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INTERACTION OF CONSERVATIVE DESIGN PRACTICES, TESTS AND INSPECTIONS IN SAFETY OF STRUCTURAL COMPONENTS By AMIT ANAND KALE 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 2005 This dissertation is dedicated to my parents ACKNOWLEDGMENTS I want to express my appreciation and special thanks to Dr. Raphael T. Haftka, chairman of my advisory committee. He has been a great mentor and constant source of inspiration and encouragement during my doctoral studies, and I want to thank him for providing me with the excellent opportunity and financial support to complete my doctoral studies under his exceptional guidance. He encouraged me to attend several conferences in the area of reliability based design optimization and helped me gain industrial experience in my research area through an internship during my doctoral studies. I am especially impressed by his unlimited zeal to explore new research areas, encourage new ideas and share his knowledge and experience with me. The interaction I have had with Dr. Haftka has helped me improve my personal and professional life. I would also like to thank the members of my advisory committee, Dr. Bhavani V. Sankar, Dr. Nam Ho Kim, Dr. Nagaraj K. Arakere and Dr. Stanislav Uryasev. I am grateful for their willingness to serve on my committee, provide me with help whenever required, involvement with my oral qualifying examination, and for reviewing this dissertation. Special thanks go to Dr. Bhavani V. Sankar for his guidance with several technical issues during my research and Dr. Nam Ho Kim for his comments and suggestions during group presentations which helped me improve my work. I would also like to thank Dr. Ben H. Thacker and Dr. Narasi Sridhar who gave me the excellent opportunity to work with them on an industrial project at Southwest Research Institute. My colleagues in the Structural and Multidisciplinary Optimization Research Group at the University of Florida also deserve thanks for their help and many fruitful discussions. Special thanks go to Dr. Melih Papila and Erdem Acar who collaborated with me on several research papers. The financial support provided by NASA CUIP (formerly URETI) Grant NCC3 994 to the Institute for Future Space Transport (IFST) at the University of Florida and NASA Grant NAG102042 is fully acknowledged. My parents deserve my deepest appreciation for their constant love and support and for encouraging me to pursue a Ph.D. TABLE OF CONTENTS ACKNOW LEDGM ENTS ............................................ iii LIST OF TABLES ................... .... ............ ............. ........ix LIST OF FIGURES .............................................. xiv KEY TO SYMBOLS ............................................ .............. xvi ABSTRACT................................. .............. xxi CHAPTER 1 INTRODUCTION ................... ...................................... ......... ....... M otivation..................... ........... ................ ............... Obj ective...................................... .................. ............... ........2 Outline ........................................ ............................... .........2 2 BACKGROUND .............. ............... ................. ............. ......5 Structural Design M ethodology......................................... ...............5 Estimating Fatigue Life and Crack Sizes............................. ............... Probabilistic Approach for Fatigue Life Prediction............................. .........8 Reliability B ased D esign ............................................................ .... ..........10 M onte C arlo Integration ...................................................................................... 11 FirstOrder Reliability Method (FORM).....................................................12 Reliability Based Inspection Scheduling .............. ................ ............................. 13 Reliability Based Design Optimization .............................. ......... 14 3 EFFICIENT RELIABILITY BASED DESIGN AND INSPECTION OF STIFFENED PANELS AGAINST FATIGUE ................. ................. ..........16 Introduction............... .......... ...... ................ .............. ......... 16 Crack Grow th and Inspection M odel.................................................................... 18 F atigu e C rack G row th .................................................................................... 18 C critical C rack Size..................... ...... .... .......... ..............20 Probability of Failure at a Given Tim e..............................................................22 Inspection M odel ............................................... ........ 24 Computational Method to Perform Reliability Based Optimization with Inspections ........................................24 Searching for Next Inspection Time Using FORM ............................................25 Updating Crack Size Distribution after Inspection using MCS .......................26 Calculation of Inspection Schedule for a Given Structure .............................29 Optimization of Structural Design............................... ............... 31 Results......................................................... ................ ........37 Summary......................................................... ......................... 43 4 TRADEOFF OF WEIGHT AND INSPECTION COST IN RELIABILITY BASED STRUCTURAL OPTIMIZATION USING MULTIPLE INSPECTION T Y P E S ....................................................... 4 4 Introduction .................. ................... ................... ............... ........ 44 Structural Design and Damage Growth Model .................. ................ ....47 Fatigue Crack G row th ................................................ ............... 47 Inspection M odel .......................................... 49 Calculating an Inspection Schedule........................................51 Estimating Crack Size Distribution after Inspection..........................................51 Calculating the Failure Probability Using the FirstOrder Reliability Method (FO RM )................................................................................. 53 Cost Model .....................................................57 O ptim ization of Inspection Types ................... ... ............. ...............58 Combined Optimization of Structural Design and Inspection Schedule....................59 SafeLife Design.......................... .. .. ........... ... 60 Cost Effectiveness of Combined Optimization ............................................60 Effect of Fuel Cost............. ............ ............... 63 Summary ............................... ...................64 5 EFFECT OF SAFETY MEASURES ON RELIABILITY OF AIRCRAFT STRUCTURES SUBJECTED TO FATIGUE DAMAGE GROWTH...................65 Introduction .......... ... .......... ....... ............... 65 Classification of Uncertainties...................... ......... ......... 67 Safety M easures.............................. ..................... ..... .........68 Simulation Procedure for Calculation of Probability .............................................70 D am age G row th M odel ................................................. ............... 72 Calculating Design Thickness ................. ...............................76 Calculating Failure Probability.....................................78 Certification Testing ... ........... .. ................. ................ 78 Service Simulation................ ..... ............... 79 Results............... .. ........ ... .............. ......... ................ ........80 Effect of Errors and Testing on Structural Safety .......................................... 80 Effect of Certification Testing W ith M achined Crack .......................................85 Effect of Variability in Material Properties on Structure Designed With all Safety M measures ................... ..... .................................. ........87 6 A PROBABILISTIC MODEL FOR INTERNAL CORROSION OF GAS PIPELIN E S ......... ..... .... ......... ........................ 92 Introdu action ......... ..... .... .................................. 92 Proposed Methodology ....................... .......... .........96 Corrosion R ate M odel ............................................... ............... 96 Inhibitor Correction M odel.................................. .................... 97 W ater A ccum ulation............................................... 98 Probabilistic Model .......... ............................. 100 Corrosion Damage ..... ........ ......... ......... ...................100 Input Uncertainties ............................................. .... .....101 Mapping Uncertainty ............... ..... ........ ...............102 Inspection Updating...................... .......... ...............103 Example 1: Determination of Critical Location Prior to Inspection......................104 Example 2: Updating Corrosion Modeling with Inspection Data .........................106 Summary ............... ................. .............. ......... 109 7 CONCLUSIONS ................ ....... ...... ..................... APPENDIX A DISPLACEMENT COMPATIBILITY ANALYSIS FOR CALCULATION OF STRESS INTENSITY OF STIFFENED PANEL....................................................114 Introduction ................................... .. ... ........... 114 Displacem ent Com patibility M ethod......................................................................115 D isplacem ent V, ............... ........ ............................. ......... ...117 Displacement V2 and V3 ............................................. ............... 18 D isplacem ent V4 .................. ....... ............................................ 119 Intact Stiffener D isplacem ent ...................................................................... 120 Broken Stiffener Displacement ............................ ............... 121 F astener D isplacem ent.......................................................................... ........ .... 12 1 Com patibility of D isplacem ents .................................................. .................121 Effectiveness of Stiffeners in Reducing Crack Tip Stress Intensity...............1...24 B CALCULATING CRACK GROWTH FOR STIFFENED PANELS USING NUMERICAL INTEGRATION AND RESPONSE SURFACE.............................126 C ACCURACY ESTIMATES OF RESPONSE SURFACE APPROXIMATIONS... 129 Response Surface Approximations for Geometric Factor Vf ...................................129 Response Surface Approximation for Reliability Index (Beta)...........................131 D COST OF STRUCTURAL WEIGHT...........................................134 E PSEUDO CODE FOR COMBINED OPTIMIZATION OF STRUCTURE AND IN SPE C TIO N SC H ED U L E ................................................................................ 136 Introduction ................... ................... ............................................ 136 O ptim ization of Inspection Types ............................................................................138 F EFFECT OF CRACK SIZE PROBABILITY DISTRIBUTION ON FAILURE PROBABILITY AND INSPECTION INTERVAL.................................................141 G WHY ARE AIRPLANES SO SAFE STRUCTURALLY? EFFECT OF VARIOUS SAFETY MEASURES ON STRUCTURAL SAFETY...................144 Introduction ...... ........ .............. .... ..............144 Stru ctu ral U n certainties ............................................................................................14 6 Safety M measures ............. .... .... .... .......... ................... 148 Panel Example Definition................ .............................. 149 Design and Certification Testing...... ............ ..... .... .......... ...............149 Effect of Certification Tests on Distribution of Error Factor e .....................153 Probability of Failure Calculation by Analytical Approximation ................154 Probability of Failure Calculation by Monte Carlo Simulations ......................156 Effect of Three Safety Measures on Probability of Failure.................. ............157 Concluding Remarks .......................................... ...... .... ....169 H CALCULATION OF CONSERVATIVE MATERIAL PROPERTIES...............1...71 I CONFLICTING EFFECTS OF ERROR AND VARIABILITY ON PROBABILITY OF FAILURE..........................................................................173 J CALCULATIONS OF P(C E), THE PROBABILITY OF PASSING CERTIFICATION TEST................... ..........................175 LIST OF REFERENCES ................ ....... ......... .........178 BIOGRAPHICAL SKETCH ................. ................... ........187 LIST OF TABLES Table page 31 Fatigue properties of 7075T651 Aluminum alloy ................................... 22 32 Structural design for fuselage................................................... 22 33 Pseudo code for updating crack size distribution after N cycles from previous inspection .......................................................27 34 Example 31, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 107 ........31 35 Cost of inspection, material and fuel............. ............. .........................32 36 Description of response surface approximations used in optimization....................33 37 Computational time spent in exact calculation of next inspection time and error due to RSA usage ........... ...... ................................34 38 Pseudo code for combined optimization of structural design and inspection schedule ...................................... .................................. ........ 36 39 SafeLife design of an unstiffened panel................. ............................ ...... 37 310 SafeLife design of a stiffened panel ........................................... 37 311 Optimum structural design and inspection schedule of an unstiffened panel ..........38 312 Optimum structural design and inspection schedule for stiffened panel...............39 313 Optimum structural design for regulations based inspections conducted at four constant interval or 8000 flights for stiffened panel ................................... 40 314 Tradeoff of inspection cost against cost of structural weight required to maintain fixed reliability level for stiffened panel ..........................................41 315 Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel w ith inspection................................................. 42 41 Fatigue properties of 7075T651 Aluminum alloy ................................... 49 42 Pseudo code for updating crack distribution after N cycles from previous inspection .......................................................53 43 Example 41, inspection schedule and crack size distribution after inspection (ah = 0.63 mm) for an unstiffened plate thickness of 2.48 mm and a threshold probability of 10 7 .................................. ........................ ...........56 44 Design details and cost factors .............................. .........57 45 Structural size required to maintain a specified reliability level without and inspection. ........................................................60 46 Optimum structural design and inspection schedule required to maintain specified threshold reliability level .............................. ............... 61 47 Comparison of optimum inspection schedule using a single inspection type for a fixed structural size .......................................... ...... ........ 62 48 Optimum structural design and inspection schedule using only a single inspection type...................... ....................... 63 49 Optimum structural design (plate thickness of 2.02 mm) and inspection schedule for Pfth = 107 . . ...................... . ............ . ................ 64 51 Uncertainty classification.............. ... ........ ..... ........68 52 Distributions of errors, design and material parameters for 7075T6 aluminum.....75 53 Nomenclature of symbols used to calculate failure probability and describe the effect of certification testing ................. .............. ................80 54 Probability of failure for 10 % COV in e and different bounds on error k using all safety measures for failsafe design for 10,000 flights ......................................81 55 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for failsafe design for 10,000 flights ......................................... 82 56 Probability of failure for 10 % COV in e for different bounds on error k using all safety measures for safelife design of 40,000 flights................... .............82 57 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for safelife design of 40,000 flights...................................82 58 Probability of failure for different bounds on error k for 10 % COV in e without any safety measures for failsafe design for 10,000 flights..................................84 59 Probability of failure for different bounds on error k for 50 % COV in e without any safety measures for failsafe design for 10,000 flights..................................84 510 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for failsafe for 10,000 flights and tested using a m machine cracked panel ............................................................. 86 511 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for safelife of 40,000 flights and tested using a m machine cracked panel ............................................................. 86 512 Probability of failure for different bounds on k and 10 % COV in e for structure designed with all safety measures for failsafe for 10,000 flights and COV in material property m reduced to 8.5% .......................................................87 513 Probability of failure for different bounds on k and 50 % COV in e for structures designed with all safety measures for failsafe criteria for 10,000 flights and COV in material property m reduced to 8.5%................. .......................87 514 Probability of failure for different bounds on k and 10 % COV in e for structures designed using only ABasis m for failsafe criteria for 10,000 flights ...................88 515 Probability of failure for different bounds on k, 10 % COV in e for structure designed using conservative properties for failsafe design for 10,000 flights........89 516 Probability of failure for different bounds on k, 50 % COV in e for structures designed using conservative properties for failsafe criteria for 10,000 flights.......89 517 Effective safety factor and measures of probability improvement in terms of individual safety measures and error bounds for structure designed using fail safe criteria of 10,000 flights............................. ...............90 61 Typical wet gas pipeline flow parameters..............................................................99 62 Typical wet gas pipeline corrosion growth parameters............... ... ...... 101 63 Updating of model weights given assumed observations corresponding to input component models................ .. .... .. ............ 107 64 Inspection locations along pipeline ....................... ...............109 Ci Bounds on design variables used to evaluate response surface approximation for safe life design..................... .............. .......... 129 C2 Error estimate of analysis response surfaces used to obtain safelife stiffened panel design ............... .... .................. ......... 130 C3 Bounds on design variables used to evaluate response surface approximation for inspection based design ............... .... ......... ......................130 C4 Error estimate of analysis response surfaces used to obtain inspection based stiffened panel design ...................... .. .... ....... .. .. .... .......... 131 C5 Error estimate of design response surfaces .....................................131 C6 Bounds on design variables used to evaluate response surface for crack sizes parameters after inspection and reliability index ...................................... 132 C7 Error estimate of crack size response surfaces used to estimate the crack size distribution parameters after the first inspection............................................133 C8 Error estimate of crack size response surfaces used to estimate the distribution after inspection ................. .... ...................... 133 C9 Error estimate of reliability index response surfaces used to schedule first inspection ......................................................133 C10 Error estimate of reliability index response surfaces, /fdRSA............................. 133 D1 Area of structural dimensions for cost calculation.................... ............... 135 Fi Inspection schedule and crack size distribution after inspection (ah = 1.27 mm) for an unstiffened plate thickness of 2.48 mm and a threshold probability of 107 141 G1 Uncertainty classification................................. ........................ .. ............147 G2 Distribution of random variables used for panel design and certification .............152 G3 Comparison of probability of failures (P/ s) for panels designed using safety factor of 1.5, mean value for allowable stress and error bound of 50%.................156 G4 Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and Abasis property for allowable stress..... ......... 158 G5 Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and mean value for allowable stress...............................160 G6 Probability of failure for different bounds on error e for safety factor of 1.0 and Abasis property for allowable stress .......................................162 G7 Probability of failure for different error bounds for panels designed using safety factor of 1.0 and mean value for allowable stress............. ...........163 G8 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 50% error bounds e and Abasis property for allowable stress ........................................................................ ........ 164 G9 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 30% error bound e and Abasis properties.............................164 G10 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 10% error bounds e and Abasis properties .........................165 LIST OF FIGURES Figure page 31 Fuselage stiffened panel geometry and applied loading in hoop direction ..............20 32 Comparison of actual and lognormally fitted CDF of crack sizes after an inspection conducted at 9288 flights.................................................. .............29 33 Example 31, Variation of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10 ...................31 41 Probability of detection curve for different inspection types from Equation 48 ....51 42 Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfh = 107 ..................................56 51 Flowchart for Monte Carlo simulation of panel design and failure.........................71 61 Uncertainty in inclination and critical angle ................. ................. ..........101 62 Probability of water formation along pipe length with highest probability observed at location 971 ................................ .........................105 63 Probability of corrosion depth exceeding critical depth along pipe length assuming water is present at all locations .........................................105 64 Total probability of corrosion exceeding critical depth along pipe length.......... 106 Ai Halfgeometry of a center cracked stiffened panel with a central broken stiffener and two intact stiffeners placed symmetrically across from crack....................115 A2 Description of applied stress and resulting fastener forces and induced stress on stiffened panel ........... ............. ...................... ............116 A3 Description of position coordinate of forces and displacement location with respect to crack centerline as y axis.................................. ...............117 A4 Description of position coordinate of forces and induced stress distribution along the crack length .................. .... ...... ................119 A5 Comparison of stress intensity factor for a panel with skin thickness = 2.34 mm and stiffener area of 2.30 x 103 meter2............. ........ ...................124 A6 Comparison of stress intensity factor for a panel with skin thickness =1.81 mm and stiffener area of 7.30 x 104 meter2.............. .......... ... .................... 125 Bi Typical response curves for effect of stiffening on geometric factor yfor a stiffener area of 1.5 mm2 and skin thickness of 1.5 mm .......................................128 Fi Probability of exceeding 2.0 for a lognormally distributed random variable with a mean of 1.0. Note that large standard deviation decreases probability .......142 F2 Comparison of failure probability (1 CDF) of two probability distributions with mean 105 and standard deviation of 2 and 10 units..............................................143 G1 Flowchart for Monte Carlo simulation of panel design and failure .......................151 G2 Initial and updated probability distribution functions of error factor e..................... 155 G3 Design thickness variation with low and high error bounds ...............................162 G4 Influence of effective safety factor, error, and variability on the probability ratio (3D view) ........................................................ 167 G5 Influence of effective safety factor, error and variability on the probability ratio (2D contour plot)................... ............................ ......... .. .. ..... .............. 167 G6 Influence of effective safety factor, error and variability on the probability difference (3D view) ............... .... ........................................ .. ....... 168 G7 Influence of effective safety factor, error and variability on the probability difference (2D contour plot) ......................................................... 169 KEY TO SYMBOLS a = Crack size, mm a, = Critical crack size, mm acH = Critical crack length due to hoop stress, mm acL = Critical crack length for transverse stress, mm acy = Critical crack length causing yield of net section of panel, mm ah = Crack size at which probability of detection is 50%, mm a, = Initial crack size, mm a1,o = Crack size due to fabrication defects, mm aN = Crack size after N cycles of fatigue loading, mm As = Area of a stiffener, meter2 ATotal = Total cross sectional area of panel, meter2 b = Panel length, meters Bk = Error bounds on error in stress, k cov = Coefficient of variation, (standard deviation divided by mean) C = Distance from neutral axis of stiffener to skin, meters Ckb = Cost of inspection schedule developed using kth inspection type, dollars C,,, = Minimum cost of inspection schedule, dollars Ctot = Total life cycle cost, dollars d = Fastener diameter, mm Paris model parameter, meters Paris model parameter, meters 2 (MPa) m e = Error in crack growth rate E = Elastic modulus, MPa F = Force at a rivet on intact stiffener, N F, = Fuel cost per pound per flight, dollars Fstifner = Maximum stress on first stiffener, MPa Fs mnadsffener= Maximum stress on second stiffener, MPa F iffesner = Maximum stress on third stiffener, MPa g = Limit state function used to determine structural failure h = Panel width, meters H1 = Fastener shear displacement parameter H2 = Fastener shear displacement parameter i = Subscript used to denote indices I = Stiffener inertia, meter4 Ic = Inspection cost, dollars Ick = Cost of inspection of kth type, Ici, Ic2,, Ic3, Ic4, dollars Ik = Inspection of kth type, k = 1...4 k = Error in stress calculation K = Stress intensity factor, MPa meters KF = Stress intensity due to fastener forces, MPameter Kic = Fracture toughness, MPa meters KTotal = Total stress intensity on stiffened panel, MPaJ meter L = Frame spacing, meters I = Fuselage length, meters m = Paris model exponent, Eq. 31 MA, = Average bending moment between the ith and i1st fastener, Nmeter M, = Material manufacturing cost per pound for aluminum, dollars n = Number of fastener on a side of crack centerline on a single stiffener N = Number of cycles of fatigue loading Nf = Fatigue life, flights (Flights, time and cycles are used interchangbly) N, = Number of Inspections Np = Number of panels N, = Number of stiffeners Nub = Number of intact stiffeners p = Fuselage pressure differential, MPa P = Force at a rivet on broken stiffener, N Pc = Probability of failure after certification testing Pd = Probability of detection pdrand = Random number for probability of detection Pf = Failure probability Pfth = Threshold probability of failure, reliability constraint Pnc = Probability of failure without certification testing r = Fuselage radius, meters r, = Distance of a point from crack leading tip, meters r2 = Distance of a point from crack tailing tip, meters xviii r3 = Parametric distance of a point ahead of y axis by a distance b, meters r4 = Parametric distance of a point behind of y axis by a distance b, meters R = Batch rejection rate s = Fastener spacing, mm SFL = Safety factor on life SF = Safety factor on load Si = Service Life (40,000 flights) S, = nth inspection time in number of cycles or flights t = Panel thickness, mm t2 = Thickness of the stiffener flange, meters tcert = Thickness of certified structures design = Thickness of designed structures ts = Stiffener thickness, mm V, = Displacement anywhere in the cracked sheet caused by the applied gross stress, meters V2 = Displacement in the uncracked sheet resulting from fastener load F, meters V3 = Displacement in the uncracked sheet resulting from broken fastener load P, meters V4 =Displacement in the cracked sheet resulting from stress applied to the crack face equal and opposite to the stresses caused by rivet loads, meters VF = Displacement at a point in and infinite plate due to a point force F W = Structural weight, lb Y = Yield stress, MPa f = Inspection parameter fd = Reliability index SD, = Stiffened displacement due to direct fastener load at ith fastener location, meters SG, = Stiffener displacement due to applied stress at ith fastener location, meters 5,, = Stiffener displacement due to bending at ith fastener location, meters 8R, = Fastener displacement due to elastic shear, meters pa,,RSA = Response surface for estimating mean of crack size distribution, mm v =Poisson's ratio = Cumulative density function of standard normal distribution y = Geometric factor due to stiffening p = Density of aluminum, lb/ft3 a = Hoop stress, MPa oaRSA = Response surface estimating standard deviation of crack size distribution, mm 0 = Angle at a point as measured from origin (The x axis lies along the crack and y axis is perpendicular to crack with origin at crack center) 01 = Angle at a point as measured from leading crack tip. 02 = Angle at a point as measured from tailing crack tip. 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 INTERACTION OF CONSERVATIVE DESIGN PRACTICES, TESTS AND INSPECTIONS IN SAFETY OF STRUCTURAL COMPONENTS By Amit Anand Kale December 2005 Chair: Raphael T. Haftka Cochair: Bhavani V. Sankar Major Department: Mechanical and Aerospace Engineering Structural safety is achieved in aerospace application and other fields by using conservative design measures like safety factors, conservative material properties, tests and inspections to compensate for uncertainty in predicting structural failure. The objective of this dissertation is to clarify the interaction between these safety measures, and to explore the potential of including the interaction in the design process so that lifetime cost can be reduced by trading more expensive safety measures for less expensive ones. The work is a part of a larger effort to incorporate the effect of error and variability control in the design process. Inspections are featured more prominently than other safety measures. The uncertainties are readily incorporated into the design process by using a probabilistic approach. We explore the interaction of variability, inspections and structural sizes on reliability of structural components subjected to fatigue damage growth. Structural sizes and inspection schedule are optimized simultaneously to reduce operational cost by trading the cost of structural weight against inspections to maintain desired safety level. Reliability analysis for fatigue cracking is computationally challenging. The high computational cost for estimating very low probabilities of failure combined with the need for repeated analysis for optimization of structural design and inspection times makes combined optimization of the inspection schedules and structural design prohibitively costly. This dissertation develops an efficient computational technique to perform reliability based optimization of structural design and inspection schedule combining Monte Carlo simulation (MCS) and firstorder reliability method (FORM). The effect of the structural design and the inspection schedule on the operational cost and reliability is explored. Results revealed that the use of inspections can be very cost effective in maintaining structural safety. Inspections can be made more effective if done at critical locations where likelihood of failure is maximum and the information obtained from inspections can be used to improve failure prediction and update reliability. This aspect is studied by developing a probabilistic model for predicting locations of maximum corrosion damage in gas pipelines. Inspections are done at these locations and failure probabilities are updated based on data obtained from inspections. CHAPTER 1 INTRODUCTION Motivation Computation of life expectancy of structural components is an essential element of aircraft structural design. It has been shown that the life of a structure cannot be accurately determined even in carefully controlled conditions because of variability in material properties, manufacturing defects and environmental factors like corrosion. Safety of aircraft structures is largely maintained by using conservative design practices to safeguard against uncertainties involved in the design process and service usage. Typically, conservative material property, scatter factor in fatigue life and conservative loads are used to design structures. This is further augmented by quality control measures like certification testing and inspections. Safety measures compensate for uncertainty in load modeling, stress analysis, material properties and factors that lead to errors in modeling structural failure. These safety measures were gradually developed based on empirical data obtained from service experience and are usually geared to target specific types of uncertainty. For example, the use of conservative material properties provide protection against variability in material properties, using machined crack for certification and conservative initial defect provide protection against flaws induced during manufacturing and fabrication, and inspections protect against uncertainty in damage growth and accidental damage that cannot be predicted during the service life. The use of multiple safety measures along with quality control measures is costly. With a view of reducing lifetime cost and maintaining structural safety, this dissertation is a step towards understanding the interaction between inspections and structural design. Inspections serve as protection against uncertainty in failure due to damage growth and reliability based design optimization is used to incorporate these uncertainties and trade the cost of inspection against structural weight to reduce overall life cycle cost. Objective The objective of this dissertation is to explore the possibility of designing safe structures at lower lifetime cost by including the interaction between safety measures and trading inspection costs against the cost of additional structural weight. With the view to reducing cost of operation of aircraft structures and maintaining low risk of structural failure, we address the problem of developing optimum structural design together with inspection schedule. The approach is based on the application of methods of structural reliability analysis. Reliability based optimization is computationally expensive when inspections are involved because crack size distribution has to be recharacterized after each inspection to simulate replacement. Typically, the crack size distribution after an inspection will not have a simple analytical form and can only be determined using expensive numerical techniques. A second objective of this dissertation is to develop an efficient computational method to estimate reliability with inspection. Outline This dissertation uses a combination of reliability methods, Monte Carlo simulation (MCS), firstorder reliability method (FORM) and response surface approximations (RSA's), to perform reliability based optimization of structural design and inspection schedule. Typical examples of aircraft structures designed for fatigue crack growth and inspection plans are used to demonstrate the application of this methodology. Most of the chapters in the dissertation are revised versions of conference or journal papers with multiple authors. The outline below gives the chapter description and an acknowledgement of the role of the other authors. Chapter 2 presents the background and a literature survey on current methods used to design aircraft structures for damage growth. Uncertainty is a critical component in aircraft structural design and probabilistic methods are used to incorporate uncertainty in designing structures. This chapter also reviews reliability based methods used to design for structural safety. Chapter 3 is close to Kale et al. (2005). It presents the simultaneous optimization of structural design and inspection schedule for fatigue damage growth. The computational methodology for efficient reliability calculation in the presence of inspections is described here. A typical aircraft structural design of fuselage stiffened panel is used to demonstrate application of the proposed method. Chapter 4 is close to Kale et al. (2004). It presents the optimization of inspection schedule with multiple inspection types which are typically used in aerospace applications. This work was done in collaboration with Dr. Melih Papila, who provided inputs on cost of inspections and structural weight. A simple unstiffened panel design is used to obtain optimal structural design and inspection sequence. A mixture of different inspection types is used to generate the inspection schedule. Chapter 5 is close to Kale et al. (2005). It presents the interaction among various safety measures recommended by the Federal Aviation Administration (FAA) to design aircraft structures for damage tolerance. Interaction among safety measures, uncertainty and certification tests is studied. In particular it sheds light on the effectiveness of certification testing for fatigue. The computational method used in this chapter was developed in collaboration with Erdem Acar. Chapter 6 is close to Kale et al. (2004). It shows how information obtained from in service inspections can be used to update failure models and reliability using Bayesian updating. The methodology is applied to reliability assessment of gas pipelines subjected to corrosion damage. Risk based inspection plans are developed to determine optimal inspection locations where probability of corrosion damage is maximum. This work was done in collaboration with Dr. Ben H. Thacker, Dr. Narasi Sridhar and Dr. Chris Waldhart at the Southwest Research Institute. CHAPTER 2 BACKGROUND Structural Design Methodology Aerospace structural design philosophy has been evolving continuously based on feedback from operational experience. The major drive in this evolution has been improving safety throughout the service life of the structure while reducing weight. Consequently, in the past few years there has been growing interest in reliabilitybased design and optimization of structures. The loss of structural integrity with service usage is associated with propagation of damage such as fatigue cracks in metal structures or delamination in composite structures. In addition, damage may be inflicted by corrosion, freezethaw cycles, and accidents such as a turbine blade tearing through the structure or damage due to impact from birds or other objects. The effect of damage may be to reduce the residual strength of the structure below what is needed to carry the flight loads (limit loads or the design load). Alternatively, the damage may be unstable and propagate quickly resulting in the destruction of structural components. In case of damage due to fatigue, a designer must consider damage initiation and damage growth. The potential for damage initiation and growth in structures has led to two concepts in structural design for safety: safelife and failsafe. Niu (1990) and Bristow (2000) have characterized the safelife and failsafe design methodologies in that, reliability of a safelife structure is maintained by replacing components if their design life is less than the service life. Inspections or repairs are not performed. In contrast, structural safety in a failsafe design is maintained by means of design for damage containment or arrestment and alternative loadpaths that preserve limitload capabilities. These mechanisms are complemented with periodic inspections and repairs. Bristow (2000) provided historical insight on the evolution of structural design philosophy from safelife in the early 50's to damagetolerance used in present time. The current practice to design structures using damage tolerance has gained widespread acceptance because of uncertainty in damage initiation and growth. Here we assume that cracks are always present in the structure due to manufacturing and fabrication and grow due to applied loads, corrosion and impacts. The Federal Aviation Administration (FAA) requires that all structures designed for damage tolerance be demonstrated to avoid failure due to fatigue, manufacturing defects and accidental damage (FAR 25.571, damage tolerance and fatigue evaluation of civil and transport category airplanes). The purpose of damage tolerant design is to ensure that cracks will not become critical until they are detected and repaired by means of periodic inspections. Inspections play an important role in maintaining structural integrity by compensating for damage that cannot be predicted or modeled during the design due to randomness in loading, accidental impact damage and environmental factors. In today's practice both safelife and failsafe structural design concepts are necessary to create a structurally safe and operationally satisfactory components. These two concepts have found application in structural design of airplanes, bridges and other engineering structures for different structural parts based on the functionalities and associated redundancy level. For instance, nose landing gear and main landing gear do not employ any redundancy and exhibit a short fatigue life. Therefore they are designated as safelife structures. Wing skinstringer and fuselage skinstringer panels have a substantial fatigue life and usually offer structural redundancy, so they are designated as failsafe structures. Estimating Fatigue Life and Crack Sizes Structural components experience numerous repetitive load cycles of normal flight conditions during their service life. In addition, less frequent but higher loads originating from strong atmospheric gusts or unexpected maneuvers during the life of aircraft are inevitable. Flaws and imperfections in the structure, such as micro cracks or delamination, may propagate under such service experience. Estimating fatigue life and crack size is a challenging task as there are no physical models available to determine crack growth as a function of the numerous factors that affect it. The load spectrum of an aircraft gives first hand information on the expected service load for which the airplane should be designed. The load history of aircraft is generated by load factor measurements from accelerometer placed at the centerof gravity. The number of times a load factor is exceeded for a given maneuver type (cruise, climb, etc.) is recorded for 1000 hours of flight. This load factor data are converted into stress histories, which can be used in fatigue calculations (Nees and Canfield, 1998; Arietta and Striz, 2000, 2005). Load histories are converted into number of cycles at given load levels and then a damage accumulation rule can be used with stressfatigue life (SN curve) to estimate fatigue life. The PalmgrenMiner linear damage accumulation rules (Miner, 1945) has been popular in aerospace application since the early 1950s to the present day. This rule computes the fatigue life as the summation of ratios of applied load cycles at a given level divided by the allowable number of load cycles to failure at the same stress level which can be obtained from SN curve (e.g., Tisseyre et al., 1994). An alternative fatigue life estimation method involves using crack propagation models obtained by fitting empirical models to experimental data. A breakthrough in damage growth rate prediction was achieved when Paris and Erdogan (1960) showed that damage grows exponentially as a function of crack tip stress intensity with each load cycle. Several modifications of the Paris model have been suggested to make the prediction more accurate and suitable for a specific set of loading condition; however the basic nature of the equations have remained unaltered. For instance Walker (1970) modified the Paris model by introducing an additional parameter to make it more accurate for variable amplitude loading when the history has both tensile and compressive stresses. Elber (1970) introduced the fatigue crack closure effect due to tensile overload effect in variable amplitude loading. Later crack growth retardation effects observed in variable amplitude loading were also introduced. Wheeler (1972) used the plastic zone size to modify the Paris model. These damage growth models have been widely used for life prediction with some modifications in structural design applications; e.g., Harkness (1994) and Tisseyre et al. (1994) used it in aerospace applications, and Enright and Frangopol (2000) used it for bridge design. Probabilistic Approach for Fatigue Life Prediction Aircraft structural design is still done by and large using codebased 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. These safety measures listed in FAR 25 for civil and transport category airplanes and Joint Service Specification Guide2006 (JSSG). Use of large safety measures increases the structural weight and operational cost. The main complexity for designing damage tolerant structures via safelife and fail safe concepts in design is due to uncertainties involved. These include uncertainty in modeling physical phenomena affecting structural integrity (e.g., loading, crack growth) and uncertainty in data (e.g., material properties). Inspection and replacement add additional uncertainty because damage detection capabilities depend on random factors such as location of the damages or labor quality and equipment. It has been demonstrated that small variations in material properties, loading and errors in modeling damage growth can produce huge scatter in fatigue life, (e.g., Harkness, 1994; Sinclair and Pierie, 1990) which makes it inevitable to use large safety measures during the design process. Uncertainties are inevitable and past service experience in the design of new structures have become a key factor in modem damage tolerant design approaches. Statistical data are collected for material properties, load histories (by the use of accelerometers) and damage initiation and growth by scheduled inspections. Then the associated uncertainties may be introduced into the design procedure by probabilistic approaches. A reliabilitybased approach towards structural design requires us to account for uncertainty in damage initiation, damage growth with time, residual strength and damage detection. In probabilistic formulation uncertainty is incorporated into the design process by representing random variables by probability distributions and unacceptable design is determined by calculating probability of failure of the damage state exceeding critical allowable state. The combination of probabilistic approach and fracture mechanics in fatigue life prediction has been demonstrated by Provan et al. (1987) and Belytschko et al. (1992). Uncertainty in damage initiation and growth has been introduced into life prediction by Rahman and Rice (1992); Harkness (1994); Brot (1994) and Backman (2001). Uncertainty in loading has been incorporated by Nees and Canfield (1998) and Arietta and Striz (2005) by using load history. Tisseyre et al. (1994) and Enright and Frangopol (2000) used reliability based formulation to predict fatigue failure of structural components subjected to uncertainty in loading, damage initiation and growth. Backman (2001) studied reliability of aircraft structures subjected to impact damage. Environmental factors like corrosion, enhance crack growth rates. The effect of environmental factors has been studied by fitting empirical models to experimental data. Weir et al. (1980) developed a linear model to describe the enhancement in fatigue crack growth in the presence of aggressive environment due to hydrogen enhanced embrittlement. Recently there has been advancement in estimating corrosionfatigue growth rates. Harlow and Wei (1998) obtained empirical model for rate of corrosion fatigue in aggressive environment by fitting experimental results to linear models. Probabilistic analysis is also very useful when there is no single model that can completely describe the crack growth phenomena for given set of conditions. When there are wide range of competing models, Bayesian updating techniques can be used to identify the most appropriate model that accurately predict the physical phenomenon. Zhang and Mahadevan (2000) used this method to determine the better of two competing crack growth models based on observed data. Reliability Based Design Fluctuations in loads, variability in material properties and errors in analytical models used for designing the structure contribute to a chance that the structure will not perform its intended function. Reliability analysis deals with the methods to calculate the probability of structural failure subjected to such uncertainty. A typical reliability analysis problem can be defined as P, (d, x)= J f(x)dx g(d,x0O where (21) g(d,x)= R S where dis the vector of design variables, x is the vector of random variables, Pf is the failure probability as function of design variables and random variables,f, is the joint probability density function of random variables and g is the performance function which decides if the structure has failed in terms of load S and resistance R. The reliability is defined as the complement of failure probability. Calculation of structural reliability is computationally expensive because many evaluations of the performance function (e.g., fatigue life, stresses or displacements) are needed for accurate computations. Ang and Tang (1975) and Madsen et al. (1986) have presented good review of various methods of structural reliability analysis. Here the two most extensively used methods, the Monte Carlo simulation (MCS) and the firstorder reliability method (FORM), are presented. Monte Carlo Integration The Monte Carlo integration is by far the simplest and potentially most accurate method to obtain failure probability, although it can be computationally very expensive. A key aspect of Monte Carlo method is random number generation which provides a basis for selecting random realization of uncertain variables in the structural model (e.g., Melchers, 1987). The event of failure is evaluated by checking if the response of the structural design for each random realization of the set of uncertain variable is greater than the allowable response defined by the performance function. If Nis the total number of simulations of random variables and Nf the number of failed simulations then the probability of structural failure is estimated by Nf P  (22) r N The accuracy of the probability calculated from Equation 22 increases with the number of simulations. An estimate of the accuracy in failure probability is obtained by calculating the standard deviation in Pf \PfPf Pf= N (23) FirstOrder Reliability Method (FORM) Monte Carlo method can be computationally very expensive for evaluating very low probabilities because large number of simulations is required for accuracy. The first order reliability method is an efficient alternative. The FORM method is presented in several references (Madsen et al., 1986 and Melchers, 1987). The key idea of FORM is to make a linear approximation to the failure surface between safe and failed realization in the standard Gaussian space (all random variables are transformed to standard normal variables). This linear approximation is made at a point where the distance of the origin of standard space and the limit surface is minimum. This point is referred to as the most probable point and the shortest distance is termed as reliability index f/. The probability of failure is the area of tail beyond /f under the standard normal distribution. Pf = o(p) (24) and 0 is the cumulative density function of standard normal distribution. This method gives accurate results when the limit state function is linear. For nonlinear function, FORM underestimates failure probability for concave function and overestimates it for convex function. Higher order method like the secondorder reliability method (SORM) can be used to improve the accuracy. Reliability Based Inspection Scheduling Designing structure for damage containment can lead to overly conservative design which will be cost prohibitive in terms of manufacturing and operation. Reliability based inspection and maintenance can be used instead to detect and repair damage at periodic intervals. Inspections serve as protection against damage that cannot be modeled or predicted during design process (e.g., environmental, accidental impacts etc.). Designing inspection schedule is challenging for two reasons. First, the ability of the inspection to detect damage is limited because of human and mechanical errors, so that probabilistic models of inspection detection are needed. The function used to represent the probability of detection represents a common characteristic that small cracks will have low chance of detection and large cracks will be almost certainly detected. Palmberg et al. (1987); Tober and Klemmt (2000); Tisseyre et al. (1994) and Rummel and Matzkanin (1997) developed/used empirical equations to model probability of detection based on experimental data. Another reason for the computational expense is that damage size distribution changes with time due to crack growth and also after inspections because components with damage are replaced by new components. Recharacterizing crack size distribution after inspections is computationally challenging. Reliability centered maintenance focuses on scheduling inspections when the failure probability exceeds a threshold probability level. The reliability level is computed by determining the probability that damage becomes too large and remains undetected in all the previous inspections. The simplest and potentially most accurate method is to use Monte Carlo simulations, MCS (e.g., Harkness et al., 1994; Enright and Frangopol, 2000). MCS is computationally expensive as it requires large samples for estimating low probability of failure. Moment based techniques have been used to reduce the computational expense of reliability calculations with inspections. The firstorder reliability method (FORM) and secondorder reliability method (SORM) have been used to obtain probability after inspection by Rahman and Rice (1992); Harkness (1994); Fujimoto et al. (1998); ToyodaMakino (1999) and Enright and Frangopol (2000). The main problem with the use of moment based method is that the damage size distribution cannot be updated explicitly after each inspection using these techniques. Some modification and simplifying assumptions have been used in the moment based methods to make the calculations less time consuming. For instance Rahman and Rice (1992) developed a methodology to update crack size distribution after inspections using Bayesian updating. Harkness (1994) modified the FORM to directly calculate reliability with inspections without updating the crack size distribution. Reliability Based Design Optimization Structural optimization is a reasonable tool for helping a designer address the challenge of designing complex structures, at least in the preliminary design stage. For instance, Nees and Canfield (1998) and Arietta and Striz (2000, 2005) optimized F16 wing panels subject to constraints on damage growth. Reliability based design optimization further increases the cost of reliability analysis because several iterations on design variables are required to obtain optimum design that will satisfy the specified reliability constraint. The main reason for the computational expense is when the objective function and\or the constraints do not have simple analytical form and have to be evaluated numerically (e.g., finite element model). In these circumstances the numerically expensive function can be replaced by an approximation or surrogate model having lower computational cost such as response surface approximation. Response surface methodology can be summarized as a collection of statistical tools and techniques for constructing an approximate functional relationship between a response variable and a set of design variables. This approximate functional relationship is typically constructed in the form of a low order polynomial by fitting it to a set of experimental or numerical data. The unknown coefficients of a response surface approximation are estimated from experimental data points by means of a process known as linear regression. These coefficients are estimated in such a way as to minimize the sum of square of the error between the experimental response and the estimated response (e.g., Myers and Montgomery, 1995). The accuracy of a response surface is expressed in terms of various error terms and statistical parameters that represent the predictive capability of the approximation. Response surfaces have been widely used in structural optimization to reduce computational cost. NESSUS (Riha et al., 2000) and DARWIN (Wu et al., 2000) use response surface approximations for reducing computational cost of probabilistic finite element analysis. Venter (1998) proposed methods to improve accuracy of response surface approximation and used them for optimizing design of composites. Papila (2001) also used response surfaces in structural optimization for estimation of structural weight. Qu (2004) used RSA's to minimize cost of reliability based optimization. CHAPTER 3 EFFICIENT RELIABILITY BASED DESIGN AND INSPECTION OF STIFFENED PANELS AGAINST FATIGUE Introduction Reliability based optimization is computationally expensive when inspections are involved because crack size distribution has to be recharacterized after each inspection to simulate replacement. Inspections improve the structural safety through damage detection and replacement. However, inspections cannot detect all damage with absolute certainty due to equipment limitations and human errors. Probabilistic model of inspection effectiveness can be used to incorporate the uncertainty associated with damage detection. Typically, the crack size distribution after an inspection will not have a simple analytical form and can only be determined numerically during reliability analysis. Exact evaluation of failure probability following an inspection can be done by Monte Carlo simulation (MCS) with large population which is computationally expensive. The high computational cost for estimating very low probabilities of failure combined with the need for repeated analysis for optimization of structural design and inspection times make MCS cost prohibitive. Harkness (1994) developed a computational methodology to calculate structural reliability with inspections without updating the crack size distribution after each inspection. He assumed that repaired components will never fail again and incorporated this assumption by modifying the first order reliability method (FORM).* This expedites reliability computations which require only the initial crack size distribution to be specified. In previous papers (Kale et al., 2003, 2004), we used the same methodology to optimize inspection schedule. When inspections are needed earlier than half the service life, repaired components can have large probability of failure. In this case Harkness's method may not be accurate enough. In this chapter we propose an approximate method to simulate inspection and repair using Monte Carlo simulation (MCS) and estimate the failure probability using the first order reliability method (FORM). MCS is computationally very expensive for evaluating low failure probabilities due to large population requirement but is very cheap for estimating probability distribution parameters (e.g., mean and standard deviation). We use the data obtained from MCS to obtain the mean and standard deviation of crack size distribution. Subsequently, FORM is used to calculate the failure probabilities between inspections. The combined MCS and FORM approach to calculate failure probability with inspection removes the computational burden associated with using MCS alone. This method is applied to combined optimization of structural design and inspection schedule of fuselage stiffened panels. Stiffened panels are popular in aerospace applications. Stiffeners improve the load carrying capacity of structures subjected to fatigue by providing alternate load path so that load gets redistributed to stiffeners as cracks progress. Typical stiffening members include stringers in the longitudinal directions and frames, failsafestraps and doublers in the circumferential direction of the fuselage. Fracture analysis of stiffened panels has been performed by * FORM is a moment based technique which calculates the failure probability using a first order approximation about the point on the limit state where failure is most probable. Swift (1984) and Yu (1988). They used displacement compatibility to obtain the stress intensity factor due to stiffening. Swift (1984) studied the effect of stiffener area, skin thickness and stiffener spacing on the stress intensity factor. He also discussed failure due to fastener unzipping and effect of stiffening on residual strength of the panel. Yu (1988) also compared the results with finite element simulation. Our previous paper Kale et al. (2003) demonstrated the combined structural design and optimization of inspection schedule of an unstiffened panel. The main objective of the present chapter is to develop a cost effective computational methodology to perform reliability based optimization of structural design and inspection schedule. The methodology is demonstrated by performing structural optimization and inspection scheduling of stiffened structures against fatigue. To reduce the computational time associated with fatigue life calculation and reliability analysis, response surface approximations are developed for tracking crack growth. Crack Growth and Inspection Model Fatigue Crack Growth The rate of fatigue crack propagation can be expressed as a function of applied stress intensity factor, crack size and material constants (which are obtained by fitting empirical model to experimental data). For the example in this chapter we use the Paris law. c = D(AK)" (31) dN where a is the crack size in meters, Nis the number of cycles of fatigue loading in flights, da/dN is the crack growth rate in meters/cycles, the stress intensity factor range AK is inMPammeters and m is obtained by fitting the crack growth model to empirical data. More complex models account for load history effects. The stress intensity factor range AK for cracked stiffened panel can be calculated using finite element or analytical method as a function of stress a and crack length a. AK = wJcm (32) The effect of stiffening on the stress intensity is characterized by the geometric factory which is the ratio of stress intensity factor for the cracked body to that of stress intensity factor at the crack tip of an infinite plate with a through the thickness center crack. The calculation of y/ usually requires detailed finite element analysis. Here, y is calculated using a method due to Swift (1984). The number of fatigue cycles accumulated in growing a crack from the initial size a, to the final size aN can be obtained by integrating Equation 31 between the initial crack a, and final crack aN. Alternatively, the final crack size aN after N fatigue cycles can be determined by solving Equation 33. This requires repeated calculation of V/ as the crack propagates. The computational approach for integrating Equation 33 is illustrated in Appendix B. aN da N= (33) a, f (AK(V), m) Here we focus on designing a fuselage panel for fatigue failure caused by hoop stresses. The hoop stress is given by Equation 34 and crack grows perpendicular to the direction of hoop stress given by o + a= (34) th + N,A where r is the fuselage radius, p is the pressure differential inside the fuselage, h is the panel width, t is panel thickness, Ns is the number of stiffeners and As is the area of single stiffener (See Figure 31). t _ 1.72  Crack 0.6 2a L IF S Applied I I load Figure 31: Fuselage stiffened panel geometry and applied loading in hoop direction (crack grows perpendicular to the direction of hoop stress) Critical Crack Size We consider optimizing the design of a typical fuselage panel for fatigue failure due to hoop stress. The failsafe stiffening members in circumferential direction such as frames, failsafe straps and doublers are modeled as equispaced rectangular rods discretely attached to the panel by fasteners. The panel size is assumed to be small compared to the fuselage radius so it is modeled as a flat panel following Swift (1984). We assume that only three stiffeners adjacent to crack centerline are effective in reducing the stress intensity factor. So we model the aircraft fuselage structure by a periodic array of throughthethickness center cracks with three stiffeners on either sides of centerline as show in Figure 31. The critical crack length ac at which failure will occur is dictated by considerations of residual strength or crack stability. Structural failure occurs if the crack size at that time is greater than critical crack. The crack length causing net section failure is given by ac, = 0.5[h h NhA ) (35) Equation 35 gives the crack length ay at which the residual strength of the panel will be less than yield stress Y and Nub is the number of intact stiffeners. K Y acH = \ I\ (36) acL = (37) 2t ) Equation 36 determines the critical crack length for failure due to hoop stress a and Equation 37 determines the critical crack length for failure due to transverse stress. This is required to prevent fatigue failure in longitudinal direction where skin is the only load carrying member (effect of stringers in longitudinal direction is not considered because hoop stress in more critical for fatigue). The critical crack length for preventing structural failure is given by Equation 38 and the fatigue life Nfof structure is determined by integrating Equation 33 between the initial crack a, and critical crack ac. ac= min(acy,acHacL) (38) Typical material properties for 7075T651 aluminumalloy most commonly used in aerospace application are presented in Table 31. The applied load due to fuselage pressurization is assumed to be 0.06 MPa (maximum pressure differential, 8.85 psi, Niu, 1990). The Joint Service Specification Guide2006 specifies design assuming a minimum initial crack of 0.127 mm to exist in structure at all times. However we consider more conservative value of initial crack distribution (mean of 0.2 mm) to account for uncertainties associated with damage initiation and growth associated with corrosion, environmental effects and accidental damage. The structural design parameters obtained for B747 series aircraft from Niu (1990) and Jane's all the world's aircraft (Jackson, 1996) are listed in Table 32. Table 31: Fatigue properties of 7075T651 Aluminum alloy Property Yield Initial crack Paris Fuselage Pressure Fracture stress a1,o meters exponent m radius, r, load, p, toughnes Y, Mean. Mean. meters (MPa) s, Kic MPa Standard Standard MPa deviation deviation Jmetes Distribution 500.0 Lognormal Lognormal 3.25 Lognormal 36.58 type 0.0002, 2.97 0.06 0.00007 1.05 0.003 (Source: Sinclair and Pierie, 1990 and Niu, 1990) Table 32: Structural design for fuselage Fastener diameter, d 4.8 mm Fastener spacing, s 3.1 cm Fuselage length, / 68.3 m Fuselage radius, r 3.25 m Number of panels, Np 1350 Number of fasteners per stiffener 20 Number of stiffeners, Ns 6 Panel length, b 0.6 m Panel width, h 1.72 m Stiffener thickness, ts 5 mm (Source: Swift, 1984; Jackson, 1996 and Niu, 1990) Probability of Failure at a Given Time The probability of failure after N cycles of loading is defined as the event that fatigue life (number of cycles accumulated in growing a crack from initial crack to critical crack) is less that N. The purpose of reliability analysis is to determine the probability that the structure will not fail for a random realization of uncertain variables (a,, m and a). The equation which define the failure boundary is known as the limit state function, g. So for our case g(a,,m,cr)= Nf(a,,m,cr) N (39) where the fatigue life Nfis determined by integrating Equation 33 between the initial crack a, and the critical crack a,. The failure probability corresponding to Equation 39 is calculated using the firstorder reliability method (FORM). In this method the limit state function is represented in the transformed standard normal variables (a,, m and a are transformed to normal distributions with mean = 0 and standard deviation = 1) and the point on the limit surface closest to origin is determined. This point is known as design point or most probable point and the shortest distance is called reliability index, fld. The calculation of reliability index is an optimization problem requiring repeated calculation of Equation 39 for several iterations in the random variables. In this chapter the MATLABC fmincon function (which employs sequential quadratic programming) is used to determine the reliability index. The main reason for using reliability index instead of directly calculating failure probability is that FORM is computationally cheaper compared to MCS. A second reason is that reliability index is more suitable for generating accurate surrogate models because it varies over much smaller range compared to the failure probability. Failure probability is determined from the reliability index using the cumulative density function, 0 of the standard normal distribution. Pf = P(d ) (310) For an unstiffened panel analytical expression of fatigue life is available; however for stiffened panels, determining fatigue life requires computationally expensive calculation of the geometric factory. The computational procedure for calculating fatigue life is described in Appendix B. Inspection Model When the structure is subjected to periodic inspections, cracks are detected and repaired or the structural part is replaced. We assume that the probability Pd, of detecting a crack of length a is given by Palmberg's equation (Palmberg et al., 1987). Pd (a)=J (311) l+(a/ah)8 Where ah is the crack size that will be detected with 50% probability and /f is inspection parameter. An approximate values of ah of 1 mm was obtained by rounding off data from the probability of detection curves in Rummel and Matzkanin (1997) for eddy current inspection. They obtained the probability of detection curves by machining artificial cracks in panels and counting the number of times they were detected after inspecting several times. The value of the other inspection parameter / of 3 was obtained by fitting Equation 311 to the inspection data in that reference and increasing it slightly (to account for improvement in inspection technology since 1997). It is assumed that once a crack is detected, the panel is replaced by newly manufactured panel with the fabrication defect distribution. Computational Method to Perform Reliability Based Optimization with Inspections When inspection and replacement of structural components are scheduled, the damage size distribution changes because defective parts are replaced with new parts having a smaller damage sizes (fabrication defects, a1,o). Reliability computation is very expensive when inspections are involved because crack size distribution has to be re characterized after each inspection to simulate replacement and exact computation of failure probability using MCS requires large sample size. Harkness (1994) developed an approximate method to expedite reliability computation with inspection by assuming that repaired components will never fail again and incorporated this assumption by modifying the first order reliability method (FORM). The failure probability at any time following an inspection is the probability that the crack size is greater than the critical crack size at that time and that it is not detected in any of the previous inspections. Using an empirical crack growth model like Equation 33 to predict crack size at any time, a probabilistic model for inspection probability of detection and a specified value of critical crack size, he calculated the structural reliability using the FORM method. The effect of inspections is incorporated into the FORM by integrating the (probability density function) PDF of undetected cracks over the failure region using numerical integration. The assumption that detected cracks are replaced and the new component will not fail during the remainder of service life greatly simplifies the numerical computation by considering only the PDF of undetected cracks. When inspections are needed earlier than half the service life, repaired components may have large probability of failure and Harkness's method may not be accurate enough. Kale et al. (2005) proposed an approximate method to simulate inspection and repair using Monte Carlo simulation (MCS) with small sample size to update the crack size mean and standard deviation after an inspection and first order reliability method (FORM) to calculate the failure probability between inspections. The procedure described below expedites the reliability calculations by removing the need of exact MCS analysis with large sample size. Searching for Next Inspection Time Using FORM The main computation associated with determining inspection schedule for a given structure is to find the next inspection time at which the structural reliability will be lower than the specified threshold value. The probability of failure after N cycles of loading since the most recent inspection is defined as the event that fatigue life Nfis less than N. Pf (N, a,, m, )= P(Nf (a,, m, u) N) (312) where a, is the crack size distribution at the beginning of inspection period and the fatigue life Nf is the number of cycles accumulated in growing a crack from initial crack a, to critical crack ac. For a given structural thickness, the next inspection time is obtained such that the probability of failure before the inspection is just equal to the maximum allowed value (Pfth, reliability constraint). The next inspection time S, for a given threshold reliability level is obtained by solving Equation 313. P(N (a,m,)N) Pfh = 0 (313) Equation 313 is solved for time interval Nby using bisection method between previous inspection time S,_1 and service life Si and for each of the bisection iteration, the first term is calculated by FORM. For an unstiffened panel FORM is very cheap; however for stiffened panel it is computationally expensive because calculation of fatigue life is expensive and additional computational burden is added because of the bisection search between previous inspection time S,_ and service life S1. Updating Crack Size Distribution after Inspection using MCS The algorithm for simulating crack growth and inspections is shown in Table 33. After obtaining the next inspection time, the crack size distribution has to be updated after that inspection. This updated crack size distribution serves as initial crack size distribution for the following inspection interval. The damage distribution after an inspection can easily be updated by using Monte Carlo simulation (MCS) with a small sample size and is computationally very cheap compared to calculating probabilities. The crack size aN after N cycle of fatigue loading is obtained by solving Equation 33. To obtain the crack size mean and standard deviation after an inspection, we produce 50,0 random numbers for each random variable in Equation 33 (a,, m, a) and obtain the final crack size aN. We then simulate the inspection by using Equation 311 with another random number for probability of detection. If the crack is detected the panel is replaced by a new one with a random crack size picked from the distribution of manufacturing defects a;,o. After all cracks are analyzed for detection, the updated crack sizes are used to fit a distribution and to obtain its mean and standard deviation. This serves as the initial crack distribution for the next inspection. For the data used in this chapter the fabrication crack distribution is lognormal, and the distribution after inspections was also found to be best approximated by lognormal distribution out of 12 analytical distributions in ARENA software (Takus and Profozich, 1997). Table 33: Pseudo code for updating crack size distribution after N cycles from previous inspection (1) Generate a panel by a random vector of uncertain variables (a,, m, and a) (2) Solve Equation 33 for crack size aN after N cycles of fatigue loading for the panel using Newton's method or bisection (if Newton's method does not converge). (3) Compute the probability of detection of crack aN from Equation 311, Pd(aN). (4) Generate a random number from a uniform distribution with bounds (0, 1) pd.d (5) IfPd (a ,> Pynd then simulate replacement of defective component by generating a random crack a1,o for a new panel and set a, = ay else keep aN (7) Store aN for fitting probability distribution to crack sizes after inspection and go back to (1) (8) Stop after 50,000 random panels have been simulated and fit distribution to crack sizes tA large sample size was used to get accurate estimate of mean and standard deviation. This makes the optimization results insensitive to MCS seed. For the unstiffened panel 100,000 samples are used. The crack size probability distribution after the inspection is estimated by fitting probability distribution to the crack size samples obtained from MCS. The goodness of fit of this distribution affects the accuracy of probability calculations. To illustrate this we calculate the actual probability of failure for two inspection times calculated for a 2.00 mm thick unstiffened panel using the proposed method. The first inspection time of 9288 flights is calculated using FORM with a lognormal initial crack size distribution with mean of 0.20 mm and coefficient of variation of 0.35. The crack size distribution after this inspection is updated by Table 33 using a crack growth time N of 9288 flights. The updated crack distribution is found to be lognormal with mean = 0.30 mm and cov = 0.86. The next inspection time of 15,540 flights is obtained from FORM using the updated crack distribution. The actual and best fitted (cumulative distribution function) CDF of crack size distribution after 9288 flights are shown in Figure 32. The corresponding pvalue is less than 0.005 indicating a bad fit; however for low failure probabilities (e.g., 107) this fit ensures accurate structural design calculation at very low computational expense. To validate this claim, failure probability is calculated for the inspection schedule (first inspection = 9,288, second inspection = 15,540 flights) using MCS with 108 samples. The exact failure probability after 9288 flights is 4.0 x 107 and after 15,540 flights is 2.7x 107 which are close to the value of 107 calculated using the proposed method. The square error between actual PDF and lognormally fitted PDF is 0.00029 and the maximum error between CDF's is 0.06 at crack size of 0.28 mm Empirical CDF  actual 0.9      lognormal fit 0.3  L.      _ 0.7 C   0.7  /     _ LL 0 .5          0.4  ^ L L  L_ 0.3  r 0.4       02 013 L   L I  0 0.5 1 1.5 2 2.5 3 crack size a in mm Figure 32: Comparison of actual and lognormally fitted CDF of crack sizes after an inspection conducted at 9288 flights Calculation of Inspection Schedule for a Given Structure For a given structural design optimum inspections are added one by one until the probability at end of service life is less than the specified threshold reliability level. Example 31 illustrates the approach described in previous sections for a 2.0 mm thick unstiffened plate and a required reliability level of 107. Solving Equation 313 for N, the first inspection time is 9288 flights. Crack growth simulation using the MCS pseudo code in Table 33 is performed with initial crack sizes a;,o and crack growth time of 9288 flights giving the updated crack size distribution after the first inspection. The lognormal distribution fitted after inspection has a mean of 0.30 mm and coefficient of variation 86.0%. This serves as the initial crack size distribution for the second inspection. Again, the second inspection time of 15,540 flights is obtained solving Equation 313. This cycle of scheduling inspections is continued until the failure probability at the end of service life is less than the specified value. Figure 33 illustrates the variation of the probability of failure with and without inspection. Table 34 presents the inspection schedule during the service life and the crack size distribution parameters after each inspection. It can be seen that inspections are very helpful in maintaining the reliability of the structure. From Table 34 it can be seen that first inspection interval is the largest. After the first inspection the repaired components are replaced with the same initial crack distribution (mean = 0.20 mm and cov = 35%); however some cracks escape detection, leading to smaller inspection intervals. From the crack size distribution parameters shown in last column of Table 34 we can conclude that the crack size distribution after each inspection essentially remains unchanged after a certain number of inspections, leading to uniform inspection intervals. We can infer that towards the end of service the rate at which unsafe cracks are introduced in the structure due to replacement is same as the rate at which cracks are detected by the inspections. 1.E01 1) 5000 10000 15000 0000 35000 40000 45000 1.E03  S No Inspections ~ 1.E05 Optimal Inspections 1.E07  0 2 1.E09  U2 i.E11  1.E13  1.E15  Flights Figure 33: Example 31, Variation of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 107 Table 34: Example 31, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 107 Number of Inspection Inspection interval Crack size distribution after inspections time, S, (flights) S,, S,, inspection (mean, mm cov) (flights) 0   Initial crack distribution (0.200, 35%) 1 9,288 9,288 (0.300, 86%) 2 15,540 6,252 (0.326, 90%) 3 20,741 5,201 (0.335, 87%) 4 26,223 5,482 (0.342, 87%) 5 31,649 5,426 (0.345, 86%) 6 37,100 5,451 (0.347, 86%) Optimization of Structural Design The cost associated with change in the structural weight for aluminum and the fuel cost is taken from Venter (1998). He assumed a fuel cost of $0.89 per gallon and that a pound of structural weight will cost 0.1 pound of fuel per flight. From this we calculated that a pound of structural weight will cost $0.015 in a flight for fuel. The structural weight is assumed to be directly proportional to the plate thickness and a pound of structural weight is assumed to cost $150 for material and manufacturing. Appendix D shows the details of material and fuel cost calculations. A typical inspection cost of about a million dollars was obtained from Backman (2001). Following Backman (2001) the service life is assumed to be 40,000 flights. The structural design parameters obtained for B747 series aircraft and cost factors are summarized in Table 35. Table 35: Cost of inspection, material and fuel Density of aluminum (p) 166 Ib ft Fuel cost per pound per flight (Fe) $ 0.015 Inspection cost (I,) $ 1,000,000 Material and manufacturing cost per lb (Me) $ 150.0 Number of panels, Np 1350 Service life, S1 40,000 flights (Source: Venter, 1998 and Backman, 2001) The life cycle cost Ctt0 for N, inspections is Ct = MW + FWS1 +N,Ic (314) Where Wis the total weight of all the panels in the fuselage, given as W = N, (N,A,b + thb )p (315) The parameters in Equations 314 and 315 are defined in nomenclature. Reliability based design optimization is computationally very expensive when inspections are involved because several iterations on structural design variables and inspection times are required to find an optimum combination of structural sizes and inspections that will minimize total cost. For an unstiffened panel, analytical expression for crack growth is available and exact computations using the combined MCS and FORM technique is very cheap. For stiffened panel, the crack growth has to be determined numerically and reliability computations are very expensive even with the combined MCS and FORM approach. The key factor responsible for computational time is the calculation of geometric factory due to stiffening, which can be determined using detailed finite element analysis or displacement compatibility method due to Swift (1984). In this chapter we used Swift's approach which takes about 0.5 second for evaluating single value of y/ for a given structural design and crack length. Table 36 explains the various RSA's used to make computations faster and Table 37 gives the breakdown of computational cost for calculation of exact inspection time and updating crack size distribution. Table 36: Description of response surface approximations used in optimization Name of Description Function of variables response surface i/ RSA Geometric factor Skin thickness ts, Stiffener area As, Crack due to stiffeners length a pa,RSA Crack size mean Skin thickness ts, Stiffener area As, mean crack after inspection length pua, mean crack length oa, time N, standard deviation in stress up a,,RSA Crack size standard Skin thickness ts, Stiffener area As, mean crack deviation after length p/a, mean crack length oa, time N, inspection standard deviation in stress up f/dRSA Reliability index Skin thickness ts, Stiffener area As, mean crack length pua, mean crack length oa, time N, standard deviation in stress up Table 37: Computational time spent in exact calculation of next inspection time and error due to V/ RSA usage Variable Computational Number of Total time, Typical error due method function seconds to use ofRSA evaluation Geometric Displacement 1 0.5 0.02 approximately factor, fi compatibility Fatigue Numerical 100 evaluation of 50 600' flights (error life, Nf integration using W in Nf due to error MATLAB's adaptive in V) Simpson 's quadrature Reliability Iterative search 100 evaluation of 5,000 0.2, average fitting index, Od using MA TLAB's Nf error from PdRSA finincon + 0.1s from error in Nf Next Bisection between SN 15 evaluation of 75,000 inspection 1 and service life Si Od (0.86 days) time, SN Computational time spent in exact updating of crack size distribution using MCS and error due to V/ RSA usage Crack size aN Iterative search 20 evaluation 1,000 after N cycles using Newton's ofNf method Crack size MCS 50,000 5,000,000,0 Less that 0.1 % distribution using evaluations of (578 days) from pa,,RSA step F a _and oa,RSA. When structural design and inspections schedule are optimized together, the computational cost will be several hundred times that shown in Table 37 because of iterative search on structural sizes and number of inspections in a schedule required to t Assuming that the structure is designed for inspection interval of 10,000 flights (typical results obtained in this chapter), the error in fatigue life calculation due to error in Vy will be 10,000/ (1.02) "' where m is the Paris law exponent. Using the mean value of m = 2.97 an error of 600 flights in fatigue life is obtained The fatigue life has an error of 600 flights because of which the inspection time has the same error. The reliability calculated at the next inspection time plus the error in NF that is S,+ 600 will have error in reliability index of about 0.1 (2%); however this translates to error in probability of 200 % for a reliability level of the order of 107. maintain the specified reliability level. To overcome this we develop surrogate models based on response surface approximations shown in Table 36. The last columns of Table 37 show the errors made by the use of RSA's in calculations. An error of 0.02 is the typical fitting error in construction of RSA foriV Table 38 gives the overview of the methodology describing the computational challenge in its implementation and explains the approach used to perform reliability based optimization of structural design and inspection schedule. The computational procedure is explained first followed by the computational challenge associated with implementation of this procedure. The surrogate models used to remove the computational burden involved in this method are also explained. It is to be noted that the main cause of using the surrogate models is because there is no analytical expression available to calculate crack growth for stiffened panel. If analytical expression for crack growth is available then exact computations will become feasible. For an unstiffened panel the computational cost for calculating crack size distribution after inspection is very low and is calculated by Monte Carlo simulation with 100,000 samples during the optimization. However, for stiffened panel, calculation of crack growth is expensive and we use response surface approximation (RSA) to estimate the crack size mean and standard deviation after an inspection. This RSA is constructed by fitting data obtained from MCS at some sampled locations in design domain (skin thickness, initial crack size mean, initial crack size standard deviation, stiffener area, standard deviation in stress and number of cycles of loading) as shown in Appendix C. Table 38: Pseudo code for combined optimization of structural design and inspection schedule (A) Optimization of structural design: For a given structural design calculate inspection schedule using step B and obtain cost of structural weight and inspections. Stop if convergence on minimum cost is obtained otherwise update the structural design. (B) Optimization of inspection schedule: Add one inspection at a time using step C, update crack size distribution using step D. Check if the number of inspections is sufficient for maintaining the given reliability level during the service using FORM, if not add one additional inspection. (C) Searching for next inspection time: Given structural sizes, probability distribution of random variables, find when the next inspection is needed by calculating the time S, at which the probability of failure equals the required reliability level Pfth using the first order reliability method (FORM). This is a computationally intensive optimization problem which requires repetitive computation of reliability index. A rough estimate of computer time is described in Table 37. To reduce the computational burden associated with repeated reliability calculation during optimization we generate design response surface to estimate reliability index. Appendix C explains the details of this response surface approximation (fd RSA). (D) Updating crack size distribution after inspection: After obtaining the next inspection time from step C, use Monte Carlo simulation (MCS) to update the crack size distribution after this inspection by growing cracks between the inspection time S, and the previous inspection time S, . The MCS method is described in Table 33 and the computational expense associated with it is described in Table 37. The computational burden associated with estimating crack size distribution parameters after an inspection is solved by using RSA's to estimate the crack size mean pa,,RSA and standard deviation oaRSA after an inspection. Appendix C explains the details of these RSA's. During the optimization the structural thickness t and the stiffener area As are changed, which changes the structural weight according to Equation 315. The optimum inspection schedule is determined for this structural design using Table 38 and the total cost of structural weight and inspection is obtained from Equation 314. The optimization iteration is stopped after a specified convergence tolerance is achieved. The convergence tolerance on minimum cost is assumed to be $10,000 in this chapter and MATLAB 0 is used to perform optimization of the design. Using the RSA's, entire calculations can be done in about 3 days on a Windows Pentium 4 processor. Results Structural design can have large effect on operational cost and weight of the structure. When inspections and maintenance are not feasible, safety can be maintained by having conservative (thick) structural design. To demonstrate this we first obtain safe life design required to maintain desired level of reliability throughout the service life for unstiffened and stiffened structures. Table 39 shows the safe life design of unstiffened panel and Table 310 shows the safelife design of a stiffened panel. Table 39: SafeLife design of an unstiffened panel Required probability Minimum required skin Life cycle Structural of failure, Pfth thickness t (mm) cost C0or weight, Ib $ x 106 10 4.08 25.42 33902 108 4.20 26.16 34880 10 4.24 26.34 35129 Table 310: SafeLife design of a stiffened panel Required Total Skin A Life cycle Structural probability of stiffener thickness, t A1t o cost, Ctor weight, Ib failure, Pfth area 103 (mm)Total $ x 106 m 2, As 107 2.23 2.31 35.85 22.42 29900 10 2.26 2.33 36.00 22.68 30248 109 2.30 2.35 36.22 22.91 30555 An unstiffened panel is a single load path structure without load transfer capability. Comparing Table 39 to Table 310, we see that if structure is designed with multiple load transfer capability then the weight and cost can be reduced by about 10 %. Stiffeners improve the load carrying capacity and reduce crack growth rates allowing greater crack length safely. This issue is further explored in Appendix A. Next we demonstrate the effect of inspections on structural safety and operational cost. Inspections improve the reliability by detecting and removing cracks. By optimizing the structural design together with inspection schedule, we can trade structural weight against inspection cost to reduce overall life cycle cost. To demonstrate the effectiveness of inspections, optimum structural design and inspection schedule were first obtained for an unstiffened panel design with results shown in Table 311. Table 311: Optimum structural design and inspection schedule of an unstiffened panel (in all cases the optimum number of inspection is 3) Required Skin thickness Optimum inspection Life cycle Structural probability of t (mm) times, S, cost C0or weight, lb failure, Pfth (flights) $ x 106 107 2.30 12346,22881,31365 17.28 19109 10 2.43 13158,23496,31496 18.15 20199 109 2.56 13927,24016,31682 18.97 21295 It can be seen that inspection and repair lower the life cycle cost by about 25% over safelife unstiffened panel design and by 20% over to stiffened panel safelife design. The corresponding reductions in structural weight are 40% and 30%, respectively. There is an additional incentive for conducting inspections in that they protect against other types of damage like that due to accidental impacts and corrosion. Next we optimize the structural design and inspection schedule for stiffened panel design (Table 312) and illustrate the tradeoff of structural weight in skin and stiffeners against inspection cost. Table 312: Optimum structural design and inspection schedule for stiffened panel Required Total Required A, Optimal Life cycle Structural probability stiffener skin A inspection cost, Ctrt weight, of failure, area, As thickness of times, $ x 10 lb Pfth x 104 ,t(mm) S meter2 (flights) 10 7.11 1.71 19.40 10844,18625, 17.20 17659 25791, 32908 10 7.30 1.81 18.95 11089,18758, 17.87 18504 25865, 32943 10 13.74 1.67 32.29 12699,22289, 18.33 20443 _____ _______31163 _____ Comparing Table 310 to Table 312 we see that inspections lower the life cycle cost of stiffened panel design by about 20% compared to safelife design. Comparing Tables 311 and 312 we see only a small gain (about 3%) in designing stiffened structures when inspections are involved and cost can be minimized by designing single load path structures (unstiffened) with inspections. Comparing Table 312 to Table 311, the increased design flexibility allows additional tradeoff of structural weight against inspections by having one additional inspection over the unstiffened panel design. About 20% to 30 % of the structural weight is transferred from skin to stiffeners. In aircraft operation the inspection intervals are dictated by practical considerations and regulation which are based on service experience. The Joint Service Specification Regulations2006 requires all airlines to conduct major depot level inspection four times during the service life. These inspections are conducted at uniform intervals. Table 313 shows the design with fixed inspection schedule. Table 313: Optimum structural design for regulations based inspections conducted at four constant interval or 8000 flights for stiffened panel Required Total Required A, 1000 Inspection Life Structural probability stiffener skin A100 times, S, cycle weight, lb of failure, area As thickness cost, Pfth 104 ,t (mm) Ctot meter2 $x 106 107 13.41 1.38 35.94 8000, 17.44 17927 16000,24000, 32000 10 13.80 1.47 35.12 8000, 18.16 18878 16000,24000, 32000 109 14.85 1.49 36.60 8000, 18.61 19491 16000,24000, 32000 It is seen that inspections done at constant interval are only marginally less cost effective than the optimized inspection schedule. From Tables 311, 312 and 313 we can conclude that when inspections are used to maintain safety there is less gain in using stiffeners for stable fatigue crack growth. However, stiffeners might be very useful in maintaining structural rigidity to resist buckling and pillowing. Also, from Table 310 when structures are designed without any inspections, stiffeners can be very helpful in reducing crack growth rate. Appendix A discusses the effect of stiffening on structural design and crack growth rates. Next we obtain optimum structural design and inspection times for fixed number of inspections. Through this we seek to demonstrate the tradeoff of inspection cost against cost of structural weight Table 314: Tradeoff of inspection cost against cost of structural weight required to maintain fixed reliability level for stiffened panel Required Number Total Required A Optimal inspection Life probability of stiffene skin A 100 times cycle y of inspection r thicknes otal (flights) cost $ x failure, n area A, s (mm) 106 Pfth x 104 0 meter2 107 5 7.05 1.60 20.26 9497,16029,22064, 17.53 28060,34036 107 4 7.11 1.71 19.40 10844,18625,25791, 17.20 32908 107 3 7.23 1.88 18.14" 12743,22435,31212 17.35 108 5 7.00 1.70 19.18 9933,16406,22363, 18.14 28271,34145 108 4 7.30 1.81 18.95 11089,18758,25865, 17.87 32943 108 3 13.29 1.63 32.04 12514,22178, 31110 18.03 109 5 7.50 1.74 19.92 10091,16428,23260, 18.53 29268, 34412 109 4 7.89 1.88 19.51 11546,19064,26064, 18.59 33044 109 3 13.74 1.67 32.29 12699,22289,31163 18.33 From Table 314 we see that the optimum structural weight decreases monotonically with the number of inspections because structural weight is traded against inspections. However, the stiffener areas show sudden jump with the required number of inspections (decreasing inspections from four to three). The main reason for this is the presence of several local minima because the inspection cost is a discrete variable and any change in number of inspection will lead to huge jump in either the skin thickness or stiffener area if total cost is minimized. In this case the stiffener areas show sudden change because of the reason that stiffeners break during crack growth reducing reliability so that when the number of inspections are large, minimum cost can be ** There exist another local minimum i\ iii. I = 14.64 mm2, t = 1.505 mm and inspection times of 12375, 22097 and 31083 flights. However this design has slightly higher cost (2%), hence it is not shown in Table 314. achieved by reducing stiffener areas and increasing structural thickness. Because of this there are several local optimums for structural sizes. The cost difference between these optimums is very small. Actual failure probability was calculated for each of the local optimums and the design whose failure probability was closest to threshold value was selected. Exact evaluation of failure probability for some designs is shown below. The ratio of stiffener area to skin area is constant at about 20% when the number of inspections is large. For lesser number of inspections about 30% of the structural weight is transferred to stiffeners. As more inspections are added the structural weight is traded against inspection cost until a minimum is reached, beyond this any further reduction in structural weight will lead to faster crack growth rate requiring frequent inspections to maintain reliability. Table 315 presents the exact evaluation of failure probability without any RSA for the optimum obtained from Table 312. This is done by calculating reliability using FORM without using fldRSA reliability index. Table 315: Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel with inspection Required Optimum design Inspection times, Actual Pf/ Pfth reliability (skin thickness mm, flights before each level, Pfth A, 1 inspection ATotal 107 1.71, 19.40 10844,18625,25791, 2.89, 2.26, 1.98 , 32908 3.90, 1.87 108 1.81, 18.95 11089,18758, 25865, 0.98, 3.75, 3.35, 32943 3.18, 3.06 109 1.67, 32.29 12699,22289, 31163 2.12, 5.27, 1.47, 1.44 It can be seen that RSA's can be used to predict the optimum with sufficient accuracy. The error in actual reliability versus calculated reliability using approximations can comes from the RSA for qf which can affect the accuracy in calculation of inspection time by 600 flights. Additional error is introduced because of convergence tolerance for calculation of reliability index (0.1 used in the chapter) and randomness in MCS seed for calculating crack size distribution. The effect of RSA on accuracy of results and computational cost is explained in detail previous section. Typically the optimum obtained from RSA will be slightly different from the true optimum because of error in RSA. To get more accurate results, optimum obtained from RSA's can be iteratively calibrated so that the actual failure probabilities are close to the threshold value. The entire calculations can be done in about 3 days on a Windows Pentium 4 processor. Summary A computational method was developed using a combination of MCS and FORM to perform combined optimization of structural design and inspection schedule. The method was found to be computationally cheap and accurate in determining structural reliability with inspection. Response surface approximations were used to obtain fatigue life to reduce computational cost associated with life calculations. Optimum combination of structural design and inspection schedule was obtained that will maintain the desired reliability level during service at minimum cost. CHAPTER 4 TRADEOFF OF WEIGHT AND INSPECTION COST IN RELIABILITYBASED STRUCTURAL OPTIMIZATION USING MULTIPLE INSPECTION TYPES Introduction The integrity of structural components is affected by damage due to fatigue, corrosion and accidental impact. Damage may reduce the residual strength of the structure below what is needed to carry the service loads. In a failsafe design, structural safety can be maintained by inspecting the components and repairing the detected damage. Alternatively, stresses can be lowered by increasing structural sizes so that damage never grows to a critical length during service life. Structural component safety checks have gained widespread acceptance because of uncertainty in damage initiation and propagation. The damage tolerance approach to structural integrity assumes that damage is present in the structure at all times and sufficient safety measures should be employed to ensure that it will not grow to a critical length during the operational life of the structure. The Federal Aviation Administration (FAA) requires that all structures designed for damage tolerance be demonstrated to avoid failure due to fatigue, environmental effects, manufacturing defects and accidental damage. It is easier to perform reliabilitybased structural optimization of safelife structures than of failsafe structures because the optimization of the former involves only structural sizes while for the latter the inspection regime also needs to be optimized. Nees and Canfield (1998) and Arietta and Striz (2000, 2005) performed safelife structural optimization of F16 wing panels to obtain the minimum structural weight for fatigue crack growth under a service load spectrum. For aircraft failsafe design, reliabilitybased design optimization has been applied to the design of inspection schedules. Harkness et al. (1994) developed an approximate method to evaluate reliability with inspections, Provan and Farhangdoost (1994) used Markovchains to estimate the failure probability of a system of components and Brot (1994) demonstrated that using multiple inspection types could minimize cost. Fujimoto et al. (1998); ToyodaMakino (1999); Enright and Frangopol (2000); Wu et al. (2000); Garbatov and Soares (2001) and Wu and Shin (2003) developed optimum inspection schedules for a given structural design to maintain a specified probability of failure. Wu and Shin (2005) developed a methodology to improve the accuracy of reliability calculations with inspections. Reliabilitybased optimization of the structural design and inspection schedules has also been applied to pipelines subjected to corrosion damage. Hellevik et al. (1999) optimized the pipeline thickness together with the inspection regime to minimize the total operational cost. Using Bayesian updating and a decision tree, they obtained the optimum inspection regime in times and types of inspection. The corrosion information obtained from the inspection was used to update the corrosion model and corrosion probabilities. Backman (2001) also used multiple inspection types to develop an optimum inspection schedule. However, he also considered the tradeoff between the cost of inspection and the cost of additional structural weight for maintaining the same probability of failure. Using an approximate relationship between structural weight and damage propagation, he concluded that increasing the structural weight is more cost effective than increasing the inspection frequency. Reliability based optimization is computationally very expensive when inspections are involved because crack size distribution has to be recharacterized after each inspection to simulate replacement. Exact computation using Monte Carlo simulation (MCS) is very expensive for estimating a low probability of failure due to the large sample size. Harkness (1994) developed a computational methodology to calculate reliability with inspections without updating the crack size distribution after each inspection. He assumed that repaired components will never fail again and incorporated this assumption by modifying the firstorder reliability method (FORM)." This leads to fast and accurate reliability computations that require only the specification of the initial crack size distribution. In previous papers (Kale et al., 2003, 2004), we used the same methodology to optimize the inspection schedule. When inspections are scheduled before half the service life, repaired components can have a large probability of failure. In this case, Harkness's method may not be accurate enough because the repaired components can fail during the remainder of the service life. In Chapter 3 we proposed and approximate method to simulate inspection and repair using Monte Carlo simulations (MCS) and calculate the failure probability using FORM. In this chapter we use the approximate method from Chapter 3 for combined optimization of structural design and inspection schedule using multiple inspection types. MCS is computationally very expensive for evaluating low failure t FORM is a moment based technique which calculates the failure probability using a first order approximation about the point on the limit state closest to origin and is computational very cheap compared to MCS probability, but is cheap for estimating probability distribution parameters (e.g., mean and standard deviation). We use the data obtained from MCS to approximate the probability distribution of crack size after inspection, and obtain the mean and standard deviation. Subsequently, FORM is used to calculate the failure probabilities between inspections. The combined MCS and FORM approach to calculate failure probability with inspection removes the computational burden associated with calculating the failure probability using MCS for low failure probabilities. The main objective of the present chapter is to use this methodology to optimize aircraft structural design together with inspection schedules using multiple inspection types. The optimization parameters are structural thickness, inspection times and inspection types. The application of the proposed methodology is demonstrated by using an example of an aircraft structure modeled as an unstiffened panel designed for fatigue. A second objective of this chapter is to demonstrate that if structures are designed together with the inspection schedule, then the cost of additional structural weight can be traded against the cost of additional inspections in order to achieve an overall minimum operational cost. Structural Design and Damage Growth Model Fatigue Crack Growth We consider fatigue crack growth in a fuselage panel with an initial crack size a, subjected to load cycles with constant amplitude. We assume that the main fatigue loading is due to pressurization, with stress varying between a maximum value of a to a minimum value of zero in one flight. One cycle of fatigue loading consist of one flight. Like many other researchers (e.g., Tisseyre et al., 1994; Harkness et al., 1994 and Lin et al., 2000), we assume that damage growth follow the Paris equation a= D(AK)m (41) dN where a is the crack size in meters, Nis the number of cycles (flights), da dNis the crack growth rate in meters/cycle, and the stress intensity factor range AK is inMPa meters. For 7075T651 aluminum alloy, D and m are material parameters related by Equation 42 obtained from Sinclair and Pierie (1990). D has units in meters 2 (MPa) m D= e(3.2m12.47) (42) The stress intensity factor range AK for a center cracked panel is calculated as a function of the stress a and the crack length a in Equation 43, and the hoop stress due to the pressure differential p is given by Equation 44 AK = .V. (43) =pr (44) t where r is the fuselage radius and t is the skin thickness. The number of cycles of fatigue loading Naccumulated in growing a crack from the initial crack size a, to the final crack aN can be obtained by integrating Equation 41 between the initial crack a, and the final crack aN Alternatively, the crack size aN after N cycle of fatigue loading can be obtained by solving Equation 45 m m "" da a 2 a 2 N= f (45) The fatigue life of the panel can then be obtained by substituting the critical crack length ac in place of aN in Equation 45 1 1 2 2 a a N, C (46) D'1 in)(J)m Here we assume that the critical crack length ac is dictated by considerations of crack stability, so that a z2e (47) and KIc is the fracture toughness of the material. Typical material properties for 7075 T651 aluminum alloy are presented in Table 41. A conservative distribution of initial defects was chosen following the Department of Defense Joint Service Specification Guide for aluminum alloys to account for uncertainties in damage initiation and growth. The applied fuselage pressure differential is 0.06 MPa, obtained from Niu (1990) and the stress is given by Equation 44. Table 41: Fatigue properties of 7075T651 Aluminum alloy Property Initial Paris Pressure, p Fracture Fuselage crack a, o, exponent, (MPa) toughness, Klc radius, r mm m MPa meters Meters Distribution Lognormal Lognormal Lognormal 36.58 3.25 type, mean, 0.2,0.07 2.97, 1.05 0.06, 0.003 Deterministic standard deviation (Source: Sinclair and Pierie, 1990 and Niu, 1990) Inspection Model When the structure is subjected to periodic inspections, cracks are detected and repaired or the structural part is replaced. We assume that the probability Pd, of detecting a crack of length a is given by Equation 48 (Palmberg et al., 1987) Pd (a) (a=ah (48) 1+(a/ah)Y Where ah is the crack size that will be detected with 50% probability and /f is inspection parameter. Values of ah of 0.63 mm, 0.80 mm and 1.27 mm were obtained from the probability of detection curves from Rummel and Matzkanin (1997) for eddy current inspection and ah of 2.00 mm was obtained from Tober and Klemmt (2000) for ultrasonic inspection. The three versions of eddy current inspections differ in terms of the instruments used and the number of operators inspecting the component. They all obtained the probability of detection curves by artificially machining cracks in panels and counting the number of times that they were detected after being inspected. The value of the other inspection parameters/, as shown in Figure 41, was obtained by fitting Equation 48 to the inspection data in these references. The probability of detection curves for different inspection types are shown in Figure 41. Type 1 is the most effective inspection followed by type 2, and so on. It is assumed that once a crack is detected, the panel is replaced by a newly manufactured panel with a smaller defect size distribution (fabrication defects). 0.8   Type 3 a = 0.8 mm 1.8 Rummel et al. 1997 S0.4 I  Type 1 a = 1.273 mm 3 = 2.0 Rummel et al. 1997 S0.1  0 1 2 3 4 5 7 9 10 crack length, mm Figure 41: Probability of detection curve for different inspection types from Equation 4 8 Calculating an Inspection Schedule Estimating Crack Size Distribution after Inspection When inspection and replacement of structural components are scheduled, the damage size distribution changes because defective parts are replaced with new parts having a smaller value of the damage size (fabrication defects). Reliabilitybased optimization is computationally very expensive when inspections are involved, because crack size distribution has to be recharacterized after each inspection to simulate replacement and exact computation using Monte Carlo simulation (MCS) is very expensive for estimating low probability of failure due to large sample size. Harkness (1994) developed a computational methodology to calculate reliability with inspections without updating the crack size distribution after each inspection. He assumed that repaired components will never fail again and incorporated this assumption by modifying the firstorder reliability method (FORM). Using this method, FORM is updated over the failure region after each inspection using numerical integration. This expedites the reliability computations, which require only that the initial crack size distribution be specified. When inspections are scheduled after half the service life, this method gives accurate results because the repaired component will not fail until the end of service. However when inspections are scheduled before half the service life, the repaired components can have a large probability of failure and Harkness's method may not be accurate enough compared to exact probability of failure obtained from MCS. Kale et al. (2005) developed an approximate method to account for inspection and repair using Monte Carlo simulation (MCS) and evaluated the failure probability using the firstorder reliability method (FORM). Evaluating low failure probability using MCS requires a large sample size, which makes reliabilitybased optimization expensive; instead MCS is used to estimate the mean and standard deviation of probability distribution using small sample size, which is computationally cheap. The data obtained from MCS are used to fit a probability distribution of crack size after inspection and then FORM is used to calculate the failure probabilities at any time following the inspection. This greatly expedites the reliability computations. Here we assume that inspections do not change the type of distribution and that damaged components are replaced by new components with damage distribution due to fabrication. The damage distribution after an inspection can easily be updated by using MCS with a small sample size. The crack size aN after N cycle of fatigue loading is obtained by solving Equation 45. To obtain the crack size mean and standard deviation after an inspection is conducted, we produce 100,000n random numbers for each random A large sample size was used to get accurate estimate of mean and standard deviation. This makes the optimization results insensitive to MCS seed. variable in Equation 45 (a,, m, a) and obtain the final crack size ay. We then simulate the inspection by using Equation 48 with another random number for probability of detection. If the crack is detected, the panel is replaced by a new one with a random crack size picked from the distribution of manufacturing defects a,o. After all cracks are analyzed for detection, the updated crack sizes are used to fit a distribution and to obtain its mean and standard deviation. This serves as the initial crack distribution for the next inspection. For the data used in this chapter, the fabrication crack distribution is lognormal, and the distribution after inspections was also found to be best approximated by lognormal distribution out of 12 analytical distributions in ARENA software (Takus and Profozich, 1997). If better accuracy is needed then a distribution with more parameters can be fitted to the data. Even though this distribution may not represent the data accurately, it provides a conservative fit to data. The algorithm for simulating crack growth and inspections is shown in Table 42. Table 42: Pseudo code for updating crack distribution after N cycles from previous inspection Generate a panel by a random vector of uncertain variables (a, ,m, a) Solve Equation 45 for crack size aN after N cycles of fatigue loading corresponding to the random vector (a,, m, a) Compute the probability of detection of crack aN from Equation 48, Pd (aN) Generate a random number from a uniform distribution with bounds (0, 1) drand Ifpd (aN) Pdand then simulate replacement of defective component by generating a random crack a,,o for a new panel and set a, = a, o else keep aN Store aN for fitting probability distribution to crack sizes after inspection Stop after 100,000 random panels have been simulated and fit distribution to crack sizes Calculating the Failure Probability Using the FirstOrder Reliability Method (FORM) It would be possible to use the same MCS procedure as described in the previous step in order to calculate the probabilities of failure needed for scheduling inspections. However, since the required probabilities of failure are of the order of 108, this would require a prohibitively large MCS. So instead we use FORM, taking advantage of the characterization of the crack distribution as lognormal, as described in the previous section and in Table 42. The probability of failure after N cycles of fatigue loading since the most recent inspection is Pf (N, a,)= P(a(N, a,)> a,) (49) where a, is the crack size distribution (either initial or updated) at the beginning of the inspection period and ac is the critical crack given by Equation 47. This probability is calculated by the firstorder reliability method (FORM). For a given structural thickness, optimum inspection times are obtained such that the probability of failure before the inspection is just equal to the maximum allowed value (Pfth, reliability constraint). The probability of failure decreases after the inspection, because cracks are detected and repaired. With the number of cycles of loading (flights), the failure probability increases until it hits the threshold value again, defining the next inspection. The nth inspection time S, is obtained by solving Equation 49 using a bisection method between the previous inspection time Sn1 and the service life S, (40,000 flights). To ascertain whether the number of inspections is adequate, the probability of failure at the end of service is calculated. If this failure probability is greater than the threshold value, additional inspections must be added. The combined MCS and FORM approach removes the computational burden associated with calculating the failure probability using MCS for very low failure probabilities. Time, cycle and flights are used interchangeably in this chapter because for one cycle of fatigue loading is equal to one flight in a fuselage and time is measured in number of flights. To demonstrate the application of the combined FORM and MCS method to calculate the failure probability we calculate the inspection time for a threshold reliability level of Pfth = 107 in Table 43 for a 2.48 mm thick panel. Calculating P(a(N, a,) > a,) using FORM in Equation 49 with a, = a,,o and solving for N, the first inspection time is 14569 flights. To update the crack size distribution after this inspection, crack growth simulation using the MCS pseudo code is performed with initial crack sizes a,,o and a crack growth time of 14569 flights. This gives the updated crack size distribution after the first inspection a, (mean = 0.264 mm, cov = 1.04). This serves as the initial crack size distribution for the second inspection. The second inspection time is obtained by solving Equation 49 using FORM with the updated initial crack size distribution a, (mean = 0.264 mm, cov = 1.04). This is continued until the failure probability at the end of service life is less than the specified value. Figure 42 illustrates the variation of the probability of failure with and without inspection. It can be seen that inspections are very helpful in maintaining the reliability of the structure. From Table 43 it can be seen that the first inspection interval is the largest. After the first inspection, damaged components are replaced with the same initial crack distribution (mean = 0.20 mm and cov = 0.35); however some cracks may have escaped detection. The fact that some cracks (larger than the initial cracks) may have escaped detection and will grow faster leads to smaller intervals. Table 43: Example 41, inspection schedule and crack size distribution after inspection (ah = 0.63 mm) for an unstiffened plate thickness of 2.48 mm and a threshold probability of 107 Number of Inspection Inspection interval Crack size distribution inspections time (flights) S, S,, after inspection (mean, (flights) mm, cov) 0   (0.200,0.35) 1 14569 14569 (0.264,1.04) 2 26053 11484 (0.271,1.11) 3 35576 9523 (0.245,1.10) * inspection times may ditter by a maximum value of 100 flights due to MCS seed; the corresponding error in probability calculation is negligible. 1.00E+00 1.00E02 1.00E04  1.00E06  1.00E08  1.00E10 1.00E12 1.00E14 1.00E16 1.00E18 1.00E20 5000 10000 15000 20000 45000  NoInspections mInspections Flights Figure 42: Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfth = 107 The previous example showed how an optimum inspection schedule can be developed for a single inspection type. The same procedure is followed for scheduling the inspection sequence with multiple inspection types. Here the probability of detection of each inspection type can be different and inspections are performed in the same order as specified in the sequence. If the specified reliability level cannot be maintained with the inspection sequence, then it is not feasible. Cost Model The cost associated with a change in the structural weight for aluminum and the fuel cost is taken from Venter (1998). He assumed a fuel cost of $0.89 per gallon and calculated that a pound of structural weight will cost 0.1 pound of fuel per flight. From this we calculate that a pound of structural weight will cost $0.015 in a flight for fuel. The structural weight is assumed to be directly proportional to the plate thickness and a pound of structural weight is assumed to cost $150 for material and manufacturing. Appendix D shows the details of material and fuel cost calculations. A typical inspection cost of about a million dollars was obtained from Backman (2001) and costs of other inspection types were adjusted such that the incentive for carrying less effective inspection decreases with the number of inspections. Thus, one inspection of the first type is more attractive than carrying two inspections of the second type, three inspections of the third type or four inspections of the fourth type. The structural design parameters for the B747 series aircraft obtained by Niu (1990) are also listed in Table 44. Following Backman (2001) the service life S1 is assumed to be 40,000 flights. Table 44: Design details and cost factors Density of aluminum p 166 lb ft Frame spacing b 0.6 m Fuel cost per pound per flight $ 0.015 F, Fuselage radius r 3.25 m Length / 68.3 m Material and manufacturing $ 150.0 cost per pound M, Number of panels Np 1350 Panel width h 1.72 m Type 1 inspection cost Ic, $ 1.35 million Type 2 inspection cost Ic2 $ 1.23 million Type 3 inspection cost Ic3 $ 0.98 million Type 4 inspection cost 1c4 $ 0.85 million (Source: Venter, 1998 and Backman, 2001) The life cycle cost is calculated as 4 C, =MW+FWS, +Z Ik (410) k=l W = N,thbp (411) During the optimization, the structural thickness t changes, this in turn changes the structural weight according to Equation 411. The optimum inspection schedule (times and types) is determined for this structural design and the total cost is obtained from Equation 410. Optimization of Inspection Types The combined optimization of inspection times to minimize the cost for the specified reliability constraint is complicated because of the large number of permutations of inspection types that can occur in an inspection schedule. To reduce the number of permutations that need to be considered, we first calculate the inspection schedule and the cost of all the single type inspections. We then use the lowest cost as a bound that allows us to eliminate many possible sequences. Appendix E provides a detailed description of the algorithm. (1) Determine optimum inspection times and costs using each of the four inspection types. This step provides (Nkb, Ckb), where Nkb is the number of required inspections of the kth type (that is if only type k is used) and Ckb is the total cost of an inspection schedule developed using only the kth inspection type. Determine the minimum cost, Cmin = Min(Ckb, k = 1...4). (2) Eliminate impossible or clearly suboptimal inspection sequences to seek a mix of inspection types with Nk inspections of type k. If we use more than one inspection type, the total number of inspections in the sequence should be at most equal to the number of inspections required by the least effective inspection in the sequence. Also, the total number should be at least equal to the number of inspections required by most effective inspection in the sequence min(Nkb)< Nk N, k 1 (413) I ck (3) Generate the cheapest inspection sequence satisfying Equations 412 and 413. (4) Generate the inspection times for the inspection sequence and check if the failure probability at the end of the service life is less than the specified reliability constraint. P, (S,) Pf, < 0 (i.e., whether the inspection sequence is feasible.) (5) Stop if the sequence is feasible otherwise generate the next cheapest inspection sequence and go back to step 4. Combined Optimization of Structural Design and Inspection Schedule Our objective is to determine the optimum combination of the structural thickness, inspection types and inspection times that lead to a minimum life cycle cost for maintaining the specified reliability level (Pfth) through the service life. To obtain the optimum thickness we first obtain the safelife thickness, which is the minimum thickness necessary to maintain the threshold probability of failure without any inspection. In order to determine whether additional inspections reduce cost, we do a one dimensional search on the thickness by reducing the thickness gradually and obtain optimum inspection schedule using the algorithm described in Appendix E until we get the optimum lifetime cost. SafeLife Design When inspections and maintenance are not feasible, safety can be maintained by having a conservative (thicker) structural design. To demonstrate this, we first obtain the safelife design required to maintain the desired level of reliability throughout the service life. Table 45 shows the safelife design. Table 45: Structural size required to maintain a specified reliability level without and inspection. Threshold Minimum Life cycle Structural % Increase probability required cost $ x weight, lb in cost of of failure, skin 106 improving Pfth thickness reliability (mm) by a factor of 10 10 4.08 25.42 33902  108 4.20 26.16 34880 2.91 109 4.24 26.34 35129 0.68 From Table 45, comparing the minimum thickness to that used in Example 41 (Table 4 3), we see that the safelife design must be very thick and heavy in order to maintain the required safety levels. Cost Effectiveness of Combined Optimization Next we demonstrate the effect of inspections on structural safety and operational cost. Inspections improve the reliability by detecting and removing cracks. If this effect is used to optimize the structural design together with the inspection schedule, then the structural weight could be traded against the inspection cost to reduce the overall operational cost. The results of combined structure and inspection optimization are shown in Table 46. Table 46: Optimum structural design and inspection schedule required to maintain specified threshold reliability level Threshold Optimum Optimum Optimum Minimu Cost factors probability plate inspection inspection m cost % y of thickness type time (flights) $ x 106 Fc Mc Ic failure, (mm) sequence Pfth 10 2.48 11, 14,13 14569,26023, 18.66 66 16 18 32706 10 2.54 11, 13, 11 14321,23780, 19.47 64 16 20 30963 109 2.66 Ii, 13, Ii 15064, 23532, 20.27 65 16 19 30023 From Table 46 we can see that if inspections are added, the structural thickness can be reduced to maintain the required reliability level at a lower cost. Inspections are very useful in maintaining the structural safety in that large cracks are detected and the damaged part is replaced with new components improving the fatigue life. In this chapter we optimize the inspection schedule for fatigue damage. However inspections are also used to detect other damage, such as tool drop, bird impact and corrosion, which makes them even more cost effective compared to the safelife design. The combined optimization of structural design and inspection schedule leads to tradeoff of the costs of structural weight against the inspection cost. Comparing Tables 4 5 and 46, we can see that adding inspection leads to a 25% saving in life cycle cost over the safelife design. Also, we can see that as the safety requirement becomes more stringent, additional and\or more effective inspections become worthwhile. For a safety level of 107, cheaper inspections can be used (14 and 13), whereas for 109 more effective inspections are useful. We can see that only a single inspection type may not be the best choice for maintaining different reliability levels. For maintaining a reliability level of 10', a structural size of 2.48 mm and three inspections of type one, four and three leads to minimum cost, but the same choice of inspection types is not suitable for a reliability level of 108. The last columns of Table 46 show the cost factors in percentage of fuel cost (Fe), manufacturing cost (Me) and inspection cost (IJ). It can be seen that the fuel cost is the major design driver and more inspections can be used to tradeoff cost if fuel cost increases. This issue is further explored below. Next we compare the optimum inspection schedule developed using only a single inspection type for the structural thickness obtained in Table 46 for a reliability level of 107 Table 47: Comparison of optimum inspection schedule using a single inspection type for a fixed structural size Threshold Optimum Optimum inspection type sequence using Minimum probability of plate a single inspection type cost failure, Pfth thickness 1i I2 13 14 $ x 106 10 2.48 14569, 14569, 14569, 14569, I : 19.17 26053, 24683, 19596, 18991, 35576 33430 29502, 25952, 35156 30128, ___________ ____38167 Inspection cost $ x 106 4.05 3.69 3.92 4.25 Table 47 shows the inspection schedules and cost for the inspection sequence generated using individual inspection types for a fixed structural size. Compar ing Tables 46 and 47, it can be seen that for a fixed structure, multiple inspection types can reduce cost. For a given structure, the advantage of multiple inspection types is partly driven by the fact that at the end of the service life, each inspection schedule leads to a different probability of failure. That is, the cost differential is partly due to different safety margins at the end of service. With combined structural and inspection optimization, the margin at the end of the service life is removed by a reduction in structural thickness. This leads to a smaller incentive for multiple inspection types, as shown in Table 48. Table 48: Optimum structural design and inspection schedule using only a single inspection type Threshold Optimum Optimum Optimum Minimum probability of plate inspection time inspection type cost failure, Pfth thickness (flights) $ x 106 10 2.39 13317, 18651, 13 18.78 26642, 32460 108 2.50 13971, 22897, 1i 19.65 31443 109 2.64 14975, 19642, 13 20.41 26230, 32670 Comparing Tables 46 and 48 we can conclude that mixing inspections lead to only a small improvement in cost over a single inspection type design (1%) when structural optimization is done with inspection scheduling. Effect of Fuel Cost Fuel cost has a large effect on the optimization of the structural design and inspection schedule. To demonstrate the effect of the increase in fuel cycle cost since 1998, we double the fuel cost to $1.8 per gallon or $0.03 per pound per flight. For the optimum design in Table 46, the fuel cost is about 60% of the total life cycle cost and inspections accounted for 20%. Optimization of the structural design and inspection schedule is performed for a reliability level of 107. The optimum plate thickness is 2.02 mm and a comparison of the optimum inspection schedule using different inspection sequences is shown in Table 49. Table 49: Optimum structural design (plate thickness of 2.02 mm) and inspection schedule for Pfh = 10 Inspection type Optimum inspection time (flights) Minimum sequence cost $ x 106 Ii 9472, 14383, 20204, 25583, 31192, 36623 30.71 I2 9472, 13431, 17315, 21659, 26191, 30784, 32.45 35359, 39917 13 9472, 11290, 17422, 20206, 25773, 29006, 30.45 34178, 37711 14 9472, 11001, 14575, 17130, 21277, 24231, 31.11 28099, 31225, 34907, 38178 Ii, 13, 13, 13, 13, 13, 13 9472, 14383, 29.84 (Optimum schedule) 18120, 22941, 26770, 31495, 35433 Comparing Table 49 to the first row of Tables 46 and 47 we can conclude that as fuel cost increases it becomes advantageous to schedule additional inspections and reduce weight to reduce the overall lifecycle cost. For the optimum structural design and inspection schedule in Table 49, the fuel cost is 66%, the manufacturing cost 8% and inspections are 26% of the total cost. It should be noted; however fatigue is not the only structural design driver, so that at lower thicknesses, other structural constraints may dominate. Summary A methodology for developing optimum inspection type sequences, time, and structural thickness was developed for fuselage panels. Uncertainty in material properties, crack sizes and loads were considered. The FORM method was used to determine the probability of failure at a given time and crack size distribution after inspection was updated using Monte Carlo simulation. Inspections and structural sizes were optimized so that a given threshold probability of failure was never exceeded. CHAPTER 5 EFFECT OF SAFETY MEASURES ON RELIABILITY OF AIRCRAFT STRUCTURES SUBJECTED TO FATIGUE DAMAGE GROWTH Introduction Safety of aircraft structures is largely maintained by using conservative design practices to compensate for the uncertainties in the design process and service usage. Typically, conservative initial defect specification, Abasis material properties, safety factor in fatigue life and conservative loads are used to design structures for fatigue crack growth. This is further augmented by quality control measures like certification testing and inspections. The main reason for using several layers of safety measures is the uncertainties involved (e.g., loading, material properties, accidental impact etc.). It has been shown that the life of a structure cannot be accurately determined even in carefully controlled conditions because of variability in material properties, manufacturing defects etc. (Sinclair and Pierie, 1990). Because of uncertainty in damage initiation and growth, a damage tolerance approach to assuring structural integrity has become popular in aerospace applications. Here we assume that damage is present in the structure at all times in the form of cracks and we make sure that these cracks will not grow to a critical length during the operational life before they are detected and removed by inspections. Inspections are scheduled at fixed time intervals to detect cracks and protect against unmodeled damage and unexpected accidental damage. The Federal Aviation Administration (FAA) requires that all structures designed for damage tolerance should be shown to avoid failure due to fatigue, manufacturing defects and accidental damage. As an example, we study here the design of fuselage structures for fatigue failure due to pressure loading and compute the service failure probability and the effect of safety measures and certification tests. Design of fuselage structures for fatigue is described in detail in Niu (1990). Ahmed et al. (2002) studied the initiation and distribution of cracks in fuselage panel by performing fatigue testing. Structural optimization and inspection scheduling of fuselage structure subjected to fatigue damage has been studied in our previous papers (Kale et al., 2003, 2004). Vermeulen and Tooren (2004) designed fuselage structures for fatigue crack growth and found that damage growth and residual strength were two main design drivers. Widespread fatigue damage in fuselage structure has been investigated by testing panels according to FAA regulations (Gruber et al., 1996, FAA/AR95/47). The combined effects of various safety measures used to design structures for static strength were studied by Acar et al. (2005). They studied the interaction of uncertainties, safety factors and certification tests on safety of aircraft structures designed for static strength. As in earlier studies, the effect of variability in geometry, loads, and material properties is readily incorporated here by the appropriate random variables. However, there is also uncertainty due to lack of knowledge epistemicc uncertainty), such as modeling errors in the analysis. 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. The objective of this chapter is to study the effect of individual safety measures used to design structures for fatigue damage growth. The effectiveness of certification testing as a function of magnitude of safety factors and errors is also explored. Classification of Uncertainties A good analysis of different sources of uncertainty is provided by Oberkampf et al. (2002). Here we simplify the classification, with a view to the question of how to control uncertainty. The classification in Table 51 distinguishes between (1) uncertainties that apply equally to the entire fleet of an aircraft model and (2) uncertainties that vary for individual aircraft. The distinction is important because safety measures usually target one or the other. While type 2 are random uncertainties that can be readily modeled probabilistically, type 1 are fixed for a given aircraft model (e.g., Boeing 737400) but they are largely unknown. For failure of a structural member this classification reflects distinction between systemic errors and variability. Systemic 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 fleetlevel uncertainties. They can reflect differences in analysis, manufacturing and operation of the aircraft from an ideal. The ideal aircraft is 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 the operating environment agrees on average with the design specifications. The other type of uncertainty reflects variability in material properties, geometry, or loading between different copies of the same structure. Table 51: Uncertainty classification Type of Spread Cause Remedies uncertainty Systemic Entire fleet of Errors in predicting Testing and error components structural failure. simulation to (Modeling designed using the improve math errors) model model. Variability Individual Variability in tooling, Improve tooling component level manufacturing process, and construction. and flying environment. Quality control. In this chapter, we focus on design of fuselage structures for fatigue damage growth due to pressure loading. To simulate the effect of these modeling errors we pretend that there are several aircraft companies designing the same airplane but each having some different stress calculations and crack growth model leading to different designs. Because these models are only approximate, the stress and crack growth rates calculated by the companies for structural design will be different from the true stress and true crack growth rate. We account for this difference between the "true" and "calculated" value by model error factors k and e for stress and crack growth, respectively. Following this, we calculate the failure probability by selecting random value of errors k and e fixed for an aircraft company. Safety Measures Aircraft structural design is still done largely using codebased 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 certification tests that can reveal inadequacies in analysis or construction. In this section, we detail some of these safety measures listed in FAR 25.571 for civil and transport airplanes and in the Department of Defense, Joint Services Specification Guide2006 (JSSG). Safety Margin on Load. Aircraft structures should be demonstrated to withstand 1.5 times the limit load without failure. For damaged fuselage structure, it should be demonstrated by tests that the structure has enough residual strength to withstand 1.15 times the differential pressure. Conservative Material Properties. In order to account for uncertainty in material properties, the FAA recommends 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 Abasis property is defined as the value of a material property exceeded by 99% of the population with 95% confidence, and the Bbasis property is that exceeded by 90% of the population with 95% confidence. For structures without redundancy, Abasis properties are used and for structures with redundancy, Bbasis properties are used. The conservative material properties considered here are ABasis values of crack growth exponent (leading to rapid crack propagation), yield stress Y and fracture toughness Kic. Appendix H describes the methodology for obtaining these properties from coupon tests. Conservative Initial Crack. The FAA requires all damage tolerant structures to be designed assuming initial flaw of maximum probable size that could exist because of manufacturing or fabrication. The JSSG2006 damage tolerance guidelines specify this value as the Bbasis crack size. This is the value that can be detected by an eddy current inspection with a 90% probability and 95% confidence. Safety Factor on Life. Typically, transport aircraft are designed with a safety factor of two on the fatigue life. Fatigue tests are conducted to validate the structural design and the FAA requires that no aircraft be operated for more than half the number of cycles accumulated on a fullscale fatigue test. Inspections. Inspections provide protection against structural failure by detecting damage. The FAA requires that inspection schedule should be in place to detect and repair damage before it grows to unacceptable size causing structural failure. The JSSG 2006 damage tolerance guidelines specify that failsafe multiple load path structures should be designed for depotlevel inspections every one quarter of service life. Component and Certification Tests. Component tests and certification tests of major structural components reduce stress and material uncertainties due to inadequate structural models. These tests are conducted in a building block procedure. First, individual coupons are tested, and then a subassembly is tested followed by a fullscale test of the entire structure. Since these tests cannot apply to every load condition, they leave uncertainties with respect to some load 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. Certification testing may be conducted in two ways. A panel may be randomly chosen out of a batch and subjected to fatigue test. Alternatively, the panel may be chosen and a larger crack can be machined so that tests become more conservative. For the latter, we simulate the panel with the Bbasis value of crack size, which is also used to design the panel. A summary of fatigue testing of fuselage panels is documented in FAA/AR95/47 (Gruber et al., 1996). Simulation Procedure for Calculation of Probability We simulate the effect of these safety measures by assuming statistical distributions of uncertainties and incorporating them in 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, etc.), with the errors being fixed for each aircraft. This creates a twolevel Monte Carlo simulation, with different aircraft models being considered at the upper level, and different instances of the same aircraft at the lower level. (A) Select random errors k and e, create a new design Failed (B) Perform certification test Reject design I Passed (C) Build a copy of the aircraft and apply service loads (D) Check if the aircraft fails under the service load and count the number of failed aircraft (E) Check ifMcopies of aircraft are built N Y (F) Check ifNVdifferent designs are created Y (C) Calculate probability of failure Figure 51: Flowchart for Monte Carlo simulation of panel design and failure We consider a generic structural component characterized by thickness t. The random error parameters k and e account for the difference between the "true" and "calculated" value of stress and crack growth, respectively, and are fixed for each aircraft manufacturing company. We have N different aircraft companies designing essentially the same airplane but with different errors in their calculations. During the design process, the only random quantities are the conservative material properties and the model errors k and e. In the outer loop of the Monte Carlo simulation (Figure 51, step A) we generate different values of k and e and obtain the structural design tdesign for each company. This is the average thickness of the structure built by the company. The actual structural thickness tact and material properties will be different because of manufacturing variability. We simulate the effect of this variability by producing M instances of the design obtained in step A (Figure 51) from assumed statistical distributions for material properties and thickness distribution. Next we simulate certification testing in step B of Figure 51 by selecting a structure from each company and testing if it fails before the design life. If the design passes the test, each of the M structures produced by that aircraft company is assumed to be used in actual service. The failure probability is calculated by applying actual random service loads in step D (Figure 51) and counting the number of failed structures. Damage Growth Model The typical structural design process involves determining structural sizes that would be sufficient to meet given strength and crack growth requirements during the design life. Prototypes of these designs are then tested for fatigue crack growth by applying design loads in a test rig. These tests continue for several months until structural failure occurs. To illustrate the procedure we consider a simple example of fuselage structure modeled as unstiffened panel and designed for fatigue under uniaxial tensile loads. The stress varies from a maximum value of a to a minimum value of zero in one flight. One cycle of fatigue loading consists of one flight. To model fatigue, we assume that crack growth follows the Paris model. Equation 51 represents the rate of crack growth da/dN in terms of stress intensity. da = D(AK)m (51) dN cacd Nda where a is the crack size in meters, Nis the number of cycles (flights), is the crack dN growth rate in meters/cycle, and the stress intensity factor range AK is in MlPa/meters For 7075T651 aluminum alloy, D and m are material parameters related by Equation 52 _m obtained from Sinclair and Pierie (1990). D has units in meters 2 (MPa) m D= e(3.2m12.47) (52) We use the subscript "calc" to note that relations, (such as Equation 51), that we use in the analysis, provide only approximations to true values. For a center cracked infinite panel with far field stress a and one cycle of fatigue loading per flight we have AK= caIcm (53) Equation 51 is integrated to obtain estimated fatigue life Na"'e 1m 1 a 2 2 Ncac = a a (54) f D 1 m ); icmalc where the computed stress ca/c (as obtained from finite element analysis) is different from the actual stress atrue in the structure. Here a, is the initial crack size and the critical crack length ac is the crack length at which crack growth becomes unstable. The critical crack length can be obtained as a function of fracture toughness Kic and a from Equation 55. ac < 2 (55) C 4r Equation 51 represents an approximate value of crack growth rate because it is obtained by fitting empirical model to observed crack growth data and calculated stress ai. The true crack growth rate is different from that estimated by Equation 51. We include and error factor e in analysis and then Equations 51 and 54 become.  =eD(AK) (56a) dN true m m 1 1 The actual stress atrue in the structure due to applied loading is different from the calculated stress ae used to design the structure. Equation 57 represents the error in the calculated stress, through an error parameter k. N true = a+ k) calc (57) Values of k and e greater than the mean values (0 for k and 1 for e) yield conservative estimates of the true stress and fatigue life and those less than the means yield unconservative estimations. Table 52: Distributio m (Source: Sinclair and Piene, 1990 and Niu, 199U. A lognormal distribution tor error e in crack growth rate is chosen to reflect the lognormal distribution of crack sizes used in literature (e.g., Harkness, 1994 and Rahman and Rice, 1992). Uniform distribution for k is chosen to reflect lack of information) Table 52 lists uncertainties in form of errors and variability in the life prediction and structural design model assumed here for 7075T6aluminum alloy. Typical service life of 40,000 flights is obtained from Backman (2001). In this chapter we demonstrate the effect of safety factors on two design criteria, (i) safelife: structure is designed for safe crack growth for the entire service life of 40,000 flights; no inspections are performed (ii) failsafe: structure is designed for safe crack growth until the next inspection (10,000 flights). The typical inspection interval of 10,000 flights was obtained from JSSG2006. Variables Distribution Mean Coefficient of variation, (standard deviation/ mean) a,, initial crack Lognormal 0.2 mm 35 % e, error in crack growth Lognormal 1.0 Variable k, error in stress Uniform 0.0 Variable Krc, fracture toughness Lognormal 30.5 MPam5 10% m, paris exponent Lognormal 2.97 17% N1, service life Deterministic 40,000 flights  Ns, design life Deterministic 10,000 flights  p, pressure load Lognormal pd 8.3 psi (0.057 2.5 % MPa) r, fuselage radius Deterministic 3.25 m  SFL safety factor on life Deterministic 2  SF, safety factor on load Deterministic 1.5  tact, actual thickness Lognormal design, mm 3% Y, yield stress Lognormal 495 MPa 5% Calculating Design Thickness This section determines the design thickness calculation in step A of Figure 51. The calculated stress o,,calc on the structure is found from Equation 58 representing hoop stress due to pressure loading. Ucalc = (58) where C is a function of the geometry, p is applied pressure differential and t is structural thickness. In subsequent calculations for stress in fuselage components in hoop direction, the parameter C was approximated for convenience by the value for a cylindrical pressure vessel. C = r (59) where r is the radius of fuselage. Combining Equations 57 and 58, the stress in the structure is calculated as U,., =(+ k) (510) To design a panel for fixed life we first obtain the stress level required to grow the initial crack a, to critical crack ac during the design life. This is obtained by solving Equation 5 6b with all the safety measures. The safety measures considered are using mA, the A Basis value of m, aB the Bbasis value of a,, the conservative value of critical crack acA obtained using Abasis value of Kic in Equation 54, ABasis value of yield stress, and SFL of 2.0, the safety factor on fatigue life. *** The actual stress may be somewhat different, but for the purpose of this chapter it is only important to model the inverse relationship between stress and thickness. SFLNseDrl mA 1 1 2 =2A 0 (511) f fai( KA Ofaugue [B Solving Equation 511 for daIgue gives the stress that should not be exceeded so that the fatigue life does not fall below twice the service life. Further, 1.5 times the maximum load in any component should not exceed the yield stress of the material to prevent static failure. We assume that the ABasis property of yield stress is used to design the structure for static strength. _yield _yield /  design = ABasis (512) Equations 511 and 512 give two different values of allowable design stress. Also the FAA requires that the damaged structure should have sufficient residual strength to withstand 1.15 times the limit load without failure. We assume that the net section area does not reduce below 80% until the crack is detected and repaired. The design thickness is determined so that all the three criteria are satisfied. For the undamaged structure, a safety factor SF = 1.5 on design load is also specified, then structural thickness is designed with a design load pd multiplied by a safety factor SF, hence the design thickness of the structural is calculated from Equation 513 as (I + k) d" fatigue design = max (1 + k1.5 (513) design max ( + k ) yield Design (l + k) 1.15 pdr 0.80 yed design For the example considered here, the second and third components of Equation 5 13 are less critical than the first. The thickness obtained from Equation 513 is the average thickness for a given aircraft model. The actual thickness will vary due to individuallevel manufacturing uncertainties, which are incorporated in calculation of failure probability using Monte Carlo simulation. Calculating Failure Probability Certification Testing After the structure has been designed (that is, design thickness determined from Equation 513), we simulate certification testing for the aircraft. We assume that the structure will not be built with complete fidelity to the design due to manufacturing and fabrication variability. To check if the structure is fit for use, we conduct two tests in step B of Figure 51 (i) randomly selected structural design from each aircraft company is fatigue tested for pressure differential equal to the design load pd for twice the design life (ii) another random structural design from the same company is loaded with 1.5 times the limit static load to check certification passage if the following inequalities are satisfied m m 1 1 2 a. 2 Fatigue certification test: SFLNs ac > 0 (514a) DI I mPdr 2 tact Static certification test: o of = dr o > 0 (514b) tact where the actual thickness tact is uniformly distributed with mean equal to design and 3% bound. If the structural design fails either test than that design is rejected. Here the thickness t, initial crack a,, fractures toughness KIc, yield stress of and Paris model constant m are random variables (see Table 52). This procedure of design and testing is 