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Aircraft structural safety

Baldwin Library of Historical Literature for Children at the University of Florida National Endowment for the Humanities ICDL CCLC UFSPEC

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AIRCRAFT STRUCTURAL SAFETY: EFFECTS OF EXPLICIT AND IMPLICIT SAFETY MEASURES AND UNCERT AINTY REDUCTION MECHANISMS By ERDEM ACAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Erdem Acar

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This dissertation is dedicated to my fa mily: my father Zuhuri Acar, my mother erife Acar, and my sister Asiye Acar.

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iv ACKNOWLEDGMENTS I would like to express special thanks a nd appreciation to Dr. Raphael T. Haftka, chairman of my advisory committee. I am gr ateful to him for providing me with this excellent opportunity and fi nancial support to complete my doctoral studies under his exceptional guidance. He encouraged me to attend several conferen ces and assisted in finding an internship during my studies. Th rough our weekly meetings and his open door policy, which I definitely over-exploited, he gr eatly contributed to this dissertation. His limitless knowledge and patience are inspiration to me. During the past three years, he was more than my PhD supervisor; he wa s a friend, and sometimes like a father. I sincerely hope we will remain in contact in the future. I would also like to thank the members of my advisory committee, Dr. Bhavani V. Sankar, co-chair of the committee, Dr. Na garaj Arakere, Dr. Nam-Ho Kim and Dr. Stanislav Uryasev. I am grateful for their wil lingness to review my Ph.D. research and to provide me with the constructive comments which helped me to complete this dissertation. In particular, I would like to extend special thanks to Dr. Bhavani V. Sankar for his guidance on the papers we co-author ed, and Dr. Nam-Ho Kim for his comments and suggestions during the meetings of the Structural and Multidisciplinary Group. I also wish to express my gratitude to my M.Sc. advi sor, Dr. Mehmet A. Akgun, who provided a large share of motivation for pursuing a doctorate degree. The experience he supplied me during my masters degree studies contributed to this dissertation.

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v I also wish to thank to my colleagues at the Structural and Multidisciplinary Group of the Mechanical and Aerospace Engineering Department of the University of Florida for their support, friendship and many technica l discussions. In partic ular, I would like to thank Dr. Melih Papila, Dr. Jaco Schutte, my soul sister Lisa Schut te, Tushar Goel and Ben Smarslok for their friendship (in the order of meeting with them). Financial support provided by NASA C ooperative Agreement NCC3-994, NASA University Research, Engineering and Tec hnology Institute and NAS A Langley Research Center Grant Number NAG1-03070 is gratefully acknowledged. Finally, my deepest appreciation goes to my family: my father Zuhuri Acar, my mother erife Acar and my sister Asiye Acar The initiation, continuation and final completion of this thesis would not have happened without their continuous support, encouragement and love. I am incredibly lucky to have them in my life.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................xi LIST OF FIGURES...........................................................................................................xv NOMENCLATURE........................................................................................................xix ABSTRACT.....................................................................................................................xxv CHAPTER 1 INTRODUCTION........................................................................................................1 Motivation.....................................................................................................................1 Objectives.....................................................................................................................5 Methodology.................................................................................................................6 Outline........................................................................................................................ ..7 2 LITERATURE REVIEW...........................................................................................12 Probabilistic vs. Deterministic Design.......................................................................12 Structural Safety Analysis..........................................................................................14 Probability of Failure Estimation........................................................................15 Analytical calculation of probability of failure............................................15 Moment-based techniques............................................................................16 Simulation techniques..................................................................................17 Separable Monte Carlo simulations.............................................................18 Response surface approximations................................................................19 Reliability-Based Design Optimization...............................................................20 Double loop (Nested) RBDO.......................................................................20 Single loop RBDO.......................................................................................21 Error and Variability...................................................................................................22 Uncertainty Classification...................................................................................22 Reliability Improvement by Error and Variability Reduction.............................23 Testing and Probabilistic Design................................................................................24

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vii 3 WHY ARE AIRPLANES SO SAFE STRUCTURALLY? EFFECT OF VARIOUS SAFETY MEASURES............................................................................28 Introduction.................................................................................................................28 Structural Uncertainties..............................................................................................30 Safety Measures..........................................................................................................32 Design of a Generic Component.................................................................................33 Design and Certification Testing.........................................................................33 Effect of Certification Tests on Distribution of Error Factor e...........................36 Probability of Failure Calculati on by Analytical Approximation.......................38 Effect of Three Safety Measur es on Probability of Failure........................................41 Summary.....................................................................................................................51 4 COMPARING EFFECTIVENESS OF MEASURES THAT IMPROVE AIRCRAFT STRUCTURAL SAFETY.....................................................................53 Introduction.................................................................................................................53 Load Safety Factor..............................................................................................54 Conservative Material Properties........................................................................54 Tests.....................................................................................................................54 Redundancy.........................................................................................................55 Error Reduction...................................................................................................55 Variability Reduction..........................................................................................55 Errors, Variability and Total Safety Factor................................................................56 Errors in Design...................................................................................................56 Errors in Construction.........................................................................................58 Total Error Factor................................................................................................59 Total Safety Factor..............................................................................................60 Variability............................................................................................................61 Certification Tests.......................................................................................................62 Probability of Failure Calculation..............................................................................65 Probability of Failure Calc ulation by Separable MCS........................................65 Including Redundancy.........................................................................................70 Results........................................................................................................................ .70 Effect of Errors....................................................................................................70 Weight Saving Due to Certificat ion Testing and Error Reduction......................73 Effect of Redundancy..........................................................................................74 Additional Safety Factor Due to Redundancy.....................................................77 Effect of Variability Reduction...........................................................................78 Summary.....................................................................................................................81 5 INCREASING ALLOWABLE FLIGHT LOADS BY IMPROVED STRUCTURAL MODELING....................................................................................82 Introduction.................................................................................................................82 Structural Analysis of a Sandwich Structure..............................................................85 Analysis of Error and Variability...............................................................................89

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viii Deterministic Design and B-basis Value Calculations...............................................93 Assessment of Probability of Failure..........................................................................96 Analyzing the Effects of Improved M odel on Allowable Flight Loads via Probabilistic Design................................................................................................99 Summary...................................................................................................................101 6 TRADEOFF OF UNCERTAINTY REDUCTION MECHANISMS FOR REDUCING STRUCTURAL WEIGHT..................................................................103 Introduction...............................................................................................................104 Design of Composite Laminates for Cryogenic Temperatures................................106 Calculation of Probability of Failure........................................................................108 Probabilistic Design Optimization............................................................................112 Probabilistic Sufficiency Factor (PSF)..............................................................112 Design Response Surface (DRS).......................................................................113 Weight Savings by Reducing Error and Employing Manufacturing Quality Control..................................................................................................................114 Choosing Optimal Uncertainty Reduction Combination..........................................118 Summary...................................................................................................................119 7 OPTIMAL CHOICE OF KN OCKDOWN FACTORS THROUGH PROBABILISTIC DESIGN.....................................................................................121 Introduction...............................................................................................................122 Testing of Aircraft Structures...................................................................................123 Quantification of Erro rs and Variability...................................................................125 Errors in Estimating Material Stre ngth Properties from Coupon Tests............125 Errors in Structural Element Tests....................................................................127 Allowable stress updating and the us e of explicit knockdown factors..............129 Current industrial practice on updatin g allowable stresses using worstcase conditions (implicit knockdown factors)........................................129 Proposal for a better way to update a llowable stresses: Using the average failure stress measured in the tests and using optimal explicit knockdown factors..................................................................................130 Error updating via element tests.................................................................134 Errors in Design.................................................................................................135 Errors in Construction.......................................................................................137 Total Error Factor.......................................................................................138 Total Safety Factor.....................................................................................138 Variability..........................................................................................................139 Simulation of Certification Test and Probability of Failure Calculation..................141 Simulation of Certification Test........................................................................141 Calculation of Probability of Failure.................................................................142 Results.......................................................................................................................144 Optimal Choice of Explicit Knockdow n Factors for Minimum Weight and Minimum Certification Failure Rate..............................................................145

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ix Optimal Choice of Explicit Knockdow n Factors for Minimum Weight and Minimum Probability of Failure....................................................................148 Effect of Coupon Tests and Structural Element Tests on Error in Failure Prediction.......................................................................................................150 Effect of number of coupon tests al one (for a fixed number of element tests, ne=3)..............................................................................................150 Effect of number of element test s alone (for a fixed number of coupon tests, nc=40)............................................................................................151 Advantage of Variable Explicit Knockdown Factors........................................153 Effect of Other Uncertainty Reduction Mechanisms........................................157 Effect of variability reduction....................................................................157 Effect of error reduction.............................................................................159 Effect of Number of Coupon Tests...................................................................161 Effect of Number of St ructural Element Tests..................................................162 Summary...................................................................................................................164 8 RELIABILITY BASED AIRCRAFT ST RUCTURAL DESIGN PAYS EVEN WITH LIMITED STATISTICAL DATA................................................................165 Introduction...............................................................................................................165 Demonstration of Gains from Reliability-Based Structural Design Optimization of a Representative Wing and Tail System...............................................................167 Problem Formulation and Simplifying Assumptions........................................167 Probabilistic Optimization with Correct Statistical Data..................................169 Effect of Errors in Information about Deterministic Design....................................174 Errors in Coefficient of Variation of Stresses...................................................174 Erroneous Mean Stresses...................................................................................177 Errors in Probability of Failure Estimates of Deterministic Design..................179 Effect of Using Wrong Probability Di stribution Type for the Stress................181 Approximate Probabilistic Design Base d on Failure Stress Distributions...............182 Application of Characteristic Stre ss Method to Wing and Tail Problem.................186 Summary...................................................................................................................189 9 CONCLUDING REMARKS....................................................................................192 APPENDIX A A-BASIS AND B-BASIS VALUE CALCULATION.............................................197 B PROBABILITY CALCULATIO NS FOR CHAPTER 3.........................................199 Calculation of Pr(CT|e), the Probability of Passing Certification Test....................199 Calculations of Mean and Standard Deviation of Probability of Failure.................200 C CONFLICTING EFFECTS OF ERROR AND VARIABILITY ON PROBABILITY OF FAILURE IN CHAPTER 3....................................................202

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x D COMPARISON OF RESULTS OF SINGLE ERROR FACTOR AND MULTIPLE ERROR FACTOR CASES..................................................................204 E DETAILS OF SEPARABLE MON TE CARLO SIMULATIONS FOR PROBABILITY OF FAILURE CALC ULATIONS IN CHAPTER 4.....................209 F CALCULATION OF THE SYSTEM FAILURE PROBABILITY USING BIVARIATE NORMAL DISTRIBUTION.............................................................212 G TEMPERATURE DEPENDENT MATER IAL PROPERTIES FOR THE CRYOGENIC LAMINATES IN CHAPTER 6.......................................................214 H DETAILS OF CONSERVATIVE CUMULATIVE DISTRIBUTION FUNCTION (CDF) FITTING..................................................................................216 I DETAILS OF DESIGN RESPONSE SURFACE FITTING FOR THE PROBABILITY SUFFICIENCY FACTOR FOR THE CRYOGENIC LAMINATES IN CHAPTER 6................................................................................218 J ASSESSMENT OF THE ERROR DUE TO LIMITED NUMBER OF COUPON TESTS.......................................................................................................................222 K PROBABILITY OF FAILURE CALCUL ATIONS FOR CHAPTER 7 USING SEPARABLE MCS..................................................................................................224 L CHANGE IN COST DUE TO INCREA SE OF THE STRUCTURAL WEIGHT..230 M RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY INDEX OF CERTIFICATION FAILURE RATE, RELIABILITY INDEX OF PROBABILITY OF FAILURE AND BUILT SAFETY FACTOR IN CHAPTER 7.............................................................................................................................. ..232 N CALCULATION OF THE MEAN AND THE C.O.V. OF THE STRESS DISTRIBUTION USING PROBABILIT Y OF FAILURE INFORMATION.........233 O RELATION OF COMPONENT WEIGHTS AND OPTIMUM COMPONENT FAILURE PROBABILITIES IN CHAPTER 8.......................................................236 P HISTORICAL RECORD FOR AIRCRAFT PROBABILITY OF FAILURE........241 LIST OF REFERENCES.................................................................................................243 BIOGRAPHICAL SKETCH...........................................................................................256

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xi LIST OF TABLES Table page 3-1 Uncertainty classification.........................................................................................31 3-2 Distribution of random va riables used for componen t design and certification......36 3-3 Comparison of probability of failure s for components designed using safety factor of 1.5, mean value for allo wable stress and error bound of 50%...................40 3-4 Probability of failure for different bounds on error e for components designed using safety factor of 1.5 and A-ba sis property for allowable stress.......................42 3-5 Probability of failure for different bounds on error e for components designed using safety factor of 1.5 and m ean value for allowable stress................................44 3-6 Probability of failure for different bounds on error e for safety factor of 1.0 and A-basis allowable stress...........................................................................................46 3-7 Probability of failure for different e rror bounds for safety factor of 1.0 and mean value for allowable stress.........................................................................................46 3-8 Probability of failure for different uncertainty in failure stress for the components designed with safety factor of 1.5, 50% error bounds e and A-basis allowable stress........................................................................................................47 3-9 Probability of failure for different uncertainty in failure stress for the components designed with safety factor of 1.5, 30% error bound e and A-basis allowable stress........................................................................................................47 3.10 Probability of failure for uncertainty in failure stress for components designed using safety factor of 1.5, 10% error bounds e and A-basis properties....................48 4-1 Distribution of erro r factors and their bounds..........................................................59 4-2 Distribution of random va riables having variability................................................61 4-3 Mean and standard deviations of the built and certified distributions of the error factor etotal and the total safety factor SF...................................................................64 4-4 Average and coefficient of variation of the probability of failure for the structural parts designed w ith B-basis properties and SFL=1.5.................................72

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xii 4-5 Reduction of the weight of structural parts by certif ication testing for a given probability of failure.................................................................................................74 4-6 Effect of redundancy on the probabilities of failure.................................................75 4-7 Effect of redundancy on the effec tiveness of certification testing...........................76 4-8 Effect of correlation coefficient on system failure probabilities and effectiveness of certification testing.........................................................................77 4-9 Additional safety factor due to redundancy.............................................................78 4-10 Comparison of system failure probabili ties corresponding to different variability in failure stress f......................................................................................................79 5-1 Deviations between measured and fitted values of average Gc and Gc with mode mixity for different designs..........................................................................90 5-2 The mean and B-basis values of the fracture toughness of the designs analyzed....94 5-3 Allowable flight load of failure of the sandwich panels designed using deterministic approach.............................................................................................96 5-4 Corresponding probabilities of failure of the sandwich panels designed using deterministic approach.............................................................................................99 5-5 Allowable flight loads of the sandw ich panels calculated via probabilistic approach.................................................................................................................101 6-1 Allowable strains for IM600/133...........................................................................107 6-2 Deterministic optimum design...............................................................................108 6-3 Coefficients of variatio n of the random variables..................................................108 6-4 Evaluation of the accuracy of the analysis response surface..................................110 6-5 Comparison of probability of failure estimations for the deterministic optimum..111 6-6 Probabilistic optimum designs for different error bounds when only error reduction is applied................................................................................................114 6-7 Probabilistic optimum designs for di fferent error bounds when both error and variability reduction are applied.............................................................................116 7-1 Distribution of erro r factors and their bounds........................................................138 7-2 Distribution of random va riables having variability..............................................140

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xiii 7-3 Mean and standard deviations of the built and certified distribution of the total safety factor SF.......................................................................................................142 7-4 Comparing explicit knockdown factors for minimum built safety factor for a specified certification failure rate...........................................................................148 7-5 Comparing explicit knockdown factors for minimum built safety factor for a specified probability of failure...............................................................................150 7-6 Comparison of constant and variab le explicit knockdow n factors case and corresponding area ratios, A/A0.............................................................................154 7-7 Comparison of constant (i.e., test independent) implicit and explicit knockdown factors and correspond ing area ratios A/A0............................................................156 7-8 Comparison of mean and coefficient of variation of total knockdown reduction at the element test level for the cases of implicit constant knockdown factor and explicit variable knockdown factors......................................................................157 7-9 Optimal explicit knockd own factors for minimum CFR when variability in failure stress is reduced by half..............................................................................159 7-10 Optimal explicit knockd own factors for minimum CFR when all errors reduced by half.....................................................................................................................161 7-11 Optimal explicit knockd own factors for minimum CFR different number of coupon tests, nc.......................................................................................................162 7-12 Optimal explicit knockdown factors for di fferent number of structural element tests, ne....................................................................................................................163 8-1 Probabilistic structural design optimization for safety of a representative wing and tail system........................................................................................................171 8-2 Probabilistic structural optimization of wing, horizontal ta il and vertical tail system.....................................................................................................................173 8-3 Errors in the ratios of failure probabi lities of the wing and tail system when the c.o.v. of the stresses under-estimated by 50%........................................................175 8-4 Errors in the ratios of failure probabi lities of the wing and tail system when the mean stresses are under-estimated by 20%............................................................178 8-5 Errors in the ratios of failure probabi lities of the wing and tail system when the probability of failure of the deterministic design is under-predicted.....................180 8-6 Errors in the ratios of failure probab ilities of wing and tail system when the probability of failure of the dete rministic design is over-predicted.......................181

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xiv 8-7 Errors in the ratios of failure probab ilities of the wing and tail system if the optimization is performed using wrong probability distribution type for the stress.......................................................................................................................182 8-8 Probabilistic design optimization for safe ty of the representative wing and tail system using the characteristic-stress method........................................................188 8-9 Effect of 20% under-estimate of k on the ratios of probability of failure estimate188 D-1 Equivalent error bounds for the SEF m odel corresponding to the same standard deviation in the MEF model...................................................................................205 D-2 Comparison of system failure proba bilities for the SEF and MEF models...........206 D-3 Comparison of the total safety factor SF used in the design of structural parts for the SEF and MEF models.......................................................................................207 E-1 Comparison of the probability of failure estimations.............................................211 I-1 The ranges of variables for the three DRS constructed for PSF calculation..........218 I-2 Accuracies of DRS fitted to PSF and Pf in terms of four design variables (t1, t2, 1 and 2) for error bounds, be, of 0, 10%, and 20%..............................................219 I-3 Ranges of design variables for the three DRS constructed for probability of failure estimation for the error and variability reduction case...............................221 M-1 Accuracy of response surfaces...............................................................................232 P-1 Aircraft accidents and probability of failure of aircraft structures ........................242

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xv LIST OF FIGURES Figure page 2-1 Building block approach..........................................................................................26 3-1 Flowchart for Monte Carlo simula tion of component design and failure................35 3-2 Initial and updated pr obability distribution func tions of error factor e ....................38 3-3 Design thickness variation with low and high error bounds....................................45 3-4 Influence of effective safety factor, er ror, and variability on the probability ratio (3-D view)................................................................................................................50 3-5 Influence of effective safety factor, error and variability on the probability ratio (2-D plot)..................................................................................................................50 3-6 Influence of effective safety factor error and variability on the probability difference (3-D view)...............................................................................................51 3-7 Influence of effective safety factor error and variability on the probability difference (2-D plot).................................................................................................51 4-1 Comparing distributions of built and certified total error etotal of SEF and MEF models......................................................................................................................63 4-2 Initial and updated distributi on of the total safety factor SF.....................................64 4-3 The variation of the probability of fa ilure with the built total safety factor.............68 4-4 Flowchart for MCS of co mponent design and failure..............................................69 4-5 Total safety factors for MEF model fo r the structural part and system after certification...............................................................................................................78 4-6 Effect of variability on failure probability...............................................................79 5-1 The model of face-sheet/core debonding in a one-dimensional sandwich panel with pressure load.....................................................................................................86 5-2 Critical energy release rate as a function of mode mixity........................................88

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xvi 5-3 Comparison of actual and fitted cumulati ve distribution functions of variability, dMM, of Gc.................................................................................................................92 5-4 Comparison of actual and fitted cumulative distribution functions of total uncertainty (error and variability, dA) of Gc.............................................................92 5-5 Fitted least square lines for fracture toughness, and derived B-basis allowables....95 6-1 Geometry and loading of the laminate with two ply angles...................................107 6-2 Comparison of CDF obtained via 1,000 MCS, the approximate normal distribution and conservative appr oximate normal distributions for 2 on 1 corresponding to the deterministic optimum..........................................................111 6-3 Reducing laminate thickness (hence weight) by error reduction (no variability reduction)...............................................................................................................115 6-4 Reducing laminate thickness by error reduction (ER) and quality control (QC)...116 6-5 Trade-off plot for the probability of failure, design thickne ss and uncertainty reduction measures.................................................................................................117 6-6 Tradeoff of probability of fa ilure and uncertainty reduction.................................119 7-1 Building-block approach for aircraft structural testing..........................................123 7-2 Simplified three level of tests.................................................................................124 7-3 Current use of knockdown factors based on worst-case scenarios........................131 7-4 Shrinkage of the failure surface.............................................................................132 7-5 The variation of the explicit knockdown factors with ratio of the failure stress measured in the test and calculated fa ilure stress with a nd without transition interval....................................................................................................................133 7-6 Proposed use of explicit knockdown factors dependent on test results.................134 7-7 Initial and updated distributi on of the total safety factor SF with and without structural element test.............................................................................................142 7-8 The variation of probability of failure of a structural part built by a single aircraft company.....................................................................................................144 7-9 Optimal choice of explicit knockdown factors kcl and kch for minimum built safety factor for specified certification failure rate................................................146 7-10 Comparing CFR and PF of the structures designed for minimum CFR and minimum PF...........................................................................................................149

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xvii 7-11 Effect of number of coupon tests on th e error in failure pr ediction for a fixed number of element test s (3 element tests)..............................................................151 7-12 Effect of number of element tests on the error in failure prediction for a fixed number of coupon tests (40 coupon tests)..............................................................152 7-13 Evolution of the mean failure stress distribution with and without Bayesian updating..................................................................................................................153 7-14 Comparison of variable and c onstant explicit knockdown factor..........................154 7-15 Comparison of Pareto fronts of certification failure rate and built safety factor for two different approaches while upda ting the allowable stress based on failure stresses measured in element tests.........................................................................155 7-16 Reducing probability of failure and cert ification failure rate using variability reduction.................................................................................................................159 7-17 Reducing certification failure rate us ing error reduction, va riability reduction and combination of error and variability reduction................................................160 7-18 Optimal explicit knockdown factors fo r different number of coupon tests for minimum CFR and PF............................................................................................162 7-19 Effect of number of structural element tests, ne.....................................................163 8-1 Stress distribution s ( ) before and after redesign in relation to failure-stress distribution f ( f)......................................................................................................168 8-2 The change of the ratios of probabilities of failure of the probabilistic design of Table 8-1 versus the error in c.o.v.( )....................................................................176 8-3 Two different stress distri butions at the wing leading to the same probability of failure of 1-7......................................................................................................177 8-4 The change of the ratios of probabilities of failure w ith respect to the error in mean stress.............................................................................................................179 8-5 Calculation of characteristic stress from probability of failure.........................185 8-6 Comparison of approximate and exact and and the resulting probabilities of failure for lognormal failure stress.....................................................................186 8-7 The variation of the ratios of probabili ties of failure with respect to error in k .....189 D-1 System failure probabilities for the SEF and MEF models after certification.......206 D-2 Total safety factors for the SEF and MEF model after certification......................208

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xviii E-1 Comparison of numerical CDF with the assumed lognormal CDF for the distribution of the re quired safety factor................................................................210 G-1 Material properties E1, E2, G12 and 12 as a function of temperature.....................214 G-2 Material properties 1 and 2 as a function of temperature....................................215 K-1 The variation of probability of fa ilure with built total safety factor......................227 K-2 Flowchart for MCS of co mponent design and failure............................................228

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xix NOMENCLATURE ARS = Analysis response surface reqA = Minimum required cross sectiona l area for the component to carry the service lo ading without failure A0 = Load carrying area if there is no variability and no safety measures 1, 2 = Coefficient of thermal expansion along and transverse to fiber direction be = Bound of error = Reliability index C = Capacity of structure, for example, failure stress CFD = Cumulative di stribution function CFR = Certification failure rate CLT = Classical lamination theory c.o.v. = Coefficient of variation DRS = Design response surface = Relative change in the characteristic stress corresponding to a relative change of in stress e = Error factor efc = Error in failure pr ediction at the coupon level eC = Error in capacity calculation efe = Error in failure prediction at the element level efs = Error in failure prediction at the structural level

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xx efT = Total error in failure prediction em = Error in materi al property prediction eP = Error in load calculation eR = Error in response calculation e = Error in stress calculation et = Error in thickness calculation etotal = Total error factor ew = Error in width calculation eA = Error in facture t oughness assessment if tradi tional (averaging) method is used eMM = Error in facture t oughness assessment if tradi tional (averaging) method is used ER = Error reduction E1, E2 = Youngs modulus along and tr ansverse to fiber direction 1, 2 = Strains in the fiber direction a nd transverse to the fiber direction f( ) = Probability density function of the failure stress F( ) = Cumulative distribution function of the failure stress FAA = Federal Aviation Administration G = Strain energy release rate GC = Fracture toughness G12 = Shear modulus 12 = Shear strain k = Error multiplier kA, kB = Tolerance coefficients for Abasis and B-basis value calculation

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xxi kdc = Knock-down factor used to calculate allowable stress KI, KII = Model I and II stress inte nsity factors, respectively M = Number of simulations in the first stage of MCS MCS = Monte Carlo simulation MEF model = Multiple error factor model N = Number of simulations in the second stage of MCS Nx, Ny = Mechanical loading in x and y directions, respectively nc = Number of coupon tests ne = Number of structural element tests pallow = Allowable flight load P = Load PDF = Probability density function PSF = Probability sufficiency factor Pd = Design load according to the FAA specifications Pf = Probability of failure of a component Pf* = Approximate probability of failure of probabilistic design Pfd = Probability of failure of deterministic design PF = Probability of failure of a system cP = Average probability of failure after certification test ncP = Average probability of failure before certification test QC = Quality control for manufacturing ret = Ratio of failure stresses measured in test and its predicted value R = Response of a structure, for example, stress

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xxii RMSE = Root mean square error RSA = Response surface approximation R2 adj = Adjusted coefficient of multiple determination = Coefficient of correlation s( ) = Probability density function of the stress SEF model = Single error factor model Sc = Additional company safety factor Scl = Additional company safety factor if the failure stress measured in element tests are lower than the predicted failure stress Sch = Additional company safety factor if the failure stress measured in element tests are higher than the predicted failure stress Sfe = Total safety factor added during structural element tests SFL = Load safety factor of 1.5 (FAA specification) SF = Total safety factor = Stress = Characteristic stress a = Allowable stress f = Failure stress t = Thickness vt = Variability in built thickness vw = Variability in built width VR = Coefficient of variation w = Width W = Weight

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xxiii Wd = Weight of the deterministic design = Cumulative distribu tion function of the sta ndard normal distribution = Mode-mixity angle Subscripts act = The value of the relevant qua ntity in actual flight conditions built = Built value of the relevant quantity, which is different than the design value due to errors in construction calc = Calculated value of the relevant quantity, which is different from the true value due to errors cert = The value of the relevant quantity after certification test d = Deterministic design design = The design value of the relevant quantity spec = Specified value of the relevant qunatity target = Target value of the relevant quantity true = The true value of the relevant quantity worst = The worst value of the relevant quantity W = Wing T = Tail Subscripts ave = Average value of the relevant quantity ini = Initial value of the relevant quantity upd = Updated value of the relevant quantity

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xxiv U = Upper limit of the relevant quantity L = Lower limit of the relevant quantity

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xxv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AIRCRAFT STRUCTURAL SAFETY: EFFECTS OF EXPLICIT AND IMPLICIT SAFETY MEASURES AND UNCERT AINTY REDUCTION MECHANISMS By Erdem Acar August 2006 Chair: Raphael T. Haftka Cochair: Bhavani V. Sankar Major Department: Mechanic al and Aerospace Engineering Aircraft structural safety is achieved by using different safety measures such as safety and knockdown factors, tests and redundancy. Safety factors or knockdown factors can be either explicit (e.g., load safety fact or of 1.5) or implicit (e.g., conservative design decisions). Safety measures protect agains t uncertainties in loading, material and geometry properties along with uncertainties in structural modeling and analysis. The two main objectives of this dissert ation are: (i) Analyzing and co mparing the effectiveness of structural safety measures and their interact ion. (ii) Allocating the resources for reducing uncertainties, instead of living with the un certainties and allocating the resources for heavier structures for the given uncertainties. Certification tests are found to be most eff ective when error is large and variability is small. Certification testing is more effective for improving safety than increased safety factors, but it cannot compete with even a small reduction in errors. Variability reduction is even more effective than error reduction for our examples.

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xxvi The effects of structural element test s on reducing uncertainty and the optimal choice of additional knockdown factors are explored. We fi nd that instead of using implicit knockdown factors based on worst-case scenar ios (current practice), using testdependent explicit knockdown f actors may lead weight savings. Surprisingly, we find that a more conservative knockdown factor should be used if the failure stresses measured in tests exceeds predicted failure stresses in order to reduce the variability in knockdown factors generated by variab ility in material properties. Finally, we perform probabilistic optim ization of a wing and tail system under limited statistical data for the stress dist ribution and show that the ratio of the probabilities of failure of the probabilistic design and determin istic design is not sensitive to errors in statistical data. We find that the deviation of the probabilistic design and deterministic design is a small perturbation, which can be achieved by a small redistribution of knockdown factors.

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1 CHAPTER 1 INTRODUCTION Motivation Traditionally, the design of aerospace stru ctures relies on a deterministic design (code-based design) philosophy, in which safety factors (bot h explicit and implicit), conservative material properties, redundancy and certification testing are used to design against uncertainties. An example of explicit safety factor is the load safety factor of 1.5 (FAR 25-303), while the conservative deci sions employed while updating the failure stress allowables based on structural element tests are examples for implicit safety factors. In the past few years, however, th ere has been growing in terest in applying probabilistic methods to design of aerospace structures (e.g., Lincoln 1980, Wirsching 1992, Aerospace Information Report of SAE 1997, Long and Narciso 1999) to design against uncertainties by e ffectively modeling them. Even though probabilistic desi gn is a more efficient wa y of improving structural safety than deterministic design, many engineers are skeptical of probability of failure calculations of structural de signs for the following reasons. First, data on statistical variability in material properties, geometry and loading distribut ions are not always available in full (e.g., joint distributions), and it has been shown that insufficient information may lead to larg e errors in probab ility calculations (e.g., Ben-Haim and Elishakoff 1990, Neal et al. 1992). Second, the magnitude of errors in calculating loads and predicting structural res ponse is not known precisely, and there is no consensus on how to model these errors in a probabilistic setting. As a result of these concerns, it is

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2 possible that transition to probability based design will be gradual. An important step in this transition is to understand the way safety is built into aircraft structures now, via deterministic design practices. One step taken in the transition to probabi listic design is in the definition of conservative material properties (A-basis or B-basis material property values depending on the failure path in the structure) by the Federal Aviation Administration (FAA) regulation (FAR 25.613). A-basis material property is one in which 99 percent of the material property distribution is better than the design value with a 95 percent level of confidence, and B-basis material property is one in which 90 per cent of the material property distribution is better th an the design value with a 95 percent level of confidence. The use of conservative material properties is intended to protect against variability in material properties. In deterministic design the safety of a stru cture is achieved through safety factors. Even though some safety factor s are explicitly specified, othe rs are implicit. Examples of explicit safety factors are the load safety factor and material pr operty knock-down values. The FAA regulations require a load safety fact or equal to 1.5 for airc raft structures (FAR 25-303). The load safety factor compensates for uncertaintie s such as un certainty in loading and errors in load calculations, stru ctural stress analysis, accumulated damage, variations in material properties due to manufacturing defects and imperfections, and variations in fabrica tion and inspection standards. Safety factors are gene rally developed from empirically based design guidelines establ ished from years of structural testing of aluminum structures. Muller and Schmid (1978) review the historical evolution of the load safety factor of 1.5 in the United Stat es. Similarly, the use of A-basis or B-basis

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3 material properties leads to a knock-down factor from the average values of the material properties measured in the tests. Note that these knock-down factors depend on the number of tests, because they compensate fo r both variability in material properties and uncertainty due to a finite number of tests. As noted earlier, an important step in tran sition to probabilistic design is to analyze the probabilistic impact of the safety measures used in deterministic design. This probabilistic analysis require s quantification of uncertain ties encountered in design, manufacturing and actual service cond itions of the aircraft structures. A good analysis of different sources of uncertainty in engineering modeling and simulations is provided by Oberkampf et al. (2000, 2002). These pape rs also supply good literature reviews on uncertainty quantification and divide the uncertainty into three types: variability, uncertainty, a nd error. In this di stinction, variability refers to aleatory uncertainty (inherent randomness), uncertainty refers to epistemic uncertainty (due to lack of knowledge), and error is defined as a recognizable deficiency in any phase or activity of modeling and simulation that is not due to lack of knowledge. To simplify the treatment of uncertainty cont rol, in this dissertation we combine the unrecognized (epistemic) and recognized e rror in the classification of Oberkampf et al. and name it error. That is, we use a simple classification that divides the uncertainty in the failure of a structural member into two types: errors and variability. Errors reflect inaccurate modeling of physical phenomena, errors in structural analys is, errors in load calculations, or deliberate use of materials a nd tooling in construction that are different from those specified by the de signer. Errors affect all th e copies of the structural components made and are therefore fleet-lev el uncertainties. Variability, on the other

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4 hand, reflects the departure of ma terial properties, geometry pa rameters or loading of an individual component from the fleet-ave rage values and he nce are individual uncertainties. Modeling and quantification of variability are much easier compared to that of error. Improvements in tooling and constructio n or application of tight quality control techniques can reduce variabilit y. Quantification of variability control can be easily done by statistical analysis of records taken thr oughout process of quality control. However, quantification of errors is not as easy, b ecause errors are larg ely not known before a structure is built. So, errors can only be quantified after the structure has been built. Errors can be controlled by improving accuracy of load and stress calculations, by using more sophisticated analysis and failure predic tion techniques or by te sting of structural components. Testing of aircraft structural components is performed in a building block type of approach starting with material characterizat ion tests, followed by testing of structural elements and including a final certification test. Te sting of structures is discussed in detail in the next chapter. The comparison of deterministic design a nd probabilistic design can be performed in many views. First of all, input and output variables of deterministic design are all deterministic values, while input and output va riables of probabilistic design are random (along with some deterministic variables, of course). Here, on the other hand, we compare probabilistic design and deterministic de sign in terms of use of safety factors. In deterministic design uniform safety factors are us ed; that is, the same safety factor is used for all components of a system. However, probabilistic design allo ws using variable

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5 safety factors through allo wing risk and reliability al location between different components. That is, instead of using the same safety factor for all components, probabilistic design allows to use higher factors for components or failure modes that can be controlled with low weight expenditure (Yang, 1989). This means the failure modes with small scatter and lightwe ight components. In addition, probabilistic de sign allows a designer to trade off uncertainty control for lower safety factors. That is, by reducing uncertainty, the designer can avoi d using high safety factors in the design and thereby can reduce the weight of the structural system. Th is design paradigm allows the designer to allocate risk and reliability between differe nt components in a rational way to achieve a safer design for a fixed weight comp ared to the deterministic design. Objectives There are two main objectives of this dissertation. The first is to analyze and compare the effectiveness of safety measure that improve structural safety such as safety factors (explicit or implicit) structural tests, redundanc y and uncertainty reduction mechanisms (e.g., improved structural analys is and failure prediction, manufacturing quality control). The second objective is to explore the advantage of uncertainty reduction mechanisms (e.g., improved structural analysis and failure prediction, tighter manufacturing quality control) ve rsus safety factors. That is we consider the possibility of allocating the resources for reducing uncertainties, in stead of living with the uncertainties and allocating the resources fo r designing the aircraft structures for the given uncertainties. We aim to analyze the effectiveness of safety measures taken in deterministic design methodology and investigat e the interaction and effec tiveness of these safety measures with one another and also with uncer tainties. In particular, the effectiveness of

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6 uncertainty reduction mechanisms is analyzed and compared. The uncertainty reduction mechanisms considered in this dissertati on are reduction of errors by improving the accuracy of structural analysis and failure prediction (analytically or through tests), and reduction of variability in failure stress as a result of tighter quality control. We explore the optimal choice of additional company safety factors used on top of the FAA regulation safety factors by using probabilistic design, which provides a rational way in the analysis. Additional company safety factors we consider are the conservative decisions of aircraft companies while updating the allowable stresses based on the results of structural element tests. We perform probabilistic design optimization for the case of limited statistical data on stress distribution and show that when th e probabilistic design is achieved by taking the deterministic design as a starting point, the ratio of probabilities of failure of the probabilistic design and deterministic design is not sensitive to errors due to limited statistical data, which would l ead to substantial errors in the probabilistic design if the probabilistic design starts from scratch. In addition, we propose a probabilistic design methodology in which the probability of failure calculation is confined only to stress limits, thereby eliminating the necessity for a ssessment of stress distribution that usually requires computationally expens ive finite element analyses. Methodology Probability of failure calculation of stru ctures can be performed by using either analytical techniques or simulation techniques. Analytical methods are more accurate but for complex systems they may not be pract ical. Simulation techniques include direct Monte Carlo simulation (MCS) as well as many variance-reduction methods including

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7 stratified sampling, importance sampling, a nd adaptive importance sampling (Ayyub and McCuen 1995). In probabilistic design of structures, the us e of inverse reliability measures helps a designer to have an easy estimate of the change in structural weight from the values of probabilistic performance meas ure and its target value as well as computational advantages (Ramu et al. 2004). Amongst those measures we use probabilist ic sufficiency factor (PSF) developed by Qu and Haftka (2003). Here we consider a simplified design problem for illustration purposes, so that the reliability analysis can be performed by analy tical means. The effect of testing then can be analyzed by using Bayesian approach. Th e Bayesian approach has special importance in engineering design where the available info rmation is limited and it is often necessary to make subjective decisions. Bayesian updating is used to obtain the updated (or posterior) distribution of a random variab le upon combining the initial (or prior) distribution with new information about the random variable. The detailed theory and procedures for applying Bayesian methods in re liability and risk anal ysis can be found in texts by Morgan (1968) and Martz and Waller (1982). Outline A literature survey on the historical e volution of probabilistic design, comparison of deterministic design and probabilistic desi gn practices, uncertainty control measures and testing of aircraft struct ures is given in Chapter 2. Chapter 3 investigates the effects of error, variability, safety measures and tests on structural safety of aircraft. A simple exampl e of point stress design and a simple error model are used to illustrate the effects of seve ral safety measures taken in aircraft design: safety factors, conservative ma terial properties, and certificati on tests. This chapter serves

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8 as the opening chapter; therefore the analysis and the number of safety measures are kept at a minimum level. For instance, only certif ication tests are included in the analysis. The effects of coupon tests and structural elemen t tests are delayed until Chapter 7. The simplifying assumptions in Chapter 3 allow us to perform analytical calculations for probability of failure and Bayesian updating. The interactions of the safety measures with one another and also with errors and variabili ties are investigated. For instance, we find that the certification tests ar e most effective when errors are large and variabilities are small. We also find that as safety measures combine to reduce the probability of failure, our confidence in the probability of failure estimat es is reduced. Chapter 4 extends the analysis presented in Chapter 3 by delivering the following refinements. The effectiveness of safety meas ures is compared with one another in terms of safety improvement and weight savings Structural redundancy, a safety measure which is omitted in Chapter 3, is also incl uded in the analysis. The simple error model used in Chapter 3 is replaced with a more detailed error model in which we consider individual error components in load calculation, stress calculation, material properties and geometry parameters including the eff ect of damage. The analysis in Chapter 4 enables us to discover that while certificat ion testing is more effective than increased safety factors for improving safety, it cannot compete with even a small reduction in errors. We also find that variability reduction is even more effective than error reduction. Realizing in Chapter 4 how powerful uncertainty reduction mechanisms are, we analyze the tradeoffs of uncertainty redu ction mechanisms, structural weight and structural safety in Chapters 5 and 6. The effect of error reduction (due to improved failure prediction model) on increasing the al lowable flight loads of existing aircraft

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9 structures is investigated in Chapter 5. Th e analysis is performed for a sandwich panel because the improved model is developed by Prof. Bhavani Sankar (co-chair of the advisory committee for this dissertation) so that we had good access to the details of experiments and computations. We find th at the improved modeling can increase the allowable load of a sandwich panel on average by about 13 pe rcent without changing the safety level of the panel when deterministic design principles is followed. The use of probabilistic design is found to double the load increase. Similarly to improvements of accuracy in fa ilure predictions, the improvements in the accuracy of structural analysis also l ead to error reduction. Th e improved structural analysis through taking the chem ical shrinkage of composite laminates is considered as the error reduction mechanism in Chapter 6. The work by Qu et al. (2003), which explored the effect of variability reduction th rough quality control, is extended in Chapter 6 to investigate the tradeoffs of error and variability reduction mechanisms for reducing the weight of the composite laminates at cryogenic temperatures Tradeoff plots of uncertainty reduction mechanisms, probability of failure and weight are generated that enable a designer to choose the optimal uncer tainty control mechanism combination to reach a target probability of failure with minimum cost. Chapter 7 finalizes the analysis of e ffects of explicit and implicit knockdown factors and uncertainty control mechanisms. In particular, Chapter 7 analyzes the optimal choice of the knockdown factors. These knockdown factors refer to conservative decisions of aircraft companies in choice of material properties and while updating the allowable stresses based on the results of struct ural element tests. We find that instead of using implicit knockdown factors based on worstcase scenarios (curre nt practice), using

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10 test-dependent explicit knockdow n factors may lead weight savings. Surprisingly, we find that a more conservative knockdown factor should be us ed if the failure stresses measured in tests exceeds predicted failure stresses in order to reduce the variability in knockdown factors generated by variability in ma terial properties. In addition, the effects of coupon tests, structural el ement tests and uncertainty co ntrol mechanisms (such as error reduction by improved structural mo deling or improved failure prediction, variability reduction by tighter quality contro l) on the choice of co mpany safety factors are investigated. Using a simple cost function in terms of stru ctural weight, we show that decisions can be made whether to invest resources on coupon tests, structural element tests, uncertainty reduction mechanis ms or extra structural weight. The analyses presented in Chapters 3-7 show how probabilistic design can be exploited to improve aircraft structural safety by allowing a rational analysis of interactions of safety a nd knockdown factors and uncertainty reduction mechanisms. There are, however, two main reasons for reluctance of engineers for pursuing the probabilistic design: the sensitiv ity of probabilistic design to limited statistical data and computational expense associated to the probabilistic design. Besides, Chapters 3-7 include analyses of a singl e aircraft structural com ponent, so in Chapter 8 the probabilistic design of an aircra ft structural system is presented. We show in Chapter 8, by use of probabilistic design of a representati ve wing and tail system, that errors due to limited statistical data affect the probability of failure of both probabilistic and deterministic designs, but the ratio of these proba bilities is quite inse nsitive to even very large errors. In addition, to alleviate the problem of computational expense, a probabilistic design optimization method is prop osed in which the probability of failure

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11 calculation is limited to failure stresses to dispense with most of the expensive structural response calculations (typica lly done via finite elemen t analysis). The proposed optimization methodology is illustrated with the design of the wing and tail system. Chapter 8 reveals that the difference betw een probabilistic design and deterministic design is a small perturbation, which ca n be achieved by choosing the additional knockdown factors through proba bilistic design, instead of choosing them based on experience. In addition, the proposed approximate method is found to lead to similar redistribution of material between structural components and similar system probability of failure. Finally, the dissertation culminates with Chapter 9, where the concluding remarks are listed.

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12 CHAPTER 2 LITERATURE REVIEW The literature review in this chapter first compares deterministic and probabilistic design methodologies. Then, we review st ructural safety analysis, followed by probability of failure estimation techniques. Next, reliability-based design optimization is reviewed. Then, uncertainty classifications available in the literature are discussed followed by our simplified classification based on simplifying the analysis of uncertainty reduction measures. Finally, the ut ilization of structural test s in probabilistic design is reviewed. Probabilistic vs. Deterministic Design Aircraft structural design still relies on the Federal Aviation Administration (FAA) deterministic design code. In deterministic design, conservative material properties are used and safety factors are introduced to protect against uncertainties. The FAA regulations (FAR-25.613) state that conservative material prop erties are characterized as A-basis or B-basis values. Detailed informa tion on these values was provided in Chapter 8 of Volume 1 of Composite Materials Ha ndbook (2002). The safety factor compensates for uncertainties such as uncertainty in loading and errors in load calculations, errors in structural stress analysis and accumulated damage, variations in material properties due to manufacturing defects and imperfections, an d variations in fabrication and inspection standards. Safety factors are generally developed from empirically based design guidelines established from years of structural testing and fli ght experience. In transport aircraft design, the FAA regulations state th e use of safety factor of 1.5 (FAR-25.303).

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13 Muller and Schmid (1978) reviewed the historical evolution of the 1.5 f actor of safety in the United States. On the other hand, probabilistic design me thodology deals with uncertainties by the use of statistical characterization of uncer tainties and attempts to provide a desired reliability in the design. The uncertainties of individual design parameters and loads are modeled by appropriate probability density functions. The credibility of this approach depends on several factors such as the accuracy of the analytical model used to predict the structural response, the accuracy of the data and the probabilistic techniques employed. Examples of the use of probabilistic design in aerospace applications include the following. Pai et al. (1990, 1991 and 1992) performed probabi listic structural analysis of space truss structures for a typical space st ation. Murthy and Chamis (1995) performed probabilistic analysis of composite aircraft structure based on first ply failure using FORM*. The probabilistic methodology has show n some success in the design of composite structures where parameter uncertainties are relatively well known. For example, the IPACS (Integrated Probabilis tic Assessment of Composite Structures) computer code was develope d at NASA Glenn Research Ce nter (Chamis and Murthy 1991). Fadale and Sues (1999) performed reli ability-based design optimization of an integral airframe structure lap joint. A proba bilistic stability analysis for predicting the buckling loads of compression loaded com posite cylinders was developed at Delft University of Technology (Arbocz et al. 2000). The FORM method is discussed later in this chapter.

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14 Although probabilistic design methodology offers the potential of safer and lighter designs than deterministic design, transition from deterministic design to probabilistic design is difficult to achieve. Zang et al. (2002) discussed the reas ons for this difficulty, and some of these reasons are given below. Industry feels comfortable with traditional design methods. Few demonstrations of the benefits of probabilistic design met hods are available. Current probabilistic design methods are more complex and computationally expensive than deterministic methods. Characterization of structural imperfections and uncertainties necessary to facilitate accurate analysis and design of the stru cture is time-consuming and is highly dependent on structural configuration, material system, and manufacturing processes. Effective approaches for characteri zing model form error are lacking. Researchers and analysts lack training in statistical methods and probabilistic assessment. Structural Safety Analysis In probabilistic design, the safety of a structure is evaluated in terms of its probability of failure Pf. The structures are designed such that the probability of failure of the structure is kept below a pre-specified level. The term reliability is defined in terms of probability of failure such that 1 f R eliabilityP (2.1) A brief history of development of the met hods for probability of failure calculation for structures was presented in a report by Wirsching (1992). As Wirsching noted, the development of theories goes back some 50 to 60 years. The modern era of probabilistic design started with the paper by Fruedenthal (1947). Most of the ingredients of structur al reliability such as probability theory,

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15 statistics, structural analysis and design, quality control existed prior to that time; however, Fruedenthal was the first to put them together in a definitive and compressive manner. The development of reliability th eory progressed in 1950s and 1960s. There are three cornerstone papers in 1960s. The firs t one is the paper by Cornell (1967), who suggested the use of a second moment method and demonstrated that Cornells safety index could be used to derive set of factors on loads and resistance. However, Cornells safety index had a problem of invariance in that it was not constant when the problem was reformulated in a mechanically equiva lent way. Hasofer and Lind (1974) defined a generalized safety index which was invariant to mechanical formulation. The third paper is the one by Turkstra (1970), who presented structural design as a problem of decision making under uncertainty and risk. More recen t papers are sophisticated extensions of these papers, and some of them are re ferenced in the following sections. Probability of Failure Estimation This section reviews the literature on pr obability of failure estimation. First, analytical calculation of probability of failure is discussed, followed by moment-based methods and simulation techniques. Analytical calculation of probability of failure In its most general form, the probabil ity of failure can be expressed as x xX xd f PG f 0 ) ( (2.2) where G x is the limit-state function whose nega tive values corresponds to failure and fXx is the joint probability density function for the vector X of random variables. The analytical calculation of this expression is challenging due to the following reasons

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16 (Melchers 1999). First, the joint probability density function fXx is not always readily obtainable. Second, for the cases when fXx is obtainable, the integration over the failure domain is not easy. The calculation of probability of failure can be made more tractable by simplifying (1) the limit-state defi nition, (2) the integration process, and (3) the integrand fXx. Moment-based techniques When the calculation of limit-state is expensive, moment-based techniques such as First Order Reliability Method (FORM) or Second Order Reliability Method (SORM) are used (Melchers, 1999). The basi c idea behind these techniques is to transfor m the original random variables into a set of uncorrelated standard normal random variables, and then approximate the limit-state function linearly (FORM) or quadratically (SORM) about the most probable failure point (MPP). The prob ability of failure of the component is estimated in terms of reliability index such that fP (2.3) where is the cumulative distribution functi on of a standard normal variable. The first paper on the use of FORM is proba bility of failure calculation appears to be Hasofer and Linds (1974). There exist enormous amount of papers on the use of FORM. The pioneer papers include Rackw itz and Fiessler (1978), Hohenbichler and Rackwitz (1983), Gollwitzer and Rackwitz (1983). FORM is usually accurate for limit stat e functions that are not highly nonlinear. SORM has been proposed to improve the re liability estimation by using a quadratic approximation of the limit state surface. So me papers on the use of SORM include

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17 Fiessler et al. (1979), Breitung (198 4), Der Kiureghian et al. (1987), Hohenbichler et al. (1987), Der Kiureghian and De Stefano ( 1991), Koyluoglu and Nielsen (1994) and Zhao and Ono (1999). Simulation techniques For most problems the number of variables in the problem definition is high, so the analytical calculation of the integral in Eq. (2.2) requires challenging multidimensional integration. Also the moment based approxi mations gives inaccurate results for high number of random variables (Melchers 1999). Under such conditions, simulation techniques such as Monte Carlo simulations (MCS) are used to compute the probability of failure. In MCS technique, samples of the random variables are generated according to their probabilistic distributions and then failu re condition is checked. The probability of failure Pf can be estimated by N N Pf f (2.4) where Nf is the number of simulations leading to failure and N is the total number of simulations. The statistical accuracy of the pr obability of failure estimation is commonly measured by its coefficient of variation ... f covP as (1) (1) ...()ff f f f fPP P N covP PNP (2.5) From Eqs. (2.4) and (2.5) it is seen that a small probability of failure will require a very large number of simulations for accepta ble accuracy. This usually results in an increase in computational cost. When li mit-state function calculations are obtained

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18 directly from analysis, then computational co st of MCS is not sens itive to the number of variables. When surrogate models are used, on the other hand, the computational cost of MCS is dependent on the number of variable s. To overcome the deficiency of MCS, several more efficient alte rnative sampling methods are introduced. Ayyub and McCuen (1995) supplied basic information and good re ferences for these sampling techniques. Some useful references taken from A yyub and McCuen (1995) are the followings: Importance sampling (Madsen et al., 1986, Melchers, 1989), stratified sampling (Law and Kelton 1982, Schuller et al. 1989), Latin hypercube sampling (Iman and Canover 1980, Ayyub and Lai 1989), adaptive importance sampling (Busher 1988, Karamchandani et al. 1989, Schuller et al. 1989), conditional expectation (Law and Kelton 1982, Ayyub and Haldar 1984), antithe tic variates (Law and Kelton 1982, Ayyub and Haldar 1984). In this study, we mainly deal with problems with simple limit-state functions. For these simple cases the integrand fXx can easily be obtained when the random variables are statistically i ndependent. The beneficial properties of normal and lognormal distributions are utilized fo r the variables with small coefficients of variations. Approximate analytical calculations of proba bility of failure are checked with Monte Carlo simulations to validate the acceptabil ity of assumptions. When limit-state functions are complex, Monte Carlo simulations are used to calculate the probability of failure. Separable Monte Carlo simulations As noted earlier, when estimating very low probabilities, the number of required samples for MCS can be high, thus MCS become s a costly process. In most structural problems, the failure condition may be writte n as response exceeding capacity. When the

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19 response and capacity are independent, it may be possible to analyze them separately with a moderate sample size, a nd still be able to estimate very low probabilities of failure. This is due to the fact that most failures do not involve extreme values of response or capacity but instead moderately high res ponse along with moderately low capacity. Therefore, to bypass the requirement of sa mpling the extreme tail of the limit-state function, the variables could be considered independently, by separating the response and the capacity, as discussed by Melchers (1999, Chapter 3). A good analysis of efficiency and accuracy of separable Monte Carlo simulations can be found in Smarslok et al. (2006). The common formulation of the structural failure condition is in the form of a stress exceeding the material limit. This fo rm, however, does not sati sfy the separability requirement. For example, the stress depends on variability in material properties as well as design area, which reflects errors in the anal ysis process. In that case, the limit-state function can still be re-formulate d in a separable form. In this dissertation we re-write the limit-state in terms of the required area (dep ends only on variabilit ies) and built area (depends only on errors) to bring the limit state to separa ble form (see Chapter 4). Response surface approximations Response surface approximations (RSA) can be used to obtain a closed-form approximation to the limit state function to faci litate reliability analysis. Response surface approximations usually fit low-order polynomials to the structural response in terms of random variables. The probability of failure can then be calculated inexpensively by Monte Carlo simulation or by FORM or SORM using the fitted polynomials. Response surface approximations can be appl ied in different ways. One approach is to construct local response surfaces in the MPP region that contributes most to the

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20 probability of failure of the structure. Bucher and Bourgund (1990), Rajashekhar and Ellingwood (1993), Koch and Kodiyalam ( 1999), Das and Zheng (2000a, 2000b) and Gayton, Bourinet and Lemaire (2003) used local response surfaces. Another approach is to construct global response surface over the entire range of random variables. The examples include Fox (1994, and 1996), Romero and Bankston (1998), Qu et al. (2003), Youn and Choi (2004) and Kale et al. (2005). Reliability-Based Design Optimization Design optimization under a probability of fa ilure constraint is usually referred as reliability-based design optimization (RBDO) The basic structure of an RBDO problem is stated as et tP P t s farg. mi n (2.6) where f is the objective function (for mo st problems it is weight), and P and Ptarget are the probabilistic performance func tion and the target value for it. The probabilistic performance function can be probability of failure Pf, reliability index or an inverse reliability measure such as probabilistic sufficiency factor, PSF. Double loop (Nested) RBDO Conventional RBDO approach is formul ated as a double-loop optimization problem, where an outer loop performs the design optimization, while an inner loop optimization is also used for estimating probability of failure (or another probabilistic performance function). The reliability index ap proach (RIA) is the most straightforward approach. In RIA, the probabil ity of failure is usually calculated via FORM, which is an iterative process an so computationa lly expensive and sometimes troubled by

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21 convergence problems (Tu et al. 1999). To reduce the computational cost of double loop approach, various techniques has been pr oposed, which can be divided into two categories: (i) techniques that improve the e fficiency of uncertainty analysis methods, such as the methods of fast probability in tegration (Wu 1994) a nd two-point adaptive nonlinear approximations (Grandhi and Wang 1998); (ii) techniques that modify the formulation of probabilistic constraints, for instance, using inverse reliability measures, such as the performance measure approach (Tu et al. 1999), probabilis tic sufficiency factor (Qu and Haftka 2003). Inverse reliability measures are based on marg in of safety or safety factors, which are safety measures in deterministic design. Th e safety factor is usually defined as the ratio of structural resistance (e.g., failure stress) to structural res ponse (e.g., stress). Safety factors permit the designer to estimate th e change in structural weight to satisfy a target safety factor requirement. In proba bilistic design, however, the difference between the probabilistic performance measure and it s target value does not provide the designer with an estimate of the required change in structural weight. Inverse safe ty measures thus help the designer to easily estimate the change in structural weight from the values of probabilistic performance measure and its targ et value and the inverse safety measures also improve the computati onal efficiency (Qu and Haft ka 2004). A good analysis and survey on safety factor and inverse reli ability measures was presented by Ramu et al. (2004). Single loop RBDO Single loop formulation avoids nested loops of optimization and reliability assessment. Some single loop formulations are based on formulating the probabilistic constraints as deterministic constraints by either approximating th e Karush-Kuhn-Tucker

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22 conditions at the MPP or defi ning a relationship between prob abilistic design and safety factors of deterministic design (e.g., Chen et al. 1997, Kuschel and Rackwitz 2000, Wu et al. 2001, Qu et al. 2004, Liang et al. 2004). Single loop formulation increases the efficiency by allowing the solution to be inf easible before convergence and satisfying the probability constraints only at the optimum. Th ere exist also singe loop formulations that performs optimization and reliability assessment sequentially (e.g., Royset et al. 2001, Du and Chen 2004). Sequential optimization an d reliability assessment (SORA) of Du and Chen (2004), for instance, decouples the optimization and reliab ility assessment by separating each random design variable into a deterministic component, which is used in a deterministic optimization, a nd a stochastic component, whic h is used in reliability assessment. Error and Variability Uncertainty Classification Over years researchers proposed many differe nt classifications for uncertainty. For instance, Melchers (1999) di vided uncertainty into seve n types: phenomenological uncertainty, decision uncertainty, modeling un certainty, prediction uncertainty, physical uncertainty, statistical uncertainty and human error. Haimes et al. (1994) and Hoffman and Hammonds (1994) distinguis hed uncertainty into two types: uncertainty (epistemic part) and variability (a leatory part). Epistemic uncertainti es arise from lack of knowledge about the behavior of a phenomenon. They may be reduced by review of literatu re, expert consultation, close examination of data a nd further research. T ools such as scoring system, expert system and fish-bone diag ram can also help in reducing epistemic uncertainties. Aleatory uncertainties arise fr om possible variation and random errors in

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23 the values of the parameters and their estima tes. They can be reduced by using reliable manufacturing tools and quality control measures. Oberkampf et al. (2000, 2002) provided a good analysis of different sources of uncertainty in engineering modeling and si mulations, supply good literature review on uncertainty quantification and divide the un certainty into three types: variability, uncertainty and error. The cla ssification provided by Oberkampf et al. is discussed in the Motivation section of Chapter 1. Reliability Improvement by Error and Variability Reduction Before designing a new structure, materi al properties and lo ading conditions are assessed. The data is collected to constitute the probability distri butions of material properties and loads. The data on material pr operties is obtained by performing tests on batches of materials and also from the materi al manufacturer. To re duce the variability in material properties quality controls may be applied. Qu et al. (2003) analyzed the effect of application of quality controls over materi al allowables in the design of composite laminates for cryogenic environments. They found that employing qual ity control reduces the probability of failure significantly, allowi ng substantial weight reduction for the same level of safety. Similarly, before a newly designed structur e is put into service, its performance under predicted operational condit ions is evaluated by collecting data. The data is used to validate the initial assumptions being ma de through the design and manufacturing processes to reduce error in those assumptions. This can be accomplished by the use of Bayesian statistical methods to modify the a ssumed probability distribut ions of error. The present author will investigate this issue on following the chapters.

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24 After the structure is put into service inspections are performed to detect the damage developed in the structure. Hence, th e inspections are another form of uncertainty reduction. The effect of inspections in the safety of structures was analyzed (among others) by Harkness et al. (1994), Provan et al. (1994), Fujimoto et al. (1998), Kale et al. (2003) and Acar et al. (2004b). Testing and Probabilistic Design In probabilistic design, models for pred icting uncertainties and performance of structures are employed. These models invo lve idealizations and approximations; hence, validation and verification of these models is necessary. The validation is done by testing of structures, and verification is done by using more detailed models. Historical development of testing of stru ctures was given in the papers by Pugsley (1944) and Whittemore (1954). A literature su rvey of load testing by Hall and Lind (1979) presented many uses for load testing in design and safety valid ation of structures. Conventional design by calculati on relies upon tensile coupon tests to estimate material strength (Hall and Tsai, 1989). Coupon testing is a de structive test to measure loads and displacements at failure. On the other hand, proo f load testing is not a destructive test in which the structure is tested at a fixed load to measure resistance level of the structure. Proof load testing in a variety of applica tions was studied by several authors such as Barnett and Herman (1965), Shinozuka ( 1969), Yang (1976), Fujino and Lind (1977), Rackwitz and Schurpp (1985) and Herbert and Trilling (2006). Jiao and Moan (1990) illustrated a met hodology for probability density function updating of structural resistance by additiona l events such as proof loading and nondestructive inspection by utilizing FORM or SORM methods. Ke (1999) proposed an approach that specifically addressed the means to design component tests satisfying

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25 reliability requirements and objectives by a ssuming that the component life distribution follows Weibull distribution. Zhang and Mahadevan (2001) deve loped a methodology that utilizes Bayesian updating to integrate testing and analysis for test plan determination of a structural components. They considered two kinds of tests: failure probability estimation and life estimation tests. Soundappan et al. (2004) presented a method for designing targeted analytical and physical tests to validated reliability of structures obtained from reliability based designs. They found that the optimum number of tests for a component is nearly proportional to the square root of probability of failure. Guidelines for testing of composite materials were presented in Volume 1, Chapter 2 of Composite Materials Handbook (2002). The following are quoted from this source (pages 2-1 and 2-2). Analysis alone is generally not considered adequate for substantiation of composite structural designs Instead, the "buildin g-block approach" to design development testing is used in concert with analysis. This approach is often considered essential to the qualification/certif ication of composite st ructures due to the sensitivity of composites to out-of-plane lo ads, the multiplicity of composite failure modes and the lack of standard analytical me thods. The building-block approach is also used to establish environmental compensation va lues applied to full-scale tests at roomtemperature ambient environment, as it is often impractical to conduct these tests under the actual moisture and temperature enviro nment. Lower-level tests justify these environmental compensation factors. Simila rly, other building-bl ock tests determine truncation approaches for fatigue spectra and compensation for fatigue scatter at the fullscale level. The building-block approach is shown schematically in Figure 2.1.

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26 Figure 2-1. Building block approach (Repri nted, with permission, from MIL 17The Composite Materials Handbook, Vol. 1, Chapter 2, copyright ASTM International, 100 Barr Harbor Dr ive, West Conshohocken, PA 19428) The approach can be summarized in the following steps: 1. Generate material basis values and preliminary design allowables. 2. Based on the design/analysis of the structur e, select critical areas for subsequent test verification. 3. Determine the most strength-critical failure mode for each design feature. 4. Select the test environment that will pr oduce the strength-critical failure mode. Special attention should be given to ma trix-sensitive failure modes (such as compression, out-of-plane shear, and bondlin es) and potential "hot-spots" caused by out-of-plane loads or stiffness tailored designs. 5. Design and test a series of test specime ns, each one of which simulates a single selected failure mode and loading condition, compare to analytical predictions, and adjust analysis models or de sign allowables as necessary. 6. Design and conduct increasingl y more complicated test s that evaluate more complicated loading situations with the possibility of failure from several potential failure modes. Compare to analytical predictions and adjust analysis models as necessary. 7. Design (including compensation factors) and conduct, as required, full-scale component static and fatigue testing for final validation of internal loads and structural integrity. Compare to analysis. As noted earlier, validation is done by tes ting of structures, a nd verification is done by using more detailed models. Detailed mo dels may reduce the errors in analysis models; however errors in the uncertainty models cannot be reduced by this approach. In

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27 addition, very detailed models can be co mputationally prohibitive. Similarly, while testing of structures reduces both the errors in response mode ls and uncertainty models, it is expensive. Therefore, the testing of structures needs to be performed simultaneously with the structural design to reduce cost wh ile still keeping a specified reliability level.

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28 CHAPTER 3 WHY ARE AIRPLANES SO SAFE STRU CTURALLY? EFFECT OF VARIOUS SAFETY MEASURES This chapter investigates the effects of error, variability, safety measures and tests on the structural safety of aircraft. A simp le point stress design problem and a simple uncertainty classification are used. Since this chapter serves as the opening chapter, the level of analysis and the number of safety m easures are kept at a minimum level. Safety measures considered in this chapter are the load safety factor of 1.5, the use of conservative material properties and certificat ion test. Other safety measures such as structural redundancy, coupon and structural element tests will be included in the following chapters. Interaction of the consider ed safety measures with one another and their effectiveness with respect to uncertainties are also explored. The work given in this chapter was also published in Acar et al. (2006a). My colleague Dr. Amit Kales contributi on to this work is acknowledged. Introduction In the past few years, there has been gr owing interest in ap plying probability methods to aircraft stru ctural design (e.g., Lincol n 1980, Wirsching 1992, Aerospace Information Report of Society of Automo tive Engineers 1997, Long and Narciso 1999). However, many engineers are skeptical of our ability to calculate the probability of failure of structural designs for the following reasons. First, data on statistical variability in material properties, geometry and loading distributions are not al ways available in full (e.g., joint distributions), and it has been show n that insufficient information may lead to

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29 large errors in probability calculations (e.g., Ben-Haim and Elishakoff 1990, Neal et al. 1992). Second, the magnitude of errors in ca lculating loads and predicting structural response is not known precisely, and there is no consensus on how to model these errors in a probabilistic setting. As a result of th ese concerns, it is possible that transition to probability based design will be gradual. In such circumstances it is important to understand the impact of exis ting design practices on safet y. This chapter is a first attempt to explore the effects of various safety measures ta ken during aircraft structural design using the deterministic design approach based on FAA regulations. The safety measures that we include in this chapter are (i) the use of safety factors, (ii) the use of conservative material properties (A-basis), and (iii) the use of final certification tests. These safety measures are re presentative rather than all inclusive. For example, the use of A-basis properties is a representative measure for the use of conservative material properties. The safety measures (e.g., stru ctural redundancy) are discussed in the following chapters. We use A-Basis value rather than B-basis because we did not include redundancy in this ch apter. FAA suggests that (FAR 25.613) when there is a single failure pat h, A-Basis properties should be employed, but in case of multiple failure paths, B-Basis properties are to be used. In next chapter, for instance, we include structural redundancy in our analysis, so we use B-basis values in Chapter 4. The effect of the three individual safety measures and their combined e ffect on the probability of structural failure of the aircraft are de monstrated. We use Monte Carlo simulations to calculate the effect of these safety measures on the probability of failure of a structural component.

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30 We start with a structural design employi ng all considered safety measures. The effects of variability in geometry, loads, a nd material properties are readily incorporated by the appropriate random variables. However, there is also uncertainty due to various errors such as modeling errors in the anal ysis. These errors are fixed but unknown for a given airplane. To simulate these epistemic uncertainties, we transform the error into a random variable by considering the design of mu ltiple aircraft models. As a consequence, for each model the structure is di fferent. It is as if we pret end that there are hundreds of companies (Airbus, Boeing, Bombardier, Embraer etc.) each designing essentially the same airplane, but each havi ng different errors in their structural analysis. This assumption is only a device to model lack of knowledge or errors in probabilistic setting. However, pretending that the di stribution represents a large number of aircraft companies helps to motivate the probabilistic setting. For each model we simulate certification tes ting. If the airplane passes the test, then an entire fleet of airplanes with the same design is assumed to be built with different members of the fleet having different geometr y, loads, and material properties based on assumed models for variability in these properties. That is, the uncertainty due to variability is simulated by considering multiple realizations of th e same design, and the uncertainty due to errors is simulated by desi gning different structures to carry the same loads. Structural Uncertainties A good analysis of different sources of uncertainty is provided by Oberkampf et al. (2000, 2002). Here we simplify the classification, with a view to the question of how to control uncertainty. We propose in Table 3-1 a classification that distinguishes between errors (uncertainties that apply equally to the entire fleet of an aircraft model) and

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31 variabilities (uncertainties that vary for th e individual aircraft). The distinction is important because safety measures usually targ et one or the other. While variabilities are random uncertainties that can be readily modeled probabilistic ally, errors are fixed for a given aircraft model (e.g., Boeing 737-400) but they are largely unknown. Errors reflect inaccurate modeling of phys ical phenomena, errors in structural analysis, errors in load calculations, or use of materials and tooli ng in construction that are different from those specified by the designe r. Systemic errors affect all the copies of the structural components made and are ther efore fleet-level uncer tainties. They can reflect differences in analysis, manufacturing and operation of the aircraft from an ideal. The ideal aircraft is an airc raft designed assuming that it is possible to perfectly predict structural loads and structural failure for a given structure, th at there are no biases in the average material properties and dimensions of the fleet with respect to design specifications, and that there exists an opera ting environment that on average agrees with the design specifications. The other type of uncertainty reflects variability in material properties, geometry, or loadi ng between different copies of the same structure and is called here individual uncertainty. Table 3-1. Uncertainty classification Type of uncertainty Spread Cause Remedies Systemic error (modeling errors) Entire fleet of components designed using the model Errors in predicting structural failure and differences between properties used in design and average fleet properties. Testing and simulation to improve math model and the solution. Variability Individual component level Variability in tooling, manufacturing process, and flying environments. Improve tooling and construction. Quality control.

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32 Safety Measures Aircraft structural design is still done, by and large, using code-based design rather than probabilistic approaches. Safety is improved through conservative design practices that include use of safety fact ors and conservative material prope rties. It is also improved by tests of components and certification tests that can reveal inadequacies in analysis or construction. In the following we detail some of these safety measures. Load Safety Factor: Traditionally all aircraft stru ctures are designed with a load safety factor to withstand 1.5 times the limit-load without failure. A-Basis Properties: In order to account for uncertain ty in material properties, the Federal Aviation Administration (F AA) states the use of conser vative material properties. This is determined by testing a specified number of coupons selected at random from a batch of material. The A-basis property is determined by calculating the value of a material property exceeded by 99% of the population with 95% confidence. Component and Certification Tests: Component tests and certification tests of major structural components reduce stress and material uncertainties for given extreme loads due to inadequate structural models. These tests are conducte d in a building block procedure. First, individual coupons are tested, and then a sub assembly is tested followed by a full-scale test of the entire st ructure. Since these tests cannot apply every load condition to the structur e, they leave uncertainties wi th respect to some loading conditions. It is possible to reduce the probabi lity of failure by performing more tests to reduce uncertainty or by extra structural weig ht to reduce stresses. If certification tests were designed together with the structure, it is possible that additional tests would become cost effective because they w ould allow reduced structural weight.

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33 We simulate the effect of these three sa fety measures by assuming the statistical distribution of the uncertainties and incorp orating them in approximate probability calculations and Monte Carlo simulation. For variability the simulation is straightforward. However, while systemic erro rs are uncertain at the time of the design, they will not vary for a single structural com ponent on a particular ai rcraft. Therefore, to simulate the uncertainty, we assume that we have a large number of nominally identical aircraft being designed (e.g., by Airbus, Boeing, Bombardier Embraer, etc.), with the errors being fixed for each aircraft. This cr eates a two-level Monte Carlo simulation, with different aircraft models being considered at the upper level, and different instances of the same aircraft at the lower level. To illustrate the procedure we consider point stress design of a small part of an aircraft structure. Aircraft structures have more complex failure modes, such as fatigue and fracture, which require substantially diffe rent treatment and the consideration of the effects of inspections (See Kale et al., 2003). However, this simple example serves to further our understanding of the interaction between vari ous safety measures. The procedure is summarized in Fig. 3-1, which is described in detail in the next section. Design of a Generic Component Design and Certification Testing We assume that we have N different aircraft models, i.e., we have N different companies producing a model with errors. We consider a generic component to represent the entire aircraft structure. The true stress (true ) is found from the equation trueP wt (3.1)

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34 where P is the applied load on the component of width w and thickness t. In a more general situation, Eq. (3.1) may apply to a sm all element in a more complex component. When errors are included in the analysis the true stress in the component is different from the calculated st ress. We include the errors by introducing an error factor e while computing the stress as (1)calctruee (3.2) Positive values of e yield conservative estimates of the true stress and negative values yield unconservative stress estimati on. The other random variables account for variability. Combining Eqs. (3.1) and (3.2), the stress in the component is calculated as (1)calcP e wt (3.3) The design thickness is determ ined so that the calculated stress in the component is equal to material allowable stress for a design load Pd multiplied by a safety factor SF, hence the design thickness of the component is calculated from Eq. (3.3) as (1)Fd design designaSP te w (3.4) where the design component width, wdesign, is taken here to be 1.0, and a is the material stress allowable obtained from testing a ba tch of coupons according to procedures that depend on design practices. Here, we assume that A-basis properties are used (see Appendix A). During the design proce ss, the only random quantities are a and e. The thickness obtained from Eq. (3.4) step A in Fig. 3-1, is the nominal thickness for a given aircraft model. The actual thickness will vary due to individuallevel manufacturing uncertainties.

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35 Figure 3-1. Flowchart for Monte Carlo simu lation of component design and failure After the component has been designed (t hat is, thickness is determined from Eq. (3.4)), we simulate certification testing for the aircraft. Here we assume that the component will not be built with complete fide lity to the design due to variability in geometry (width and thickness). The compone nt is then loaded with the design axial force of SF times Pd, and the stress in the component is recorded. If this stress exceeds the failure stress (itself a random variable, se e Table 3-2) then the design is rejected, otherwise it is certified for use. That is, the ai rplane is certified (step B in Fig. 3-1) if the following inequality is satisfied 0Fd ffSP wt (3.5) and we can build multiple copies of the ai rplane. We subject the component in each airplane to actual random maxi mum (over a lifetime) service loads (step D in Fig. 3-1) and decide whether it fails using Eq. (3.6).

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36 f PCtw (3.6) Here, P is the applied load, and C is the load carrying capacity of the structure in terms of the width w thickness t and failure stress f. A summary of the distributions for the random variables used in design and certification is list ed in Table 3-2. Table 3-2. Distribution of random variables used for component de sign and certification Variables Distribution Mean Scatter Plate width ( w ) Uniform 1.0 (1%) bounds Plate thickness ( t ) Uniform designt (3%) bounds Failure stress (f) Lognormal 150.0 8 % coefficient of variation Service Load ( P ) Lognormal 100.0 10 % coefficient of variation Error factor ( e ) Uniform 0.0 10% to 50% This procedure of design and testin g is repeated (steps A-B) for N different aircraft models. For each new model, a different random error factor e is picked for the design, and different allowable properties are genera ted from coupon testing (Appendix A). Then in the testing, different thicknesses and widths and different failure stresses are generated at random from their distributions. Effect of Certification Tests on Distribution of Error Factor e One can argue that the way certification tests reduce the probability of failure is by changing the distribution of the error factor e. Without certification testing, we assume symmetric distribution of this error fact or. However, designs based on unconservative models are more likely to fail certificati on, and so the distribution of e becomes conservative for structures that pass certifica tion. In order to quantify this effect, we calculated the updated distribut ion of the error factor e The updated distribution is calculated analytically by Bayesian upda ting by making some approximations, and Monte Carlo simulations are conducted to ch eck the validity of those approximations.

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37 Bayesian updating is a comm only used technique to ob tain updated (or posterior) distribution of a random variable upon obtai ning new information about the random variable. The new information here is that th e component has passed th e certification test. Using Bayes Theorem, the upda ted (posterior ) distribution ) (Uf of a random variable is obtained from the initial (prior) distribution ) (If based on new information as d f f fI I U) ( ) | Pr( ) ( ) | Pr( ) ( (3.7) where Pr(|) is the conditional probability of observing the experimental data given that the value of the random variable is For our case, the posterior distribution ) ( e fUof the error factor e is given as Pr(|)() () Pr(|)()I U b I bCTefe fe CTefede (3.8) where CT is the event of passing certification, and Pr (CT|e) is the probability of passing certification for a given e Initially, e is assumed to be uniformly distributed. The procedure of calculation of Pr(CT|e) is described in Appendix B, where we approximate the distribution of the geometrical variables, t and w as lognormal, taking advantage of the fact that their coefficient of variation is small compared to that of the failure stress (see Table 3-2). We illustrate the effect of certification tests for the components designed with ABasis material properties. An initial and updated distribution plot of error factor e with 50 % bound is shown in Fig. 3-2. Monte Carlo simu lation with 50,000 aircraft models is also

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38 shown. Figure 3-2 shows that the certificat ion tests greatly reduce the probability of negative error, hence eliminating most unconser vative designs. As seen from the figure, the approximate distribution calculated by th e analytical approach matches well the distribution obtained from Monte Carlo simulations. Initial and Updated Distribution of Error Factor e and Comparison of Analytical Approx. with Monte Carlo Simulations0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 error factor, eInitial and Updated Distributions Monte Carlo, initial Monte Carlo, updated Analytical Approx., initial Analytical Approx., updated Figure 3-2. Initial and upda ted probability distribution functions of error factor e Error bound is 50% and Monte Carlo simulation done with sample size of 50,000. Probability of Failure Calculation by Analytical Approximation The stress analysis represented by Eq. (3.1) is trivial, so that the computational cost of Monte Carlo simulation of the probability of failure is not high. However, it is desirable to obtain also analytic al probabilities that may be used for more complex stress analysis and to check the Monte Carlo simulations. In order to take advantage of simplifying approximations of the distribution of the geometry parameters, it is convenient to perf orm the probability calculation in two stages, corresponding to the inner and out er loops of Fig. 3-1. That is, we first obtain expressions

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39 for the probability of failure of a single aircraft model (that is, given e and allowable stress). We then calculate the probability of failure over all aircraft models. The mean value of the probability of failure over all aircraft models is calculated as ()() f fdesigndesigndesignPPtftdt (3.9) where designt is the non-deterministic distribution parameter, and ()designft is the probability density function of designt. It is important to have a measure of variab ility in this probability from one aircraft model to another. The standard deviation of failure probability gives a measure of this variability. In addition, it provides informa tion on how accurate is the probability of failure obtained from Monte Carlo simulations The standard deviation can be calculated from 1/2 2()()Pfdesignfdesigndesign fPtPftdt (3.10) Probability of Failure Calculatio n by Monte Carlo Simulations The inner loop in Fig. 3-1 (steps C-E) re presents the simulation of a population of M airplanes (hence components) that all have the same design. However, each component is different due to variability in geometry, fa ilure stress, and loading (step D). We subject the component in each airplane to actual random maximum (over a lifetime) service loads (step E) and calculate whethe r it fails using Eq. (3.6). For airplane model that pass certificatio n, we count the number of components failed. The failure probability is calculate d by dividing the number of failures by the

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40 number of airplane models that passed certif ication, times the numbe r of copies of each model. The analytical approximation for the probability of failure suffers due to the approximations used, while the Monte Carlo simulation is subject to sampling errors, especially for low probabiliti es of failure. Using large sa mples, though, can reduce the latter. Therefore, we compared the two met hods for a relatively large sample of 10,000 aircraft models with 100,000 instances of each model. In addition, the comparison is performed for the case where mean material properties (rather than A-basis properties) are used for the design, so that the probab ility of failure is high enough for the Monte Carlo simulation to capture it accurately. Table 3-3 shows the results for this case. Table 3-3. Comparison of probability of fa ilures for components designed using safety factor of 1.5, mean value for a llowable stress and error bound of 50% Value Analytical Approximation Monte Carlo Simulation* % error Average Value of Pf without certification (Pnt) 1.715-1 1.726-1 0.6 Standard Deviation of Pnt 3.058-1 3.068-1 0.3 Average Value of Pf with certification (Pt) 3.166-4 3.071-4 3.1 Standard Deviation of Pt 2.285-3 2.322-3 1.6 Average Value of Initial error factor (ei) 0.0000 -0.00024 --Standard Deviation of eI 0.2887 0.2905 0.6 Average Value of Updated error factor (eup) 0.2468 0.2491 0.9 Standard Deviation of eup 0.1536 0.1542 0.4 N = 10,000 and M = 100,000 is used in the Monte Carlo Simulations The last column of Table 3-3 shows the percent error of the analytical approximation compared to Monte Carlo simulations. It is seen that the analytical approximation is in good agreement with the values obtained through Monte Carlo simulations. It is remarkable that the standard deviation of the probability of failure is almost twice the average value of the probability (the ratio, the coefficient of variation, is about 178%) before certification, and about seven times larger after certification. This

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41 indicates huge variability in the probability of failure for different aircraft models, and this is due to th e large error bound, be=50% With 10,000 different aircraft models (N), the standard deviation in the Monte Carlo estimates is about 1%, and the differences between the Monte Carlo simulation and the anal ytical approximation are of that order. Effect of Three Safety Measures on Probability of Failure We next investigate the effect of other safety measures on failure probability of the components using Monte Carlo simulations. We performed the simulation for a range of variability in error factor e for 5000 airplane models ( N samples in outer loop) and 20,000 copies of each airplane model ( M samples in inner loop). Here, we compare the probability of failure of a structure designed with three safety measures (safety factor, conservative material property and certification testing) to th at of a structure designed without safety measures. Table 3-4 presents the results when all safety measures are used for different bounds on the error. The second column sh ows the mean and standard deviation of design thicknesses generated for components that passed certification. These components correspond to the outer loop of Fig. 3-1. The va riability in design thickness is due to the randomness in the error e and in the stress allowable. The average thickness before certification was 1.269, so that the column sh ows the conservative e ffect of certification testing. When the error bound is 10%, 98.8% of the components pass certification (third column in Table 3-4), and the average thickness is increased by only 0.24% due to the certification process. On the other hand, when the error bound is 50%, 29% of the components do not pass certification, and this raises the average thickness to 1.453. Thus, the increase in error bound has two opposite effects. Without cer tification testing, increasing the error bound greatly increases the probability of failure. For example, when

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42 the error bound changes from 30% to 50%, th e probability of failure without certification changes from 0.00091 to 0.0449 or by a factor of 49. On the other hand, with the increased average thickness, after certifica tion the probability increases only from 1.343x10-4 to 1.664x10-4. Table 3-4. Probability of failure for different bounds on error e for components designed using safety factor of 1.5 and A-basi s property for allowable stress. Numbers in parenthesis denote the coefficient of variation of the quantity. Average design thickness without certification is 1.271. Error Bound be Average design thickness after certificati on* Certifi cation failure rate % Probability of failure after certification (Pt)10-4 Probability of failure without certification (Pnt)10-4 Probability ratio (Pt/Pnt) Probabilit y difference (Pnt-Pt) 50% 1.453 (0.19) 29.3 1.664 (7.86) 449.0 (2.74) 3.706-3 4.473-2 40% 1.389 (0.17) 24.3 1.586 (6.92) 89.77 (3.22) 1.767-2 8.818-3 30% 1.329 (0.15) 16.3 1.343 (5.28) 9.086 (3.46) 1.479-1 7.742-4 20% 1.283 (0.12) 6.2 0.304 (4.81) 0.477 (3.51) 6.377-1 1.727-5 10% 1.272 (0.07) 1.2 0.027 (4.71) 0.029 (4.59) 9.147-1 2.490-7 *Average over N=5000 models The effectiveness of the certification test s can be expressed by two measures of probability improvement. The first measure is th e ratio of the probability of failure with the test, Pt, to the probability of failure without tests, Pnt. The second measure is the difference of these probabilities. The ratio is a more useful indicator for low probabilities of failure, while the difference is more me aningful for high proba bilities of failure. However, when Pt is high, the ratio can mislead. That is, an improvement from a probability of failure of 0.5 to 0.1 is more substantial than an improvement in probability of failure of 0.1 to 0.01, because it saves more airplanes. However, the ratio is more useful when the probabilities are small, a nd the difference is not very informative.

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43 Table 3-4 shows that certification testing is more important for large error bounds e For these higher values the number of co mponents that did not pass certification is higher, thereby reducing the failure probabilit y for those that passed certification. While the effect of component tests (bu ilding block tests) is not simula ted, their main effect is to reduce the error magnitude e This is primarily due to th e usefulness of component tests in improving analytical models and revealin g unmodeled failure modes. With that in mind, we note that the failure probabil ity for the 50% error range is 1.7-4, and it reduces to 2.7-6 for the 10% error range-that is, by a factor of 63. The actual failure probability of aircraft co mponents is expected to be of the order of 10-8 per flight, much lower than the best num ber in the fourth column of Table 3-4. However, the number in Table 3-4 is for a lifetime for a single structural component. Assuming about 10,000 flights in the life of a component an d 100 independent structural components, this 10-5 failure probability for a component will translate to a per flight probability of failure of 10-7 per airplane. This factor of 10 discrepancy is exacerbated by other failure modes like fatigue that have not been considered. However, other safety measures, such as conservative load specifi cations may account for this discrepancy. Table 3-5 shows results when average rather than conservative material properties are used. It can be seen from Table 3-5 that the average thickness determined using the mean value of allowable stress is lower than that determined using the A-basis value of allowable stress (Table 3-4). This is equivalent to adding an additional safety factor over an already existing safety factor of 1.5. For the distribution (lognormal with 8% coefficient of variation) and number of batc h tests (40 tests) considered here, a typical value of the safety factor due to A-Basis property is around 1.27.

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44 Table 3-5. Probability of failure for different bounds on error e for components designed using safety factor of 1.5 and mean value for allowable stress. Numbers in parenthesis denote the coefficient of va riation of the quantity. Average design thickness without cert ification is 1.000. Error bound be Average design thickness after certification Certificat ion failure rate+ % Probability of Failure after certification (Pt)-4 Probability of failure without certification (Pnt)-4 Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 50% 1.243 (0.13) 50.1 3.420 (5.82) 1681 (1.81) 2.035-3 1.677-1 40% 1.191 (0.11) 50.1 4.086 (6.78) 969.0 (1.99) 4.217-3 9.649-2 30% 1.139 (0.09) 50.8 5.616 (5.45) 376.6 (2.00) 1.495-2 3.700-2 20% 1.086 (0.07) 50.7 6.253 (3.19) 92.67 (1.83) 6.748-2 8.642-3 10% 1.029 (0.05) 51.0 9.209 (1.70) 19.63 (1.25) 4.690-1 1.043-3 *Average over N =5000 models +With only 5000 models, the standard deviation in the certification failure rate is about 0.71%. Thus, all the numbers in this column are about 50% as may be expected when mean material properties are used. Without the A-basis properties, the stress in the certification test is approximately equal to the average ultimate service stress, so that about 50% of the components fail certification. When the errors are large, this raises substantially the average thickness of the components that pass certif ication, so that for an erro r bound of 50% the certification test is equivalent to a safety factor of 1.243. Large errors produce some super-strong and some super-weak components (see Fig. 3-3b ). The super-weak components are mostly caught by the certification te sts, leaving the super-strong components to reduce the probability of failure. Another way of looking at this effect is to note that when there are no errors, there is no point to the tests. I ndeed, it can be seen that the probability of failure without certifi cation tests improves with reduced error bound e but that the reduced effect of the ce rtification tests reverses the trend. Thus for this case we obtain the counter-intuitive results that larger errors produce safer designs. Comparing the first row of Table 3-5 to that of Table 3-3 we see the effect of the smaller sample for the Monte Carlo simulations. Table 3-3 was obtained with 10,000

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45 models and 100,000 copies per model, while Table 3-5 was obtained with 5000 models, and 20,000 copies per model. The differe nce in the probability of failure after certification between the two tables is about 11 percent. However, the two values straddle the analytical approximation. The effects of building block type of test s that are conducted before certification are not included in this study. These tests reduce the errors in analytical models. For instance, if there is 50% error in the analytical model the buil ding block type of tests may reduce this error to lower values. Hence, th e difference between the rows of Table 3-4, may be viewed as indicating the benefits of reducing the error by building block tests. Figure 3-3. Design thickness variation with low and high error bounds. Note that after certification testing only the designs above the minimum thickness are built and flown. Those on the right have a much higher average design thickness than those on the left. Table 3-6 shows the effect of not using a safety factor. Although certification tests improve the reliability, again in a general tre nd of high improvement with high error, the lack of safety factor of 1.5 limits the impr ovement. Comparing Tabl es 3-4 and 3-6 it can be seen that the safety factor reduces the probability of failure by two to four orders of magnitudes. It is interesting to note that the effect of the e rror bound on the probability of

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46 failure after certification is not monotoni c, and this phenomenon is discussed in Appendix C. Table 3-6. Probability of failure for different bounds on error e for safety factor of 1.0 and A-basis allowable stress. Numbers in parenthesis denote the c.o.v. of the quantity. Average design thickness without certification is 0.847. Error bound be Average design thickness after certification Certificat ion failure rate % Failure probability after certification (Pt)-2 Failure probability with no certification (Pnt)-2 Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 50% 0.969 (0.19) 29.4 6.978 (2.12) 29.49 (1.31) 2.366-1 2.251-1 40% 0.929 (0.17) 25.0 7.543 (1.98) 24.56 (1.38) 3.071-1 1.702-1 30% 0.886 (0.15) 16.6 8.923 (1.73) 17.11 (1.43) 5.216-1 8.184-2 20% 0.855 (0.11) 5.7 8.171 (1.40) 9.665 (1.34) 8.454-1 1.494-2 10% 0.847 (0.06) 1.3 4.879 (0.97) 4.996 (0.97) 9.767-1 1.163-3 *Average over N=5000 models Table 3-7, shows results when the only safety measure is certification testing. Certification tests can reduce the probabi lity of failure of components by 38%, the highest improvement corresponds to the high est error. As can be expected, without certification tests and safety measures, the prob ability of failure is near 50%. Tables 3-4 through 3-7 illustrate the probability of failure for a fixed 8 % coefficient of variation in failure stress. The general conclu sion that can be drawn from th ese results is that the error bound e is one of the main parameters affect ing the efficacy of certification tests to improve reliability of components. Table 3-7. Probability of failure for different error bounds for safety factor of 1.0 and mean value for allowable stress. Average design thickness without certification is 0.667. Error bound be Average design thickness after certification Certific ation failure rate % Probability of Failure after certification (Pt) Probability of failure without certification (Pnt) Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 50% 0.830 (0.12) 50.1 0.125 (1.39) 0.505 (0.83) 2.463-1 3.808-1 40% 0.796 (0.11) 50.2 0.158 (1.20) 0.504 (0.79) 3.140-1 3.459-1 30% 0.761 (0.09) 50.4 0.205 (0.92) 0.503 (0.72) 4.075-1 2.981-1 20% 0.727 (0.08) 50.9 0.285 (0.64) 0.503 (0.58) 5.653-1 2.189-1 10% 0.686 (0.05) 50.7 0.412 (0.34) 0.500 (0.34) 8.228-1 8.869-2 *Average over N=5000 models

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47 Next, we will explore how another parameter, variability, influences the efficacy of tests. This is accomplished by changing the coefficient of variation of failure stress f between 0% and keeping the error bound constant. Table 3-8. Probability of failure for different uncertainty in failure stress for the components designed with safety factor of 1.5, 50% error bounds e and Abasis allowable stress. Coeffi cient of variati on of f Average design thickness without certificati on Average design thickness after certificati on Certific ation failure rate % Probability of failure after certificatio n (Pt)-4 Probability of failure without certificatio n (Pnt)-4 Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 0 % 0.998 (0.29) 1.250 (0.11) 50.2 0.017 (6.85) 1699 (1.87) 1.004-5 1.698-1 4% 1.127 (0.29) 1.347 (0.15) 38.9 0.087 (7.20) 970.4 (2.35) 8.973-5 9.703-2 8 % 1.269 (0.29) 1.453 (0.19) 29.3 1.664 (7.86) 449.0 (2.74) 3.706-3 4.473-2 12 % 1.431 (0.29) 1.574 (0.22) 20.9 13.33 (7.71) 206.1 (3.08) 6.469-2 1.927-2 16% 1.616 (0.30) 1.723 (0.25) 14.1 22.52 (5.54) 107.3 (3.24) 2.100-1 8.476-3 *Average over N=5000 models Table 3-9. Probability of failure for different uncertainty in failure stress for the components designed with safety factor of 1.5, 30% error bound e and A-basis allowable stress. Coeffi cient of variati on of f Average design thickness without certificati on Average design thickness after certificati on Certific ation failure rate % Probability of failure after certificatio n (Pt)-4 Probability of failure without certificatio n (Pnt)-4 Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 0 % 1.001 (0.17) 1.148 (0.08) 50.1 0.026 (4.79) 223.8 (2.50) 1.163-4 2.238-2 4 % 1.126 (0.17) 1.232 (0.11) 31.6 0.146 (6.03) 35.25 (2.97) 4.149-3 3.511-3 8 % 1.269 (0.17) 1.329 (0.15) 16.3 1.343 (5.28) 9.086 (3.46) 1.479-1 7.742-4 12 % 1.431 (0.18) 1.459 (0.17) 7.2 2.404 (3.87) 4.314 (3.45) 5.572-1 1.911-4 16% 1.617 (0.18) 1.630 (0.18) 3.3 2.513 (3.73) 3.102 (3.54) 8.099-1 5.896-5 Average over N=5000 models

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48 Table 3.10. Probability of failure for uncertain ty in failure stress for components designed using safety factor of 1.5, 10% error bounds e and A-basis properties Coeffi cient of variati on of f Average design thickness without certificati on Average design thickness after certificati on Certific ation failure rate % Probabili ty of failure after certificati on (Pt)-4 Probability of failure without certificatio n (Pnt)-4 Probability ratio (Pt/Pnt) Probability difference (Pnt-Pt) 0 % 1.000 (0.06) 1.048 (0.03) 50.3 0.075 (2.91) 1.745 (1.78) 4.304-2 1.669-4 4 % 1.126 (0.06) 1.131 (0.06) 5.9 0.053 (3.85) 0.070 (3.56) 7.548-1 1.716-6 8% 1.269 (0.06) 1.272 (0.07) 1.2 0.027 (4.71) 0.029 (4.59) 9.147-1 2.490-7 12 % 1.431 (0.07) 1.432 (0.07) 0.8 0.049 (4.30) 0.051 (4.23) 9.623-1 1.926-7 16% 1.623 (0.08) 1.624 (0.08) 0.5 0.085 (3.50) 0.083 (3.55) 9.781-1 1.853-7 *Average over N=5000 models The increase in the variability in failure stress has a large effect on the allowable stress because A-basis properties specify an allowable that is below 99% of the sample. Increased variability reduces the allowable stress and therefore increases the design thickness. It is seen from Tables 3-8 through 3-10 that when the variability increases from 0% to 16%, the design thickness increas es by more than 60%. This greatly reduces the probability of failure with out certification. However, th e probability of failure with certification still deteriorates That is, the use of A-basis properties fails to fully compensate for the variability in material properties. This oppos ite behavior of the probability of failure before and after cer tification is discussed in more detail in Appendix C. The variability in failure stress greatly changes the effect of certification tests. Although the average design th icknesses of the components in crease with the increase in variability, we see that when the variability is large, the value of the tests is reduced because the tested aircraft can be greatly di fferent from the airplanes in actual service.

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49 We indeed see from the Tables 3-8, 3-9 and 310 that the effect of certification tests is reduced as the variability in the failure stress increases. Recall that the effect of certification tests is also reduced when the error e decreases. Indeed, Table 3-8 shows a much smaller effect of the tests than Table 3-10. Comparing the second and third columns of Tables 3-8, 3-9 and 3-10 we s ee that as the bound of error decreases, the change in the average value of design thic knesses of the components become less which is an indication of loss in th e efficacy of certification tests. Up to now, both the probability difference (Pnt-Pt) and the probability ratio (Pt/Pnt) seem to be good indicators of efficacy of te sts. To allow easy visualization, we combined the errors and the variability in a single ratio (Bound of e) / VR( / f) ratio (ratio of error bound e to the coefficient of variation of th e stress ratio). The denominator accounts for the major contributors to the variability. The value in the denominator is a function of four variables; service load P width w thickness t and failure stress f. Here, P and f have lognormal distributions but w and t are uniformly dist ributed. Since the coefficient of variations of w and t is very small, they can also be treated as lognormally distributed to make calculation of the denominator easy while plotting the graphs. Since the standard deviations of the variables are small, the de nominator is now the square root of the sum of the squares of coefficient of variations of the four variables mentioned above, that is ) ( ) ( ) ( ) ( ) (2 2 2 2 f R R R R f RV t V w V P V V (3.11) The effective safety factor is the ratio of the design thickness of the component when safety measures (such as usage of A-basi s values for material properties and safety factor) are applied to the thickness of the co mponent when no safety measures are taken.

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50 Figures 3-4 and 3-5, present the Pt/Pnt ratio in visual formats. It can be seen that as expected, the ratio decreases as the (Bounds on e )/VR( / f) ratio increases. However, these two figures do not give a clear indication of how certi fication tests are influenced by the effective safety factor. Figure 3-4. Influence of effective safety factor, error, and variability on the probability ratio (3-D view) Figure 3-5. Influence of effective safety factor, error and variability on the probability ratio (2-D plot) Figures 3-6 and 3-7 show the probability difference, Pnt-Pt. In these cases, the dependence on the effective safety factor is mono tonic. As expected, it is seen that as the effective safety factor increases, the improvem ent in the safety of component decreases; meaning that the certification tests become le ss useful. The probability difference is more descriptive as it is proportional to th e number of aircraft failures prevented by certification testing. The probability ratio lack s such clear physical interpretation, even though it is a more attractive measure when the probability of failure is very small. Considering the results presented by Fi gures 3-4 through 3-7, the probability difference (Pnt-Pt) is the more appropriate choice for expressing the effectiveness of tests.

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51 Figure 3-6. Influence of effective safety factor, error and variability on the probability difference (3-D view) Figure 3-7. Influence of effective safety factor, error and variability on the probability difference (2-D plot) Summary We have used a simple example of point stress design for yield to illustrate the effects of several safety measures taken in aircraft design: safety factors, conservative material properties, and certification tests. Analytical calculations and Monte Carlo simulation were performed to account for both fl eet-level uncertainties (such as errors in analytical models) and individual uncertainties (such as variability in material properties). It was seen that an increase of the systemic errors in the analysis causes an increase in the probability of failure. We found that the systemic errors can be reduced by the use of certification tests, thereby reducing the probability of failure. Also we found that design thicknesses of the components increas ed as the bounds of systemic errors increased. We found that the effect of certification tests is most important when errors in analytical models are high and when the variab ility between airplanes is low. This leads to the surprising result that in some situations larger error variability in analytical models reduces the probability of failure if certi fication tests are conducted. For the simple

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52 example analyzed here, the use of conser vative (A-basis) material properties was equivalent to a safety factor of up to 1.6, depending on the scatter in failure stresses. The effectiveness of the certification tests is expressed by two measures of probability improvement. The ratio of th e probability of failure with the test, Pt, to the probability of failure without tests, Pnt, is useful when Pt is small. The difference is more meaningful when the probability is high. Us ing these measures we have shown that the effectiveness of certification tests increases when the ratio of error to variability is large and when the effective safety factor is small. The effect of building-block type tests that are conducted before certification was not assessed here. However, these tests reduce the errors in the analytical models, and on that basis we determined that they can re duce the probability of failure by one or two orders of magnitude. The calculated probabilities of failure w ith all the considered safety margins explain why passenger aircraft are so safe st ructurally. They were still somewhat high about 10-7compared to the probability of failu re of actual aircraft structural componentsabout 10-8. This may be due to additional safety measures, such as conservative design loads or to the effect of design against additional failure modes.

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53 CHAPTER 4 COMPARING EFFECTIVENESS OF MEASURES THAT IMPROVE AIRCRAFT STRUCTURAL SAFETY Chapter 3 explored how safety measures compensate for errors and variability. The major finding of that chapter was that certif ication tests are most effective when errors are large, variability is low, and the overall sa fety factor is low. Chapter 3 mainly focused on the effectiveness of certification testing, but the relative effectiveness of safety measures was not addressed. The present chapte r takes a further step and aims to discover how measures that improve airc raft structural safety compar e with one anot her in terms of weight effectiveness. In addition, structural redundanc yanother safety measureis included in the analysis. In addition the simple error model of Chapter 3 is replaced by a more detailed error model. Comparison of the effectiveness of error and variability reduction with other safety measures is also given. The research presented in this chapter is submitted for publication (Acar et al 2006d). My colleague Dr. Amit Kales contribution to this work is acknowledged. Introduction As noted earlier, aircraft structural desi gn is still carried ou t by using code-based design, rather than probabilistic design. Sa fety is improved through conservative design practices that include the use of safety f actors and conservative material properties. Safety is also improved by testing of co mponents, redundancy, improved modeling to reduce errors and improved manufacturing to reduce variability. The following gives brief description of these safety measures.

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54 Load Safety Factor In transport aircraft design, FAA regulations state the use of a load safety factor of 1.5 (FAR 25.303). That is, aircraft structures are designed to withst and 1.5 times the limit load without failure. Conservative Material Properties In order to account for uncertainty in mate rial properties, FAA regulations state the use of conservative material properties (FAR 25.613). The conservative material properties are characterized as A-basis and Bbasis material property values, and the use of A-basis or B-basis values depends on the re dundancy. If there is si ngle failure path in the structure, A-basis values are used, while for the case of multiple failure paths (i.e., redundant structures), B-basis values are us ed. Detailed information on these values is provided in Chapter 8 of Volume 1 of the Composite Materials Handbook (2000). The basis values are determined by testing a numb er of coupons selected at random from a material batch. The A-basis value is determined by calculating the value of a material property exceeded by 99% of the populati on with 95% confidence, while the B-basis value is the value of a material property exceeded by 90% of the population with 95% confidence. Here, we take the redundancy of the structure into account, so we use B-basis values (see Appendix A for the B-basis value calculation). Th e number of coupon tests is assumed to be 40. Tests Tests of major structural components redu ce stress and material uncertainties for given extreme loads due to in adequate structural models. These tests are conducted in a building block procedure (Composite Materi als Handbook (2000), Volume 1, Chapter 2). First, individual coupons are tested, and then a sub-assembly is tested, followed by a full-

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55 scale test of the entire structure. Here, we on ly consider the final certification test for an aircraft. Other tests are assumed to be erro r reduction measures and their effect is analyzed indirectly by considerin g the effect of error reduction. Redundancy Transport airliners are designed with doubl e and triple redundancy features in all major systems to minimize the failure probabil ity. Redundancy is intended to ensure that a single component failure does not lead to catastrophic failure of the system. In the present work, we assume that an aircraft stru cture will fail if two local failures occur in the structure. Error Reduction Improvements in the accuracy of struct ural analysis and failure prediction of aircraft structures reduce errors and enhance the level of safety of the structures. These improvements may be due to better modeling techniques developed by researchers, more detailed finite element models made possi ble by faster computer s, or more accurate failure predictions due to extensive testing. Variability Reduction Examples of mechanisms that reduce vari ability in material properties include quality control and improved manufacturing pr ocesses. Variability in damage and ageing effects is accomplished through inspections and structural health monitoring. Variability in loads may be reduced by better pilot trai ning and information that allows pilots to more effectively avoid regions of high turbulence. Here we investigate only the effect of reduced variability in material properties. The next section of this chapter discusses th e more detailed error model used in this chapter, along with variability and total safety factor. Next, the effect of certification tests

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56 on error distribution is analyzed. Then, detail s of the calculation of the probability of failure via separable Monte Carlo simulati ons (MCS) are given. Finally, the chapter finalizes with the results and summary. Errors, Variability and Total Safety Factor The simplified uncertainty classification used in Chapter 3 is also used in this chapter, where errors are uncerta inties that apply equally to th e entire fleet of an aircraft model and variabilities are unce rtainties that vary for the in dividual aircraft (see Table 31, Chapter3). This section first discusses the errors in design and c onstruction. Next, total error factor and total safety factor are introduced, finally, simulation of variability is discussed. Errors in Design We consider static point stress design for simplicity. Other types of failures such as fatigue, corrosion or crack instability are not taken into account. We assume that an aircraft structure will fail only if two local fa ilure events occur. For example, we assume that the wing will fail structurally if two local failures occur at the wing panels. The correlation coefficient between the probabilities of these two events is assumed to be 0.5. Before starting the structural design, aer odynamic analysis needs to be performed to determine the loads acting on the aircraft. However, the calculated design load value, Pcalc, differs from the actual loading Pd under conditions corresponding to FAA design specifications (e.g., gust-strength specifica tions). Since each company has different design practices, the error in load calculation, ep, is different from one company to another. The calculated design load Pcalc is expressed in terms of the true design load Pd as

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57 d P calcP e P ) 1 ( (4.1) Besides the error in load calculation, an ai rcraft company may also make errors in stress calculation. We consider a small regi on in a structural part, characterized by a thickness t and width w, that resists the load in that region. The value of the stress in a structural part calculated by the stress analysis team, calc, can be expressed in terms of the load values calcula ted by the load team Pcalc, the design width wdesign, and the thickness t of the structural part by introducing the term e representing error in the stress analysis t w P edesign calc calc) 1 ( (4.2) Equation (4.3) is used by a structural designer to calculat e the design thickness tdesign required to carry the calculated de sign load times the safety factor SFL. That is, 111FLcalcFLd designP designadesigna calccalcSPSP teee ww (4.3) where a calc is the value of allowable stress for th e structure used in the design, which is calculated based on coupon te sts using failure models such as Tresca or von Mises. Since these failure theories are not exact, we have 1afa calctruee (4.4) where ef is the error associated with failure pr ediction. Moreover, the errors due to the limited amount of coupon testing to determ ine the allowables, and the differences between the material properties used by the designer and the average true properties of the material used in production are included in this error. Note that the formulation of Eq. (4.4) is different to that of Eqs. (4.1) and (4.2 ) in that the sign in front of the error factor

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58 ef is negative, because we consistently formul ate the expressions such that positive error implies a conservative decision. Combining Eqs. (4.3) and (4.4), we can express the design value of the load carrying area as 11 1P FLd designdesigndesign fa trueee SP Atw e (4.5) Errors in Construction In addition to the above errors, there wi ll also be construc tion errors in the geometric parameters. These construction e rrors represent the difference between the values of these parameters in an average ai rplane (fleet-average) built by an aircraft company and the design values of th ese parameters. The error in width, ew, represents the deviation of the design widt h of the structural part, wdesign, from the average value of the width of the structural part built by the company, wbuilt. Thus, (1)builtwdesignwew (4.6) Similarly, the built thickness value will differ from its design value such that (1)builttdesigntet (4.7) Then, the built load carrying area Abuilt can be expressed using the first equality of Eq. (4.5) as (1)(1)builttwdesignAeeA (4.8) Table 4-1 presents nominal values for the errors assumed here. In the results section of this chapter we will vary these e rror bounds and investigate the effects of these changes on the probability of failure. As seen in Table 4-2, the error having the largest

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59 bound in its distribution is the error in failure prediction ef, because we use it to model also the likelihood of unexpected failure modes. Table 4-1. Distribution of error factors and their bounds Error factors Distribution Type Mean Bounds Error in stress calculation, e Uniform 0.0 5% Error in load calculation, eP Uniform 0.0 10% Error in width, ew Uniform 0.0 1% Error in thickness, et Uniform 0.0 2% Error in failure prediction, ef Uniform 0.0 20% The errors here are modeled by uniform di stributions, following the principle of maximum entropy. For instance, the error in the built thickness of a structural part ( et) is defined in terms of the error bound t builtb via Eq. (4.9). 0,tt builteUb (4.9) Here U indicates that the distribution is un iform and (zero) is the average value of et. Table 4-1 shows that t builtb = 0.02. Hence, the lower bound for the thickness value is the averag e value minus 2% of the av erage and the upper bound for the thickness value is the average value plus 2% of the averag e. Commonly av ailable random number generators provide random numbers uniformly distribute d between 0 and 1. Then, the error in the built thickness can be calc ulated from Eq. (4.10) using such random numbers r as 21tt builterb (4.10) Total Error Factor The expression for the built load carrying area, Abuilt, of a structural part can be reformulated by combining Eqs. (4.5) and (4.8) as 1FLd builttotal a trueSP Ae (4.11)

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60 where 1111 1 1Ptw total feeee e e (4.12) Here etotal represents the cumulative eff ect of the individual errors ( e, eP, ) on the load carrying capacity of the structural part. Total Safety Factor The total safety factor, SF, of a structural part represen ts the effects of all safety measures and errors on the built structural pa rt. Without safety measures and errors, we would have a load carrying area, A0, required to carry the design load 0d f P A (4.13) where f is the average value of the failure stress. Then, the total safety factor of a built structural component can be defined as the ratio of Abuilt/A0 01f built FtotalFL built a trueA SeS A (4.14) Here we take SFL = 1.5 and conservative material prope rties are based on B-basis values. Certification tests add another layer of safety. Structures with large negative etotal (unconservative) fail certification, so the certi fication process adds safety by biasing the distribution of etotal. Denoting the built area after certification (or certified area) by Acert, the total safety factor of a certified structural part is 0cert F certA S A (4.15)

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61 Variability In the previous sections, we analyzed the different types of errors made in the design and construction stages, representing the differences between the fleet average values of geometry, material and loadi ng parameters and their corresponding design values. For a given design, these parameters vary from one aircraft to another in the fleet due to variabilities in tooling, construction, flying environment, etc. For instance, the actual value of the thickness of a structural pa rt, tact, is defined in terms of its fleet average built value, tbuilt, by 1acttbuilttvt (4.16) We assume that vt has a uniform distribution w ith 3% bounds (see Table 4-2). Then, the actual load carrying area Aact can be defined as 1111actactacttbuiltwbuilttwbuiltAtwvtvwvvA (4.17) where vw represents effect of the variability on the built width. Table 4-2 presents the assumed distribut ions for variabilities. Note that the thickness error in Table 4-1 is uniformly distributed w ith bounds of %. Thus the difference between all thicknesses over the fleets of all companies is up to %. However, the combined effect of the uniform ly distributed error a nd variability is not uniformly distributed. Table 4-2. Distribution of rando m variables having variability Variables Distribution Mean Scatter Actual service load, Pact Lognormal Pd = 100 10% c.o.v. Actual structural part width, wact Uniform wbuilt 1% bounds Actual structural part thickness, tact Uniform tbuilt 3% bounds Failure stress, f Lognormal 150 8% c.o.v. Variability in built width, vw Uniform 0 1% bounds Variability in built thickness, vt Uniform 0 3% bounds c.o.v.= coefficient of variation

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62 Certification Tests After a structural part has been built with random errors in stress, load, width, allowable stress and thickness, we simulate cer tification testing for the structural part. Recall that the structural part will not be manufactured with complete fidelity to the design due to variability in the geometric prop erties. That is, the actual values of these parameters wact and tact will be different from their fleet-average values wbuilt and tbuilt due to variability. The structur al part is then loaded w ith the design axial force of SFL times Pcalc, and if the stress exceeds the failure stress of the structure f, then the structure fails and the design is rejected; otherwise it is certif ied for use. That is, th e structural part is certified if the following inequality is satisfied 0FLcalc ff actactSP wt (4.18) The total safety factor (see Eq (4.14)) depends on the load safety factor, the ratio of the failure stress to the B-basis allowable stre ss and the total error factor. Note that the Bbasis properties are affected by the number of coupon tests. As th e number of tests increases, the B-basis value is also increases, so a lower total safety factor is used. Amongst the terms in the total safety factor ex pression, the er ror term is subject to the largest change due to certification testing. Certification tests reduce the probability of failure by mainly changing the di stribution of the error factor etotal. Without certification testing, we assume uniform distributions fo r all the individual errors. However, since designs based on unconservative models ar e more likely to fail certification, the distribution of etotal becomes conservative for structures that pass certification. In order to quantify this effect, we calculated the updated distribution of the error factor etotal by Monte Carlo Simulation (MCS) of a sample size of 1,000,000.

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63 In Chapter 3, we represented the overall error with a single error factor e, hereinafter termed the Single Error Factor model (SEF model), and we used uniform distribution for the initial (i.e., built) distribution of this error. In the present work, we use a more complex representation of error with individual error factors, hereinafter termed the Multiple Error Factor model (MEF model), and we represent the initial distribution of each individual error factor with uniform distribution. In this case, the distribution of the total error is no longer uniform. Figure 4-1 shows how certificat ion tests update the distribution of the total error for the SEF and MEF models. For both models the initial distribution is updated such that the likelihood of conservative values of the total error is increased. This is due to the fact that st ructures designed with unconservative (negative) errors are likely to be rejected in cer tification tests. Notice that the SEF model exaggerates the effectiveness of certification testing. The reader is referred to Appendix D for a detailed comparison of the two error models. Figure 4-1. Comparing distributions of built and certified total error etotal of SEF and MEF models. The distributions are obt ained from simulation of 1,000,000 structural parts. The lower and upper bounds for the single error are taken as 22.3% and 25.0%, respectively, to matc h the mean and sta ndard deviation of the total error factor in the MEF model (see Table D-1 of Appendix D).

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64 Figure 4-2 shows the distributions of the bu ilt and certified total safety factors of the MEF model. Notice that the structural part s designed with low total safety factors are likely to be rejected in the certification testi ng. The mean and standard deviations of built and certified distributions of the error factor and the total safety factor are listed in Table 4-3. Comparing the mean and standard deviati on of the built and cer tified total error (and similarly the total safety factor), we see th at the mean is increased and the standard deviation is reduced due to certification testing. Figure 4-2. Initial and updated distri bution of the total safety factor SF. The distributions are obtained via Monte Carlo Simulati ons with 1,000,000 structural part models. Table 4-3. Mean and standard deviations of the built and certified distributions of the error factor etotal and the total safety factor SF shown in Figures 4-1 and 4-2. The calculations are performed with 1,000,000 MCS. Mean Std. dev. Built total error 0.0137 0.137 Certified total error 0.0429 0.130 Built safety factor 1.747 0.237 Certified safety factor 1.799 0.226

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65 Probability of Failure Calculation As noted earlier, we assume that struct ural failure requires the failure of two structural parts. In this secti on, we first describe the probabi lity of failure calculations of a single structural part by using separable MC S. Then, we discuss the calculation of the system probability of failure. Probability of Failure Calculation by Separable MCS To calculate the probability of failure we first incorporate the statistical distributions of errors and variability in a M onte Carlo simulation. Errors are uncertain at the time of design, but do not change for individual realizations (in actual service) of a particular design. On the other hand, all individual realizations of a pa rticular design are different from each other due to variability. In Chapter 3, we implemented this through a two-level Monte Carlo simulation. At the u pper level we simulate d different aircraft companies by assigning random errors to ea ch, and at the lower level we simulated variability in dimensions, material prope rties, and loads related to manufacturing variability and variability in service conditions. This prov ided not only the overall probability of failure, but also its variatio n from one company to another (which we measured by the standard deviation of the pr obability of failure). This variation is important because it is a measure of the conf idence in the value of the probability of failure due to the epistemic uncertainty (lack of knowledge) in the errors. However, the process requires trillions of simulations for good accuracy. In order to address the computational bur den, we turned to the separable Monte Carlo procedure (e.g., Smarslok and Haftka (2006)). This procedure applies when the failure condition can be expressed as g1( x1)> g2( x2), where x1 and x2 are two disjoint sets of random variables. To take advantage of this proced ure, we need to formulate the

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66 failure condition in a separable form, so that g1 will depend only on variabilities and g2 only on errors. The common formul ation of the structural failu re condition is in the form of a stress exceeding the material limit. This form, however, does not satisfy the separability requirement. For example, the stress depends on variability in material properties as well as design area, which reflects errors in the analysis process. To bring the failure condition to the right form, we instead formulate it as the required cross sectional area reqA being larger than the built area Abuilt, as given in Eq. (4.19) 11req builtreq twA AA vv (4.19) where reqA is the cross-sectional area required to carry the actual load ing conditions for a particular copy of an aircraft model, and req A is what the built ar ea (fleet-average) needs to be in order for the particular copy to have the required area after allowing for variability in width and thickness. reqfAP (4.20) The required area depends only on variability while the built area depends only on errors. When certificatio n testing is taken into account, the built area, Abuilt, is replaced by the certified area, Acert, which is the same as the built area for companies that pass certification. However, companies that fail ar e not included. That is, the failure condition is written as Failure without certification tests: 0reqbuiltAA (4.21-a) Failure with certification tests: 0reqcertAA (4.21-b)

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67 Equation (4.21) can be normalized by dividing the terms with A0 (load carrying area without errors or safety measures, Eq. (4.13)). Since Abuilt/ A0 or Acert/ A0 are the total safety factors, Eq. (4.21) is equivalent to the requirement that failure occurs when the required safety factor is larger than the built one. Failure without certification tests: 0FF reqbuiltSS (4.22-a) Failure with certification tests: 0FF reqcertSS (4.22-b) where F builtS and F certS are the built and certified total safety factors given in Eqs. (4.14) and (4.15), and the required total safety factor F reqS is calculated from 0req F reqA S A (4.23) For a given F builtS we can calculate the probabil ity of failure, Eq. (4.22.a), by simulating all the variabilities with MCS. Figure 4-3 shows the dependence of the probability of failure on the total safety factor using MCS with 1,000,000 variability samples. The zigzagging in Figure 4-3 at high safety factor values is due to the limited MCS sample. Note that the probability of failu re for a given total safety factor is one minus the cumulative distribution function (CDF) of the total required safety factor. This required safety factor depends on the four random variables Pact, f, vt and vw. Among them Pact and f have larger variabilities compared to vt and vw (see Table 4-2). We found that F reqS is accurately represented with a lognormal distribution, since Pact and f follow lognormal distributions. Figure 4-3 also shows the probability of failure from the lognormal distribution with the same mean and standard deviation. Note that the nominal load safety factor of 1.5 is associated with a probability of failure of about 10-3, while the

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68 probabilities of failure obse rved in practice (about 10-7) correspond to a total safety factor of about two. Figure 4-3. The variation of th e probability of failure with the built total safety factor. Note that Pf is one minus the cumulativ e distribution function of F reqS. Figure 4-4 represents flowchart of a sepa rable MCS procedure. Stage-1 represents the simulation of variab ilities in the actual serv ice conditions to generate the probability of failure as shown in Figure 4-3. This probability of failure is one minus the cumulative distribution function (CDF) of the required safety factor F reqS. In Stage-1, M =1,000,000 simulations are performed and CDF of F reqS is assessed. A detailed discussion on CDF assessment for F reqS is given in Appendix E.

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69 Figure 4-4. Flowchart for MCS of component design and failure In Stage-2, N =1,000,000 designs are generated for N different aircraft companies. For each new design, different random error factors e, eP, ew, et and ef are picked from their corresponding distributions to generate the built safety factor, F builtS Then, each design is subjected to certification testing. If it passes, we obtain the probability of failure from the distribution obtained in Stage-1 (Figure 4-3). We calculate the average and coefficient of variation (c.o.v.) of the failu re probability over all designs and explore the effects of error, variability, and safety measures on these values in Results section. The separable Monte Carlo procedure redu ces the computational burden greatly. For instance, if the probability of failure is 2.510-5, a million simulations varying both errors and variability simultaneously estimate this probability with 20% error. We found for our problem that the use of the separable Monte Carl o procedure requires only 20,000

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70 simulations (10,000 simulations for Stage-1 a nd 10,000 for Stage-2) for the same level of accuracy. Including Redundancy The requirement of two failure events is modeled here as a parallel system. We assume that the limit-states of the both failure events follow normal distribution to take advantage of known properties of the bivariat e normal distribution. For a parallel system of two elements with equal failure probabilities, Eq. (4.24) is used to calculate the system probability of failure PFS (see Appendix F for details) 2 2 2 011 exp 21 1FSfPPdz z z (4.24) where Pf is the probability of failur e of a single structural part, is the correlation coefficient of the two limit-states and is the reliability index for a single structural part, which is related to Pf through Eq. (4.25) fP (4.25) Results In this section, the effectiveness of safety measures is investigated and the results are reported. First, we discuss the eff ects of error reduction. Then, the relative effectiveness of error reducti on and certification is compared Next, the effectiveness of redundancy is explored. Finally, the effectivenes s of variability reduc tion is investigated. Effect of Errors We first investigate the effect of errors on the probability of failure of a single structural part For the sake of simplicity, we scale all error components with a single multiplier, k replacing Eq. (4.12) by

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71 1111 1 1Ptw total fkekekeke e ke (4.26) and explore the effect of k on the probability of failure. Table 4-4 presents the average and coeffici ent of variation of the probability of failure of a single structural part. The coeffici ent of variation of the failure probability is computed to explore our confidence in the proba bility of failure estimate, since it reflects the effect of the unknown errors. Columns 5 and 6 of Table 4-4 show a very high coefficient of variation for the failure probabili ties (variability in the probability of failure for different aircraft models). We see that as the error grows (i.e., k increases), the coefficient of variation of fa ilure probabilities after certification also grows. Comparing the failure probabilities before certification (c olumn 5) and after cer tification (column 6), we notice that even though certif ication tests reduce the mean failure probability, they increase the variability in failure probability. Table 4-4 shows that for nominal error (i.e., k =1) the total safety factor before certification is 1.747, which is translated into a probability of failure of 8.83-4. When the certification testing is included, the safety factor is increased to 1.799, which reduces the probability of failure to 3.79-4. Notice also that the coeffi cient of variation of the safety factor is reduced from 13.6% to 12.5% which is a first glimpse of an indication that the certification testing is more effective than simply increasing the safety factor with an increased built area. A detailed analysis of the effectiveness of certification testing is given in the next subsection. Column 2 of Table 4-4 shows a rapid increase in the certification failure rate with increasing error. This is reflected in a rapi d increase in the average safety factor of

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72 certified designs in column 4, F certS This increased safety factor manifests itself in the last column of Table 4-4 that presents the effect of certifi cation tests on failure probabilities. As we can see from that column when the error increas es, the ratio of the two failure probabilities decreases, demonstratin g that the certification tests become more effective. This trend of the increase of th e design areas and the pr obability ratios is similar to the one observed in Chapter 3. Note, however, that even the average safety factor before certification (F builtS in column 3) increases with the error due to the asymmetry of the initial total error distribution (see Figure 4-1). Table 4-4. Average and coefficient of varia tion of the probability of failure for the structural parts designed w ith B-basis properties and SFL=1.5. The numbers inside the parentheses represent the coe fficient of variation of the relevant quantity. k CFR(a) (%) F builtS (b) F certS (b) Pnc (c)/10-4 Pc (c)/10-4 /cncPP 0.25 6.4 1.725 (4.2%) 1.728 (4.1%) 0.244 (148%) 0.227 (148%) 0.930 0.50 9.3 1.730 (6.9%) 1.741 (6.7%) 0.763 (247%) 0.609 (257%) 0.798 0.75 13.4 1.737 (10.2%) 1.764 (9.7%) 2.70 (324%) 1.66 (357%) 0.616 0.82 14.7 1.740 (11.2%) 1.773 (10.6%) 3.79 (340%) 2.13 (384%) 0.561 1 18.0 1.747 (13.6%) 1.799 (12.5%) 8.83 (371%) 3.79 (450%) 0.430 1.5 26.0 1.779 (20.5%) 1.901 (17.8%) 60.0 (385%) 11.5 (583%) 0.191 (a) CFR: Certification failure rate. (b) F builtS and F certS are the total safety factors before and after certification testing, respectively. (c) Pnc and Pc are the probabilities of failure before and after certification testing, respectively. Table 4-4 shows the huge waste of weight due to errors. For instance, for the nominal error (i.e., k=1.0), an average built total safe ty factor of 1.747 corresponds to a probability of failure of 8.83-4 according to Table 4-4, but we see from Figure 4-3 that a safety factor of 1.747 appr oximately corresponds to a pr obability of failure of 7-6, two orders of magnitude lowe r. This discrepancy is du e to the high value of the coefficient of variation of th e safety factor. For the nominal error, the coefficient of variation of the total safety factor is 14%. Tw o standard deviations below the mean safety

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73 factor is 1.272, and two standard deviations above the mean safety factor is 2.222. The probability of failure corresponding to the safe ty factor of 1.272 (from Figure 4-3) is about 2.98-2, while the safety of 1.985 the probability of failure is es sentially zero. So even though about 0.8% of the designs a have safety factor belo w 1.272 (Figure 4-2), these designs have a huge impact on the probability of failure Reducing the error by half (i.e., k=0.50), reduces the weight by 1%, while at the same time the probability of failure is reduced by a factor of 3. Weight Saving Due to Certificat ion Testing and Error Reduction We have seen in Table 4-4 that since structures built with unconservative errors are eliminated by certification testing; the tests increase the average safety factor of the designs and therefore reduce the average probability of failure. Since certification testing is expensive, it is useful to check if the sa me decrease in the probability of failure can be achieved by simply increasing the load carrying area by the same amount (i.e., by increasing the safety factor) without certifica tion testing. Column 2 of Table 4-5 shows that the required area with no certification testing, Ar,nc, is greater than the certified area, Acert, (i.e., area after certif ication testing) shown in column 3. The last column shows that the weight saving by using certif ication test instead of a me re increase of the safety factor. We notice that weight saving increases rapidly as the error increases. For instance, when k=0.25 the weight saving is very small. Co lumns 4 and 5 show that even though we match the average probability of failure, there are small differences in the coefficients of variation.

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74 Table 4-5. Reduction of the weight of structur al parts by certification testing for a given probability of failure. The numbers in side the parentheses represent the coefficient of variation of the relevant quantity. k Ar,nc/A0 (a) Acert/A0 Pnc (b)/10-4 Pc (b)/10-4 % A (c) 0.25 1.7285 (4.2%) 1.7283 (4.1%) 0.227 (148%) 0.227 (148%) -0.01 0.50 1.743 (6.9%) 1.741 (6.7%) 0.609 (252%) 0.609 (257%) -0.14 0.75 1.770 (10.3%) 1.764 (9.7%) 1.66 (342%) 1.66 (357%) -0.36 1 1.815 (13.7%) 1.799 (12.5%) 3.79 (416%) 3.79 (450%) -0.87 1.5 1.961 (20.7%) 1.901 (17.8%) 11.5 (530%) 11.5 (583%) -3.09 (a) Ar,nc is the required area with no certification testing, the area required to achieve the same probability of failure as certification. (b) Pnc and Pc are the probabilities of failure before and after certification testing, respectively. (c) A = ( AcertAr,nc)/ Ar,nc indicates weight saving due to testing while keeping the same level of safety We notice from Table 4-5 that, for the nominal error (i.e., k=1.0), certification testing reduces the weight by 0.87% for the same probab ility of failure (3.79-4). The same probability of failure could have b een attained by reduci ng the error bounds by 18%, that is by reducing k from 1.0 to 0.82. This reduction would be accompanied by an F builtS =1.740 (see Table 4-4). Compared to the 1.799 reduction F builtS this represents a reduction of 4.13% in average weight, so error reduction is much more effective than certification testing in reducing weight. Effect of Redundancy To explore the effect of redundancy, we fi rst compare the failure probability of a single structural part to that of a structural system that fails due to failure of two structural parts. Certification testing is simulated by m odeling the testing of one structural part and certifying the structural system based on this test. Table 4-6 shows that while the average failure probability is reduced through structur al redundancy, the coefficients of variation of the failure probabilities are increased. That is, even thoug h the safety is improved, our confidence in the failure probability estimation is reduced. This behavior is similar to the

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75 effect of certifica tion (Table 4-4). In addition, we also notice that as the error grows, the benefit of redundancy also diminishes. This resu lt reflects the fact th at high errors result in high probabilities of failu re, and redundancy is more effective for smaller probabilities of failure. This behavior, however, is opposite to that resulting from certification testing. We notice that even though one safety meas urecertification testingis more effective when errors are high, another safety m easureredundancyis more effective when errors are low. So the level of uncertainty in the problem may decide on the efficient use of safety measures. Comparing the reduction probabilities of failu re before and after certification listed in the columns 4 and 7 of Table 4-6, we noti ce that the effect of redundancy is enhanced through certification testing. Table 4-6. Effect of redundancy on the probabi lities of failure. The numbers inside the parentheses represent the coefficient of variation of the relevant quantity. The coefficient correlation between failures of structural parts is taken as 0.5. Before certification After certification Part System Part System k Pnc (a)/10-4 Pnc (a)/10-4 reducti on Pc (a)/10-4 Pc (a)/10-4 reduct ion 0.25 0.244 (148%) 0.005 (230%) 52.1 0.227 (148%) 0.004 (230%) 53.5 0.50 0.763 (247%) 0.029 (388%) 26.3 0.609 (257%) 0.022 (408%) 28.0 0.75 2.70 (324%) 0.195 (503%) 13.8 1.66 (357%) 0.106 (568%) 15.6 1.0 8.83 (371%) 1.11 (563%) 7.9 3.79 (450%) 0.390 (718%) 9.7 1.5 60.0 (385%) 17.2 (549%) 3.5 11.5 (583%) 2.21 (945%) 5.2 (a) Pnc and Pc are the probabilities of failure before and after certification testing, respectively. (b) The ratio of probabilities of failure of the st ructural part and the system of two parts Next, we investigate the interaction of two safety measures: redundancy and certification testing. Comparing the probability ratios in Tabl e 4-7, we see that including redundancy improves the effectiven ess of certification testing. Mathematically, this can be explained with the following example. For a nominal error, k=1.0, the probabilities of failure before and after certificatio n of a structural part are 8.83-4 and 3.79-4,

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76 respectively. The system probabilities of fa ilure before and after certification are calculated by using Eq. (4.24) as 1.31-4 and 0.39-4, respectively. Notice that the system failure probability ratio is smaller than the component probability ratio, because redundancy is more effective for small probabili ties of failure. Physically, the reason for the increase in the effectiveness certification is that in the certification test, failure of a single part leads to rejection of the design of structural sy stem, while under actual service conditions, two failure events are needed for the failure of the structure. Thus, modeling redundancy is equivalent to modeling a relative ly more severe certification testing. This result is similar to the finding of Kale a nd Haftka (2003), who explored the effect of safety measures on aircraft structures desi gned for fatigue. They f ound that certification testing of an aircraft structure with a larg e machined crack of B-basis initial size was more effective than testing the stru cture with a rando m (natural) crack. Table 4-7. Effect of redundancy on the eff ectiveness of certification testing. The coefficient correlation between failures of structural parts is taken as 0.5. k /cncPP (part) /cncPP (system) 0.25 0.930 0.905 0.50 0.798 0.749 0.75 0.616 0.543 1 0.430 0.350 1.5 0.191 0.129 ncPand cP are the mean values of probabilities of failure before and after certification testing, respectively. Effect of the correlation coefficient Recall that the correlation co efficient of the probabilities of failure of the two structural parts was assumed to be 0.5. Table 4-8 shows that as the correlation coefficient decreases, the probability of fa ilure of the system also decrea ses, but at the same time our confidence in the probability estimation also reduces. The last column of Table 4-8 shows that as the correlation coefficient decreases, certification testing becomes more effective,

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77 which can be explained as follows. As the coefficient of correlation decreases, the structural parts behave more independently. Applying certif ication testing based on the failure of a single structural part means us ing more severe certification testing. This reminds us that as with any redundant system it pays to reduce the co rrelation coefficient of duplicate hardware (e.g., to use a back up part made by a different company). It is intriguing to speculate on the possible applica tion to structural design. Is it feasible, for example, to buy structural materials from different vendors for skin and stiffeners? Table 4-8. Effect of correlation coefficient on system failure probabilities and effectiveness of certification testing. The numbers inside the parentheses represent the coefficient of variatio n of the relevant quantity. The error multiplier k is taken as 1.0. Pnc/10-4 Pc/10-4 /cncPP 0.3 0.506 (678%) 0.161 (885%) 0.319 0.4 0.761 (615%) 0.255 (794%) 0.335 0.5 1.11 (563%) 0.390 (718%) 0.350 0.6 1.60 (519%) 0.583 (655%) 0.365 0.7 2.27 (480%) 0.859 (516%) 0.378 Additional Safety Factor Due to Redundancy Recall that the results given in Tabl e 4-6 show how redundancy reduces the probability of failure. For instance, for k =1.0 the average probability of failure before certification, Pnc, is reduced from 8.83-4 to 1.11-4. This reduction in probability of failure leads to an increase in the total safety factor. For each error multiplier k value, we calculate the additional safety factor require d to reduce the probabi lity of failure of a structural part to that of th e structural system. The second and third columns of Table 4-9 show two opposing effects on th e additional safety factor. As the error grows, the probabilities of failure before and after certification increase, so the effect of redundancy decreases because the redundancy is more effect ive for lower failure probabilities. Hence, the additional safety factor due to redundancy decreases with increas ed error (see also

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78 Figure 4-5). However, as indicated in the last column of Table 4-9 the ratio of safety factors after and before certification testing increases with increa sed error because the certification is more effective for high errors. Table 4-9. Additional safety factor due to redundancy k Fadd ncS Fadd cS % increase due certification 0.25 1.120 1.120 0.0 0.50 1.111 1.112 0.1 0.75 1.101 1.103 0.2 1 1.093 1.096 0.3 1.5 1.078 1.085 0.7 Figure 4-5. Total safety factor s for MEF model for the struct ural part and system after certification Effect of Variability Reduction Finally, we investigate the effect of va riability reduction on the average safety factor, design area and system probability of failure. We obs erve from Table 4-10 that the average safety factor and design area increase with the increase of variability in failure stress. In addition, we observe from the Pf ratio given in the last column of Table 4-10 that certification testing becomes less effec tive as variability increases. Figure 4-6 also shows the reduced efficiency of testing with increased vari ability. The second column of Table 4-10 shows that the certification testing failure rate (CFR) reduces with increased

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79 variability. As variability is increased, th e built load carrying area is also increased (column 5), so CFR is reduced accordingly. Table 4-10. Comparison of system failure probabilities corresponding to different variability in failure stress f. c.o.v. ( f) CFR(a) (%) Average Abuilt/A0 (b) Average Acert/A0 (b) ncP /10-4 cP /10-4 Pf Ratio 0 50.0 1.521 1.676 9.27 0.001 0.001 4% 32.3 1.629 1.727 2.00 0.008 0.040 8% 18.0 1.747 1.799 1.11 0.390 0.350 12% 11.6 1.878 1.910 1.19 0.737 0.619 (a)CFR: Certification failure rate (b) Abuilt/ A0 and Acert/ A0 are the total safety factors before and after certification testing, respectively. Figure 4-6. Effect of variability on failure probability Table 4-10 shows two opposing effects of variability on the two failure probabilities (before certification, ncP and after certification, cP see columns 7 and 8). When the coefficient of variation in the failu re stress is increased from 0% to 8%, the safety factor before certification (column 3) increases from 1.521 to 1.676, because a smaller B-basis value is used for the allowable failure stress. Note that the initial safety factor for no variability woul d be 1.5 if the error distribu tion (hence the safety factor distribution) is symmetric, but since the distributions are skewed (see Figures 4-1 and 4-

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80 2) the safety factor is 1.521. The increase in the safety factor with increased error leads to a reduction in the probability of failure be fore certification (column 5). However, for higher coefficients of variati on, the probability of failure before certification increases again, because the increased safety factor is not enough to compensate for the large variation in airplanes. However, once certifica tion is included, the picture is different. For no variability, even though the safety factor is increased by 10% (from 1.521 to 1.676, see columns 3 and 4), the probability of failure reduces four order of magnitudes (columns 5 and 6) due to the high effectiveness of certification testing at low variability. As variability increases, the effectiveness of certification testing re duces (column 7), so the probability of failure after certification is still high. Table 4-10 also indicates the advant age of reducing variability. Reducing variability from 8% to 4% redu ces the weight by 4%, while at the same time reducing the probability of failure by a factor of 50. However, the certification failure rate is unacceptably increased from 18% to 32%. To compensate for this, however, a company may reduce the weight gain back to an addi tional safety factor of 1.747/1.629=1.072, and have a reduced system probability of failure of 3.64-6 (compared to 3.90-5) and a reduced certification failure rate of 14.7% (compared to 18.0%). However, in reality companies reduce the chance of certification failure by structural element tests and conservative interpretation of the results of th ese tests. These are not analyzed in this chapter. The effects of structural element tests will be discussed in detail in Chapter 7. In addition, Table 4-10 reveals that variabil ity reduction is more effective than error reduction. For example, reducing all errors by ha lf (i.e., reducing k from 1 to 0.5) leads to reducing the built safety fact or from 1.747 to 1.730 (Table 4-4), along with reducing the

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81 system probability of failure from 3.90-5 to 2.20-6 (Table 4-6). On the other hand, reducing variability by half (that is, reducing c.o.v. of the failure stress from 8% to 4%) leads to reducing the built safety factor from 1.747 to 1.629, along with reducing the system probability of failure from 3.90-5 to 8.0-7 (Table 4-10). That is, variability reduction leads to more weight saving and pr obability of failure reduction than error reduction. Summary The relative effectiveness of safety measur es taken during aircraft structural design is demonstrated in this chapter. The safety factor, conservative material properties, certification testing, redundancy, error and va riability reduction were included in this study and the following was observed. While certification testing is more effective for improving safety rather than increased safety factors, it cannot compet e with even a small reduction in errors. Variability reduction is even more effectiv e than error reduction, but it needs to be accompanied by additional knockdown factors to compensate for the increase in the B-basis value. Our probabilities of failure are still high compared with the historical record (probability of failure of 10-7). This is probably due to the effect of building block tests, which we will address in Chapter 7. One safety measure, certification testing, is more effective when errors are large, while another safety measure, redundancy, is more effective when errors are low. Certification testing is more effective when the variability is low. At a low variability level, redundancy accompanied w ith certification testing is effective. Adding redundancy is equivalent to using an additional safety f actor of about 1.1.

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82 CHAPTER 5 INCREASING ALLOWABLE FLIGHT LOADS BY IMPROVED STRUCTURAL MODELING In this chapter we analyze the tradeoffs of allowable flight loads and safety of aerospace structures via deterministic and pr obabilistic design methodologies. The design methodologies are illustrated by performing a llowable flight load calculation of a sandwich panel used in aerospa ce structures. We explore th e effect of using a more accurate prediction technique for interf acial fracture toughness, which combines interfacial fracture toughness with mode-mixity instead of using th e traditional model that disregards mode-mixity, on increasin g the allowable design load of existing structures. The work presented in this chapter was also published in Acar et al (2006b). Mr. Xueshi Qiu is acknowledged for his contribution to this work. Introduction Structural design of aerospace structures is still performed with deterministic design philosophy. Researchers are constantly improving the accuracy of structural analysis and failure prediction. This improvement in accuracy reduces uncertainty in aircraft design and can therefore be used to enhance safety. However, since the record of structural safety in civilian transport aircra ft is very good, it makes sense to ask how to translate the reduced uncertainty to increased fl ight loads or weight re duction if safety is to be maintained at a specified level. The te rm allowable flight load here refers to the maximum allowable load that can be carried by the structure for a specific failure mode.

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83 Currently, there is no accepted way to transl ate the improvements in accuracy to weight savings or increased allowable flight loads. The objective of this chapter is to take the first step in this direction by utilizing probabilistic design methodology. Haftka (2005) describes how works of Star nes and colleagues (e.g., Li et al. 1997, Arbocz and Starnes 2002) to model variability in buckling of circul ar cylinders inspired work in his research group on using variability control on reduci ng the weight of composite liquid hydrogen tanks. Qu et al (2003) showed that for fixed probabi lity of failure, small reductions in variability can be translated to substantial we ight savings. Here we seek to investigate the potential of improved structural modeling. Some commercial aircraft which entered service in 1970s or 1980s are expected to reach their design service life soon. Ho wever, since researchers are constantly improving the accuracy of structural analys is and failure prediction, the maximum allowable flight loads of those aircraft can be recalculated to utilize the full potential of their structures. Motivated by this goal, we co nsider a given aerospace structure that is already designed and we aim to re-calculate the allowable flight load of the structure due to improved analysis. We expect that for so me designs, lower allowable loads will be predicted by the improved analys is, whereas for others higher allowable flight loads will be predicted. However, because improved m odels reduce uncertainty, we may expect an average increase of the allow-able flight loads over all design s. An important focus of the chapter is to show that mode ling error can masquerade as observed variability, which can be reduced (or even elimin ated) by better understanding of the physical phenomenon. Here, we chose a sandwich panel as an example because the improved model was developed by one of the aut hors and we had good access to the details of experiments and

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84 computations. Sandwich structures are used in aerospace vehicles due to their low areal density and high stiffness. Debonding of core from the face sheet is a common failure mode in sandwich construction, and the interf acial fracture is tradi tionally characterized by a single fracture toughness parameter. Howe ver, in reality the fracture toughness is a function of the relative amount of mode II to mode I (mode-mixity) acting on the interface (e.g., Suo 1999). Stiffness of sandwic h structures depends very much on the integrity of the face sheet/core bonding. Even a small disbond can significantly reduce the load carrying capacity, especially when the structure is unde r compressive loads (Avery and Sankar 2000, Sanka r and Narayanan 2001). Grau et al (2005) measured the interfacial fracture toughness as a function of mode-mix ity to characterize the propagation of the disbond between the f ace sheet and the core. They performed asymmetric double cantilever beam fracture te sts to determine the interfacial fracture toughness of the sandwich composite, and then de monstrated its application in predicting the performance of a sandwich structure cont aining a disbond. The use of mode-mixity dependent fracture toughness led to improvement in the accuracy of failure prediction of the debonded structure. We perform probabili stic analysis of th e debonded sandwich structure analyzed by determ inistic approach by Grau et al (2005) to expl ore a possible increase in the allowable flight load of the structure. The following section discusses the design of a sandwich structure used as an illustration. Next, the analysis of structural uncertainties (error and variability), with the main perspective of how to control uncertainty, is presen ted. Then, discussion on calculation of B-basis propert ies and allowable flight load calculation for sandwich structures by deterministic design are given. Ne xt, the assessment of probability of failure

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85 of sandwich structures is presented, followed by discussion of the tradeoffs of accuracy and allowable flight load vi a probabilistic design. The chapter is finalized with concluding remarks given in Summary section. Structural Analysis of a Sandwich Structure Sandwich panels are suscepti ble to debonding of the face sheet from the core. This is similar to the phenomenon of delamination in laminated composites. Disbonds can develop due to poor manufacturin g or during service, for exam ple, due to foreign object impact damage. Evaluation of damage and pred iction of residual strength and stiffness of debonded sandwich panels is critical because the disbonds can grow in an unstable manner and can lead to catastrophic failure. Stiffness of sandwich structures depends very much on the integrity of the face sh eet/core bonding. Even a small disbond can significantly reduce the load carrying capacity, when the st ructure is under compressive loads (Avery and Sankar 2000, Sankar and Narayanan 2001), because the debonded face sheet can buckle and create conditions at th e crack tip that are conducive for unstable propagation of the disbond. This problem has become very significant after the historic failure of X-33 vehicle fuel tank made of sa ndwich panels of polym er matrix composite face sheets and honeycomb core. Fracture at the interface between dissimila r materials is a cr itical phenomenon in many multi-material systems including sa ndwich construction. Traditionally, in engineering practice, the interfacial fract ure was characterized by a single fracture toughness parameter obtained by averaging the in terfacial fracture t oughness, hereinafter termed as average Gc or Gc A, obtained for some number of KI and KII combinations, where KI and KII are the mode I and mode II stress inte nsity factors, respectively. Later, studies have indicated, e.g. Suo (1999), th at for these multi-material systems, the

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86 interfacial fracture is a strong function of the relative amount of mode II to mode I acting on the inter-face, hereinafter termed as Gc with mode-mixity or Gc MM. The criterion for initiation of crack advance at the inter-face can be stated as I II cK K G G1tan (5.1) where G is the strain energy release rate and Gc is the interfacial fracture toughness, which depends on the mode-mixity angle In bimaterial fracture, KI and KII are the real and imaginary parts of the comp lex stress intensity factor K The toughness of interface cG can be thought of as an effective surface energy that depends on the mode of loading. Grau et al (2005) analyzed a debonded sandwich panel, and determined the maximum internal gas pressure in the core before the disbond could propagate. They used interfacial fracture mechanics concepts to anal yze this problem. The main premise here is that the crack will propagate when the ener gy release rate equals the fracture toughness for the core/face-sheet inte rface. The load and boundary conditions for the model problem are depicted in Figure 5-1. Figure 5-1. The model of face-sheet/core de bonding in a one-dimensional sandwich panel with pressure load. Note that due to symmetry only half of the structure is modeled.

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87 The maximum allowable pressure for a gi ven disbond length is calculated from the energy release rate G0 for a unit applied pressure. The energy release rate G is proportional to the square of the applied load or 2 0GGp (5.2) where p is the applied pressure. This failure assessment is a good approximation within the limits of a linear analysis. We assume that the epistemic uncertainty related to this failure function is negligible compared to the uncertainty in fracture toughness. The critical pressure pmax can be obtained using max 0cG p G (5.3) where Gc is the interfacial fracture toughness of the sandwich material system obtained from testing. Grau (2003) conducted asymmetric Doubl e Cantilever Beam (DCB) tests to determine the interfacial fr acture toughness of the sandwic h composite. The face sheet material was A50TF266 S6 Class E, Fi ber designation T800HB-12K-40B, matrix 3631 and the core sheet material was Euro-Compos ites aramid (ECA) fiber type honeycomb. Grau et al (2005) performed finite element analys es to compute the mode-mixity angle corresponding to designs tested in experi ments. The average interfacial fracture toughness prediction and the fracture toughness in terms of mode-mixity angle based on their work are presented in Figure 5-2.

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88 Figure 5-2. Critical energy releas e rate as a function of mode mixity. The continuous line denotes average Gc (Gc A) and the dashed line denotes a linear least square to fit to Gc (Gc MM) as a function of mode-mixity angle. The linear fit has R2 adj=0.473, erms=121.6 N/m. As shown in Fig.5-2, a simple way of de termining the interfaci al fracture toughness parameter is to perform fracture toughness test s for different core thickness, face sheet thickness and crack length combinations, wh ich correspond to different mode-mixity values, and to take the average fracture toughne ss value. However, as seen from Fig. 5-2 that the critical energy release rate is asse ssed better as a functi on of mode-mixity. Grau et al (2005) represent the critical energy releas e rate as a linear function of the modemixity (which they calculate from finite el ement analysis), that improves the accuracy of estimate of Gc. From Fig. 5-2 we note that w ithout the mode-mixity model, Gc would exhibit huge scatter (443 N/m to 1047 N/m). The mode-m ixity model reduces the scatter, because instead of a constant, Gc is now predicted to vary from 513 N/m to 875 N/m. That is, the simplicity of the average Gc model causes error in that model to masquerade as variability. For instance, the model of constant gravity acceleration (c onstant g) will lead to a scatter when measured in different town s partially due to difference in altitude. A

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89 model that takes altitude into account will show less scatter around the computed value of g. The uncertainty reduction in turn can be used to increase the safety or the effectiveness of the structure. Analysis of Error and Variability As in the previous chapters, we classify the uncertainties into two as errors and variability. The uncertainties that affect the entire fleet are ca lled here errors. They reflect inaccurate modeling of physical phenomena, errors in structural analys is, errors in load calculations, or use of material s and tooling in construction th at are different from those specified by the designer. The variability (a leatory uncertainty) reflects variability in material properties, geometry, or loading betw een different copies of the same structure. For the sake of simplicity, we assume that with mode-mixity there are no remaining errors in the predicted value of Gc for given mode-mixity angl e calculated from finite element analysis. Adding an estimate of th e remaining error can be easily accommodated by the analysis below. However, we assume that the scatter of Gc around mode-mixity dependent Gc represents variability. The experiment al values given in Table 5-1 are the mean values of the fracture toughness m easured through five experiments in Grau et al (2005) for each mode-mixity. We assume that the use of these mean values eliminates most of the measurement variability and leav es out only the material variability. On the other hand, the scatter around the average Gc represents combined error and variability. The deviations of experimentally measur ed fracture toughness values from the two fits dA and dMM (the deviations from the constant fit and from the linear fit, see Fig. 5-2) given in Table 5-1 ar e calculated from dA = Gc EXPGc A, dMM = Gc EXPGc MM (5.4)

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90 where Gc EXP is the experimentally m easured fracture toughness, Gc A and Gc MM are the fracture toughness from th e constant fit and linea r fit, respectively. Table 5-1. Deviations between measured and fitted values of average Gc and Gc with mode mixity for different designs. Th e superscript A denotes the average fracture toughness and MM indicate s mode-mixity dependent fracture toughness, and d represents the deviati on of experimental values from the constant fit or from the linear fit. Specimen (deg) Gc EXP (N/m) Gc A (N/m) Gc MM (N/m) dA (N/m) dMM (N/m) 1 16.52 609.4 746.6 513.2 -137.1 96.2 2 17.53 443.1 746.6 552.2 -303.5 -109.1 3 18.05 577.9 746.6 572.3 -168.7 5.6 4 18.50 628.7 746.6 589.7 -117.9 39.0 5 22.39 565.7 746.6 739.5 -180.9 -173.8 6 23.89 711.0 746.6 797.1 -35.6 -86.1 7 24.50 863.4 746.6 820.6 116.8 42.8 8 24.89 956.2 746.6 835.9 209.6 120.3 9 23.48 679.5 746.6 781.4 -67.1 -101.9 10 24.98 707.5 746.6 839.3 -39.1 -131.7 11 25.55 767.1 746.6 861.1 20.5 -94.1 12 25.90 817.8 746.6 874.8 71.3 -56.9 13 22.65 702.3 746.6 749.3 -44.3 -47.1 14 23.69 903.7 746.6 789.5 157.1 114.1 15 24.15 964.9 746.6 807.2 218.4 157.7 16 24.54 1047.3 746.6 822.3 300.7 224.9 Standard deviation 162.2 0 115.6 162.2 113.8 Each row of Table 5-1 corresponds to a di fferent specimen. Each specimen has a different core thickness, face sheet thickness and crack length, thus having a different mode-mixity angle (calculated through finite element analysis). The sixth column of Table 5-1 presents the deviations of Gc values obtained through experiments from their average values. These deviations combine va riability and error. Errors are due to neglecting the effect of mode-mixity in Gc. We assume that these are the only errors so that dMM represents only variability. Approximate cumulative distribution functi on (CDF) for the vari ability is obtained by using ARENA software (Kelton et al 1998). The distribution parameters and

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91 goodness-of-fit statistics for the distribut ions are as follows. For variability, dMM, ARENA found the best distribution as the nor mal distribution with a mean value of zero and a standard deviation of 113.8 N/m. Fo r obtaining goodness-of-f it statistics, Chisquare and Kolmogorov-Smirnov tests are the commonly used. For our case the number of data points is low; hence, the Chi-square test does not provide reliable statistics, therefore ARENA uses the Kolmogorov-Smirnov te st to decide if a sample comes from a population with a specific di stribution. The p-value of Kolmogorov-Smirnov test is greater than 0.15. For total uncertainty, dA, ARENA found the best distribution as the normal distribution with a mean value of zero and a standard deviation of 162.2 N/m. The p-value for Kolmogorov-Smirnov test is agai n greater than 0.15. The corresponding pvalue is a measure for goodness of the fit. Larger p-values indicate better fits (Kelton et al 1998), with p-values less than about 0.05 indicating poor fit. Figures 5-3 and 5-4 show the comparison of the actual and fitted CDFs of the variability (Fig. 5-3) and the total uncertainty (Fig. 5-4) of the average fracture toughness, respectively. In the figures, x-axis represents the fitted CDF while y-axis represents the actual CDF. If the fits were exact, they w ould follow the linear lines shown in the figures. We see in Figs. 5-3 and 5-4 that the deviat ions from the linear lines are not high; and hence the fitted distributions are acceptable.

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92 Figure 5-3. Comparison of actual and fitte d cumulative distribution functions of variability, dMM, of Gc. Figure 5-4. Comparison of actual and fitted cu mulative distribution functions of total uncertainty (error and variability, dA) of Gc. In addition to variability in Gc predictions, there is also variability in the pressure p We assume that the maxi-mum lifetime loading p follows lognormal distribution with mean value of pallow and coefficient of variation (c.o.v.) of 10%.

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93 Deterministic Design and B-basis Value Calculations In deterministic design, the only use of pr obabilistic (or statistical) information is via conservative material properties, which are determined by statistical analysis of material tests. FAA regulat ions (FAR-25.613) state that the conservative material properties are characterized as A-basis and B-basis material prope rty values. A-basis values are used when there is a single failu re path in the structure, while the B-basis values are used when there are multiple failure paths in the structure. Detailed information on these values is provided in Ch apter 8 of Volume 1 of Composite Materials Handbook (2002). Here we use B-basis Gc, which is defined as the value exceeded by 90% of the population (of material batches) with 95% confidence. This is given by B B basisXks (5.5) where X is the sample average, s is the sample standard deviation and kB is the tolerance coefficient needed to achieve the 90% set-o ff and the 95% confidence. If infinitely many material characterization tests were carried out, there would be no issue of confidence, and for normal distribution 90% of the population will be exceeded by 282 1 1 01 0 z kB where is the CDF of the standard normal distribution. With a finite sample of N tests, this is adjusted as N z z b N z a a ab z z kB 2 05 0 2 1 0 2 05 0 2 1 0 1 0; ) 1 ( 2 1 (5.6) where z0.1 = 1 0 is the critical value of normal di stribution that is exceeded with a probability of 10%.

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94 Grau et al. (2005) used the fracture toughness valu es obtained from experiments to calculate the predicte d failure load of a debonded sandw ich structure s hown earlier in Fig. 5-2. They used different core thic kness, face sheet thickness and crack length combinations and compared the predicted failu re load of the struct ures designed via the use of average Gc and mode-mixity dependent Gc. In their failure load calculation, Grau et al (2005) used the mean values for the frac ture toughness and they did not use a safety factor. Here, we use B-basis values for fracture toughness and a safety factor of 1.4 for loading to assess the allowable flight load of the same sandwich designs used by Grau et al (2005). To calculate B-basi s values, we use the standard deviations for fracture toughness given in Table 5-1. The mean valu es and the corresponding B-basis values of the fracture toughness for the thirteen de signs given in the example in Grau et al (2005) are given in Table 5-2. Table 5-2. The mean and B-basis values of the fracture toughness of the designs analyzed (total 13 designs). The B-basis values are calculated assuming that the improvements in accuracy a ffect the B-basis values Design number Mode-mixity angle (deg) A mean cG (N/m) MM mean cG (N/m) A Bbasis cG (N/m) MM Bbasis cG (N/m) 1 16.24 638.8 498.4 308.8 266.9 2 17.15 638.8 529.0 308.8 297.5 3 18.95 638.8 589.6 308.8 358.1 4 21.08 638.8 661.3 308.8 429.8 5 22.27 638.8 701.3 308.8 469.8 6 18.32 638.8 568.4 308.8 336.9 7 20.18 638.8 630.9 308.8 399.4 8 22.27 638.8 701.4 308.8 469.9 9 23.41 638.8 739.7 308.8 508.2 10 18.28 638.8 567.1 308.8 335.6 11 19.86 638.8 620.2 308.8 388.7 12 21.57 638.8 708.6 308.8 477.1 13 16.24 638.8 498.4 308.8 266.9

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95 Even though the scatter around the average Gc is combination of error and variability, for deterministic design following FAA regulations it is treated as variability. The reduced standard deviation of the mode-mixity dependent Gc allows then increasing the B-basis allowable. Figure 5-5 shows the fitted and B-basis values of the two approaches. Figure 5-5. Fitted least square lines fo r fracture toughness, and derived B-basis allowables While calculating the B-basis values for fracture toughness given in Table 5-2, we use N =16, which increases kB to 2.035. Recall that the standard deviations ( ) of designs are obtained in the previous section. For exam ple, for the first design the mean value is 638.8. The corresponding B-basis value is calculated as 638.8.035.2 = 308.8. After obtaining the B-basis va lues in Table 5-2, we compute the allowable flight load pallow by deterministic design philosophy. As not ed earlier, besides the use of B-basis material properties, a safety factor of 1.4 is also used for loads. Hence, Eq. (5.3) is modified to calculate the allowable flight loads for thirteen different designs as 01 1.4c B basis allowG p G (5.7)

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96 The calculated pallow values corresponding to the use of Gc A and Gc MM are given in Table 5-3. The last column of Table 5-3 shows the percent change in the allowable flight load by using Gc MM instead of Gc A. We see that allowable flight load is increased by 13.1% on average by using the mode-mixity based B-basis properties. This is the improvement in allowable flight load using a deterministic approach. As shown in the next section, this increase in allowable flight load is accompanied by a reduction in probability of failure, so that the additional gains may be realized by using probabilistic approach. Table 5-3. Allowable flight load of failu re of the sandwich panels designed using deterministic approach. The superscr ipt A denotes the use of average fracture toughness of experiments and MM indicates the use of mode-mixity dependent fracture toughness Design number pallow A (kPa) pallow MM (kPa) % p 1 51.2 47.6 -7.0 2 267.0 262.0 -1.9 3 158.6 170.8 7.7 4 77.1 90.9 18.0 5 45.2 55.8 22.3 6 247.2 258.2 4.4 7 154.1 175.3 13.7 8 73.1 90.2 23.4 9 42.8 54.8 28.3 10 247.2 257.7 4.2 11 146.2 164.1 12.2 12 70.1 84.3 20.2 13 40.8 50.7 24.3 Average 13.1 Assessment of Probability of Failure The probability of failure of a structural component can be expressed in terms of its structural response R and its capacity C corresponding to that response by PrfPCR (5.8)

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97 For the sandwich structure analyzed here the response R = G is the energy release rate (Eq. 5.2), and the capacity C = Gc is the interfacial fracture toughness. G depends on structural dimensions through G0 (see Eq. (5.2)). Both response G and the capacity Gc have variability that needs to be included in the calculation of the probability of failure. We assume that the variability in G is mainly due to the variability in load p rather than G0. Besides variability, there exist errors in assessing G and Gc (e.g., errors in load, G0, and material property calculations). The general equation for probability of failu re given in Eq. (5.8) can be expressed in this problem as 2 0PrfcPGGp (5.9) Then, the probability of failure can be written in a functional form as 0,,,,,ffcGGallowP ccPPGeVARpVARG (5.10) where cG is the mean value of Gc, c Ge is the error in Gc predictions (that we reduce by using mode-mixity dependent Gc instead of average Gc), c GVAR is the variability in Gc, allowp is the allowable flight load (or mean value of the loading p ), PVAR is the variability in p and 0G is the strain energy release rate corresponding to unit pressure that we assume to be deterministic. Since th e limit-state function for this problem, 2 0cgGGP, is a simple function with only two random variables, we easily calculate the probability of failure by analytical means as follows The probability distribution function (PDF) of a function Z of two random variables X and Y Z =h( X Y ) can be calculated as (Ang and Tang (1975), p.170)

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98 dy z x y x f z fY X Z ) (, (5.11) where fX,Y(x,y) is the joint probabil ity distribution function of X and Y We can write the limit-state function for the sandwich panel problem as 2 0p G G gc (5.12) Therefore, to calculate the PDF of g from Eq. (5.11), we replace Z with g X with Gc, Y with p and also we have 2 0p G g Gc so 1 g G z xc. After these substitutions and noting that p only takes positive values, we get from Eq. (5.11) that 0(),GGpc c f gfGpdp (5.13) Here we assume that Gc and p are statistically independent, hence the joint distribution in Eq. (5. 13) is calculated as p f p G g f p G fp c G c p c G 2 0 ,, (5.14) Then, the cumulative distribution function (CDF) of g is calculated as ()(')'g GGFgfgdg (5.15) which allows us to compute the probability of failure simply as Pf = FG(0). Table 5-4 shows the probabi lities of failure corresponding to deterministic allowable flight loads. We observe that in addition to the 13.1% average increase in allowable flight load, the average probability of failure was reduced by about a factor of five.

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99 Table 5-4. Corresponding probabilities of failu re of the sandwich pa nels designed using deterministic approach. The superscr ipt A denotes the use of average fracture toughness of experiments and MM indicates the use of mode-mixity dependent fracture toughness. The B-basi s values are calcul ated considering that the improvements in accuracy affect the B-basis values (adjusted B-basis values). Design number Pf A (10-3) Pf MM (10-3) 1 1.869 1.064 2 1.869 0.762 3 1.869 0.407 4 1.869 0.211 5 1.869 0.153 6 1.869 0.504 7 1.869 0.275 8 1.869 0.153 9 1.869 0.117 10 1.869 0.511 11 1.869 0.304 12 1.869 0.184 13 1.869 0.145 Average 1.869 0.369 Notice that the probabilities of failure given in Table 5-4 are high. These probabilities of failure correspond to component failure probabilities. The probability of the actual structure will be much smaller due to the redundancy in the structure. For example, if we define the failure of th e structure as simultaneous failures of two components having a correlation coefficient (o f probability of failure) of 0.5, then component probabilities of failure 1.869-3 and 0.369-3 given in the last row of Table 5-4 correspond to system probabilities of failure 1.28-4 and 1.39-5, respectively. Analyzing the Effects of Improved Mo del on Allowable Flight Loads via Probabilistic Design As seen from Eq. (5.10), there are four distinct ways to increase the allowable flight load of a structure: (a) Use a different material to increase cG; (b) Develop more accurate solutions that reduce c Ge (such as the use of mode-mixity dependent Gc instead of

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100 average Gc); (c) Improve quality control an d manufacturing processes to reduce variability c GVAR or employ measures to reduce PVAR; (d) Use a heavier design to reduce G0. For a structure that is already built, only option (b) is available. The previous section showed how reductions in variability can increase allowable flight load using deterministic design. For probabilistic design, the mode-mixity approach is treated as accuracy improvement and we calcu late its effect on the safe allowable flight load. For a target probability of failure, targetfP, the allowable flight load can be calculated from 0 target,,,,,ccffcGGallowPfPPGeVARpVARGP (5.16) Thus, given the target probability of failure, the allowable flight loads corresponding to different error factors on fracture toughness, c Ge, can be calculated from Eq. (5.17). 1212 target,,ccfGallowfGallowfPepPepP (5.17) For the present calculation, the target probability of failure is taken 1.869-3, which is the probability of failure with the deterministic allowable flight load using the average Gc (see Table 5-4). Table 5-5 shows the comp arison of allowable flight load for the average Gc and mode-mixity dependent Gc approaches in case of probabilistic design. We see in Table 5-5 that by fixing the probabi lity of failure rather than adjusting the Bbasis properties, the average al lowable flight load can be in creased by 26.5%. It must be noted, however, that for some structures th e improved analysis may indicate a small reduction in allowable flight loads. With the deterministic approach, this applied to

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101 Designs 1 and 2 (Table 5-3), while with the probabilistic approach, only Design 1 suffers (a small) load reduction. Table 5-5. Allowable flight loads of the sa ndwich panels calculat ed via probabilistic approach. The superscript A denotes th e use of average fracture toughness of experiments and MM indi cates the use of mode-mixity dependent fracture toughness. The probabilities of failure of the all designs are 1.869-3. Design number pallow A (kPa) pallow MM (kPa) % p 1 51.2 50.5 -1.3 2 266.9 283.9 6.4 3 158.6 190.1 19.9 4 77.1 102.6 33.1 5 45.2 63.2 39.8 6 247.3 285.2 15.3 7 154.2 196.9 27.7 8 73.1 102.3 39.9 9 42.8 62.4 45.8 10 247.3 284.4 15.0 11 146.2 184.0 25.8 12 70.1 95.4 36.1 13 40.8 57.6 41.1 Average 26.5 Summary The effect of improved model for fractur e toughness on allowable flight load was investigated using both de terministic and probabilist ic design methodologies. For deterministic allowable flight load calcula tion, the improved model reduces scatter and allows increase in the fracture toughness allowable calculated by B-basis properties, while for probabilistic allowable flight load calculation, the reduced error is incorporated into the calculation of probability of failure. We find that the deterministic approach leads to 13.1% increase on average in the allowable flight load and reduction of the probability of failure by a factor of five. The use of Bbasis properties in the deterministic de-sign does not permit translating the full potential of improved modeling to increase allowable

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102 flight loads. In contrast, the probabilistic approach allows 26.5% increase on average in the allowable flight load, wh ile still maintaining the orig inal probability of failure.

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103 CHAPTER 6 TRADEOFF OF UNCERTAINTY REDUCTION MECHANISMS FOR REDUCING STRUCTURAL WEIGHT Inspired by work on allocating risk between the different components of a system for a minimal cost, we explore the optimal allocation of uncertainty in a single component. The tradeoffs of uncertainty reducti on measures on the we ight of structures designed for reliability are explored. The uncertainties in the problem are broadly classified as error and variability. Probabilistic design is carried out to analyze the effect of reducing error and variabili ty on the weight. As a demons tration problem, the design of composite laminates at cryogenic temperatures is chosen because the design is sensitive to uncertainties. For illustration, variability reductio n takes the form of quality control, while error is redu ced by including the effect of chemical shrinkage in the analysis. Tradeoff plots of uncertainty redu ction measures, probabi lity of failure and weight are generated that c ould allow choice of optimal uncertainty reduction measure combination to reach a target probability of failure with minimum cost. In addition, we also compare response surface approximations to direct approximation of a probability distribution for efficient estimation of reliability. The research presented in this chapter will also be published in Acar et al. (2006c). Dr. Theodore F. Johnson of NASA Langley Re search Center is acknowledged for his contribution to this work.

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104 Introduction For systems composed of multiple component s, system failure probability depends on the failure probabilities of the components, and the co st of changing the failure probability may vary from one component to a nother. The risk or reliability allocation problem can be defined (Mohamed et al 1991, Knoll 1983, Hurd 1980) as determining the optimal component reliabiliti es such that the system objec tive function (e.g., cost) is optimized and all design constraints (e.g., system reliability level) are met. Several researchers applied risk and re liability allocation methods to optimize the total cost of nuclear power plants by allocating the risk a nd reliability of individual subsystems such that a specified reliability goal is met (Gokcek et al 1978, Cho et al 1986, Yang et al 1989 and Yang et al 1999). Ivanovic (2000) applied reliabili ty allocation to design of a motor vehicle. The vehicle reliability is a llocated to its elements for minimum vehicle cost while keeping the reliabil ity of the vehicle at a specif ied level. Acar and Haftka (2005) investigated relia bility allocation between the wing and tail of a transport aircraft. The concept of risk allocation is also used in finance applications, where risk allocation is defined as the process of a pportioning individual risks rela ting to projects and service delivery to the party best placed to manage each risk. Risks are allocated across the supply chain that is, between the departme nt, its customers, its suppliers and their subcontractors. Vogler (1997), Bing et al (2005) and Niehaus (2003) are some examples of numerous publications on risk al location in finance applications. Instead of considering a system of multip le components, we may also consider multiple sources of uncertainty for a single co mponent. Again, the probability of failure can be reduced by reducing the uncertainty from each source, with different cost associated with each. That is the probability constraints can be satisfied by reducing

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105 different types of uncertainti es. The objective of this chapter is to demonstrate this approach for reliability based design optimization (RBDO) of structures. As in earlier chapters, uncertainty is divided into error and variability, to distinguish between uncertainties that apply equally to an en tire fleet of a structural model (error) and the uncertainti es that vary for an individua l structure (variability). In aircraft structural de sign there are different play ers engaged in uncertainty reduction. Researchers reduce errors by develo ping better models of failure prediction and this leads to safer structures (see Chapter 5). Aircraft companies constantly improve finite element models, thus reducing errors in structural response. The Federal Aviation Administration (FAA) leads to further reduction in error through the process of certification testing. Aircraft makers also constantly improve manufacturing techniques and quality control procedure to reduce variability between airplanes. Airlines reduce variability in structural failure due to ope rating conditions by conducting inspections, and the FAA contributes to reduced variability by licensing pilots, ther eby reducing the risk that incompetent pilots may subject airplanes to excessively high loads. These uncertainty reduction mechanisms are costly, and their cost can be traded against the cost of making the structur e safer by increasing its weight. Kale et al 2005 investigated the tradeoff of inspection cost against the cost of structural weight, and found that inspections are qu ite cost effective. Qu et al (2003) analyzed the effect of variability reduction on the weight savings from composite laminates under cryogenic conditions. They found that employing quality control to -2sigma for the transverse failure strain may reduce the weight of composite laminates operating at cryogenic

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106 temperatures by 25% marking such laminates as a structure where weight is sensitive to the magnitude of uncertainties. Here, we use the example of this composite laminate to explore tradeoffs between the variability reducti on, considered by Qu et al (2003), and error re duction in the form of improved accuracy of structural analysis. The chapter is organized as follows. The design of compos ite laminates for cryogenic temperatures is discussed in the ne xt section. Then, probability of failure estimation of the laminates is described, fo llowed by the formulation of the probabilistic design optimization for our problem. Next, we ight savings using error and variability reduction mechanisms are given. Finally, th e optimum use of uncertainty reduction mechanisms is discussed, followed by the summary of the chapter. Design of Composite Laminate s for Cryogenic Temperatures We consider the design of a composite panel at cryogenic temperatures as demonstration for trading off uncertainty re duction mechanisms. The definition of the problem is taken directly from Qu et al (2003). The laminate (Fig. 6-1) is subject to mechanical loading ( Nx is 4,800 lb/inch and Ny is 2,400 lb/inch) and thermal loading due to the operating temperature -423F, wher e the stress-free temperature is 300F. The objective is to optimize the weight of laminates with two ply angles 12/ s The design variables are the ply angles 1, 2 and ply thicknesses t1, t2. The material used in the laminates is IM600/ 133 graphite-epoxy of ply thickness 0.005 inch. The minimum thickness necessary to prevent hydrogen leakage is assumed to be 0.04 inch. The geometry and loading conditions are shown in Fig. 6-1. Temperature-dependent material properties ar e given in Appendix G.

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107 The deterministic design optimizati on of the problem was solved by Qu et al (2003). They used continuous design variable s and rounded the thicknesses to integer multiples of the basic ply thickness 0.005 inches In the deterministic optimization, they multiplied the strains by a safety factor of SF=1.4. The deterministic optimization problem is formulated as 121212 ,,, 1112221212 12min4 suchthat,, ,0.005tt L ULUU FFFhtt SSS ttin (6.1) where the allowable strains are given in Table 6-1. Table 6-1. Allowable strains for IM600/133 1 L 1 U 2 L 2 U 12 U -0.0109 0.0103 -0.013 0.0154 0.0138 Figure 6-1. Geometry and loading of the laminate with two ply angles. Note that x-is the hoop direction and y is the axial direction. Since designs must be feasible for the entire range of temperatures, strain constraints were applied at twenty-one different temperat ures, which were uniformly distributed from 77F to F. Qu et al (2003) found the optimum design given in Table 6-2.

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108 Table 6-2. Deterministic optimum design*. The number in parentheses denotes the unrounded design thicknesses. 1 (deg) 2 (deg) t1 (in) t2 (in) h (in) 27.04 27.04 0.010 0.015 0.100 (0.095) taken from Qu et al (2003) Calculation of Probability of Failure The failure of the laminates is assessed based on the first ply failure according to the maximum strain failure criterion. The strain allowables listed in Table 6-1 are the mean values of the failure strains according to Qu et al (2003). The first step in the calculation of the probability of failure is to quantify uncertainties included in the problem. As in earlier chapters, we use a simple classification for uncertainty to distinguish be tween the uncertainties that apply equally to the entire fleet of a structural model (erro rs) and the uncertainties that vary for an individual structure (variability). Since errors are epistemic, they are often modeled using fuzzy numbers or possibility analysis (Anton sson and Otto 1995, Nikolaidis et al 2004 and Vanegas and Labib 2005). We model the erro rs probabilistically by usi ng uniform distributions to maximize the entropy. Variability refers to the departure of a quantity in individual laminates that have the same desi gn. Here, the elastic properties ( E1, E2, G12, and 12), coefficients of thermal expansion ( 1 and 2), failure strains ( 1 L, 1 U, 2 L, 2 U, and 12 U) and the stress-free temperature (Tzero) have variability. These random variables are all assumed to follow uncorrelated normal distributions with coefficients of variations listed in Table 6-3. Table 6-3. Coefficients of variation of the random variables (assumed uncorrelated normal distributions) E1, E2, G12, and 12 1 and 2 Tzero 1 L and 1 U 2 L, 2 U, and 12 U 0.035 0.035 0.03 0.06 0.09

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109 We also use a simple error model, assuming that calculated values of failure strains differ from actual values due to experiment al or measurement errors. Using standard classical lamination theory (CLT) for ply strain calculation leads to errors in part, because standard CLT does not take chemical shrinkage into account. We relate the actual values of the strains to their calc ulated values via Eq. (6.2) true calce 1 (6.2) where e is the representative error factor that in cludes the effect of all error sources on the values of strains and failure st rains. For example, if the estimated failure strain is 10% too high, this is approximately equivalent to the strain being calculated 10% too low. For the error factor e, we use a uni form distribution with bounds of be. This error bound can be reduced by using more accurate failure models For example, the cure reference method (Ifju et al 2000) may be used to account for the shrinkage due to a chemical process. In Sections 6-4 and 6-5, we will investigate the effect of reducing be on the probability of failure and the weight. To calculate the probability of failure, we use Monte Carlo Simulation (MCS). For acceptable accuracy, sufficient strain analys es (simulations) must be obtained through standard CLT analysis. However, this is computationally expensive and needs to be repeated many times during the optimization. In order to reduce the computational cost, Qu et al [19] used response surface approximations for strains ( 1 in 1, 1 in 2, 2 in 1, 2 in 2, 12 in 1, and 12 in 2). They fitted quadratic res ponse surface approximations to strains in terms of f our design variables (t1, t2, 1, and 2), material parameters (E1, E2, G12, 12, 1, and 2) and service temperature Tserv. These response surfaces were called the analysis response surfaces (A RS), because they replace th e CLT analysis. A quadratic

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110 response surface approximation in terms of 12 va riables includes 91 coefficients, so they used 182 realizations from Latin hypercube sampling (LHS) design. As seen from Table 6-4 the root mean square error predictions are less than 2% of the mean responses, so the accuracies of the ARS is good. Table 6-4. Evaluation of the accuracy of the analysis response surface(a). Note that the strains are in millistrains. 1 in 1 1 in 2 2 in 1 2 in 2 12 in 1 12 in 2 R2 adj (b) 0.9977 0.9978 0.9956 0.9961 0.9991 0.9990 RMSE Predictor(c) 0.017 0.017 0.060 0.055 0.055 0.060 Mean of response 1.114 1.108 8.322 8.328 -3.13 -3.14 (a) taken from Qu et al (2003) (b) adjusted coefficient of multiple determination (c) root mean square error predictor We found, however, that even small errors in strain values may lead to large errors in probability of failure calculations, so we considered approximate cumulative distribution functions (CDF) of strains instead of ARS. We assume normal distributions for strains and estimate the mean and the sta ndard deviation of strains conservatively by MCS. That is, the mean and standard devia tion of the assumed dist ribution are found so that the CDF of the approximated distribution is smaller than or equal to (i.e., more conservative) the CDF calculated via MCS, excep t for strain values very near the tail of the distribution. Detailed information on conservative CDF fitting is given in Appendix H. We use 1,000 MCS simulations, which are accu rate to a few percent of the standard deviation for estimating the mean and stan dard deviation. Cumu lative distribution function obtained through 1,000 MCS, the approximate normal distribution and the conservative approximate normal distributions for 2 corresponding to one of the deterministic optimum are compared in Figs. 6-2(a) and 6-2(b).

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111 Next, we compare the accuracy of the analysis response surface and approximate CDF approaches by using 1,000,000 MCS in Table 6-5. We can see that the use of approximate CDFs for strains leads to more accurate probability of failure estimations than the use of ARS. Furthermore, the case of conservative fit to CDF leads to overestimation of the probability of failure However, the approximate CDFs were obtained by performing 1,000 MCS, while the ARS were constrained by using only 182 MCS. In addition, the approximate CDF needs to be repeatedly calculated for each design. It is possible that so me combination of ARS with approximate CDF may be more efficient and accurate than either using AR S or approximate CDF alone, and this might be explored in future work. Figure 6-2. Comparison of CDF obtained via 1,000 MCS, the approximate normal distribution and conservative appr oximate normal distributions for 2 on 1 corresponding to the deterministic optimum. Table 6-5. Comparison of probability of fa ilure estimations for the deterministic optimum(a). Samples size of MCS is 1,000,000. Approach followed Probability of Failure, Pf (-4) Standard error in Pf due to limited sampling (-4) MCS with CLT (exact analysis) 10.21 0.320 MCS with ARS* of strains 16.83 0.410 MCS with approximation to CDF of strains 11.55 0.340 (a) Taken from Qu et al (2003) (b) ARS: Analysis response surface approximation

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112 Probabilistic Design Optimization The laminates are designed for a target failure probability of 10-4. The optimization problem can be formulated as given in Eq. (6.3). The design variables are the ply thicknesses and angles. 121212 ,,, 12min4 suchthat ,0.005tt ff targethtt PP ttin (6.3) For this optimization, we need to fit a design response surface (DRS) to the probability of failure in terms of the design variables. The accuracy of the DRS may be improved by using an inverse safety measure. We use the probabilistic sufficiency factor (PSF) developed by Qu and Haftka (2004). Probabilistic Sufficiency Factor (PSF) The safety factor S is defined as the ratio of the capacity GC of the structure to the structural response GR. The PSF is the probabilistic inte rpretation of the safety factor S with its CDF defined as ProbC S RG Fss G (6.4) Given a target probability of failure, f targetP PSF is defined as the solution to argProbProbC Sf tet RG FsPSFSPSFP G (6.5) That is, the PSF is the safety factor obt ained by equating the CDF of the safety factor to the target failure probabi lity. The PSF takes values such that

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113 1 1 1ff target ff target ff targetifPP PSFifPP ifPP (6.6) When MCS are used, the PSF can be estimated as the nth smallest safety factor over all MCS, where n = N f targetP Using the PSF, the optimization problem can be formulated as 121212 ,,, 12min4 suchthat1 ,0.005tthtt PSF ttin (6.7) The optimization problem given in Eq. (6.7) is solved by using Sequential Quadratic Programming (SQP) in MATLAB. Design Response Surface (DRS) We have three components of strain for each angle: 1, 2 and 12. The strain 2 and 12 are more critical than 1. The mean and standard deviation of four strains ( 2 in 1, 2 in 2, 12 in 1 and 12 in 2) are computed by using MCS of sample size 1,000 and fitted with conservative normal distributions as show n in Fig. 6-2. These di stributions are used in MCS using 1,000,000 simulations at each de sign point to compute PSF. In order to perform the optimization, we need to appr oximate the PSF in terms of the design variables by a design response surface (DRS). We fit three DRS of the PSF as function of the four design variables (t1, t2, 1, and 2) for three different error bound (be) values of 0, 10%, and 20%. As shown in Appendix I, the use of the PSF leads to much more accurate estimate of the safety margin than fitting a DRS to the probability of failure.

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114 Weight Savings by Reducing Error and Em ploying Manufacturing Quality Control As noted earlier, the probabilistic design optimizations of the composite laminates were performed for three different values of the error bound, be, namely 0, 10%, and 20%. Schultz et al (2005) have shown th at neglecting chemical shrinkage leads to substantial errors in strain calculations. Based on Schultz et al (2005), we assume that using the standard CLT without chemical sh rinkage leads to 20% errors in strain calculations, while using the modified CLT (i.e ., CLT that takes chemical shrinkage into account) leads to the reduction of error bounds from 20% to 10%. As noted earlier, the errors are assumed to have uniform dist ribution, which corresponds to maximum entropy. For the error bounds discussed, we solve the optimization problem given in Eq. (6.7). The results of the optimization are presented in Table 6-6 and the weight (proportional to thickness) savings due to e rror reduction are shown in Fig. 6-3. We see that reducing the error bounds from 20% to 10% leads to 12.4% weight saving. In addition, reducing error from 20% to 0 (clearly only a hypothetical case) leads to weight saving of 23.1%. Table 6-6. Probabilistic optimum designs fo r different error bounds when only error reduction is applied. The PSF and Pf given in the last two columns are calculated via Monte Carlo simulatio ns (sample size of 10,000,000) where the strains are directly comp uted via standard CLT analysis. The numbers in parentheses under PSF and Pf show the standard erro rs due to limited Monte Carlo sampling. Error bound 1 2 (deg) t1 t2 (in) h (in) [h* (%)] PSF Pf (1-4) 0 25.47 26.06 0.0156 0.0137 0.1169 [23.1] 0.9986 (0.0030) 1.017 (0.032) 10% 25.59 25.53 0.0167 0.0167 0.1332 [12.4] 1.018 (0.0035) 0.598 (0.024) 20% 23.71 23.36 0.0189 0.0191 0.1520 [0.0] 0.9962 (0.0035) 1.111 (0.105) The optimum laminate thickness for 20% error bound is taken as the basis for h computations

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115 Figure 6-3. Reducing laminate thickness (hence weight) by error reduction (no variability reduction) We have shown that it is possible to reduce the lamina te thickness by 12.4% through reducing the error from 20% to 10% Now, we combine error reduction with variability reduction and analyze the overa ll benefit of both uncertainty reduction mechanisms. An example of variability reductio n is testing a set of composite laminates and rejecting the laminates having lower failure strains as a form of quality control. The test can involve a destructive evaluation of a small coupon cut out from laminate used to build the structure. Alternativ ely, it can involve a non-destru ctive scan of laminates to detect flaws known to be associated with lowered strength. We study the case where specimens that have transverse failure strain s lower than two standard deviations below the mean are rejected (2.3% rejection rate). We cons truct three new DRS for PSF corresponding to error bounds of 0, 10% and 20%. The probabilistic design optimizations of composite laminates for three different values of error bound ( be) are performed and the results ar e presented in Table 6-7 and in Fig. 6-4. We note that when th is form of variability reduction is applied, the laminate

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116 thickness can be reduced by 19.5%. If the error bound is reduced from 20% to 10% together with the variability reduction, the laminate thickness can be reduced by 36.2%. Table 6-7. Probabilistic optimum designs for different error bounds when both error and variability reduction are applied. PSF and Pf given in the last two columns are calculated via MCS (sample size of 10,000,000) where the strains are directly computed via the standard CLT analysis. The numbers in parentheses under PSF and Pf show the standard errors due to limited sample size of MCS. Error bound 1 2 (deg) t1 t2 (in) h (in) [h* (%)] PSF Pf (-4) 0 28.52 28.71 0.0089 0.0114 0.0813 [-46.6] 0.9965 (0.0014) 1.255 (0.035) 10% 27.34 27.37 0.0129 0.0114 0.0970 [-36.2] 1.0016 (0.0015) 0.906 (0.030) 20% 25.57 25.66 0.0168 0.0138 0.1224 [-19.5] 0.9968 (0.0015) 1.190 (0.109) The optimum laminate thickness for the 20% error bound given in Table 6-6 (i.e. h=0.1520 in) is taken as the basis for h computations Figure 6-4. Reducing laminate thickness by error reduction (ER) and quality control (QC). The numbers in the last two columns of Table 6-7 show the PSF and Pf calculated by using the 10,000,000 MCS where strains are di rectly calculated th rough the standard CLT analysis. The design values for PSF and Pf of the optimum designs are expected to be 1.0 and 10-4. Discrepancies can be the result of the following.

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117 Error due to the use of normal distribu tions for strains which may not exactly follow normal distributions. Error due to limited sample size of MCS while calculating the mean and standard distribution of strains. Error due to limited sample size of MC S while computing the probabilistic sufficiency factor PSF. Error associated with th e use of response surface approximations for PSF. Next, a plot for the probability of fa ilure (calculated via 1,000,000 MCS), weight and error reduction measures is shown in Fi g. 6-5. The optimum ply angles for the case with 20% error bound and no variability reduc tion are 25.59 and 25.5 3. Here we take both ply angles at 25. We note from Fig. 6-5 that for our problem, the error reduction is a more effective way of reducing weight compar ed to the specified variability reduction when the target probability of failure of the laminates is higher than 210-4 and quality control is more effective for lower probabilities. Figure 6-5. Trade-off plot for the probability of failure, design thic kness and uncertainty reduction measures. ER: error reduction (reducing from 20% to 10%), QC: quality control to -2 sigma.

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118 Choosing Optimal Uncertainty Reduction Combination Obviously, when it comes to a decision of what uncertainty reduction mechanisms to use, the choice depends on the cost of the uncertainty reduc tion measures. For a company, the costs of small error reduction may be moderate, since they may involve only a search of the literature for the best models available. Subs tantial error reduction may entail the high cost of doing additional research. Similarly, small improvements in variability, such as improve d quality control may entail using readily available nondestructive testing methods while large improvements may entail developing new methods, or acquiring expensive new equipmen t. To illustrate this, we assume a hypothetical cost function in quadratic form 223 CostAERBQC (6.8) where A and B are cost parameters, ER repres ents the error reduction and QC stands for the number of standard deviations that are th e threshold achieved by quality control. We generated hypothetical cost c ontours by using Eq. (6.8) as shown in Fig. 6-6. The nominal value of error is taken as 20% and we assume that the quality control to -3 sigma is associated with no cost. For example, if error is reduced from 20% to 15%, ER=0.200.15=0.05. Similarly, if quality control to -2.5 sigma is employed, then QC+3 takes the value of 0.5. As an example we take A=$20 million and B=$100,000. Next, we generated trade-off plot for probability of failu re and uncertainty reduction measures for laminates of thickness t1=0.010 in and t2=0.015 in as shown in Fig. 6-6. The optimum ply angles are calculated such that they minimize the probability of failure. The probabilities of failure are calculated via MCS (sample size of 106). The hypothetical cost contours for th e uncertainty reduction measur es given in Fig. 6-6 enable

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119 a designer to identify the optimal uncertainty reduction selection. We see in Fig. 6-6 that for high probabilities quality control is not cost effective, while for low failure probabilities quality control becomes more ef fective and a proper combination of error reduction and quality control leads to a minimum cost. Figure 6-6. Tradeoff of probability of failure and uncertainty reduction. Probabilities of failure are calculated via MCS (sam ple size of 1,000,000). The crosses in the figure indicate the optimal uncertainty reduction combination that minimizes the cost of uncertainty reduction for a specified probability of failure. Summary The tradeoffs of uncertainty reduction meas ures for minimizing structural weight were investigated. Inspired by the allocatio n of the risk between the components of a system for minimal cost, the optimal allocation of uncertainty as error and variability was analyzed. As a demonstration problem, the de sign of composite laminates at cryogenic temperatures is chosen because the design is very sensitive to uncertainties. Quality control was used as a way to reduce variability, and its effect was compared to the effect

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120 of reducing error in the analysis. Tradeoff plots of uncertainty reduction measures, probability of failure and weight were generated that would enable a designer to choose the optimal uncertainty reduction measure comb ination to reach a target probability of failure with minimum cost. For this specific example problem we observed the following Reducing errors from 20% to 10% led to 12% weight reduction Quality control to -2 sigma led to 20% weight reduction The use combined of error reduction and quality control mechanisms reduced the weight by 36%. Quality control was more effective at low required failure probabilities, while the opposite applied for higher requ ired probabilities of failure. In addition, a computational procedure fo r estimating the probability of failure based on approximating the cumulative distributi on functions for strains in a conservative manner was developed. We found that this appr oach led to more accurate probability of failure estimates than response surf ace approximations of the response.

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121 CHAPTER 7 OPTIMAL CHOICE OF KNOCKDOWN FA CTORS THROUGH PROBABILISTIC DESIGN Structural design of aircraft components still relies on deterministic (or code-based) approach governed by the Federal Aviation Ad ministration (FAA) regulations. The use of a load safety factor of 1.5 and conservative material properties in design accompanied with certification testing of aircraft are re quired to follow the FAA requirements. On top of the FAA requirements aircraft compan ies add their own knockdown factors, for example while updating the allowable stresses ba sed on the results of structural element tests. These knockdown factors are mostly ba sed on worst-case scenarios, so they are implicit rather than explicit and because of material variability they depend on chance. This paper aims to show, however, that these knockdown factor s can be selected explicitly by taking advantage of probabilistic anal ysis based on structural element test results. The knockdown factors can be chosen so as to minimize the chance of failure in certification or proof tests. We find that selection to minimi ze certification or proof test failure rate provides a choice that is also close to the optimum choice that minimizes structural failure in flight. We show that explicit knockdown factors can reduce weight for the same level of safety and they are also less variable than worst-case knockdown factors. In addition, the effect s of coupon tests, structural element tests and uncertainty reduction mechanisms (such as error reduc tion by improved structural modeling or improved failure prediction, variab ility reduction by tighter quali ty control) on structural weight are investigated. In part icular, since structural tests are expensive, the effect of

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122 number of tests on the struct ural weight is translated to life-time cost by considering manufacturing cost and fuel cost. Prof. Peter J. Ifju of University of Fl orida and Dr. Theodore F. Johnson of NASA Langley Research Center are acknowledged for their contributions for the research presented in this chapter. Introduction In Chapters 3 and 4, we analyzed the eff ects of measures that improve aircraft structural safety and compared the relative e ffectiveness of safety measures taken during aircraft structural design. The safety measur es that we included were the load safety factors of 1.5, conservative material proper ties, redundancy, certifi cation test, and error and variability reduction. The most common form of error redu ction is conducting structural element tests, which is the focu s of the present work. Similarly, one possible form of variability reduction is employing tight er quality control. Structural element tests are usually used conservativel y, by taking the worst resu lt of a batch of nominally identical tests. This constitutes an implicit knockdown factor beca use of material and test-condition variabil ity. Here we aim to show, however, that it is better to use average test results and add explicit safety factors selected by using probabilistic optimization to reduce certification or pr oof test failure rates. This chapter is organized as follows. The next section briefly discusses the building block approach followed in testing of aircra ft structures. Then, the quantification and simulation of uncertainties, the allowable stre ss updating using the results of structural element tests through explicit kno ckdown factors and formulation of total safety factor is discussed. Next, a brief discussion on simulati on of certification testing and probability of failure calculation are given. Then, the re sults of the optimal choice of knockdown

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123 factors based on minimum certification failure rate and minimum probability of failure along with the analysis of effects of safety measures on the optimal choice of knockdown factors are presented. Finally, the chapter cu lminates with the concluding remarks are given in the Summary section. Testing of Aircraft Structures Testing of aircraft struct ures is performed following a building-block approach similar to that of composite structures as s hown in Fig. 7-1. First, generic specimens are tested where coupon tests are followed by element tests. Then, non-generic specimens are tested where details, subcomponents and compon ents are tested. The last level of testing is the full scale certification te sting of the structural system. Figure 7-1. Building-block appro ach for aircraft structural testing (Reprinted, with permission, from MIL 17The Composite Materials Handbook, Vol. 1, Chapter 2, copyright ASTM Internati onal, 100 Barr Harbor Drive, West Conshohocken, PA 19428) Here we simplify the pyramid of tests depict ed in Fig. 7-1 to three levels as shown in Fig. 7-2. The first level is the coupon tes ting level, where coupons (i.e., material samples) are tested. The FAA regulation FAR 25-651 requires aircraft companies to perform enough tests to establis h design values of material st rength properties (A-basis

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124 or B-basis value). As the number of coupon te sts increases, the errors in the assessment the material properties are reduced. However, since testing is co stly, the number of coupon tests is limited to about 100 to 300 fo r A-basis calculation and 30+ (i.e., more than 30) for B-basis value calculation. In our analysis, the nominal value of the number of coupon tests is taken 40, and in the Results se ction of the chapter we analyze the effect of number of coupon tests on the optimal choice of the explicit knockdown factors. Figure 7-2. Simplified three level of tests At the second level of testing, structural elements a nd details are tested. In this chapter, we refer to tests at this at this level as structural element tests. The main target of structural element tests is to reduce errors related to failure theo ries (e.g., Tresca, von Mises) used in assessing the failure load of the structural details/elements. In this chapter, the nominal value of the number of st ructural element tests is taken as 3. At the uppermost level, certif ication testing of the overa ll structure is conducted. This final certification testing is intended to reduce errors in the structural analysis of the overall structure (e.g., errors in finite element analysis, errors in failure mode prediction).

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125 Quantification of Errors and Variability This section first discusses the errors in estimating the material strength properties due to limited number of coupon tests. Then, errors due to inaccuracies of the failure theories used to predict the failure load in the structural elemen t tests are introduced. Next, the explicit knockdown f actors employed on the allowabl e stresses using the results of structural element tests are discussed. Then, the errors in certification tests are presented. All the errors are combined to form a total error factor, and a total safety factor is defined. Finally, the variabilities in loadi ng, geometry parameters and failure stress are discussed. Errors in Estimating Material St rength Properties from Coupon Tests As noted earlier, the first level in testing sequence is coupon testing to assess the statistical characterizatio n of material strength properties, such as failure stress, and their corresponding design values (A-basis or B-ba sis). Since a finite number of coupon tests are performed, statistical charac terization of the material prop erties involves errors. For example, the error f ce in assessing the mean value of the failure stress at the coupon level, f c relates the average value of the failure stress calculated from nc coupon tests, fc calc, to the true mean value of the failure stress (mean of the infinite size population), fc true 1fcfcfc calctruee (7.1) where the subscript c stands for coupon level tests. The population average fc calc is estimated as the sample average ave fc obtained from nc coupon tests

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126 11c in ave f cfcfc calc c in (7.2) where i f c is the failure stress calculate d from each coupon test. The error f ce is due to variability of the failure stress, hence the standard deviation of fc calc (and of efc) is cn times smaller than the standard deviation of f c Details of assessment of the error f ce is given in Appendix J. No tice that in Eq. (7.1) the sign in front of the error f ce is negative, because we consistently formulate the expressions such that a positive error implies a conservative decision. The error f ce may have several components sinc e the failure of many materials is characterized by several parameters and related tests, such as tensile limit, compressive limit, shear limit, etc. However, we ma ke here a simplifying assumption that all components of f c have the same coefficients of va riation (c.o.v.) and are based on the same number of coupon tests, so they have the same error f ce, since f ce depends only on the c.o.v. and number of coupon tests (see Appendix J). The allowable stress at the coupon level, ac can be computed from the failure stress calculated at the coupon level, fc calc, by using a knockdown factor, dck as acdcfc calck (7.3) The knockdown factor dck is specified by the FAA regulations (FAR). For instance, for a redundant struct ure, FAR-25.613 states that the allowable stress must be the B-basis value of the failure stress, that is, 90% of the failure stresses (measured in coupon tests) must exceed the allowable stress with 95% confidence. The requirement of

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127 90% probability and 95% confidence is responsible for the knockdown factor dck in Eq. (7.3). For normal distribution, the knockdow n factor depends on the number coupon tests and the c.o.v. of the failure stress as 1dcBfc calckkc (7.4) where fc calcc is the c.o.v. of failure stress calc ulated from coupon tests. For lognormal distribution the formulation is more co mplicated, but can be derived utilizing a logarithmic transformation. The tolerance coefficient kB is a function of the number of coupon tests nc as given in the Composite Materi al Handbook (2000, Volume 1, Chapter 8, page 84) as 3.19 1.282exp0.9580.520ln()Bc ckn n (7.5) Similar to the error f ce, the knockdown factor kdc also has several components since f c has several components. With the assumption of the same c.o.v. and same number of coupon tests for different components of f c a single kdc can be used. Errors in Structural Element Tests The second level in the test ing sequence is the structural element testing level, where structural elements/details are tested to validate the a ccuracy of the failure criterion used (e.g., von Mises, Tsai-Wu). Here, we a ssume that structural element tests are conducted for a specified combination of loads corresponding to critical loading. For this load combination, the failure surface can be boiled down to a si ngle failure stress f e (see Fig. 7-3), where the subscript e stands for structural element tests.

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128 If the failure theory used to predict th e failure was perfect, and we performed thousands of coupon tests (that essentially reduces the error f ce to zero), then we could obtain the true element failure stress at the structural element test from fefefc truetrue (7.6) where ferefers to the failure criterion functi onal used (e.g., von Mises, Tsai-Wu). So, the calculated value of failure stress, fe calc, is fefefc calccalc (7.7) To relate fe calc to fe true we make a simplifying assumption that the failure criterion fe is a homogenous functional of or der one, such as von Mises or TsaiHill. Then, 1fefcfcfefc calctruee (7.8) Since there are errors in failure prediction in structural element testing level due to limitations of the failure theory used, we write 1fefcfefe truetruee (7.9) Combining Eqs. (7.7), (7.8) and (7.9), the calculated failure stress at the structural element level can be related to its true value as 11fefcfefe calctrueee (7.10) The structural element test is repeated en times yielding a sample average failure stress ave fe test, which is used to esti mate the population average fe test. The

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129 population average is now used to update the allowable stresses for the element design. We assume that similarly to the knockdown factor dck mandated by the FAA, the designer may add additional knockdown factor f ek to compensate for the uncertainty in the element test. The value of f ek is likely depend on whether the failure stress at the element test exceeds or falls below the predicted failure stress fe calc. That is, the allowable stress based on the element test is calculated from aefefe testk (7.11) Allowable stress updating and the us e of explicit knockdown factors This section describes updating of the allowable stress based on the results of structural element tests. First, we discu ss updating using worstcase approach, which amounts to implicit knockdown factors. Next, allowable stress updating through using explicit knockdown factors is e xplained. Then, updating of erro r in failure prediction in structural element tests is discussed Current industrial practice on updating allowable stresses using worst-case conditions (implicit knockdown factors) Current practice followed in industry is using the smallest of the failure stresses measured in structural element tests, worst fe test, is used to update the allowable stresses. We define the ratio of failure stress measured in the tests and calculated (or predicted) failure stress, fe calc, as worst fe worst test et fe calcr (7.12)

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130 Then, according to this practice th e allowable stress is updated as worstworst aecaeet updkr (7.13) where kc is an additional knockd own factor corresponding to worst-case operational conditions (e.g., high temperature, humidity). Note here that kc does not depend on the results of element tests, that is, ccetkkr. Figure 7-3 shows three aircraft companies A, B and C performing three structural element test s. Since we are interested in the implicit knockdown factor associated with worst-case conditions, we assume that all three companies use the same failure theory. That is, the difference in test results is entirely due to material variability Under these conditions, worst etr constitutes an implicit knockdown factor. For the examples in this pape r, we assume 8% variability in failure stress (lognormal distribution) This translates to an av erage knockdown fact or of 0.932 with a standard deviation of 0.0598. The re d knockdown factors in Fig. 7-3 corresponds to worst etr in Eq. (7.13), blue knockdown factor represents additional knockdown factor kc (notice in Figure 7-3 that kc does not depend on element test results). Proposal for a better way to update allowa ble stresses: Using the average failure stress measured in the te sts and using optimal explicit knockdown factors Instead of using the smallest failure stresse s measured in tests, we propose to use average value of the failure stresses measur ed in the tests accomp anied by an explicit knockdown factor selected based on probabilistic anal ysis. Now, we define the failure stress ratio in the structural elements, ave etr, as the ratio of the average failure stress measured in the element test a nd the calculated failure stress, fe calc, as

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131 Figure 7-3. Current use of knockdown factors based on worstcase scenarios. The red knockdown factors are due to updating the allowable stress using the worst failure stress measured in the test, and the blue knockdown factors are test independent and due to testing structur al elements at worst-case conditions (e.g., high temperature, high humidity). For a lognormal distribution and 8% coefficient of variation, the implic it knockdown factor is 0.932 with a standard deviation of 0.0598. fe ave test et fe calcr (7.14) The use of an explicit knockdown factor is required because a limited number of structural element tests are performed, so th e value of failure stress ratio ret can have substantial variability In addition, these knockdown fact ors may be used to reduce the likelihood of failing certification or proof te st. The updated failure surfaces with and without additional knockdown fact ors are depicted Fig. 7-4.

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132 Figure 7-4. Shrinkage of the failure surface. Original failure surface, updated failure surface without explicit knockdown f actor and the failure surface updated with an explicit knock factor kcl. Asterisk shows the failure at the structural element test. In general, we may expect that an op timal explicit knockdown factor will be a function of ret. Since we assume that the failure prediction error efe has zero mean, when ave etr is smaller or larger than one, it is likely that this is caused by variability. Since we increase the allowable stress when ave etr> 1, we run a chance that this increase is dangerous. So as we will see later, the optimal kc is smaller for ave etr>1 than for ave etr<1. We require that the explic it knockdown factor does not in crease so fast that the allowable stress may decrease with increasing ret. So for the allowable stress to be monotonic with respect to failure stress ratio ret we require 0ae upd caeet etetd d kr drdr (7.15) which leads to 0cc etetkdk rdr (7.16)

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133 Here we assume that kc( ret) has the simple form of two constants connected by a ramp, with the angle selected to satisfy Eq. (7.16) as depicted in Fig. 7-5. That is, 1.0 1 1.0 1clet et cclchcleth h chethkifr r kkkkifrr r kifrr (7.17) The parameter defining the transition interval, rh, is also taken constant in our analysis for the sake of simplicity. Our num erical studies showed that the use of rh = 1.10 is an acceptable value. Figure 7-5. The variation of the explicit knockdown factors with ratio of the failure stress measured in the test and calculated fa ilure stress with a nd without transition interval Figure 7-6 illustrates this approach for the three aircraft companies A, B and C performing three structural element tests shown earlier in Fig. 7-3. The red knockdown factors in Fig. 7-6 correspond to updating the allowable stress using the average failure stress measured in tests (Eq. (7.14)), while bl ue knockdown factor repr esents an explicit knockdown factor kc (notice in Fig. 7-6 that kc depends on element test results as depicted in Fig. 7-5).

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134 Figure 7-6. Proposed use of exp licit knockdown factors dependent on test results. The red knockdown factors are due to updating the allowable stress using the average failure stress measured in the test, and the blue knockdown factors are test dependent explicit safety factor. Error updating via element tests The main effect of structural element tests is to reduce the error in failure prediction at the element level, efe. The error efe after element tests are updated as follows. After the element tests, the calculated failu re stress is updated using tests results as upd feetfe calccalcr (7.18) Hence, the updated error efe can be calculated from 1111upd feetfe updini calccalc feetfe fefe truetruer ere (7.19) and the updated allowable st ress can be written as upd upd aefefe calck (7.20) where the total knockdown factor used in setting the allowable stress, f ek, is defined as f ecdckkk (7.21)

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135 Errors in Design For structural design, first the loads acting on the struct ure are calculated. For an aircraft structure, the design load Pd is computed by following the FAA design specifications (e.g., gust-strength specifications ). However, the calculated design load value, Pcalc, differs from the true design loading Pd. Since each aircraft company has different design practices, th e error in lo ad calculation, ep, is different from one company to another. The cal culated design load Pcalc is expressed in terms of true value Pd as (1)calcPdPeP (7.22) Here, we examine a small part of the overa ll structure, which can be characterized by its width, w and thickness, t If stress calculations are performed without errors, the true value of the stress in the structure is calc trueP wt (7.23) However, an aircraft company may also commit errors in stre ss calculation. The calculated value of stress, calc, can be expressed by introducing the error in the stress analysis, e, as (1)calc calcP e wt (7.24) If there were no errors in failure prediction of the overall structure, failure prediction at the structural element testing le vel, and failure stresses calculated at the coupon level, then the true failure st ress at the structural design level, fs true, could be related to true failure stress at element testing level, fe true via Eq. (7.25) (as in the

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136 case of relating the true failure stress at the el ement testing level to the failure stress at the coupon level, Eq. (7.6)). fsfsfe truetrue (7.25) where fs is the failure functional used to predict the overall structural failure in terms of the failure stress results of struct ural element test. Since the above mentioned errors do all exist, the failure stress calculated at structural design level, fs calc, differs from its true value, fs true. So, we have 1111upd fsfsfefcfefsfsfTfs calccalctruetrueeeee (7.26) where f se includes the errors in predicting the structural failure of the overall structure (e.g., errors in predicting the failure mode). As before, Eq. (7.26) assumes that the overall structural failure is a homogenous functional of order one in terms of the element failure stress. The allowable stress at the structural design level, as can be computed from the failure stress calculated at the element level, fs calc, by using the knockdown factor kfe as asfefs calck (7.27) Combining Eqs. (7.26) and ( 7.27), the allowable stress at the structural design level, as can be related to the true mean failure stress at the structur al design level as 1asfefTfs trueke (7.28)

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137 A structural designer uses Eq. (7.29) to calculate the design thickness tdesign required to carry the calculated design load, Pcalc, times the safety factor, SFL. That is, 11 1 1P FLcalcFLd design designas fTdesignfefs trueee SPSP te w ewk (7.29) where wdesign is the design width of th e part. Then, the design va lue of the load carrying area can be expressed as 11 1P FLd designdesigndesign fTfefs trueee SP Atw ek (7.30) Errors in Construction In addition to the above errors, there wi ll also be construction errors in the geometric parameters. These construction e rrors represent the difference between the values of these parameters in an average ai rplane (fleet-average) built by an aircraft company and the design values of these parameters. For the structural part, errors in width, ew, represent the deviation of the values of structural part wi dth designed by an individual aircraft company, wdesign, from the average value of the width built by the company, wbuilt. Thus, we have 1builtwdesignwew (7.31) Similarly, the built thickness value will differ from its design value such that 1builttdesigntet (7.32) Then, the built load carrying area builtA can be expressed as 11builttwdesignAeeA (7.33)

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138 Table 7-1 presents nominal values for the er ror factors. In the Re sults section of the chapter we will vary these e rror bounds and investigate the effects of reducing these errors on the built area, probability of failure, etc. Table 7-1. Distribution of error factors and their bounds Error factors Distribution Type Mean Scatter Error at the coupon level, f ce Normal 0 Std f cc n Error in failure prediction in structural element test, f ee Uniform 0 10% Error in failure prediction of the overall structure, f se Uniform 0 10% Error in stress calculation, e Uniform 0 5% Error in load calculation, eP Uniform 0 10% Error in width, ew Uniform 0 1% Error in thickness, et Uniform 0 2% Total Error Factor The expression for the built area, builtA of a structural part can be reformulated to Eq. (7.34) by combining Eqs. (7.30) and (7.33) as 1FLd builttotal fefs trueSP Ae k (7.34) where 1111 1 1Ptw total fTeeee e e (7.35) Here etotal represents the cumulative effect of the individual errors ( e, eP, ) on the load carrying area of the built structural part. Total Safety Factor The total safety factor, SF, of a structural part represents the effects of all errors and the safety measures on the built structural part. W ithout errors and safety measures, we would calculate the load car rying area by simply dividing the design load by the mean

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139 value of the true failure stress at the structural design level, fs true. That is, the load carrying area without safety measures, A0, is calculated from 0d fs trueP A (7.36) Then, the total safety factor can be defined as the ratio of Abuilt/ A0. Using Eqs. (7.34) and (7.36), we can write the total safety factor as 01built FL Ftotal f eA S Se Ak (7.37) Variability In the previous sections we analyzed the different types of errors committed during design and construction stages, representin g the differences between the true and calculated values of the fleet average of the material properties, the geometry parameters and the loading. These parameters (the materi al properties, the geometry parameters and the loading), however, vary from one aircraft to another in the fleet due to variabilities in tooling, construction, flying environment, etc. For instance, the actual value of the thickness of a structural part, tact, is defined in terms of its fleet average value, tbuilt, by ,actactbuiltttUtb (7.38) Here U indicates that the distribution is uniform, tbuilt is the average value of thickness (fleet average) and acttb defines the bounds for the variability in thickness. Table 7-2 shows that the scatter in tact is taken here to be 3%, that is t actb=0.03. Hence, the lower bound for thickness value is the average value minus 3% of the average and the upper bound for thickness va lue is the average value plus 3% of the average. Then, the actual thickness can be calculated from Eq. (7.39)

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140 1acttbuilttvt (7.39) where 21actttvbr represents effect of the variability on built thickness and r is a uniformly distributed random number between 0 and 1. Then, the actual load carrying area, Aact, can be defined as 11111d FL actactacttwbuilttwtotal fe fs trueP S AtwvvAvve k (7.40) where vw represents effect of the variability on built width, actwb= 0.01, and the second equality is obtained by using Eq. (7.34). Note that the thickness error in Table 71 is uniformly distributed with bounds of %. Thus the difference between all thicknesses over the fleets of all companies is up to %. However, the combination of error and variability which is the sum of two uniformly distributed components is not unifo rmly distributed. Table 7-2 presents the assumed distributions for variabil ities. The actual service loading Pact is assumed to follow extreme value distribution type I, since we consider the maximum (over a lifetime) loading. The failure stress is assumed to follow lognormal distribution. Table 7-2. Distribution of rando m variables having variability Variables Distribution type Mean Scatter Actual service load, Pact Extreme type I Pd = 100 10% c.o.v. Actual width, wact Uniform wbuilt 1% bounds Actual thickness, tact Uniform tbuilt 3% bounds Variability in built width, vw Uniform 0 1% bounds Variability in built thickness, vt Uniform 0 3% bounds Actual failure stress, fs Lognormal f s =150 0.08fc (i.e., 8% c.o.v.)

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141 Simulation of Certification Test an d Probability of Failure Calculation Simulation of Certification Test After a structural part is built with all the errors, variabilities and safety factors, we simulate certification testing for the structural part. That is, the struct ural part with crosssectional area of Aact, Eq. (7.40), is loaded with the design axial force of SF times Pcalc, and if the stress exceeds the failure stress f s then the structure fails and the design is rejected; otherwise it is certified for use. That is, the structural part is certified if the following inequality is satisfied 0FLcalc fsfs actSP A (7.41) Figure 7-7(a) shows the distri butions of the built and certi fied total safety factors. Notice that the structural parts designed with low total safety factors are likely to be rejected in the certification testing. The mean and standard deviations of built and certified total safety factor ar e listed in Table 7-3, which sh ows that the mean is increased and the standard deviation is reduced due to cer tification testing. Noti ce that the effect of certification test is very small. If there were no structural element tests, on the other hand, the effect would be more significant (Fig. 7-7(b)).

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142 0 0.4 0.8 1.2 1.6 2 11.522.53 Safety factor, SF Pdf built certified 0 0.4 0.8 1.2 1.6 11.522.53 Safety factor, SF Pdf built certified (a) with structural element tests (b) without structural elements tests Figure 7-7. Initial and updated distri bution of the total safety factor SF. (a) with structural element tests, (b) without structural element test. The distributions are obtained via Monte Carlo Simulations w ith 1,000,000 structural part models. Note that in (a) three st ructural element tests, forty coupon tests, and company safety factors kcl=0.9, kch=0.83 and rh=1.1 are used. See Tables 7-1 and 7-2 for error and variability data. Table 7-3. Mean and standard deviations of the built and certified distribution of the total safety factor SF. The calculations are performed with 1,000,000 MCS. Mean Std. dev. Built safety factor 1.861 0.193 Certified safety factor 1.871 0.188 Calculation of Probability of Failure To calculate the probability of failure first we incorporate the statistical distributions of errors and variability in a M onte Carlo simulation. Errors are uncertain at the time of design, but do not change for individual realizations (in actual service) of a particular design. On the other hand, all individual realizations of a pa rticular design are different from each other due to variability. Errors and variability could be simu lated through a double-loop Monte Carlo simulation (as in Chapter 3), in the upper lo op we could simulate different aircraft companies assigning random errors to each, a nd in the lower loop we could simulate variability in dimensions, material propertie s, loads related to manufacturing variability

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143 and variability in service conditions. However, this process would require trillions of simulations for good accuracy. In order to addr ess the computational burden we turned to the separable Monte Carlo procedure (e.g., Smarslok et al 2006), which consists of a single loop that comprises two stages. The first stage is fo r simulation of variabilities only, and the second stage is for simulating errors and te sts (coupon, structural element and certification). The discussi on on separable MCS is given in Appendix K. To achieve separable form, the failure condition is written as Failure without certification tests: 0FF reqbuiltSS (7.42) Failure with certification tests: 0FF reqcertSS (7.43) where F builtS and F certS are the built and certified total safety factors, and F reqS is the required safety factor necessary to account for the variabilities. For a given F builtS we can calculate the probability of failure, Eq. (7.42), by simulating all the variabilities with an MCS. Figure 7-8 shows the dependence of the probability of failure on the total safety factor using MCS with 1,000,000 variability sample s. Note that the probability of failure, Pf, presented here is the pr obability of failure of a structural pa rt built by a single aircraft company with a total safety factor of F builtS This probability of failure is different from the average probability of failure over all companies. We see from Fig. 7-8 that the nominal load safety factor of 1.5 is associat ed with a probability of failure of about 10-3, while the probabilities of failu re observed in practice (about 10-7) correspond to a total safety factor of about 2.4. The average probability of failure over all companies, PF, is

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144 calculated by performing Monte Carlo simula tions with first stage sample size of M =1,000,000 and second stage sample size of N =1,000,000. Figure 7-8. The variation of probability of failure of a structural part built by a single aircraft company. Note that Pf is one minus the cumulative distribution function of F reqS. See Tables 7-1 and 7-2 for e rror and variability data used for this case. Results In this section we first analyze the optim al choice of explicit knockdown factors for minimum weight, minimum likelihood of failure in certification (or proof) test, and minimum probability of failure under actual flig ht loads. Next, the effects of coupon tests and element tests on reducing the errors in failure predic tion are explored. Then, the effect of using different a pproaches and different formul ation for knockdown factors are analyzed. Finally, the effects of uncertainty reduction mechanisms (error and variability reduction), number of coupon test s and number of structural element tests on the optimal choice of knockdown factors and on the pr obability of failure are investigated.

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145 Optimal Choice of Explicit Knockdow n Factors for Minimum Weight and Minimum Certification Failure Rate Even though using smaller knockdown factors reduces th e likelihood of failure of the structure in the certification test (certification failure rate, CFR ), the weight of the structure increases accordingly. The cost of weight increase of aircraft structure is discussed in Appendix L. Since a company aims to have both minimal CFR and minimum weight, the optimal choice of kcl and kch is formulated here as a multi-objective optimization problem with two objective functions: the first being the built area Abuilt (or the built safety factor F builtS ) and the second is the certification failure rate CFR We seek to find the Pareto front, the curve of optimal trade-off between the two objectives. There are two popular ways to obtain th e Pareto front. On e is to optimize a weighted sum of the objectives for different combinations of weights. Another approach is to add one of the objective functions as th e constraint and change the constraint margin to generate the front. Here we follow the seco nd approach. We minimize the built safety factor, F builtS for a series of specified certificati on failure rates. So the optimization problem can be stated as 0 ,min suchthat ,1clchF built kk s pec clchAAS CFRCFR kk (7.44) Solving Eq. (7.44) requires calculation of CFR many times, which is computationally very expensive. To alleviate the computational cost, we estimate CFR and F builtS by using response surface approximations (RSA) for the relia bility index of CFR CFR which is defined as

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146 CFRCFR (7.45) where is the cumulative distribution function of the standard normal distribution. We use fifth-order polynomial RSAs for the built safety factor, F builtS and reliability index of CFR CFR The average (over all aircraft comp anies) probability of failure of a structural part PF is also approximated with RSA for its reliability index FP FPFP (7.46) The accuracy of RSAs is discussed in A ppendix M. Figure 7-8 depicts the tradeoff of safety ( CFR and corresponding PF) against weight (built safety factor) when 40 coupons tests and 3 structural element tests are employed. Each point in the trade-off plot is associated with a different kcl and kch combination as given in Table 7-4. We see from Fig. 7-9 that as lower CFR are specified, larger built safe ty factors are needed. We also see that as expected larger safety fact ors leads to lower fa ilure probabilities. Figure 7-9. Optimal choice of explicit knockdown factors kcl and kch for minimum built safety factor for specified certificat ion failure rate. The number of coupons tests is 40, and the number of structural element tests is 3. See Tables 7-1 and 7-2 for error and variability data.

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147 The optimal values of kcl and kch for minimum built safety factor for specified CFR are given in Table 7-4. When CFR is reduced, smaller knockdown factors kcl and kch are used. We also notice that as expected kch is smaller than kcl. This may appeal to a designer when the tests allow weight reduction trading some of the weight for some extra margin as insurance against variability in the results of st ructural element tests. The monotonicity constraint, Eq. (7.16), is active for CFR =10-3 and 10-4, indicating that the parameter defining the tran sition interval, rh, is important. Investigation of the effect of this parameter is left for a future work. The comparison of the certification failu re rates and probabilities of failure obtained from RSAs and calcula ted using MCS of 1,000,000 sample size is also given in Table 7-4. We see that the values obtained from RSAs are very cl ose to MCS values and the differences are within the limits of MCS e rror due to finite sample size. For example, while predicting a probability of failure 8.14-5, two-stage separable MCS with 106 samples (for each stages) is equivalent to about 108 crude MCS, so the error for predicting PF of 8.14-5 within two standard deviation is about 2.2%, which larger than the actual error 1.6% that we see in the first row of Table 7-4. We also see in Table 7-4 that as CFR (or PF) reduce the error grows. There are tw o contributions to the error: (i) error due to limited MCS sample size (which grows as CFR reduces), (ii) error in RSA (that can be high if RSA is performed near design boundaries). However, even for CFR =10-4, the error in CFR is only 6%.

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148 Table 7-4. Comparing explicit knockdown fact ors for minimum built safety factor for a specified certification failure rate. Note that CFR and PF are calculated using MCS of 1,000,000 sample size, while CFR-RS and PF-RS are obtained from the RSAs. CFR-RS CFR A/A0 kcl kch PF-RS PF 10-1 9.95-2 1.750 1.000 0.963 8.01-5 8.14-5 10-2 1.04-2 1.987 0.892 0.820 4.09-6 4.25-6 10-3 1.03-3 2.193 0.809 0.742 3.46-7 3.58-7 10-4 9.40-4 2.372 0.747 0.687 3.99-8 4.29-8 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built Optimal Choice of Explicit Knockdow n Factors for Minimum Weight and Minimum Probability of Failure Instead of designing the structure for minimum certification failure rate, the structure can be designed for minimum pr obability of failure (i.e., performing probabilistic optimization). In that case the optimization problem can be stated as 0 ,min suchthat ,1clchF built kk FF s pec clchAAS PP kk (7.47) where F s pecP is the specified probability of failu re. The Pareto front can be obtained by varying the specified probability of failure, F s pecP. Surprisingly Fig. 7-10(a) shows that the CFR of the structures designed for minimum CFR and CFR of the structures designed for minimum PF are very close to each other. Th e same observation is also true for the PF of the structures designed either for minimum CFR or for minimum PF (Fig. 710(b)). So choosing the explicit knockdown factors to minimize the failure in certification test offers a rational way of choosing the explicit knockdown factor.

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149 (a) CFR comparison (b) PF comparison Figure 7-10. Comparing CFR and PF of the structures designed for minimum CFR and minimum PF (a) Comparison of CFR of the structures designed for minimum CFR and minimum PF, (b) Comparison of PF of the structures designed for minimum CFR and minimum PF The optimal values of kcl and kch for minimum built safety factor for specified PF are given in Table 7-5. We s ee the trend in the explicit knockdow n factor is similar to that given in Table 7-4 in that kcl is larger than kch. We see that CFR s and PFs obtained from RSAs and MCS of 1,000,000 sample size are very close to each other. So from this point on we only present the CFR and PF values obtained from RSAs.

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150 Table 7-5. Comparing explicit knockdown fact ors for minimum built safety factor for a specified probability of failure PF-RS PF A/A0 kcl kch CFR-RS CFR 10-4 1.00-5 1.739 1.000 0.984 1.12-1 1.11-1 10-5 1.04-5 1.914 0.926 0.853 2.13-2 2.21-2 10-6 1.03-6 2.105 0.843 0.773 2.80-3 2.89-3 10-7 1.06-7 2.295 0.773 0.709 2.75-4 2.73-4 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built Effect of Coupon Tests and Structural Elem ent Tests on Error in Failure Prediction As noted earlier, coupon tests and structur al element tests contribute to structural safety mainly by reducing the error in failure prediction. Here we analyze the effect of number of these tests on reducing the error in failure prediction. First, we analyze the effect of number of coupon tests alone for a fixed number of element tests (here we take the fixed number of element tests as three). Th en, the effect of structural element tests are explored for a fixed number of coupon tests (here we take fixed number of coupon tests as forty). Effect of number of coupon tests alone (for a fixed number of element tests, ne=3) The effect of number of coupon tests, nc, on error in failure prediction at the coupon level, efc, and total error in failure prediction, efT, is depicted in Fig. 7-11(a) and 7-11(b), respectively. Here we consider increasing nc from 10 to 40 and from 40 to 100. Note that nc=10 is not realistic, but include d here for the sake of illustra tion. We see in Fig. 7-11(a) that as nc is increased from 10 to 40, and then from 40 to 100, standard deviation of efc is reduced significantly. The effect of nc on efT, on the other hand, shows a different trend (Fig. 7-11(b)). Even though we see a signifi cant reduction of standard deviation of efT when nc is increased from 10 to 40, the ch ange of standard deviation of efT is barely noticeable when nc is further increased to 100.

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151 (a) error in failure prediction at the coupon level, efc (b) total error in failure prediction, efT Figure 7-11. Effect of number of coupon tests on the error in failure prediction for a fixed number of element tests (3 element tests). The probability densities are obtained through MCS of 10,000 sample size. Effect of number of element tests alon e (for a fixed number of coupon tests, nc=40) The effect of number of element tests, ne, on error in failure prediction at the element level, efe, and total error in failure prediction, efT, is depicted in Fig. 7-12(a) and 7-12(b), respectively. Here we consider no element tests, a single element test, three element tests and five element tests. Note that the scatter in error efe is smaller when no element test is performed compared to pe rforming a single element tests. That is, performing only a single element test is not effective due to variab ility in the failure stress. We see that as ne is increased from 1 to 3, and then from 3 to 5, standard deviation of efe is reduced significantly (F ig. 7-12(a)) and the mean error is increased. The increase of the mean error makes sense since it indica tes the tendency towards conservative error. The effect of nc on efT, on the other hand, shows a different trend (Fig. 7-12(b)) in view of standard deviation. Even though we see a sign ificant reduction of standard deviation of efT when ne is increased from one to three, th e reduction of standard deviation of efT diminishes when ne is further increased from three to five.

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152 (a) error in failure prediction at the element level, efe (b) total error in failure prediction, efT Figure 7-12. Effect of number of element te sts on the error in failure prediction for a fixed number of coupon tests (40 coupon te sts). The probability densities are obtained through MCS of 10,000 sample size. Figure 7-12 showed us that performing a single structural element test is not enough to reduce the error in fail ure prediction due to variability in failure stress. One way to solve this deficiency is to use Bayesi an updating that can be successfully used to update the error distribution. Figure 7-13(a) shows the evolution of mean failure stress using the test results. Initial mean stress di stribution is taken unifo rm with % bounds around mean value, which is 150. When a single test is performed, due to the variability in failure stress, the updated distribution has a wider range compared to the initial one. The evolution of the mean stress requires more element tests to be performed as seen in Fig. 7-13(a). On the other hand, Bayesian updating allows a mo re effective updating process. We see in Fig. 7-13(b) that even a single test is effective now. Discussion of Bayesian updating is not analyz ed in this chapter, but more information on error updating using Bayesian updating can be found in Chapter 3.

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153 (a) without Bayesian updati ng (b) with Bayesian updating Figure 7-13. Evolution of the mean failure stress distributi on with and without Bayesian updating Advantage of Variable Explicit Knockdown Factors To see the advantage of using a variable explicit knockdown factor that depends on the test results, we compare its Pareto front with that of a cons tant explicit knockdown factor. Figure 7-14 shows comparison of Pare to fronts for certification failure rate and normalized area (Fig. 7-14(a)) a nd Pareto fronts for probability of failure (Fig. 7-14(b)). Table 7-6 presents the knockdown factors and the normalized areas. We see that for a specified CFR or PF, lower weight (area) is require d if a variable explicit knockdown factor is used. For instance, for CFR of 10-3, using variable explicit knockdown factor leads to 0.77% lighter stru cture, which corresponds to about $520,000 life-time cost saving for a typical large tran sport aircraft such as Boei ng 777 (see Appendix L for the cost model used).

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154 (a) CFR comparison (b) PF comparison Figure 7-14. Comparison of variable a nd constant explicit knockdown factor Table 7-6. Comparison of constant and variable explicit knoc kdown factors case and corresponding area ratios, A/A0. Constantaverage Variable (Eq. (7.17)) CFR A/A0 kcl=kch A/A0 kcl kch 10-1 1.752 0.988 1.750 1.000 0.963 10-2 2.008 0.862 1.987 0.892 0.820 10-3 2.211 0.783 2.193 0.809 0.742 10-4 2.413 0.717 2.372 0.747 0.687 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built We next examine the effect of using th e implicit knockdown factor associated with worst (i.e., smallest) failure stress measured in structural element tests instead of using the average. The comparison of Pareto fronts of certification failure rate and total safety factor is shown in Fig. 7-15. The knoc kdown factors and total failure stress corresponding to these two approa ches are shown in Table 7-7. We find that the use of the implicit safety factor carries a weight pena lty. For example, for a certification failure rate of one in a thousand, the use of the sma llest of the failure stress measured in element tests leads to 0.95% heavier structure compar ed to the use of the average failure stress measured in element test. The cost function in Appendix L indicates that the cost of structure is increased by about $640,000. The ratio of th e knockdown factors

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155 corresponding to use of average and worst cas e failure stress result s are also listed in Table 7-7 (the last column). We see that inst ead of using a worst-ca se failure stresses, a company might use average failure stresses measured in element tests and by adding a knockdown factor on top of it more efficient design decisions can be made. The combined effect of using a variab le explicit knockdown factor, Eq. (7.17), along is seen by comparing results gi ven in Tables 7-6 and 7-7. For CFR =10-3 for instance, we see that 1.75% of structural weight can be corresponding to lifetime cost reduction of about $ 1.2 million. (a) comparing constant explicit and constant implicit knockdown factors (b) comparing variable explicit, constant explicit and constant implicit knockdown factors Figure 7-15. Comparison of Pareto fronts of certification failure rate and built safety factor for two different approaches while updating the allo wable stress based on failure stresses measured in element tests. One approach uses the smallest of the failure stresses measured in te sts while the other used the average failure stress while updati ng the allowable stresses.

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156 Table 7-7. Comparison of constant (i.e., test independent) implicit and explicit knockdown factors and corres ponding area ratios A/A0. One approach uses the smallest of the failure stresses measured in the tests while the other used the average failure stress while updating the allowable stresses. Constant explicit knockdown factor Constant implicit knockdown factor Ratio of knockdown factors CFR A/A0 kc=kcl=kch A/A0 kc=kcl=kch (kc)exp/(kc)imp 10-1 1.752 0.988 1.856 1.000** 0.989** 10-2 2.008 0.862 2.017 0.921 0.939 10-3 2.211 0.783 2.232 0.832 0.942 10-4 2.413 0.717 2.460 0.755 0.951 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built ** Note that using kc=1.0 leads to CFR=0.049 so the ratio is different compared to other CFR. The main reason that the variable explicit safety factor is more efficient is that it reduces the element of chance introduced by material variability on the total knockdown factor. For lognormal failure stress with 8% coefficient of variation used here, the average implicit knockdown fact or has a mean value of 0.932, and a coefficient of variation of 0.0642. The average failure stress, on the other hand, is 1.0 (as expected) and its coefficient of variation is 0.0467. That is the average is 27% le ss variable than the worst value. In addition, the variable knockdow n factor reduces the s catter further since it is higher for high test results than for low test results. This is demonstrated by calculating the mean and coefficient of variation of the actual knockdown f actor obtained from a Monte Carlo simulation of one million comp anies. Table 7-8 shows that using test dependent (i.e., variable, Eq. (7.17)) knockd own factor further indeed reduces the coefficient of variation of the knockdown factor. For CFR =10-3, for instance, the total reduction in the coeffici ent variation is 36%.

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157 Table 7-8. Comparison of mean and coeffici ent of variation of total knockdown reduction at the element test level for the cases of implicit constant knockdown factor and explicit variable knockdown factors A) Total knockdown factor for use of worst failure stress and constant additional knockdown B) Total knockdown factor for use of average failure stress and variable (i.e., test dependent, Eq. (7.17)) additional knockdown Ratios (A / B) Ratios (A / B) CFR mean c.o.v. mean c.o.v. mean c.o.v. 10-2 0.859 0.0639 0.871 0.0409 1.02 0.64 10-3 0.776 0.0642 0.790 0.0409 1.02 0.64 10-4 0.704 0.0642 0.730 0.0394 1.04 0.61 Effect of Other Uncertainty Reduction Mechanisms This section provides the e ffects of other uncertainty reduction mechanisms on and the optimum values of the explicit knockdown factors kcl and kch, the probabilities of failure and the certification failure rates. Effect of variability reduction Recall in Chapter 4, we found that the reduction va riability in the fa ilure stresses is a very efficient way of reducing probability of failure, but it leads to increased certification failure rates. In that chapter, we argued that when companies reduce the variability in failure stresses, they also must employ additiona l knockdown factors to reduce the increased certification failure rates. In a view to analyze that argument, we explore the effect of reduced variability on the optimal choice of kcl and kch. We reduce the coefficient of variation of the failure stre ss by half of its nominal value, that is, we reduce it from 8% to 4%. We kept the numbe r coupon tests and the number of structural element tests at their nominal values 40 and 3, respectively. Figure 7-14 shows that as the coefficient of variation is reduced to 4%, the Pareto fronts move to the left, allowing the use of smaller built safety factor for specified CFR

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158 and PF, or alternatively allowing smaller CFR and PF for a given built safety factor. We see that the variability reduction is most e ffective at low probability of failure (and certification failure rate). Fi gure 7-16 also shows that CFR is more sensitive to variability reduction than PF. The optimal values of the e xplicit knockdown factors for minimum CFR are given in Table 7-9. We see that the reduced variability allows using larger (i.e., less conservative) knockdown factors kcl and kch for the same CFR In Chapter 4, we found the opposite trend when element tests we re not performed. That is, without the element tests variability reduction needed to be accompanied by using more conservative knockdown factors for acceptable cer tification failure rates. T hus the reduced uncertainty due to structural element tests allows us to take fuller advantage of variability reduction. Table 7-9 shows that, for a certification fa ilure rate of one in a thousand, the variability reduction allows 11.9% lower tota l built safety factor That is, the same certification failure rate can be attained with 11.9% lighter structure if the variability in failure stress can be reduced by half. The co st model in Appendix L indicates us that variability reduction leads to saving about 8 million dollars. Hence, the company might compare the cost of variability reduction (reducing the coefficient of variation of the failure stress by half) with the cost saving due to reduced structural weight and decide whether paying more for more uniform material is worth it.

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159 (a) minimum CFR (b) minimum PF Figure 7-16. Reducing (a) certific ation failure rate and (b) probability of failure using variability reduction. The number of coupons tests is 40, and the number of structural element tests is 3. Table 7-9. Optimal explicit knockdown factors for minimum CFR when variability in failure stress is reduced by half c.o.v.=8%* c.o.v.=4% CFR A/A0 ** kcl kch A/A0 ** kcl kch 10-1 1.750 1.000 0.963 1.657 0.964 1.000 10-2 1.987 0.892 0.820 1.811 0.905 0.850 10-3 2.193 0.809 0.742 1.932 0.845 0.805 10-4 2.372 0.747 0.687 2.028 0.809 0.757 From Table 7-4 ** A/A0 is the ratio of the built cross sectional ar ea, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built Effect of error reduction Similar to variability reduction, error reduction is anothe r powerful way of reducing certification failure rate and probability of failu re. We consider a hypothetical error control mechanism that would reduce all the errors by half. So we scale all error components with a single multiplier, k replacing Eq. (7.35) by 1111 1 111Ptw total fcfefskekekeke e kekeke (7.48)

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160 and explore the effect of reducing it from its nominal value 1.0 to 0.5. Figure 7-17 shows that indeed error reduction is also effective in reducing CFR and PF. We also see that variability reduction is more effective than error reduction, especially when CFR (and PF) are low. Note also that CFR is more sensitive to error reduction than PF. The company safety factors for error reduction are listed in Table 7-10, which shows that as error is reduced a specified certification failure rate can be attained by using lower built safety factors. For instance, for certi fication failure rate of one in a thousand, we see that the error reduction leads to 4.9% lower total safety factor. That is, the same certification failure rate can be attained with 4.9% lighter weight if the errors are reduced by half. The cost model in Appendix L indicates us that error reduction leads to about 3.3 million dollars cost saving. Hence, the company might compare the cost of error reduction with the cost saving due to reduced structural we ight and decide whethe r investing resources on error reduction mechanisms is profitable. The superior effect of variability reduction over error reduction is all the more remark able, since only one component of the variqabilities was reduced, compared to all errors. Figure 7-17. Reducing certificat ion failure rate using error reduction, variability reduction and combination of error and variab ility reduction. The number of coupons tests is 40, and the number of structural element tests is 3.

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161 Table 7-10. Optimal explicit knockdown factors for minimum CFR when all errors reduced by half. k=1.0 k=0.5 CFR A/A0 kcl kch A/A0 kcl kch 10-1 1.750 1.000 0.963 1.728 1.000 1.000 10-2 1.987 0.892 0.820 1.906 0.923 0.846 10-3 2.193 0.809 0.742 2.084 0.844 0.774 10-4 2.372 0.747 0.687 2.241 0.785 0.720 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built Effect of Number of Coupon Tests Next, we analyze the effect of number of coupon tests on the optimal choice of knockdown factors. Figure 7-18 shows the Pare to fronts corresponds to three different numbers of coupon tests: 10 test s, 40 tests and 100 tests. Note here that performing only 10 coupon tests is not realistic, but it is included here for il lustrative purposes. We see that increasing number of tests from 10 to 40 leads Pareto front to shift to left substantially, whereas the Pareto fronts for 40 coupon te sts and 100 coupon tests are quite close. Table 7-11 shows that the company sa fety factors (and hence the total safety factor) corresponding to 40 coupon tests and 100 coupon tests are very close. The optimal kcl and kh values are more conservative (that is, smaller) when 100 coupon tests are used for small CFR s. This is because as the number of coupon tests increases the B-basis value grows, so the total safety factor reduces. Th en the company needs to compensate for that by introducing explicit knockdown factors to reach a desired le vel of certification failure rate. When CFR is large (e.g., CFR =0.1), on the other hand, the error reduction due to large number of coupon tests overcomes the change in B-basis. Table 7-11 shows that, for certif ication failure rate of 10-3, the total built safety factor can be reduced by 1.3% if 100 coupon tests are performed rather than 40. The cost model given in Appendix L indicates that these additional coupon te sts leads to about

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162 $890,000 cost saving. Hence, the company migh t compare the cost of extra coupon tests with the cost saving due to reduced struct ural weight and decide whether performing these additional coupon tests is profitable. (a) minimum CFR (b) minimum PF Figure 7-18. Optimal explicit knockdown factors for different number of coupon tests for minimum CFR and PF. The number of structural elem ent tests is 3. Number of coupon tests is indicated as nc in the figure. Table 7-11. Optimal explicit knockdown factors for minimum CFR different number of coupon tests, nc. nc=40 nc=100 CFR A/A0 kcl kch A/A0 kcl kch 10-1 1.750 1.000 0.963 1.748 0.996 0.934 10-2 1.987 0.892 0.820 1.978 0.885 0.814 10-3 2.193 0.809 0.742 2.165 0.809 0.742 10-4 2.372 0.747 0.687 2.328 0.752 0.691 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built Effect of Number of Structural Element Tests Finally, we explore the effect of structural element test s. Figure 7-19 shows that as the number of tests increases from one to thr ee, the probability of failure and certification failure rates are reduced by a bout one order of magnitude. However, as the number of tests further increased to five, for instance, the effectiveness of tests reduces. Table 7-12 shows that as the number of element tests increases, the knockdown factors kcl and kch

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163 increase and hence the total safety factor reduce. That is, the additional element tests possess a hidden safety factor by reducing the e rror in failure prediction. For example, for certification failure rate of 10-3, the total built safety factor must be increased by 3.0% if only one structural element test is performed rather than th ree structural element tests. According to the cost model given in Appendi x L, performing three st ructural tests leads to about 2 million dolla rs cost saving. Hence, the compan y might compare the cost of two extra structural elements with the cost saving due to reduced structur al weight and decide whether these additional element tests are preferable. (a) minimum CFR (b) minimum PF Figure 7-19. Effect of number of structural element tests, ne Table 7-12. Optimal explicit knockdown factors for different number of structural element tests, ne. ne = 1 nb = 3 CFR A/A0 kcl kch A/A0 kcl kch 10-1 1.774 1.000 0.936 1.750 1.000 0.963 10-2 2.038 0.876 0.803 1.987 0.892 0.820 10-3 2.259 0.790 0.724 2.193 0.809 0.742 10-4 2.586 0.675 0.667 2.372 0.747 0.687 A/A0 is the ratio of the built cross sectional area, A, and the area without safety measures, A0. Note that the area ratio is equal to the total safety factor (SF)built

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164 Summary The effects of optimal choi ce of explicit knockdow n factors, structural tests, and uncertainty reduction mechanisms were analyzed From the results obtained in this study, we drew the following conclusions Companies can minimize the probabilities of the failure of the aircraft structures by rationally choosing explicit knockdown fact ors that are based on test results. Currently these knockdown f actors are implicit and base d on worst-case scenarios (e.g., testing the structural elements at worst-case operational conditions, using the smallest measured failure stress in design) We found that optimally chosen explicit knockdown factors may lead to 1 or 2% we ight savings, which can be translated into cost savings in the order of a million dollars over the lifetime of a typical airliner. Surprisingly, we found that a lower (i.e ., more conservative) knockdown factor should be used if the failure stresses m easured in tests exceeds predicted failure stresses, because good test resu lts can simply be due to luck. The use of variable explicit safety factor based on averag e test results was found to reduce the variability in knockdown factor generated by variability in ma terial properties by about 36%. It is th is reduction in variability that is responsible for the weight savings. Selecting explicit knockdow n factor to minimize certification failure rate, CFR leads to designs very near to probabilistic optimum (minimum probability of failure, PF). Uncertainty reduction mechanisms are powerful ways of increasing aircraft structural safety. They reduce CFR and PF by one or more orders of magnitudes for a given safety factor. When the effectiv eness of error reduction and variability reduction is compared, we found that reducing variability in failure stress to half of its nominal value is more effective than reducing all errors by half. We found that the efficiency of tests depends on the number of tests conducted. As the number of tests increases, the cost effectiveness of tests diminishes. Given a cost function for coupon tests, struct ural element tests, error and variability reduction mechanisms, the optimum choice of explicit knockdow n factors can be carried out in a rational way to minimize the lifetime cost of aircraft without changing the aircraft structures probability of failure in the certification tests and under the actual flight loads.

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165 CHAPTER 8 RELIABILITY BASED AIRCRAFT STRUCT URAL DESIGN PAYS EVEN WITH LIMITED STATISTICAL DATA Probabilistic structural desi gn tends to apply higher safety factors to inexpensive or light-weight components, because it is a more efficient way to achieve a desired level of safety. In this chapter we show that even with limited knowledge about stress probability distributions we can increase the safety of an airplane by followi ng this paradigm. The structural optimization for safety of a repr esentative system composed of a wing, a horizontal tail and a vertical tail is used to demonstrate the paradi gm. In addition, to alleviate the problem of computational expe nse we propose an approximate probabilistic design optimization method, where the probabili ty of failure calculation was confined only to failure stresses to dispense with mo st of the expensive structural response calculations (typically done via finite elem ent analysis). The proposed optimization methodology is illustrated with the de sign of the wing and tail system. The work presented in this chapter is also submitted for publication (Acar et al 2006e). Prof. Efstratios Nikolai dis is acknowledged for his va luable suggestions for this work. Introduction The FAA design code is based on uniform safety factors (that is, the same safety factor is used for all compone nts). Probabilistic design derive s an important part of its advantage over deterministic design by allo wing the use of substantially non-uniform safety factors and hence there is growing interest in replacing safety factors by

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166 probabilistic design (e.g., Li ncoln 1980, Wirsching 1992, Aerospace Information Report of SAE 1997, Long and Narciso 1999). Howeve r, with only partial information on statistical distributions of variabilities, a nd guesswork on reasonab le distributions for errors, engineers are reluctant to pursue proba bilistic design. Also, it has been shown that insufficient information may lead to large e rrors in probability calculations (e.g., BenHaim and Elishakoff 1990, Neal, et al. 1992). Th e main objective of this chapter is to show that we can increase the safety of an airplane without increas ing its weight even with the limited data available today follo wing the design paradigm mentioned (higher safety factors for light-weight components). Our approach utilizes two statistical data that are well understood. The first is the statistical di stribution of failure stress, which is required by the FAA for choosing A-basis or B-basis allowabl es. The second is a special property of the normal distribut ion: when large number of un certainties contributes to a distribution, it tends to become similar to a no rmal distribution. This applies to the stress estimation because it is influenced by larg e number of error and variability sources. Finally, we show that even though the limited st atistical data may subs tantially affect the probabilities of failure of both the probabilis tic design and the code -based deterministic design, the ratio of probabilitie s of failure of the probabilistic design and the deterministic design is insensitive to even la rge errors due to limited data. The chapter is structured as follows. First, reliability-based design optimization of a representative wing and tail system with perfect and limited statistical data is given. Next, we discuss the effects of erro rs in statistical information for the deterministic design. Then, we propose an approximate method that allows probabilistic design based only on

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167 probability distribution of failu re stresses, followed by the application of the method to the wing and tail system. Finally, the concl uding remarks are listed in Summary section. Demonstration of Gains from Reliability-Based Structural Design Optimization of a Representative Wing and Tail System Problem Formulation and Simplifying Assumptions Calculating the probability of stress failure in a struct ure can be done by generating the probability distribution functions (PDF) () sof the stress and the PDF () f f of the failure stress f (see solid lines in Fig. 8-1). Once these distributions are available, calculating the probability of failure can be accomplished by simple integration, as discussed later. The distributi on of failure stress is typically available from experiments, and so it does not require much computation. For materials used in aircraft design, the FAA regulations mean that statistical information on failure stresses is often available quite accurately. On the other hand, the PDF for the stress requires data such as analysis error distributions that are difficult to estimate, and it al so requires expensive finite element computations. Fortunately, though, the PDF of the stress contains contributions from large number of parameters such as va riabilities and errors in material properties, geometry and loading, and so it is likely to be well represented by a normal distribution. However, estimating well the mean and standa rd deviation of that normal distribution is difficult due to limited data. First, we momentarily disregard this difficulty in order to demonstrate the advantage of probabilistic design deriving from the use of higher safety factors for lighter structural components. Then the effects of limited statistical data will be addressed. We consider a representative wing and tail system. In general, in order to perform reliability-based design, we need to re-calcu late the stress PDF as we change the design.

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168 For the sake of simplicity, we assume that st ructural redesign changes the entire stress distribution as shown in Fig. 8-1 by a simple scaling of to (1+ ). This assumption will be accurate when the uncertainties are in the loading, and the relative errors in stress calculation are not sensitive to the redesign. Figure 8-1. Stress distribution s ( ) before and after redesign in relation to failure-stress distribution f ( f). We assume that re-designs scales the entire stress distribution. We denote the failure probabilities of wi ng and tail obtained from deterministic design by ( PfW)d and ( PfT)d, respectively. If failure of th e two components is uncorrelated, the probability that at least one of them will fail is Pf0 = 1-[1 ( PfW)d][1 ( PfT)d] (8.1) If the two failure probabilities are correlate d, the calculation is still simple for a given correlation coefficient. For the purpos e of demonstration, we make simplifying assumptions that the structure is approximately fully stressed, and that the stresses are inversely proportional to weight. This allows us to treat the wing and the tail as a system with a single stress level and perform the demonstratio n without resorting to finite element modeling and analysis of these two component s. That is, denoti ng the stresses in the wing and the tail by W and T, respectively, we use

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169 dW WdW WW W dT TdT TW W (8.2) where WW and WT are the wing and the tail structur al weights, respectively, and the subscript d denotes the values of stresses and st ructural weights for the deterministic design. Using Eq. (8.2) we can now formul ate the following probabi listic design problem to minimize the probability of failure for constant weight ,min,111 suchthatWTfWTfWWfTT WW WTdWdTPWWPWPW WWWW (8.3) In the following subsection, we first perf orm probabilistic optimization for safety of the wing and tail system. Next, the effect of adding a vertical tail to the wing and horizontal tail system on the overall safe ty enhancement will be explored. Probabilistic Optimization with Correct Statistical Data For a typical transport aircraft, the structural weight of the horizontal tail is about 20% of that of the wing. So the weights of the wing and the tail before probabilistic optimization are taken as 100 and 20 units, re spectively. We assume that wing and the tail are built from the same material and the failure stress of the material follows lognormal distribution with a mean value f of 100 and coefficient of variation (c.o.v), f c, of 10%. The c.o.v. of the stresses in the wing and the tail, c is assumed to be 20%. This may appear large in that stress calcu lation is quite accurate. However, there is substantial uncertainty in loading and geometry changes due to damage. For illustrative purpose, we assume that the historical record showed that the wing had lifetime probability of failure of 110-7. Since the deterministic design uses uniform safety factors (that is, the same safety factor is used for all components), and we assume that the wing

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170 and tail are made from the same material and have the same failure mode (point stress failure here), it is reasonable to assume that the probability of failure for the deterministic design of the tail is also 110-7. As indicated earlier, we as sume that the stresses follow normal distribution, which is characterized by only two parameters, mean and standard deviation. Therefore, given full informati on on the failure-stress distribution (lognormal with f =100 and f c=10%), the probability of failure (Pf=10-7) and the c.o.v. of the stress (c =20%), the mean stresses in the wing and the tail are calculated as 39.77. The reader is referred to Appendix N for details of calculation of the unknown mean stresses in the wing and the tail for given probability of failure and c.o.v. of the stresses. With the simple relation of stress to weight, Eq. (8.2), and the assumptions on the distributions of stresses and fa ilure stresses, the probability of failure can be easily calculated by a variety of methods. One of th ese methods will be discussed later. Then the probabilistic design optimization is so lved assuming a zero correlation coefficient between the probabilities of failure of wing and tail. We see from Table 8-1 that the probabilistic design and deterministic design are very close in that probabilistic design is achieved by a small perturbati on of deterministic design (moving 0.75% of wing weight to tail, see columns 2 and 3). Table 8-1 show s that by moving 0.75% of the wing material to the tail, the probability of failure of the wing is increased by 31% of its original value. On the other hand, the weight of the tail is increased by 3.77% and thereby its probability of failure is reduced by 74%. The overall pr obability of failure of the wing and tail system is reduced by 22%.

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171 Table 8-1. Probabilistic struct ural design optimization for sa fety of a representative wing and tail system. In the optimization only the mean stresses are changed, the c.o.v. of the stresses are fixed at c.o.v. =0.20. Probability of failure of the wing and the tail for deterministic design are both 110-7. W0 W Pf ratio (a) Mean stress before optim. Mean stress after optim. Wing 100 99.25 1.309 39.77 40.07 Hor. Tail 20 20.75 0.257 39.77 38.32 System 120 120 0.783 (a) Pf ratio is the ratio of the probabilities of failure of the probabilistic design and deterministic design The mean stresses and the c.o.v. of the st resses in the wing and the tail before and after probabilistic optimization are also listed in Table 8-1. We note that the mean stress in the wing is increased (by 0.76%) while the mean stress in the tail is reduced (by 3.6%). That is, a higher safety factor is used for th e tail than the wing, but the difference is not large. For example, if the safety factor for the both structur es was 1.5 for the deterministic design, it would be 1. 49 for the wing and 1.55 for the tail. Since aircraft companies often use additional knockdown fact ors on top of those required by the FAA code, they can slightly increase the knockd own factor for the wing to achieve the probabilistic design that satis fies all the FAA requirements fo r a determinis tic design! A striking result in Table 8-1 is that the ratio of probabilities of failure of the tail and the wing is about 1/5. Recall that the ratio of the tail weight and the wing weight is also 1/5. That is, at optimum the ratio of th e probabilities of failure of the components is almost equal to the ratio of their weights. This optimum probability ratio depends on the following parameters: the target probability of failure, mean and c.o.v. of the stress and c.o.v. of the failure stress. We checked a nd found that the ratio of probabilities falls between 4.5 and 6.5 for a wide range of these parameters. Appendix O provides analytical proof that the ratios of the wei ghts and probabilities ar e indeed approximately the same.

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172 Recall that for deterministic design we assu med that the probabilities of failure of the wing and tail are the same because the co mponents are designed with the same safety factor. So, it is worthwhile chec king the historical record. Cowan et al. (2006) reports that according to the historical result, 18 out of 717 aircraft accidents between 1973 and 2003 were due to wing structural failure, while 9 of the accidents were due to tail structural failure (see also Appendix P). Even though wing an d tail are designed with the same nominal safety factors, the large weight differential may lead to different actual safety factors. Designers may intuitively atte mpt to reduce the structural weight of the heavier wing by squeezing out the weight down to the limit, while they may be laxer with the tail. This, for example, may happen if more approximate methods, with higher safety margins of safety are used for the tail. Th e probabilistic design s upports this incentive and indicates that the design paradigm of using higher safety factors for inexpensive components can further be exploited to incr ease the structural safety of aircraft. Adding a Vertical Tail to the System Next, we added a vertical tail to the wing and horizontal tail system. For a typical transport aircraft, the structural weight of th e vertical tail is about 10% of that of wing. The weights of the wing, the horizontal tail and the vertical tail of our representative system before probabilistic optimization are taken as 100, 20 and 10 units, respectively. The probability of failure of the deterministic designs of the wing, and the tails are taken 110-7 each. The results of structural optimization for safety are listed in Table 8-2. By moving material from the wing to the tails the probability of fa ilure of the wing is increased by 53%, but the probabi lities of failure of the horizontal tail and the vertical tail are reduced by 70% and 85%, respectively. Tabl e 8-2 demonstrates that by including the

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173 vertical tail in the system, the system probab ility of failure is reduced by 34% compared to 22% with two-component (T able 8-1). An increase in nu mber of components may thus increase the safety improvement of the system. Table 8-2 shows a similar finding of Table 81, in that at optimum the ratio of the probabilities of failure of the co mponents are nearly 10:2:1, which is the same ratio of the weights of the components. This optimum proba bility ratio is obtained by using different safety factors for the different components. The mean stresses and the c.o.v. of the stresses in the wing, the horizontal tail and th e vertical tail before and after probabilistic optimization are also listed in Table 8-2. The mean stress in the wing is increased by 1.2%, while the mean stress in th e horizontal tail and the mean stress in the vertical tail are reduced by 3.2% and 5.0%, respectivel y. Again, the substantial reduction in probability of failure is accomplished with a sm all perturbation of the safety factor. So a company that employs an additional knockdown factor of just a few percent would be able to reduce it for the wing, and fully comp ly with the FAA regulations while achieving superior safety. Table 8-2. Probabilistic structural optimization of wing, horizontal tail and vertical tail system. In optimization only the mean st resses are changed, the c.o.v. of the stresses are fixed at c.o.v.=0.20. Probabi lity of failure of the wing and the tails for deterministic design are all 110-7. W0 W Pf Ratio (a) Mean stress before optim. Mean stress after optim. Wing 100 98.80 1.531 39.77 40.25 Hor. Tail 20 20.67 0.300 39.77 38.48 Ver. Tail 10 10.53 0.149 39.77 37.78 System 130 130 0.660 (a) Pf ratio is the ratio of the probabilities of failure of the probabilistic design and deterministic design

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174 Effect of Errors in Informat ion about Deterministic Design The demonstration of the pay-off from pr obability-based desi gn in the previous section was based on assumptions on the stress di stribution and probability of failure. It is known that the calculation of pr obability of failure can be ve ry sensitive to errors in distribution (Ben-Haim a nd Elishakoff 1990, Neal et al. 1992). So here we seek to demonstrate that because we merely seek to ob tain a design with the same weight as the deterministic design, and because the probabilistic design is close to the deterministic design, the effect of errors on the ratio of the probabilities of failure of the probabilistic design and the deterministic design is minimal. In measuring the effects of error in the statistical data, we distinguish between loss of accuracy and loss of opportunity. That is, we report on the accuracy of our estimate of the improvement in the probability of failure compared to the deterministic design. We also report on the missed opportunity to make the design even safer if we had the correct statistical data. Errors in Coefficient of Variation of Stresses We first assume that we under-estimated th e c.o.v. of stresses in the wing and the tail by 50% and performed the optimization usi ng the wrong c.o.v. of stresses. That is, even though the true values of c.o.v. of st resses for the wing and the tail are both 40%, we performed the optimization based on 20% c.o.v. and obtained the de sign shown in Table 8-1. With the overall probability of failure of the deterministic design being fixed, an under-estimate of the c.o.v mu st go with an over-estimate of the mean. Following the procedure in Appendix N, we find that the mean is over estimated by about 45% (actual mean is 31% lower than value used in Table 8-1). Table 8-3 shows bo th the error in the estimation of the probability gain and the lo ss of opportunity to make the design safer. For the wing, we over-estimate the Pf ratio (ratio of PfW of the probabilistic and the

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175 deterministic designs) by 4.2% and under-estimate the Pf ratio for the tail by 19%. However, that the system probability of failu re is under-estimated by only -0.5%, because the two errors canceled each other out (see Fig. 8-2). We also see from the table that the probability of failure of the true optimum is ve ry close to our estimate of the probability of failure for the optimum obtained based on the erroneous data. This is a well known result for the effect of a parameter on the optimum of an unconstrained problem (e.g., Haftka and Grdal 1992, Section 5.4). That is the loss of accu racy is approximately equal to the opportunity loss for sm all changes in the design. Table 8-3. Errors in the ratios of failure proba bilities of the wing and tail system when the c.o.v. of the stresses under-estimated by 50%. The estimated values of c.o.v. of stresses for wing and tail are both 20%, while their actual values are both 40%. Note that the under-estimate of th e c.o.v corresponds to an overestimate of the mean stress, so that its actual va lue is 31% percent lower than the value given in Table 8-1. Optimization based on erroneous data True optimum Optimi zed weight (a) Estimat ed (a) Pf ratio Actual (b) Pf ratio % Error in Pf estim. Mean stress before optim. Mean stress after optim. (assumed) True optimal weight True optimal Pf ratio Mean stress after optim. (true) Wing 99.25 1.309 1.256 4.2 27.42 27.62 99.11 1.309 27.66 H. Tail 20.75 0.257 0.317 -19.0 27.42 26.42 20.89 0.257 26.24 System 120 0.783 0.786 -0.5 0.783 (a) From Table 8-1. (b) Note that the Pf given here is the actual Pf of the assumed optimum (obtained via erroneous c.o.v. of the stress), which is differe nt than the true optimum corresponding to the use of true c.o.v. of the stress The variation of the component and system probability-of-failure ratios with the error in c.o.v. of stresses in the wing and th e tail are shown in Fig. 8-2. We see that for negative errors (under-estimated c.o.v. of stress) the Pf ratio of the wing is over-estimated while the Pf ratio of the tail is unde r-estimated. The two errors mostly cancel each other out and error in the system Pf ratio (and hence the opportuni ty loss) is very small. Similarly, for positive errors (over-estim ated c.o.v. of stress) even though the Pf ratio of

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176 the wing is under-estimated and the Pf ratio of the tail is over-estimated, the estimate of system Pf ratio is quite accurate ov er a wide range of error magnitude. As important is that we lose very little in terms of the pot ential improvement in the probability of failure due to the error. The smallne ss of the opportunity loss is a ma nifestation of the fact that the optimum ratio of the probabilities of failure is insensitive to the coefficient of variation of the stress. Figure 8-2. The change of the ratios of probab ilities of failure of the probabilistic design of Table 8-1 versus the error in c.o.v.( ). Negative errors indicate underestimate, while positive errors indicate over-estimate. We have this remarkable insensitivity of ra tio of probabilities of failure to errors because the probabilistic design is close to the deterministic design. For a given probability of failure of the deterministic desi gn, errors in the mean lead to compensating errors in the standard deviation as shown in Fig. 8-3, which shows two different possible distributions of the stresses in the wing (one with c.o.v. =0.20% and the other with c.o.v.=0.40%) leading to the sa me probability of failure, 1-7. We observe in Fig. 8-3 that when c.o.v. is 20%, the mean stress is 39.77, while the mean stress is lower, 27.42, for a higher c.o.v.=40% so that they both l ead to the same probability of failure. Of

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177 course, these errors can grea tly affect the probability of failure. Had we performed a probabilistic design for a given pr obability of failure, these erro rs could have caused us to get a design which was much less safe than the deterministic design. To complete the investigation, we also discuss the effect of er roneous estimates of proba bility of failure of deterministic design. Figure 8-3. Two different stress distributions at the wing lead ing to the same probability of failure of 1-7. Erroneous Mean Stresses Instead of erroneous estimates for c.o.v. of stresses, we now check the effect of errors in estimates of the mean stresses in the wing and the tail. We first assume that we under-estimated the mean stresses in the wi ng and the tail by 20% for the wing and the tail system of Table 8-1. That is, even though the true values of mean stresses in the wing and the tail are both 49.71, we under-estimated them as 39.77 to obtain the design of Table 8-1. Since the overall probability of failu re of the deterministic design is fixed, an under-estimate of the mean stress must go with an over-estimate of the coefficient of variation. Following the procedure in Appe ndix N, we find that the c.o.v is over

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178 estimated by about 193% (actual c.o.v. is 48% lower than valu e used in Table 8-1). Table 8-4 shows that under-estimation of mean stresses leads to under-estimating the wing Pf ratio by 4.9%, and over-estimating the tail Pf ratio by 27.6%. On the other hand, the system probability of failure ratio is estimat ed with a very small error, because the two errors mostly cancel each other out. Compar ing Table 8-1 and Tabl e 8-4 we see that under-estimate of the mean stresses led to an ov er-estimate of c.o.v. of the stresses and thus compensated for the errors in probability of failure estimations. Table 8-4. Errors in the ratios of failure proba bilities of the wing and tail system when the mean stresses are under-estimated by 20% The estimated values of mean and c.o.v. of stresses for wing and tail are both (39.77, 20%). Note that the underestimate of the mean stress corresponds to an overestimate of the coefficient of variation, so that its actual value is 48% percent lower than the value given in Table 8-1 (c.o.v.=0.1038). Optimization based on erroneous data True optimum Optimi zed weight (a) Estimat ed (a) Pf ratio Actual (b) Pf ratio % Error in Pf estim. Mean stress before optim. Mean stress after optim. (assumed) True optimal weight True optimal Pf ratio Mean stress after optim. (true) Wing 99.25 1.309 1.377 -4.9 49.71 50.09 99.36 1.309 50.03 H. Tail 20.75 0.257 0.198 29.4 49.71 47.91 20.64 0.257 48.18 System 120 0.783 0.788 -0.6 0.783 (a) From Table 8-1 (b) Note that the Pf given here is the actual Pf of the assumed optimum (obtained via erroneous c.o.v. of the stress), which is differe nt than the true optimum corresponding to the use of true c.o.v. of the stress Figure 8-4 shows that negativ e errors (under-estimated m ean stress) lead to overestimated probability of failure ratio of the wing, under-estim ated probability of failure ratio of the tail. However, the two errors ar e mostly cancelled and the error in the system failure probability ratio estimation is very sm all. Positive errors ha ve the opposite effect.

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179 Figure 8-4. The change of the ra tios of probabilities of failure with respect to the error in mean stress. The negative errors indi cate under-estimate, while the positive errors indicate over-estimate. Errors in Probability of Failure Estimates of Deterministic Design Mansour (1989) showed that there can be a significant variation in the failure probabilities of designs constructed using th e same deterministic code. Supposedly, this reflects the effect of errors in predicting stru ctural failure that may be different between designers, companies, or materials. This mean s that the probability of failure estimate of the deterministic design which we used in previous calculations may be inaccurate. To address this issue, we explore the sensitivity of the ratio of the probabilities of failure of the probabilistic design and the deterministic design to erroneous estimates of probability of failure of the deterministic design. Note, how ever, that we still assume that the several structural components are made from the same material and are designed for the same failure mode, so that they have approximately the same probability of failure in the deterministic design. We consider an under-estimate of the probabi lity of failure of deterministic design by two orders of magnitudes. That is, we assu me that we performed probabilistic design

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180 by taking the probability of failure of deterministic design as 10-7 instead of using true value 10-5. Table 8-5 shows th at under-estimated Pf of deterministi c design leads to transferring lower amount of material (0.75% column 6) than optimum (0.95%; column 5) from the wing to the tail. However, even though the wing is designed to be 5.4% safer than the true optimum and the tail is desi gned to be 32.9% less safe, the actual system probability of failure ratio is only 0.8% larg er than its estimated value (columns 2-4). Table 8-5. Errors in the ratios of failure proba bilities of the wing and tail system when the probability of failure of the determinis tic design is under-predicted. The actual Pf is 10-5, while it is predicted as 10-7. Note that the c.o.v. of the stress is 20%. Optimum weight Estimated= Actual (a) Pf ratio Mean stress before optim. Mean stress after optim. (assumed) True optimal weight True optimal Pf ratio % safety loss Mean stress after optim. (true) Wing 99.25 1.240 45.91 46.26 99.05 1.310 -5.4 46.35 H. Tail 20.75 0.336 45.91 44.24 20.95 0.253 32.9 43.83 System 120 0.788 0.782 0.8 (a) Estimated an actual probabilities of failure of the assumed optimum are the same, since the mean and c.o.v. of the stress do not involve any error Similarly, we checked what happens when we over-estimate the probability of failure of the deterministic design by two orde rs of magnitudes. Table 8-6 shows similar results as Table 8-5, but this time a larger am ount of material is tran sferred from the wing to the tail compared to the true optimum (columns 5 and 6). This time, the wing is designed to be 5.1% less safe and the tail is designed to be 22% sa fer than their optimum values, the probability of failure ratio is only 0.6% greater than its estimated value (columns 2-4).

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181 Table 8-6. Errors in the ratios of failure pr obabilities of wing and tail system when the probability of failure of the deterministic design is over-predicted. The actual Pf is 10-9, while it is predicted as 10-7. Optimum weight Estimated =Actual (a) Pf ratio Mean stress before optim. Mean stress after optim. (assumed) True optimal weight s True optimal Pf ratio % safety loss Mean stress after optim. (true) Wing 99.25 1.374 35.33 35.60 99.36 1.306 5.1 35.56 Hor. Tail 20.75 0.202 35.33 34.05 20.64 0.261 -22.4 34.24 System 120 0.788 0.783 0.6 (a) Estimated an actual probabilities of failure of the assumed optimum are the same, since the mean and c.o.v. of the stress do not involve any error Effect of Using Wrong Probability Distribution Type for the Stress Apart from the parameters we investigated (c.o.v. of the stresses, the mean stresses, and the probability of failure deterministic desi gn), the distribution t ype of the stress also affects the results of probabilistic design. Here we explore the sensitivity of the probabilistic design to using wr ong distribution type for the st ress. We assume that even though the stress follows the lognormal distribu tion, the optimization is performed using a normal probability distribution to obtain the result s in Table 8-1. Table 8-7 shows that if th e true stress probability di stribution is lognormal, the Pf ratio of the wing is over-estimated and the Pf ratio of the tail is under-estimated, so smaller amount of material is moved from the wing to the tail compared to the true optimum design. The error in the total system Pf ratio estimate is only 2.3%. Also, as in Tables 8-3 to 8-6 the loss of accuracy is a pproximately equal to the opportunity loss since the changes in the design are small. The loss of optimality reflects the fact that with lognormal distribution of the stress it is advant ageous to transfer more material from the wing to the tail than with the normal distribu tion. Of course, the true distribution may be different from lognormal, however, the insens itivity is still encouraging. It is also interesting to note that even with the lognormal distribution the optimal probabilities of

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182 failure are still proportional to the weight of the two components. Analytically, we have obtained a proof of this phenomenon only fo r the normal distribution (see Appendix O). Table 8-7. Errors in the ratios of failure proba bilities of the wing and tail system if the optimization is performed using wrong probability distribution type for the stress. The probability of failure of the deterministic design is 10-7. The c.o.v. of the stress is 20%. Optimization assuming stress is normal Optimization using the correct distribution type: lognormal Opti mize d weig ht (a) Estimat ed (a) Pf ratio Actual (b) Pf ratio % Error in Pf estim. Mean stress before optim. Mean stress after optim. (assumed) True optimal weight True optimal Pf ratio Mean stress after optim. (true) Wing 99.25 1.309 1.201 9.0 32.04 32.28 98.90 1.307 32.39 H. Tail 20.75 0.257 0.402 -36.1 32.04 30.87 21.10 0.265 30.36 System 120 0.783 0.801 -2.3 0.786 (a) From Table 8-1. (b) The actual Pf of the optimum obtained via erroneous stress distribution type Approximate Probabilistic Design Ba sed on Failure Stress Distributions One of the main barriers to the application of probabilistic struct ural optimization is computational expense. Probabilistic stru ctural optimization is expensive because repeated stress calculations (typically FEA) are required for updating probability calculation as the structure is being changed. That is, the simplified approach that we used in Eq. (8.2) is replaced by costly FEAs. Traditionally, reliability based design op timization (RBDO) is performed based on a double-loop optimization scheme, where the outer loop is used for design optimization while the inner loop performs a sub-optimization for reliability analysis using methods such as First Order Reliability Method (FOR M). Since this traditional approach is computationally expensive, even prohibitive for problems that re quire complex finite element analysis (FEA), alternative methods have been proposed by many researchers (e.g., Lee and Kwak 1987, Kiureghian et al. 1994, Tu et al. 1999, Lee et al. 2002, Qu and

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183 Haftka 2004 and Du and Chen 2004). Thes e methods replace the probabilistic optimization with sequential deterministic optim ization using inverse reliability measures to reduce the computational expense. The down side of these approaches is that they do not necessarily converge to the optimum desi gn. We note, however, that most of the computational expense is associated with re peated stress calculati on and we have just demonstrated insensitivity to the details of the stress distribution. This allows us to propose an approximate probabili stic design approach that mi ght lead to a design nearer the optimum (depending, of course, on the accuracy of the approximation). Structural failure, using most failure criteria, occurs when a stress at a point exceeds a failure stress f. For a given deterministic stress the probability of failure is ProbffPF (8.4) where F is the cumulative distributi on function of the failure stress f. For random stress, the probability of failure is calculated by inte grating Eq. (8.4) for all possible values of the stress fPFsd (8.5) where s is the probability density function of the stress. For the calculations of the probability of failure in the preceding section, numerical in tegration of Eq. (8.5) was performed. It is clear from Eq. (8.5) that accurate estimation of probability of failure requires accurate assessments of the pr obability distributio ns of the stress and the failure stress f. For the failure stress f, the FAA requires aircraft builders to perform characterization tests, use them to construct a statistical model, and then select failure allowables (A-basis or B-basis values) ba sed on this model. Hence, the statistical characterization of the failure stress is soli d. On the other hand, the probability density

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184 function of the stress, s(), is poorly known, because it depends on the accuracy of structural and aerodynamic calculations, the knowledge of the state of the structure, damage progression and pilot actions. As we di scussed earlier, it is reasonable to assume that stress is normally distributed, because a large number of sources contribute to the uncertainty in stress, such as errors in load and stress calculations, variabilities in geometry, loads and material properties. Reca ll that more detailed discussion on the sources of uncertainty in stress is discussed in Chapter 4. By using the mean value theorem, Eq. (8.5) can be re-written as ** fPFsdF (8.6) where the second equality is obtained by using the fact that the integral of s() is one. Equation (8.6) basically states th at the effect of the poorly ch aracterized probability of the stress can be boiled down to a single characteristic stress value *. This value can be obtained by estimating s() and integrating as specified in Eq. (8.6). However, it is equally possible to use historic al data on probabilities of fa ilure of aircraft structural components to do the reverse. That is, given an estimate of the probab ility of failure, we can obtain the characteristic stress that corresponds to this hi storical aircraft accident data when airplanes are designed using the deterministic FAA process (see Fig. 8.5).

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185 Figure 8-5. Calculation of characteristic stress from probability of failure Recall that in probabilistic design of the wi ng and tail system we deviate from the deterministic process by reducing the struct ural margin on the wing and increase the margin on the tail, assuming that the structural redesign changes the stress distribution by simple scaling of to (1+). Under this simple stress scaling, the characteristic stress will change from to *(1+*) to allow probabilistic desi gn with a minimum number of stress analyses. We assume here that the re lative change in the characteristic stress, *, is proportional to the relative change in the stress, That is, = k (8.7) The value of k depends on the mean and c.o.v. of the stress and the failure stress. For lognormally distributed failure stress w ith mean = 100 and c.o.v.=10%, and normally distributed stress with mean = 39.77 and c.o. v.=20% (the values from our representative example), Figure 8-6(a) shows the relation of and *. Notice that the variation is almost linear. Figure 8-6(b) sh ows the effect of the approximation on the probability of failure. We see that the linearity assumption is quite accurate over the range -10% 10%.

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186 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 actual approx. (a) -10 -8 -6 -4 -2 0 2 4 6 8 10 10-9 10-8 10-7 10-6 10-5 Pf actual approx. (b) Figure 8-6. (a) Comparison of approximate and exact and and (b) the resulting probabilities of failure for lognormal failure stress (with mean=100 and c.o.v.=10%) and normal stress (w ith mean=39.77 and c.o.v.=20%) Application of Characteristic Stres s Method to Wing and Tail Problem In this section, we apply the probability of failure estimation to the wing and tail problem. The weights of the wing and the tail before probabilistic optimization are taken as 100 and 20 units, respectively. The probability of failure of the wi ng and the tail are both taken as 110-7. The failure stress of the wing and tail materials is assumed to follow lognormal distribution with a mean value of 100 and 10% c.o.v. The coefficients of

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187 variations of the stresses in the wing and the tail are assumed 20%. The correlation coefficient for probabilities of failure of wing and tail is assumed zero. As we discussed earlier, some material is taken from the wing and added to the tail so that stresses in the wing and the tail are scaled by (1+W) and (1+T), respectively. Similarly, the characteristic stresses in the wing and the tail, W and T *, are scaled by (1+W *) and (1+T *), respectively. Then, the probabi listic design optimization problem stated earlier in Eq. (8.3) can now be re-formulated as *** ,min,11 suchthatWTfWTfWWfTT WW WTdWdTPWWPWPW WWWW (8.8) The weights of the components and charac teristic stresses are related via **1WWWdWk, **1TTTdTk (8.9) where the relative changes in th e stresses are calculated from 1dW W WW W 1dT T TW W (8.10) The probabilistic optimization problem stat ed in Eq. (8.8) is solved, and the probabilities of failure are com puted. Table 8-8 shows that the Pf ratios of the wing and the tail are estimated as 1.307 and 0.263, inst ead of their actual values 1.305 and 0.261. So, the characteristic-stress method estimates the system Pf ratio as 0.785, while the actual Pf ratio corresponding to the redesign is 0.783, which is the same system Pf ratio in Table 8-1.

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188 Table 8-8. Probabilistic design optimization for safety of the representative wing and tail system using the characteristic-stress met hod. The c.o.v. of the stresses in the wing and tail are both 20%. Table 8-1 Proposed method W Pf ratio W (a) Pf ratio Actual Pf ratio Wing -0.75 1.309 -0.75 1.307 1.305 Hor. Tail 3.77 0.257 3.73 0.263 0.261 System 0.0 0.783 0.0 0.785 0.783 (a) % W is percent changes in weight Table 8-8 shows that the error associated with the approximation of the characteristic stress in Eq. (8.7) is small. This is expected based on Fig. 8-6 that shows that the approximation of is very good. However, the main issue here is to show what happens if we commit e rrors in evaluating the k value in Eq. (8.7) due errors in the distribution parameters in the stresses and the fa ilure stress. We investigated the effects of over-estimating and under-estimating k with 20% error. Table 8-9 shows that 20% underestimate of k leads to designing the wing for a higher Pf ratio (1.392 instead of 1.306) and designing the tail for a lower Pf ratio (0.187 instead of 0.269). The overall system Pf ratio, however, is increased by onl y 0.4%. The variation of Pf ratio of the wing, the tail and the system with the error in k is depicted in Fig. 8-7. It is seen that the effect is small for a wide range of errors. Table 8-9. Effect of 20% under-estimate of k on the ratios of probability of failure estimate k Table 8-1 Characteristic-stress method with erroneous k correct With 20% underestimate W Pf Ratio W (a) Pf ratio Actual Pf ratio Wing 0.664 0.532 -0.75 1.309 -0.93 1.306 1.392 Hor. Tail 0.664 0.532 3.77 0.257 4.64 0.269 0.187 System 0.0 0.783 0.0 0.787 0.790 (a) % W is percent changes in weight

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189 Figure 8-7. The variation of the ratios of probabilities of failure with respect to the error in k. The negative errors indicate under-estimate, while the positive errors indicate over-estimate. For a more complicated problem, when th e stresses are calculated via FEA, the application the proposed method is as follows After calculating the stresses from FEA, the relative changes in the stresses (that is, the values) are calculated. Then, the characteristic stresses are updated by using Eqs. (8.7) and (8.9). Finally, the probabilities of failure of the components are updated us ing Eq. (8.6). The computational expense regarding with probability calculations are reduced greatly, and the probabilistic optimization problem is reduced to a se mi-deterministic optimization problem. Summary Probabilistic structural de sign achieves better perfor mance than deterministic design by applying higher safety factors to lower -weight components. This was demonstrated on a design problem of distribut ing structural material between the wing, horizontal tail and vertical tail of a typical airliner. While deterministic design leads to similar probabilities of failure for the three components, the probabilistic design led to probabilities of failure that were approximate ly proportional to stru ctural weight. This

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190 result has shown to be a property of the nor mal distribution. Remarkably, even though the ratios of weights and probability of failure ratios of the three components were 10:2:1, this was accomplished by reducing the safe ty factor on the wing by only about one percent and using the material to increase the safety factor on the horizontal and vertical tails by 3 and 5 percent, respec tively. This led to a reduction of 34% in the probability of failure for the same total weight. The small perturbation of the safety factor can be accommodated by the additional knockdown factors that aircraft companies often use on top of those required by the FAA code. So the aircraft companies can slightly increase these additional knockdown factors for the wing to achieve the probabilistic design that satisfies all the FAA requirements for a deterministic design! We used estimates of the probability of fa ilure of the deterministic design (obtained from historical record) as st arting point of the probabilist ic optimization. Because the exact values of the probability of failure of the deterministic design and the parameters of the probability distribution of structural response are ra rely known, we checked the sensitivity of the ratio of probabilities of the probabilistic design and the deterministic design to large inaccuracies in the paramete rs of the stress distri bution, the type of distribution, and probability of failure estimate of the deterministic design. In particular, 50% errors in the standard deviation of the st ress, or 20% error in the mean stress led to less than 1% difference in the probabili ty of failure ratios (i.e., ratio of Pfs of the probabilistic and the deterministic designs). We also found that two orders of magnitude of errors in the probability of failure estimat e of the deterministic design led to less than 1% difference in the system probability of failure ratio.

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191 Finally, these results inspired us to offer an approximate characteristic-stress method that dispenses with most of the e xpensive structural re sponse calculations (typically done via finite element analysis). We showed that this approximation still leads to similar re-distribution of material betw een structural components and similar system probability of failure.

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192 CHAPTER 9 CONCLUDING REMARKS The two primary objectives of this dissert ation were the following: (i) Analyze and compare the effectiveness of safety measures that improve structural safety such as safety factors (can be explicit or implicit), stru ctural tests, redundancy and uncertainty reduction mechanisms (e.g., improved structural anal ysis and failure prediction, manufacturing quality control). (ii) Explore the advantag e of uncertainty reduction mechanisms (e.g., improved structural analysis and failure pred iction, tighter manufacturing quality control) versus safety factors. That is, we considered the possibility of allo cating the resources for reducing uncertainties, instead of living with the uncertainties and allocating the resources for heavier aircraft structur es designed for the given uncertainties. We started with a point stress design analysis of an aircraft structure by incorporating the uncertainties and safety measures that protect against these uncertainties. The uncertainties are classified as error and variabil ity. Errors reflect inaccurate modeling of physical phenomena, errors in structural analysis, errors in load calculations, or deliberate use of materials a nd tooling in constructi on that are different from those specified by the designer. Variabil ity, on the other hand, reflects the departure of material properties, geometry parameters or loading of an individual component from the fleet-average values. The safety measures included in this dissertation were the load safety factor of 1.5, conservative material properties, structural redundancy, coupon tests, structural element tests, certification tests and error and variability reduction mechanisms (e.g., improving the accuracy of structural anal ysis and failure predic tion to reduce error,

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193 employing tighter manufacturing quality cont rols to reduce variability in material properties). First a simple analysis was performed in Chapter 3 to understand the basics. A simple error model was used, and coupon tests a nd element tests are excluded at first. We found that (1) The load safety factor of 1.5 accomp anied with conservative material properties, redundancy, and certification testing raised the actual safety factor to about 2. Next, the analysis was refined in Chapter 4 by using a more detailed error model and modeling the structural redundancy. We found in this analysis that (2) While certification testing was more effective than increased safety factors for improving safety, it cannot compete with even a small reduction in errors. (3) Variability reduction was even more ef fective than error reduction, but it needed to be accompanied by increased knockdown factors to compensate for the increase in the B-basis value. Discovering the power of error and variabil ity reduction in increasing the structural safety, we were motivated to analyze the tr adeoffs of uncertainty reduction mechanisms, structural weight and structural safety. Th e effect of error reduction (due to improved failure prediction model) on increasing the al lowable flight loads of existing aircraft structures was investigated in Chapter 5. We found that (4) The allowable flight load of existi ng aircraft structures can be substantially increased on average by improve d failure prediction modeling. Next, the improved structural analysis th rough taking the chem ical shrinkage of composite laminates was considered in Chapter 6 as the error reduction mechanism to

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194 investigate the tradeoffs of error and vari ability reduction mechan isms for reducing the weight of the composite laminates at cr yogenic temperatures. Tradeoff plots of uncertainty reduction mechanisms, probability of failure and weight were generated that enable a designer to choose the optimal uncer tainty control mechanism combination to reach a target probability of failure with minimum cost. Investigation of the interact ion and effectiveness of safety factors was culminated with our final analysis in Chapter 7, which included coupon tests and element tests that we disregarded in our first an alysis. In particular, emphasi s was placed on analyzing the optimal choice of additional knockdown fact ors. These knockdown factors refer to conservative decisions of aircraft companie s while updating the allowable stresses based on the results of structural element tests. Currently these knockdown factors are implicit and based on worst-case scenar ios (e.g., testing the structural elements at worst-case operational conditions, using the smallest meas ured failure stress in design). Here, we proposed use of explicit knockdown factors, which depend on structural element test results. The effects of coupon tests, struct ural element tests and uncertainty control mechanisms on the choice of company safety fa ctors were investigated. The Pareto fronts of structural weight and lik elihood of structures failure in certification testing are generated. The following observations were drawn. (5) Instead of using imp licit knockdown factors based on wo rst-case scenarios, the use of test-dependent explicit knockdown factors may lead to 1 or 2% weight savings. Surprisingly, we found that a lower (i.e ., more conservative) knockdown factor should be used if the failure stresses measured in tests exceeds predicted failure stresses in order to reduce the variab ility in knockdown fact or generated by

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195 variability in material properties. A thirty -six percent reduction in variability was observed, and it is likely to be re sponsible for the weight savings. (6) Selecting additiona l knockdown factors to minimi ze certification failure rate provides a choice that is al so very close to the optim um choice that minimizes structural failure in flight. (7) Using a simple cost function in term s of structural weight, we have shown that decisions can be made whether to inve st resources on coupon tests, structural element tests, uncertainty reduction mech anisms or extra structural weight. The analyses mentioned earlier showed how probabilistic design could be exploited to improve aircraft structural safety. A first step was taken towards the two main barriers in front of the probabilistic design in Chapter 8. These barriers are the sensitivity of the probabilistic design to li mited statistical data and the com putational expense associated to the probabilistic design. Probabilistic desi gn optimization of a representative wing and tail system was performed with limited statistical data. We showed that (8) Errors in statistical data affect the probability of failure of both probabilistic and deterministic designs, but the ratio of these probabilities is quite insensitive to even very large errors. (9) The probabilistic design was found to be a small perturbation of deterministic design. This small perturbation could be achieved by a small redistribution of additional knockdown factors. To alleviate the problem of computati onal expense an appr oximate probabilistic design optimization method was proposed, where the probability of failure calculation was confined only to failure stresses to dispen se with most of the expensive structural

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196 response calculations (typica lly done via finite elemen t analysis). The proposed optimization methodology is illustrated with th e design of the wing an d tail system. We showed that this approximation still leads to similar re-distributi on of material between structural components and similar system probability of failure. More detailed conclusions corresponding to each stage of the dissertation can be found in the Summary sections of individual chapters.

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197 APPENDIX A A-BASIS AND B-BASIS VALUE CALCULATION A-basis value is the value exceede d by 99% of the population with 95% confidence. Similarly, B-basis value is the value exceeded by 90% of the population with 95% confidence. The basis values are calculated by B asisXks (A1) where X is the sample mean, s is the sample standard deviation and k is the tolerance coefficient for normal distribution given by Eq.(A2) 2 22 11 11 2 1,1, 2(1)pp pzzab zz kabz aNN (A2) where N is the sample size and z1-p is the critical value of normal distribution that is exceeded with a probability of 1-p (for A-basis value p=0. 99 while for B-basis value p=0.90). Similarly, 1z is the critical value of normal di stribution that is exceeded with a probability of 1( =0.95 for both A-basis and B-basis va lues). The tolerance coefficient k for a lognormal distribution is obtaine d by first transforming the lognormally distributed variable to a norma lly distributed variable. Equa tions (A1) and (A2) can be used to obtain an intermediate value. Th is value is then converted back to the lognormally distributed variable using inverse transformation. In order to obtain the A-basi s or B-basis values, we as sume that 40 panels are randomly selected from a batch. Here the un certainty in material property is due to allowable stress. The mean and standard de viation of 40 random values of allowable

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198 stress is calculated and used in determining the A-basis value of allowable stress. For instance, when the failure stress is lognorma l with 8% coefficien t of variation and 40 tests are performed, the coefficient of vari ation of A-basis value is about 3 percent.

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199 APPENDIX B PROBABILITY CALCULATIONS FOR CHAPTER 3 Calculation of Pr(CT|e), the Probability of Passing Certification Test The probability that the structure will pass the certification testing is calculated from Pr(|)Pr()PrPr Pr()Fd f ffFdSP CTewtSP wt CR (B.1) where fCtw is the load carrying capacity of structure and Fd R SP is the applied load in the certification testing. Since the coefficients of variati ons of the geometry parameters t and w are small compared to the coefficient of variation of f we assume t and w can be assumed lognormal, so the capacity C can be treated as lognormal with distribution parameters C and C given as ()()Ctw fee and 2222Ctw f (B.2) where 2()ln(())0.5tdesigntete (B.3) Recall that the design thickness designt is defined earlier in Eq (3-4, Chapter 3) as (1)Fd design designaSP te w (B.4)

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200 Since the response Rc is a deterministic value, the probability Pr(CT|e) can be calculated as 2() Pr(|)Pr()()CFd CeSP CTeCRe (B.5) where is the cumulative distribution function of the standard normal distribution. Calculations of Mean and Standard De viation of Probability of Failure Failure is predicted to occur when the load carrying capacity of the structure C is less than the applied load P. So the probability of failure is given as PrfPCP (B.6) The load P is lognormally distributed, and as ex plained in above in this appendix, the distribution of capacity C can also be approximated by a lognormal distribution, which allows us to immediately obtain the probability of failure of a single aircraft model. To calculate the probability of failure over all aircraft models, we take into account the fact that that designt is a random variable. Then, the e xpected value of probability of failure is given as ()() f fdesigndesigndesignPPtftdt (B.7) where designt is the non-deterministic di stribution parameter, and ()designft is the probability density function (PDF)of designt. The standard deviation of failure probability can be calculated from 1/2 2()Pffff fPPfPdP (B.8)

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201 where () f fdesignPPt ()()design fdesign fdt fPft dP (B.9) 1 f design designfdPdt dtdP Hence, Eq. (B.8) can be re-written as 1/2 2()()Pfdesignfdesigndesign fPtPftdt (B.10) As seen from Eqs. (B.8) and (B.10), the mean and standard deviation of the probability of failure can be expressed in terms of the ()designft. Therefore, we can perform the failure probability esti mations to after calculating the ()designft. The random variables cont ributing to designt are e w and a (see Eq. (B.4)). Since the variations of w and a are small compared to error e we neglect the cont ribution of w and and a and calculate the ()designft from ()designe designde ftfe dt (B.11) where e f e is the PDF of e

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202 APPENDIX C CONFLICTING EFFECTS OF ERROR AND VARIABILITY ON PROBABILITY OF FAILURE IN CHAPTER 3 As explained in the discu ssion of Table 3-5 and Figure 3-3, large errors coupled with certification tests can improve the average (over all companies) safety of an aircraft model. This was most apparent when mean ma terial properties are used for design (Table 3-5) because for this case airplanes would be tested at their average failure load, so that fifty percent will fail certification. A large error bound means a wide variation in design thicknesses. Certification testing fails most of the airplane models with unconservative designs and passes a group of ai rplane models with high av erage thickness (that is, overdesigned planes). When the additional safety factor of conservative material properties is used, as in Table 3-6, the picture is more complex. Certif ication is still done at the same loads, but the test airplane is designed for higher loads because of the conservative material properties. For high errors, many airplanes will st ill fail certification, but small errors will be masked by the conservative properties. Thus in Table 3-6, the certif ication failure rate varies from 29.4% for the largest errors to 1.3% for the smallest errors. At the highest error bound (50%), the certification process incr eases the average thic kness from 0.847 to 0.969, and this drops to 0.866 for 30% erro r bound. This substantial drop in average certified model thicknesses in creases the probability of fa ilure. Below an error bound of 30%, the change in thickness is small, and then reducing errors reduces the probability of

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203 failure. This is because small negative errors are not caught by certification, but they still reduce the effective safety factor. A similar phenomenon is observed when the va riability is changed in Tables 3-8 to 3-10. When the coefficient of variation in failure stress is increased from 0% to 16%, the average design thickness before certificati on increases by about 60% and so the probability of failure without certification is reduced by f actors of 16-70. Note, however, that for the smallest error bound (Table 310), the drop occurs from zero to 8% coefficient of variation. At th e higher coefficients of variat ion the probability of failure before certification increases again as the increased design thickne ss does not suffice to compensate for the large variat ion between airplanes .Once certification is included in the process, variability is mostly detrimental. Certification doe s not amount to much for large variability, because the certifie d airplane can be very different from the production aircraft. For large error bounds (Table 3-8) th ere are large errors that can be masked during certification by the high material safety factor. Thus in Table 3-8, while the probability of failure without cer tification is reduced by a fact or of 16, the probability of failure with certification is incr eased by a factor of 1320 as th e coefficient of variation in the failure stressed is increased from 0 to 16% For small errors (Table 3-10) the picture is more mixed as the non-monotonic behavior without certificati on is mirrored with certification.

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204 APPENDIX D COMPARISON OF RESULTS OF SING LE ERROR FACTOR AND MULTIPLE ERROR FACTOR CASES In Chapter 3, we used a single error f actor model (SEF model), where the overall error is represented with a single error factor e and uniform distribution is used for the initial distribution of this error. On the ot her hand, Chapter 4 utilizes a multiple error factor model (MEF model), which uses a mo re complex representa tion of error with individual error factors and wher e initial distributions of each individual error factor are represented with uniform distribu tion. In this case, the distribution of the total error is no longer uniform. We find that the SEF model exaggerates the effectiveness of certification testing (see Figure 4-1). This is due to the fact that the SEF mode l does not consider the fact that errors in load calcul ation affect the load used in ce rtification testing. In the SEF model (Chapter 3), the certificat ion testing is assumed to be performed with the average value of the actual load ( Pd), while in the MEF model certi fication testing is performed with the calculated load ( Pcalc). Therefore, one component of the error cannot lead to failure in certification testing and this reduces the effectiveness of certification testing. Note that the single error of the SEF model is symmetric. On the other hand, even though the individual errors of MEF model are symmetric, the total error has a bellshaped distribution with a positive, hence conservative, mean. One of the interesting differences between the SEF and MEF models is that we have a built-i n safety factor due to asymmetric error distribution. This asymmetry is due mostly to the term 11 f e in

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205 Eq. (4.12). While ef is symmetrically distributed (-0.2, 0.2), 11 f e varies in (0.833, 1.25). The conservative tilt of the total error may be serendipitous because it will partially account for designer bias response to th e building block tests used to reduce ef. Tests that show that the failure model is slightly c onservative typically do not lead to updating of the model. In contrast, tests showing even small unconservative bias typically lead to correction of the failure model. In order to compare the e ffect of the two models on the probability of failure calculations, we match the mean and standard deviation values of the total error distribution (MEF model) and those of a uniform distribution (SEF model). Then, the upper and lower bounds ( lb and ub ) for the uniformly distributed error factor can be calculated via Eq. (D.1), where e and e are the mean and standard deviation of the total error, respectively. 3eelb 3eeub (D.1) Using the equivalent error bounds of the SE F model given in the right-hand side of Table D-1 we calculate the probabilities of failu re before and after certification testing for the SEF model and we compare them in Table D-2 with corresponding failure probabilities of the MEF model from Table 4-8. Table D-1. Equivalent error bounds for the SEF model corresponding to the same standard deviation in the MEF model. The average and standard deviation is calculated via 1,000,000 MCS. k Average ini totale Standard deviation of ini totale Lower bound for ini totale Upper bound for ini totale 0.25 0.0009 0.033 -0.057 0.059 0.50 0.0034 0.067 -0.113 0.119 0.75 0.0076 0.101 -0.168 0.183 1.0 0.0137 0.137 -0.223 0.250 1.5 0.0317 0.212 From MEF model to SEF model -0.336 0.400

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206 The comparison of the probability of failures after certification for the two models is presented in Fig. D-1. Figure D-1. System failure pr obabilities for the SEF and ME F models after certification The total safety factor for the SEF model is defined as 01designf FFL design aA SeS A (D.2) Similarly, the design area for the SEF model is expressed as 1FLd design aSP Ae (D.3) Using the SEF model, we repeat the calculation of the probabilities of failures. The comparison of the SEF and MEF models probability of failure calculations are given in Table D-2. Table D-2. Comparison of syst em failure probabilities for the SEF and MEF models. The coefficient correlation between failures of structural parts is taken 0.5. k M EF ncP/10-4 M EF cP/10-4 Pf Ratio* SEF ncP/10-4 SEF cP/10-4 Pf Ratio(a) 0.25 0.0 0.0 0. 0. 0. 0. 0.50 0.029 0.022 0.749 0.026 0.018 0.689 0.75 0.195 0.106 0.543 0.165 0.069 0.419 1 1.11 0.390 0.350 1.03 0.186 0.181 1.5 17.2 2.21 0.129 27.7 0.311 0.112 (a) Pf is the ratio of the average failure proba bilities before and after certification testing; /cncPP

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207 When we compare the probability of failure before certification, the mean values of the failure probabilities are higher for the SE F model than those for the MEF model at high errors (see columns 2 and 5, Table D-2) due to the use of uniform distribution for the total error factor. Comparing the failure pr obabilities after certification, we notice that the MEF model leads to higher probabil ity of failure values, and higher /cncPP ratios (less effective certification testing). Recall that this is due to the fact that in the MEF model error in load calculation is also included in the certification testing. This effect is also apparent when we compare the total safe ty factor values for these two models in Table D-3 and in Fig. D-2. The single error factor after certification of failure probabilities in Table D-2 also indicates that the effect of the error bound on the probability of failure after certification is not monotonic. One possible explanation for this behavior are the competing effects of error and the total safety factor. For the highe st error bound, the tota l safety factor is increased to 2.108 (see Table D-3), which ove rcomes the effect of high error on the probability of failure. Table D-3. Comparison of the total safety factor SF used in the design of structural parts for the SEF and MEF models k M EF F builtS M EF F certS SF Ratio(a) SEF F builtS SEF F certS SF Ratio(a) 0.25 1.725 1.728 1.002 1.725 1.729 1.002 0.5 1.730 1.741 1.007 1.730 1.745 1.009 0.75 1.737 1.764 1.016 1.737 1.776 1.023 1 1.747 1.799 1.030 1.747 1.825 1.044 1.5 1.779 1.901 1.069 1.779 1.954 1.099 (a) SF is the ratio of total safety factors before and after certification

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208 Figure D-2. Total safety f actors for the SEF and MEF model after certification Comparing the total safety factors, SF, after certification corresponding to the MEF and SEF models (columns 3 a nd 6, Table D-3), we see that the total safety factor corresponding to the SEF model is larger, whic h will in turn lead to a smaller probability of failure (see Table D-2). Columns 4 and 7 of Table D-3 exhibit the expected trend of an increase in the total safety factor ratio with increasing erro r bounds, reflecting more effective certification testing. In short, the effect of using a more de tailed error model can be summarized as follows: The uniformly distributed individual e rror components add up to a bell-shaped representative total error. This total error has an asymmetric distribution and this asymmetry results in a built-in safety factor. The single error model exaggerates the effectiveness of certification testing, because it does not include the fact that error in load calculation is also present in the certification process. The single erro r model inflates the design area after certification, thereby leadi ng to under-estimation of pr obabilities of failures.

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209 APPENDIX E DETAILS OF SEPARABLE MONTE CARL O SIMULATIONS FOR PROBABILITY OF FAILURE CALCULATIONS IN CHAPTER 4 The separable MCS procedure applies when the failure condition can be expressed as g1( x1)> g2( x2), where x1 and x2 are two disjoint sets of ra ndom variables. For that case, the probability of failure can be written as 211fPftFtdt (E.1) where f2 is the probability density function of g2 and F1 is the cumulative distribution function of g1. Since the two sets of random variab les are disjoint, we can perform one Monte Carlo simulation with x1 to calculate F1 and then perform a second Monte Carlo simulation on x2 to calculate Pf from Eq. (E.1). Note that 1F1 in Eq. (E.1) is the probability of failure if g2 takes the value t, and the second Monte Carlo simulation calculates the average of this probability over all possible values of g2. For our problem, f2 is the probability density func tion of the built safety factor, 0/built A A, and F1 is the cumulative distribution function of the required safety factor,0/req A A Since f1( ) and F2( ) depend on different sets of random variables, we separate the MCS into two stages. In the first stage, the cumulative distribu tion function of the re quired safety factor, 0/req A A is assessed. We use 1,000,000 MCS for th is purpose. It is possible to assess CDF numerically by di viding the range of 0/req A A into a number of bi ns (for instance,

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210 1,000 bins) and calculating the CDF for each bi n. Then, in the second stage, the CDF value can be obtained by interpolation. On the other hand, we notice for our problem that the dominant terms in 0/req A A are Pact and f, since they have much larger variabilities than vt and vw (see Table 4-2, Chapter 4). Since Pact and f follow the lognormal distribution, it is possible to represent 0/req A A with lognormal distribution. We ind eed found that numerical CDF is in good agreement with the assumed lognormal as shown in Fig. E-1. Figure E-1. Comparison of numerical CDF with the assumed lognormal CDF for the distribution of the re quired safety factor To ensure that the assumed lognormal distribution leads to an accurate probability of failure estimations, we performed the following study. Five different sets of 0/req A A values are obtained from fi ve different MCS with 1,000,000 sample size. Then, the probabilities of failure are calculated usi ng the same second-stage random numbers for both numerical CDF and assumed lognormal CDF. Table E-1 shows th at the probability

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211 of failure estimation using a ssumed lognormal CDF is accurate to the third digit and also has a smaller standard deviation indicati ng that the numerical noise is reduced. Table E-1. Comparison of the proba bility of failure estimations Pf estimation using numerical CDF (-4) Pf estimation using assumed lognormal CDF (-4) MCS 1 8.961 8.855 MCS 2 8.902 8.807 MCS 3 8.901 8.825 MCS 4 8.734 8.856 MCS 5 8.859 8.816 Average 8.871 8.832 Std. dev. 0.085 0.023

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212 APPENDIX F CALCULATION OF THE SYSTEM FAILURE PROBABILITY USING BIVARIATE NORMAL DISTRIBUTION Bivariate normal distribution describes th e joint behavior of two random variables X1 and X2, for which the marginal distribu tions are normally distributed and correlated through the correlation coefficient The probability density function is defined as (Melchers, 1999) 12 1222 12 2112 ,,exp 22 1XX XXhkhk fxx (F.1) where 11 1x h and 22 2x k 1 and 1 are the mean and standard deviation of variable X1, and 2 and 2 are the mean and standard deviation of variable X2. The joint cumulative distribution is defined as 2 12 12212 1212 1,,Pr,,,,xx XXiiXX iFxxXxfuvdudvxx (F.2) In addition, 2 can be reduced to a si ngle integral (Owen, 1956) 22 2 2 2 01112 ,,exp 22 1 1 hkhk hkdzhk z z (F.3) where is the standard normal cumu lative distribution function. The two local failure events requirement of our problem is modeled as a parallel system. Thus we aim at computing the prob ability of failure of a parallel system composed of two elements having equal failure probabilities. We assume that the limit-

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213 state functions for these two elements follo w normal distribution. Thus we can use the bivariate normal distri bution to calculate the system pr obability of failure. Since the failure probabilities are identical, the reli ability indices are also identical (i.e., h = k =). Then Eq. (F.3) can further be simplified into Eq. (F.4). Thus, given the probability of failure of a single element and the correlation coefficient Eq. (F.4) can be used to calculate system failure probability PFS. 2 2 2 2 011 ,,exp 21 1FSfPPdz z z (F.4) where Pf and are the probability of failure and the reliability index for a single element, respectively, which are related to each other through Eq. (F.5). fP (F.5)

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214 APPENDIX G TEMPERATURE DEPENDENT MATERIAL PROPERTIES FOR THE CRYOGENIC LAMINATES IN CHAPTER 6 Since we analyze the problem that was addressed by Qu et al (2003), the geometry, material parameters and the loadin g conditions are taken from that paper. Qu et al (2003) obtained the temp erature dependent propertie s by using the material properties of IM600/133 given in Aoki et al (2000) and fitted with smooth polynomials in order to be used in cal culations. The reader is refe rred to Appendix 1 of Qu et al (2003) for the details. The temperature de pendent material properties are shown in Figures G-1 and G-2. Figure G-1. Materi al properties E1, E2, G12 and 12 as a function of temperature

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215 Figure G-2. Materi al properties 1 and 2 as a function of temperature

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216 APPENDIX H DETAILS OF CONSERVATIVE CUMULATIVE DISTRIBUTION FUNCTION (CDF) FITTING In Chapter 6, we assume normal distributi ons for strains in the cryogenic laminates and estimate the mean and the standard deviation of the assumed distributions conservatively. Conservative fitting is assess ed as follows. We first perform Monte Carlo simulations with sample size of 1,000 and calcu late the mean and the standard deviation of the strains. Then we assume that th e strains follow normal distribution with the calculated mean and standard de viation. We see in Fig. 6-2 (of Chapter 6) that the normal CDF is smaller than the empirical CDF for some strain values, and larger for other strain values. That is, the normal CDF leads to conservative estimates for some strain values, while it leads to unconservative estimates for ot her strain values. It is desirable to have conservative estimates for all strain values, th at is, to have a conservative CDF fit which is smaller than the empirical CDF for all strain values. However, the tails of the distribution are volatile and fitting conservati ve CDF including these values can lead to over conservative results. Accordingly, we do not apply constraints to the first 5 points (out of 1,000 points) of the left tail and last 5 points of the right tail. Out of the remaining 990points, we choose uniformly spaced 100 point s and calculate the maximum deviation of the normal CDF fit from the empirical CDF fit at these 100 points. The maximum deviation of the fitted CDF for the empi rical CDF is called the Kolmogorov-Smirnov distance. We shift the mean value of th e fitted CDF to close the Kolmogorov-Smirnov distance between the normal fit and empirica l fit. The normal distribution with this

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217 shifted mean and original standard deviation is our conservative normal fit. As we see in Fig. 6-2 (of Chapter 6) that the conservati ve normal CDF lies below the empirical CDF for all strains except near the tails. A better conservative fit can possibly be obtained by varying the mean and standard deviation at the same time. Detailed investig ation on conservative estimation of CDF for probability of failure calculati ons is provided in Picheny et al (2006).

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218 APPENDIX I DETAILS OF DESIGN RESPONSE SURF ACE FITTING FOR THE PROBABILITY SUFFICIENCY FACTOR FOR THE CRYOGENIC LAMINATES IN CHAPTER 6 Qu et al (2003) showed that using the co mbination of face centered central composite design (FCCCD) and Latin hypercube sampling (LHS) designs gives accurate results, so we follow the same procedure. The ranges for design variab les for design response surf ace (DRS) are decided as follows. The initial estimates of the ranges for design variables were taken from Qu et al (2003). When we used these ranges, we fo und that the prediction variances at the optimum designs were unacceptably large. The ranges for DRS were then reduced by zooming around the optimum designs obtained fro m the wider ranges. After zooming, the prediction variances at the optimum designs were found to be smaller than the RMSE predictors. The final ranges for respon se surfaces are given in Table I-1. Table I-1. The ranges of variables for the three DRS constructed for PSF calculation t1 and t2 (in) 1 and 2 (deg) be=0 0.012-0.017 24-27 be=10% 0.013-0.018 24-26 be=20% 0.015-0.022 22-25 Qu et al. (2003) used a fifth-order DRS for the probability of failure, and found it to be quite accurate. We also use a fifth-orde r DRS. A fifth-order response surface in terms of four variables has 126 coefficients. Following Qu et al (2003), we used 277 design points, 25 correspond to FCCCD and 252 are generated by LHS. In addition to response surfaces for probability sufficiency factor, three more DRS were also fitted to the probability of failure for comparison purpose. The comparison of the accuracies of DRS

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219 for PSF and DRS for Pf are shown in Table I-2. For instance, for error bound of 20%, the root mean square error predictions of DRS for PSF and DRS for Pf are 3.610-3 and 7.664-4, respectively. Since PSF and Pf are not of the same order of magnitude, we cannot compare these errors directly. One possi bility is to compare the ratios of RMSE and mean of the response. When we comp are the ratios for erro r bound of 20%, we see that the ratio of RMSE and mean of the response DRS for Pf is 0.1868, while the same ratio of DRS for PSF is 0.0042. It is an indicati on that DRS for PSF is more accurate than DRS for Pf. Table I-2. Accuracies of DRS fitted to PSF and Pf in terms of four design variables (t1, t2, 1 and 2) for error bounds, be, of 0, 10%, and 20% Mean of response RMSE predictor* Ratio of RMSE to the mean of response Equivalent error in Pf Equivalent error in PSF PSF 1.077 4.655-3 4.332-3 5.397-7 ( < 9.4470-4) --be=0% Pf 8.081-4 9.447-4 1.196 --4.205-2 ( > 4.6550-3) PSF 0.9694 4.645-3 4.792-3 4.615-6 ( < 8.2810-4) --be=10% Pf 1.340-3 8.281-4 0.6180 --1.862-2 ( > 4.6450-3) PSF 0.8621 3.610-3 4.187-3 6.308-5 ( < 7.6640-4) --be=20% Pf 4.103-3 7.664-4 0.1868 --1.013-2 ( > 3.6100-3) Another way of comparing the accuracies is to calculate equivalent errors of DRS for PSF to those of DRS for Pf. That is, the equi valent error in Pf due to error in DRS for PSF can be compared to the equivalent erro r in PSF due to error in DRS for Pf. The standard errors in calculation of PSF and Pf due to limited MCS sample size are given in the last two columns of Table I-2. The standard error for Pf is calculated from

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220 1f f f PPP N (I.1) The standard error in PSF is calculated as illustrated in the following example. Assume that for calculating a pr obability of failure of 1-4, we use sample size of 106 in MCS. Then, the number of simulations failed is 100 and the standard error for Pf calculation from Eq. (I.1) is 1-5. Thus, 10 simulations out of 100 represent the standard error. The standard error in PSF can be approximated as the difference between the 105th smallest safety factor and 95th sma llest safety factor. A better estimation for PSF can be obtained by utilizing th e CDF of the safety factor S The equivalent error in Pf due to the error in DRS for PSF for error bound of 20% for instance, can be approximated as foll ows. The mean of response and RMSE prediction of DRS for PSF are =0.8621 and =3.610-3, respectively. We calculate the Pf values corresponding to PSF values of /2 and /2 as 4.605-4 and 3.974-4, respectively. The difference between these two Pf values, 6.31-5, gives an approximation for the equivalent error in Pf. We see that this equivalent error in Pf is smaller than the error in DRS for Pf, 7.664-4, indicating that the DRS for PSF has better accuracy than DRS for Pf The equivalent error in PSF due to errors in DRS for Pf can be computed in a similar manner. The equivalent error in PSF (1.013-2) due to error in DRS for Pf is larger than the error in DRS for PSF (3.610-3) indicating that the DRS for Pf does not have good accuracy. The errors in DRS for Pf are clearly unacceptable in view that the require d probability of failure is 1-4.

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221 DRS for error and quality control case Table I-1 showed the ranges of design va riables for DRS when only error reduction was of interest. When quality control is al so considered, we changed the ranges of the design variables. All properties such as th e design of experiments and the degree of polynomial were kept the same for the new re sponse surfaces; the only change made was the ranges of design variables. The new ranges of design variables used while constructing the new response surfaces are gi ven in Table I-3. Notice that the ranges for laminates thicknesses are reduced and ranges for ply angles ar e increased, the safety of the laminates are further improved by addition of quality control. Table I-3. Ranges of design variables for th e three DRS constructed for probability of failure estimation for the error and variability reduction case t1 and t2 (in) 1 and 2 (deg) be=0 0.008-0.012 27-30 be=10% 0.009-0.014 26-29 be=20% 0.013-0.018 24-27

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222 APPENDIX J ASSESSMENT OF THE ERROR DUE TO LIMITED NUMBER OF COUPON TESTS Since the (mean) failure stress of a materi al is estimated based on a finite number of coupon tests, the estimate invol ves error. Recall that the true mean of the failure stress (i.e., the population mean) fc true and the estimated mean failure stress fc calc are related to each other via 1fcfcfc calctruee (J.1) Now, the error term can be written as 1fc calc fc fc truee (J.2) Even though the true mean failure stress is a deterministic value, the calculated mean failure stress is random due to lim ited number of coupon tests. The mean and standard deviation of the calculated mean failure stress are given by fsfs calctrueE (J.3) fs true fs calcStd Std n (J.4) where E and Std denote the expected value and the standard deviation, respectively. Then, the mean and standard deviation of efc can be estimated as follows. The mean value of efc can be estimated by using first order Taylor series expansion as

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223 1fc calc fc fc trueE Ee (J.5) Since the mean value of the calculated mean and true mean are the same, Eq. (J.3), we have 0fcEe (J.6) Similarly, the standard deviation of efc can be calculated by using uncertainty propagation equation as fc fcfc calc fc calce StdeStd (J.7) where 11m fcfcfc calctruecalce E Hence, combining Eqs. (J.4) and (J.7) we can obtain the standard deviation of efc as c.o.v.fc calc fcStde n (J.8) where c.o.v. denotes the co efficient of variation.

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224 APPENDIX K PROBABILITY OF FAILURE CALCULATIONS FOR CHAPTER 7 USING SEPARABLE MCS As noted earlier, the separable Monte Ca rlo procedure applies when the failure condition can be expressed as g1( x1)> g2( x2), where x1 and x2 are two disjoint sets of random variables. To take advantage of this fo rmulation, we need to formulate the failure condition in a separable form, so that g1 will depend only on variabilities and g2 only on errors. The separable MCS procedure applies when the failure condition can be expressed as g1( x1)> g2( x2), where x1 and x2 are two disjoint sets of ra ndom variables. For that case, the probability of failure can be written as 211FPftFtdt (K.1) where f2 is the probability density function of g2 and F1 is the cumulative distribution function of g1. Since the two sets of random variab les are disjoint, we can perform one Monte Carlo simulation with x1 to calculate F1 and then perform a second Monte Carlo simulation on x2 to calculate PF from Eq. (K.1). Note that 1F1 in Eq. (K.1) is the probability of failure if g2 takes the value t and the second Monte Carlo simulation calculates the average of this probability over all possible values of g2. The common formulation of structural failure condition is in the form of a stress exceeding the material limit. This form, however, does not satisfy the separability requirement. For example, the stress depends on variability in material properties as well as design area which reflects errors in the anal ysis process. To bring the failure condition

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225 to the right form, we formulate it inst ead as the required cross section area reqA being larger than the built area Abuilt as given in Eq. (K.2) 11req builtreq twA AA vv (K.2) where Areq is the cross-sectional area required to calculate the actual service load, which is defined as reqfAP (K.3) The required area depends only on variabi lity, while the built area only on errors. When certification testing is ta ken into account, the built area, Abuilt, is replaced by the certified area, Acert, which is the same as the built area for companies that pass certification, but companies that fail are not included. That is, the failure condition is written as Failure without certification tests: 0reqbuiltAA (K.4a) Failure with certification tests: 0reqcertAA (K.4b) Equations (K.4a) and (K4.b) can be normalized by diving the terms with A0 (load carrying area without safe ty measures). Since Abuilt/A0 or Acert/A0 are the total safety factors, Eq. (K.5) states that failure occurs wh en the required safety factor is larger than the built one. Failure without certification tests: 0FF reqbuiltSS (K.5a) Failure with certification tests: 0FF reqcertSS (K.5b)

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226 So for our problem, f2 is the probability density function of the built safety factor, F builtS and F1 is the cumulative distribution function of the required safety factor, F reqS. So we can re-write the probability of failure as 1FbuiltFbuiltbuilt builtreqfSFSFFPfSFSdS (K.6) Here the probability of failure for a single company is 1F reqfSPF and integration of Pf over all aircraft companies gives the average probability of failure PF. For a given F builtS we can calculate the probability of failure, Eq. (K.5a), by simulating all the variabilities with an MC S. Figure K-1 shows the dependence of the probability of failure on the total safety factor obtained using MCS with 1,000,000 variability samples. The zigza gging in Figure K-1 at high safe ty factor values is due to limited sample of MCS. The dependence on log10( Pf) on F builtS can be represented with response surface approximations (RSA), which al so eliminates the noise at high failure probabilities. The fitted RSA for log10( Pf) in terms of the required safety factor F builtS is given in Eq. (K.7). 432 10100.8 log5.78318.0613.291.552.8860.81.3 4.8014.6571.3F built fFFFFF builtbuiltbuiltbuiltbuilt FF builtbuiltforS PSSSSforS SforS (K.7)

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227 Figure K-1. The variation of probability of failure with built total safety factor. Note that Pf is one minus the cumulative distribution function of F reqS. Figure K-2 represents flowchart of sepa rable MCS procedure. In Stage-1 of separable MCS, we first assess th e cumulative distribution function F reqSF numerically, that is we calculate the empi rical CDF. Then, we compute 1Fbuilt reqSFFS which is equal to the probability of failure for a single company. As noted earlier we find that the dependence on log10( Pf) on F builtS can be represented accura tely with response surface approximations. We use 1,000,000 MCS for genera ting the RSA. Using Fig. K-1 (that is, Eq. (K.7), we calculate probabi lity of failure for a given F builtS for an individual aircraft company. The integration of Pf over all companies to estimate the average probability of failure over all companies, PF, is performed in Stage-2.

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228 Figure K-2. Flowchart for MCS of component design and failure In Stage-2, N designs are generated for N different aircraft companies. For each new design, different random errors are pick ed from their corres ponding distributions. The testing of designs is performed in a building-block type of approach. In this sequence, first simulate coupon tests (i.e., unidi rectional laminate test s) that reduce errors in the material constants and failure limits. The nominal value of the number of tests is taken as 40, but the number of tests is varied to see their effect on the results. Then, we simulate structural element tests and materi al allowable stress is updated based on these tests. Finally, certification testing is simulated in this stage. The separable Monte Carlo procedure re duces the computati onal burden greatly. For instance, if the probability of failure is 2.5-5, a million MCS estimates this probability with 20% error. We found for our problem that the use of separable Monte

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229 Carlo procedure requires only 20,000 simulati ons (10,000 simulations for Stage-1 and 10,000 for Stage-2) for the same level of accuracy. Once the probability of failure of a single structural part is calculated, the probability of failure of the system can be calculated from 2 2 2 011 exp 21 1FSfPPdz z z (K.8) Calculation of system probability of failure utilizing bivariate normal distribution is discussed in detail in Appendix E.

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230 APPENDIX L CHANGE IN COST DUE TO INCREASE OF THE STRUCTURAL WEIGHT The cost associated with a ch ange in the structural weight and the fuel cost is taken from PhD thesis of Kale (2005). Kale refers to Venter (1999) who assumed a jet fuel cost of $0.89 per gallon and calculated that a pound of structural wei ght costs 0.1 pound of fuel per flight. Using this information Kale (2005) calculated a pound of structural weight costs $0.015 per flight for fuel Here, in this paper, we update the fuel cost by simply doubling it to take the recent fuel price increase into account. Th at is, we assume that fuel cost is $1.78 per gallon. So, a pound of stru ctural weight costs $0.03 per flight. The material and manufacturing co st per pound of structural wei ght is taken as $150 as in Kale (2005). Following the cost function formul ation of Kale (2005), we write the cost function as ccfstrucCostMFNW (L.1) where Mc is material and manufacturing co st per pound of structural weight, Fc is fuel cost per flight per pound of structural weight, Nf is the number of fli ghts (taken as 40,000 following Kale (2005)) and Wstruc is the structural weight. Using Mc=$150/lb, Fc=$0.03 / (lb-flight) and Nf=40,000, the cost function in Eq. (L.1) becomes 15012001350 s trucstrucCostWW (L.2) where Cost is in dollars when Wstruc is in pounds. Notice from Eq. (L.2) that fuel cost dominates over material and manufacturing cost (fuel cost is eight times larger than material and manufacturing cost). The struct ural weight of a t ypical large transport

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231 aircraft is about 100,000 pounds. We assume that half of th e structure is designed for point stress failure. So, half of the structural weight will be aff ected from the simple analysis presented herein. For example, wh en our analysis results in 1% weight reduction, it translates into $675,000 cost saving. Note that we use a simple cost function with representative cost parameters, but still it helps to translate the weight saving due to of structural element tests a nd uncertainty reduction mechanisms to lifetime cost saving.

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232 APPENDIX M RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY INDEX OF CERTIFICATION FAILURE RATE, RELIABILITY INDEX OF PROBABILITY OF FAILURE AND BUILT SAFETY FACTOR IN CHAPTER 7 In order to alleviate the computational co st of the optimization problems stated in Eqs. (7.44) and (7.47) we use response su rface approximations, RSA. The reliability index of CFR, CFR reliability index of PF, FP and built safety factor, ( SF)built, are approximated with fifth-order polynomial RSAs. For design of experiments, we use combination of face centered composite central design (FCCCD) and Latin hypercube sampling (LHS). A fifth-order pol ynomial in two variables (here Scl and Sch) have 21 coefficients. FCCCD provides 9 designs for tw o variables, and we generate 33 designs via LHS so that the number of designs is twice the number of coefficients. Table M-1 presents evaluation of the accuracies of RSAs for the nominal case (i.e., number of coupon tests is 40, number of element tests is 3, error bounds are at their nominal values and coefficient of variation of the failure stre ss is 8% as given in Tables 7-1 and 7-2 of Chapter 7). We see that accuracies of RSAs are at acceptable level. Table M-1. Accuracy of response surfaces RSA for CFR RSA for f P RSA for ( SF)built R2 adj 0.9991 0.9988 1.000 RMSE predictor/mean of the response 0.95% 0.54% 0.01% erms in 10 tests points/mean of the response* 2.62% 0.39% 0.01% Test points are generated via Latin hypercube sampling

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233 APPENDIX N CALCULATION OF THE MEAN AND THE C.O.V. OF THE STRESS DISTRIBUTION USING PROBABILITY OF FAILURE INFORMATION The probability of failure is defined in terms of the probability distribution functions of the stress and the failure stress in Chapter 8, Eq. (8.5) fPFsd We assume that the stress follows normal distribution. The parameters of the normally distributed stress are the mean, and standard deviati on (or coefficient of variation, c can also be used instead of standard deviation). We assume that material characterization tests provide us an accurate failu re stress distribution. If the failure stress also follows normal distribut ion with mean value of f and coefficient of variation of f c then Eq. (8.5) can be reduced to 1fP (N.1) where is the cumulative distribution function of the standard nor mal distribution and is the reliability index, which is calculated as 2222,f ffc cc (N.2) Now consider the reverse situ ation. Given the estimate of the probability of failure, Pf given, and the coefficient of variation of the stress c we can compute the mean value of the stress from

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234 24 2 B BAC A (N.3) where 221givenAc, 2 f B 2221fgivenfCc and 11 g ivenfgivenP. Similarly, if the mean value of the stress is known, then the coefficient of variation of the stress can be calculated from Eq. (N.1). When the failure stress follows lognormal distribution, then the probability of failure is calculated via the integral given in Eq. (8.5). Hence, given the distribution parameters of lognormally distributed failure stress f and f probability of failure is a function of the mean and coefficient of vari ation of the stress as given in Eq. (N.4) ,,,,,fffPcFscd (N.4) So given the estimate of the probability of failure, f givenP, the mean and coefficient of variation of the stress distri bution can be calculated from ,0ffgivenPcP (N.5) Equation (N.5) is nonl inear in terms of and c When c is known, then can be calculated using Bisect ion method or Newtons method. Alternatively, if the mean value of the stress is known, then the coefficient of variation of th e stress can be calculated. The computations are performed using MATLAB which has the following built-in functions for numerical computations. Equatio n (N.5) can be solved for the mean or coefficient of variation of the stress using the function fzero, which uses a combination of bisection, secant, and inverse quadratic interpolation methods. The integral given in Eq. (N.4) can be computed using the function quadl, which numerically evaluates the

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235 integral using adaptive Lobatto quadrature technique. The integrand of Eq. (N.4) can be easily calculated using MATLAB. For norm ally distributed stress, the function normpdf can be used to compute the probability density function s( ) and for lognormally distributed failu re stress the function logncdf can be used to compute the cumulative distribution function F ( ).

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236 APPENDIX O RELATION OF COMPONENT WEIGHTS AND OPTIMUM COMPONENT FAILURE PROBABILITIES IN CHAPTER 8 In Chapter 8, we found that the ratio of probabilities of failure of the wing and the tail are very close to the ratio of their weight s. Here we aim to provide an analytical proof by utilizing some approximations. The probability of failure of the wing and tail system is defined as 111 f fWfTPPP (O.1) Let w be the weight transferred from the wing to the tail as a result of probabilistic optimization. The optimality condition of optimization for safety is 110ffTfWfTfW fWfTPPPPP PP wwwww (O.2) Noting that f TTPF and *fWWPF, where F is the cumulative distribution function of the failu re stress, and using the chain rule the partial derivatives in Eq. (O.2) can be written as ** *fTfT TT T TPP f www ** *fWfW WW W WPP f www (O.3) where and T f and W f are the values of probability de nsity function (PDF) of the failure stress evaluated at *T and *W respectively. That is, *TTffand *WWff, where f is the PDF of the failure stress. Now, combining (O.2) and (O.3) we get

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237 *0W T TWff ww or *W T W Tw f f w (O.4) Recall that we assumed the stresses and wei ghts are inversely propor tional. That is, *** dWdW WdWdW WdWWW WWw *** dTdT TdTdT TdTWW WWw (O.5) Then, the ratio of partial derivatives *Ww and *Tw can be approximated as 2 * 2 dW dW dW WdT dT dW T dT dTW Ww wW W W w Ww (O.6) where we the second equality holds true since the moved material w is much smaller than both component weights dTW and dWW, and the deterministic characteristic stresses are equal (since they are made of the same mate rial and designed for the same probability of failure). Then, Eqs. (O.4) and (O.6) can be combined to yield dT T WdWW f fW (O.7) Now, we need to relate the ratio of PDFs to the ratio of probabilities of failure. The probability of failure is defined in term s of the PDF of the failure stress as *fPfxdx (O.8) If we assume that the failure stress is normally distributed, then the probability of failure can be written as 22s x fPcedx (O.9)

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238 where c is a constant and s is the characteristic stress scaled with the mean and standard deviation of the failure stress as given in Eq. (O.10) 1 2 c * f f s std (O.10) Here f and f s td are the mean and standard deviation of the failure stress. Since the normal distribution is symmet ric, the probability of fail ure, Eq. (O.9), can be rewritten as 22 *22s xx f sPcedxcedx (O.11) The probability of failure can be re-formulated by using the equality, Eq. (O.12), given in Abramovitz and Stegun (p. 298, eq. 7.1.14) 2211/213/22 20zx zeedxz zzzzz (O.12) where 11/213/22 zzzzz is a continued fraction (lets denote as CF(z)) which can also be written as 11/213/221 1/2 1 3/2 2 CFz zzzzz z z z z z (O.13) Then, Eq. (O.12) is re-written as 222 x z zCFz edxe (O.14) or

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239 22221 22 x z zz edxCFe (O.15) Then, the probability of failure, Eq. (O.11), is reduced to 2 ** 222s fcs PCFe (O.16) Since the probabilities of failure of airc raft structures are on the orders of 10-7, the scaled characteristic stress is negative and its absolute value is much smaller than one. That is, *1s *0 s (O.17) Based on Eq. (O.17), the cont inued fraction in Eq. (O.13) can be approximated as 1 CFz z Then, the probability of failure, Eq. (O.16), becomes 2 ** 2 **s f f s c Pe ss (O.18) Thus, from Eq. (O.18) the probabilities of failure of the wing and the tail are * T fT Tf ffs f P s std * W W fW W Wf ffs f P s std (O.19) which leads to * f TWf T fWW TfP f Pf (O.20) Note that the stresses at the deterministic design are close to stre sses at probabilistic design, so *1W T Then, Eq. (O.20) can be simplified to

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240 fT T f WWP f Pf (O.21) Finally, combining Eqs. (O.7) and (O.21) we find that the ratio of failure probabilities is approximately equal to the ratio of weights. That is, fT dT T f WWdWP W f PfW (O.22)

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241 APPENDIX P HISTORICAL RECORD FOR AIRCRAFT PROBABILITY OF FAILURE Since aircraft structural de sign still relies on deterministic optimization, we first look at the historical record on the probability of failure of traditionally (i.e., via deterministic design) designed aircraft structures. Tong (2001) performed a thorough literature review on aircraft stru ctural risk and relia bility analysis. Tong (2001) refers to the paper by Lincoln (1996) that reports the overall failure rate for all systems due to structural faults is one aircraft lost in more than ten million flight hours, i.e. Pf =10-7 per flight hours. The Boeing Compa ny publishes the Statistical Summary of Commercial Jet Aviation Accidents each year, and provides data back to 1959 to indicate trends. The number of accidents that occurred between 1959 and 2001 due to structural failure, the total number of accidents and the accident ra te corresponding to different aircraft generations are listed in Table P-1. Table P1 shows that failure probability per departure of second generation airplanes is 4.31 10-8, whereas the failure probability of early widebody airplanes and current generation airplanes are 2.0 10-7 and 1.86 10-8, respectively. Cowan et al (2006) presented data on commer cial jet plane accidents involving aircraft operated by U.S. Air Carriers betw een 1983 and 2003, and listed the number of accidents that caused failure of structural co mponents. The number of accidents resulted in wing failure is given as 18, while the numbe r of accidents due to ta il failure is 9. This indicates that the probability of failure of the tail is about half of that of the wing.

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242 Table P-1. Aircraft accidents a nd probability of failure of airc raft structures. Examples of first generation airplanes are Comet 4, 707, 720, DC-8. Boeing 727, Trident, VC-10, 737-100/-200 are examples of second generation airplanes. Early widebody airplanes are 747-100/-200 /-300/SP, DC-10, L-1011 and A300. Examples of current generation airp lanes are MD-80/-90, 767, 757, A310, A300-600, 737-300/-400/-500, F-70, F-100, A320/319/321. Aircraft Generation* Accident Rate per million departures* (A) Total Number of accidents* (B) Accidents due to structural failure* (C) Structural failure rate per departure (AC / B) First 27.2 49 0 0 Second 2.8 130 2 4.31 10-8 Early widebody 5.3 53 2 2.00 10-7 Current 1.5 161 2 1.86 10-8 Total --393 6 --* These columns are taken from the Boeing accident report (2001)

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243 LIST OF REFERENCES Abramovitz, M., and Stegun, I. (1970), Handbook of Mathematical Functions Dover, New York. Acar, E., Kale, A., and Akgn, M.A. (2004a), Reliability-Based Design and Inspection Schedule Optimization of an Aircraft St ructure Containing Multiple Site Damage. Proceedings of New Trends in Fatigue and Fracture IV Paper no. 19, Aleppo, Syria, 10-12 May 2004. Acar, E., Kale, A., and Haftka, R.T. (2004b), Effects of Error, Variability, Testing and Safety Factors on Aircraft Safety. Proceedings of the NSF workshop on Reliable Engineering Computing Savannah, Georgia, September 2004, pp. 103-118. Acar, E., Kale, A., Haftka, R.T., and Stroud, W. J. (2006a), Structural Safety Measures for Airplanes. Journal of Aircraft Vol. 43, No. 1, pp. 30-38. Acar, E., Haftka, R.T., Sankar, B.V., and Qu i, X. (2006b), Increasing Allowable Flight Loads by Improved Structural Modeling. AIAA Journal Vol. 44, No. 2, pp. 376381. Acar, E., Haftka, R.T., and Johnson, T.F. (2006c), Tradeoff of Uncertainty Reduction Mechanisms for Reducing Structural Weight. ASME Journal of Mechanical Design in press. Acar, E., Kale, A., and Haftka, R.T. (2006d), Comparing Effectiveness of Measures that Improve Aircraft Structural Safety. ASCE Journal of Aerospace Engineering submitted. Acar, E., and Haftka, R.T. (2006e), Reliabil ity Based Aircraft St ructural Design Pays Even with Limited Statistical Data, Journal of Aircraft submitted. Aerospace Information Report, (1997), Integr ation of Probabilistic Methods into the Design Process. Society of Automotive Engineers Report No: AIR-5080. Ang, A.H-S., and Tang, W.H. (1975), Probability Concepts in Engineering Planning and Design, Volume I: Basic Principles John Wiley & Sons, New York. Antonsson, E.K., and Otto, K.N. (1995) Imprecision in Engineering Design. ASME Journal of Mechanical Design Vol. 117 B, pp. 25-32.

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256 BIOGRAPHICAL SKETCH Erdem Acar was born in Ankara, Republic of Trkiye, in 1977. He received his Bachelor of Science in aeronautical engine ering from Middle East Technical University in June 1999. Mr. Acar started his graduate stud ies as a graduate research assistant in the same institution. He did his masters on the subject of thermo-mechanical fatigue life assessment of jet engine components. His inte rest in conducting research motivated him to join the Structural and Multidisciplinary Optimization Group of Professor Haftka at the University of Florida, in July 2003, to pur sue his PhD degree in aerospace engineering. During his PhD study, he did internship in EMBRAER aircraft company in Brazil, where he worked on multidisciplinary design optimi zation of tail boom of an aircraft.


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Permanent Link: http://ufdc.ufl.edu/UFE0015222/00001

Material Information

Title: Aircraft structural safety : effects of explicit and implicit safety measures and uncertainty reduction mechanisms
Physical Description: Mixed Material
Language: English
Creator: Acar, Erdem ( Dissertant )
Haftka, Raphael T. ( Thesis advisor )
Sankar, Bhavani V. ( Thesis advisor )
Arakere, Nagaraj ( Reviewer )
Kim, Nam-Ho ( Reviewer )
Uryasev, Stanislav ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2006
Copyright Date: 2006

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, Ph. D.
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
theses   ( marcgt )

Notes

Abstract: Aircraft structural safety is achieved by using different safety measures such as safety and knockdown factors, tests and redundancy. Safety factors or knockdown factors can be either explicit (e.g., load safety factor of 1.5) or implicit (e.g., conservative design decisions). Safety measures protect against uncertainties in loading, material and geometry properties along with uncertainties in structural modeling and analysis. The two main objectives of this dissertation are: (i) Analyzing and comparing the effectiveness of structural safety measures and their interaction. (ii) Allocating the resources for reducing uncertainties, instead of living with the uncertainties and allocating the resources for heavier structures for the given uncertainties. Certification tests are found to be most effective when error is large and variability is small. Certification testing is more effective for improving safety than increased safety factors, but it cannot compete with even a small reduction in errors. Variability reduction is even more effective than error reduction for our examples. The effects of structural element tests on reducing uncertainty and the optimal choice of additional knockdown factors are explored. We find that instead of using implicit knockdown factors based on worst-case scenarios (current practice), using test-dependent explicit knockdown factors may lead weight savings. Surprisingly, we find that a more conservative knockdown factor should be used if the failure stresses measured in tests exceeds predicted failure stresses in order to reduce the variability in knockdown factors generated by variability in material properties. Finally, we perform probabilistic optimization of a wing and tail system under limited statistical data for the stress distribution and show that the ratio of the probabilities of failure of the probabilistic design and deterministic design is not sensitive to errors in statistical data. We find that the deviation of the probabilistic design and deterministic design is a small perturbation, which can be achieved by a small redistribution of knockdown factors.
Subject: aircraft, certification, composite, conservative, cumulative, density, effectiveness, error, failure, function, knockdown, optimization, probabilistic, proof, reduction, redundancy, reliability, safety, sandwich, strain, stress, structural, test, uncertainty, variability
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 282 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2006.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003614515
System ID: UFE0015222:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015222/00001

Material Information

Title: Aircraft structural safety : effects of explicit and implicit safety measures and uncertainty reduction mechanisms
Physical Description: Mixed Material
Language: English
Creator: Acar, Erdem ( Dissertant )
Haftka, Raphael T. ( Thesis advisor )
Sankar, Bhavani V. ( Thesis advisor )
Arakere, Nagaraj ( Reviewer )
Kim, Nam-Ho ( Reviewer )
Uryasev, Stanislav ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2006
Copyright Date: 2006

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, Ph. D.
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
theses   ( marcgt )

Notes

Abstract: Aircraft structural safety is achieved by using different safety measures such as safety and knockdown factors, tests and redundancy. Safety factors or knockdown factors can be either explicit (e.g., load safety factor of 1.5) or implicit (e.g., conservative design decisions). Safety measures protect against uncertainties in loading, material and geometry properties along with uncertainties in structural modeling and analysis. The two main objectives of this dissertation are: (i) Analyzing and comparing the effectiveness of structural safety measures and their interaction. (ii) Allocating the resources for reducing uncertainties, instead of living with the uncertainties and allocating the resources for heavier structures for the given uncertainties. Certification tests are found to be most effective when error is large and variability is small. Certification testing is more effective for improving safety than increased safety factors, but it cannot compete with even a small reduction in errors. Variability reduction is even more effective than error reduction for our examples. The effects of structural element tests on reducing uncertainty and the optimal choice of additional knockdown factors are explored. We find that instead of using implicit knockdown factors based on worst-case scenarios (current practice), using test-dependent explicit knockdown factors may lead weight savings. Surprisingly, we find that a more conservative knockdown factor should be used if the failure stresses measured in tests exceeds predicted failure stresses in order to reduce the variability in knockdown factors generated by variability in material properties. Finally, we perform probabilistic optimization of a wing and tail system under limited statistical data for the stress distribution and show that the ratio of the probabilities of failure of the probabilistic design and deterministic design is not sensitive to errors in statistical data. We find that the deviation of the probabilistic design and deterministic design is a small perturbation, which can be achieved by a small redistribution of knockdown factors.
Subject: aircraft, certification, composite, conservative, cumulative, density, effectiveness, error, failure, function, knockdown, optimization, probabilistic, proof, reduction, redundancy, reliability, safety, sandwich, strain, stress, structural, test, uncertainty, variability
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 282 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2006.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003614515
System ID: UFE0015222:00001


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AIRCRAFT STRUCTURAL SAFETY: EFFECTS OF EXPLICIT AND IMPLICIT
SAFETY MEASURES AND UNCERTAINTY REDUCTION MECHANISMS
















By

ERDEM ACAR


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

Erdem Acar

































This dissertation is dedicated to my family: my father Zuhuri Acar, my mother Serife
Acar, and my sister Asiye Acar.















ACKNOWLEDGMENTS

I would like to express special thanks and appreciation to Dr. Raphael T. Haftka,

chairman of my advisory committee. I am grateful to him for providing me with this

excellent opportunity and financial support to complete my doctoral studies under his

exceptional guidance. He encouraged me to attend several conferences and assisted in

finding an internship during my studies. Through our weekly meetings and his open door

policy, which I definitely over-exploited, he greatly contributed to this dissertation. His

limitless knowledge and patience are inspiration to me. During the past three years, he

was more than my PhD supervisor; he was a friend, and sometimes like a father. I

sincerely hope we will remain in contact in the future.

I would also like to thank the members of my advisory committee, Dr. Bhavani V.

Sankar, co-chair of the committee, Dr. Nagaraj Arakere, Dr. Nam-Ho Kim and Dr.

Stanislav Uryasev. I am grateful for their willingness to review my Ph.D. research and to

provide me with the constructive comments which helped me to complete this

dissertation. In particular, I would like to extend special thanks to Dr. Bhavani V. Sankar

for his guidance on the papers we co-authored, and Dr. Nam-Ho Kim for his comments

and suggestions during the meetings of the Structural and Multidisciplinary Group.

I also wish to express my gratitude to my M.Sc. advisor, Dr. Mehmet A. Akgun,

who provided a large share of motivation for pursuing a doctorate degree. The experience

he supplied me during my master's degree studies contributed to this dissertation.









I also wish to thank to my colleagues at the Structural and Multidisciplinary Group

of the Mechanical and Aerospace Engineering Department of the University of Florida

for their support, friendship and many technical discussions. In particular, I would like to

thank Dr. Melih Papila, Dr. Jaco Schutte, my soul sister Lisa Schutte, Tushar Goel and

Ben Smarslok for their friendship (in the order of meeting with them).

Financial support provided by NASA Cooperative Agreement NCC3-994, NASA

University Research, Engineering and Technology Institute and NASA Langley Research

Center Grant Number NAG1-03070 is gratefully acknowledged.

Finally, my deepest appreciation goes to my family: my father Zuhuri Acar, my

mother Serife Acar and my sister Asiye Acar. The initiation, continuation and final

completion of this thesis would not have happened without their continuous support,

encouragement and love. I am incredibly lucky to have them in my life.
















TABLE OF CONTENTS



A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .............. ............................................... ........ xi

LIST OF FIGURES .................................................... ............... ......... ... xv

N O M E N C L A T U R E .................................................. ................................................ xix

A B STR A C T .............................. ........................................................ xxv

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

M o tiv atio n .......................................................... 1
O bj ectiv e s ................................................................... ................................. . 5
M methodology ............................................................................................................. 6
O u tlin e ............................................................................ . 7

2 LITERATURE REVIEW .............................................................. ...............12

Probabilistic vs. D eterm inistic D esign ....................................................... 12
Structural Safety A analysis ............................................................. 14
Probability of Failure Estimation ..................................... .......... 15
Analytical calculation of probability of failure ...........................................15
Moment-based techniques ........................................... ..................16
Sim ulation techniques ............................................................................ 17
Separable Monte Carlo simulations ..................................................18
Response surface approximations .......................................................... 19
Reliability-Based Design Optimization .......................................... .......... .........20
Double loop (Nested) RBDO ..................................... ...............20
Single loop RB D O .................................. ....................................... 21
Error and V ariability ...................... ................. ..... .......................... 22
U uncertainty Classification ........................................................ ...............22
Reliability Improvement by Error and Variability Reduction..............................23
Testing and Probabilistic D esign ................................................... ............... ... 24









3 WHY ARE AIRPLANES SO SAFE STRUCTURALLY? EFFECT OF
V ARIOU S SAFETY M EA SURES ................................................. .....................28

In tro du ctio n ...................................... ................................................ 2 8
Structural U uncertainties ........................................... ................................. 30
Safety M measures .......................................... ........... ...... ....... .......... 32
D esign of a G eneric Com ponent........................................... .......................... 33
Design and Certification Testing.......... ............ ......... .............. 33
Effect of Certification Tests on Distribution of Error Factor e ..........................36
Probability of Failure Calculation by Analytical Approximation.....................38
Effect of Three Safety Measures on Probability of Failure.................... ........ 41
S u m m a ry .......................... ............... .... .. .................... ................ 5 1

4 COMPARING EFFECTIVENESS OF MEASURES THAT IMPROVE
AIRCRAFT STRUCTURAL SAFETY .......................................... ............... 53

In tro d u ctio n ........................................................................................5 3
L oad Safety Factor ............................ .. .... .............................. ......... 54
Conservative M material Properties ............................................. ............... 54
T e st s ............................................................................................................... 5 4
R e d u n d a n cy ................................................................................................... 5 5
Error R education .................................. ... .. ......... ...............55
Variability Reduction ................................ ..... .. ....... .... .. ............. 55
Errors, Variability and Total Safety Factor ..................................... ............... 56
E rrors in D esign ............................................... ................ 56
Errors in Construction ................... .... .......... ............. .... ........58
T total E rror F actor ...................................................... ...... .................. .... 59
T otal Safety F actor ......................... .. .................... ......... ........... 60
V ariability ................... ................................................................................. 6 1
C ertificatio n T ests............. ........................................................................... .. ...... .. 6 2
Probability of Failure Calculation ................................................... .....................65
Probability of Failure Calculation by Separable MCS .......................................65
In clu ding R edu n dan cy ............................ .........................................................70
R e su lts ............................................................................................ 7 0
E effect of E errors ............................ ................... ................................... 70
Weight Saving Due to Certification Testing and Error Reduction................. 73
Effect of R edundancy ............................ ..... ............. ................................. 74
Additional Safety Factor Due to Redundancy...............................................77
Effect of V ariability Reduction ..........................................................................78
S u m m ary ................................ ........................................................ 8 1

5 INCREASING ALLOWABLE FLIGHT LOADS BY IMPROVED
STRUCTURAL MODELING............................................... 82

Introduction ........................................................... ..... ......................... 82
Structural Analysis of a Sandwich Structure.............................................................85
Analysis of Error and Variability ................................................... ..................89









Deterministic Design and B-basis Value Calculations.............................................93
A ssessm ent of Probability of Failure..................................................... ............... 96
Analyzing the Effects of Improved Model on Allowable Flight Loads via
P rob ab ilistic D e sig n ...................................................................... .......... .. .. 9 9
S u m m a ry .......................................................................................1 0 1

6 TRADEOFF OF UNCERTAINTY REDUCTION MECHANISMS FOR
REDUCING STRUCTURAL WEIGHT......................................................103

Introduction...................................................... ................. ............ ..... 104
Design of Composite Laminates for Cryogenic Temperatures ............................ 106
Calculation of Probability of Failure .................................. ........... ................... 108
Probabilistic Design Optim ization...................................................................... 112
Probabilistic Sufficiency Factor (PSF).............................................................112
D esign Response Surface (DR S)................. ..................... ....................... 113
Weight Savings by Reducing Error and Employing Manufacturing Quality
C o n tro l .................... ................. ........... ... .............. ............... 1 14
Choosing Optimal Uncertainty Reduction Combination .............. .... ...............118
Sum m ary .............. ........................................ ..............................119

7 OPTIMAL CHOICE OF KNOCKDOWN FACTORS THROUGH
PROBABILISTIC DESIGN ......................................................... ............. 121

Introduction .................................122................................................
Testing of A aircraft Structures .............................................................. ............... 123
Quantification of Errors and Variability................................ .... ............... 125
Errors in Estimating Material Strength Properties from Coupon Tests ............125
Errors in Structural Elem ent Tests ..................... .. ..... ..... .................... 127
Allowable stress updating and the use of explicit knockdown factors............129
Current industrial practice on updating allowable stresses using worst-
case conditions (implicit knockdown factors) ..................................129
Proposal for a better way to update allowable stresses: Using the average
failure stress measured in the tests and using optimal explicit
knockdow n factors...................... .. .. ............................. ............... 130
Error updating via elem ent tests............................... .. ............. ........... 134
Errors in D design .................................. ..............................................135
E rrors in C construction .............................................. ............................. 137
Total Error Factor ........... ........... ........ ................. ............. 138
Total Safety Factor ............ ........................... ........... ...... ........ .. 138
Variability ......................................139
Simulation of Certification Test and Probability of Failure Calculation.................. 141
Simulation of Certification Test ...... ................. ...............141
Calculation of Probability of Failure................................. ............. ........... 142
R results ............................................. ............. ................... 144
Optimal Choice of Explicit Knockdown Factors for Minimum Weight and
M minimum Certification Failure Rate ............ ....... ............. ..........145









Optimal Choice of Explicit Knockdown Factors for Minimum Weight and
M inim um Probability of Failure ................................................................148
Effect of Coupon Tests and Structural Element Tests on Error in Failure
Prediction .................... ... .. ...... ........ ... ..... .......... ..................... 150
Effect of number of coupon tests alone (for a fixed number of element
te sts, en = 3 ) .................................... .... .... .. ............................. 1 5 0
Effect of number of element tests alone (for a fixed number of coupon
te sts, n c= 4 0 ) .................... .. .. .................... .........................15 1
Advantage of Variable Explicit Knockdown Factors ............. ... .................153
Effect of Other Uncertainty Reduction Mechanisms ....................................... 157
Effect of variability reduction ............... ............................................. 157
Effect of error reduction ................................... ............................. ....... 159
Effect of Number of Coupon Tests ............... .............................................. 161
Effect of Number of Structural Element Tests .............................................162
Sum m ary ................ ................................... ...........................164

8 RELIABILITY BASED AIRCRAFT STRUCTURAL DESIGN PAYS EVEN
WITH LIMITED STATISTICAL DATA............................................ ..........165

Introduction ......... .. ...... ......... ...... .. ....... ........ ......................... 165
Demonstration of Gains from Reliability-Based Structural Design Optimization of
a R representative W ing and Tail System ........................................ ....................167
Problem Formulation and Simplifying Assumptions ............ ... .................167
Probabilistic Optimization with Correct Statistical Data ...............................169
Effect of Errors in Information about Deterministic Design.................................. 174
Errors in Coefficient of Variation of Stresses ..................................................174
Erroneous M ean Stresses........................................... ....... ........................ 177
Errors in Probability of Failure Estimates of Deterministic Design................ 179
Effect of Using Wrong Probability Distribution Type for the Stress ..............181
Approximate Probabilistic Design Based on Failure Stress Distributions .............182
Application of Characteristic Stress Method to Wing and Tail Problem...............86
Sum m ary ............................................................... ..... ..... ......... 189

9 CONCLUDING REM ARKS ........................................................ ............. 192

APPENDIX

A A-BASIS AND B-BASIS VALUE CALCULATION...........................................197

B PROBABILITY CALCULATIONS FOR CHAPTER 3 ......................................199

Calculation of Pr(CTle), the Probability of Passing Certification Test ....................199
Calculations of Mean and Standard Deviation of Probability of Failure .................200

C CONFLICTING EFFECTS OF ERROR AND VARIABILITY ON
PROBABILITY OF FAILURE IN CHAPTER 3 ................................... .........202









D COMPARISON OF RESULTS OF SINGLE ERROR FACTOR AND
MULTIPLE ERROR FACTOR CASES ....................... ......... .........2...............204

E DETAILS OF SEPARABLE MONTE CARLO SIMULATIONS FOR
PROBABILITY OF FAILURE CALCULATIONS IN CHAPTER 4..................209

F CALCULATION OF THE SYSTEM FAILURE PROBABILITY USING
BIVARIATE NORMAL DISTRIBUTION .................................. ............... 212

G TEMPERATURE DEPENDENT MATERIAL PROPERTIES FOR THE
CRYOGENIC LAMINATES IN CHAPTER 6 ............................. ................214

H DETAILS OF CONSERVATIVE CUMULATIVE DISTRIBUTION
FUN CTION (CDF) FITTIN G ........................................................ ............... 216

I DETAILS OF DESIGN RESPONSE SURFACE FITTING FOR THE
PROBABILITY SUFFICIENCY FACTOR FOR THE CRYOGENIC
LA M IN A TE S IN CH A PTER 6..................................................... .....................218

J ASSESSMENT OF THE ERROR DUE TO LIMITED NUMBER OF COUPON
T E S T S ..................................................... .... ................. 2 2 2

K PROBABILITY OF FAILURE CALCULATIONS FOR CHAPTER 7 USING
SE P A R A B L E M C S ......................................................................... ...................224

L CHANGE IN COST DUE TO INCREASE OF THE STRUCTURAL WEIGHT ..230

M RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY INDEX OF
CERTIFICATION FAILURE RATE, RELIABILITY INDEX OF
PROBABILITY OF FAILURE AND BUILT SAFETY FACTOR IN CHAPTER
7 ............................................................................... 2 3 2

N CALCULATION OF THE MEAN AND THE C.O.V. OF THE STRESS
DISTRIBUTION USING PROBABILITY OF FAILURE INFORMATION .........233

O RELATION OF COMPONENT WEIGHTS AND OPTIMUM COMPONENT
FAILURE PROBABILITIES IN CHAPTER 8 ............................. ................236

P HISTORICAL RECORD FOR AIRCRAFT PROBABILITY OF FAILURE ........241

L IST O F R E F E R E N C E S ...................................................................... .....................243

B IO G R A PH IC A L SK E T C H ........................................ ............................................256















LIST OF TABLES


Table p

3-1 U uncertainty classification......................................................... ............... 31

3-2 Distribution of random variables used for component design and certification ......36

3-3 Comparison of probability of failures for components designed using safety
factor of 1.5, mean value for allowable stress and error bound of 50% ...................40

3-4 Probability of failure for different bounds on error e for components designed
using safety factor of 1.5 and A-basis property for allowable stress .....................42

3-5 Probability of failure for different bounds on error e for components designed
using safety factor of 1.5 and mean value for allowable stress.............................44

3-6 Probability of failure for different bounds on error e for safety factor of 1.0 and
A -basis allow able stress ........................ ......... .... ...... ............ 46

3-7 Probability of failure for different error bounds for safety factor of 1.0 and mean
v alu e for allow ab le stress .............................................................. .....................4 6

3-8 Probability of failure for different uncertainty in failure stress for the
components designed with safety factor of 1.5, 50% error bounds e and A-basis
allow able stress. .................................................... ................. 47

3-9 Probability of failure for different uncertainty in failure stress for the
components designed with safety factor of 1.5, 30% error bound e and A-basis
allow able stress. .................................................... ................. 47

3.10 Probability of failure for uncertainty in failure stress for components designed
using safety factor of 1.5, 10% error bounds e and A-basis properties....................48

4-1 Distribution of error factors and their bounds ........... ............ .......................59

4-2 Distribution of random variables having variability ...........................................61

4-3 Mean and standard deviations of the built and certified distributions of the error
factor etotal and the total safety factor SF........... .............................. ...............64

4-4 Average and coefficient of variation of the probability of failure for the
structural parts designed with B-basis properties and SFL=1.5................................ 72









4-5 Reduction of the weight of structural parts by certification testing for a given
probability of failure ............................................. .................. .... 74

4-6 Effect of redundancy on the probabilities of failure...............................................75

4-7 Effect of redundancy on the effectiveness of certification testing ...........................76

4-8 Effect of correlation coefficient p on system failure probabilities and
effectiveness of certification testing ......... ................... ....... .... ..................... 77

4-9 Additional safety factor due to redundancy .................................. ............... 78

4-10 Comparison of system failure probabilities corresponding to different variability
in failure stress of. .......... ..... ................. ................................. 79

5-1 Deviations between measured and fitted values of"average Ge" and "Go with
m ode m ixity" for different designs ............................................... ............... 90

5-2 The mean and B-basis values of the fracture toughness of the designs analyzed....94

5-3 Allowable flight load of failure of the sandwich panels designed using
determ inistic approach ....................... .. ...................... .... ...... .... ............ 96

5-4 Corresponding probabilities of failure of the sandwich panels designed using
determ inistic approach ....................... .. ...................... .... ...... .... ............ 99

5-5 Allowable flight loads of the sandwich panels calculated via probabilistic
ap p ro ach ...........................................................................10 1

6-1 A llow able strains for IM 600/133 ........................................ ....................... 107

6-2 D eterm inistic optim um design .......................................................... ............... 108

6-3 Coefficients of variation of the random variables............................. 108

6-4 Evaluation of the accuracy of the analysis response surface ..............................110

6-5 Comparison of probability of failure estimations for the deterministic optimum.. 111

6-6 Probabilistic optimum designs for different error bounds when only error
reduction is applied .................. .............................. ........ .. ........ .. 114

6-7 Probabilistic optimum designs for different error bounds when both error and
variability reduction are applied................................................................. ...... 116

7-1 Distribution of error factors and their bounds....................................................... 138

7-2 Distribution of random variables having variability .............................................140









7-3 Mean and standard deviations of the built and certified distribution of the total
safety factor SF ..................................................................................142

7-4 Comparing explicit knockdown factors for minimum built safety factor for a
specified certification failure rate.................................. .............................. ........ 148

7-5 Comparing explicit knockdown factors for minimum built safety factor for a
specified probability of failure ................................... .............................. ........ 150

7-6 Comparison of constant and variable explicit knockdown factors case and
corresponding area ratios, A/Ao. ........................................ ........................ 154

7-7 Comparison of constant (i.e., test independent) implicit and explicit knockdown
factors and corresponding area ratios A/A o.................................... .....................156

7-8 Comparison of mean and coefficient of variation of total knockdown reduction
at the element test level for the cases of implicit constant knockdown factor and
explicit variable knockdow n factors ........................................... ............... 157

7-9 Optimal explicit knockdown factors for minimum CFR when variability in
failure stress is reduced by half........................................ ........... ............... 159

7-10 Optimal explicit knockdown factors for minimum CFR when all errors reduced
by half. ................................... ............................ ........ .......... 161

7-11 Optimal explicit knockdown factors for minimum CFR different number of
coupon tests, n ........................................................................................... 162

7-12 Optimal explicit knockdown factors for different number of structural element
tests, n ......................... .......... ........................................... 163

8-1 Probabilistic structural design optimization for safety of a representative wing
and tail system ................................................................... ..........171

8-2 Probabilistic structural optimization of wing, horizontal tail and vertical tail
sy stem ...................................... .....................................................17 3

8-3 Errors in the ratios of failure probabilities of the wing and tail system when the
c.o.v. of the stresses under-estimated by 50%.................................. ........... 175

8-4 Errors in the ratios of failure probabilities of the wing and tail system when the
m ean stresses are under-estim ated by 20% ...........................................................178

8-5 Errors in the ratios of failure probabilities of the wing and tail system when the
probability of failure of the deterministic design is under-predicted.....................180

8-6 Errors in the ratios of failure probabilities of wing and tail system when the
probability of failure of the deterministic design is over-predicted .......................181









8-7 Errors in the ratios of failure probabilities of the wing and tail system if the
optimization is performed using wrong probability distribution type for the
stress ............................................................... ...... ...... ........ 182

8-8 Probabilistic design optimization for safety of the representative wing and tail
system using the characteristic-stress m ethod ............... .............. ..................... 188

8-9 Effect of 20% under-estimate of k on the ratios of probability of failure estimate 188

D-1 Equivalent error bounds for the SEF model corresponding to the same standard
deviation in the M EF model ........... .... .................... .................. .... 205

D-2 Comparison of system failure probabilities for the SEF and MEF models ..........206

D-3 Comparison of the total safety factor SF used in the design of structural parts for
the SEF and M EF m odels............................................. .............................. 207

E-1 Comparison of the probability of failure estimations ................ ...............211

I-1 The ranges of variables for the three DRS constructed for PSF calculation..........218

I-2 Accuracies of DRS fitted to PSF and Pf in terms of four design variables (ti, t2,
01 and 02) for error bounds, be, of 0, 10% and 20% .............................................219

1-3 Ranges of design variables for the three DRS constructed for probability of
failure estimation for the error and variability reduction case .............................221

M -1 Accuracy of response surfaces ...................................................... ....................232

P-1 Aircraft accidents and probability of failure of aircraft structures.........................242
















LIST OF FIGURES


Figure page

2-1 B building block approach ................................................ .............................. 26

3-1 Flowchart for Monte Carlo simulation of component design and failure ..............35

3-2 Initial and updated probability distribution functions of error factor e ..................38

3-3 Design thickness variation with low and high error bounds ..................................45

3-4 Influence of effective safety factor, error, and variability on the probability ratio
(3 -D v iew ) ........................................................................... 50

3-5 Influence of effective safety factor, error and variability on the probability ratio
(2 -D p lot) ................................................................................................ .... 5 0

3-6 Influence of effective safety factor, error and variability on the probability
difference (3-D view ) ....................................................... .... .. ...... .... 51

3-7 Influence of effective safety factor, error and variability on the probability
difference (2-D plot)........................................... .........51

4-1 Comparing distributions of built and certified total error etotal of SEF and MEF
m o d els .............................................................................. 6 3

4-2 Initial and updated distribution of the total safety factor SF............... .................64

4-3 The variation of the probability of failure with the built total safety factor............. 68

4-4 Flowchart for MCS of component design and failure............................................69

4-5 Total safety factors for MEF model for the structural part and system after
c ertific atio n .................................................... ................ 7 8

4-6 Effect of variability on failure probability ...................................................79

5-1 The model of face-sheet/core debonding in a one-dimensional sandwich panel
w ith pressure load ............. ........................................................................ . ...... 86

5-2 Critical energy release rate as a function of mode mixity ........................................88









5-3 Comparison of actual and fitted cumulative distribution functions of variability,
dMM, of G .. ...........................................................................92

5-4 Comparison of actual and fitted cumulative distribution functions of total
uncertainty (error and variability, Jd ) of G ................................. .................92

5-5 Fitted least square lines for fracture toughness, and derived B-basis allowables ....95

6-1 Geometry and loading of the laminate with two ply angles................................107

6-2 Comparison of CDF obtained via 1,000 MCS, the approximate normal
distribution and conservative approximate normal distributions for 82 on 01
corresponding to the deterministic optimum ................................ .................. 111

6-3 Reducing laminate thickness (hence weight) by error reduction (no variability
reduction) .............. .................................................. .. .. .. ....... .. 115

6-4 Reducing laminate thickness by error reduction (ER) and quality control (QC)...116

6-5 Trade-off plot for the probability of failure, design thickness and uncertainty
reduction m measures .............................................. .. ...... ................ 117

6-6 Tradeoff of probability of failure and uncertainty reduction .............................119

7-1 Building-block approach for aircraft structural testing ............... ................123

7-2 Sim plified three level of tests ................................................................................. 124

7-3 Current use of knockdown factors based on worst-case scenarios ........................131

7-4 Shrinkage of the failure surface ........................................ ......................... 132

7-5 The variation of the explicit knockdown factors with ratio of the failure stress
measured in the test and calculated failure stress with and without transition
interval ....................................................................... ........... 133

7-6 Proposed use of explicit knockdown factors dependent on test results ...............134

7-7 Initial and updated distribution of the total safety factor SF with and without
structural elem ent test ..................................... .................. .......... ... ...... .... 142

7-8 The variation of probability of failure of a structural part built by a single
aircraft com pany ............................................. .... .. ........ .. ........ .... 144

7-9 Optimal choice of explicit knockdown factors kcl and kch for minimum built
safety factor for specified certification failure rate .............. ...... ....................146

7-10 Comparing CFR and PF of the structures designed for minimum CFR and
minimum PF ................... ... ... ..................... ............ ........... 149









7-11 Effect of number of coupon tests on the error in failure prediction for a fixed
number of element tests (3 element tests) .............. ...................... ......... ...... 151

7-12 Effect of number of element tests on the error in failure prediction for a fixed
number of coupon tests (40 coupon tests).................. .. ............ ...... ....... 152

7-13 Evolution of the mean failure stress distribution with and without Bayesian
u p d atin g ................................................................................................... 1 5 3

7-14 Comparison of variable and constant explicit knockdown factor.......................154

7-15 Comparison of Pareto fronts of certification failure rate and built safety factor
for two different approaches while updating the allowable stress based on failure
stresses m measured in elem ent tests .............................................. ............... 155

7-16 Reducing probability of failure and certification failure rate using variability
reduction ............... ........... .......................... ............................159

7-17 Reducing certification failure rate using error reduction, variability reduction
and combination of error and variability reduction.......................................... 160

7-18 Optimal explicit knockdown factors for different number of coupon tests for
minimum CFR and PF ........................................... ...................... 162

7-19 Effect of number of structural element tests, ne ............................................163

8-1 Stress distribution s(a) before and after redesign in relation to failure-stress
distributionJf(a) ............ ...... .................. ........ .... ........ 168

8-2 The change of the ratios of probabilities of failure of the probabilistic design of
Table 8-1 versus the error in c.o.v.(o)........................................... .................. 176

8-3 Two different stress distributions at the wing leading to the same probability of
failure of lx 107. ............ .............................. ....... ............177

8-4 The change of the ratios of probabilities of failure with respect to the error in
m ean stress ................................................................ 179

8-5 Calculation of characteristic stress o* from probability of failure ....................185

8-6 Comparison of approximate and exact A and A* and the resulting probabilities
of failure for lognormal failure stress.............................................. .............186

8-7 The variation of the ratios of probabilities of failure with respect to error in k .....189

D-1 System failure probabilities for the SEF and MEF models after certification .......206

D-2 Total safety factors for the SEF and MEF model after certification................208









E-1 Comparison of numerical CDF with the assumed lognormal CDF for the
distribution of the required safety factor ................ ......... ..... ........ .......... 210

G-1 Material properties El, E2, G12 and v12 as a function of temperature.....................214

G-2 Material properties a, and a2 as a function of temperature...............................215

K-1 The variation of probability of failure with built total safety factor ......................227

K-2 Flowchart for MCS of component design and failure................. ....................228


xviii















NOMENCLATURE

ARS = Analysis response surface

Aeq = Minimum required cross sectional area for the component to carry
the service loading without failure

Ao = Load carrying area if there is no variability and no safety measures

c a2 = Coefficient of thermal expansion along and transverse to fiber direction

be = Bound of error

P = Reliability index

C = Capacity of structure, for example, failure stress

CFD = Cumulative distribution function

CFR = Certification failure rate

CLT = Classical lamination theory

c.o.v. = Coefficient of variation

DRS = Design response surface

A = Relative change in the characteristic stress corresponding to a relative
change of A in stress a

e = Error factor

efc = Error in failure prediction at the coupon level

ec = Error in capacity calculation

efe = Error in failure prediction at the element level

ef, = Error in failure prediction at the structural level









ejp = Total error in failure prediction

em = Error in material property prediction

ep = Error in load calculation

eR = Error in response calculation

e, = Error in stress calculation

et = Error in thickness calculation

total = Total error factor

ew = Error in width calculation

eA = Error in facture toughness assessment if traditional (averaging) method
is used

eM = Error in facture toughness assessment if traditional (averaging) method
is used

ER = Error reduction

El, E2 = Young's modulus along and transverse to fiber direction

E1, -2 = Strains in the fiber direction and transverse to the fiber direction

f() = Probability density function of the failure stress

F() = Cumulative distribution function of the failure stress

FAA = Federal Aviation Administration

G = Strain energy release rate

Gc = Fracture toughness

G12 = Shear modulus

712 = Shear strain

k = Error multiplier

kA, kB = Tolerance coefficients for A-basis and B-basis value calculation










kdc

Ki, K11

M

MCS

MEF model

N

Nx, Ny

fle

iHe

allow

P

PDF

PSF

Pd

Pf

Pf*

Pfd

PF

Pc


Pnc

QC

ret

R


Knock-down factor used to calculate allowable stress

Model I and II stress intensity factors, respectively

Number of simulations in the first stage of MCS

Monte Carlo simulation

Multiple error factor model

Number of simulations in the second stage of MCS

Mechanical loading in x and y directions, respectively

Number of coupon tests

Number of structural element tests

Allowable flight load

Load

Probability density function

Probability sufficiency factor

Design load according to the FAA specifications

Probability of failure of a component

Approximate probability of failure of probabilistic design

Probability of failure of deterministic design

Probability of failure of a system

Average probability of failure after certification test

Average probability of failure before certification test

Quality control for manufacturing

Ratio of failure stresses measured in test and its predicted value

Response of a structure, for example, stress










RMSE

RSA

R2adj

P

s( )

SEF model

S,

sc'


Sch


Sfe

SFL

SF

aU

a"*


ca




t

Vt

Vw

VR

w

W


Root mean square error

Response surface approximation

Adjusted coefficient of multiple determination

Coefficient of correlation

= Probability density function of the stress

Single error factor model

Additional company safety factor

Additional company safety factor if the failure stress measured in
element tests are lower than the predicted failure stress

Additional company safety factor if the failure stress measured in
element tests are higher than the predicted failure stress

Total safety factor added during structural element tests

Load safety factor of 1.5 (FAA specification)

Total safety factor

Stress

Characteristic stress

Allowable stress

Failure stress


Thickness

Variability in built thickness

Variability in built width

Coefficient of variation

Width

Weight









Weight of the deterministic design

Cumulative distribution function of the standard normal distribution

Mode-mixity angle


Subscripts

act

built


calc


cert

d

design

spec

target

true

worst

W

T



Subscripts

ave

ini

upd


xxiii


The value of the relevant quantity in actual flight conditions

Built value of the relevant quantity, which is different than the design
value due to errors in construction

Calculated value of the relevant quantity, which is different from the
true value due to errors

The value of the relevant quantity after certification test

Deterministic design

The design value of the relevant quantity

Specified value of the relevant qunatity

Target value of the relevant quantity

The true value of the relevant quantity

The worst value of the relevant quantity

Wing

Tail






Average value of the relevant quantity

Initial value of the relevant quantity

Updated value of the relevant quantity










U = Upper limit of the relevant quantity

L = Lower limit of the relevant quantity


xxiv















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

AIRCRAFT STRUCTURAL SAFETY: EFFECTS OF EXPLICIT AND IMPLICIT
SAFETY MEASURES AND UNCERTAINTY REDUCTION MECHANISMS

By

Erdem Acar

August 2006

Chair: Raphael T. Haftka
Cochair: Bhavani V. Sankar
Major Department: Mechanical and Aerospace Engineering

Aircraft structural safety is achieved by using different safety measures such as

safety and knockdown factors, tests and redundancy. Safety factors or knockdown factors

can be either explicit (e.g., load safety factor of 1.5) or implicit (e.g., conservative design

decisions). Safety measures protect against uncertainties in loading, material and

geometry properties along with uncertainties in structural modeling and analysis. The two

main objectives of this dissertation are: (i) Analyzing and comparing the effectiveness of

structural safety measures and their interaction. (ii) Allocating the resources for reducing

uncertainties, instead of living with the uncertainties and allocating the resources for

heavier structures for the given uncertainties.

Certification tests are found to be most effective when error is large and variability

is small. Certification testing is more effective for improving safety than increased safety

factors, but it cannot compete with even a small reduction in errors. Variability reduction

is even more effective than error reduction for our examples.









The effects of structural element tests on reducing uncertainty and the optimal

choice of additional knockdown factors are explored. We find that instead of using

implicit knockdown factors based on worst-case scenarios (current practice), using test-

dependent explicit knockdown factors may lead weight savings. Surprisingly, we find

that a more conservative knockdown factor should be used if the failure stresses

measured in tests exceeds predicted failure stresses in order to reduce the variability in

knockdown factors generated by variability in material properties.

Finally, we perform probabilistic optimization of a wing and tail system under

limited statistical data for the stress distribution and show that the ratio of the

probabilities of failure of the probabilistic design and deterministic design is not sensitive

to errors in statistical data. We find that the deviation of the probabilistic design and

deterministic design is a small perturbation, which can be achieved by a small

redistribution of knockdown factors.


xxvi














CHAPTER 1
INTRODUCTION

Motivation

Traditionally, the design of aerospace structures relies on a deterministic design

(code-based design) philosophy, in which safety factors (both explicit and implicit),

conservative material properties, redundancy and certification testing are used to design

against uncertainties. An example of explicit safety factor is the load safety factor of 1.5

(FAR 25-303), while the conservative decisions employed while updating the failure

stress allowables based on structural element tests are examples for implicit safety

factors. In the past few years, however, there has been growing interest in applying

probabilistic methods to design of aerospace structures (e.g., Lincoln 1980, Wirsching

1992, Aerospace Information Report of SAE 1997, Long and Narciso 1999) to design

against uncertainties by effectively modeling them.

Even though probabilistic design is a more efficient way of improving structural

safety than deterministic design, many engineers are skeptical of probability of failure

calculations of structural designs for the following reasons. First, data on statistical

variability in material properties, geometry and loading distributions are not always

available in full (e.g., joint distributions), and it has been shown that insufficient

information may lead to large errors in probability calculations (e.g., Ben-Haim and

Elishakoff 1990, Neal et al. 1992). Second, the magnitude of errors in calculating loads

and predicting structural response is not known precisely, and there is no consensus on

how to model these errors in a probabilistic setting. As a result of these concerns, it is









possible that transition to probability based design will be gradual. An important step in

this transition is to understand the way safety is built into aircraft structures now, via

deterministic design practices.

One step taken in the transition to probabilistic design is in the definition of

conservative material properties (A-basis or B-basis material property values depending

on the failure path in the structure) by the Federal Aviation Administration (FAA)

regulation (FAR 25.613). A-basis material property is one in which 99 percent of the

material property distribution is better than the design value with a 95 percent level of

confidence, and B-basis material property is one in which 90 percent of the material

property distribution is better than the design value with a 95 percent level of confidence.

The use of conservative material properties is intended to protect against variability in

material properties.

In deterministic design the safety of a structure is achieved through safety factors.

Even though some safety factors are explicitly specified, others are implicit. Examples of

explicit safety factors are the load safety factor and material property knock-down values.

The FAA regulations require a load safety factor equal to 1.5 for aircraft structures (FAR

25-303). The load safety factor compensates for uncertainties such as uncertainty in

loading and errors in load calculations, structural stress analysis, accumulated damage,

variations in material properties due to manufacturing defects and imperfections, and

variations in fabrication and inspection standards. Safety factors are generally developed

from empirically based design guidelines established from years of structural testing of

aluminum structures. Muller and Schmid (1978) review the historical evolution of the

load safety factor of 1.5 in the United States. Similarly, the use of A-basis or B-basis









material properties leads to a knock-down factor from the average values of the material

properties measured in the tests. Note that these knock-down factors depend on the

number of tests, because they compensate for both variability in material properties and

uncertainty due to a finite number of tests.

As noted earlier, an important step in transition to probabilistic design is to analyze

the probabilistic impact of the safety measures used in deterministic design. This

probabilistic analysis requires quantification of uncertainties encountered in design,

manufacturing and actual service conditions of the aircraft structures.

A good analysis of different sources of uncertainty in engineering modeling and

simulations is provided by Oberkampf et al. (2000, 2002). These papers also supply good

literature reviews on uncertainty quantification and divide the uncertainty into three

types: variability, uncertainty, and error. In this distinction, variability refers to aleatory

uncertainty (inherent randomness), uncertainty refers to epistemic uncertainty (due to

lack of knowledge), and error is defined as a recognizable deficiency in any phase or

activity of modeling and simulation that is not due to lack of knowledge.

To simplify the treatment of uncertainty control, in this dissertation we combine the

unrecognized epistemicc) and recognized error in the classification of Oberkampf et al.

and name it error. That is, we use a simple classification that divides the uncertainty in

the failure of a structural member into two types: errors and variability. Errors reflect

inaccurate modeling of physical phenomena, errors in structural analysis, errors in load

calculations, or deliberate use of materials and tooling in construction that are different

from those specified by the designer. Errors affect all the copies of the structural

components made and are therefore fleet-level uncertainties. Variability, on the other









hand, reflects the departure of material properties, geometry parameters or loading of an

individual component from the fleet-average values and hence are individual

uncertainties.

Modeling and quantification of variability are much easier compared to that of

error. Improvements in tooling and construction or application of tight quality control

techniques can reduce variability. Quantification of variability control can be easily done

by statistical analysis of records taken throughout process of quality control. However,

quantification of errors is not as easy, because errors are largely not known before a

structure is built. So, errors can only be quantified after the structure has been built.

Errors can be controlled by improving accuracy of load and stress calculations, by using

more sophisticated analysis and failure prediction techniques or by testing of structural

components.

Testing of aircraft structural components is performed in a building block type of

approach starting with material characterization tests, followed by testing of structural

elements and including a final certification test. Testing of structures is discussed in detail

in the next chapter.

The comparison of deterministic design and probabilistic design can be performed

in many views. First of all, input and output variables of deterministic design are all

deterministic values, while input and output variables of probabilistic design are random

(along with some deterministic variables, of course). Here, on the other hand, we

compare probabilistic design and deterministic design in terms of use of safety factors. In

deterministic design uniform safety factors are used; that is, the same safety factor is used

for all components of a system. However, probabilistic design allows using variable









safety factors through allowing risk and reliability allocation between different

components. That is, instead of using the same safety factor for all components,

probabilistic design allows to use higher factors for components or failure modes that can

be controlled with low weight expenditure (Yang, 1989). This means the failure modes

with small scatter and lightweight components. In addition, probabilistic design allows a

designer to trade off uncertainty control for lower safety factors. That is, by reducing

uncertainty, the designer can avoid using high safety factors in the design and thereby can

reduce the weight of the structural system. This design paradigm allows the designer to

allocate risk and reliability between different components in a rational way to achieve a

safer design for a fixed weight compared to the deterministic design.

Objectives

There are two main objectives of this dissertation. The first is to analyze and

compare the effectiveness of safety measure that improve structural safety such as safety

factors (explicit or implicit), structural tests, redundancy and uncertainty reduction

mechanisms (e.g., improved structural analysis and failure prediction, manufacturing

quality control). The second objective is to explore the advantage of uncertainty

reduction mechanisms (e.g., improved structural analysis and failure prediction, tighter

manufacturing quality control) versus safety factors. That is, we consider the possibility

of allocating the resources for reducing uncertainties, instead of living with the

uncertainties and allocating the resources for designing the aircraft structures for the

given uncertainties.

We aim to analyze the effectiveness of safety measures taken in deterministic

design methodology and investigate the interaction and effectiveness of these safety

measures with one another and also with uncertainties. In particular, the effectiveness of









uncertainty reduction mechanisms is analyzed and compared. The uncertainty reduction

mechanisms considered in this dissertation are reduction of errors by improving the

accuracy of structural analysis and failure prediction (analytically or through tests), and

reduction of variability in failure stress as a result of tighter quality control.

We explore the optimal choice of additional company safety factors used on top of

the FAA regulation safety factors by using probabilistic design, which provides a rational

way in the analysis. Additional company safety factors we consider are the conservative

decisions of aircraft companies while updating the allowable stresses based on the results

of structural element tests.

We perform probabilistic design optimization for the case of limited statistical data

on stress distribution and show that when the probabilistic design is achieved by taking

the deterministic design as a starting point, the ratio of probabilities of failure of the

probabilistic design and deterministic design is not sensitive to errors due to limited

statistical data, which would lead to substantial errors in the probabilistic design if the

probabilistic design starts from scratch. In addition, we propose a probabilistic design

methodology in which the probability of failure calculation is confined only to stress

limits, thereby eliminating the necessity for assessment of stress distribution that usually

requires computationally expensive finite element analyses.

Methodology

Probability of failure calculation of structures can be performed by using either

analytical techniques or simulation techniques. Analytical methods are more accurate but

for complex systems they may not be practical. Simulation techniques include direct

Monte Carlo simulation (MCS) as well as many variance-reduction methods including









stratified sampling, importance sampling, and adaptive importance sampling (Ayyub and

McCuen 1995).

In probabilistic design of structures, the use of inverse reliability measures helps a

designer to have an easy estimate of the change in structural weight from the values of

probabilistic performance measure and its target value as well as computational

advantages (Ramu et al. 2004). Amongst those measures we use probabilistic sufficiency

factor (PSF) developed by Qu and Haftka (2003).

Here we consider a simplified design problem for illustration purposes, so that the

reliability analysis can be performed by analytical means. The effect of testing then can

be analyzed by using Bayesian approach. The Bayesian approach has special importance

in engineering design where the available information is limited and it is often necessary

to make subjective decisions. Bayesian updating is used to obtain the updated (or

posterior) distribution of a random variable upon combining the initial (or prior)

distribution with new information about the random variable. The detailed theory and

procedures for applying Bayesian methods in reliability and risk analysis can be found in

texts by Morgan (1968) and Martz and Waller (1982).

Outline

A literature survey on the historical evolution of probabilistic design, comparison

of deterministic design and probabilistic design practices, uncertainty control measures

and testing of aircraft structures is given in Chapter 2.

Chapter 3 investigates the effects of error, variability, safety measures and tests on

structural safety of aircraft. A simple example of point stress design and a simple error

model are used to illustrate the effects of several safety measures taken in aircraft design:

safety factors, conservative material properties, and certification tests. This chapter serves









as the opening chapter; therefore the analysis and the number of safety measures are kept

at a minimum level. For instance, only certification tests are included in the analysis. The

effects of coupon tests and structural element tests are delayed until Chapter 7. The

simplifying assumptions in Chapter 3 allow us to perform analytical calculations for

probability of failure and Bayesian updating. The interactions of the safety measures with

one another and also with errors and variabilities are investigated. For instance, we find

that the certification tests are most effective when errors are large and variabilities are

small. We also find that as safety measures combine to reduce the probability of failure,

our confidence in the probability of failure estimates is reduced.

Chapter 4 extends the analysis presented in Chapter 3 by delivering the following

refinements. The effectiveness of safety measures is compared with one another in terms

of safety improvement and weight savings. Structural redundancy, a safety measure

which is omitted in Chapter 3, is also included in the analysis. The simple error model

used in Chapter 3 is replaced with a more detailed error model in which we consider

individual error components in load calculation, stress calculation, material properties

and geometry parameters including the effect of damage. The analysis in Chapter 4

enables us to discover that while certification testing is more effective than increased

safety factors for improving safety, it cannot compete with even a small reduction in

errors. We also find that variability reduction is even more effective than error reduction.

Realizing in Chapter 4 how powerful uncertainty reduction mechanisms are, we

analyze the tradeoffs of uncertainty reduction mechanisms, structural weight and

structural safety in Chapters 5 and 6. The effect of error reduction (due to improved

failure prediction model) on increasing the allowable flight loads of existing aircraft









structures is investigated in Chapter 5. The analysis is performed for a sandwich panel

because the improved model is developed by Prof. Bhavani Sankar (co-chair of the

advisory committee for this dissertation) so that we had good access to the details of

experiments and computations. We find that the improved modeling can increase the

allowable load of a sandwich panel on average by about 13 percent without changing the

safety level of the panel when deterministic design principles is followed. The use of

probabilistic design is found to double the load increase.

Similarly to improvements of accuracy in failure predictions, the improvements in

the accuracy of structural analysis also lead to error reduction. The improved structural

analysis through taking the chemical shrinkage of composite laminates is considered as

the error reduction mechanism in Chapter 6. The work by Qu et al. (2003), which

explored the effect of variability reduction through quality control, is extended in Chapter

6 to investigate the tradeoffs of error and variability reduction mechanisms for reducing

the weight of the composite laminates at cryogenic temperatures. Tradeoff plots of

uncertainty reduction mechanisms, probability of failure and weight are generated that

enable a designer to choose the optimal uncertainty control mechanism combination to

reach a target probability of failure with minimum cost.

Chapter 7 finalizes the analysis of effects of explicit and implicit knockdown

factors and uncertainty control mechanisms. In particular, Chapter 7 analyzes the optimal

choice of the knockdown factors. These knockdown factors refer to conservative

decisions of aircraft companies in choice of material properties and while updating the

allowable stresses based on the results of structural element tests. We find that instead of

using implicit knockdown factors based on worst-case scenarios (current practice), using









test-dependent explicit knockdown factors may lead weight savings. Surprisingly, we

find that a more conservative knockdown factor should be used if the failure stresses

measured in tests exceeds predicted failure stresses in order to reduce the variability in

knockdown factors generated by variability in material properties. In addition, the effects

of coupon tests, structural element tests and uncertainty control mechanisms (such as

error reduction by improved structural modeling or improved failure prediction,

variability reduction by tighter quality control) on the choice of company safety factors

are investigated. Using a simple cost function in terms of structural weight, we show that

decisions can be made whether to invest resources on coupon tests, structural element

tests, uncertainty reduction mechanisms or extra structural weight.

The analyses presented in Chapters 3-7 show how probabilistic design can be

exploited to improve aircraft structural safety by allowing a rational analysis of

interactions of safety and knockdown factors and uncertainty reduction mechanisms.

There are, however, two main reasons for reluctance of engineers for pursuing the

probabilistic design: the sensitivity of probabilistic design to limited statistical data and

computational expense associated to the probabilistic design. Besides, Chapters 3-7

include analyses of a single aircraft structural component, so in Chapter 8 the

probabilistic design of an aircraft structural system is presented. We show in Chapter 8,

by use of probabilistic design of a representative wing and tail system, that errors due to

limited statistical data affect the probability of failure of both probabilistic and

deterministic designs, but the ratio of these probabilities is quite insensitive to even very

large errors. In addition, to alleviate the problem of computational expense, a

probabilistic design optimization method is proposed in which the probability of failure









calculation is limited to failure stresses to dispense with most of the expensive structural

response calculations (typically done via finite element analysis). The proposed

optimization methodology is illustrated with the design of the wing and tail system.

Chapter 8 reveals that the difference between probabilistic design and deterministic

design is a small perturbation, which can be achieved by choosing the additional

knockdown factors through probabilistic design, instead of choosing them based on

experience. In addition, the proposed approximate method is found to lead to similar re-

distribution of material between structural components and similar system probability of

failure.

Finally, the dissertation culminates with Chapter 9, where the concluding remarks

are listed.














CHAPTER 2
LITERATURE REVIEW

The literature review in this chapter first compares deterministic and probabilistic

design methodologies. Then, we review structural safety analysis, followed by

probability of failure estimation techniques. Next, reliability-based design optimization is

reviewed. Then, uncertainty classifications available in the literature are discussed

followed by our simplified classification based on simplifying the analysis of uncertainty

reduction measures. Finally, the utilization of structural tests in probabilistic design is

reviewed.

Probabilistic vs. Deterministic Design

Aircraft structural design still relies on the Federal Aviation Administration (FAA)

deterministic design code. In deterministic design, conservative material properties are

used and safety factors are introduced to protect against uncertainties. The FAA

regulations (FAR-25.613) state that conservative material properties are characterized as

A-basis or B-basis values. Detailed information on these values was provided in Chapter

8 of Volume 1 of Composite Materials Handbook (2002). The safety factor compensates

for uncertainties such as uncertainty in loading and errors in load calculations, errors in

structural stress analysis and accumulated damage, variations in material properties due

to manufacturing defects and imperfections, and variations in fabrication and inspection

standards. Safety factors are generally developed from empirically based design

guidelines established from years of structural testing and flight experience. In transport

aircraft design, the FAA regulations state the use of safety factor of 1.5 (FAR-25.303).









Muller and Schmid (1978) reviewed the historical evolution of the 1.5 factor of safety in

the United States.

On the other hand, probabilistic design methodology deals with uncertainties by the

use of statistical characterization of uncertainties and attempts to provide a desired

reliability in the design. The uncertainties of individual design parameters and loads are

modeled by appropriate probability density functions. The credibility of this approach

depends on several factors such as the accuracy of the analytical model used to predict

the structural response, the accuracy of the data and the probabilistic techniques

employed. Examples of the use of probabilistic design in aerospace applications include

the following.

Pai et al. (1990, 1991 and 1992) performed probabilistic structural analysis of

space truss structures for a typical space station. Murthy and Chamis (1995) performed

probabilistic analysis of composite aircraft structure based on first ply failure using

FORM The probabilistic methodology has shown some success in the design of

composite structures where parameter uncertainties are relatively well known. For

example, the IPACS (Integrated Probabilistic Assessment of Composite Structures)

computer code was developed at NASA Glenn Research Center (Chamis and Murthy

1991). Fadale and Sues (1999) performed reliability-based design optimization of an

integral airframe structure lap joint. A probabilistic stability analysis for predicting the

buckling loads of compression loaded composite cylinders was developed at Delft

University of Technology (Arbocz et al. 2000).


The FORM method is discussed later in this chapter.









Although probabilistic design methodology offers the potential of safer and lighter

designs than deterministic design, transition from deterministic design to probabilistic

design is difficult to achieve. Zang et al. (2002) discussed the reasons for this difficulty,

and some of these reasons are given below.

* Industry feels comfortable with traditional design methods.

* Few demonstrations of the benefits of probabilistic design methods are available.

* Current probabilistic design methods are more complex and computationally
expensive than deterministic methods.

* Characterization of structural imperfections and uncertainties necessary to facilitate
accurate analysis and design of the structure is time-consuming and is highly
dependent on structural configuration, material system, and manufacturing
processes.

* Effective approaches for characterizing model form error are lacking.

* Researchers and analysts lack training in statistical methods and probabilistic
assessment.

Structural Safety Analysis

In probabilistic design, the safety of a structure is evaluated in terms of its

probability of failure Pf. The structures are designed such that the probability of failure of

the structure is kept below a pre-specified level. The term reliability is defined in terms of

probability of failure such that


Reliability = 1 P (2.1)

A brief history of development of the methods for probability of failure calculation

for structures was presented in a report by Wirsching (1992). As Wirsching noted, the

development of theories goes back some 50 to 60 years.

The modern era of probabilistic design started with the paper by Fruedenthal

(1947). Most of the ingredients of structural reliability such as probability theory,









statistics, structural analysis and design, quality control existed prior to that time;

however, Fruedenthal was the first to put them together in a definitive and compressive

manner. The development of reliability theory progressed in 1950s and 1960s. There are

three cornerstone papers in 1960's. The first one is the paper by Cornell (1967), who

suggested the use of a second moment method and demonstrated that Cornell's safety

index could be used to derive set of factors on loads and resistance. However, Cornell's

safety index had a problem of invariance in that it was not constant when the problem

was reformulated in a mechanically equivalent way. Hasofer and Lind (1974) defined a

generalized safety index which was invariant to mechanical formulation. The third paper

is the one by Turkstra (1970), who presented structural design as a problem of decision

making under uncertainty and risk. More recent papers are sophisticated extensions of

these papers, and some of them are referenced in the following sections.

Probability of Failure Estimation

This section reviews the literature on probability of failure estimation. First,

analytical calculation of probability of failure is discussed, followed by moment-based

methods and simulation techniques.

Analytical calculation of probability of failure

In its most general form, the probability of failure can be expressed as


Pf = Gix f fx(x)dx (2.2)
G(x)
where G(x) is the limit-state function whose negative values corresponds to failure and


fx (x) is the joint probability density function for the vector X of random variables. The

analytical calculation of this expression is challenging due to the following reasons









(Melchers 1999). First, the joint probability density function fx (x) is not always readily

obtainable. Second, for the cases when fx (x) is obtainable, the integration over the

failure domain is not easy. The calculation of probability of failure can be made more

tractable by simplifying (1) the limit-state definition, (2) the integration process, and (3)

the integrand fx (x) .

Moment-based techniques

When the calculation of limit-state is expensive, moment-based techniques such as

First Order Reliability Method (FORM) or Second Order Reliability Method (SORM) are

used (Melchers, 1999). The basic idea behind these techniques is to transform the original

random variables into a set of uncorrelated standard normal random variables, and then

approximate the limit-state function linearly (FORM) or quadratically (SORM) about the

most probable failure point (MPP). The probability of failure of the component is

estimated in terms of reliability index P such that


Pf = D(- /) (2.3)

where D is the cumulative distribution function of a standard normal variable.

The first paper on the use of FORM is probability of failure calculation appears to

be Hasofer and Lind's (1974). There exist enormous amount of papers on the use of

FORM. The pioneer papers include Rackwitz and Fiessler (1978), Hohenbichler and

Rackwitz (1983), Gollwitzer and Rackwitz (1983).

FORM is usually accurate for limit state functions that are not highly nonlinear.

SORM has been proposed to improve the reliability estimation by using a quadratic

approximation of the limit state surface. Some papers on the use of SORM include









Fiessler et al. (1979), Breitung (1984), Der Kiureghian et al. (1987), Hohenbichler et al.

(1987), Der Kiureghian and De Stefano (1991), Koyluoglu and Nielsen (1994) and Zhao

and Ono (1999).

Simulation techniques

For most problems the number of variables in the problem definition is high, so the

analytical calculation of the integral in Eq. (2.2) requires challenging multidimensional

integration. Also the moment based approximations gives inaccurate results for high

number of random variables (Melchers 1999). Under such conditions, simulation

techniques such as Monte Carlo simulations (MCS) are used to compute the probability

of failure.

In MCS technique, samples of the random variables are generated according to

their probabilistic distributions and then failure condition is checked. The probability of

failure Pfcan be estimated by



N
Pf=- (2.4)


where Nf is the number of simulations leading to failure and N is the total number of

simulations. The statistical accuracy of the probability of failure estimation is commonly

measured by its coefficient of variation c.o.v.(Pf) as


Pf (1 Pf )
c.o.v.(Pf) N -) (2.5)
Pf N Pf

From Eqs. (2.4) and (2.5) it is seen that a small probability of failure will require a

very large number of simulations for acceptable accuracy. This usually results in an

increase in computational cost. When limit-state function calculations are obtained









directly from analysis, then computational cost of MCS is not sensitive to the number of

variables. When surrogate models are used, on the other hand, the computational cost of

MCS is dependent on the number of variables. To overcome the deficiency of MCS,

several more efficient alternative sampling methods are introduced. Ayyub and McCuen

(1995) supplied basic information and good references for these sampling techniques.

Some useful references taken from Ayyub and McCuen (1995) are the followings:

Importance sampling (Madsen et al., 1986, Melchers, 1989), stratified sampling (Law

and Kelton 1982, Schuller et al. 1989), Latin hypercube sampling (Iman and Canover

1980, Ayyub and Lai 1989), adaptive importance sampling (Busher 1988,

Karamchandani et al. 1989, Schuller et al. 1989), conditional expectation (Law and

Kelton 1982, Ayyub and Haldar 1984), antithetic variates (Law and Kelton 1982, Ayyub

and Haldar 1984).

In this study, we mainly deal with problems with simple limit-state functions. For

these simple cases the integrand fx (x) can easily be obtained when the random

variables are statistically independent. The beneficial properties of normal and lognormal

distributions are utilized for the variables with small coefficients of variations.

Approximate analytical calculations of probability of failure are checked with Monte

Carlo simulations to validate the acceptability of assumptions. When limit-state functions

are complex, Monte Carlo simulations are used to calculate the probability of failure.

Separable Monte Carlo simulations

As noted earlier, when estimating very low probabilities, the number of required

samples for MCS can be high, thus MCS becomes a costly process. In most structural

problems, the failure condition may be written as response exceeding capacity. When the









response and capacity are independent, it may be possible to analyze them separately

with a moderate sample size, and still be able to estimate very low probabilities of failure.

This is due to the fact that most failures do not involve extreme values of response or

capacity but instead moderately high response along with moderately low capacity.

Therefore, to bypass the requirement of sampling the extreme tail of the limit-state

function, the variables could be considered independently, by separating the response and

the capacity, as discussed by Melchers (1999, Chapter 3). A good analysis of efficiency

and accuracy of separable Monte Carlo simulations can be found in Smarslok et al.

(2006).

The common formulation of the structural failure condition is in the form of a

stress exceeding the material limit. This form, however, does not satisfy the separability

requirement. For example, the stress depends on variability in material properties as well

as design area, which reflects errors in the analysis process. In that case, the limit-state

function can still be re-formulated in a separable form. In this dissertation we re-write the

limit-state in terms of the required area (depends only on variabilities) and built area

(depends only on errors) to bring the limit state to separable form (see Chapter 4).

Response surface approximations

Response surface approximations (RSA) can be used to obtain a closed-form

approximation to the limit state function to facilitate reliability analysis. Response surface

approximations usually fit low-order polynomials to the structural response in terms of

random variables. The probability of failure can then be calculated inexpensively by

Monte Carlo simulation or by FORM or SORM using the fitted polynomials.

Response surface approximations can be applied in different ways. One approach is

to construct local response surfaces in the MPP region that contributes most to the









probability of failure of the structure. Bucher and Bourgund (1990), Rajashekhar and

Ellingwood (1993), Koch and Kodiyalam (1999), Das and Zheng (2000a, 2000b) and

Gayton, Bourinet and Lemaire (2003) used local response surfaces.

Another approach is to construct global response surface over the entire range of

random variables. The examples include Fox (1994, and 1996), Romero and Bankston

(1998), Qu et al. (2003), Youn and Choi (2004) and Kale et al. (2005).

Reliability-Based Design Optimization

Design optimization under a probability of failure constraint is usually referred as

reliability-based design optimization (RBDO). The basic structure of an RBDO problem

is stated as


min f
minf (2.6)
s.t. P < Ptarget

wherefis the objective function (for most problems it is weight), and P and Ptarge are the

probabilistic performance function and the target value for it. The probabilistic

performance function can be probability of failure Pf, reliability index /f or an inverse

reliability measure such as probabilistic sufficiency factor, PSF.

Double loop (Nested) RBDO

Conventional RBDO approach is formulated as a double-loop optimization

problem, where an outer loop performs the design optimization, while an inner loop

optimization is also used for estimating probability of failure (or another probabilistic

performance function). The reliability index approach (RIA) is the most straightforward

approach. In RIA, the probability of failure is usually calculated via FORM, which is an

iterative process an so computationally expensive and sometimes troubled by









convergence problems (Tu et al. 1999). To reduce the computational cost of double loop

approach, various techniques has been proposed, which can be divided into two

categories: (i) techniques that improve the efficiency of uncertainty analysis methods,

such as the methods of fast probability integration (Wu 1994) and two-point adaptive

nonlinear approximations (Grandhi and Wang 1998); (ii) techniques that modify the

formulation of probabilistic constraints, for instance, using inverse reliability measures,

such as the performance measure approach (Tu et al. 1999), probabilistic sufficiency

factor (Qu and Haftka 2003).

Inverse reliability measures are based on margin of safety or safety factors, which

are safety measures in deterministic design. The safety factor is usually defined as the

ratio of structural resistance (e.g., failure stress) to structural response (e.g., stress).

Safety factors permit the designer to estimate the change in structural weight to satisfy a

target safety factor requirement. In probabilistic design, however, the difference between

the probabilistic performance measure and its target value does not provide the designer

with an estimate of the required change in structural weight. Inverse safety measures thus

help the designer to easily estimate the change in structural weight from the values of

probabilistic performance measure and its target value and the inverse safety measures

also improve the computational efficiency (Qu and Haftka 2004). A good analysis and

survey on safety factor and inverse reliability measures was presented by Ramu et al.

(2004).

Single loop RBDO

Single loop formulation avoids nested loops of optimization and reliability

assessment. Some single loop formulations are based on formulating the probabilistic

constraints as deterministic constraints by either approximating the Karush-Kuhn-Tucker









conditions at the MPP or defining a relationship between probabilistic design and safety

factors of deterministic design (e.g., Chen et al. 1997, Kuschel and Rackwitz 2000, Wu et

al. 2001, Qu et al. 2004, Liang et al. 2004). Single loop formulation increases the

efficiency by allowing the solution to be infeasible before convergence and satisfying the

probability constraints only at the optimum. There exist also singe loop formulations that

performs optimization and reliability assessment sequentially (e.g., Royset et al. 2001,

Du and Chen 2004). Sequential optimization and reliability assessment (SORA) of Du

and Chen (2004), for instance, decouples the optimization and reliability assessment by

separating each random design variable into a deterministic component, which is used in

a deterministic optimization, and a stochastic component, which is used in reliability

assessment.

Error and Variability

Uncertainty Classification

Over years researchers proposed many different classifications for uncertainty. For

instance, Melchers (1999) divided uncertainty into seven types: phenomenological

uncertainty, decision uncertainty, modeling uncertainty, prediction uncertainty, physical

uncertainty, statistical uncertainty and human error. Haimes et al. (1994) and Hoffman

and Hammonds (1994) distinguished uncertainty into two types: uncertainty epistemicc

part) and variability aleatoryy part). Epistemic uncertainties arise from lack of knowledge

about the behavior of a phenomenon. They may be reduced by review of literature, expert

consultation, close examination of data and further research. Tools such as scoring

system, expert system and fish-bone diagram can also help in reducing epistemic

uncertainties. Aleatory uncertainties arise from possible variation and random errors in









the values of the parameters and their estimates. They can be reduced by using reliable

manufacturing tools and quality control measures.

Oberkampf et al. (2000, 2002) provided a good analysis of different sources of

uncertainty in engineering modeling and simulations, supply good literature review on

uncertainty quantification and divide the uncertainty into three types: variability,

uncertainty and error. The classification provided by Oberkampf et al. is discussed in the

Motivation section of Chapter 1.

Reliability Improvement by Error and Variability Reduction

Before designing a new structure, material properties and loading conditions are

assessed. The data is collected to constitute the probability distributions of material

properties and loads. The data on material properties is obtained by performing tests on

batches of materials and also from the material manufacturer. To reduce the variability in

material properties quality controls may be applied. Qu et al. (2003) analyzed the effect

of application of quality controls over material allowables in the design of composite

laminates for cryogenic environments. They found that employing quality control reduces

the probability of failure significantly, allowing substantial weight reduction for the same

level of safety.

Similarly, before a newly designed structure is put into service, its performance

under predicted operational conditions is evaluated by collecting data. The data is used to

validate the initial assumptions being made through the design and manufacturing

processes to reduce error in those assumptions. This can be accomplished by the use of

Bayesian statistical methods to modify the assumed probability distributions of error. The

present author will investigate this issue on following the chapters.









After the structure is put into service inspections are performed to detect the

damage developed in the structure. Hence, the inspections are another form of uncertainty

reduction. The effect of inspections in the safety of structures was analyzed (among

others) by Harkness et al. (1994), Provan et al. (1994), Fujimoto et al. (1998), Kale et al.

(2003) and Acar et al. (2004b).

Testing and Probabilistic Design

In probabilistic design, models for predicting uncertainties and performance of

structures are employed. These models involve idealizations and approximations; hence,

validation and verification of these models is necessary. The validation is done by testing

of structures, and verification is done by using more detailed models.

Historical development of testing of structures was given in the papers by Pugsley

(1944) and Whittemore (1954). A literature survey of load testing by Hall and Lind

(1979) presented many uses for load testing in design and safety validation of structures.

Conventional "design by calculation" relies upon tensile coupon tests to estimate material

strength (Hall and Tsai, 1989). Coupon testing is a destructive test to measure loads and

displacements at failure. On the other hand, proof load testing is not a destructive test in

which the structure is tested at a fixed load to measure resistance level of the structure.

Proof load testing in a variety of applications was studied by several authors such as

Barnett and Herman (1965), Shinozuka (1969), Yang (1976), Fujino and Lind (1977),

Rackwitz and Schurpp (1985) and Herbert and Trilling (2006).

Jiao and Moan (1990) illustrated a methodology for probability density function

updating of structural resistance by additional events such as proof loading and non-

destructive inspection by utilizing FORM or SORM methods. Ke (1999) proposed an

approach that specifically addressed the means to design component tests satisfying









reliability requirements and objectives by assuming that the component life distribution

follows Weibull distribution. Zhang and Mahadevan (2001) developed a methodology

that utilizes Bayesian updating to integrate testing and analysis for test plan determination

of a structural components. They considered two kinds of tests: failure probability

estimation and life estimation tests. Soundappan et al. (2004) presented a method for

designing targeted analytical and physical tests to validated reliability of structures

obtained from reliability based designs. They found that the optimum number of tests for

a component is nearly proportional to the square root of probability of failure.

Guidelines for testing of composite materials were presented in Volume 1, Chapter

2 of Composite Materials Handbook (2002). The following are quoted from this source

(pages 2-1 and 2-2). Analysis alone is generally not considered adequate for

substantiation of composite structural designs. Instead, the "building-block approach" to

design development testing is used in concert with analysis. This approach is often

considered essential to the qualification/certification of composite structures due to the

sensitivity of composites to out-of-plane loads, the multiplicity of composite failure

modes and the lack of standard analytical methods. The building-block approach is also

used to establish environmental compensation values applied to full-scale tests at room-

temperature ambient environment, as it is often impractical to conduct these tests under

the actual moisture and temperature environment. Lower-level tests justify these

environmental compensation factors. Similarly, other building-block tests determine

truncation approaches for fatigue spectra and compensation for fatigue scatter at the full-

scale level. The building-block approach is shown schematically in Figure 2.1.











COMPONENTS
w
u c
uI SUB-COMPONENTS c
-

z CDETAILl \


0 ---------------------------
ELEMENTS





-- -- ---- --n ----

Figure 2-1. Building block approach (Reprinted, with permission, from MIL 17- The
Composite Materials Handbook, Vol. 1, Chapter 2, copyright ASTM
International, 100 Barr Harbor Drive, West Conshohocken, PA 19428)

The approach can be summarized in the following steps:
1. Generate material basis values and preliminary design allowables.
2. Based on the design/analysis of the structure, select critical areas for subsequent
test verification.
3. Determine the most strength-critical failure mode for each design feature.
4. Select the test environment that will produce the strength-critical failure mode.
Special attention should be given to matrix-sensitive failure modes (such as
compression, out-of-plane shear, and bondlines) and potential "hot-spots" caused
by out-of-plane loads or stiffness tailored designs.
5. Design and test a series of test specimens, each one of which simulates a single
selected failure mode and loading condition, compare to analytical predictions, and
adjust analysis models or design allowables as necessary.
6. Design and conduct increasingly more complicated tests that evaluate more
complicated loading situations with the possibility of failure from several potential
failure modes. Compare to analytical predictions and adjust analysis models as
necessary.
7. Design (including compensation factors) and conduct, as required, full-scale
component static and fatigue testing for final validation of internal loads and
structural integrity. Compare to analysis.

As noted earlier, validation is done by testing of structures, and verification is done

by using more detailed models. Detailed models may reduce the errors in analysis

models; however errors in the uncertainty models cannot be reduced by this approach. In






27


addition, very detailed models can be computationally prohibitive. Similarly, while

testing of structures reduces both the errors in response models and uncertainty models, it

is expensive. Therefore, the testing of structures needs to be performed simultaneously

with the structural design to reduce cost while still keeping a specified reliability level.














CHAPTER 3
WHY ARE AIRPLANES SO SAFE STRUCTURALLY? EFFECT OF VARIOUS
SAFETY MEASURES

This chapter investigates the effects of error, variability, safety measures and tests

on the structural safety of aircraft. A simple point stress design problem and a simple

uncertainty classification are used. Since this chapter serves as the opening chapter, the

level of analysis and the number of safety measures are kept at a minimum level. Safety

measures considered in this chapter are the load safety factor of 1.5, the use of

conservative material properties and certification test. Other safety measures such as

structural redundancy, coupon and structural element tests will be included in the

following chapters. Interaction of the considered safety measures with one another and

their effectiveness with respect to uncertainties are also explored.

The work given in this chapter was also published in Acar et al. (2006a). My

colleague Dr. Amit Kale's contribution to this work is acknowledged.

Introduction

In the past few years, there has been growing interest in applying probability

methods to aircraft structural design (e.g., Lincoln 1980, Wirsching 1992, Aerospace

Information Report of Society of Automotive Engineers 1997, Long and Narciso 1999).

However, many engineers are skeptical of our ability to calculate the probability of

failure of structural designs for the following reasons. First, data on statistical variability

in material properties, geometry and loading distributions are not always available in full

(e.g., joint distributions), and it has been shown that insufficient information may lead to









large errors in probability calculations (e.g., Ben-Haim and Elishakoff 1990, Neal et al.

1992). Second, the magnitude of errors in calculating loads and predicting structural

response is not known precisely, and there is no consensus on how to model these errors

in a probabilistic setting. As a result of these concerns, it is possible that transition to

probability based design will be gradual. In such circumstances it is important to

understand the impact of existing design practices on safety. This chapter is a first

attempt to explore the effects of various safety measures taken during aircraft structural

design using the deterministic design approach based on FAA regulations.

The safety measures that we include in this chapter are (i) the use of safety factors,

(ii) the use of conservative material properties (A-basis), and (iii) the use of final

certification tests. These safety measures are representative rather than all inclusive. For

example, the use of A-basis properties is a representative measure for the use of

conservative material properties. The safety measures (e.g., structural redundancy) are

discussed in the following chapters. We use A-Basis value rather than B-basis because

we did not include redundancy in this chapter. FAA suggests that (FAR 25.613) when

there is a single failure path, A-Basis properties should be employed, but in case of

multiple failure paths, B-Basis properties are to be used. In next chapter, for instance, we

include structural redundancy in our analysis, so we use B-basis values in Chapter 4. The

effect of the three individual safety measures and their combined effect on the probability

of structural failure of the aircraft are demonstrated. We use Monte Carlo simulations to

calculate the effect of these safety measures on the probability of failure of a structural

component.









We start with a structural design employing all considered safety measures. The

effects of variability in geometry, loads, and material properties are readily incorporated

by the appropriate random variables. However, there is also uncertainty due to various

errors such as modeling errors in the analysis. These errors are fixed but unknown for a

given airplane. To simulate these epistemic uncertainties, we transform the error into a

random variable by considering the design of multiple aircraft models. As a consequence,

for each model the structure is different. It is as if we pretend that there are hundreds of

companies (Airbus, Boeing, Bombardier, Embraer, etc.) each designing essentially the

same airplane, but each having different errors in their structural analysis. This

assumption is only a device to model lack of knowledge or errors in probabilistic setting.

However, pretending that the distribution represents a large number of aircraft companies

helps to motivate the probabilistic setting.

For each model we simulate certification testing. If the airplane passes the test, then

an entire fleet of airplanes with the same design is assumed to be built with different

members of the fleet having different geometry, loads, and material properties based on

assumed models for variability in these properties. That is, the uncertainty due to

variability is simulated by considering multiple realizations of the same design, and the

uncertainty due to errors is simulated by designing different structures to carry the same

loads.

Structural Uncertainties

A good analysis of different sources of uncertainty is provided by Oberkampf et al.

(2000, 2002). Here we simplify the classification, with a view to the question of how to

control uncertainty. We propose in Table 3-1 a classification that distinguishes between

errors (uncertainties that apply equally to the entire fleet of an aircraft model) and









variabilities (uncertainties that vary for the individual aircraft). The distinction is

important because safety measures usually target one or the other. While variabilities are

random uncertainties that can be readily modeled probabilistically, errors are fixed for a

given aircraft model (e.g., Boeing 737-400) but they are largely unknown.

Errors reflect inaccurate modeling of physical phenomena, errors in structural

analysis, errors in load calculations, or use of materials and tooling in construction that

are different from those specified by the designer. Systemic errors affect all the copies of

the structural components made and are therefore fleet-level uncertainties. They can

reflect differences in analysis, manufacturing and operation of the aircraft from an ideal.

The ideal aircraft is an aircraft designed assuming that it is possible to perfectly predict

structural loads and structural failure for a given structure, that there are no biases in the

average material properties and dimensions of the fleet with respect to design

specifications, and that there exists an operating environment that on average agrees with

the design specifications. The other type of uncertainty reflects variability in material

properties, geometry, or loading between different copies of the same structure and is

called here individual uncertainty.

Table 3-1. Uncertainty classification
Type of Spread Cause Remedies
uncertainty
Errors in predicting
Ente ft o structural failure and Testing and
Entire fleet of
Systemic error cs d differences between simulation to improve
components designed i i .1
(modeling errors) properties used in design math model and the
using the model
and average fleet solution.
properties.
Variability in tooling, Improve tooling and
., Individual component
Variability individual component manufacturing process, construction.
levelironments. Quality control.
and flying environments. Quality control.









Safety Measures

Aircraft structural design is still done, by and large, using code-based design rather

than probabilistic approaches. Safety is improved through conservative design practices

that include use of safety factors and conservative material properties. It is also improved

by tests of components and certification tests that can reveal inadequacies in analysis or

construction. In the following we detail some of these safety measures.

Load Safety Factor: Traditionally all aircraft structures are designed with a load

safety factor to withstand 1.5 times the limit-load without failure.

A-Basis Properties: In order to account for uncertainty in material properties, the

Federal Aviation Administration (FAA) states the use of conservative material properties.

This is determined by testing a specified number of coupons selected at random from a

batch of material. The A-basis property is determined by calculating the value of a

material property exceeded by 99% of the population with 95% confidence.

Component and Certification Tests: Component tests and certification tests of

major structural components reduce stress and material uncertainties for given extreme

loads due to inadequate structural models. These tests are conducted in a building block

procedure. First, individual coupons are tested, and then a sub assembly is tested

followed by a full-scale test of the entire structure. Since these tests cannot apply every

load condition to the structure, they leave uncertainties with respect to some loading

conditions. It is possible to reduce the probability of failure by performing more tests to

reduce uncertainty or by extra structural weight to reduce stresses. If certification tests

were designed together with the structure, it is possible that additional tests would

become cost effective because they would allow reduced structural weight.









We simulate the effect of these three safety measures by assuming the statistical

distribution of the uncertainties and incorporating them in approximate probability

calculations and Monte Carlo simulation. For variability the simulation is

straightforward. However, while systemic errors are uncertain at the time of the design,

they will not vary for a single structural component on a particular aircraft. Therefore, to

simulate the uncertainty, we assume that we have a large number of nominally identical

aircraft being designed (e.g., by Airbus, Boeing, Bombardier, Embraer, etc.), with the

errors being fixed for each aircraft. This creates a two-level Monte Carlo simulation, with

different aircraft models being considered at the upper level, and different instances of

the same aircraft at the lower level.

To illustrate the procedure we consider point stress design of a small part of an

aircraft structure. Aircraft structures have more complex failure modes, such as fatigue

and fracture, which require substantially different treatment and the consideration of the

effects of inspections (See Kale et al., 2003). However, this simple example serves to

further our understanding of the interaction between various safety measures. The

procedure is summarized in Fig. 3-1, which is described in detail in the next section.

Design of a Generic Component

Design and Certification Testing

We assume that we have N different aircraft models, i.e., we have N different

companies producing a model with errors. We consider a generic component to represent

the entire aircraft structure. The true stress (o-t ) is found from the equation


Ore = (3.1)
wt









where P is the applied load on the component of width w and thickness t. In a more

general situation, Eq. (3.1) may apply to a small element in a more complex component.

When errors are included in the analysis, the true stress in the component is

different from the calculated stress. We include the errors by introducing an error factor e

while computing the stress as

caic = (1+ e) atre (3.2)

Positive values of e yield conservative estimates of the true stress and negative

values yield unconservative stress estimation. The other random variables account for

variability. Combining Eqs. (3.1) and (3.2), the stress in the component is calculated as


calc = (1+ e) (3.3)
w t

The design thickness is determined so that the calculated stress in the component is

equal to material allowable stress for a design load Pd multiplied by a safety factor SF,

hence the design thickness of the component is calculated from Eq. (3.3) as


tdesgn = (1+ e) SFPd (3.4)
Wdeslgn'a

where the design component width, wdesgn, is taken here to be 1.0, and oa is the material

stress allowable obtained from testing a batch of coupons according to procedures that

depend on design practices. Here, we assume that A-basis properties are used (see

Appendix A). During the design process, the only random quantities are o- and e. The

thickness obtained from Eq. (3.4), step A in Fig. 3-1, is the nominal thickness for a given

aircraft model. The actual thickness will vary due to individual-level manufacturing

uncertainties.











FF Select random e and create a new design

A t mB h a Reject
SPerform certification test Fafled Des
Passed
Build a copy of the aircraft
and
Apply service loads
D I
Check if the aircraft fails under the service loads
and
Count the number of aircraft failed
E 1
N--- Check if M number of copies are built
No
SYes
--- Check if N number of different designs are created
No
Yes
Calculate Probability of Failure


Figure 3-1. Flowchart for Monte Carlo simulation of component design and failure

After the component has been designed (that is, thickness is determined from Eq.

(3.4)), we simulate certification testing for the aircraft. Here we assume that the

component will not be built with complete fidelity to the design due to variability in

geometry (width and thickness). The component is then loaded with the design axial

force of SF times Pd, and the stress in the component is recorded. If this stress exceeds the

failure stress (itself a random variable, see Table 3-2) then the design is rejected,

otherwise it is certified for use. That is, the airplane is certified (step B in Fig. 3-1) if the

following inequality is satisfied


a a = -SFPd < 0 (3.5)
wt


and we can build multiple copies of the airplane. We subject the component in each

airplane to actual random maximum (over a lifetime) service loads (step D in Fig. 3-1)

and decide whether it fails using Eq. (3.6).









P > C =twcf (3.6)

Here, P is the applied load, and C is the load carrying capacity of the structure in

terms of the width w, thickness t and failure stress of. A summary of the distributions for

the random variables used in design and certification is listed in Table 3-2.

Table 3-2. Distribution of random variables used for component design and certification
Variables Distribution Mean Scatter
Plate width (w) Uniform 1.0 (1%) bounds
Plate thickness (t) Uniform design (3%) bounds
Failure stress (cf) Lognormal 150.0 8 % coefficient of variation
Service Load (P) Lognormal 100.0 10 % coefficient of variation
Error factor (e) Uniform 0.0 10% to 50%

This procedure of design and testing is repeated (steps A-B) for N different aircraft

models. For each new model, a different random error factor e is picked for the design,

and different allowable properties are generated from coupon testing (Appendix A). Then

in the testing, different thicknesses and widths, and different failure stresses are generated

at random from their distributions.

Effect of Certification Tests on Distribution of Error Factor e

One can argue that the way certification tests reduce the probability of failure is by

changing the distribution of the error factor e. Without certification testing, we assume

symmetric distribution of this error factor. However, designs based on unconservative

models are more likely to fail certification, and so the distribution of e becomes

conservative for structures that pass certification. In order to quantify this effect, we

calculated the updated distribution of the error factor e. The updated distribution is

calculated analytically by Bayesian updating by making some approximations, and

Monte Carlo simulations are conducted to check the validity of those approximations.









Bayesian updating is a commonly used technique to obtain updated (or posterior)

distribution of a random variable upon obtaining new information about the random

variable. The new information here is that the component has passed the certification test.

Using Bayes' Theorem, the updated (posterior) distribution f/ (0) of a random

variable 0is obtained from the initial (prior) distribution f1 () based on new information

as

f (0) = Pr()f() (3.7)
f Pr(e 10) f (O)dO


where Pr(E 0) is the conditional probability of observing the experimental data E given

that the value of the random variable is 0.

For our case, the posterior distribution fU (e) of the error factor e is given as

fU (e)= b Pr(CT e) f (e) (3.8)

SPr(CT I e) f (e)de
-b

where CT is the event of passing certification, and Pr(CT e) is the probability of passing

certification for a given e. Initially, e is assumed to be uniformly distributed. The

procedure of calculation of Pr(CT e) is described in Appendix B, where we approximate

the distribution of the geometrical variables, t and w as lognormal, taking advantage of

the fact that their coefficient of variation is small compared to that of the failure stress

(see Table 3-2).

We illustrate the effect of certification tests for the components designed with A-

Basis material properties. An initial and updated distribution plot of error factor e with 50

% bound is shown in Fig. 3-2. Monte Carlo simulation with 50,000 aircraft models is also











shown. Figure 3-2 shows that the certification tests greatly reduce the probability of


negative error, hence eliminating most unconservative designs. As seen from the figure,


the approximate distribution calculated by the analytical approach matches well the


distribution obtained from Monte Carlo simulations.


Initial and Updated Distribution of Error Factor e
and
Comparison of Analytical Approx. with Monte Carlo Simulations
16

S4 Monte Carlo, initial
S Monte Carlo, updated
-P-Analytical Approx initial
S 1 2 --X-Analyt ca Approx, updated


08

06

04

02

0
-05 -04 -03 -02 -01 00 01 02 03 04 05
error factor, e


Figure 3-2. Initial and updated probability distribution functions of error factor e. Error
bound is 50% and Monte Carlo simulation done with sample size of 50,000.

Probability of Failure Calculation by Analytical Approximation

The stress analysis represented by Eq. (3.1) is trivial, so that the computational cost


of Monte Carlo simulation of the probability of failure is not high. However, it is


desirable to obtain also analytical probabilities that may be used for more complex stress


analysis and to check the Monte Carlo simulations.


In order to take advantage of simplifying approximations of the distribution of the


geometry parameters, it is convenient to perform the probability calculation in two stages,


corresponding to the inner and outer loops of Fig. 3-1. That is, we first obtain expressions









for the probability of failure of a single aircraft model (that is, given e and allowable

stress). We then calculate the probability of failure over all aircraft models.

The mean value of the probability of failure over all aircraft models is calculated as


f= f P(tdesign) f (design) dtdesign (3.9)


where tdesign is the non-deterministic distribution parameter, and f(tdesign) is the

probability density function of design .

It is important to have a measure of variability in this probability from one aircraft

model to another. The standard deviation of failure probability gives a measure of this

variability. In addition, it provides information on how accurate is the probability of

failure obtained from Monte Carlo simulations. The standard deviation can be calculated

from

( 1/2
Pf1 = [P designg) Pf f Qdesign) dtdeign (3.10)


Probability of Failure Calculation by Monte Carlo Simulations

The inner loop in Fig. 3-1 (steps C-E) represents the simulation of a population of

M airplanes (hence components) that all have the same design. However, each component

is different due to variability in geometry, failure stress, and loading (step D). We subject

the component in each airplane to actual random maximum (over a lifetime) service loads

(step E) and calculate whether it fails using Eq. (3.6).

For airplane model that pass certification, we count the number of components

failed. The failure probability is calculated by dividing the number of failures by the









number of airplane models that passed certification, times the number of copies of each

model.

The analytical approximation for the probability of failure suffers due to the

approximations used, while the Monte Carlo simulation is subject to sampling errors,

especially for low probabilities of failure. Using large samples, though, can reduce the

latter. Therefore, we compared the two methods for a relatively large sample of 10,000

aircraft models with 100,000 instances of each model. In addition, the comparison is

performed for the case where mean material properties (rather than A-basis properties)

are used for the design, so that the probability of failure is high enough for the Monte

Carlo simulation to capture it accurately. Table 3-3 shows the results for this case.

Table 3-3. Comparison of probability of failures for components designed using safety
factor of 1.5, mean value for allowable stress and error bound of 50%
Vae Analytical Monte Carlo %
ValueApproximation Simulation error
Average Value of Pf without certification (Pnt) 1.715x10-1 1.726x 101 0.6
Standard Deviation of Pnt 3.058x 101 3.068x 101 0.3
Average Value of Pf with certification (Pt) 3.166x 10-4 3.071 x 10-4 3.1
Standard Deviation of Pt 2.285x 10-3 2.322x 10-3 1.6
Average Value of Initial error factor (e') 0.0000 -0.00024 ---
Standard Deviation of e' 0.2887 0.2905 0.6
Average Value of Updated error factor (eup) 0.2468 0.2491 0.9
Standard Deviation of e1p 0.1536 0.1542 0.4
N = 10,000 and M = 100,000 is used in the Monte Carlo Simulations

The last column of Table 3-3 shows the percent error of the analytical

approximation compared to Monte Carlo simulations. It is seen that the analytical

approximation is in good agreement with the values obtained through Monte Carlo

simulations. It is remarkable that the standard deviation of the probability of failure is

almost twice the average value of the probability (the ratio, the coefficient of variation, is

about 178%) before certification, and about seven times larger after certification. This









indicates huge variability in the probability of failure for different aircraft models, and

this is due to the large error bound, be=50%. With 10,000 different aircraft models (N),

the standard deviation in the Monte Carlo estimates is about 1%, and the differences

between the Monte Carlo simulation and the analytical approximation are of that order.

Effect of Three Safety Measures on Probability of Failure

We next investigate the effect of other safety measures on failure probability of the

components using Monte Carlo simulations. We performed the simulation for a range of

variability in error factor e for 5000 airplane models (N samples in outer loop) and 20,000

copies of each airplane model (M samples in inner loop). Here, we compare the

probability of failure of a structure designed with three safety measures (safety factor,

conservative material property and certification testing) to that of a structure designed

without safety measures.

Table 3-4 presents the results when all safety measures are used for different

bounds on the error. The second column shows the mean and standard deviation of

design thicknesses generated for components that passed certification. These components

correspond to the outer loop of Fig. 3-1. The variability in design thickness is due to the

randomness in the error e and in the stress allowable. The average thickness before

certification was 1.269, so that the column shows the conservative effect of certification

testing. When the error bound is 10%, 98.8% of the components pass certification (third

column in Table 3-4), and the average thickness is increased by only 0.24% due to the

certification process. On the other hand, when the error bound is 50%, 29% of the

components do not pass certification, and this raises the average thickness to 1.453. Thus,

the increase in error bound has two opposite effects. Without certification testing,

increasing the error bound greatly increases the probability of failure. For example, when









the error bound changes from 30% to 50%, the probability of failure without certification

changes from 0.00091 to 0.0449, or by a factor of 49. On the other hand, with the

increased average thickness, after certification the probability increases only from

1.343x]0-4 to 1.664x10-4.

Table 3-4. Probability of failure for different bounds on error e for components designed
using safety factor of 1.5 and A-basis property for allowable stress. Numbers
in parenthesis denote the coefficient of variation of the quantity. Average
design thickness without certification is 1.271.
Average r Probability Probability of Probabilit
Error n C of failure failure Pobl
Bound thickness cation after without Probability y
Bound after without
be after failure certification certification ratio (Pt/Pnt) difference
certificate rate % (p -4 (Pt) 4 (Pnt-Pt)
on
1.453
50% (0.19 29.3 1.664 (7.86) 449.0 (2.74) 3.706x 10- 4.473 10-2
(0.19)
1.389
40%389 24.3 1.586 (6.92) 89.77 (3.22) 1.767x10-2 8.818 10-3
(0.17)
1.329
30% (0.1 16.3 1.343 (5.28) 9.086 (3.46) 1.479x10-1 7.742x10-4
(0.15)
1.283
20% .12 6.2 0.304 (4.81) 0.477 (3.51) 6.377x10- 1.727x10-
(0.12)
1.272
10% .07 1.2 0.027 (4.71) 0.029 (4.59) 9.147x10-1 2.490x10-7
(0.07)
*Average over N=5000 models

The effectiveness of the certification tests can be expressed by two measures of

probability improvement. The first measure is the ratio of the probability of failure with

the test, Pt, to the probability of failure without tests, Pnt. The second measure is the

difference of these probabilities. The ratio is a more useful indicator for low probabilities

of failure, while the difference is more meaningful for high probabilities of failure.

However, when Pt is high, the ratio can mislead. That is, an improvement from a

probability of failure of 0.5 to 0.1 is more substantial than an improvement in probability

of failure of 0.1 to 0.01, because it "saves" more airplanes. However, the ratio is more

useful when the probabilities are small, and the difference is not very informative.









Table 3-4 shows that certification testing is more important for large error bounds

e. For these higher values the number of components that did not pass certification is

higher, thereby reducing the failure probability for those that passed certification. While

the effect of component tests (building block tests) is not simulated, their main effect is to

reduce the error magnitude e. This is primarily due to the usefulness of component tests

in improving analytical models and revealing unmodeled failure modes. With that in

mind, we note that the failure probability for the 50% error range is 1.7x10-4, and it

reduces to 2.7x 10-6 for the 10% error range-that is, by a factor of 63.

The actual failure probability of aircraft components is expected to be of the order

of 10-8 per flight, much lower than the best number in the fourth column of Table 3-4.

However, the number in Table 3-4 is for a lifetime for a single structural component.

Assuming about 10,000 flights in the life of a component and 100 independent structural

components, this 10-5 failure probability for a component will translate to a per flight

probability of failure of 10-7 per airplane. This factor of 10 discrepancy is exacerbated by

other failure modes like fatigue that have not been considered. However, other safety

measures, such as conservative load specifications may account for this discrepancy.

Table 3-5 shows results when average rather than conservative material properties

are used. It can be seen from Table 3-5 that the average thickness determined using the

mean value of allowable stress is lower than that determined using the A-basis value of

allowable stress (Table 3-4). This is equivalent to adding an additional safety factor over

an already existing safety factor of 1.5. For the distribution (lognormal with 8%

coefficient of variation) and number of batch tests (40 tests) considered here, a typical

value of the safety factor due to A-Basis property is around 1.27.









Table 3-5. Probability of failure for different bounds on error e for components designed
using safety factor of 1.5 and mean value for allowable stress. Numbers in
parenthesis denote the coefficient of variation of the quantity. Average design
thickness without certification is 1.000.
Average Probability of
TAverage Certificat Probability of Probability of
Error design F a failure Probability Probability
ion Failure after
bound thickness ailue without ratio difference
failure certification
be after e c% P) 1-4 certification (Pt/Pnt) (Pnt-Pt)
certification rate () 1 (Pnt) x 10-4
50% 1.243 (0.13) 50.1 3.420 (5.82) 1681 (1.81) 2.035x10-3 1.677x10-1
40% 1.191 (0.11) 50.1 4.086 (6.78) 969.0 (1.99) 4.217x10-3 9.649x10-2
30% 1.139 (0.09) 50.8 5.616 (5.45) 376.6 (2.00) 1.495x 102 3.700x10-2
20% 1.086 (0.07) 50.7 6.253 (3.19) 92.67 (1.83) 6.748x10-2 8.642x10-3
10% 1.029 (0.05) 51.0 9.209 (1.70) 19.63 (1.25) 4.690x10-1 1.043 x 103
*Average over N=5000 models
+With only 5000 models, the standard deviation in the certification failure rate is about
0.71%. Thus, all the numbers in this column are about 50% as may be expected when mean
material properties are used.

Without the A-basis properties, the stress in the certification test is approximately

equal to the average ultimate service stress, so that about 50% of the components fail

certification. When the errors are large, this raises substantially the average thickness of

the components that pass certification, so that for an error bound of 50% the certification

test is equivalent to a safety factor of 1.243. Large errors produce some super-strong and

some super-weak components (see Fig. 3-3b). The super-weak components are mostly

caught by the certification tests, leaving the super-strong components to reduce the

probability of failure. Another way of looking at this effect is to note that when there are

no errors, there is no point to the tests. Indeed, it can be seen that the probability of

failure without certification tests improves with reduced error bound e, but that the

reduced effect of the certification tests reverses the trend. Thus for this case we obtain the

counter-intuitive results that larger errors produce safer designs.

Comparing the first row of Table 3-5 to that of Table 3-3 we see the effect of the

smaller sample for the Monte Carlo simulations. Table 3-3 was obtained with 10,000










models and 100,000 copies per model, while Table 3-5 was obtained with 5000 models,

and 20,000 copies per model. The difference in the probability of failure after

certification between the two tables is about 11 percent. However, the two values straddle

the analytical approximation.

The effects of building block type of tests that are conducted before certification

are not included in this study. These tests reduce the errors in analytical models. For

instance, if there is 50% error in the analytical model the building block type of tests may

reduce this error to lower values. Hence, the difference between the rows of Table 3-4,

may be viewed as indicating the benefits of reducing the error by building block tests.

(b)
Designs of
high error bound
(a)
Designs of
Design low error bound
Thickness I Minimum thickness that can
I pass certification test





Figure 3-3. Design thickness variation with low and high error bounds. Note that after
certification testing only the designs above the minimum thickness are built
and flown. Those on the right have a much higher average design thickness
than those on the left.

Table 3-6 shows the effect of not using a safety factor. Although certification tests

improve the reliability, again in a general trend of high improvement with high error, the

lack of safety factor of 1.5 limits the improvement. Comparing Tables 3-4 and 3-6 it can

be seen that the safety factor reduces the probability of failure by two to four orders of

magnitudes. It is interesting to note that the effect of the error bound on the probability of









failure after certification is not monotonic, and this phenomenon is discussed in

Appendix C.

Table 3-6. Probability of failure for different bounds on error e for safety factor of 1.0
and A-basis allowable stress. Numbers in parenthesis denote the c.o.v. of the
quantity. Average design thickness without certification is 0.847.
Average Certificat Failure Failure
Error design probability probability Probabity Probability
ion Probability
bound thickness n after with no difference
S ae failure ratio (Pt/Pnt)
be after certification certification (Pnt-Pt)
certification rate (P) x 10-2 (Pt) x 10-2
50% 0.969 (0.19) 29.4 6.978 (2.12) 29.49 (1.31) 2.366x10-1 2.251x10-1
40% 0.929 (0.17) 25.0 7.543 (1.98) 24.56 (1.38) 3.071x10-1 1.702x10-1
30% 0.886 (0.15) 16.6 8.923 (1.73) 17.11 (1.43) 5.216x10-1 8.184x10-2
20% 0.855 (0.11) 5.7 8.171 (1.40) 9.665 (1.34) 8.454x10-' 1.494x10-2
10% 0.847 (0.06) 1.3 4.879 (0.97) 4.996 (0.97) 9.767x10-1 1.163x10-3
*Average over N=5000 models

Table 3-7, shows results when the only safety measure is certification testing.

Certification tests can reduce the probability of failure of components by 38%, the

highest improvement corresponds to the highest error. As can be expected, without

certification tests and safety measures, the probability of failure is near 50%. Tables 3-4

through 3-7 illustrate the probability of failure for a fixed 8 % coefficient of variation in

failure stress. The general conclusion that can be drawn from these results is that the error

bound e is one of the main parameters affecting the efficacy of certification tests to

improve reliability of components.

Table 3-7. Probability of failure for different error bounds for safety factor of 1.0 and
mean value for allowable stress. Average design thickness without
certification is 0.667.
Certific Probability of Probability of
Error Average design Certific Probability of Probability of Probability Probability
bound t s ar ation Failure after failure without ro
bound thickness after ratio difference
S failure certification certification (Pt/Pnt) (Pnt-Pt)
be certification (P % PP) (Pnt-Pt)
rate % (Pr) (P-t)
50% 0.830 (0.12) 50.1 0.125 (1.39) 0.505 (0.83) 2.463 x 10-' 3.808x10-1
40% 0.796 (0.11) 50.2 0.158 (1.20) 0.504 (0.79) 3.140x10-1 3.459x10-1
30% 0.761 (0.09) 50.4 0.205 (0.92) 0.503 (0.72) 4.075 x 10-' 2.981x10-1
20% 0.727 (0.08) 50.9 0.285 (0.64) 0.503 (0.58) 5.653x10-' 2.189x10-1
10% 0.686 (0.05) 50.7 0.412 (0.34) 0.500 (0.34) 8.228x10-1 8.869x10-2
*Average over N=5000 models









Next, we will explore how another parameter, variability, influences the efficacy of

tests. This is accomplished by changing the coefficient of variation of failure stress of

between 0-16% and keeping the error bound constant.

Table 3-8. Probability of failure for different uncertainty in failure stress for the
components designed with safety factor of 1.5, 50% error bounds e and A-
basis allowable stress.
Coeffi Average Average Probability Probability
cient design design Certific of failure of failure Probability Probability
r-. Probability Probability
of thickness thickness ation after without
ratio difference
variati without after failure certification certification ( ) (P-
(Pt/Pnt) (Pnt-Pt)
on of certificate certificate rate % n n
-f on on *(Pt) 10-4 (Pnt) x 10-4
0.998 1.250 0.017 1699
0% 0.998 1.250 50.2 0.017 1699 1.004x105- 1.698x10-1
(0.29) (0.11) (6.85) (1.87)
4 1.127 1.347 380.087 970.4 8.9731 9.703x102
(0.29) (0.15) (7.20) (2.35)
8o% 1.269 1.453 293 1.664 449.0 3.706x 13 4.473 10-2
(0.29) (0.19) (7.86) (2.74)
1.431 1.574 13.33 206.1 2 2
12% 1.431 1.574 20.9 13.33 206.1 6.469x 10-2 1.927x 10-2
(0.29) (0.22) (7.71) (3.08)
1 1.616 1.723 14.1 22.52 107.3 2.100x 10- 8.476103
(0.30) (0.25) (5.54) (3.24)
*Average over N=5000 models

Table 3-9. Probability of failure for different uncertainty in failure stress for the
components designed with safety factor of 1.5, 30% error bound e and A-basis
allowable stress.
Coeffi Average Average Probability Probability
cient design design Certific of failure of failure Probability Probability
of thickness thickness ation after without
ratio difference
variati without after failure certification certification (P/) (
(Pt/Pnt) (Pnt-Pt)
on of certificate certificate rate % n n
-f on on *(Pt) 10-4 (Pnt) x 10-4
1.001 1.148 0.026 223.8
0% 1.001 1.148 50.1 0.026 223.8 1.163x10-4 2.238x 10-2
(0.17) (0.08) (4.79) (2.50)
4% 1.126 1.232 316 0.146 35.25 4.149 10 3 3.511 10-3
(0.17) (0.11) (6.03) (2.97)
8 1.269 1.329 161.343 9.086 1.479x10- 774210-4
(0.17) (0.15) (5.28) (3.46)
12 1.431 1.459 7.2 2.404 4.314 5.572x101 1.911x104
(0.18) (0.17) (3.87) (3.45)
1.617 1.630 2.513 3.102
16% 1.617 1.630 3.3 2.513 3.102 8.099x10-1 5.896x10-5
(0.18) (0.18) (3.73) (3.54)
Average over N=5000 models









Table 3.10. Probability of failure for uncertainty in failure stress for components designed
using safety factor of 1.5, 10% error bounds e and A-basis properties
Probabili
Coeffi Average Average Probability
ty of
cient design design Certific fa e of failure Pro
failure Probability
of thickness thickness action without Probability
after difference
variati without after failure certification ratio (P/Pnt)
certificate (Pnt-Pt)
on of certificate certificate rate % n
On ( 10-4on
of on on (Pt)x 104 (Pnt)10
000 1.000 1.048 50.3 0.075 1.745 4.304 102 1.669 104
(0.06) (0.03) (2.91) (1.78)
4% 1.126 1.131 0.053 0.070 7.548x10-1 1.716 10-6
(0.06) (0.06) (3.85) (3.56)
8% 1.269 1.272 12 0.027 0.029 914710-1 2490x107
(0.06) (0.07) (4.71) (4.59)
12 1.431 1.432 8 0.049 0.051 9.623101 1.926x107
(0.07) (0.07) (4.30) (4.23)
16% 1.623 1.624 05 0.085 0.083 9.781-10 1 1.853-107
(0.08) (0.08) (3.50) (3.55)
*Average over N=5000 models

The increase in the variability in failure stress has a large effect on the allowable

stress because A-basis properties specify an allowable that is below 99% of the sample.

Increased variability reduces the allowable stress and therefore increases the design

thickness. It is seen from Tables 3-8 through 3-10 that when the variability increases

from 0% to 16%, the design thickness increases by more than 60%. This greatly reduces

the probability of failure without certification. However, the probability of failure with

certification still deteriorates. That is, the use of A-basis properties fails to fully

compensate for the variability in material properties. This opposite behavior of the

probability of failure before and after certification is discussed in more detail in

Appendix C.

The variability in failure stress greatly changes the effect of certification tests.

Although the average design thicknesses of the components increase with the increase in

variability, we see that when the variability is large, the value of the tests is reduced

because the tested aircraft can be greatly different from the airplanes in actual service.









We indeed see from the Tables 3-8, 3-9 and 3-10 that the effect of certification tests is

reduced as the variability in the failure stress increases. Recall that the effect of

certification tests is also reduced when the error e decreases. Indeed, Table 3-8 shows a

much smaller effect of the tests than Table 3-10. Comparing the second and third

columns of Tables 3-8, 3-9 and 3-10 we see that as the bound of error decreases, the

change in the average value of design thicknesses of the components become less which

is an indication of loss in the efficacy of certification tests.

Up to now, both the probability difference (Pnt-Pt) and the probability ratio (Pt/Pnt)

seem to be good indicators of efficacy of tests. To allow easy visualization, we combined

the errors and the variability in a single ratio (Bound of e) / VR(o/cf) ratio (ratio of error

bound e to the coefficient of variation of the stress ratio). The denominator accounts for

the major contributors to the variability. The value in the denominator is a function of

four variables; service load P, width w, thickness t, and failure stress af. Here, P and af

have lognormal distributions but w and t are uniformly distributed. Since the coefficient

of variations of w and t is very small, they can also be treated as lognormally distributed

to make calculation of the denominator easy while plotting the graphs. Since the standard

deviations of the variables are small, the denominator is now the square root of the sum

of the squares of coefficient of variations of the four variables mentioned above, that is

VR (;I ) V V(P) + V] (w) + V] (t) + V] (f ) (3.11)


The effective safety factor is the ratio of the design thickness of the component

when safety measures (such as usage of A-basis values for material properties and safety

factor) are applied to the thickness of the component when no safety measures are taken.










Figures 3-4 and 3-5, present the Pt/Pnt ratio in visual formats. It can be seen that as

expected, the ratio decreases as the (Bounds on e)/VR(oC/f) ratio increases. However,

these two figures do not give a clear indication of how certification tests are influenced

by the effective safety factor.


o 0 10o


0,,
: 10" 10-
SI 10 10



VR 10 s 2 102
SFE 10S'"






Bound of
VR ("/-,)
VR(/f)
Figure 3-4. Influence of effective safety Figure 3-5. Influence of effective safety
factor, error, and variability on the factor, error and variability on the
probability ratio (3-D view) probability ratio (2-D plot)

Figures 3-6 and 3-7 show the probability difference, Pnt-Pt. In these cases, the

dependence on the effective safety factor is monotonic. As expected, it is seen that as the

effective safety factor increases, the improvement in the safety of component decreases;

meaning that the certification tests become less useful. The probability difference is more

descriptive as it is proportional to the number of aircraft failures prevented by

certification testing. The probability ratio lacks such clear physical interpretation, even

though it is a more attractive measure when the probability of failure is very small.

Considering the results presented by Figures 3-4 through 3-7, the probability

difference (Pnt-Pt) is the more appropriate choice for expressing the effectiveness of tests.















035
03 ;

025

0*


035

03

055

02Pfl-P, SFE


25



2-


01


0.35

03

025

0. P55 P
0.15


"' i Y1


SF.-E 2 Bound ofe 0
05 0 15 2 25 3 3,5 4 45 5
VR (brd) Bound ofe

Figure 3-6. Influence of effective safety Figure 3-7. Influence of effective safety
factor, error and variability on the factor, error and variability on
probability difference (3-D view) probability difference (2-D plo

Summary

We have used a simple example of point stress design for yield to illustrate the


effects of several safety measures taken in aircraft design: safety factors, conservative


material properties, and certification tests. Analytical calculations and Monte Carlo


simulation were performed to account for both fleet-level uncertainties (such as errors in


analytical models) and individual uncertainties (such as variability in material properties).


It was seen that an increase of the systemic errors in the analysis causes an increase


in the probability of failure. We found that the systemic errors can be reduced by the use


of certification tests, thereby reducing the probability of failure. Also we found that


design thicknesses of the components increased as the bounds of systemic errors


increased.


We found that the effect of certification tests is most important when errors in


analytical models are high and when the variability between airplanes is low. This leads


to the surprising result that in some situations larger error variability in analytical models


reduces the probability of failure if certification tests are conducted. For the simple


the
t)


r









example analyzed here, the use of conservative (A-basis) material properties was

equivalent to a safety factor of up to 1.6, depending on the scatter in failure stresses.

The effectiveness of the certification tests is expressed by two measures of

probability improvement. The ratio of the probability of failure with the test, Pt, to the

probability of failure without tests, Pnt, is useful when Pt is small. The difference is more

meaningful when the probability is high. Using these measures we have shown that the

effectiveness of certification tests increases when the ratio of error to variability is large

and when the effective safety factor is small.

The effect of building-block type tests that are conducted before certification was

not assessed here. However, these tests reduce the errors in the analytical models, and on

that basis we determined that they can reduce the probability of failure by one or two

orders of magnitude.

The calculated probabilities of failure with all the considered safety margins

explain why passenger aircraft are so safe structurally. They were still somewhat high -

about 107--compared to the probability of failure of actual aircraft structural

components-about 10-8. This may be due to additional safety measures, such as

conservative design loads or to the effect of design against additional failure modes.














CHAPTER 4
COMPARING EFFECTIVENESS OF MEASURES THAT IMPROVE AIRCRAFT
STRUCTURAL SAFETY

Chapter 3 explored how safety measures compensate for errors and variability. The

major finding of that chapter was that certification tests are most effective when errors

are large, variability is low, and the overall safety factor is low. Chapter 3 mainly focused

on the effectiveness of certification testing, but the relative effectiveness of safety

measures was not addressed. The present chapter takes a further step and aims to discover

how measures that improve aircraft structural safety compare with one another in terms

of weight effectiveness. In addition, structural redundancy-another safety measure-is

included in the analysis. In addition the simple error model of Chapter 3 is replaced by a

more detailed error model. Comparison of the effectiveness of error and variability

reduction with other safety measures is also given.

The research presented in this chapter is submitted for publication (Acar et al.

2006d). My colleague Dr. Amit Kale's contribution to this work is acknowledged.

Introduction

As noted earlier, aircraft structural design is still carried out by using code-based

design, rather than probabilistic design. Safety is improved through conservative design

practices that include the use of safety factors and conservative material properties.

Safety is also improved by testing of components, redundancy, improved modeling to

reduce errors and improved manufacturing to reduce variability. The following gives

brief description of these safety measures.









Load Safety Factor

In transport aircraft design, FAA regulations state the use of a load safety factor of

1.5 (FAR 25.303). That is, aircraft structures are designed to withstand 1.5 times the limit

load without failure.

Conservative Material Properties

In order to account for uncertainty in material properties, FAA regulations state the

use of conservative material properties (FAR 25.613). The conservative material

properties are characterized as A-basis and B-basis material property values, and the use

of A-basis or B-basis values depends on the redundancy. If there is single failure path in

the structure, A-basis values are used, while for the case of multiple failure paths (i.e.,

redundant structures), B-basis values are used. Detailed information on these values is

provided in Chapter 8 of Volume 1 of the Composite Materials Handbook (2000). The

basis values are determined by testing a number of coupons selected at random from a

material batch. The A-basis value is determined by calculating the value of a material

property exceeded by 99% of the population with 95% confidence, while the B-basis

value is the value of a material property exceeded by 90% of the population with 95%

confidence. Here, we take the redundancy of the structure into account, so we use B-basis

values (see Appendix A for the B-basis value calculation). The number of coupon tests is

assumed to be 40.

Tests

Tests of major structural components reduce stress and material uncertainties for

given extreme loads due to inadequate structural models. These tests are conducted in a

building block procedure (Composite Materials Handbook (2000), Volume 1, Chapter 2).

First, individual coupons are tested, and then a sub-assembly is tested, followed by a full-









scale test of the entire structure. Here, we only consider the final certification test for an

aircraft. Other tests are assumed to be error reduction measures and their effect is

analyzed indirectly by considering the effect of error reduction.

Redundancy

Transport airliners are designed with double and triple redundancy features in all

major systems to minimize the failure probability. Redundancy is intended to ensure that

a single component failure does not lead to catastrophic failure of the system. In the

present work, we assume that an aircraft structure will fail if two local failures occur in

the structure.

Error Reduction

Improvements in the accuracy of structural analysis and failure prediction of

aircraft structures reduce errors and enhance the level of safety of the structures. These

improvements may be due to better modeling techniques developed by researchers, more

detailed finite element models made possible by faster computers, or more accurate

failure predictions due to extensive testing.

Variability Reduction

Examples of mechanisms that reduce variability in material properties include

quality control and improved manufacturing processes. Variability in damage and ageing

effects is accomplished through inspections and structural health monitoring. Variability

in loads may be reduced by better pilot training and information that allows pilots to

more effectively avoid regions of high turbulence. Here we investigate only the effect of

reduced variability in material properties.

The next section of this chapter discusses the more detailed error model used in this

chapter, along with variability and total safety factor. Next, the effect of certification tests









on error distribution is analyzed. Then, details of the calculation of the probability of

failure via separable Monte Carlo simulations (MCS) are given. Finally, the chapter

finalizes with the results and summary.

Errors, Variability and Total Safety Factor

The simplified uncertainty classification used in Chapter 3 is also used in this

chapter, where errors are uncertainties that apply equally to the entire fleet of an aircraft

model and variabilities are uncertainties that vary for the individual aircraft (see Table 3-

1, Chapter3). This section first discusses the errors in design and construction. Next, total

error factor and total safety factor are introduced, finally, simulation of variability is

discussed.

Errors in Design

We consider static point stress design for simplicity. Other types of failures such as

fatigue, corrosion or crack instability are not taken into account. We assume that an

aircraft structure will fail only if two local failure events occur. For example, we assume

that the wing will fail structurally if two local failures occur at the wing panels. The

correlation coefficient between the probabilities of these two events is assumed to be 0.5.

Before starting the structural design, aerodynamic analysis needs to be performed

to determine the loads acting on the aircraft. However, the calculated design load value,

Pcalc, differs from the actual loading Pd under conditions corresponding to FAA design

specifications (e.g., gust-strength specifications). Since each company has different

design practices, the error in load calculation, ep, is different from one company to

another. The calculated design load Pcal, is expressed in terms of the true design load Pd

as









Pcal = (l+ep)Pd (4.1)

Besides the error in load calculation, an aircraft company may also make errors in

stress calculation. We consider a small region in a structural part, characterized by a

thickness t and width w, that resists the load in that region. The value of the stress in a

structural part calculated by the stress analysis team, ocalc, can be expressed in terms of

the load values calculated by the load team Pcalc, the design width Wdes gn, and the

thickness t of the structural part by introducing the term e, representing error in the stress

analysis


acalc =(1+e) calc (4.2)
Design t

Equation (4.3) is used by a structural designer to calculate the design thickness

design required to carry the calculated design load times the safety factor SFL. That is,


design (1+e) SFPcalc =(1+e)(1+e) SFLPd (4.3)
Design ()a )calc w design (0a )calc

where (oa)calc is the value of allowable stress for the structure used in the design, which

is calculated based on coupon tests using failure models such as Tresca or von Mises.

Since these failure theories are not exact, we have

(a )calc = 1-e (a )true (4.4)

where ef is the error associated with failure prediction. Moreover, the errors due to the

limited amount of coupon testing to determine the allowables, and the differences

between the material properties used by the designer and the average true properties of

the material used in production are included in this error. Note that the formulation of Eq.

(4.4) is different to that of Eqs. (4.1) and (4.2) in that the sign in front of the error factor









ef is negative, because we consistently formulate the expressions such that positive error

implies a conservative decision.

Combining Eqs. (4.3) and (4.4), we can express the design value of the load

carrying area as

( +e,)(1+ep) SFLPd
Adesig~, = t. ,.," design (4.5)
1 ( a )true

Errors in Construction

In addition to the above errors, there will also be construction errors in the

geometric parameters. These construction errors represent the difference between the

values of these parameters in an average airplane (fleet-average) built by an aircraft

company and the design values of these parameters. The error in width, ew, represents the

deviation of the design width of the structural part, wdesgn, from the average value of the

width of the structural part built by the company, iw ,,,, Thus,

Built =(1 +ew)Wdesign (4.6)

Similarly, the built thickness value will differ from its design value such that

built (1+ et) design (4.7)

Then, the built load carrying area Ab,,lt can be expressed using the first equality of

Eq. (4.5) as

Abuilt = ( + e)( + ew)Adesign (4.8)

Table 4-1 presents nominal values for the errors assumed here. In the results

section of this chapter we will vary these error bounds and investigate the effects of these

changes on the probability of failure. As seen in Table 4-2, the error having the largest









bound in its distribution is the error in failure prediction ef, because we use it to model

also the likelihood of unexpected failure modes.

Table 4-1. Distribution of error factors and their bounds
Error factors Distribution Type Mean Bounds
Error in stress calculation, e, Uniform 0.0 + 5%
Error in load calculation, ep Uniform 0.0 + 10%
Error in width, ew Uniform 0.0 + 1%
Error in thickness, e, Uniform 0.0 + 2%
Error in failure prediction, ef Uniform 0.0 + 20%

The errors here are modeled by uniform distributions, following the principle of

maximum entropy. For instance, the error in the built thickness of a structural part (et) is

defined in terms of the error bound (bt)built via Eq. (4.9).


et = U[O,(bt)built (4.9)

Here 'U' indicates that the distribution is uniform and '0 (zero)' is the average

value of et. Table 4-1 shows that (bt)built = 0.02. Hence, the lower bound for the

thickness value is the average value minus 2% of the average and the upper bound for the

thickness value is the average value plus 2% of the average. Commonly available random

number generators provide random numbers uniformly distributed between 0 and 1.

Then, the error in the built thickness can be calculated from Eq. (4.10) using such random

numbers r as

et = (2r -1)(bt)built (4.10)

Total Error Factor

The expression for the built load carrying area, Abwlt, of a structural part can be

reformulated by combining Eqs. (4.5) and (4.8) as


Abuilt =(1+e"tot S (4.11)
(-a true









where

et(tal +e)(1+ep)(+et)(+ ) 1 (4.12)
total = 1 (4.12)
1-e

Here etotal represents the cumulative effect of the individual errors (er, ep, ...) on the load

carrying capacity of the structural part.

Total Safety Factor

The total safety factor, SF, of a structural part represents the effects of all safety

measures and errors on the built structural part. Without safety measures and errors, we

would have a load carrying area, Ao, required to carry the design load

Ao:= d (4.13)
Of

where 5f is the average value of the failure stress. Then, the total safety factor of a built

structural component can be defined as the ratio of Abuilt/Ao


(S built Abuilt = (+ total ) SFL (4.14)
Ao (Ca)true

Here we take SFL = 1.5 and conservative material properties are based on B-basis values.

Certification tests add another layer of safety. Structures with large negative total

unconservativee) fail certification, so the certification process adds safety by biasing the

distribution of etotal. Denoting the built area after certification (or certified area) by Acert,

the total safety factor of a certified structural part is


(SF) Acei (4.15)
A0c









Variability

In the previous sections, we analyzed the different types of errors made in the

design and construction stages, representing the differences between the fleet average

values of geometry, material and loading parameters and their corresponding design

values. For a given design, these parameters vary from one aircraft to another in the fleet

due to variabilities in tooling, construction, flying environment, etc. For instance, the

actual value of the thickness of a structural part, tact, is defined in terms of its fleet

average built value, tbuilt, by

tact = (1+ v) tbuilt (4.16)

We assume that vt has a uniform distribution with 3% bounds (see Table 4-2).

Then, the actual load carrying area Aact can be defined as

Aact = act act = (1 + v) tbuilt (1 + Vw builtt = (l + v )(l + w ) Abuilt (4.17)

where Vw represents effect of the variability on the built width.

Table 4-2 presents the assumed distributions for variabilities. Note that the

thickness error in Table 4-1 is uniformly distributed with bounds of 2%. Thus the

difference between all thicknesses over the fleets of all companies is up to 5%.

However, the combined effect of the uniformly distributed error and variability is not

uniformly distributed.

Table 4-2. Distribution of random variables having variability
Variables Distribution Mean Scatter
Actual service load, Pact Lognormal Pd = 100 10% c.o.v.
Actual structural part width, wact Uniform 'o,,, 1% bounds
Actual structural part thickness, tact Uniform tbuilt 3% bounds
Failure stress, of Lognormal 150 8% c.o.v.
Variability in built width, Vw Uniform 0 1% bounds
Variability in built thickness, vt Uniform 0 3% bounds
c.o.v.= coefficient of variation









Certification Tests

After a structural part has been built with random errors in stress, load, width,

allowable stress and thickness, we simulate certification testing for the structural part.

Recall that the structural part will not be manufactured with complete fidelity to the

design due to variability in the geometric properties. That is, the actual values of these

parameters wact and tact will be different from their fleet-average values inw,,,,, and tbuilt due

to variability. The structural part is then loaded with the design axial force of SFL times

Pcalc, and if the stress exceeds the failure stress of the structure of, then the structure fails

and the design is rejected; otherwise it is certified for use. That is, the structural part is

certified if the following inequality is satisfied


-- SFLcalc -f < 0 (4.18)
Wacttact

The total safety factor (see Eq. (4.14)) depends on the load safety factor, the ratio of

the failure stress to the B-basis allowable stress and the total error factor. Note that the B-

basis properties are affected by the number of coupon tests. As the number of tests

increases, the B-basis value is also increases, so a lower total safety factor is used.

Amongst the terms in the total safety factor expression, the error term is subject to the

largest change due to certification testing. Certification tests reduce the probability of

failure by mainly changing the distribution of the error factor ettal. Without certification

testing, we assume uniform distributions for all the individual errors. However, since

designs based on unconservative models are more likely to fail certification, the

distribution of total becomes conservative for structures that pass certification. In order to

quantify this effect, we calculated the updated distribution of the error factor total by

Monte Carlo Simulation (MCS) of a sample size of 1,000,000.










In Chapter 3, we represented the overall error with a single error factor e,

hereinafter termed the "Single Error Factor model (SEF model)", and we used uniform

distribution for the initial (i.e., built) distribution of this error. In the present work, we use

a more complex representation of error with individual error factors, hereinafter termed

the "Multiple Error Factor model (MEF model)", and we represent the initial distribution

of each individual error factor with uniform distribution. In this case, the distribution of

the total error is no longer uniform. Figure 4-1 shows how certification tests update the

distribution of the total error for the SEF and MEF models. For both models the initial

distribution is updated such that the likelihood of conservative values of the total error is

increased. This is due to the fact that structures designed with unconservative (negative)

errors are likely to be rejected in certification tests. Notice that the SEF model

exaggerates the effectiveness of certification testing. The reader is referred to Appendix

D for a detailed comparison of the two error models.

3 -
7 -*- -MEF_e_buit
2.5 MEFe_certified
S- SEFebuilt
-5 SEF_e_certified

0 .5


I 0.5


-0.4 -0.2 0 0.2 0.4 0.6
error


Figure 4-1. Comparing distributions of built and certified total error etotal of SEF and
MEF models. The distributions are obtained from simulation of 1,000,000
structural parts. The lower and upper bounds for the single error are taken as -
22.3% and 25.0%, respectively, to match the mean and standard deviation of
the total error factor in the MEF model (see Table D-1 of Appendix D).










Figure 4-2 shows the distributions of the built and certified total safety factors of

the MEF model. Notice that the structural parts designed with low total safety factors are

likely to be rejected in the certification testing. The mean and standard deviations of built

and certified distributions of the error factor and the total safety factor are listed in Table

4-3. Comparing the mean and standard deviation of the built and certified total error (and

similarly the total safety factor), we see that the mean is increased and the standard

deviation is reduced due to certification testing.


1.6

1.2 ----------------- ---- --- built
certified

P d f 0 .8 --------------- <--------------------- ------------------
Pdf 0.8 .... ......




0. 4. ... *.
0

1 1.5 2 2.5 3
Safety factor


Figure 4-2. Initial and updated distribution of the total safety factor SF. The distributions
are obtained via Monte Carlo Simulations with 1,000,000 structural part
models.

Table 4-3. Mean and standard deviations of the built and certified distributions of the
error factor total and the total safety factor SF shown in Figures 4-1 and 4-2.
The calculations are performed with 1,000,000 MCS.
Mean Std. dev.
Built total error 0.0137 0.137
Certified total error 0.0429 0.130
Built safety factor 1.747 0.237
Certified safety factor 1.799 0.226









Probability of Failure Calculation

As noted earlier, we assume that structural failure requires the failure of two

structural parts. In this section, we first describe the probability of failure calculations of

a single structural part by using separable MCS. Then, we discuss the calculation of the

system probability of failure.

Probability of Failure Calculation by Separable MCS

To calculate the probability of failure, we first incorporate the statistical

distributions of errors and variability in a Monte Carlo simulation. Errors are uncertain at

the time of design, but do not change for individual realizations (in actual service) of a

particular design. On the other hand, all individual realizations of a particular design are

different from each other due to variability. In Chapter 3, we implemented this through a

two-level Monte Carlo simulation. At the upper level we simulated different aircraft

companies by assigning random errors to each, and at the lower level we simulated

variability in dimensions, material properties, and loads related to manufacturing

variability and variability in service conditions. This provided not only the overall

probability of failure, but also its variation from one company to another (which we

measured by the standard deviation of the probability of failure). This variation is

important because it is a measure of the confidence in the value of the probability of

failure due to the epistemic uncertainty (lack of knowledge) in the errors. However, the

process requires trillions of simulations for good accuracy.

In order to address the computational burden, we turned to the separable Monte

Carlo procedure (e.g., Smarslok and Haftka (2006)). This procedure applies when the

failure condition can be expressed as gl(xl)>g2(x2), where xl and x2 are two disjoint sets

of random variables. To take advantage of this procedure, we need to formulate the









failure condition in a separable form, so that gi will depend only on variabilities and g2

only on errors. The common formulation of the structural failure condition is in the form

of a stress exceeding the material limit. This form, however, does not satisfy the

separability requirement. For example, the stress depends on variability in material

properties as well as design area, which reflects errors in the analysis process. To bring

the failure condition to the right form, we instead formulate it as the required cross

sectional area Aeq being larger than the built area Abuilt, as given in Eq. (4.19)

Are
Abuilt < ( Aeq (4.19)
(1+vt)(1+vj)

where Areq is the cross-sectional area required to carry the actual loading conditions for a

particular copy of an aircraft model, and Aeq is what the built area (fleet-average) needs

to be in order for the particular copy to have the required area after allowing for

variability in width and thickness.

Areq = P f (4.20)

The required area depends only on variability, while the built area depends only on

errors. When certification testing is taken into account, the built area, Abu lt, is replaced by

the certified area, Acert, which is the same as the built area for companies that pass

certification. However, companies that fail are not included. That is, the failure condition

is written as

Failure without certification tests: Aeq Abuilt > 0 (4.21-a)


Failure with certification tests:


Areq Acert > 0


(4.21-b)









Equation (4.21) can be normalized by dividing the terms with Ao (load carrying

area without errors or safety measures, Eq. (4.13)). Since AbudtlAo or Acert/Ao are the total

safety factors, Eq. (4.21) is equivalent to the requirement that failure occurs when the

required safety factor is larger than the built one.

Failure without certification tests: (SF)req -(SF)bilt > 0 (4.22-a)

Failure with certification tests: (SF)req -(SF)cert > 0 (4.22-b)

where (SF)built and (SF)cer are the built and certified total safety factors given in Eqs.

(4.14) and (4.15), and the required total safety factor (SF)req is calculated from


(SF)r = req (4.23)
( r )req ./10

For a given (SF)built we can calculate the probability of failure, Eq. (4.22.a), by

simulating all the variabilities with MCS. Figure 4-3 shows the dependence of the

probability of failure on the total safety factor using MCS with 1,000,000 variability

samples. The zigzagging in Figure 4-3 at high safety factor values is due to the limited

MCS sample. Note that the probability of failure for a given total safety factor is one

minus the cumulative distribution function (CDF) of the total required safety factor. This

required safety factor depends on the four random variables Pact, af, Vt and vw. Among

them Pact and f have larger variabilities compared to vt and vw (see Table 4-2). We found

that (SF) q is accurately represented with a lognormal distribution, since Pact and af

follow lognormal distributions. Figure 4-3 also shows the probability of failure from the

lognormal distribution with the same mean and standard deviation. Note that the nominal

load safety factor of 1.5 is associated with a probability of failure of about 10-3, while the










probabilities of failure observed in practice (about 10-7) correspond to a total safety factor

of about two.


10-1 r "-
-- Lognormal approx.
10.2

10-,




104
10"
10"-

10


1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
SF =Abuilt'AO


Figure 4-3. The variation of the probability of failure with the built total safety factor.
Note that Pf is one minus the cumulative distribution function of (SF )req


Figure 4-4 represents flowchart of a separable MCS procedure. Stage-1 represents

the simulation of variabilities in the actual service conditions to generate the probability

of failure as shown in Figure 4-3. This probability of failure is one minus the cumulative


distribution function (CDF) of the required safety factor (SF)req In Stage-1,


M=1,000,000 simulations are performed and CDF of (SF),,q is assessed. A detailed


discussion on CDF assessment for (SF)req is given in Appendix E.










Simulate M different realizations of the variabilities
related to the actual service conditions
Calculate the required safety factor, Are q/A0
G STAGE 1
Generate the CDF of the (see Fig. 3)
required safety factor


Simulate N different error and variabilities related to
design and construction phases
Calculate the built safety factor Abuilt /A0
STAGE 2

Perform certification testing
-- Reject the design in case of failure


Calculate probability of failure for each design


Calculate the average and coefficient of variation of
probability of failure


Figure 4-4. Flowchart for MCS of component design and failure

In Stage-2, N=1,000,000 designs are generated for N different aircraft companies.

For each new design, different random error factors e6, ep, ew, et and ef are picked from

their corresponding distributions to generate the built safety factor, (SF)buil Then, each


design is subjected to certification testing. If it passes, we obtain the probability of failure

from the distribution obtained in Stage-1 (Figure 4-3). We calculate the average and

coefficient of variation (c.o.v.) of the failure probability over all designs and explore the

effects of error, variability, and safety measures on these values in Results section.

The separable Monte Carlo procedure reduces the computational burden greatly.

For instance, if the probability of failure is 2.5x 105, a million simulations varying both

errors and variability simultaneously estimate this probability with 20% error. We found

for our problem that the use of the separable Monte Carlo procedure requires only 20,000









simulations (10,000 simulations for Stage-1 and 10,000 for Stage-2) for the same level of

accuracy.

Including Redundancy

The requirement of two failure events is modeled here as a parallel system. We

assume that the limit-states of the both failure events follow normal distribution to take

advantage of known properties of the bivariate normal distribution. For a parallel system

of two elements with equal failure probabilities, Eq. (4.24) is used to calculate the system

probability of failure PFS (see Appendix F for details)


PFS = P + 1 exp dz (4.24)


where Pf is the probability of failure of a single structural part, p is the correlation

coefficient of the two limit-states and 8 is the reliability index for a single structural part,

which is related to Pf through Eq. (4.25)

Pf = (-fp) (4.25)

Results

In this section, the effectiveness of safety measures is investigated and the results

are reported. First, we discuss the effects of error reduction. Then, the relative

effectiveness of error reduction and certification is compared. Next, the effectiveness of

redundancy is explored. Finally, the effectiveness of variability reduction is investigated.

Effect of Errors

We first investigate the effect of errors on the probability of failure of a single

structural part. For the sake of simplicity, we scale all error components with a single

multiplier, k, replacing Eq. (4.12) by









(etoal (l+ke,)(l+ke +ke)(l+ke,,) (4.26)
etotal = 1 (4.26)
-1key

and explore the effect of k on the probability of failure.

Table 4-4 presents the average and coefficient of variation of the probability of

failure of a single structural part. The coefficient of variation of the failure probability is

computed to explore our confidence in the probability of failure estimate, since it reflects

the effect of the unknown errors. Columns 5 and 6 of Table 4-4 show a very high

coefficient of variation for the failure probabilities (variability in the probability of failure

for different aircraft models). We see that as the error grows (i.e., k increases), the

coefficient of variation of failure probabilities after certification also grows. Comparing

the failure probabilities before certification (column 5) and after certification (column 6),

we notice that even though certification tests reduce the mean failure probability, they

increase the variability in failure probability.

Table 4-4 shows that for nominal error (i.e., k=l) the total safety factor before

certification is 1.747, which is translated into a probability of failure of 8.83 x 10-4. When

the certification testing is included, the safety factor is increased to 1.799, which reduces

the probability of failure to 3.79x10-4. Notice also that the coefficient of variation of the

safety factor is reduced from 13.6% to 12.5%, which is a first glimpse of an indication

that the certification testing is more effective than simply increasing the safety factor with

an increased built area. A detailed analysis of the effectiveness of certification testing is

given in the next subsection.

Column 2 of Table 4-4 shows a rapid increase in the certification failure rate with

increasing error. This is reflected in a rapid increase in the average safety factor of









certified designs in column 4, (SF)cert This increased safety factor manifests itself in the

last column of Table 4-4 that presents the effect of certification tests on failure

probabilities. As we can see from that column, when the error increases, the ratio of the

two failure probabilities decreases, demonstrating that the certification tests become more

effective. This trend of the increase of the design areas and the probability ratios is

similar to the one observed in Chapter 3. Note, however, that even the average safety

factor before certification ((SF)built in column 3) increases with the error due to the

asymmetry of the initial total error distribution (see Figure 4-1).

Table 4-4. Average and coefficient of variation of the probability of failure for the
structural parts designed with B-basis properties and SFL=1.5. The numbers
inside the parentheses represent the coefficient of variation of the relevant
quantity.
CFR(a) (bc (b) )/10-4 p c7 / P /P
(%) (SF )built (b) SFcert ) Pnc ()/104 ()/104 Pc
0.25 6.4 1.725 (4.2%) 1.728 (4.1%) 0.244 (148%) 0.227 (148%) 0.930
0.50 9.3 1.730 (6.9%) 1.741 (6.7%) 0.763 (247%) 0.609 (257%) 0.798
0.75 13.4 1.737 (10.2%) 1.764 (9.7%) 2.70 (324%) 1.66 (357%) 0.616
0.82 14.7 1.740 (11.2%) 1.773 (10.6%) 3.79(340%) 2.13 (384%) 0.561
1 18.0 1.747 (13.6%) 1.799 (12.5%) 8.83 (371%) 3.79 (450%) 0.430
1.5 26.0 1.779 (20.5%) 1.901 (17.8%) 60.0 (385%) 11.5 (583%) 0.191
(a) CFR: Certification failure rate.
(b) (SF )built and (SF )cert are the total safety factors before and after certification testing,
respectively.
(c) Pn and Pc are the probabilities of failure before and after certification testing, respectively.

Table 4-4 shows the huge waste of weight due to errors. For instance, for the

nominal error (i.e., k=1.0), an average built total safety factor of 1.747 corresponds to a

probability of failure of 8.83 x 10-4 according to Table 4-4, but we see from Figure 4-3 that

a safety factor of 1.747 approximately corresponds to a probability of failure of 7x10-6,

two orders of magnitude lower. This discrepancy is due to the high value of the

coefficient of variation of the safety factor. For the nominal error, the coefficient of

variation of the total safety factor is 14%. Two standard deviations below the mean safety









factor is 1.272, and two standard deviations above the mean safety factor is 2.222. The

probability of failure corresponding to the safety factor of 1.272 (from Figure 4-3) is

about 2.98x10-2, while the safety of 1.985 the probability of failure is essentially zero. So

even though about 0.8% of the designs a have safety factor below 1.272 (Figure 4-2),

these designs have a huge impact on the probability of failure. Reducing the error by half

(i.e., k=0.50), reduces the weight by 1%, while at the same time the probability of failure

is reduced by a factor of 3.

Weight Saving Due to Certification Testing and Error Reduction

We have seen in Table 4-4 that since structures built with unconservative errors are

eliminated by certification testing; the tests increase the average safety factor of the

designs and therefore reduce the average probability of failure. Since certification testing

is expensive, it is useful to check if the same decrease in the probability of failure can be

achieved by simply increasing the load carrying area by the same amount (i.e., by

increasing the safety factor) without certification testing. Column 2 of Table 4-5 shows

that the required area with no certification testing, Arnc, is greater than the certified area,

Acert, (i.e., area after certification testing) shown in column 3. The last column shows that

the weight saving by using certification test instead of a mere increase of the safety

factor. We notice that weight saving increases rapidly as the error increases. For instance,

when k=0.25 the weight saving is very small. Columns 4 and 5 show that even though we

match the average probability of failure, there are small differences in the coefficients of

variation.









Table 4-5. Reduction of the weight of structural parts by certification testing for a given
probability of failure. The numbers inside the parentheses represent the
coefficient of variation of the relevant quantity.
k Ar, ncAo(a) AcerAo Pnc ()/10-4 P (b)/10-4 AA (c)
0.25 1.7285 (4.2%) 1.7283 (4.1%) 0.227 (148%) 0.227 (148%) -0.01
0.50 1.743 (6.9%) 1.741 (6.7%) 0.609 (252%) 0.609 (257%) -0.14
0.75 1.770 (10.3%) 1.764 (9.7%) 1.66 (342%) 1.66 (357%) -0.36
1 1.815 (13.7%) 1.799 (12.5%) 3.79 (416%) 3.79 (450%) -0.87
1.5 1.961 (20.7%) 1.901 (17.8%) 11.5 (530%) 11.5 (583%) -3.09
(a) A,nc is the required area with no certification testing, the area required to achieve the
same probability of failure as certification.
(b) Pnc and Pc are the probabilities of failure before and after certification testing,
respectively.
(c) AA = (Ace,,-A,nc)/A,nc indicates weight saving due to testing while keeping the same
level of safety

We notice from Table 4-5 that, for the nominal error (i.e., k=-.0), certification

testing reduces the weight by 0.87% for the same probability of failure (3.79x10-4). The

same probability of failure could have been attained by reducing the error bounds by

18%, that is by reducing k from 1.0 to 0.82. This reduction would be accompanied by an

(SF)built =1.740 (see Table 4-4). Compared to the 1.799 reduction (SF)built this

represents a reduction of 4.13% in average weight, so error reduction is much more

effective than certification testing in reducing weight.

Effect of Redundancy

To explore the effect of redundancy, we first compare the failure probability of a

single structural part to that of a structural system that fails due to failure of two structural

parts. Certification testing is simulated by modeling the testing of one structural part and

certifying the structural system based on this test. Table 4-6 shows that while the average

failure probability is reduced through structural redundancy, the coefficients of variation

of the failure probabilities are increased. That is, even though the safety is improved, our

confidence in the failure probability estimation is reduced. This behavior is similar to the