridensen et al. 2001; Malkov and Toropov 1991; Rijpkema et al. 2000; Van Keulen et al. 2000) have been proposed to use sensitivity information in constructing response surface. Vervenne (Vervenne 2005) proposed a gradientenhanced response surface method based on above mentioned methods. He developed a twostep approach is proposed: first, different response surfaces using function values and derivatives are constructed separately; Second, these response surfaces are combined together to form a single response surface which fits as good as possible for both function value and response surfaces. In his study, several types of response surface and different combination scheme have been compared. ReliabilityBased Design Optimization As mentioned in the previous section, FORM/SORM performs reliability analysis through linear/quadratic approximation of the performance function at MPP. Thus, searching MPP is the main task for momentbased RBDO. However, most advanced MPP search methods such as two point adaptive nonlinear approximation method (TPA) (Grandhi and L.P. 1998; Wang and Grandhi 1995; Xu and Grandhi 1998) or hybrid mean value (HMV) method(Youn 2001, Dec; Youn et al. 2003) can not make significant improvement of efficiency in the computational cost(Du and Chen 2002b). In conventional RBDO, the probability constraint is described by the reliability index, which in FORM is the shortest distance from the origin to the limit state in standard normal space. This approach is called reliability index approach (RIA). By modifying the formulation of probabilistic constraints, Tu proposed an inverse measure approach, called Performance Measure Approach (PMA) (Tu 1999; Tu and Choi 1997; Tu et al. 1999; Tu 2001) which is proved to be consistent with the RIA but is inherently robust and more efficient if the probabilistic constraint is inactive. Both RIA and PMA employ double loop strategy with analysis loop (inner loop for reliability analysis) nested within the synthesis loop (outer loop for design optimization). Due to the nature of double loop optimization, the computational cost is usually high. A couple of new strategies were proposed to improve the efficiency (Yang and Gu 2004). Sequential Optimization and Reliability Assessment (SORA) method (Du and Chen 2002b) decouples optimization loop from the reliability analysis loop and each deterministic optimization loop followed by a series of MPP searches. This method shifts the boundaries of violated constraints to the feasible direction based on the reliability results obtained in the previous cycle. Thus it improves design quickly from cycle to cycle and ameliorates the computational efficiency. Other single loop methods(Chen et al. 1997; Kwak and Lee 1987; Liang et al. 2004; Wang and Kodiyalam 2002) are also developed to provide efficient RBDO. In this method, the relationship between random variables and its mean is found through the KarushKuhnTucher (KKT) optimality condition. The double loop RBDO formulation is transformed to a single loop deterministic optimization problem and expensive MPP search is avoided. However, there is no guarantee that an active reliability constraint converges to its own MPP, and the required reliability may not be satisfied. Sensitivity in Reliability Analysis When RBDO problems are solved using gradientbased optimization algorithms, sensitivities of reliability or probability of failure with respect to the design parameters are required. Probability sensitivity can be used to identify those insignificant random variables during the design stage. In the momentbased approaches such as FORM, the sensitivity can be obtained accompanied by the reliability analysis without extra function evaluation once MPP is located (Karamchandani and Cornell 1992; Yu et al. 1997). Wu(Wu 1994) proposed an adaptive importance sampling(AIS) method to calculate reliability and AISbased reliability sensitivity coefficients. Liu et al(Liu et al. 2004) compare four widelyused probability sensitivity analysis(PSA) methods, which include Sobol' indices, Wu's sensitivity coefficients, the MPP based sensitivity coefficients and the KullbackLeibler entropy based method. The merits behind each method are reviewed in details. Dimension Reduction Strategy In reliability analysis, the computational cost of multidimensional integration is high. Xu and Rahman(Rahman and Xu 2004; Xu and Rahman 2004) use series expansions to decompose the multidimensional problem to lower dimensional integration, such as univariate and bivariate integration. Compared to multidimensional integration, the total computation of univarate integration is much lower. Recent development in statistics introduces global sensitivity analysis (GSA)(Saltelli et al. 2000; Saltelli et al. 1999; Sobol 1993; Sobol 2001), which studies how the variance in the output of a computational model can be apportioned, qualitatively and quantitatively, to different sources of variation. Considering the contribution of the variance of design variables to performance variances are not of same importance, Kim et al proposed an adaptive reduction method using total sensitivity indices to reduce the problem dimensions (Kim et al. 2004a). Robust Design Robust design, known as Taguchi parameter design (Taguchi 1986; Taguchi 1987), is to design a product in such a way that the performance variance is insensitive to variation of design variables which is beyond the control of designer. Wang & Kodiyalam(Wang and Kodiyalam 2002) formulated robust design as an optimization problem by minimizing the variation of system response. Since the material cost has to be considered as well as manufacturing cost, Chen and Du's formulation compromises cost reduction with performance variance control(Du and Chen 2002a). A robust design can also be achieved by using traditional optimization techniques to minimize the performance sensitivities. Chen & Choi formulated the robust design by minimizing a total cost function and sum of squares of magnitudes of firstorder design sensitivities(Chen and Choi 1996), which requires the evaluation of secondorder sensitivity analysis. This is a different philosophy compared to the variance based approach. It is more focus on the local behavior of the system performance and can achieve local robustness. The final design by minimizing local sensitivity cannot guarantee the robustness of system globally if the input variances are considerable. By summarizing approaches popularly applied in robust design, Park et al. (Park et al. 2006) define robust design methodologies into two different category: Taguchi method and robust optimization. Under the context of multiscale and multidisciplinary applications, Allen et al.(Allen et al. 2006) reviewed robust design methods and categorizes robust design into four different types based on the sources of variability. Fatigue Life Prediction In 1829, Albert found that a metal subjected to a repeated load will fail at a stress level lower than that required to cause failure on a single application of the load. Then the question comes out: how parts fail under timevarying or nonstatic conditions? Such phenomenon is called fatigue. The first approach developed to carry out fatigue analysis is the nominal stress method, which is still widely used in applications where the applied stress varies with constant amplitude within the elastic range of the material and the number of cycles to failure is large. The nominal stress method works well in high cycle fatigue analysis but does not fit for the low cycle fatigue analysis where the material has a significant part in the plastic region. August Wohler(Wohler 1860) carried out experiments to obtain a plot of cyclic stress level versus the logarithm of life in mid19th century, which is well known as SN curve. Basquin proposed a stresslife(SN) relationship(Basquin 1910) which can be plotted as a straight line using log scales. SN approach is applicable to situations where cyclic loading is essentially elastic, so the SN curve should be confined on the life axis to numbers greater than about 105 cycles in order to ensure no significant plasticity occurs. Most basic fatigue data are collected in the laboratory by testing procedures which employ fully reversed loading. However, most realistic service loads involve nonzero mean stresses. Therefore, the influence of mean stress on fatigue life should be considered so that the fully reversed laboratory data can be used in the evaluation of real service life. Since the tests required to determine the influence of mean stress are quite expensive, several empirical relationships(Gerber 1874; Goodman 1899; Soderberg 1939) which related alternating stress amplitude to mean stress have been developed. Among the proposed relationships, two are widely used, which are based on Goodman(Goodman 1899) and Gerber(Gerber 1874). SN approach works well when the cyclic loading is essentially elastic, which means in high cycle fatigue life evaluation. While using this method, it assumes that most of the life is consumed by nucleating cracks (around 0.01 mm) and nominal stresses and material strength control fatigue life. Accurate determinations of miscellaneous effects factor Kf for each geometry and material are also required. The advantage of SN approach is apparent since changes in material and geometry can easily be evaluated and large empirical database for steel with standard notch shape is available. However, the limitation should also be accounted. This method does not consider the effects of plasticity, and mean stress effect evaluation is often in error. As the matter of fact, the requirement of empirical Kf for good results is also a kind of disadvantage. As mentioned above, when the cyclic loads are relatively large and have a significant amount of plastic deformation, the components will suffer relatively short lives. This type of fatigue behavior is called lowcycle fatigue or straincontrolled fatigue. The analytical procedure in dealing with straincontrolled fatigue is called the strainlife, local stressstrain or critical location approach. In 1950's, Coffin and Manson(Coffin 1954; Manson 1954) suggested that the plastic strength component of a fatigue cycle may also be considered in fatigue life prediction by a simple power law. In order to account for the mean stress effects, two correction methods are proposed by Morrow and Smith, Topper & Watson (STW)(Smith et al. 1970), respectively. Local StrainLife (eN) method assumes that the local stresses and strains control the fatigue behavior. In this method, the plastic effects and mean stress effects are considered well. The limitation is that it also needs the empirical Kf. In the local strain life approach, the most of the life is consumed by microcrack growth (0.11mm). To account for macrocrack growth(>lmm), the fracture mechanicsbased crack propagation method is proposed(Hoeppner and Krupp 1974; Paris 1964; Paris and Erdogan 1963). In this method, major assumption is that nominal stress and crack size control the fatigue life and the initial crack size is determined accurately. It is the only method to directly deal with cracks. However, the complex sequence effects and accurate initial crack size are difficult to be determined. Linear elastic fracture mechanics (LEFM) is a new branch of engineering. The earliest work was done by Inglis(Inglis 1913) but the major developments were carried out by Griffth(Griffith 1921) at Royal Aircraft Factory(RAF,UK) in 1921 and Irwin(Irwin 1956) in the USA in 1956. In LEFM theory, the driving force for a crack to extend is not the stress or strain but the stress intensity factor, known as K. The stress intensity factor uniquely describes the crack tip stress field independent of global geometry by embodying both the stress and the crack size. The relationship of the crack growth in the sense that the rate of crack growth, da/dN, with respect to the cyclic range of the stress intensity factor, AK, was derived by Paul C. Paris(Paris et al. 1961) in 1961, known as Paris Law. In reality, mechanical component are seldom subjected to purely constant amplitude loading history. The irregular stress history must be counted as a series of constant amplitude stresses. In addition, it is difficult to define a cycle in an irregular stress history. Since the reverse of stress curve can be easily found according to the sign change of the stress history curve, cycle counting techniques such as rainflow counting method(Matsuishi and Endo 1968) are developed to combine reversals to form cycles. After that, cumulative damages can be calculated by Miner's law(Miner 1945). For most realistic structures or components, stress or strain fields are multiaxial. Fatigue life prediction methods for multiaxial loading also have been developed(Bannatine et al. 1990; Fuchs and Stephens 1980; Miller et al. 1966). In addition to some traditional method such as maximum principle stress/strain method, maximum shear stress/strain method and Von Mises' effective stress/strain method; Miller and Brown formulized the critical plane approach(Brown and Miller 1973) from the observation that the stress and strain normal to the plane with maximum shear has been recognized to strongly influence the development of fatigue crack. No consensus has been reached on the methods of multiaxial fatigue life prediction. All these methods have their own advantages in the specific application. So far, fatigue life analysis has been separated into two categories, (a) crack initiation, including SN and eN method, and (b) crack propagation. The criteria for the fatigue life of a component in engineering design depend on material properties or work conditions. In general, the automotive industry usually applies crack initiation criteria because of the nature of the product and use. On the other hand, the aircraft industry mainly uses crack propagation criteria by periodic inspection and fatigue crack monitoring to achieve and maintain structural safety. CHAPTER 3 UNCERTAINTY ANALYSIS USING STOCHASTIC RESPONSE SURFACE Introduction Uncertainty modeling and reliability analysis are the key issues in the reliability based design process. Uncertainty modeling can be decomposed into three fundamental steps: i) uncertainty characterization of model inputs, ii) propagation of uncertainty, and iii) uncertainty management/decision making. The uncertainty in model inputs can be represented in terms of standardized normal random variables (srv) with mean zero and variance equal to one. The selection is supported by the fact that they are widely used and wellbehaved. For other types of random variables, an appropriate transformation must be employed. It is assumed that the model inputs are independent so each one is expressed directly as a function of a srv through a proper transformation. Devroye(Devroye 1986) presents the required transformation techniques and approximations for a variety of probability distributions. More arbitrary probability distributions can be approximated using algebraic manipulations or by series expansions. For uncertainty propagation, Monte Carlo Simulation (MCS) may be the most common choice because of the accuracy and robustness, but the dilemma of MCS is that the required large number of samples that makes it impractical for computationally demanding models. There are several remedies to reduce the number of samples in MCS, such as importance sampling(Melchers 2001) and separable MCS(Smarslok and Haftka 2006). However, they require special knowledge of the problem or special form of the response. Several computationally efficient methods were proposed in last two decades with reasonable accuracy in many structural problems, such as first and second reliability method (FORM/SORM), and response surface method (RSM). The stochastic response surface (SRS) can be viewed as an extension of classical deterministic response surfaces for model outputs constructed using uncertain inputs and performance data collected at heuristically selected collocation points. The polynomial expansion uses Hermite polynomial bases for the space of squareintegrable probability density function (PDF) and provides a closed form solution of model outputs from a significant lower number of model simulations than those required by conventional methods such as modified Monte Carlo methods and Latin hypercube sampling. In this chapter, a surrogatebased uncertainty model using stochastic response surface (SRS) is introduced. Reliability analysis using Monte Carlo simulation on this surrogate model shows promising results in terms of accuracy and efficiency. The proposed method is compared with the firstorder reliability method (FORM) and MCS. Description of Uncertainty Model When the inputs of a system are uncertain or described as random variables/parameters, the output or response from this system will have a stochastic behavior as well. Let us assume that these random inputs are given in an ndimensional vector X with continuous joint distribution function fx(x). As shown in Figure 31, the system state can have a Boolean description such that the system fails when the limit state G(X) < 0 The probability of failure Pfcan then be defined as a cumulative distribution function (CDF) over the failure region, as P,= fx(x)dx. (3.1) G(X)<0 X2 G(X)<0 Failure region Limit state G(x)=0 G(X)>0 Safe region x1 Figure 31. Limit state function divides the safe region from the failure region Equation (3.1) is called the reliability integral. Since the integral domain defined by limit state function G(X) is complex in the multidimensional random space, the reliability integral is difficult to calculate. As introduced in the previous section, by transforming random variables from the original random space to the standard normal space, the limit state function can be expressed as a function of a set of srvs {uf} Then, Pfcan be expressed in standard Gaussian space as Pf = p, (u)du (3.2) G(U)<0 where p(*) is the standard normal PDF and U is the vector of standard random variables. The transformation between X and U is denoted as U=T(X). In FORM, the probability level of a system is usually represented by the reliability index or safety index /Y. For instance, if ((*) is the CDF of the standard random variable, the failure probability can often be represented by the reliability index 8/ = ( '(P ). Stochastic Response Surface Method (SRSM) Polynomial Chaos Expansion (PCE) in Gaussian Space Orthogonal polynomials have many useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials associated with the generalized polynomial chaos (Askey Chaos) are different according to different weight functions. The type of polynomials is decided by the match between the specific weight function and the standard probability density function (PDF). The corresponding type of polynomials and their associated random variables are listed in Table 31. Table 31. The type of polynomials and corresponding random variables for different AskeyChaos (N>0 denotes a finite integer) Random variable Orthogonal Support range polynomials Gaussian Hermite (00,00) Continuous Gamma Laguerre [0, co) Beta Jacobi [a,b] Uniform Legendre [a,b] Poisson Charlier {0,1,2,... } Discrete Binomial Krawtchouk {0,1,2,...,N} Negative Binomial Meixner {0,1,2,... } Hypergeometric Hahn {0,1,2,...,N} For example, in Table 31, Hermite polynomial chaos expansion requires the weight functions to be Gaussian probability density function, and it satisfies the following orthogonal relation: fJk fk(xk)Fk(xk)F (xk)dk =C, Vi, j (3.3) where fk(xk) is Gaussian PDF for variable xk, Fk(xk) is the Hermite polynomial basis, and upper indices i,j denote for two different polynomials. In this research, the uncertainty propagation is based on stochastic response surfaces (polynomial chaos expansion). The SRS(Isukapalli et al. 1998; Webster et al. 1996) can be view as an extension of classical deterministic response surfaces(Khuri and Cornell 1996; Myers and Montgomery 1995) for model outputs constructed using uncertain inputs and performance data collected at heuristically selected collocation points. The polynomial expansion uses Hermite polynomial bases for the space of square integrable probability density function (PDF) and provides a closed form solution of model outputs from a significant lower number of model simulations than those required by conventional methods such as modified Monte Carlo methods and Latin hypercube sampling. Let n be the number of random variables and p the order of polynomial. The model output can then be expressed in terms of the srv {u,} as: G= al+ afI,(u,)+ 1 a2 +I, I) I a 3(, ,k)+ (3.4) 1=1 1 j= 1 i1 ]=1 k=1 where GP is the model output, the ap, as,... are deterministic coefficients to be estimated, and the Fp(ui,...,Up) are multidimensional Hermite polynomials of degree: P (ut_) ( 1=)Pe 12UU _9P e 1/2UU (3.5) ( ..Ou t ( ) u O& 9 ...Ou/p where u is a vector ofp independent and identically distributed normal random variables selected among the n random variables that represent the model input uncertainties. Equation (3.4) is also called polynomial chaos expansion. The Hermite polynomials Frp(u,..., up) are set of orthogonal polynomials with weighting function e /2, which has the same form with the PDF of standard random variables. In this research, a modified version ofHermite polynomial(Isukapalli et al. 1998) is used. The first four terms are u, u2 1, u3 3u, and u4 6u2 + 3, when a single random variable is involved. The use of the Hermite polynomials has two purposes: (1) they are used to determine the sampling points, and (2) they are used as bases for polynomial approximation. In general, the approximation accuracy increases with the order of the polynomial, which should be selected reflecting accuracy needs and computational constraints. The expressions for the 2nd and 3rdorder polynomials in n dimensions (later used in the numerical examples) are: 2ndorder: n n n1 n G2(u) (2) + 2) + (2) (u2 )+) 2) (3.6) 1=1 =1 1=1 j>1 3rdorder: G(3(u) =a(3) + 3)u + a3 (u 21) =1 1=1 n n1 n + a) (u3 3u)+ ( 3) (3.7) =1 z=1 y>1 n n n2 n1 n 1=1 j=1,ji iZ=1 j>i k>j The number of unknown coefficients is determined by dimension of the design space n. For 2nd and 3rd order expansion, if the number of unknowns is denoted by 2), M3), respectively: N(2) = 1+ 2n + n(n 1) (3.8) 2 N(3) = 1+3n+ 3n(n 1) n(n 1)(n 2) (3.9) 2 6 For n = 2, 4, and 8, for example, M2) = 6, 15, and 45; and N3) = 10, 35, and 165, respectively. The coefficients in the polynomial chaos expansion are calculated using the least square method, considering samples of input/output pairs. Since all inputs are represented using srv, more accurate estimates for the coefficients can be expected, in the sense of statistics, if the probability distribution of the u,'s is considered. The idea of Gaussian Quadrature of numerical integral can be borrowed to generate collocation points(Webster et al. 1996). In Gaussian Quadrature, the function arguments are given by the roots of the next higher polynomial. Similarly, the roots of the next higher order polynomial are used as the points at which the approximation being solved, which is proposed as the orthogonal collocation method by Villadsen and Michelsen (Villadsen and Michelsen 1978). For example, to solve for a three dimensional second order polynomial chaos expansion, the roots of the third order Hermite polynomial, /r 0 and ,,r3 are used, thus the possible collocation points are (0,0,0),( 3 x/3 ),(/3 ,0, f3 ),etc.. There are 27 possible collocation points in this case. However, in equation(3.9), there are only 10 unknown coefficients. Similarly, for higher dimensional systems and higher order approximations, the number of available collocation points is always greater than the number of unknowns, which introduces a problem of selecting the appropriate collocation points. For a good approximation in polynomial chaos expansion, the choice of collocation points is critical. Hence, a set of points near the high probability region is heuristically selected among the roots of the oneorder higher polynomial under restrictions of symmetry and closeness to the mean. Since the origin always corresponds to the highest probability in standard Gaussian space, the exclusion of the origin as a collocation point could potentially lead to a poor estimation. Thus, when the roots of highorder polynomial do not include zero, it is added in addition to the standard orthogonal collocation method. The Hermite polynomials orthogonall with respect to the Gaussian PDF) provide several attractive features, namely, more robust estimates of the coefficients with respect to those obtained using nonorthogonal polynomials(Gautschi 1996); it converges to any process with finite second order moments(Cameron and Martin 1947); and the convergence is optimal (exponential) for Gaussian processes(Xiu et al. 2002). In addition, the selected SRS approach includes a sampling scheme collocationn method) designed to provide a good approximation of the model output (inspired by the Gaussian Quadrature approach) in the higher probability region with limited observations. Once the coefficients are calculated, statistical properties of the prediction, such as mean and variance can be analytically obtained, and sensitivity analyses can be readily conducted. Numerical Example of Stochastic Response Surface As an illustration of the efficiency and convergence properties of the SRS approach, consider the construction of the PDF associated with a simple analytical function represented by: y(x)= ex (3.10) with x being a normally distributed random variable, as N(0,0.42). Note that in this case the analytical expression of the PDF is known. The SRS for 2nd and 3rdorder polynomials are shown in Eqs.(3.11) and (3.12), respectively. y2) = 1.0833+0.4328u + 0.0833(u2 1) (3.11) (3 = 1.0843 + 0.4333u + 0.0863(u2 1) + 0.0112(u3 3u) (3.12) f0.8 00.6 0.4 0.2 0 1 2 3 4 5 y=ex xN(0,0.42) Figure 32. PDF of performance function y(x) = ex In this particular example, the accuracy of the proposed SRS is compared with the analytical solution. Figure 32 shows the PDF obtained from MCS applied to the SRS and from the exact solution. A good agreement is observed in the PDF approximation, and the root mean square errors decreases with higher order polynomials, showing the convergence of the proposed SRS (Table 32). Table 32. Root mean square error of PDF compared with the exact PDF of performance function y=ex Polynomial order Errors 2 0.03835 3 0.00969 To illustrate the effectiveness of the SRS in the application to the structural problem, consider a torquearm model in Figure 33 that is often used in shape optimization(Kim et al. 2003). The locations of boundary curves have uncertainties due to manufacturing tolerances, modeled as probabilistic distributions. Thus, the relative locations of corner points of the boundary curves are defined as random variables with x,~N(d,, 0.12). The mean values d, of these random variables are chosen as design parameters, while the standard deviation remains constant during the design process. X6 5 A X87 2 2789N C 5066N Symmetric Design Figure 23: Shape design parameters for the torque Figure 33. Shape design parameters for the torque arm The torque arm model consists of eight random variables. In order to show how the SRS is constructed and the PDF of the model output is calculated, we choose the three random parameters (x2, x6, and x8) that contribute most significantly to the stress performance at points A and B in Figure 33. In the deterministic analysis with mean value, the maximum stress of A = 319MPa occurs at location A. The stress limit is established to be Gmax = 800MPa. In the reliability analysis the performance function is defined such that G < 0 is considered a failure. Thus, the performance function can be defined as G(x) = omax A(X). The number of unknown coefficients is a function of the dimension n of the random variables. For 2nd and 3rdorder expansion, the numbers of coefficients, denoted by N2 and N3, are 10 and 20, respectively. There are 27 possible collocation points and 10 unknown coefficients in the case of 2ndorder expansion. For robust estimation, the number of collocation points in general should be at least twice the number of coefficients. In this particular example, all possible collocation points are used. After coefficients are obtained, MCS with 100,000 samples is used to obtain the PDF. Figure 34 shows the PDF associated with G(x) when different orders of polynomial approximations are used. The PDF obtained from the direct MCS with 100,000 sample points is also plotted. It is clear that the PDF from the 3rdorder is much closer than that of the 2ndorder to the PDF from the MCS. 0.035 1 SRS p=2(27pts) 03 SRS p=3(125pts) 03 MCS 0 025 0.02 I 0 0015 0.01 400 420 440 460 480 500 520 540 560 580 600 Response(Mpa) Figure 34. PDF of performance function G(x) torque arm model In order to compare the accuracy of the probability estimation through proposed SRS, let us check the probability of response being larger than 520MPa. In Table 33, the probability obtained from MCS is regarded as the reference. The relative error (e) of failure probability from MCS estimation with sample size of N can be calculated using the following equation: s = k  p(3.13) where k denotes the confidence level, for confidence level of 95%, k=1.96, which can be verified from standard normal table. Thus, in Table 33, number of MCS sample is 100,000, the error in Pfwill be less than 5% with 95% confidence. As shown in Table 33, it is clear that the SRS provides a convergent probability result as the order increases. With third order SRS, the accuracy of reliability analysis is significantly improved, compared to FORM. Table 33. Comparison of probability of G>520MPa obtained from different uncertainty analysis methods (Full sampling without using local sensitivity) Method FORM 2nd order 3rd order SRS MCS SRS Prob. of G>520MPa 1.875% 2.061% 1.682% 1.566% Relative error* 19.732% 31.609% 7.407%  *Relative error: prob(approx.) prob(MCS) x 100% prob (MCS) Improving Efficiency of SRS Using Local Sensitivity Information In the proposed SRS, the number of sampling points depends on the number of unknown coefficients. Although the proposed method is accurate and robust, we have to address the curse of dimensionality: as the number of random variables increases, the number of coefficients rapidly increases, as can be seen in Eqs. (3.8) and (3.9). In addition to the efficient collocation method, the number of simulations can be reduced even further when local sensitivity is available. Recently, Isukapalli et al.(Isukapalli et al. 2000) used an automatic differentiation program to calculate the local sensitivity of the model output with respect to random variables and used them to construct the SRS. Their results showed that local sensitivity can significantly reduce the number of sampling points as more information is available. The computational cost of the automatic differentiation, however, is often higher than that of direct analysis(Van Keulen et al. 2004). However, in the application to the structural analysis, local sensitivity can be obtained at a reasonable computational cost. At each sampling point, the local sensitivity is a partial derivative of the limit state with respect to random variables. Hence, if local sensitivity information is available, then n+1 data at each sampling point can be used for constructing the proposed SRS, which significantly reduces the required number of sampling points. ContinuumBased Design Sensitivity Analysis In this research, the continuumbased design sensitivity analysis(Choi and Kim 2004a) is utilized to calculate the gradient of the performance function with respect to random variables. Even if the idea can be used in a broader context, only structural problems are considered in this research. Let z be the displacement and ibe the displacement variation that belongs to the space Z of kinematically admissible displacements. For given body force f and surface traction force t, the variational equation in the continuum domain Q is formulated as a(z, ) = I(T), (3.14) for all i e Z. In Eq. (3.14), the structural bilinear and load linear forms are defined, respectively, as a(z,' Y) = ,, j(z)E, (z)dQ (3.15) l() = ffdQ + t, dF (3.16) where e,, are components of the engineering strain tensor, and a,, are components of the stress tensor. In linear elastic materials, the constitutive relation can be given as 0 (z) = C"k , (Z) (3.17) where the constitutive tensor cjki is constant. The summation rule is used for the repeated indices. In order to solve Eq.(3.14) numerically, the finiteelementbased method or the meshfree method can be employed, which ends up solving the following form of matrix equation: [K]{D} = {F} (3.18) where [K] is the stiffness matrix, {F} the discrete force vector, and {D} the vector of nodal displacements. The major computational cost in solving Eq.(3.18) is related to LU factorization of the coefficient matrix. As will be shown later, the efficiency of sensitivity calculation comes from the fact that sensitivity analysis uses the same coefficient matrix that is already factorized when Eq.(3.18) is solved. In design sensitivity analysis, the variational Eq.(3.14) is differentiated with respect to design variables. In shape design, the design variable does not appear explicitly in the governing equation. Rather, the shape of the domain that a structural component occupies is treated as a design variable. Thus, a formal procedure is required to obtain the design sensitivity expression. As shown in Figure 35, suppose that the initial structural domain Q is changed into the perturbed domain QO in which the parameter T controls the shape perturbation amount. By defining the design changing direction to be V(x), the material point at the perturbed design can be denoted as x, = x + TV(x). The solution z,(x,) of the structural problem is assumed a differentiable function with respect to shape design. The sensitivity ofz,(x,) at X, is defined as S= lim z(x + rV(x)) z(x) (3.19) TryO z Initial domain Perturbed domain /r V(x) rT Figure 35. Variation of a structural domain according to the design velocity field V(x) The design sensitivity equation is obtained by taking the material derivative of the variational equation(3.14) The derivative of the structural energy form then becomes d d aQ (z ,T) 0 = a0(z, ) +aa(z,i) (3.20) The first term on the righthand side represents an implicit dependence on the design through the state variable, while the second term, the structural fictitious load, denotes an explicit dependence on the design velocity V(x), defined as a (z, z) = [J<(1)g (z) + (z)cy (z) + ()7,(z)divV]dQ (3.21) where S Z) OZVk + jV(3.22) If the applied load is independent of displacement, i.e., conservative, then +(I)V ffQ [; 0 L, V +zf/ ]dQ S J[( + Vtx ] dF (3.23) is the external fictitious load form, where Vn is the normal component of the design velocity on the boundary, and K is the curvature of the boundary. The design sensitivity equation is obtained from Eq. (3.20) to (3.23) as a(z,z) = () a(z,7) (3.24) for all e Z. Note that by substituting z into z, the lefthand side of the design sensitivity equation (3.24) takes the same form as that of the response analysis in Eq.(3.14). Thus, the same stiffness matrix [K] can be used for sensitivity analysis and response analysis, with a different righthand side. Once the sensitivity z of the field vector is calculated, the sensitivity of the performance function with respect to design variable u, can be calculated using the chain rule of differentiation, as dy(z;x) _y(z;x) y(z;x) (3 duA au z When finite element analysis is used, the sensitivity equation can be solved inexpensively because the coefficient matrix is already factorized when solving Eq.(3.14) and the sensitivity equation uses the same coefficient matrix. The computational cost of sensitivity analysis is usually less than 20% of the original analysis cost. The computational efficiency of the uncertainty propagation approach is critical to RBDO since as previously stated at each design cycle the updated version of the PDF for the constraint function (related to model outputs) is required. Constructing SRS Using Local Sensitivity In SRS, the system response can be approximated as polynomial expansion when k sampling data are available, the linear regression equation can be written as Y1 1 ui u 2 ** ao 2r a, {y} = [A]{a} (3.26) yk U 3 23 The above equation is the standard form for linear regression to solve for unknown coefficients {a}. When the sensitivity information is available, additional information at each sampling point can be used in calculating the coefficients. By differentiating Eq.(3.4) with respect to random variable ui and by applying the same regression process in Eq.(3.26), we have {dy = A {a} (3.27) du, au, Equations (3.26) and (3.27) can be combined to obtain the following regression equations: y dy dul dy du ,T{ Let {Y}= y d A OA u. {a} (3.28) OA AA aA = A A T, Eq. (3.28) can be 9u, ou" written as {Y} = [B]{a} (3.29) Thus, the coefficients of SRS can be obtained using least square regression, such {a} = ([B]' [B]) [Bl] {Y} (3.30) Note that the sensitivity can be calculated using the transformation of au, random variables, as Y (3.31) As introduced in the previous section, the local sensitivity c9y / 9x can be obtained implicitly through Eq. (3.25), where design variable is represented by u, instead of x, since notation x has been used as space coordinate. Since the transformation between srv and variables with other types of distribution are also mathematically well developed, cx, /au, can be obtained explicitly. Therefore, Eq.(3.30) provides an explicit solution for obtaining coefficients of SRS. Numerical Example Torque Arm Model In order to show the effectiveness of the proposed SRS with local sensitivity, the same torque arm problem with previous example is used. All conditions are the same as before. By taking advantage of using sensitivity information to build stochastic response surface, the number of collocation points is reduced significantly. Here for the second order polynomial chaos expansion, 7 points have been used, while 31 points for the third order case. At each sampling point, the function value and sensitivity information are used to construct the SRS. The PDF obtained from the SRS with sensitivity is plotted in Figure 36 along with that from MCS with 100,000 samples. In the case of 2ndorder, the SRS with sensitivity is more accurate than the SRS without sensitivity (Figures 34 & 36). In order to calculate the accuracy, the probability of G > 520MPa is calculated using FORM, second and thirdorder SRS (Table 34). Since no analytical solution is available, MCS with 100,000 samples is used as a reference. Both SRS are more accurate than FORM. 0.035 SRSSEN p=2(7 pts)  SRSSEN p=3(31 pts) 03  MCS(100,000 pts) 0.025 S0.02  & 0015 \ S001 0.005 \ 400 420 440 460 480 500 520 540 560 580 600 Response (MPa) Figure 36. PDF of performance function G(x) Torque model at initial design (SRS with sensitivity) Table 34. Comparison of probability of G>520MPa obtained from different uncertainty analysis methods (reduced sampling using local sensitivity); Method FORM 2nd order SRS 3rd order SRS MCS Prob. of G>520MPa 1.875% 1.520% 1.545% 1.566% Relative error* 19.732% 2.937% 1.341%  *Relative error: prob(approx.) prob(MCS) x 100% prob (MCS) Table 35 compares the probability of G>520 MPa of second order SRS with/without using local sensitivity with that of MCS, which is regarded as the reference of exact value. With local sensitivity and seven sampling points, SRS provides more accurate probabilistic result than that without utilizing local sensitivity and twentyseven sampling points. The accuracy is improved by using local sensitivity while computational cost is reduced. Table 35. Comparison of probability of G>520MPa obtained with/without local sensitivity (7/27 sampling points) using 2nd order SRS 2nd order SRS 2rd order SRS using 27 sampling using 7 sampling MCS Method points without points with (100,000 samples) sensitivity sensitivity Prob. of G>520MPa 0.2061% 1.520% 1.566% Relative error* 31.6091% 2.937%  *Relative error: prob(approx.) prob(MCS) x 100% prob(MCS) Summary In this chapter, a stochastic response surface method (SRSM) using polynomial chaos expansion is used in calculating structural reliability. Compared with FORM, which is based on the linear approximation at the most probability point, it provides more accurate result for nonlinear responses. In addition, orthogonal polynomial basis provide 38 a convergent behavior as the order of polynomial is increased. A nonlinear function has been used as numerical example to show its accuracy and convergence. Since continuum based sensitivity results were obtained during structure analysis, the computational cost is further reduced by providing gradient information while fitting response surface. SRSM has been applied on a structural problem to show its effectiveness. When sensitivity information is provided, numerical results show that even lower number of sampling point can provide more accurate result. CHAPTER 4 RELIABILITYBASED DESIGN OPTIMIZATION Although statistical methods of uncertainties quantification have been studied intensively for decades, traditional deterministic design optimization still takes no advantage in these scientific advances and compensates uncertainties based on experience; for example, the safety factor. Such an optimization scheme usually yields either unsafe or too conservative design due to the lack of uncertainty quantification. In order to impose existing knowledge of uncertainty to engineering design process, reliabilitybased design optimization (RBDO) methodologies have been proposed and developed(Chandu and Grandi 1995; Chen et al. 1997; Du and Chen 2002b; Enevoldsen and Sorensen 1994; Grandhi and L.P. 1998; Kim et al. 2004b; Kwak and Lee 1987; Liang et al. 2004; Tu 1999; Tu and Choi 1997; Youn et al. 2003), where the system reliability or probability of failure is used to evaluate the system performance. Compared to the deterministic optimization, RBDO provides margins on reliability by quantifying the uncertainty in the response of structural system due to input uncertainty. General RBDO Model Design optimization has been introduced to structural engineering for decades(Arora 2004; Haftka and Gurdal 1991; Vanderplaats 2001). Its methodologies have been well developed mathematically, and applications in product development are flourishing. The underlying design philosophy is to reduce the cost by pushing the design to its performance margin. In traditional deterministic design, an optimization problem is formulated as minimize Cost(d) subject to G, (d) < Gaowable, j = 1, 2,..., np (4.1) dL _ corresponding maximum constraint allowable; and d denotes the vector of the deterministic design variables. The objective is to minimize the cost while meeting the system constraints. A system design depends on the system design variables. In deterministic optimization, both objective and constraints only depend on the design vector d which contains all deterministic design variables d,. In reliabilitybased design, design is based on a randomly distributed system vector, e.g., denoted by X, which contains random variable Xi. In RBDO, the mean value ti or the standard deviation ci of the system variable Xi can be used as the design variable. In some cases, uncontrollable random variables may contribute to the uncertainty of the performance. Instead of directly setting constraints on deterministic performance, the RBDO problem(Chandu and Grandi 1995; Enevoldsen and Sorensen 1994; Grandhi and L.P. 1998; Wu and Wang 1996) can generally be defined by setting constraints to be uncertainty measures, such as probability of failure. It is then formulated as minimize Cost(d) subject to P(GJ(x) f, is the prescribed failure probability limit for thejth constraint. Reliability Index Approach (RIA) and Performance Measure Approach (PMA) In the RBDO formulation described in the previous section, each prescribed failure probability limit Pf is often represented by the reliability target index as, 8 = >(73) . Hence, any probabilistic constraint in Eq. (4.2) can be rewritten using equation as FG (0) _< (/6) (4.3) where FG(O)=P(G<0) is the cumulative distribution function(CDF) of G at the failed region. Equation(4.3) can also be expressed in another way through inverse transformations S= (FG (0)) > f (4.4) where fls is traditionally called the reliability index. The expression of probability constraint in Eq. (4.4) leads to the so called reliability index approach (RIA)(Tu and Choi 1997; Tu 2001; Youn 2001). The two forms of constraint described in equations (4.2)and (4.4) are basically the same. In FORM/SORM based RBDO, an inner loop optimization is used to find the most probability point (MPP) in the standard Gaussian space. RIA may cause singularity because /, approaches infinity or negative infinity when the failure probability is zero or one. In that case, inner loop optimizer may fail to find the MPP. There is an alternative way to avoid singularity(Tu and Choi 1997) based on a different concept of reliability measure. For any given target probability, a certain level of performance can be reached to meet the reliability requirement. Tu(Tu and Choi 1997) proposed an inverse measure approach called performance measure approach (PMA) based on FORM by transforming Eq.(4.4) to g* = FGI ((fl)) > 0 (4.5) where g* is named the target probabilistic performance measure. In PMA, Eq.(4.5) is used as probabilistic constraint of RBDO. PMA has been proved to be consistent with RIA in prescribing the probabilistic constraint, but their differences in probabilistic constraint evaluation can be significant (Tu 1999). PMA is more robust in FORM/SORM than RIA based on the fact that RIA may yield singularity; that is, 8f, approaches infinity or negative infinity. In addition, for an inactive probabilistic constraint, PMA is more efficient than RIA. Known as an inverse measure approach, PMA can also be implemented on the sampling based uncertainty estimation method. For example, in MCS, a performance measure that meets reliability requirement can be obtained from the order statistics of sampled performance values. Figure 41 shows the general numerical procedure of RBDO. The effect and efficiency of inverse measure approach has been investigated for FORM/SORM(Tu 1999; Tu and Choi 1997). In this research, RIA and PMA as two different philosophies for probability constraint evaluation are also addressed for SRSbased RBDO. FORM/SORM: inner loop optimization for MPP search Reliability Analysis: RIA I PMA SRSM: DOE I Sensitivity Analysis of Cost and Constraints Update Design in Optimizer No Converge? Yes Figure 41. Flow chart for reliabilitybased design optimization Probability Sensitivity Analysis (PSA) Similar to the traditional design sensitivity, where sensitivity quantifies the effect of deterministic design variable to the structure response, probability sensitivity provides the quantitative estimation of the changing of failure probability or reliability with respect to the changes of random parameters, such as means or standard deviations of random design variables. In RBDO, the gradient based optimizer needs sensitivity information to carry out optimization. Automatic differentiation using finite differentiation leads to a significantly extra computational cost, especially when there are many design variables. In RBDO, if constraints are set with the probability of failure being less than a certain threshold, the gradient of probability with respect to the random input is required. In this research, probability sensitivity analysis is utilized to calculate the gradient information. First, the probability sensitivity calculation in FORM is introduced by taking advantage of structural sensitivity analysis. It can be shown that one can obtain accurate probability sensitivity without extra simulation cost. Since SRSM shows an advantage for nonlinear response, sampling based probability sensitivity is also introduced. For inverse measure approach, sensitivity for both FORM and sampling based RBDO can be obtained. Probability Sensitivity Analysis in FORM In first order reliability method (FORM), reliability index (3) can be obtained by following equation /7 (UTTU*)12 (4.6) where U* is the vector of MPP. The derivative of failure probability with respect to the design variables in FORM can then be written as 0r7 0)7 0,8 0r7 0t7 where (p(*) is the PDF of the standard random variable. Thus, the sensitivity of the failure probability is directly related to that of the reliability index, which can be obtained by =. (VuT I2 _ (4.8) 0o7 0Qa /7 0a For a random variable 7 = 0, /p 1 .T FT(X*,0) aT(X*,0) aX' , U8 + X 0 = T(x ) I ((4.9) 1 U.T OT(X',0) Since the reliability index and the most probable point are available from the reliability analysis, the sensitivity can be easily obtained. If the computationally expensive structure analysis code does not come with sensitivity analysis, a finite difference method is widely used to provide gradient information for searching the most probability point (MPP). The computational cost of finite difference method is proportional to the number of design variables. Using design sensitivity analysis, we can avoid the finite difference calculation and provide more accurate gradient information to the line search for MPP. In the finite difference method, the gradient of the limit state in the standard normal space is defined as Vg(U) = lim g(U + AU)g(U) (4.10) AUO AU Every iteration in line search needs to perturb each design variable to evaluate the gradient. If the sensitivity information can be obtained from a structural analysis code, there is a more efficient way to obtain the gradient information for MPP search. The gradient Vg(U) can be computed as Vg(U) = Vg(X) T (U) (4.11) where T: X>U. The transformation T from original random design space to the standard Gaussian space can usually be obtained explicitly, and the gradient Vg(X*) is provided by design sensitivity analysis. In this section, the torque arm model described in Chapter 3 is used to evaluate the accuracy of the probability sensitivity analysis using FORM. At the initial design, the probabilistic parameters of eight random variables are considered as design variables. Each random variable is assumed to be normally distributed with a mean of zero and a standard deviation 0.1. The sensitivity of reliability index is calculated based on Eq.(4.9). Since the transformation T is an explicit function of probabilistic parameters, the sensitivity can easily be calculated with reliability analysis. Table 41 shows the sensitivity results with respect to mean (0f/ /p, ) and standard deviation (6,8/ /cr ). The accuracy of the sensitivity is compared with that of the finite difference method with 1% perturbation size. Table 41. Probability sensitivity with respect to random parameters (unit: centimeter) design af A,8 A,/8 Ap, x 100% A8 Af Au, x 100% A9u, A/, C9,8 C9, Au, C9 , xi 0.376 0.377 100.26 0.030 0.030 100.00 x2 5.243 5.243 100.00 5.773 5.775 100.03 x3 0.034 0.034 100.00 0.000 0.000 100.00 x4 0.106 0.106 100.00 0.002 0.002 100.00 x5 0.055 0.055 100.00 0.001 0.001 100.00 x6 7.244 7.244 100.00 11.022 11.011 99.90 x7 0.140 0.140 100.00 0.004 0.004 100.00 x8 4.457 4.457 100.00 4.171 4.173 100.05 In Table 41, the first column represents eight random variables that have normal distributions. All random variables are assumed to be independent. Since the mean value and the standard deviation are considered as probabilistic parameters, there are 16 cases in the sensitivity calculations. The second and fifth columns represent the sensitivity results obtained from the analytical derivative, while the third and sixth columns are sensitivity results from the finite difference method. A very good agreement between the two methods is observed. Table 42 shows the computational efficiency of the proposed analytical sensitivity calculation. The gradient information is provided from design sensitivity analysis in MPP search in the standard HLRF method(Hasofer and Lind 1974; Liu and Kiereghian 1991). The computational savings are about 90% compared to the case when only the function values are provided. Once the reliability analysis is finished, the sensitivity of reliability index requires additional 17 function evaluations for the finite difference method, while only a single analysis is enough for the proposed method because the analytical expression in Eq.(4.9) and (4.11) is used. Table 42. Computational efficiency of analytical method for probability sensitivity Finite differential method Analytical method number of analyses in MPP search 90 10 number of analyses in sensitivity calculation 17 1 Total number of analysis 107 11 Probability Sensitivity Analysis Using SRSM In RBDO, the probability of failure can be formulated as P, = f (x)dx (4.12) G(X)<0 where G(X) < 0 is the failure region and f(*) is the joint probability density function(PDF). By introducing an indication function I(G(X) < 0) such that I=1 if G(X) < 0 and I=0 otherwise, Eq.(4.12) can be rewritten as Pf = f/(G(x) < 0)f (x)dx (4.13) where QX denotes the entire random design space. Since Eq.(4.12) is used as a constraint in RBDO, the sensitivity of Pfis required. The derivative of failure probability can be written as P= J I(G(x) < 0) x = J I(G(x) < 0) f(X) f (x)dx x 0 x (4.14) f I(G(u) <0). cf(x) p(u)du f(x)aO x=T 1(u) where Qu denotes the entire standard normal space. Explicit expression of Eq.(4.14) for different distribution types and numerical examples are derived in Appendix A. The accuracy of the sensitivity results are also presented in Appendix A for the case of various distribution types. ReliabilityBased Design Optimization Using SRSM Although RIA and PMA are theoretically consistent in prescribing the probability constraint, there are still significant differences in probabilistic constraint evaluation. The RBDO based on RIA and PMA may lead to either different convergence or efficiency. In this section, an RBDO problem is formulated for the same torquearm model in Chapter 3 using the concepts of RIA and PMA. A 3rdorder SRS is constructed for uncertainty analysis for both RIA and PMA. When the reliability index is used as a constraint in RBDO, it sometimes experiences numerical difficulty because it can have a value of infinity for very safe design. When SRSM is used in evaluating the probabilistic constraint in RBDO, the problem of singularity can be avoided naturally since the value of failure probability can always be obtained from MCS. The accuracy and convergence of SRSM have been illustrated in the previous chapter. Although SRSM usually requires more performance evaluation compared to FORM, it is still an affordable and applicable approach to obtain more accurate results for the highly nonlinear system. RBDO with RIA In this section, the RBDO problem of the torque arm model is solved using RIA. Stochastic response surface is used in uncertainty analysis to evaluate probability constraints. RBDO formulation of Eq. (4.2) can be used straightforwardly to solve the problem. For the torquearm problem, the objective is to minimize the weight while meeting the requirement of reliability constraint. If we define that the structure fails when stresses in this structure reach yield stress, such that G,(x)= a,(x)c <0 (4.15) where x is the random input variables, cr (x) is stress response for ith constraint, ar, denotes yield stress. The RBDO problem is then defined as Minimize Mass(d) subject to P(G,(x) where ft is the target reliability index and 1( ) is the cumulative density function of srv. During the optimization, a fit = 3 is used, which corresponds to 99.87% reliability. Since the maximum stress location can shift, the probabilities of failure at eight different locations are chosen as constraints in Eq. (4.16). The constraints can be evaluated using the SRS. The SRS needs to be reconstructed at each design cycle. Table 43 shows the properties of the random variables and the lower and upper bounds of their mean values (design variables). Note that the design variables are the relative change of the corner points and the initial values of all design variables are zero. The lower and upper bounds are chosen such that the topology of the boundary is maintained throughout the whole design process. Table 43. Definition of random design variables and their bounds. The values of design variables at optimum design are shown in the 5th column (unit: centimeter). Random dL d dU dopt Standard Distribution Variables (Initial) (optimum) Deviation type di 3.0 0.0 1.0 2.5226 0.1 Normal d2 0.5 0.0 1.0 0.4583 0.1 Normal d3 1.0 0.0 1.0 0.9978 0.1 Normal d4 2.7 0.0 1.0 2.4663 0.1 Normal d5 5.5 0.0 1.0 2.1598 0.1 Normal d6 0.5 0.0 2.0 1.9274 0.1 Normal d7 1.0 0.0 7.0 2.3701 0.1 Normal d8 0.5 0.0 1.0 0.0619 0.1 Normal The design optimization problem is solved using the sequential linear programming technique. The optimization process converges after 14 design cycles and 27 performance evaluations where the relative convergence criterion has been met in two consecutive designs. The optimum values of the design variables are shown in the 5th column in Table 43. Figure 42 shows the optimum design and analysis results at the mean values. The major changes occur at design parameters d4, d/5, d and cd. Even if the maximum stress constraint is set to 800MPa, the active constraint at optimum design converges to a lower value so that the variance of the input parameters can be accounted for. In Figure 42 the maximum value shows 739MPa, which is the extrapolated nodal stress. The actual element stress at the active constraint is about 618MPa, which is much lower than the extrapolate stress show on the figure. Figure 43 provides the optimization history of the cost function. As a result of the optimization process, the mass of the structure is reduced from 0.878kg to 0.497kg (a reduction of about 43.4 %). 7 39+00 Nt$'rUO 211 700 211,UOO 1 bR+DOE 1 06+00 5 04003 Figure 42. Optimum design and stress distribution of the torque arm model with 8 random variables. 09 0.85  08  0.75  07  S0.65 06  0.55  05  0.45  04 0 2 4 6 B 10 12 14 iterations Figure 43. Optimization history of cost function (mass) for the torque arm model with 8 random variables. In order to observe the impact of the accuracy of the uncertainty propagation procedure at the optimum design, a 3rdorder SRS is considered at the optimum design (Figure 44). Table 44 shows the values of the reliability indices for the active constraint at the optimum design obtained from MCS withl00,000 samples, the proposed SRS approach, and the FORM. The MCS result is used as a reference mark to compare the other two methods, which has about 1.5% error in estimating reliability index with confidence level of 95%. The proposed SRS approach exhibits a lower error than the corresponding to the FORM and compares very well with the exact result (namely, 3) and that obtained using MCS (0.6% error) 10 iPU {P ot vespoPsP .t Optimum Figure 44. PDF of the performance function at the optimum for the torquearm problem Table 44. Reliability Index of active constraint at optimal design Reliability Index Error (%) MCS 3.0307  SRS 3.0115 0.633 FORM 2.9532 2.556 RBDO with Inverse Measure As we discussed in section 3.2, enlightened by PMA, an inverse measure approach can also be applied in SRS based RBDO. In this section, the inverse measure approach is applied on reliability based design optimization for the torque arm model. As discussed in section 3.2, the design problem can be formulated as Minimize Mass(d) subject to G,(u) > 0, i= 1,...,NC (4.17) dL < d < is N and allowed maximum probability of failure is Pf, then G, can be found by ordering samples and selecting pth smallest sample. p=NxPf (4.18) Thus, the evaluation of reliability constraints is transformed to find the pth order statistic of sampling. One of the advantages of this approach is that the sensitivity of performance based constraint measure can be obtained directly through structure sensitivity analysis: 8G(u) a8G(x*) 8G(x*) QT (u*;d) (4.19) 00 Q0 ax, Q0 For example, if 0 = /,, and X ~ Normal(u, cr2) OG(u) _G(x*) _G(x*) QaT(u*;d) (4.20) G(x*) Summary In this chapter, reliability based design optimization using stochastic response surface is discussed. Procedures for both RIA and PMA are investigated and formulated. A torque arm problem shown in Chapter 3 has been used to demonstrate the feasibility of RBDO using SRS. Since accurate, sensitivity calculation is important to the convergence of gradient based optimizer, probability sensitivity using FORM and SRS is presented. It is shown that probability sensitivity in SRS based sampling approach can be also obtained with minimal increase of computational cost. If the SRS is accurate enough, the accuracy of sensitivities obtained also have convergent with respect to the increasing of the sampling size. CHAPTER 5 GLOBAL SENSITIVITY ANALYSIS FOR EFFICIENT RBDO Introduction In industrial applications, a system usually involves considerable number of random variables. As stated in the previous two chapters, the increasing number of random variables boosts the computational cost of reliability analysis significantly. Structural reliability analysis which involves a computationally demanding model is limited by the relatively high number of required function analyses for uncertainty. Even if the local sensitivity information described in chapter 3 can reduce the number of required simulations, the dimension of the SRS will still increase according to the number of random variables. Design engineers want to reduce the number of variables based on their contribution to the output performance. However, it is challenging to identify the importance of a random variable in the process of uncertainty propagation. Those random variables with extremely low contribution to the performance variance can be filtered out to reduce the computational cost of uncertainty propagation. Recent development in statistics introduces global sensitivity analysis (GSA)(Saltelli et al. 2000; Saltelli et al. 1999; Sobol 1993; Sobol 2001), which studies how the variance in the output of a computational model can be apportioned, qualitatively and quantitatively, to different sources of variations. In this chapter, global variance based sensitivity analysis has been applied on structural model to illustrate different roles of random variables in uncertainty propagation. Effort has been made to reduce the dimension of random space in the RBDO process. To reduce the number of simulations required to construct the SRS even further, unessential random variables are fixed during the construction of the SRS. A random variable is considered unessential (and hence it is fixed) if its contribution to the variance of the model output is below a given threshold. Global sensitivity indices considering only main factors are calculated to quantify the model input contributions to the output variability hence establishing which factors influence the model prediction the most so that: i) resources can be focused to reduce or account for uncertainty where it is most appropriate, or ii) unessential variables can be fixed without significantly affecting the output variability. The latter application is the one of interest in the context of this chapter. The RBDO problem in the previous chapter was solved with all random variables. However, some random variables did not significantly contribute to the variance of the stress function. Thus, a large amount of computational cost can be saved if the random variables whose contribution to the variance of the output is small are considered as deterministic variables at their mean values. This section describes how the global sensitivity indices considering only main factors can be used for deciding unessential random variables during the construction of stochastic response surfaces. Sensitivity Analysis As defined by Saltelli(Saltelli et al. 2000), sensitivity analysis studies the relationships between information flowing in and out of the model. In engineering design application, sensitivity usually refers to the derivative of performance measure with respect the input design variable. This derivative is used directly in deterministic design as sensitivity information is required by the gradientbased optimizer. In this research, this derivative is called local sensitivity, which has been applied in constructing SRS in the previous chapters. Local sensitivity analysis concentrates on the local impact of the design variables. It is carried out by computing partial derivatives of the output performance with respect to the input variable at the current design point. In structural optimization, substantial research has been done on the local sensitivity(Choi and Kim 2004a; Choi and Kim 2004b). Another choice of sensitivity measure explores what happens to the performance variance if all design variables are allowed a finite variation. Global sensitivity techniques apportion the output uncertainty to the uncertainty in the input factors. A couple of techniques have been developed in recent two decades. In this research, Sobol's sensitivity indices(Sobol 1993; Sobol 2001), which is based on the decomposition of function into summands of increasing dimension, will be discussed and applied to the constraint evaluation of RBDO. VarianceBased Global Sensitivity Analysis (GSA) Variancebased methods are the most rigorous and theoretically sound approaches (Chen et al. 2005; Saltelli et al. 2000; Saltelli et al. 1999; Sobol 1993; Sobol 2001)for global sensitivity calculations. This section describes the fundamentals of the variance based approach and illustrates how the polynomial chaos expansions are particularly suited for this task. The variance based methods: (i) decompose the model output variance as the sum of partial variances, and then, (ii) establish the relative contribution of each random variable (global sensitivity indexes) to the model output variance. In order to accomplish step (i), the model output is decompose as a linear combination of functions of increasing dimensionality as described by the following expression: n n n '1 lj>1 subject to the restriction that the integral of the weighted product of any two different functions is zero. Formally, f .. f p(x)flx,..., ...,x,)fJl,"(_xJl ( ...,xJ)dx = 0, for il,...,i, j i,..., j (5.2) where p(x) is the joint PDF of input random variable x. If, for example, the weighting function is the uniform distribution for the random variables or the Gaussian probability distribution, the functions of interest can be shown to be Legendre and Hermite orthogonal polynomials, respectively. The model output variance can now be calculated using a wellknown result in statistics. The result establishes that the variance of the linear combination of random variables (U,) can be expressed as: V bo0 + bU1 = Eb2V(U,) + 2EECOV(U7,UJ) (5.3) Hence, the model output variance can be shown to be: V(f)= aV(f) a jaV(fj) +... a2...(2...) (5.4) where the terms represent partial variances and each V( ) may be found by definition as: V(f) [ f () E(f (x))]2p(x)dx (5.5) In the above formula, f(x) represents the function under consideration and the symbol E( ) denotes expected value. There are no covariance terms in Eq.(5.4) because of the orthogonality property shown in Eq.(5.2). The global sensitivity index S, that considers only main factor is called main sensitivity index, which associated with each of the random variables which is represented by Eq. (5.4): A sensitivity index that considers the interaction of two or more factors is called interaction sensitivity index. Thus, as denoted by Chan and Saltelli(Saltelli et al. 2000), the summation of all sensitivity indices, involving both main and interaction effect of ith random variable, is called total sensitivity index. Sobol(Sobol 2001) suggested to use total sensitivity indices to fix unessential variables. If total sensitivity index for certainty variable extremely small compare to 1, that means the contribution of the variable is neglectable and the variable can be considered as a deterministic one. Global Sensitivity Analysis Using Polynomial Chaos Expansion The polynomial chaos expansion is particularly suited for computing global sensitivity indices because: (1) The model output is already decomposed as a sum of functions of increasing dimensionality; (2) the functions are orthogonal with respect to the Gaussian measure (Hermite polynomials); and (3) the variance of the bases are analytically available. For example, the variances of the functions associated with a two dimensional chaos expansion of order 2 are shown in Table 51. Table 51. Variances of the Hermite bases up to the second order Function f V(f) Xi 1 X2 1 2 Xi2 1 2 X1X2 1 2 X22 1 2 In addition, given the polynomial chaos expansion (i.e., the coefficients of the linear combination of Hermite polynomials), the model output variance and global sensitivity indices can be easily computed using Eqs. (5.4) and (5.6), respectively. In both equations, the variances V( ), are readily available for polynomial chaos expansions of arbitrary order and number of variables. In order to show the effect of the global sensitivity, let us consider the torque arm model presented in chapter 3. The eight random design variables are normally distributed with standard deviation of 0.1. A cubic stochastic response surface with eight variables in standard normal space has been constructed to approximate the stress response. Global sensitivity indices for main factor have been calculated to quantify the contribution of each random variable to performance variability. Figure 51 shows that three design variables(x2,X6,x8) makes most contribution to the total variance, influences from other variables are extremely small. That means, at the initial design, that the randomness of those variables with low global sensitivity indices can be ignorable. If only three variables are considered in constructing SRS instead of eight, the computational cost will be saved significantly, while maintaining the same level of variability. 1.595% 0.05% 0.001%. 0.021% 15.784% 15% EX1i 0.000% EX2 OX3 OX4 EX5 EX6 EX7 82.332% 1 X8 Figure 51. Global sensitivity indices for torque arm model at initial design Adaptive Reduction of Random Design Space Using GSA in RBDO The idea of adaptive reduction of random variables is based on the main factor of each random variable. If it is smaller than a threshold, it is fixed in constructing SRS. For that purpose, a linear polynomial chaos expansion is enough to obtain the main factors of GSI. The current algorithm with linear polynomial chaos expansion for the reduction of random variables can be modified to use a sensitivity index that accounts for interactions. These interactions will only appear in higher order polynomial chaos expansions. The choice of a nonlinear polynomial chaos expansion would reduce the computational efficiency of the proposed approach with unclear significant advantages. As stated at the beginning of this section, once the global sensitivity indices are calculated, variables that have the least influence on the model prediction (unessential variables) can be identified and eventually held fixed without significantly affecting the output variability. The procedure is adaptive because the global sensitivity indices are calculated at each design iteration and as a result different sets of random variables may be fixed throughout the RBDO process. A flow chart of RBDO using this strategy is shown in Figure 52. Figure 52. Adaptive reduction of unessential random design variables using global sensitivity indices in RBDO. Loworder SRS is used for global sensitivity analysis, while a highorder SRS is used to evaluate the reliability of the system. At the initial design stage, a lowerorder stochastic response surface is constructed using all random variables. In this particular example the firstorder SRS is constructed using 17 sampling points. At the initial design, the firstorder SRS with eight random variables can be expressed as, G1 a0 + alul + a22 +' + a44 + a5u5 a6u6 a7u7 + a8u8 = 4.95 + 0.0063u, + 0.117u2 + 0.Onnn( ,. 0.0019u4 (5.7) +0.01121 ., 0.052u6 0.0002u7 0.016us One useful aspect of the polynomial chaos expansion is that the coefficients in Eq. (5.7) are a measure of the contribution of the corresponding random variable to the variation of the output, and these coefficients will not change significantly in higher order SRS. On the other hand, typically the global sensitivity index associated with a particular variable is responsible for most of its contribution to the output variance. Thus, evaluating the global sensitivity indices using the firstorder SRS can be justified. Since all random variables are transformed into standard normal random variables, the variance of G1 can be evaluated analytically. Using Eq. (5.6) and (5.7) and assuming the design variables are independent, the global sensitivity index can be calculated as: a2 S= (5.8) j=1 Note that the denominator in Eq. (5.8) is the total variance of G1 using the first order approximation. Thus, the global sensitivity index, S,, is the ratio of the contribution of ith random variable to the total variance. If the global sensitivity index of a specific variable is less than a threshold value, the variable is considered as deterministic and fixed at its mean value. In order to show the advantage of fixing unessential random variables, the global sensitivity indices of the torquearm model are calculated. Table 52 shows the global sensitivity indices of the torquearm model using the firstorder SRS at the initial design. The total variance of stress function is 1.670x 102. Based on the global sensitivity indices, there are only three random variables whose contribution is greater than 1.0%; i.e., u2, U6, and u8. Thus in the reliability analysis, only these three random variables are used in constructing the thirdorder SRS, which now requires only 19 sampling points for 10 unknown coefficients. All other random variables are considered as deterministic variables at their mean values. If the total number of sampling points for both lower (17) and higherorder (19) polynomial expansions are compared with the higherorder SRS using all random variables (89), a significant reduction of the number of sampling points was achieved. Table 52. Global sensitivity indices considering only main factors for the torque arm model at the initial design. Only three random variables (u2, u6, and u8) are preserved when a threshold value of 1.0% is in place. SRV Variance GSI (%) ui 3.916 x 105 0.235 u2 1.369x 102 82.0 u3 6.403 x 109 0.00003834 U4 3.667 x 106 0.02197 us 6.864x106 0.04109 u6 2.702x103 16.179 u7 4.818x108 0.0002885 u8 2.538x 104 1.519 The RBDO problem, defined in Eq.(5.2) in chapter 4 is now solved using the proposed adaptive reduction of random variables. The optimization algorithm converges after the 17th iteration. As shown in Figure 53, the optimum design using the adaptively reduced SRS is slightly different from that with all random variables in chapter 4. The former has a longer interior cutout than the latter. This can be explained from the fact that the model with reduced random variables has less variability than the full model. Furthermore, the optimum value achieved using the adaptively reduced SRS converges to a lower value than the one without adaptive reduction). The total mass of the torque arm is reduced by 54.8%. The difference between the two approaches is approximately 1.8%. The number of active random variables associated with the modeling of the first constraint during the design iterations are listed in Table 53. On average, four random variables were preserved, which implies that only 29 sampling points were required for constructing the SRS. This is three times less than the SRS approach without adaptive reduction (89 sampling points). Figure 53. Optimum designs for the full SRS (solid line) and adaptively reduced SRS (dotted line). Because some variables are fixed, the interior cutout of the design from the adaptively reduced SRS is larger than that from the full SRS. Table 53. Comparison of the number of random variables in each design cycle. The threshold of 1.0% is used. The first constraint is listed. Iter Full SRS Reduced SRS 1 8 3 2 8 3 3 8 3 4 8 3 5 8 4 6 8 4 7 8 5 8 8 4 17 8 4 Summary In this chapter, a dimension reduction technique using global sensitivity indices is introduced. Since the variances of the Hermite polynomial bases are analytically available, the SRS is suitable to compute global sensitivity indices. Its application to RBDO is also presented. In the RBDO procedure, the global sensitivity indices that are calculated using the lowerorder SRS are used to fix unessential random variables, a higherorder SRS with reduced dimension is then used in evaluating the probability constraint. Fixing the unessential random variables accelerates the design optimization process. The RBDO result obtained in this way is compared to that from previous chapter, which shows little difference because of the loss of variability in fixing random variables. CHAPTER 6 FATIGUE RELIABILITYBASED LOAD TOLERANCE DESIGN Introduction Traditional reliabilitybased structural design usually makes assumptions on randomness of factors involved in modeling a structural system such as design variables, material properties, etc. These parameters are relatively well controlled so that the variability is usually small. However, it is also important to consider the capacity of the system subject to working conditions, e.g., uncertain loadings, because the uncertainty in load or force is much larger than that of others. The variability of the load is often ignored in the design stage and is difficulty to quantify it. Without knowing the accurate uncertainty characteristics of input load, it is hard to rely on the reliability of the output. In this chapter, a different approach from the traditional RBDO is taken by asking how much load a system can support. The amount of load, which a structural system can support, becomes an important information for evaluating a design. As an illustration, the fatigue reliabilitybased load tolerance of the front loader frame of CAT 994D wheel loader is studied. Besides the uncertainty in the material properties, which can be incorporated in SN curve(Ayyub et al. 2002; Chopra and Shack), uncertainties are also investigated on both mean and amplitude of a given dynamic load. Either the variation of load amplitude or mean may affect the fatigue life of the structural system. This research presents a reliability based load design method, which provides the load envelope for a structure subject to fatigue failure mode. Both one dimensional and multidimensional problems are addressed. Since service loads are subjective such that the load characteristic of one operator may completely different from that of others. In order to perform reliability analysis, it is necessary to know uncertainty characteristics of inputs. However, distribution type and parameters of loads are often unknown. In this chapter, instead of modeling variability in parameters by assuming specific type of random distribution, the effect of different distribution types on the system response is investigated by introducing the concept of conservative distribution type, which provides a safer way to model uncertainties. Fatigue Life Prediction Recent developments in the computeraided analysis provide a reasonable simulation for fatigue life prediction at early design stage for components under complex dynamic loads. For most automotive components, fatigue analysis means to find crack initiation fatigue life. Figure 61 illustrates the procedure to the crack initiation fatigue life prediction. FE Model Quasixtic Stress FE analysis .. Inference Indexes < Dynamic / Dynamic / Loading ) Superposition  Stress History Peak & Value Rainflow Multiaxial Life Editing Counting Prediction Figure 61. Flow chart for fatigue life prediction Crack Initiation Fatigue Life Prediction Two major crack initiation fatigue life prediction methods are stressbased and strainbased methods. The stresslife (S N) approach employs relationship between the stress amplitude and the fatigue life. This method is based primarily on linear elastic stress analysis. The advantage of stresslife approach is apparent since changes in material and geometry can easily be evaluated and large empirical database for steel with standard notch shape is available. However, the effects of plasticity are not considered in this method. The local strainlife( N ) method assumes that the local strains control the fatigue behavior. The plastic effects are considered well in this method. It is similar to the stresslife approach in that it uses E N curve instead of S N curve, but differs in that the strain is the variable related to the life, and also in that plastic deformation effects are specifically considered. Machine parts are usually required to be durable and able to undertake high numbers of life cycles. The front loader frame of CAT 994D wheel loader is one of such case. The critical position of fatigue failure is usually at welding joints. Because the stresslife method is works well for the brittle material and provides a reasonable approximation for a high cycle fatigue crack initiation life, by taking advantage of the availability of a large amount of available uniaxial fatigue data, stresslife method is employed in this chapter. Since the crack is usually initiated along the component surface, for saving unnecessary computation, FE based fatigue analysis chooses element along surface to calculate the fatigue life. For multiaxial application, the principal stress method has been applied using the planes perpendicular to the surface. Fatigue lives are calculated on eighteen planes spaced at 10 degree increment. On each plane the principal stresses are used to calculate the time history of the stress normal to the plane. It has been shown that this method should only be used for fatigue analysis of 'brittle' metals like cast iron and very high strength steels, as it provides nonconservative results for most ductile metals. Based on the factor that material in our application is cast iron and the interested region is welding joints, the principal stress algorithm can offer the fatigue life calculation with reasonable accuracy. Using superposition of dynamic loadings and the quasistatic FE analysis, the dynamic stresses in the component are used to analyze multiaxial fatigue, based on principal stress using conventional SN curve(Fesafe 2004). Most basic fatigue data are collected in the laboratory by testing procedures which employ fully reversed loading. However, realistic service loading usually involves nonzero mean stresses. Therefore, the influence of mean stress on fatigue life should be considered so that the fully reversed laboratory data can be used in the evaluation of real service life. Since the tests required to determine the influence of mean stress are quite expensive, several empirical relationships which related alternating stress amplitude to mean stress have been developed. Among the proposed relationships, two are widely used, which are Goodman and Gerber models. Goodman: (S/ S)+ + (SmS) = 1 Gerber: (Sa/Se)+(Sm/S)2 =1 where S,: Alternating stress amplitude; S,: Endurance stress limit Sm: Mean stress; S,: Ultimate strength 70 Experience shows that test data tend to fall between the Goodman and Gerber curves. In the application of fatigue life prediction of front loader frame of CAT 994D, Goodman relation is applied to address the mean stress effect. Variable Amplitude Loading and Cumulative Damage In real application, components are usually subject to complex dynamic loading which has variable amplitude. It requires identifying cycles and assessing fatigue life for each cycle. The rainflow counting method(Matsuishi and Endo 1968) is the most commonly used cycle counting technique. This method defined cycles as closed stress strain hysteretic loops as shown in the figure below: Stress D 0 1 T im e Figure 62. Rainflow and hysteresis An algorithm of rainflow counting can be developed based on ASTM standard description. Although the rainflow counting method is not based on an exact physical concept to account for fatigue damage accumulation, it is expected to provide a more realistic representation of the loading history. Cumulative damage of each cycle can be obtained by the PalmgrenMiner hypothesis, which is referred to as the linear damage rule: D= (6.1) in ith cycle. In Eq. (6.1), D, is the fraction of the damage; n, is the counted number of cycles forjth stress range; N, is the cycles to fail; and n is the total number of stress ranges counted from rain flow. Failure is predicted to occur if N D >1 (6.2) where N is the number of cycles. Thus, the fatigue life can be calculated as the number of applied load cycle until the cumulative damage reaches 1: Life cycles = N (6.3) YD, 1=1 Model Preparation for Fatigue Reliability Analysis Finite Element Model Figure 63 shows the component of a front loader frame of CAT 994D wheel loader, which is subjected to 26 channels of dynamic loading. As show in Figure 64, the finite element model consists of 49,313 grid points and 172,533 elements (24 beam, 280 gap, 952 hexagon, 1016 pentagon, 226 quadrilateral, 160,688 tetrahedron, 9,144 triangular, 203 rigid body elements). In order to apply for the displacement boundary 72 conditions and loads, pins are modeled using beam and gap elements. The existence of gap elements makes the problem nonlinear. However, if the gap status does not change during analysis, we can still consider the problem to be linear. 1 APin Lf X 2 APin Lf Y 3 APin Lf Z Pos Upper Hitch 4 APin Lf Z Neg 5 APin Rt X 6 APin Rt Y 7 APin Rt Z Pos 8 APin Rt Z Neg 9 GPin Lf X 10 GPin Lf Y Steer 11 GPin Rt X ylinder Pin 12 GPin Rt Y Lower Hitch 13 YPin Lf X 14 YPin Lf Y Figure 63. Front loader frame of CAT 994D wheel loader (subject to 26 channels of dynamic loading) GPin Bore APin Bore Axle Pad Casi,... YPin Bore . 15 YPin Rt X 16 YPin Rt Y 17 UHitch X 18 UHitch Z 19 LHitch X 20 LHitch Z 21 Steer Rt X 22 Steer Rt Z 23 Steer Lf X 24 Steer Lf Z 25 LHitch Y Pos 26 LHitch_Y Neg Clamped Clamped Figure 64. Finite element model for front frame Dynamic Load History In Figure 63, a total of 26 degreesoffreedom are chosen for the application of dynamic loads. All loads are located in the center of the pins and the hitches. Even if the dynamic loadf(t) is applied to the system, it is assumed that the inertia is relatively small and the method of superposition can be applied. Thus, only a linear static analysis is enough with the unit load applied to each degreeoffreedom. The stress value from the unit load is called the stress influence coefficient. The dynamic stress can be obtained by multiplying these stress influence coefficients with the dynamic load history. Measured data of 26 channels are used for the dynamic loads with the duration of 46 minutes. This duration is defined as a working cycle. The dynamic load is sampled such that 9,383 data points are available for each channel. Uncertainty in Material Properties and SN Curve Interpolation Based on available material properties and the component's working conditions, principle stress analysis using the Goodman model is used as the algorithm for fatigue life prediction. From superposition of quasistatic linear finite element analysis and dynamic loading, the stress data are obtained for each element. These stresses can be regard as 'true stress', which means SN curve can be applied directly on principle stress life method without considering the stress concentrate factor. The SN curve can be interpolated from nominal stresslife data. Considering the uncertainties of material properties, this interpolation will be implemented in a random manner. A lognormal distribution in SN curve can be assumed to simplify the randomness. Although there is no rigorous statistical evaluation was performed, but this assumption seems reasonable empirically(Ayyub et al. 2002; Chopra and Shack). Figure 65 shows the SN curve obtained from stress life data for cast iron used in the front frame. Solid line is the nominal SN curve and two dashed lines represent the variation in SN curve. 104 SN Curve(Logarithmic scale) I 6MP MPa UP a Figure 65. Material SN curve with uncertainty Uncertainty Modeling of Dynamic Loadings Dynamic loadings are usually very complicated and may involve a lot of uncertainties. Figure 66 shows one channel of the dynamic load. The mean and amplitude of dynamic loading usually plays the most important role in fatigue life estimation. Therefore, for the purpose of illustration and simplification, uncertainties can be model based on these two quantities. Figure 66. Illustration of one channel of dynamic loads By combining the effects of the randomness of mean and amplitude of the loads, two load capacity coefficients a and y are defined for the mean and amplitude, respectively. The dynamic load can be parameterized as f(t) = mean + y(f,(t) fmean) (6.4) where a y are random parameters to describe the uncertainties of the loads called load capacity coefficient (LCC). In Eq. (6.4), f0(t) is the original dynamic loads and /fa is the mean value of the initial loads. Due to the random parameters, the dynamic load f(t) shows probabilistic behavior. Equation (6.4) provides a simple two dimensional model of uncertainty in dynamic load history. Note that when both a and y equal to one and fixed, the original deterministic loading history f (t) is recovered. In the following section, onedimensional problem will first be investigated by fixing one of them. For example, if we fix a at 1 and treat y as a random variable, then uncertainty is modeled for the amplitude of the load. Linear Estimation of Load Tolerance The major challenge of the research is to estimate the load tolerance with respect to the reliability of fatigue life performance, which depends on the load history and uncertainty characterization. Identifying the load distribution is one of the most difficult tasks in the uncertainty analysis because different operating conditions will yield completely different distribution types. At this point, it is assumed that the load has a specific uncertainty characteristic (distribution type and corresponding parameters). When the variance of the load is fixed, for example, it is possible to construct the safety envelope by gradually changing the mean value of the applied load, which requires a large number of reliability analyses. When nonlinearity of the system is small, it is possible to estimate the safety envelope using the sensitivity information at the current load without requiring further reliability analyses. This estimation is based on the first order Taylor series expansion method. For illustration, onedimensional models (only considering single random variable) for the effect of amplitude and mean are separately investigated to meet the reliability requirement of fatigue life under uncertainty. In these one dimensional cases, the variation in SN curve is ignored. Variability of Dynamic Load Amplitude In order to consider the variability of the dynamic load, the mean value of the load is first assumed to remain constant, while the amplitude of the load is varied randomly. From Eq. (6.4), the uncertainty caused by the amplitude can be represented using the following decomposition of the dynamic load: f(t) = fn + (fO(t) /mean) (6.5) When /= 1, the original load history is recovered. When y= 0, the dynamic load becomes a static load with the mean value. In this definition, cannot take a negative value. Since yis a random variable, it is necessary to assume the distribution type and distribution parameters for y First we assume that y is normally distributed with the mean of one and the standard deviation of 0.25 (COV=0.25). The standard deviation is estimated from the initial dynamic load history. Since the firstorder reliability analysis is performed using the standard random variable, we convert into the standard random variable u by (6.6) =1+0.25u where u ~ N(0,12), y N(1,0.252), / = mean, u= standard deviation. For any given sample point u corresponding ycan be obtained from Eq. (6.6), and using ya new dynamic load history can be obtained from Eq. (6.5). By applying this dynamic load history, we can calculate the fatigue life of the system. Since this procedure includes multiple steps, we can construct a (stochastic) response surface for the fatigue life and then perform the reliability analysis using the response surface. Since the fatigue life changes in several orders of magnitudes, it would be better to construct the response surface for the logarithmic fatigue life. In one dimensional case, five collocation points are available from the DOE introduced in chapter 3. Using these five sampling points, a cubic stochastic response surface is constructed as a surrogate model for the logarithmic fatigue life as L(y) Loglo(Life) = 5.7075 0.7223u 0.0581(u2 1) + 0.0756(u3 3u) (6.7) Note that in one dimensional case, the five collocation points available from previous DOE scheme are sometimes not enough to construct a high fidelity SRS, a couple of complementary sampling points can be chosen to construct a new SRS, which spread evenly between the original collocation points, i.e., four more point in the middle of intervals of the original five points have been chosen. The corresponding SRS logarithmic fatigue life becomes L(y) Logl(Life)= 5.6976 0.6826u0.0541(u2 1)+0.0617(u3 3u) (6.8) Various quantities for estimating the quality of SRS are shown in Table 61 for both five and nine sampling points scheme. The table shows nine points DOE schemes fit the data better based on significant improvement of PRESS (prediction error sum of squares). Table 62 lists the tstatistics for the evaluation of each coefficient in the above response surface. Although using more sampling points can increase the fidelity of estimation, it also increases the computational cost. In our specific problem, considering the saving of computation, the five sampling DOE scheme is sufficient. Table 61. Quality of response surface Error RMSE SSe R2 R2adj PRESS statistics 5 sampling 0.0406 0.0082 0.9980 0.9921 8.5671 DOE 9 sampling 0.0511 0.0235 0.9965 0.9943 0.1503 DOE Table 62. Tstatistic of the coefficients coefficient 1 2 3 4 tstatistics of 5 122.7590 15.9279 3.5759 4.0828 sampling DOE tstatistics of 9 229.1431 33.9519 4.9313 6.4894 sampling DOE The response surface in Eq. (6.7) shows that the mean of logarithmic fatigue life is 5.6976 and the standard deviation is about 0.6826. It also shows that the contribution of the higherorder terms is relatively small, compared with the constant and linear terms. Thus, we can conclude that the performance is mildly nonlinear with respect to the random variable. Since the required life of the working component is 60,000 hours and each cycle corresponding to 46 minutes, the target of the fatigue life can be written in the logarithmic scale by Ltarget = Loglo (60,000 hours) = Logo, (78,261 cycles). (6.9) &4.9 The system is considered to be failure when the predicted life from Eq. (6.7) is less than the target life in Eq. (6.9). Accordingly, we can define the probability of failure as Pf,[L(ca)Ltarget <0]< Parget, (6.10) where Ptarget is the target probability of failure. For example, when Ptarget = 0.1, the probability of failure should be less than 10%. Even though the interpretation of Eq. (6.10) is clear, it is often inconvenient because the probability changes in several orders of magnitudes. In reliability analysis, it is common to use the reliability index, which uses the notion of the standard random variable. Equation (6.10) can be rewritten in terms of the reliability index as Pf (t) < target target), (6.11) where /fis called the reliability index and D is the cumulative distribution function of the standard random variable. When Ptarget = 0.1, /target 1.3. The advantage of using the reliability index will be clear in the following numerical results. With the response surface in Eq. (6.7), reliability analysis is carried out using the firstorder reliability method (FORM) at /,= 1. The results of reliability analysis are as follows: P =17.81% = 0.922456 (6.12) =_ 3.972 where 08//80r is the sensitivity of the reliability index with respect to the mean value of y. Since Ptarget = 0.1 and targett = 1.3, the current system does not satisfy the reliability requirement. From the mild nonlinear property of the response, we can estimate the mean value of y that can satisfy the required reliability. The linear approximation of the mean value can be obtained from /esmte = 1 ( target)/ = 0.9049, (6.13) which means that the mean value of y needs to be decreased about 10% from the original load amplitude in order to satisfy the required reliability. In order to verify the accuracy of the estimated result, several sampling points are taken and reliability analyses are performed. Figure 67 shows the reliability index with respect to u,, while Figure 68 shows the probability of failure Pfwith respect to /r. The solid line is linearly approximated reliability using sensitivity information. The reliability 81 index is almost linear and the estimation using sensitivity is close to the actual reliability index. When the target probability of failure is 0.1 and y has the distribution of N(/u,0.252), the safety envelope can be defined as 0_< y_< 0.9049. (6.14) Effect of Load Amplitude 25 1 7 Linear predition 2      A Exact value 1 CI I I 15   0.7 OB 0.9 1 1.1 12 1.3 1.4 1.5 1.6 Figure 67. Reliability index /?with respect to random parameter wr 100 Effect of load amplitude P  Linear prediction      A Exact value  105   :::::::::: ::::::: 10ge 07 08 0.9 1 11 12 13 14 1.5 16 Figure 68. Probability of failure Pfwith respect to random parameter [r The result means that current design, considering 25% standard deviation in the load amplitude, is not enough to achieve 90% reliability. The structure should work under milder working condition, which means either lower the mean of the load amplitude by about 10% or provide more accurate estimation of the initial load to reduce the variance. Variability of Mean of Dynamic Load Since both mean and amplitude are used to describe the dynamic load history, both of their effects under uncertainty are studied separately. When the mean value of the load is assumed to be varied randomly and the load amplitude remains as the initial load, the uncertainty in load can be modeled as: f(t) = afmean +(fo (t) fmean) (6.15) Same as the case of load amplitude, when a= 1, the applied load is identical to the original load history. When a= 0, the applied load has the same amplitude with the original load history but the mean value is zero. In this definition, a can be a negative value. Since a is a random variable, it is necessary to assume the distribution type and distribution parameters for a. First we assume that a is normally distributed with the mean of one and the standard deviation of 0.25 (COV=0.25). Since the firstorder reliability analysis is performed using the standard random variable, we convert a into the standard random variable u by ,a =U+ (6.16) = 1+0.25u where u~ N(0,12), a N(1,0.252), p= mean, a= standard deviation. By following the same procedure with previous section, using nine sampling points, we can construct a cubic stochastic response surface as a surrogate model for the logarithmic fatigue life as L(a) Loglo(Life) =5.6906 0.0905u 0.0013(u2 1)0.0003(u33u). (6.17) If we compare the response surface in Eq. (6.17) with the case of amplitude change in Eq.(6.7), the mean values of the both cases are close but the standard deviations are quite different. From this result, we can conclude that the variance of the mean value does not contribute significantly to the variance of the fatigue life. Using the response surface in Eq. (6.17), reliability analysis is carried out using the firstorder reliability method (FORM) at / = 1. The results of reliability analysis are as follows: Pf, 08 ,8 6.3435 (6.18) 8= 4.012 Since Ptarget = 0.1 and Aarget = 1.3, the current system satisfies the reliability requirement. The linear approximation of the mean value can be obtained from estimate = 1(/f 1 target )/0=1 2.257, (6.19) which means that the system satisfies the reliability requirement even if the mean value of a is increased up to 225% from the original load. This observation is consistent with the conventional notion of fatigue analysis in which the effect of the amplitude is significant while that of the mean is not. In order to verify the accuracy of the estimated result, several sampling points are taken and reliability analyses are performed. Figure 69 shows the reliability index with respect to /a, while Figure 610 shows the probability of failure Pfwith respect to /e. The solid line is linearly approximated reliability using sensitivity information. The reliability index is almost linear and the estimation using sensitivity is close to the actual 84 reliability index. When the target probability of failure is 0.1 and a has the distribution of N(/,a,0.252), the safety envelope can be defined as 0 < /l < 2.257. (6.20) Effect of mean of load Figure 69. Reliability index /?with respect to random parameter [a effect of mean lU Pf 10,5 rI S i target Linear estimation    A Exact value 10 25 I i I i I i I 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 610. Probability of failure Pf with respect to random parameter ta Safety Envelope Concept for Load Tolerance Design The safety envelope is defined as the magnitudes of the input design variables when the system fails. When design variables are loads, it is called the load envelope. In one dimensional case, this is simply the range of the allowed loads, e.g., the range of mean value of uc or y in the previous section. In multidimensional case, the combination of various loads constitutes an envelope, which is convex in linear systems. Such information is very useful as a capacity of the current design, a future reference for design upgrade, maintenance and control. Figure 611 shows a schematic illustration of the safety envelope when two variables are involved. In such a complex situation, a systematic way of searching the boundary of the safety envelope needs to be developed. S Safety envelope SSafe N Figure 611. Safety envelope for two variables When the relationship between the safety of the system and the applied loads is linear or mildly nonlinear, linear approximation can produce a very effective way of estimating the safety envelope. In context of reliability based safety measure, the target of safety envelope is that failure probability cannot reach over the prescribed value. Numerical Path Following Algorithm According to the reliability based safety envelope concept introduced above, when target reliability has been specified, a safety envelope can be constructed using numerical path following algorithm to search the boundary of the safety envelope(Allgower and Georg 1990). In this research, a systematic way of searching the boundary of the safety envelope is proposed using a predictorcorrector method, which is similar to the Euler Newton continuation method(Allgower and Georg 1990; Kwak and Kim 2002). When the relationship between the safety of the structure and the applied loads is linear or mildly nonlinear, this approach can produce an efficient way of estimating the safety envelope. In the context of reliabilitybased safety measure, the boundary of the safety envelope is the location where the probability of failure is equal to the target probability. (k+l1 k+1) B Boundary ,3 = constant Figure 612. Predictorcorrector algorithm The predictorcorrector algorithm(Figure 612) is explained below with two random variables, a and y First, the distribution type of random variables is assumed. The effect of different distribution types on the safety envelope can be investigated by following the same procedure as in the previous section. It is clear that the two parameters must have nonnegative values, which means that the safety envelope only exists in the first quadrant. The capacity of the structure with respect to ([ta, [ty) is interesting. If the required probability of failure is Ptarget (i.e., target t = '(Ptarget)), the following steps can been taken to construct the safety envelope: 