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## Material Information- Title:
- Interaction of Conservative Design Practices, Tests and Inspections in Safety of Structural Components
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- Aircraft ( jstor )
Aircraft design ( jstor ) Corrosion ( jstor ) Design optimization ( jstor ) Fatigue ( jstor ) Material properties ( jstor ) Modeling ( jstor ) Safety factors ( jstor ) Standard deviation ( jstor ) Structural design ( jstor )
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- University of Florida
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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 NAG1-02042 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 First-Order 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 First-Order Reliability Method (FO RM )................................................................................. 53 Cost Model .....................................................57 O ptim ization of Inspection Types ................... ... ............. ...............58 Combined Optimization of Structural Design and Inspection Schedule....................59 Safe-Life 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 3-1 Fatigue properties of 7075-T651 Aluminum alloy ................................... 22 3-2 Structural design for fuselage................................................... 22 3-3 Pseudo code for updating crack size distribution after N cycles from previous inspection .......................................................27 3-4 Example 3-1, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 107 ........31 3-5 Cost of inspection, material and fuel............. ............. .........................32 3-6 Description of response surface approximations used in optimization....................33 3-7 Computational time spent in exact calculation of next inspection time and error due to -RSA usage ........... ...... ................................34 3-8 Pseudo code for combined optimization of structural design and inspection schedule ...................................... .................................. ........ 36 3-9 Safe-Life design of an unstiffened panel................. ............................ ...... 37 3-10 Safe-Life design of a stiffened panel ........................................... 37 3-11 Optimum structural design and inspection schedule of an unstiffened panel ..........38 3-12 Optimum structural design and inspection schedule for stiffened panel...............39 3-13 Optimum structural design for regulations based inspections conducted at four constant interval or 8000 flights for stiffened panel ................................... 40 3-14 Tradeoff of inspection cost against cost of structural weight required to maintain fixed reliability level for stiffened panel ..........................................41 3-15 Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel w ith inspection................................................. 42 4-1 Fatigue properties of 7075-T651 Aluminum alloy ................................... 49 4-2 Pseudo code for updating crack distribution after N cycles from previous inspection .......................................................53 4-3 Example 4-1, 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 4-4 Design details and cost factors .............................. .........57 4-5 Structural size required to maintain a specified reliability level without and inspection. ........................................................60 4-6 Optimum structural design and inspection schedule required to maintain specified threshold reliability level .............................. ............... 61 4-7 Comparison of optimum inspection schedule using a single inspection type for a fixed structural size .......................................... ...... ........ 62 4-8 Optimum structural design and inspection schedule using only a single inspection type...................... ....................... 63 4-9 Optimum structural design (plate thickness of 2.02 mm) and inspection schedule for Pfth = 10-7 . . ...................... . ............ . ................ 64 5-1 Uncertainty classification.............. ... ........ ..... ........68 5-2 Distributions of errors, design and material parameters for 7075-T6 aluminum.....75 5-3 Nomenclature of symbols used to calculate failure probability and describe the effect of certification testing ................. .............. ................80 5-4 Probability of failure for 10 % COV in e and different bounds on error k using all safety measures for fail-safe design for 10,000 flights ......................................81 5-5 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for fail-safe design for 10,000 flights ......................................... 82 5-6 Probability of failure for 10 % COV in e for different bounds on error k using all safety measures for safe-life design of 40,000 flights................... .............82 5-7 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for safe-life design of 40,000 flights...................................82 5-8 Probability of failure for different bounds on error k for 10 % COV in e without any safety measures for fail-safe design for 10,000 flights..................................84 5-9 Probability of failure for different bounds on error k for 50 % COV in e without any safety measures for fail-safe design for 10,000 flights..................................84 5-10 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for fail-safe for 10,000 flights and tested using a m machine cracked panel ............................................................. 86 5-11 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for safe-life of 40,000 flights and tested using a m machine cracked panel ............................................................. 86 5-12 Probability of failure for different bounds on k and 10 % COV in e for structure designed with all safety measures for fail-safe for 10,000 flights and COV in material property m reduced to 8.5% .......................................................87 5-13 Probability of failure for different bounds on k and 50 % COV in e for structures designed with all safety measures for fail-safe criteria for 10,000 flights and COV in material property m reduced to 8.5%................. .......................87 5-14 Probability of failure for different bounds on k and 10 % COV in e for structures designed using only A-Basis m for fail-safe criteria for 10,000 flights ...................88 5-15 Probability of failure for different bounds on k, 10 % COV in e for structure designed using conservative properties for fail-safe design for 10,000 flights........89 5-16 Probability of failure for different bounds on k, 50 % COV in e for structures designed using conservative properties for fail-safe criteria for 10,000 flights.......89 5-17 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 6-1 Typical wet gas pipeline flow parameters..............................................................99 6-2 Typical wet gas pipeline corrosion growth parameters............... ... ...... 101 6-3 Updating of model weights given assumed observations corresponding to input component models................ .. .... .. ............ 107 6-4 Inspection locations along pipeline ....................... ...............109 C-i Bounds on design variables used to evaluate response surface approximation for safe life design..................... .............. .......... 129 C-2 Error estimate of analysis response surfaces used to obtain safe-life stiffened panel design ............... .... .................. ......... 130 C-3 Bounds on design variables used to evaluate response surface approximation for inspection based design ............... .... ......... ......................130 C-4 Error estimate of analysis response surfaces used to obtain inspection based stiffened panel design ...................... .. .... ....... .. .. .... .......... 131 C-5 Error estimate of design response surfaces .....................................131 C-6 Bounds on design variables used to evaluate response surface for crack sizes parameters after inspection and reliability index ...................................... 132 C-7 Error estimate of crack size response surfaces used to estimate the crack size distribution parameters after the first inspection............................................133 C-8 Error estimate of crack size response surfaces used to estimate the distribution after inspection ................. .... ...................... 133 C-9 Error estimate of reliability index response surfaces used to schedule first inspection ......................................................133 C-10 Error estimate of reliability index response surfaces, /fd-RSA............................. 133 D-1 Area of structural dimensions for cost calculation.................... ............... 135 F-i 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 10-7 141 G-1 Uncertainty classification................................. ........................ .. ............147 G-2 Distribution of random variables used for panel design and certification .............152 G-3 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 G-4 Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and A-basis property for allowable stress..... ......... 158 G-5 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 G-6 Probability of failure for different bounds on error e for safety factor of 1.0 and A-basis property for allowable stress .......................................162 G-7 Probability of failure for different error bounds for panels designed using safety factor of 1.0 and mean value for allowable stress............. ...........163 G-8 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 50% error bounds e and A-basis property for allowable stress ........................................................................ ........ 164 G-9 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 30% error bound e and A-basis properties.............................164 G-10 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 10% error bounds e and A-basis properties .........................165 LIST OF FIGURES Figure page 3-1 Fuselage stiffened panel geometry and applied loading in hoop direction ..............20 3-2 Comparison of actual and lognormally fitted CDF of crack sizes after an inspection conducted at 9288 flights.................................................. .............29 3-3 Example 3-1, Variation of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10 ...................31 4-1 Probability of detection curve for different inspection types from Equation 4-8 ....51 4-2 Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfh = 107 ..................................56 5-1 Flowchart for Monte Carlo simulation of panel design and failure.........................71 6-1 Uncertainty in inclination and critical angle ................. ................. ..........101 6-2 Probability of water formation along pipe length with highest probability observed at location 971 ................................ .........................105 6-3 Probability of corrosion depth exceeding critical depth along pipe length assuming water is present at all locations .........................................105 6-4 Total probability of corrosion exceeding critical depth along pipe length.......... 106 A-i Half-geometry of a center cracked stiffened panel with a central broken stiffener and two intact stiffeners placed symmetrically across from crack....................115 A-2 Description of applied stress and resulting fastener forces and induced stress on stiffened panel ........... ............. ...................... ............116 A-3 Description of position coordinate of forces and displacement location with respect to crack centerline as y axis.................................. ...............117 A-4 Description of position coordinate of forces and induced stress distribution along the crack length .................. .... ...... ................119 A-5 Comparison of stress intensity factor for a panel with skin thickness = 2.34 mm and stiffener area of 2.30 x 10-3 meter2............. ........ ...................124 A-6 Comparison of stress intensity factor for a panel with skin thickness =1.81 mm and stiffener area of 7.30 x 10-4 meter2.............. .......... ... .................... 125 B-i 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 F-i 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 F-2 Comparison of failure probability (1- CDF) of two probability distributions with mean 10-5 and standard deviation of 2 and 10 units..............................................143 G-1 Flowchart for Monte Carlo simulation of panel design and failure .......................151 G-2 Initial and updated probability distribution functions of error factor e..................... 155 G-3 Design thickness variation with low and high error bounds ...............................162 G-4 Influence of effective safety factor, error, and variability on the probability ratio (3-D view) ........................................................ 167 G-5 Influence of effective safety factor, error and variability on the probability ratio (2-D contour plot)................... ............................ ......... .. .. ..... .............. 167 G-6 Influence of effective safety factor, error and variability on the probability difference (3-D view) ............... .... ........................................ .. ....... 168 G-7 Influence of effective safety factor, error and variability on the probability difference (2-D 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, MPa-meter 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. 3-1 MA, = Average bending moment between the ith and i-1st fastener, N-meter 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 o-a-RSA = 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 first-order 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 re-characterized 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), first-order 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 reliability-based 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, freeze-thaw 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: safe-life and fail-safe. Niu (1990) and Bristow (2000) have characterized the safe-life and fail-safe design methodologies in that, reliability of a safe-life 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 fail-safe design is maintained by means of design for damage containment or arrestment and alternative load-paths that preserve limit-load capabilities. These mechanisms are complemented with periodic inspections and repairs. Bristow (2000) provided historical insight on the evolution of structural design philosophy from safe-life in the early 50's to damage-tolerance 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 safe-life and fail-safe 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 safe-life structures. Wing skin-stringer and fuselage skin-stringer panels have a substantial fatigue life and usually offer structural redundancy, so they are designated as fail-safe 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 center-of 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 stress-fatigue life (S-N curve) to estimate fatigue life. The Palmgren-Miner 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 S-N 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 code-based design rather than probabilistic approaches. Safety is improved through conservative design practices that include use of safety factors and conservative material properties. It is also improved by tests of components and certification tests that can reveal inadequacies in analysis or construction. These safety measures listed in FAR 25 for civil and transport category airplanes and Joint Service Specification Guide-2006 (JSSG). Use of large safety measures increases the structural weight and operational cost. The main complexity for designing damage tolerant structures via safe-life 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 reliability-based 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 corrosion-fatigue 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 (2-1) 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 first-order 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 -- (2-2) r N The accuracy of the probability calculated from Equation 2-2 increases with the number of simulations. An estimate of the accuracy in failure probability is obtained by calculating the standard deviation in Pf \Pf-Pf Pf= N (2-3) First-Order 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) (2-4) 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 second-order 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. Re-characterizing 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 first-order reliability method (FORM) and second-order reliability method (SORM) have been used to obtain probability after inspection by Rahman and Rice (1992); Harkness (1994); Fujimoto et al. (1998); Toyoda-Makino (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 F-16 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 re-characterized 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, fail-safe-straps 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)" (3-1) 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 (3-2) 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 3-1 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 3-3. This requires repeated calculation of V/ as the crack propagates. The computational approach for integrating Equation 3-3 is illustrated in Appendix B. aN da N= (3-3) 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 3-4 and crack grows perpendicular to the direction of hoop stress given by o + a=- (3-4) 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 3-1). t _ 1.72 - Crack 0.6 2a L IF S Applied I I load Figure 3-1: 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 fail-safe stiffening members in circumferential direction such as frames, fail-safe 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 through-the-thickness center cracks with three stiffeners on either sides of centerline as show in Figure 3-1. 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- ) (3-5) Equation 3-5 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\ (3-6) acL = (3-7) 2t ) Equation 3-6 determines the critical crack length for failure due to hoop stress a and Equation 3-7 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 3-8 and the fatigue life Nfof structure is determined by integrating Equation 3-3 between the initial crack a, and critical crack ac. ac= min(acy,acHacL) (3-8) Typical material properties for 7075-T651 aluminum-alloy most commonly used in aerospace application are presented in Table 3-1. 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 Guide-2006 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 3-2. Table 3-1: Fatigue properties of 7075-T651 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 3-2: 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 (3-9) where the fatigue life Nfis determined by integrating Equation 3-3 between the initial crack a, and the critical crack a,. The failure probability corresponding to Equation 3-9 is calculated using the first-order 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 3-9 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 ) (3-10) 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 (3-11) 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 3-11 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 3-3 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) (3-12) 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 3-13. P(N (a,m,)-N)- Pfh = 0 (3-13) Equation 3-13 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 3-3. 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 3-3. To obtain the crack size mean and standard deviation after an inspection, we produce 50,0 random numbers for each random variable in Equation 3-3 (a,, m, a) and obtain the final crack size aN. We then simulate the inspection by using Equation 3-11 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 3-3: 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 3-3 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 3-11, 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 3-3 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 3-2. The corresponding p-value is less than 0.005 indicating a bad fit; however for low failure probabilities (e.g., 10-7) 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 10-7 which are close to the value of 10-7 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 3-2: 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 3-1 illustrates the approach described in previous sections for a 2.0 mm thick unstiffened plate and a required reliability level of 10-7. Solving Equation 3-13 for N, the first inspection time is 9288 flights. Crack growth simulation using the MCS pseudo code in Table 3-3 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 3-13. This cycle of scheduling inspections is continued until the failure probability at the end of service life is less than the specified value. Figure 3-3 illustrates the variation of the probability of failure with and without inspection. Table 3-4 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 3-4 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 3-4 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.E-01 1) 5000 10000 15000 0000 35000 40000 45000 1.E-03 - S- No Inspections ~ 1.E-05 ---Optimal Inspections 1.E-07 - 0 2 1.E-09 - U2 i.E-11 - 1.E-13 - 1.E-15 - Flights Figure 3-3: Example 3-1, Variation of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7 Table 3-4: Example 3-1, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 10-7 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 3-5. Table 3-5: 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 (3-14) Where Wis the total weight of all the panels in the fuselage, given as W = N, (N,A,b + thb )p (3-15) The parameters in Equations 3-14 and 3-15 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 3-6 explains the various RSA's used to make computations faster and Table 3-7 gives the breakdown of computational cost for calculation of exact inspection time and updating crack size distribution. Table 3-6: 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/d-RSA Reliability index Skin thickness ts, Stiffener area As, mean crack length pua, mean crack length oa, time N, standard deviation in stress up Table 3-7: 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 Pd-RSA 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 3-7 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 10-7. maintain the specified reliability level. To overcome this we develop surrogate models based on response surface approximations shown in Table 3-6. The last columns of Table 3-7 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 3-8 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 3-8: 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 3-7. 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 3-3 and the computational expense associated with it is described in Table 3-7. 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 o-a-RSA 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 3-15. The optimum inspection schedule is determined for this structural design using Table 3-8 and the total cost of structural weight and inspection is obtained from Equation 3-14. 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 3-9 shows the safe life design of unstiffened panel and Table 3-10 shows the safe-life design of a stiffened panel. Table 3-9: Safe-Life 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 10-8 4.20 26.16 34880 10- 4.24 26.34 35129 Table 3-10: Safe-Life 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 10-3 (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 10-9 2.30 2.35 36.22 22.91 30555 An unstiffened panel is a single load path structure without load transfer capability. Comparing Table 3-9 to Table 3-10, 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 3-11. Table 3-11: 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 10-9 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 safe-life unstiffened panel design and by 20% over to stiffened panel safe-life 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 3-12) and illustrate the tradeoff of structural weight in skin and stiffeners against inspection cost. Table 3-12: 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 10-4 ,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 3-10 to Table 3-12 we see that inspections lower the life cycle cost of stiffened panel design by about 20% compared to safe-life design. Comparing Tables 3-11 and 3-12 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 3-12 to Table 3-11, 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 Regulations-2006 requires all airlines to conduct major depot level inspection four times during the service life. These inspections are conducted at uniform intervals. Table 3-13 shows the design with fixed inspection schedule. Table 3-13: 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 10-4 ,t (mm) Ctot meter2 $x 106 10-7 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 10-9 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 3-11, 3-12 and 3-13 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 3-10 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 3-14: 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 10-4 0 meter2 10-7 5 7.05 1.60 20.26 9497,16029,22064, 17.53 28060,34036 10-7 4 7.11 1.71 19.40 10844,18625,25791, 17.20 32908 10-7 3 7.23 1.88 18.14" 12743,22435,31212 17.35 10-8 5 7.00 1.70 19.18 9933,16406,22363, 18.14 28271,34145 10-8 4 7.30 1.81 18.95 11089,18758,25865, 17.87 32943 10-8 3 13.29 1.63 32.04 12514,22178, 31110 18.03 10-9 5 7.50 1.74 19.92 10091,16428,23260, 18.53 29268, 34412 10-9 4 7.89 1.88 19.51 11546,19064,26064, 18.59 33044 10-9 3 13.74 1.67 32.29 12699,22289,31163 18.33 From Table 3-14 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 3-14. 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 3-15 presents the exact evaluation of failure probability without any RSA for the optimum obtained from Table 3-12. This is done by calculating reliability using FORM without using fld-RSA reliability index. Table 3-15: 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 10-9 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 RELIABILITY-BASED 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 fail-safe 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 reliability-based structural optimization of safe-life structures than of fail-safe 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 safe-life structural optimization of F-16 wing panels to obtain the minimum structural weight for fatigue crack growth under a service load spectrum. For aircraft fail-safe design, reliability-based 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 Markov-chains 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); Toyoda-Makino (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. Reliability-based 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 re-characterized 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 first-order 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 (4-1) 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 7075-T651 aluminum alloy, D and m are material parameters related by Equation 4-2 obtained from Sinclair and Pierie (1990). D has units in meters 2 (MPa) m D= e(-3.2m-12.47) (4-2) 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 4-3, and the hoop stress due to the pressure differential p is given by Equation 4-4 AK = .V. (4-3) =pr (4-4) 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 4-1 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 4-5 m m "" da a 2 -a 2 N= f (4-5) The fatigue life of the panel can then be obtained by substituting the critical crack length ac in place of aN in Equation 4-5 1- 1- 2 2 a -a N, C -(4-6) D'-1 in)(J)m Here we assume that the critical crack length ac is dictated by considerations of crack stability, so that a z-2e (4-7) and KIc is the fracture toughness of the material. Typical material properties for 7075- T651 aluminum alloy are presented in Table 4-1. 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 4-4. Table 4-1: Fatigue properties of 7075-T651 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 4-8 (Palmberg et al., 1987) Pd (a) (a=ah (4-8) 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 4-1, was obtained by fitting Equation 4-8 to the inspection data in these references. The probability of detection curves for different inspection types are shown in Figure 4-1. 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 4-1: 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). Reliability-based optimization is computationally very expensive when inspections are involved, because crack size distribution has to be re-characterized 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 first-order 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 first-order reliability method (FORM). Evaluating low failure probability using MCS requires a large sample size, which makes reliability-based 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 4-5. 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 4-5 (a,, m, a) and obtain the final crack size ay. We then simulate the inspection by using Equation 4-8 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 4-2. Table 4-2: 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 4-5 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 4-8, 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 First-Order 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 10-8, 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 4-2. The probability of failure after N cycles of fatigue loading since the most recent inspection is Pf (N, a,)= P(a(N, a,)> a,) (4-9) 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 4-7. This probability is calculated by the first-order 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 4-9 using a bisection method between the previous inspection time Sn-1 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 = 10-7 in Table 4-3 for a 2.48 mm thick panel. Calculating P(a(N, a,) > a,) using FORM in Equation 4-9 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 4-9 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 4-2 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 4-3 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 4-3: Example 4-1, 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 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 1.00E-02 1.00E-04 - 1.00E-06 - 1.00E-08 - 1.00E-10 1.00E-12 1.00E-14 1.00E-16 1.00E-18 1.00E-20 5000 10000 15000 20000 45000 - No-Inspections -m-Inspections Flights Figure 4-2: Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7 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 4-4. Following Backman (2001) the service life S1 is assumed to be 40,000 flights. Table 4-4: 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 (4-10) k=l W = N,thbp (4-11) During the optimization, the structural thickness t changes, this in turn changes the structural weight according to Equation 4-11. The optimum inspection schedule (times and types) is determined for this structural design and the total cost is obtained from Equation 4-10. 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 (4-13) I ck (3) Generate the cheapest inspection sequence satisfying Equations 4-12 and 4-13. (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 safe-life 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. Safe-Life 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 safe-life design required to maintain the desired level of reliability throughout the service life. Table 4-5 shows the safe-life design. Table 4-5: 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 10-9 4.24 26.34 35129 0.68 From Table 4-5, comparing the minimum thickness to that used in Example 4-1 (Table 4- 3), we see that the safe-life 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 4-6. Table 4-6: 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 10-9 2.66 Ii, 13, Ii 15064, 23532, 20.27 65 16 19 30023 From Table 4-6 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 safe-life 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 4-6, we can see that adding inspection leads to a 25% saving in life cycle cost over the safe-life 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 10-7, cheaper inspections can be used (14 and 13), whereas for 10-9 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 10-8. The last columns of Table 4-6 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 4-6 for a reliability level of 10-7 Table 4-7: 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 4-7 shows the inspection schedules and cost for the inspection sequence generated using individual inspection types for a fixed structural size. Compar ing Tables 4-6 and 4-7, 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 4-8. Table 4-8: 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 10-8 2.50 13971, 22897, 1i 19.65 31443 10-9 2.64 14975, 19642, 13 20.41 26230, 32670 Comparing Tables 4-6 and 4-8 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 4-6, 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 10-7. The optimum plate thickness is 2.02 mm and a comparison of the optimum inspection schedule using different inspection sequences is shown in Table 4-9. Table 4-9: 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 4-9 to the first row of Tables 4-6 and 4-7 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 4-9, 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, A-basis 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/AR-95/47). 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 5-1 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 737-400) 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 fleet-level 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 5-1: 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 code-based design rather than probabilistic approaches. Safety is improved through conservative design practices that include use of safety factors and conservative material properties. It is also improved by 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 Guide-2006 (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 A-basis property is defined as the value of a material property exceeded by 99% of the population with 95% confidence, and the B-basis property is that exceeded by 90% of the population with 95% confidence. For structures without redundancy, A-basis properties are used and for structures with redundancy, B-basis properties are used. The conservative material properties considered here are A-Basis 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 JSSG-2006 damage tolerance guidelines specify this value as the B-basis 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 full-scale 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 fail-safe multiple load path structures should be designed for depot-level 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 sub-assembly is tested followed by a full-scale 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 B-basis value of crack size, which is also used to design the panel. A summary of fatigue testing of fuselage panels is documented in FAA/AR-95/47 (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 two-level Monte Carlo simulation, with different aircraft models being considered at the upper level, and different instances of the same aircraft at the lower level. (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 5-1: 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 5-1, 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 5-1) from assumed statistical distributions for material properties and thickness distribution. Next we simulate certification testing in step B of Figure 5-1 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 5-1) 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 5-1 represents the rate of crack growth da/dN in terms of stress intensity. da = D(AK)m (5-1) 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 7075-T651 aluminum alloy, D and m are material parameters related by Equation 5-2 -_m obtained from Sinclair and Pierie (1990). D has units in meters 2 (MPa)- m D= e(-3.2m-12.47) (5-2) We use the subscript "calc" to note that relations, (such as Equation 5-1), 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 (5-3) Equation 5-1 is integrated to obtain estimated fatigue life Na"'e 1-m 1- a 2 2 Ncac = a a (5-4) 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 5-5. ac < 2 (5-5) C -4r- Equation 5-1 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 5-1. We include and error factor e in analysis and then Equations 5-1 and 5-4 become. -- =eD(AK) (5-6a) 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 5-7 represents the error in the calculated stress, through an error parameter k. N true = a+ k) calc (5-7) 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 5-2: 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 5-2 lists uncertainties in form of errors and variability in the life prediction and structural design model assumed here for 7075-T6-aluminum 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) safe-life: structure is designed for safe crack growth for the entire service life of 40,000 flights; no inspections are performed (ii) fail-safe: 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 JSSG-2006. 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 MPa-m5 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 5-1. The calculated stress o,,calc on the structure is found from Equation 5-8 representing hoop stress due to pressure loading. Ucalc =- (5-8) 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 (5-9) where r is the radius of fuselage. Combining Equations 5-7 and 5-8, the stress in the structure is calculated as U,., =(+ k) (5-10) 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 B-basis value of a,, the conservative value of critical crack acA obtained using A-basis value of Kic in Equation 5-4, A-Basis 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 (5-11) f fai( KA Ofaugue [B Solving Equation 5-11 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 A-Basis property of yield stress is used to design the structure for static strength. _yield _yield / - design = A-Basis (5-12) Equations 5-11 and 5-12 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 5-13 as (I + k) d" fatigue design = max (1 + k1.5 (5-13) 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 5-13 is the average thickness for a given aircraft model. The actual thickness will vary due to individual-level 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 5-13), 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 5-1 (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 (5-14a) DI I mPdr 2 tact Static certification test: o of = dr- o- > 0 (5-14b) 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 5-2). This procedure of design and testing is |

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PAGE 1 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 FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 PAGE 2 This dissertation is dedicated to my parents PAGE 3 iii ACKNOWLEDGMENTS I want to express my appreciation and sp ecial thanks to Dr. Raphael T. Haftka, chairman of my advisory committee. He has b een a great mentor and constant source of inspiration and encouragement during my doctoral studies, a nd 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 base d 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 un limited 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 a nd 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. Thack er and Dr. Narasi Sridhar who gave me the excellent opportunity to work with th em on an industrial pr oject at Southwest Research Institute. PAGE 4 iv My colleagues in the Structural and Multidisciplinary Optimization Research Group at the University of Florida also dese rve thanks for their he lp 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 NCC3994 to the Institute for Future Space Transpor t (IFST) at the University of Florida and NASA Grant NAG1-02042 is fully acknowledged. My parents deserve my deepest appreciati on for their constant love and support and for encouraging me to pursue a Ph.D. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.........................................................................................................xiv KEY TO SYMBOLS.......................................................................................................xvi ABSTRACT.....................................................................................................................xxi CHAPTER 1 INTRODUCTION........................................................................................................1 Motivation.....................................................................................................................1 Objective...................................................................................................................... .2 Outline........................................................................................................................ ..2 2 BACKGROUND..........................................................................................................5 Structural Design Methodology....................................................................................5 Estimating Fatigue Life and Crack Sizes......................................................................7 Probabilistic Approach for Fatigue Life Prediction......................................................8 Reliability Based Design.....................................................................................10 Monte Carlo Integration......................................................................................11 First-Order Reliability Method (FORM).............................................................12 Reliability Based Inspection Scheduling....................................................................13 Reliability Based Design Optimization......................................................................14 3 EFFICIENT RELIABILITY BASE D DESIGN AND INSPECTION OF STIFFENED PANELS AGAINST FATIGUE..........................................................16 Introduction.................................................................................................................16 Crack Growth and Inspection Model..........................................................................18 Fatigue Crack Growth.........................................................................................18 Critical Crack Size...............................................................................................20 Probability of Failure at a Given Time................................................................22 Inspection Model.................................................................................................24 PAGE 6 vi Computational Method to Perform Re liability 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 Sche dule for a Given Structure..................................29 Optimization of Structural Design.......................................................................31 Results........................................................................................................................ .37 Summary.....................................................................................................................43 4 TRADEOFF OF WEIGHT AND INSPE CTION COST IN RELIABILITYBASED STRUCTURAL OPTIMIZATION USING MULTIPLE INSPECTION TYPES........................................................................................................................44 Introduction.................................................................................................................44 Structural Design and Damage Growth Model..........................................................47 Fatigue Crack Growth.........................................................................................47 Inspection Model.................................................................................................49 Calculating an Inspection Schedule............................................................................51 Estimating Crack Size Distri bution after Inspection...........................................51 Calculating the Failure Probability Using the First-Order Reliability Method (FORM)............................................................................................................53 Cost Model..........................................................................................................57 Optimization of Inspection Types.......................................................................58 Combined Optimization of Structur al Design and Inspection Schedule....................59 Safe-Life 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 Measures..........................................................................................................68 Simulation Procedure for Calculation of Probability.................................................70 Damage Growth Model..............................................................................................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 With Machined Crack........................................85 Effect of Variability in Material Pr operties on Structure Designed With all Safety Measures...............................................................................................87 PAGE 7 vii 6 A PROBABILISTIC MODEL FOR INTERNAL CORROSION OF GAS PIPELINES.................................................................................................................92 Introduction.................................................................................................................92 Proposed Methodology...............................................................................................96 Corrosion Rate Model.........................................................................................96 Inhibitor Correction Model..................................................................................97 Water Accumulation............................................................................................98 Probabilistic Model...................................................................................................100 Corrosion Damage.............................................................................................100 Input Uncertainties............................................................................................101 Mapping Uncertainty.........................................................................................102 Inspection Updating...........................................................................................103 Example 1: Determination of Critic al Location Prior to Inspection.........................104 Example 2: Updating Corrosion Modeling with Inspection Data............................106 Summary...................................................................................................................109 7 CONCLUSIONS......................................................................................................111 APPENDIX A DISPLACEMENT COMPATIBILITY ANALYSIS FOR CALCULATION OF STRESS INTENSITY OF STIFFENED PANEL....................................................114 Introduction...............................................................................................................114 Displacement Compa tibility Method........................................................................115 Displacement V1................................................................................................117 Displacement V2 and V3.....................................................................................118 Displacement V4................................................................................................119 Intact Stiffener Displacement............................................................................120 Broken Stiffener Displacement.........................................................................121 Fastener Displacement.......................................................................................121 Compatibility of Displacements........................................................................121 Effectiveness of Stiffeners in Reducing Crack Tip Stress Intensity..................124 B CALCULATING CRACK GROWTH FOR STIFFENED PANELS USING NUMERICAL INTEGRATION A ND RESPONSE SURFACE.............................126 C ACCURACY ESTIMATES OF RESPO NSE SURFACE APPROXIMATIONS...129 Response Surface Approximations for Geometric Factor ...................................129 Response Surface Approximation for Reliability Index (Beta)................................131 D COST OF STRUCTURAL WEIGHT......................................................................134 E PSEUDO CODE FOR COMBINED OP TIMIZATION OF STRUCTURE AND INSPECTION SCHEDULE.....................................................................................136 PAGE 8 viii Introduction...............................................................................................................136 Optimization 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 Structural Uncertainties............................................................................................146 Safety 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 Calculati on by Analytical Approximation.....................154 Probability of Failure Calcula tion by Monte Carlo Simulations.......................156 Effect of Three Safety Measur es on Probability of Failure......................................157 Concluding Remarks................................................................................................169 H CALCULATION OF CONSERVATI VE MATERIAL PROPERTIES..................171 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 PAGE 9 ix LIST OF TABLES Table page 3-1 Fatigue properties of 7075-T651 Aluminum alloy..................................................22 3-2 Structural design for fuselage...................................................................................22 3-3 Pseudo code for updating cr ack size distribution after N cycles from previous inspection.................................................................................................................27 3-4 Example 3-1, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 10-7........31 3-5 Cost of inspection, material and fuel........................................................................32 3-6 Description of response surface a pproximations used in optimization....................33 3-7 Computational time spent in exact calc ulation of next inspection time and error due to -RSA usage................................................................................................34 3-8 Pseudo code for combined optimizati on of structural design and inspection schedule....................................................................................................................36 3-9 SafeLife design of an unstiffened panel.................................................................37 3-10 SafeLife design of a stiffened panel.......................................................................37 3-11 Optimum structural design and inspec tion schedule of an unstiffened panel..........38 3-12 Optimum structural design and insp ection schedule for stiffened panel..................39 3-13 Optimum structural design for regulati ons based inspections conducted at four constant interval or 8000 flights for stiffened panel................................................40 3-14 Tradeoff of inspection cost against cost of structural weight required to maintain fixed reliability level for stiffened panel..................................................................41 3-15 Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel with inspection.................................................................................42 4-1 Fatigue properties of 7075-T651 Aluminum alloy..................................................49 PAGE 10 x 4-2 Pseudo code for updating crack distribution after N cycles from previous inspection.................................................................................................................53 4-3 Example 4-1, inspection schedule and cr ack 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 4-4 Design details and cost factors.................................................................................57 4-5 Structural size required to maintain a specified reliability level without and inspection.................................................................................................................60 4-6 Optimum structural design and inspec tion schedule required to maintain specified threshold reliability level..........................................................................61 4-7 Comparison of optimum inspection sche dule using a single in spection type for a fixed structural size..................................................................................................62 4-8 Optimum structural design and insp ection schedule using only a single inspection type..........................................................................................................63 4-9 Optimum structural design (plate thickness of 2.02 mm ) and inspection schedule for Pfth = 10-7............................................................................................................64 5-1 Uncertainty classification.........................................................................................68 5-2 Distributions of errors, design and ma terial parameters for 7075-T6 aluminum.....75 5-3 Nomenclature of symbols used to calc ulate failure probability and describe the effect of certification testing....................................................................................80 5-4 Probability of failure for 10 % COV in e and different bounds on error k using all safety measures for fail-safe design for 10,000 flights.......................................81 5-5 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for fail-safe design for 10,000 flights................................................82 5-6 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 5-7 Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for safe-life design of 40,000 flights..................................................82 5-8 Probability of failure for different bounds on error k for 10 % COV in e without any safety measures for fail-safe design for 10,000 flights......................................84 5-9 Probability of failure for different bounds on error k for 50 % COV in e without any safety measures for fail-safe design for 10,000 flights......................................84 PAGE 11 xi 5-10 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for fa il-safe for 10,000 flights and tested using a machine cracked panel..........................................................................................86 5-11 Probability of failure for different bounds on k and 10 % COV in e for structures designed with all safety measures for sa fe-life of 40,000 flights and tested using a machine cracked panel..........................................................................................86 5-12 Probability of failure for different bounds on k and 10 % COV in e for structure designed with all safety measures fo r fail-safe for 10,000 flights and COV in material property m reduced to 8.5%.......................................................................87 5-13 Probability of failure for different bounds on k and 50 % COV in e for structures designed with all safety measures for fail-safe criteria for 10,000 flights and COV in material property m reduced to 8.5%..........................................................87 5-14 Probability of failure for different bounds on k and 10 % COV in e for structures designed using only A-Basis m for fail-safe criteria for 10,000 flights...................88 5-15 Probability of failure for different bounds on k, 10 % COV in e for structure designed using conservative properties for fail-safe design for 10,000 flights........89 5-16 Probability of failure for different bounds on k 50 % COV in e for structures designed using conservative properties fo r fail-safe criteria for 10,000 flights.......89 5-17 Effective safety factor and measures of probability improvement in terms of individual safety measures and erro r bounds for structure designed using failsafe criteria of 10,000 flights....................................................................................90 6-1 Typical wet gas pipeline flow parameters................................................................99 6-2 Typical wet gas pipeline corrosion growth parameters..........................................101 6-3 Updating of model weights given as sumed observations corresponding to input component models..................................................................................................107 6-4 Inspection locations along pipeline........................................................................109 C-1 Bounds on design variables used to ev aluate response surface approximation for safe life design........................................................................................................129 C-2 Error estimate of analysis response su rfaces used to obtain safe-life stiffened panel design............................................................................................................130 C-3 Bounds on design variables used to ev aluate response surface approximation for inspection based design..........................................................................................130 PAGE 12 xii C-4 Error estimate of analysis response surfaces used to obtain inspection based stiffened panel design.............................................................................................131 C-5 Error estimate of design response surfaces............................................................131 C-6 Bounds on design variables used to ev aluate response surface for crack sizes parameters after inspection and reliability index...................................................132 C-7 Error estimate of crack size response su rfaces used to estimate the crack size distribution parameters af ter the first inspection....................................................133 C-8 Error estimate of crack size response su rfaces used to estimate the distribution after inspection.......................................................................................................133 C-9 Error estimate of reliability index re sponse surfaces used to schedule first inspection...............................................................................................................133 C-10 Error estimate of reliabil ity index response surfaces, d-RSA...............................133 D-1 Area of structural dimensions for cost calculation.................................................135 F-1 Inspection schedule and crack si ze distribution after inspection ( ah = 1.27 mm ) for an unstiffened plate thickness of 2.48 mm and a threshold probability of 10-7141 G-1 Uncertainty classification.......................................................................................147 G-2 Distribution of random variables us ed for panel design and certification.............152 G-3 Comparison of probability of failures ( Pfs) for panels designed using safety factor of 1.5, mean value for allo wable stress and error bound of 50%.................156 G-4 Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and A-basis property for allowable stress...............................158 G-5 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 G-6 Probability of failure for different bounds on error e for safety factor of 1.0 and A-basis property for allowable stress.....................................................................162 G-7 Probability of failure for different e rror bounds for panels designed using safety factor of 1.0 and mean value for allowable stress..................................................163 G-8 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 50% error bounds e and A-basis property for allowable stress.......................................................................................................................164 G-9 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 30% error bound e and A-basis properties.............................164 PAGE 13 xiii G-10 Probability of failure for uncertainty in failure stress for panels designed using safety factor of 1.5, 10% error bounds e and A-basis properties...........................165 PAGE 14 xiv LIST OF FIGURES Figure page 3-1 Fuselage stiffened panel geometry and applied loading in hoop direction..............20 3-2 Comparison of actual and lognormally fitted CDF of crack sizes after an inspection conducted at 9288 flights........................................................................29 3-3 Example 3-1, Variation of failure proba bility with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7....................31 4-1 Probability of detection curve for diff erent inspection types from Equation 4-8....51 4-2 Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7...................................56 5-1 Flowchart for Monte Carlo simula tion of panel design and failure.........................71 6-1 Uncertainty in inclina tion and critical angle..........................................................101 6-2 Probability of water formation along pipe length with highest probability observed at location 971.........................................................................................105 6-3 Probability of corrosion depth exc eeding critical depth along pipe length assuming water is present at all locations..............................................................105 6-4 Total probability of corrosion exc eeding critical depth along pipe length.............106 A-1 Half-geometry of a center cracked stiffe ned panel with a central broken stiffener and two intact stiffeners placed symmetrically across from crack.........................115 A-2 Description of applied stress and resu lting fastener forces and induced stress on stiffened panel........................................................................................................116 A-3 Description of position coordinate of forces and displacement location with respect to crack centerline as y axis........................................................................117 A-4 Description of position coor dinate of forces and indu ced stress distribution along the crack length......................................................................................................119 A-5 Comparison of stress intensity factor for a panel with skin thickness = 2.34 mm and stiffener area of 2.30 10-3 meter2..................................................................124 PAGE 15 xv A-6 Comparison of stress intensity factor for a panel with skin thickness = 1.81 mm and stiffener area of 7.30 10-4 meter2..................................................................125 B-1 Typical response curves for effect of stiffening on geometric factor for a stiffener area of 1.5 mm2 and skin thickness of 1.5 mm .........................................128 F-1 Probability of exceeding 2.0 for a lognor mally distributed random variable with a mean of 1.0. Note that large st andard deviation decreases probability.......142 F-2 Comparison of failure probability (1CD F) of two probability distributions with mean 10-5 and standard deviation of 2 and 10 units...............................................143 G-1 Flowchart for Monte Carlo simula tion of panel design and failure.......................151 G-2 Initial and updated pr obability distribution func tions of error factor e ..................155 G-3 Design thickness variation with low and high error bounds..................................162 G-4 Influence of effective safety factor, er ror, and variability on the probability ratio (3-D view)..............................................................................................................167 G-5 Influence of effective safety factor, error and variability on the probability ratio (2-D contour plot)...................................................................................................167 G-6 Influence of effective safety factor error and variability on the probability difference (3-D view).............................................................................................168 G-7 Influence of effective safety factor error and variability on the probability difference (2-D contour plot).................................................................................169 PAGE 16 xvi KEY TO SYMBOLS a = Crack size, mm ac = Critical crack size, mm acH = Critical crack length due to hoop stress, mm acL = Critical crack le ngth for transverse stress, mm acY = Critical crack length causi ng yield of net section of panel, mm ah = Crack size at which pr obability of detection is 50%, mm ai = Initial crack size, mm ai,0 = 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 variat ion, (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 Cmin = Minimum cost of inspection schedule, dollars Ctot = Total life cycle cost, dollars d = Fastener diameter, mm PAGE 17 xvii D = Paris model parameter, m MPa m meters 2 1 e = Error in crack growth rate E = Elastic modulus, MPa F = Force at a rivet on intact stiffener, N Fc = Fuel cost per pound per flight, dollars max ener FirstStiffF = Maximum stress on first stiffener, MPa max fener SecondStifF= Maximum stress on second stiffener, MPa max ener ThirdStiffF = Maximum stress on third stiffener, MPa g = Limit state f unction 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, Ic1, Ic2,, Ic3, Ic4, dollars Ik = Inspection of kth type, k = 1 k = Error in stress calculation K = Stress intensity factor, MPa meters KF = Stress in tensity due to fastener forces, meter MPa KIC = Fracture toughness, MPa meters KTotal = Total stress intensity on stiffened panel, meter MPa PAGE 18 xviii L = Frame spacing, meters l = Fuselage length, meters m = Paris model exponent, Eq. 3-1 iAM = Average bending moment between the ith and i -1st fastener, N-meter Mc = Material manufacturi ng cost per pound for aluminum, dollars n = Number of fastener on a side of crack cente rline on a single stiffener N = Number of cycles of fatigue loading Nf = Fatigue life, flights (Flights, time and cycl es are used interchangbly) Ni = Number of Inspections Np = Number of panels Ns = 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 af ter certification testing Pd = Probability of detection rand dP = Random number for probability of detection Pf = Failure probability Pfth = Threshold probability of failure, reliability constraint Pnc = Probability of failure wi thout certifi cation testing r = Fuselage radius, meters r1 = Distan ce of a point from crack leading tip, meters r2 = Distan ce of a point from crack tailing tip, meters PAGE 19 xix 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 Sl = Service Life (40,000 flights) Sn = 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 tdesign = Thickness of designed structures ts = Stiffener thickness, mm V1 = 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 cau sed by rivet loads, meters VF = Displacement at a point in and infinite plate due to a point force F PAGE 20 xx W = Structural weight, lb Y = Yield stress, MPa = Inspection parameter d = Reliability index iD = Stiffened displaceme nt due to direct fastener load at ith fastener location, meters iG = Stiffener displacement due to applied stress at ith fastener location, meters iM = Stiffener displacement due to bending at ith fastener location, meters iR = Fastener displacement due to elastic shear, meters ai-RSA = Response surface for estimati ng mean of crack size distribution, mm = Poissons ratio = Cumulative density function of standard normal distribution = Geometric factor due to stiffening = Density of aluminum, lb/ft3 = Hoop stress, MPa ai-RSA = Response surface estimating standard deviation of crack size distribution, mm = 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) 1 = Angle at a point as measured from leading crack tip. 2 = Angle at a point as measured from tailing crack tip. PAGE 21 xxi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 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: Mechanic al and Aerospace Engineering Structural safety is achieved in aerosp ace application and other fields by using conservative design measures like safety fact ors, conservative materi al properties, tests and inspections to compensate for uncertain ty in predicting structural failure. The objective of this dissertation is to clarify the interaction be tween these safety measures, and to explore the potential of including the interaction in the de sign 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 th e effect of error and variability control in the desi gn process. Inspections are feat ured more prominently than other safety measures. The uncertainties are readily incorporat ed into the design process by using a probabilistic approach. We e xplore the interaction of va riability, inspections and structural sizes on reliability of structural components subjected to fatigue damage growth. Structural sizes and inspection schedule are optimized simultaneously to reduce PAGE 22 xxii operational cost by trading the cost of structur al 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 optimizati on of structural design and inspection times makes combined optimization of the insp ection schedules and structural design prohibitively costly. This disse rtation develops an efficien t computational technique to perform reliability based optimization of structural design and inspection schedule combining Monte Carlo simulation (MCS) a nd first-order reliability method (FORM). The effect of the structural design and the inspection schedule on th e 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 effectiv e if done at critical locations where likelihood of failure is maximum and the info rmation obtained from inspections can be used to improve failure prediction and update reliability. This aspect is studied by developing a probabilistic model for predicti ng 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. PAGE 23 1 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 lif e of a structure cannot be accurately determined even in carefully controlled conditions because of variability in material properties, manufacturing defect s and environmental factors like corrosion. Safety of aircraft structures is largel y maintained by using conservative design practices to safeguard agains t uncertainties involved in the design process and service usage. Typically, conservative material prope rty, scatter factor in fatigue life and conservative loads are used to design struct ures. This is further augmented by quality control measures like certific ation testing and inspections. Safety measures compensate for uncertainty in load modeling, stress analysis material properties a nd factors that lead to errors in modeling structur al failure. These safety measures were gradually developed based on empirical data obtained from service experience and ar e usually geared to target specific types of uncertainty. For example, the use of conservative material properties provide protection against vari ability in material propertie s, using machined crack for certification and conservative initial defect provide protection against flaws induced during manufacturing and fabrication, and in spections protect agai nst uncertainty in damage growth and accidental damage that cannot be predicted during the service life. The use of multiple safety measures along w ith quality control measures is costly. With a view of reducing lifetime cost and main taining structural safe ty, this dissertation PAGE 24 2 is a step towards understanding the interacti on 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 a dditional structural weight. With the view to reducing cost of operation of aircraft structures and mainta ining low risk of structural failure, we address the problem of developi ng optimum structural design together with inspection schedule. The approach is based on the application of methods of structural reliability analysis. Reliability based optim ization is computationally expensive when inspections are involved because crack size di stribution has to be re-characterized after each inspection to simulate replacement. T ypically, 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 di ssertation is to develop an efficient computational method to es timate reliability with inspection. Outline This dissertation uses a co mbination of reliab ility methods, Monte Carlo simulation (MCS), first-order reliability method (F ORM) and response surface approximations (RSAs), to perform reliability based optim ization of structural design and inspection schedule. Typical examples of aircraft stru ctures designed for fati gue crack growth and inspection plans are used to demonstr ate the application of this methodology. PAGE 25 3 Most of the chapters in the dissertati on are revised versions of conference or journal papers with multiple authors. The ou tline below gives the chapter description and an acknowledgement of the ro le of the other authors. Chapter 2 presents the bac kground and a literature survey on current methods used to design aircraft structures for damage grow th. Uncertainty is a cr itical component in aircraft structural design and probabilistic methods are used to incorporate uncertainty in designing structures. This chapter also review s 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 reliab ility calculation in the pr esence 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 collabo ration with Dr. Melih Papila, who provided inputs on cost of inspections a nd 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 ge nerate the inspection schedule. Chapter 5 is close to Kale et al. (2005). It presents th e interaction among various safety measures recommended by the Federa l Aviation Administration (FAA) to design aircraft structures for damage tolerance. Interaction among safety measures, uncertainty and certification tests is studie d. In particular it sheds lig ht on the effectiveness of PAGE 26 4 certification testing for fati gue. 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 info rmation obtained from inservice inspections can be used to update fa ilure models and reliability using Bayesian updating. The methodology is applied to reliabil ity assessment of gas pipelines subjected to corrosion damage. Risk based inspection plans are developed to determine optimal inspection locations where probability of co rrosion damage is maximum. This work was done in collaboration with Dr. Ben H. Th acker, Dr. Narasi Sridhar and Dr. Chris Waldhart at the Southwes t Research Institute. PAGE 27 5 CHAPTER 2 BACKGROUND Structural Design Methodology Aerospace structural design philosophy ha s 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 reliability-based design and optimization of structures. The loss of structural integrity with servic e usage is associated with propagation of damage such as fatigue cracks in metal structures or delamination in composite structures. In addition, damage may be in flicted by corrosion, freeze-thaw cycles, and accidents such as a turbine blade tearing thr ough 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 uns table and propagate qui ckly 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 safe ty: safe-life and fail-safe. Niu (1990) and Bristow (2000) have characterized the safe -life and fail-safe de sign methodologies in that, reliability of a safe-lif e structure is maintained by replacing components if their design life is less than the service life. In spections or repairs are not performed. In PAGE 28 6 contrast, structural safety in a fail-safe design is maintained by means of design for damage containment or arrestment and altern ative load-paths that preserve limit-load capabilities. These mechanisms are complement ed with periodic inspections and repairs. Bristow (2000) provided historical insight on the evolution of structural design philosophy from safe-life in the early 50s to damage-tolerance 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 stru ctures designed for damage tolerance be demonstrated to avoid failure due to fati gue, manufacturing defects and accidental damage (FAR 25.571, damage tolerance and fa tigue 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 peri odic inspections. Inspections play an important role in maintaining stru ctural integrity by compensating for damage that cannot be predicted or modeled duri ng the design due to randomness in loading, accidental impact damage and environmental f actors. In todays practice both safe-life and fail-safe structural design concepts are n ecessary to create a structurally safe and operationally satisfactory components. These two concepts have found application in structural design of airplanes, bridges and other engineering stru ctures for different structural parts based on the functionalitie s and associated redundancy level. For instance, nose landing gear and main landing gear do not employ any redundancy and PAGE 29 7 exhibit a short fatigue life. Therefore they are designated as safelife structures. Wing skin-stringer and fuselage skin-stringer panels have a substantial fa tigue life and usually offer structural redundancy, so they are designated as fail-safe structures. Estimating Fatigue Life and Crack Sizes Structural components experien ce numerous repetitive load cycles of normal flight conditions during their service life. In addi tion, less frequent but hi gher loads originating from strong atmospheric gusts or unexpected maneuvers during the life of aircraft are inevitable. Flaws and imperfections in th e structure, such as micro cracks or delamination, may propagate unde r such service experience. Estimating fatigue life and crack size is a challenging task as there ar e no physical models available to determine crack growth as a function of the numerous factors that affect it. The load spectrum of an aircraft give s first hand information on the expected service load for which the airp lane should be designed. The lo ad history of aircraft is generated by load factor measurements from accelerometer placed at the center-of 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 calculat ions (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 accumula tion rule can be used with stress-fatigue life (S-N curve) to estimate fatigue life. The Palmgren-Miner linear damage accumulation rules (Miner, 1945) has been popular in aerosp ace application since the early 1950s to the present day. This rule computes the fatigue lif e as the summation of ratios of applied load cycles at a given level divided by the allowabl e number of load cycles to failure at the same stress level which can be obt ained from S-N curve (e.g., Tisseyre et al. 1994). PAGE 30 8 An alternative fatigue life estimati on 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 sp ecific set of loading condition; however the basic nature of the equations have rema ined 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 amplitu de loading. Later crack growth retardation effects observed in variable amplitude loadi ng were also introduced. Wheeler (1972) used the plastic zone size to modify the Paris mode l. These damage growth models have been widely used for life prediction with some modi fications in structural design applications; e.g., Harkness (1994) and Tisseyre et al. (1994) used it in aerospace applications, and Enright and Frangopol (2000) us ed it for bridge design. Probabilistic Approach for Fatigue Life Prediction Aircraft structural design is still done by and large using code-based design rather than probabilistic approaches. Safety is im proved through conservative design practices that include use of safety fact ors and conservative material pr operties. It is also improved by tests of components and certifi cation tests that can reveal in adequacies in analysis or construction. These safety measures listed in FAR 25 for civil and transport category airplanes and Joint Service Specification Guide-2006 (JSSG). Use of large safety measures increases the structur al weight and operational cost. PAGE 31 9 The main complexity for designing damage to lerant structures via safe-life and failsafe concepts in design is due to uncertain ties involved. These in clude uncertainty in modeling physical phenomena affecting struct ural integrity (e.g., loading, crack growth) and uncertainty in data (e.g., material pr operties). Inspection and replacement add additional uncertainty because damage dete ction capabilities depend on random factors such as location of the damages or labor quality and equipments. It has been demonstrated that small variations in materi al properties, loading and errors in modeling damage growth can produce huge scatter in fatigue life, (e.g., Ha rkness, 1994; Sinclair and Pierie, 1990) which makes it inevitable to use large safety measures during the design process. Uncertainties are inevitable and past se rvice experience in the design of new structures have become a key factor in m odern damage tolerant design approaches. Statistical data are collected for material properties, load hist ories (by the use of accelerometers) and damage initiation and growth by scheduled inspections. Then the associated uncertainties may be introduced into the design pro cedure by probabilistic approaches. A reliability-based approach towards stru ctural design requires us to account for uncertainty in damage initiation, damage grow th with time, residual strength and damage detection. In probabilistic formul ation uncertainty is incorpor ated 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 probab ilistic approach and fracture mechanics in fatigue life prediction has been demonstrated by Provan et al. (1987) and Belytschko et PAGE 32 10 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 inco rporated by Nees and Canfield (1998) and Arietta and Striz (2005) by us ing load history. Tisseyre et al. (1994) and Enright and Frangopol (2000) used reliability based formulati on to predict fatigue failure of structural components subjected to uncertainty in lo ading, damage initiation and growth. Backman (2001) studied reliability of aircraft structures subjected to impact damage. Environmental factors like corrosion, enhan ce 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 desc ribe the enhancement in fatigue crack growth in the presence of aggressive environment due to hydrogen enhanced embrittlement. Recently there has been advancement in estimating corrosion-fatigue growth rates. Harlow and Wei (1998) obtai ned empirical model for rate of corrosion fatigue in aggressive environment by fitting experimental results to linear models. Probabilistic analysis is al so very useful when there is no single model that can completely describe the crack growth phenom ena for given set of conditions. When there are wide range of competing models, Baye sian 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 mate rial properties and errors in analytical models used for designing the structure contri bute to a chance that the structure will not perform its intended function. Reliability analys is deals with the methods to calculate the PAGE 33 11 probability of structural failure subjected to such uncertainty. A typical reliability analysis problem can be defined as S R x d g where dx x f x d Px d g x f ,0 (2-1) where d is the vector of design variables, x is the vector of random variables, Pf is the failure probability as function of de sign variables and random variables, fx is the joint probability density function of random variables and g is the performance function which decides if the structure ha s failed in terms of load S and resistance R The reliability is defined as the complement of failure probabilit y. Calculation of structural reliability is computationally expensive because many eval uations 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 revi ew of various methods of structural reliability analysis. Here the tw o most extensively used methods, the Monte Carlo simulation (MCS) and the first-order reliability method (FOR M), are presented. Monte Carlo Integration The Monte Carlo integration is by far the simplest and potentially most accurate method to obtain failure proba bility, 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 realiz ation of uncertain variables in the structural model (e.g., Melchers, 1987). The event of failure is ev aluated 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 defi ned by the performance function. If N is the total number PAGE 34 12 of simulations of random variables and Nf the number of failed simulations then the probability of structural failure is estimated by N N Pf f (2-2) The accuracy of the probability calculated fr om Equation 2-2 increases with the number of simulations. An estimate of the accur acy in failure probability is obtained by calculating the standard deviation in Pf N P Pf f Pf 1 (2-3) First-Order Reliability Method (FORM) Monte Carlo method can be computationally very expensive for evaluating very low probabilities because large number of si mulations is required for accuracy. The firstorder reliability method is an efficient alte rnative. 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 surf ace between safe and failed realization in the standard Gaussian space (all random vari ables are transformed to standard normal variables). This linear approxima tion 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 distan ce is termed as reliability index The probability of failure is the area of tail beyond under the standard normal distribution. fP (2-4) and 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 fo r concave function and overestimates it for PAGE 35 13 convex function. Higher order method like th e second-order 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 ma nufacturing 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 reason s. First, the ability of the inspection to detect damage is limited because of human a nd 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 al most 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 co mputational expense is that damage size distribution changes with time due to crack growth and also after inspections because components with damage are replaced by ne w components. Re-characterizing crack size distribution after inspections is computationally challenging. Reliability centered maintenance focuses on scheduling inspections when the failure probability exceeds a threshold probabilit y level. The reliability level is computed by determining the probability that damage becomes too large and remains undetected in all the previous inspections. Th e simplest and potentially most accurate method is to use Monte Carlo simulations, MCS (e.g., Harkness et al., 1994; Enright and Frangopol, PAGE 36 14 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 first-order reliability method (FORM) and second-order re liability method (SORM) have been used to obtain probability afte r inspection by Rahman and Rice (1992); Harkness (1994); Fujimoto et al. (1998); Toyoda-Makino (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 consum ing. For instance Rahman and Rice (1992) developed a methodology to update crack si ze distribution after inspections using Bayesian updating. Harkness ( 1994) modified the FORM to di rectly calculate reliability with inspections without updati ng 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 Ar ietta and Striz (2000, 2005) optimized F-16 wing panels subject to constraints on damage growth. Reliability based design optimization further increases the cost of reli ability analysis because several iterations on design variables are required to obtain optim um 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 el ement model). In these circumstances the numerically expensive function can be replaced by an approximation or surrogate model PAGE 37 15 having lower computational cost such as response surface approximation. Response surface methodology can be summarized as a colle ction of statistical tools and techniques for constructing an approximate functional re lationship between a re sponse variable and a set of design variables. This approximate f unctional relationship is typically constructed in the form of a low order polynomial by fitti ng it to a set of experi mental 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 wi dely used in structural optimization to reduce computational cost. NESSUS (Riha et al. 2000) and DARWIN (Wu et al. 2000) use response surface a pproximations for reducing computational cost of probabilistic finite element analysis. Vent er (1998) proposed methods to improve accuracy of response surface approximation a nd used them for optimizing design of composites. Papila (2001) also used respons e surfaces in structural optimization for estimation of structural wei ght. Qu (2004) used RSAs to minimize cost of reliability based optimization. PAGE 38 16 CHAPTER 3 EFFICIENT RELIABILITY BASED DESI GN AND INSPECTION OF STIFFENED PANELS AGAINST FATIGUE Introduction Reliability based optimization is computat ionally expensive when inspections are involved because crack size distribution has to be re-characterized after each inspection to simulate replacement. Inspections improve the structural safety through damage detection and replacement. However, inspecti ons cannot detect all da mage with absolute certainty due to equipment limitations and human errors. Probabilistic model of inspection effectiveness can be used to incorporate the un certainty associated with damage detection. Typically, the crack size di stribution after an inspection will not have a simple analytical form and can only be determined numerically during reliability analysis. Exact evaluation of failure proba bility following an inspection can be done by Monte Carlo simulation (MCS) with larg e population which is computationally expensive. The high computational cost for es timating very low probabilities of failure combined with the need for repeated analys is for optimization of structural design and inspection times make MCS cost prohibitive. Harkness (1994) developed a computationa l methodology to calculate structural reliability with inspections without upda ting the crack size distribution after each inspection. He assumed that repaired com ponents will never fail again and incorporated PAGE 39 17 this assumption by modifying the firs t 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 servic e life, repaired components can have large probability of failure. In th is case Harknesss method may not be accurate enough. In this chapter we propose an appr oximate method to simulate inspection and repair using Monte Carlo simu lation (MCS) and estimate the failure probability using the first order reliability met hod (FORM). MCS is computati onally very expensive for evaluating low failure probabilities due to la rge 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 th e mean and standard deviation of crack size distribution. Subsequently, FORM is used to calculate the failure probabilities between inspections. The combined MCS and FORM a pproach 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 pane ls 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 s tiffening members include stringers in the longitudinal directions and frames, fail-saf e-straps 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. PAGE 40 18 Swift (1984) and Yu (1988). They used displace ment 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 streng th 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 th e example in this chapter we use the Paris law. mK D dN da (3-1) where a is the crack size in meters, N is the number of cycles of fatigue loading in flights, da/dN is the crack growth rate in meters/cycles, the stress intensity factor range K is in meters MPaand m is obtained by fitting the crack growth model to empirical data. PAGE 41 19 More complex models account for load history effects. The stress in tensity factor range K for cracked stiffened panel can be calculat ed using finite element or analytical method as a function of stress and crack length a. a K (3-2) The effect of stiffening on the stress in tensity is characterized by the geometric factor which is the ratio of stre ss intensity factor for the cr acked 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 usually requires detailed finite element analysis. Here, is calculated using a method due to Swift (1984). The number of fatigue cycles accumulated in growing a crack from the initial size ai to the final size aN can be obtained by integrating Equation 3-1 between the initial crack ai and final crack aN. Alternatively, the final crack size aN after N fatigue cycles can be determin ed by solving Equation 3-3. This requires repeated calculation of as the crack propagates. The computational approach for integrating Equation 3-3 is illustrated in Appendix B. N ia am K f da N, (3-3) Here we focus on designing a fuselage pane l for fatigue failure caused by hoop stresses. The hoop stress is given by Equation 3-4 and cr ack grows perpendicular to the direction of hoop stress given by s sA N th rph (3-4) 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 3-1). PAGE 42 20 Figure 3-1: Fuselage stiffened panel geomet ry and applied loadi ng in hoop direction (crack grows perpendicular to the direction of hoop stress) Critical Crack Size We consider optimizing the design of a t ypical fuselage panel for fatigue failure due to hoop stress. The fail-safe stiffening memb ers in circumferential direction such as frames, fail-safe straps and doublers are modeled as equispaced rectangular rods discretely attached to the panel by fastener s. 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 ai rcraft fuselage struct ure by a periodic array of through-the-thickness center crac ks with three stiffeners on either sides of centerline as show in Figure 3-1. The critical crack length ac at which failure will occur is dictated by considerations of residual strengt h or crack stability. Structural failure occurs if the crack size at that time is greater than critical cr ack. The crack length causing net section failure is given by t A N Yt rph h as ub cY5 0 (3-5) PAGE 43 21 Equation 3-5 gives the crack length acY at which the residual streng th of the panel will be less than yield stress Y and Nub is the number of intact stiffeners. 2 IC cHK a (3-6) 22 t pr K aIC cL (3-7) Equation 3-6 determines the critical cr ack length for failure due to hoop stress and Equation 3-7 determines the critical crack length for failure due to transverse stress. This is required to prevent fatigue failure in longitudinal direction wh ere skin is the only load carrying member (effect of stringers in longitudinal direction is not considered because hoop stress in more critical for fati gue). The critical crack length for preventing structural failure is given by Equation 3-8 and the fatigue life Nf of structure is determined by integrating Equation 3-3 between the initial crack ai and critical crack ac. cL cH cY ca a a a, min (3-8) Typical material properties for 7075-T651 aluminum-alloy most commonly used in aerospace application are presen ted in Table 3-1. 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 initia tion and growth associ ated with corrosion, environmental effects and accidental damage. The structural design parameters obtained PAGE 44 22 for B747 series aircraft from Niu (1990) and Janes all th e worlds aircraft (Jackson, 1996) are listed in Table 3-2. Table 3-1: Fatigue propertie s of 7075-T651 Aluminum alloy (Source: Sinclair and Pierie, 1990 and Niu, 1990) Table 3-2: Structural design for fuselage Fastener diameter, d 4.8 mm Fastener spacing, s 3.1 cm Fuselage length, l 68.3 m Fuselage radius, r 3.25 m Number of panels, Np 1350 Number of fasteners per stiffener20 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 accumulate d 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 (ai, m and ). The equation which define the failu re boundary is known as the limit state function, g. So for our case Property Yield stress Y, MPa Initial crack ai,0 meters Mean. Standard deviation Paris exponent m Mean. Standard deviation Fuselage radius, r, meters Pressure load, p, (MPa) Fracture toughnes s, KIC MPa meters Distribution type 500.0 Lognormal 0.0002, 0.00007 Lognormal 2.97 1.05 3.25 Lognormal 0.06 0.003 36.58 PAGE 45 23 N m a N m a gi f i , , (3-9) where the fatigue life Nf is determined by integrating Equation 3-3 between the initial crack ai and the critical crack ac. The failure probability co rresponding to Equation 3-9 is calculated using the first-orde r reliability method (FORM). In this method the limit state function is represented in the tran sformed standard normal variables ( ai, m and 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 shorte st distance is called reliability index, d. The calculation of reliability index is an optimi zation problem requiring repeated calculation of Equation 3-9 for several iterations in the random variables. In this chapter the MATLAB fmincon function (w hich employs sequential quadratic programming) is used to determine the reliability index. The main reason for using reliability index instead of directly calculating failu re 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 becau se it varies over much smaller range compared to the failure probability. Failure probability is determined from the reliability index using the cumulative density function, of the standard normal distribution. d fP (3-10) For an unstiffened panel analytical expressi on of fatigue life is available; however for stiffened panels, determining fatigue life requires comput ationally expensive calculation of the geometric factor The computational procedure for calculating fatigue life is described in Appendix B. PAGE 46 24 Inspection Model When the structure is subjected to peri odic 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 Palmbergs equation (Palmberg et al., 1987). h h da a a a a P/ 1 / (3-11) Where ah is the crack size that will be detected with 50% probability and is inspection parameter. An approximate values of ah of 1 mm was obtained by rounding off data from the probability of detection curves in Ru mmel and Matzkanin ( 1997) for eddy current inspection. They obtained the probability of detection curves by machining artificial cracks in panels and counting th e number of times they were detected after inspecting several times. The value of the other inspection parameter of 3 was obtained by fitting Equation 3-11 to the inspection data in that reference and increasing it slightly (to account for improvement in inspection technolog y 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 stru ctural components are scheduled, the damage size distribution changes because def ective parts are replaced with new parts having a smaller damage sizes (fabrication defects, ai,0). Reliability computation is very expensive when inspections are involved b ecause crack size distribution has to be recharacterized after each insp ection to simulate replacement and exact computation of failure probability using MCS requires large sample size. PAGE 47 25 Harkness (1994) developed an approxima te method to expedite reliability computation with inspection by assuming that repaired components wi ll never fail again and incorporated this assumption by modifyi ng the first order reli ability method (FORM). The failure probability at any time following an inspection is the probability that the crack size is greater than the cr itical crack size at that time a nd that it is not detected in any of the previous inspections. Using an em pirical crack growth model like Equation 3-3 to predict crack size at any time, a probabi listic model for inspection probability of detection and a specified value of critical crack size, he calcul ated the structural reliability using the FORM method. The effect of inspections is incorporated into the FORM by integrating the (proba bility density function) PDF of undetected cracks over the failure region using numeri cal integration. The assumpti on that detected cracks are replaced and the new component will not fail during the remainder of service life greatly simplifies the numerical computation by cons idering only the PDF of undetected cracks. When inspections are needed earlier than half the servic e life, repaired components may have large probability of failure and Harknesss method may not be accurate enough. Kale et al. (2005) proposed an approximate method to simulate inspection and repair using Monte Carlo simu lation (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 probab ility 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 de termining inspection schedule for a given structure is to find the next inspection time at which the structural reliability will be lower PAGE 48 26 than the specified threshold value. The probability of failure after N cycles of loading since the most recent inspection is de fined as the event that fatigue life Nf is less than N. N m a N P m a N Pi f i f , , (3-12) where ai is the crack size distribution at the be ginning of inspection period and the fatigue life Nf is the number of cycles accumulated in growing a crack from initial crack ai 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 Sn for a given threshold reliability level is obtained by solving Equation 3-13. 0 , th i fPf N m a N P (3-13) Equation 3-13 is solved for time interval N by using bisection method between previous inspection time Sn-1 and service life Sl and for each of the bi section iteration, the first term is calculated by FORM. For an unstiffened panel FORM is very cheap; however for stiffened panel it is computationa lly expensive because calculation of fatigue life is expensive and additional computational burden is added because of the bisection search between previous inspection time Sn-1 and service life Sl. Updating Crack Size Distribut ion after Inspection using MCS The algorithm for simulating crack growth a nd inspections is shown in Table 3-3. After obtaining the next inspection time, th e 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 inte rval. The damage distribution after an inspection can easily be updated by using M onte Carlo simulation (MCS) with a small sample size and is computationally very cheap compared to calculating probabilities. The PAGE 49 27 crack size aN after N cycle of fatigue loading is obta ined by solving Equation 3-3. To obtain the crack size mean and standard deviation after an in spection, we produce 50,000 random numbers for each random variable in Equation 3-3 (ai, m, ) and obtain the final crack size aN. We then simulate the inspec tion by using Equation 3-11 with another random number for probability of detec tion. 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 ai,0. After all cracks are analyzed for detection, the updated crack sizes are used to fit a distri bution and to obtain its mean a nd 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 th e distribution after inspections was also found to be best a pproximated by lognormal distribution out of 12 analytical distributions in ARENA soft ware (Takus and Profozich, 1997). Table 3-3: Pseudo code for updati ng crack size distribution after N cycles from previous inspection (1) Generate a panel by a random vector of uncertain variables (ai, m, and ) (2) Solve Equation 3-3 for crack size aN after N cycles of fatigue loading for the panel using Newtons method or bisection (i f Newtons method doe s not converge). (3) Compute the probability of detection of crack aN from Equation 3-11, Pd(aN). (4) Generate a random number from a uni form distribution with bounds (0, 1) rand dP (5) Ifrand d N dP a P then simulate replacement of defective component by generating a random crack ai,0 for a new panel and set 0 i Na a 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 A 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. PAGE 50 28 The crack size probability distribution after the inspection is estimated by fitting probability distribution to the crack size sa mples 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 FO RM 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 3-3 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 ar e shown in Figure 3-2. The corresponding p-value is less than 0.005 indicating a bad fit; however for low failure probabilities (e.g., 10-7) this fit ensures accurate structural design calculati on 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 af ter 9288 flights is 4.0 -7 and after 15,540 flights is 2.7-7 which are close to the value of 10-7 calculated using the proposed method. The square error between actual PDF and lognormally f itted PDF is 0.00029 and the maximum error between CDFs is 0.06 at crack size of 0.28 mm PAGE 51 29 Figure 3-2: Comparison of actual and lognorma lly fitted CDF of crack sizes after an inspection conducted at 9288 flights Calculation of Inspection Sc hedule 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 3-1 illustrates the approach described in previous sections for a 2.0 mm thick unstiffened plate and a require d reliability level of 10-7. Solving Equation 3-13 for N, the first inspection time is 9288 flights. Crack gr owth simulation using the MCS pseudo code in Table 3-3 is performed with initial crack sizes ai,0 and crack growth time of 9288 flights giving the updated crack size distribution after the fi rst 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 3-13. This cycle of scheduling inspections is continued until th e failure probability at the end of service life is less than the specified value. PAGE 52 30 Figure 3-3 illustrates the va riation of the probability of failure with and without inspection. Table 3-4 presents the inspecti on schedule during the service life and the crack size distribution parameters after each insp ection. It can be seen that inspections are very helpful in maintaining the reliability of the structure. From Table 3-4 it can be seen that first inspection interval is the larges t. After the first inspection the repaired components are replaced with the same initial crack distri bution (mean = 0.20 mm and cov = 35%); however some cracks escape de tection, leading to smaller inspection intervals. From the crack size distribution parameters shown in last column of Table 3-4 we can conclude that the crack size distribut ion after each inspecti on essentially remains unchanged after a certain number of inspections leading to uniform inspection intervals. We can infer that towards the end of se rvice 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. PAGE 53 31 1.E-15 1.E-13 1.E-11 1.E-09 1.E-07 1.E-05 1.E-03 1.E-01 050001000015000200002500030000350004000045000 FlightsFailure probability No Inspections Optimal Inspections Figure 3-3: Example 3-1, Variat ion of failure probability with number of cycles for a 2.00 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7 Table 3-4: Example 3-1, Inspection schedule and crack size distribution after inspection for an unstiffened plate thickness of 2.00 mm and a threshold probability of 10-7 Number of inspections Inspection time, Sn (flights) Inspection interval (flights) 1 n nS S Crack size distribution after inspection (mean, mm cov) 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 st ructural weight for aluminum and the fuel cost is taken from Venter (1998). He assume d 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 wei ght will cost $0.015 in a fli ght for fuel. The structural PAGE 54 32 weight is assumed to be directly proportional to th e plate thickness and a pound of structural weight is assume d to cost $150 for material and manufacturing. Appendix D shows the details of material and fuel cost calculations. A typical in spection cost of about a million dollars was obtained from B ackman (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 fact ors are summarized in Table 3-5. Table 3-5: Cost of inspection, material and fuel Density of aluminum () 166 lb/ft3 Fuel cost per pound per flight (Fc) $ 0.015 Inspection cost (Ic) $ 1,000,000 Material and manufacturing cost per lb (Mc)$ 150.0 Number of panels, Np 1350 Service life, Sl 40,000 flights (Source: Venter, 1998 and Backman, 2001) The life cycle cost Ctot for Ni inspections is Ctot = Mc W + Fc W Sl + Ni Ic (3-14) Where W is the total weight of all the panels in the fuselage, given as thb b A N N Ws s p (3-15) The parameters in Equations 3-14 and 3-15 are defined in nomenclature. Reliability based design optimization is computationally very expensive when inspections are involved because several ite rations on structural design variables and inspection times are required to find an optim um 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 stiffe ned panel, the crack growth has to be determined numerically and reliability comput ations are very expensive even with the combined MCS and FORM approach. The key factor responsible for computational time PAGE 55 33 is the calculation of geometric factor due to stiffening, which can be determined using detailed finite element analysis or disp lacement compatibility method due to Swift (1984). In this chapter we used Swifts approach whic h takes about 0.5 second for evaluating single value of for a given structural desi gn and crack length. Table 3-6 explains the various RSAs used to make co mputations faster a nd Table 3-7 gives the breakdown of computational cost for calcu lation of exact insp ection time and updating crack size distribution. Table 3-6: Description of response surface approximations used in optimization Name of response surface Description Function of variables -RSA Geometric factor due to stiffeners Skin thickness ts, Stiffener area As, Crack length a ai-RSA Crack size mean after inspection Skin thickness ts, Stiffener area As, mean crack length a, mean crack length a, time N, standard deviation in stress p ai-RSA Crack size standard deviation after inspection Skin thickness ts, Stiffener area As, mean crack length a, mean crack length a, time N, standard deviation in stress p d-RSA Reliability index Skin thickness ts, Stiffener area As, mean crack length a, mean crack length a, time N, standard deviation in stress p PAGE 56 34 Table 3-7: Computational time spent in ex act calculation of next inspection time and error due to -RSA usage Variable Computational method Number of function evaluation Total time, seconds Typical error due to use of RSA Geometric factor, Displacement compatibility 1 0.5 0.02 approximately Fatigue life, Nf Numerical integration using MATLABs adaptive Simpsons quadrature 100 evaluation of 50 600 flights (error in Nf due to error in ) Reliability index, d Iterative search using MATLABs fmincon 100 evaluation of Nf 5,000 0.2, average fitting error from d-RSA + 0.1 from error in Nf Next inspection time, SN Bisection between SN -1 and service life Sl 15 evaluation of d 75,000 (0.86 days) -Computational time spent in exact updati ng of crack size distribution using MCS and error due to -RSA usage Crack size aN after N cycles Iterative search using Newtons method 20 evaluation of Nf 1,000 -Crack size distribution using step F MCS 50,000 evaluations of aN 5,000,000,0 (578 days) Less that 0.1 % from ai-RSA and ai-RSA. When structural design and inspections schedule are optimized together, the computational cost will be several hundred times that shown in Table 3-7 because of iterative search on structural sizes and number of inspections in a schedule required to 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 will be 10,000/ (1.02) m 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 insp ection time has the same error. The reliability calculated at the next inspection time plus the error in NF that is Sn+ 600 will have error in reliability index of about 0.1 (2%); however this translat es to error in probability of 200 % for a reliability level of the order of 10-7. PAGE 57 35 maintain the specified reliability level. To overcome this we develop surrogate models based on response surface approximations shown in Table 3-6. The last columns of Table 3-7 show the errors made by the use of RSAs in calculations. An error of 0.02 is the typical fitting error in construction of RSA for Table 3-8 gives the overview of the methodology describing the computational chal lenge in its implementation and explains the approach used to perform reliability based optimization of structural design and inspection schedule. The computational procedure is explai ned first followed by the computational challenge associated with implementation of this procedure. The surrogate models used to remove the computational burden involved in th is method are also expl ained. It is to be noted that the main cause of using the surroga te models is because there is no analytical expression available to calculat e crack growth for stiffened pa nel. If analytical expression for crack growth is available then exact computations will become feasible. For an unstiffened panel the computa tional 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. Howeve r, for stiffened panel, calculation of crack growth is expensive and we use res ponse surface approximation (RSA) to estimate the crack size mean and standard deviation af ter 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 cy cles of loading) as shown in Appendix C. PAGE 58 36 Table 3-8: Pseudo code for combined optimi zation of structural design and inspection schedule (A) Optimization of structural design: For a given structural design calculate inspection schedule using step B and obt ain cost of struct ural 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 reliabi lity 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 variab les, find when the next inspection is needed by calculating the time Sn at which the probability of failure equals the required reliability level Pfth using the first order re liability method (FORM). This is a computationally intensive optim ization problem which requires repetitive computation of reliability index. A rough es timate of computer time is described in Table 3-7. To reduce the computational burden associated with repeated reliability calculation during optimization we generate design re sponse surface to es timate reliability index. Appendix C explains the details of this response surface approximation (dRSA). (D) Updating crack size distribution after inspection: After obtaining the next inspection time from step C, use Monte Ca rlo simulation (MCS) to update the crack size distribution after this inspection by growing cracks between the inspection time Sn and the previous inspection time Sn-1. The MCS method is described in Tabl e 3-3 and the computational expense associated with it is described in Table 3-7. The computational burden associated w ith estimating crack size distribution parameters after an inspection is solved by using RSAs to estimate the crack size mean ai-RSA and standard deviation ai-RSA after an inspection. Appendix C explains the details of these RSAs. During the optimization the structural thickness t and the stiffener area As are changed, which changes the structural wei ght according to Equation 3-15. The optimum inspection schedule is determined for this st ructural design using Table 3-8 and the total PAGE 59 37 cost of structural weight a nd inspection is obtained from Equation 3-14. The optimization iteration is stopped after a specified converg ence tolerance is achieved. The convergence tolerance on minimum cost is assumed to be $10,000 in this ch apter and MATLAB is used to perform optimization of the design. Us ing the RSAs, 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) st ructural design. To demonstrat e this we first obtain safelife 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 3-10 shows the safelife design of a stiffened panel. Table 3-9: SafeLife design of an unstiffened panel Required probability of failure, Pfth Minimum required skin thickness t (mm) Life cycle cost Ctot $ 106 Structural weight, lb 10-7 4.08 25.42 33902 10-8 4.20 26.16 34880 10-9 4.24 26.34 35129 Table 3-10: SafeLife design of a stiffened panel Required probability of failure, Pfth Total stiffener area 10-3 m2, As Skin thickness t (mm) 100Total sA A% Life cycle cost, Ctot $ 106 Structural weight, lb 10-7 2.23 2.31 35.85 22.42 29900 10-8 2.26 2.33 36.00 22.68 30248 10-9 2.30 2.35 36.22 22.91 30555 An unstiffened panel is a single load path st ructure without load transfer capability. Comparing Table 3-9 to Table 3-10, we see that if structure is designed with multiple PAGE 60 38 load transfer capability then the weight and co st 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 fu rther explored in Appendix A. Next we demonstrate the effect of inspect ions on structural safety and operational cost. Inspections improve the reliability by detecting and removing cracks. By optimizing the structural design together with inspecti on schedule, we can trade structural weight against inspection cost to reduce overall life cycle cost. To demonstr ate the effectiveness of inspections, optimum structural design and inspection schedule were first obtained for an unstiffened panel design with results shown in Table 3-11. Table 3-11: Optimum structural design and in spection schedule of an unstiffened panel (in all cases the optimum nu mber of inspection is 3) Required probability of failure, Pfth Skin thickness t (mm) Optimum inspection times, Sn (flights) Life cycle cost Ctot $ 106 Structural weight, lb 10-7 2.30 12346,22881,31365 17.28 19109 10-8 2.43 13158,23496,31496 18.15 20199 10-9 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 safe-life unstiffened panel design and by 20% ove r to stiffened panel safe-life design. The corresponding reductions in structural weight are 40% and 30%, respectively. There is an additional incentive for conducting inspections in that they pr otect against other types of damage like that due to accidental impacts a nd corrosion. Next we optimize the structural design and inspection schedule for stiffened pa nel design (Table 3-12) and illustrate the tradeoff of structural weight in skin and stiffeners agains t inspection cost. PAGE 61 39 Table 3-12: Optimum structural design a nd inspection schedule for stiffened panel Required probability of failure, Pfth Total stiffener area, As 10-4 meter2 Required skin thickness t (mm) 100Total sA A % Optimal inspection times, Sn (flights) Life cycle cost Ctot $ 106 Structural weight, lb 10-7 7.11 1.71 19.40 10844,18625, 25791, 32908 17.20 17659 10-8 7.30 1.81 18.95 11089,18758, 25865, 32943 17.87 18504 10-9 13.74 1.67 32.29 12699,22289, 31163 18.33 20443 Comparing Table 3-10 to Tabl e 3-12 we see that inspections lower the life cycle cost of stiffened panel design by about 20% compared to safe-life design. Comparing Tables 3-11 and 3-12 we see only a small gain (about 3%) in designing stiffened structures when inspections are involved a nd cost can be minimized by designing single load path structures (unstiffe ned) with inspections. Compari ng Table 3-12 to Table 3-11, the increased design flexibility allows additional tradeoff of structural weight against inspections by having one additional inspecti on over the unstiffened panel design. About 20% to 30 % of the structural weight is transferred from skin to stiffeners. In aircraft operation the inspection interval s are dictated by practical considerations and regulation which are based on service e xperience. The Joint Service Specification Regulations-2006 requires all airlines to conduc t major depot level inspection four times during the service life. These inspections ar e conducted at uniform intervals. Table 3-13 shows the design with fixed inspection schedule. PAGE 62 40 Table 3-13: Optimum structural design for regulations based inspections conducted at four constant interval or 8000 flights for stiffened panel Required probability of failure, Pfth Total stiffener area AS 10-4 meter2 Required skin thickness t ( mm ) 100Total sA A % Inspection times, Sn Life cycle cost, Ctot $ 106 Structural weight, lb 10-7 13.41 1.38 35.94 8000, 16000,24000, 32000 17.44 17927 10-8 13.80 1.47 35.12 8000, 16000,24000, 32000 18.16 18878 10-9 14.85 1.49 36.60 8000, 16000,24000, 32000 18.61 19491 It is seen that inspections done at constant interval are only ma rginally less cost effective than the optimized inspection sche dule. From Tables 3-11, 3-12 and 3-13 we can conclude that when inspections are used to maintain safety there is less gain in using stiffeners for stable fatigue crack growth. Ho wever, stiffeners might be very useful in maintaining structural rigidity to resist buckling and pillowing. Also, from Table 3-10 when structures are designed without any insp ections, stiffeners can be very helpful in reducing crack growth rate. Appendix A discu sses the effect of s tiffening on structural design and crack growth rates. Next we obtain optimum stru ctural 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 PAGE 63 41 Table 3-14: Tradeoff of inspection cost agains t cost of structural weight required to maintain fixed reliability level for stiffened panel Required probabilit y of failure, Pfth Number of inspectio n Total stiffene r area As 10-4 meter2 Required skin thicknes s ( mm ) 100Total sA A % Optimal inspection times (flights) Life cycle cost $ 106 10-7 5 7.05 1.60 20.26 9497,16029,22064, 28060,34036 17.53 10-7 4 7.11 1.71 19.40 10844,18625,25791, 32908 17.20 10-7 3 7.23 1.88 18.14** 12743,22435,31212 17.35 10-8 5 7.00 1.70 19.18 9933,16406,22363, 28271,34145 18.14 10-8 4 7.30 1.81 18.95 11089,18758,25865, 32943 17.87 10-8 3 13.29 1.63 32.04 12514,22178, 31110 18.03 10-9 5 7.50 1.74 19.92 10091,16428,23260, 29268, 34412 18.53 10-9 4 7.89 1.88 19.51 11546,19064,26064, 33044 18.59 10-9 3 13.74 1.67 32.29 12699,22289,31163 18.33 From Table 3-14 we see that the op timum structural weight decreases monotonically with the number of inspections be cause 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 stiffe ners break during crack growth reducing reliability so that when the number of inspections are large, minimum cost can be ** There exist another local minimum with As = 14.64 mm2, ts = 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 3-14. PAGE 64 42 achieved by reducing stiffener areas and increasi ng structural thickness. Because of this there are several local optimums for structural sizes. The cost difference between these optimums is very small. Actual failure proba bility was calculated for each of the local optimums and the design whose failure probabi lity 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 insp ections about 30% of the structural weight is transferred to stiffeners. As more inspecti ons are added the structur al 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 re quiring frequent inspections to maintain reliability. Table 3-15 presents the exact evaluation of failure probability without any RSA for the optimum obtained from Table 3-12. This is done by calculating reliability using FORM without using d-RSA reliability index. Table 3-15: Exact evaluation of structural reliability for optimum obtained from RSA for stiffened panel with inspection Required reliability level, Pfth Optimum design (skin thickness mm 100Total sA A %) Inspection times, flights Actual Pf / Pfth before each inspection 10-7 1.71, 19.40 10844,18625,25791, 32908 2.89, 2.26, 1.98 3.90, 1.87 10-8 1.81, 18.95 11089,18758, 25865, 32943 0.98, 3.75, 3.35, 3.18, 3.06 10-9 1.67, 32.29 12699,22289, 31163 2.12, 5.27, 1.47, 1.44 It can be seen that RSAs can be used to predict the optimum with sufficient accuracy. The error in actual re liability versus calculated reliability using approximations can comes from the RSA for which can affect the accuracy in calculation of inspection PAGE 65 43 time by 600 flights. Additional error is intr oduced 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 RSAs can be iteratively calibrated so that the actual failure probabi lities 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 us ing a combination of MCS and FORM to perform combined optimization of stru ctural design and insp ection 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. PAGE 66 44 CHAPTER 4 TRADEOFF OF WEIGHT AND INSPECTI ON COST IN RELIABILITY-BASED STRUCTURAL OPTIMIZATION US ING MULTIPLE INSPECTION TYPES Introduction The integrity of structural components is affected by damage due to fatigue, corrosion and accidental impact. Damage ma y reduce the residual strength of the structure below what is needed to carry the se rvice loads. In a fail-safe design, structural safety can be maintained by inspecting th e components and repairing the detected damage. Alternatively, stresses can be lowe red 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 propaga tion. The damage tole rance 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 st ructure. 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 reliability-based stru ctural optimization of safe-life structures than of fail-safe structures because the optim ization of the former involves only structural sizes while for the latter the inspection regi me also needs to be optimized. Nees and Canfield (1998) and Arietta and Striz (2000, 2005) performed safe-life structural PAGE 67 45 optimization of F-16 wing panels to obtain the minimum structural weight for fatigue crack growth under a serv ice load spectrum. For aircraft fail-safe design, reliabilitybased design optimization has been applied to the design of inspection schedules. Harkness et al. (1994) developed an approximate method to evaluate reliabili ty with inspections, Provan and Farhangdoost (1994) used Markov-chains to estimate the failure probabi lity of a system of components and Brot (1994) demonstrated that using multiple inspection types could minimize cost. Fujimoto et al (1998); Toyoda-Makino (1999); En right 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. Reliability-based optimization of the stru ctural design and insp ection schedules has also been applied to pipelines subj ected 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 in spection. The corrosion information obtained from the inspection was used to update the corrosion model and corro sion 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 PAGE 68 46 damage propagation, he concl uded that increasing the structural weight is more cost effective than increasing the inspection frequency. Reliability based optimization is computat ionally very expensive when inspections are involved because crack size distributi on has to be re-characterized 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 upda ting the crack size distribution after each inspection. He assumed that repaired com ponents will never fail again and incorporated this assumption by modifying the firs t-order 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 ha lf the service life, repaired components can have a large probability of failure. In this case, Harknesss method may not be accurate enough because the repaired compone nts 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 (M CS) and calculate the failure probability using FORM. In this chapter we use the approximate method from Chapter 3 for combined optimization of structural de sign and inspection schedule using multiple inspection types. MCS is computationally very expensive for evaluating low failure 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 PAGE 69 47 probability, but is cheap for estimating proba bility distribution parameters (e.g., mean and standard deviation). We use the data obtained from MCS to approximate the probability distribution of crack size after insp ection, and obtain the mean and standard deviation. Subsequently, FORM is used to calculate the failure probabilities between inspections. The combined MCS and FORM a pproach to calculate failure probability with inspection removes the computational bur den associated with calculating the failure probability using MCS for lo w 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 applic ation of the proposed methodol ogy 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 de monstrate that if st ructures are designed together with the inspection schedule, then the cost of additional stru ctural weight can be traded against the cost of additional inspecti ons 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 fu selage panel with an initial crack size ai subjected to load cycles with constant am plitude. We assume that the main fatigue loading is due to pressurization, with st ress varying between a maximum value of to a minimum value of zero in one flight. One cycl e 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 PAGE 70 48 mK D dN da (4-1) where a is the crack size in meters N is the number of cycles (flights), da/dN is the crack growth rate in cycle meters /, and the stress inte nsity factor range K is in meters MPa For 7075-T651 aluminum alloy, D and m are material parameters related by Equation 4-2 obtained from Sinclair and Pierie (1990). D has units in m MPa m meters 2 1 47 12 2 3 me D (4-2) The stress intensity factor range K for a center cracked pa nel is calculated as a function of the stress and the crack length a in Equation 4-3, and the hoop stress due to the pressure differential p is given by Equation 4-4 a K (4-3) t pr (4-4) where r is the fuselage radius and t is the skin thickness. The number of cycles of fatigue loading N accumulated in growing a crack from the initial crack size ai to the final crack aN can be obtained by integrating E quation 4-1 between the initial crack ai and the final crack aN Alternatively, the crack size aN after N cycle of fatigue loading can be obtained by solving Equation 4-5 m m i m N a a mm D a a a D da NN i 2 12 1 2 1 (4-5) The fatigue life of the panel can then be obt ained by substituting the critical crack length ac in place of aN in Equation 4-5 PAGE 71 49 m m i m c fm D a a N 2 12 1 2 1 (4-6) Here we assume that the critical crack length ac is dictated by considerations of crack stability, so that 2 IC cK a (4-7) and KIC is the fracture toughness of the materi al. Typical material properties for 7075T651 aluminum alloy are presented in Table 4-1. A conservative distribution of initial defects was chosen following the Department of Defense Joint Service Specification Guide for aluminum alloys to account for un certainties in damage initiation and growth. The applied fuselage pre ssure differential is 0.06 MPa obtained from Niu (1990) and the stress is given by Equation 4-4. Table 4-1: Fatigue propertie s of 7075-T651 Aluminum alloy (Source: Sinclair and Pierie, 1990 and Niu, 1990) Inspection Model When the structure is subjected to peri odic inspections, cracks are detected and repaired or the structural part is re placed. We assume that the probability Pd, of detecting a crack of length a is given by Equation 4-8 (Palmberg et al ., 1987) Property Initial crack ai,0, mm Paris exponent, m Pressure, p ( MPa ) Fracture toughness, KIC MPa meters Fuselage radius, r Meters Distribution type, mean, standard deviation Lognormal 0.2,0.07 Lognormal 2.97, 1.05 Lognormal 0.06, 0.003 36.58 Deterministic 3.25 PAGE 72 50 h h da a a a a P / 1 / (4-8) Where ah is the crack size that will be detected with 50% probability and 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 Rumm el 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 opera tors 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 parameter, as shown in Figure 4-1, was obtained by fitting Equation 4-8 to the inspection data in these referen ces. The probability of detection curves for different inspection types are s hown in Figure 4-1. 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 w ith a smaller defect size distribution (fabrication defects). PAGE 73 51 Figure 4-1: Probability of detection curve for different inspection types from Equation 48 Calculating an Inspection Schedule Estimating Crack Size Di stribution after Inspection When inspection and replacement of stru ctural components are scheduled, the damage size distribution changes because def ective parts are repla ced with new parts having a smaller value of the damage size (fabrication defects). Reliability-based optimization is computationally very expens ive when inspections are involved, because crack size distribution has to be re-charact erized after each inspection to simulate replacement and exact computation usi ng 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 first-order reliabi lity method (FORM). Using this method, FORM is updated over the failure region after each inspection using num erical integration. This expedites the PAGE 74 52 reliability computations, which require only that the initial crack size distribution be specified. When inspections are scheduled after ha lf the service life, this method gives accurate results because the repaired compone nt will not fail until the end of service. However when inspections are scheduled befo re half the service life, the repaired components can have a large probability of failure and Harknesss 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 simu lation (MCS) and evaluated the failure probability using the first-order reliability method (FORM). Ev aluating low failure probability using MCS requires a large sample size, which makes reliability-based 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 distri bution of crack size afte r 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 ch ange the type of di stribution and that damaged components are replaced by new comp onents with damage distribution due to fabrication. The damage dist ribution after an in spection 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 4-5. To obtain the crack size m ean and standard deviation after an inspection is co nducted, we produce 100,000 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. PAGE 75 53 variable in Equation 4-5 ( ai, m, ) and obtain the final crack size aN. We then simulate the inspection by using Equation 4-8 with a nother random number for probability of detection. If the crack is detected, the pane l is replaced by a new one with a random crack size picked from the distribu tion of manufacturing defects ai,0. After all cracks are analyzed for detection, the updated crack sizes are used to fit a dist ribution 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 chap ter, the fabrication crack distribution is lognormal, and the distribution after inspections was also f ound 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 fi t to data. The algorithm for simulating crack growth and inspections is shown in Table 4-2. Table 4-2: Pseudo code for upda ting crack distribution after N cycles from previous inspection Generate a panel by a random vect or of uncertain variables ( ai ,m ) Solve Equation 4-5 for crack size aN after N cycles of fatigue loading corresponding to the random vector ( ai, m, ) Compute the probability of detection of crack aN from Equation 4-8, Pd (aN) Generate a random number from a un iform distribution with bounds (0, 1) rand dP Ifrand d N dP a P then simulate replacement of defective component by generating a random crack ai,0 for a new panel and set 0 ,i Na a else keep aN Store aN for fitting probability distributi on to crack sizes after inspection Stop after 100,000 random panels have been simulated and fit distribution to crack sizes Calculating the Failure Probability Usin g the First-Order 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. PAGE 76 54 However, since the required probabilities of failure are of the order of 10-8, this would require a prohibitively large MCS. So inst ead we use FORM, taking advantage of the characterization of the crack distribution as lognormal, as described in the previous section and in Table 4-2. The probability of failure after N cycles of fatigue loading since the most recent inspection is c i i fa a N a P a N P , (4-9) where ai is the crack size distribu tion (either initial or update d) at the beginning of the inspection period and ac is the critical crack given by Equation 4-7. This probability is calculated by the first-order reliability method (FORM). For a given structural thickness, optimum inspection times are obtained such th at 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 in spection, because cracks are detected and repaired. With the number of cycles of loadi ng (flights), the failure probability increases until it hits the threshold value again, defining the next inspection. The nth inspection time Sn is obtained by solving Equation 4-9 using a bisection method between the previous inspection time Sn-1 and the service life Sl (40,000 flights). To ascertain whether the number of inspections is adequate, the probabi lity of failure at the end of service is calculated. If this failure probability is gr eater than the threshold value, additional inspections must be added. The combin ed MCS and FORM approach removes the computational burden associated with calcula ting the failure probability using MCS for very low failure probabilities. Time, cycle and flights are used interchangeably in this chapter because for on e cycle of fatigue loading is equal to one flight in a fuselage and time is measured in number of flights. PAGE 77 55 To demonstrate the application of th e combined FORM and MCS method to calculate the failure probability we calculate the inspection time for a threshold reliability level of Pfth = 10-7 in Table 4-3 for a 2.48 mm thick panel. Calculating c ia a N a P using FORM in Equation 4-9 with ai = ai,0 and solving for N, the first inspection time is 14569 flights. To update the crack size distribu tion after this inspec tion, crack growth simulation using the MCS pseudo code is performed with in itial crack sizes ai,0 and a crack growth time of 14569 flights. This gi ves the updated crack size distribution after the first inspection ai (mean = 0.264 mm, cov = 1.04). This serves as the initial crack size distribution for the second insp ection. The second inspecti on time is obtained by solving Equation 4-9 using FORM with the upda ted initial crack size distribution ai (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 4-2 illustrates the va riation 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 4-3 it can be seen th at the first inspection interval is the largest. After the first inspection, damaged component s are replaced with th e same initial crack distribution (mean = 0.20 mm and cov = 0.35); however some cracks may have escaped detection. The fact that some cracks (large r than the initial crac ks) may have escaped detection and will grow faster leads to smaller intervals. PAGE 78 56 Table 4-3: Example 4-1, 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 Number of inspections Inspection time (flights) Inspection interval (flights) 1 n nS S Crack size distribution after inspection (mean, 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 differ by a maximum value of 100 flights due to MCS seed; the corresponding error in probability calculat ion is negligible. 1.00E-20 1.00E-18 1.00E-16 1.00E-14 1.00E-12 1.00E-10 1.00E-08 1.00E-06 1.00E-04 1.00E-02 1.00E+00 050001000015000200002500030000350004000045000 FlightsFailure probability No-Inspections Inspections Figure 4-2: Variation of failure probability with number of cycles for a 2.48 mm thick unstiffened panel with inspections scheduled for Pfth = 10-7 The previous example showed how an optimum inspection schedule can be developed for a single inspection type. The sa me procedure is followed for scheduling the inspection sequence with multiple inspection ty pes. Here the probability of detection of each inspection type can be different and insp ections are performed in the same order as specified in the sequence. If th e specified reliability level cannot be maintained with the inspection sequence, then it is not feasible. PAGE 79 57 Cost Model The cost associated with a change in th e structural weight for aluminum and the fuel cost is taken from Ve nter (1998). He assumed a fuel cost of $0.89 per gallon and calculated that a pound of structur al weight will co st 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 assu med to be directly proportiona l 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 insp ections. 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 insp ections of the fourth type. The structural design parameters for the B747 series aircraft obtained by Niu (1990) are also listed in Table 4-4. Following Backman (2001) the service life Sl is assumed to be 40,000 flights. Table 4-4: Design detail s and cost factors Density of aluminum 166 lb/ft3 Frame spacing b 0.6 m Fuel cost per pound per flight Fc $ 0.015 Fuselage radius r 3.25 m Length l 68.3 m Material and manufacturing cost per pound Mc $ 150.0 Number of panels Np 1350 Panel width h 1.72 m Type 1 inspection cost Ic1 $ 1.35 million Type 2 inspection cost Ic2 $ 1.23 million Type 3 inspection cost Ic3 $ 0.98 million Type 4 inspection cost Ic4 $ 0.85 million (Source: Venter, 1998 and Backman, 2001) PAGE 80 58 The life cycle cost is calculated as 4 1k k ck l c c totN I WS F W M C (4-10) thb N Wp (4-11) During the optimization, the structural thickness t changes, this in turn changes the structural weight according to Equation 4-11. The optimum inspection schedule (times and types) is determined for this structural design and the total co st is obtained from Equation 4-10. Optimization of Inspection Types The combined optimization of inspecti on times to minimize the cost for the specified reliability constraint is complicated because of the large number of permutations of inspection types that can o ccur 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 th en 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). (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 PAGE 81 59 inspections required by the least effective in spection in the sequence. Also, the total number should be at least equal to the numbe r of inspections required by most effective inspection in the sequence 4 1 0 0max mink b k N k b k NN N Nk k (4-12) ck min kI C N (4-13) (3) Generate the cheapest inspection seque nce satisfying Equations 4-12 and 4-13. (4) Generate the inspection times for the in spection sequence and check if the failure probability at the end of the service life is less than the specified reliability constraint.0 th l fPf S P(i.e., whether the inspection sequence is feasible.) (5) Stop if the sequence is f easible otherwise generate the next cheapest inspection sequence and go back to step 4. Combined Optimization of Structur al 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 safe-life thickness, which is the minimum thickness necessary to maintain the thres hold probability of failure without any inspection. In order to determine whether ad ditional inspections reduce cost, we do a onedimensional search on the thickness by re ducing the thickness gradually and obtain optimum inspection schedule using the algorithm described in Appendix E until we get the optimum lifetime cost. PAGE 82 60 Safe-Life Design When inspections and maintenance are not feasible, safety can be maintained by having a conservative (thicker) structural design. To demonstr ate this, we first obtain the safe-life design required to maintain the desi red level of reliability throughout the service life. Table 4-5 shows the safe-life design. Table 4-5: Structural size re quired to maintain a specified reliability level without and inspection. Threshold probability of failure, Pfth Minimum required skin thickness (mm) Life cycle cost $ 106 Structural weight, lb % Increase in cost of improving reliability by a factor of 10 10-7 4.08 25.42 33902 -10-8 4.20 26.16 34880 2.91 10-9 4.24 26.34 35129 0.68 From Table 4-5, comparing the minimum thickne ss to that used in Example 4-1 (Table 43), we see that the safe-life design must be ve ry thick and heavy in order to maintain the required safety levels. Cost Effectiveness of Combined Optimization Next we demonstrate the effect of inspect ions on structural safety and operational cost. Inspections improve the reliability by det ecting and removing cracks. If this effect is used to optimize the structural design togeth er with the inspection schedule, then the structural weight could be traded against the inspecti on cost to re duce the overall operational cost. The results of combined st ructure and inspection optimization are shown in Table 4-6. PAGE 83 61 Table 4-6: Optimum structural design a nd inspection schedule re quired to maintain specified threshold reliability level Cost factors % Threshold probabilit y of failure, Pfth Optimum plate thickness (mm) Optimum inspection type sequence Optimum inspection time (flights) Minimu m cost $ 106 Fc Mc Ic 10-7 2.48 I1, I4, I3 14569, 26023, 32706 18.66 66 16 18 10-8 2.54 I1, I3, I1 14321, 23780, 30963 19.47 64 16 20 10-9 2.66 I1, I3, I1 15064, 23532, 30023 20.27 65 16 19 From Table 4-6 we can see that if inspec tions are added, the structural thickness can be reduced to maintain the required reliab ility level at a lower cost. Inspections are very useful in maintaining the structural safe ty in that large crack s are detected and the damaged part is replaced with new components improving the fatigue life. In this chapter we optimize the inspection schedule for fati gue 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 co mpared to the safe-life design. The combined optimization of structural design and inspectio n schedule leads to tradeoff of the costs of structural weight ag ainst the inspection cost. Comparing Tables 45 and 4-6, we can see that adding inspection l eads to a 25% saving in life cycle cost over the safe-life 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 10-7, cheaper inspections can be used (I4 and I3), whereas for 10-9 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 PAGE 84 62 10-7, 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 inspec tion types is not suitable for a reliability level of 10-8. The last columns of Tabl e 4-6 show the cost factor s in percentage of fuel cost (Fc), manufacturing cost (Mc) and inspection cost (Ic). 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 4-6 for a reliability level of 10-7. Table 4-7: Comparison of optimum inspection schedule using a single inspection type for a fixed structural size Optimum inspection type sequence using a single inspection type Threshold probability of failure, Pfth Optimum plate thickness (mm) I1 I2 I3 I4 Minimum cost $ 106 10-7 2.48 14569, 26053, 35576 14569, 24683, 33430 14569, 19596, 29502, 35156 14569, 18991, 25952, 30128, 38167 I2 : 19.17 Inspection cost $ 106 4.05 3.69 3.92 4.25 Table 4-7 shows the inspection schedule s and cost for the inspection sequence generated using individual insp ection types for a fixed structural size. Compar ing Tables 4-6 and 4-7, 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 different ial is partly due to different safety margins at the end of service. With combined structur al and inspection optimization, the margin at PAGE 85 63 the end of the service life is removed by a re duction in structural thickness. This leads to a smaller incentive for multiple inspec tion types, as shown in Table 4-8. Table 4-8: Optimum structural design and inspection schedule using only a single inspection type Threshold probability of failure, Pfth Optimum plate thickness (mm) Optimum inspection time (flights) Optimum inspection type Minimum cost $ 106 10-7 2.39 13317, 18651, 26642, 32460 I3 18.78 10-8 2.50 13971, 22897, 31443 I1 19.65 10-9 2.64 14975, 19642, 26230, 32670 I3 20.41 Comparing Tables 4-6 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 op timization 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 4-6, 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 10-7. The optimum plate thickness is 2.02 mm and a comparison of the optimum inspec tion schedule using different inspection sequences is shown in Table 4-9. PAGE 86 64 Table 4-9: Optimum structural design (plate thickness of 2.02 mm) and inspection schedule for Pfth = 10-7 Inspection type sequence Optimum inspection time (flights) Minimum cost $ 106 I1 9472, 14383, 20204, 25583, 31192, 36623 30.71 I2 9472, 13431, 17315, 21659, 26191, 30784, 35359, 39917 32.45 I3 9472, 11290, 17422, 20206, 25773, 29006, 34178, 37711 30.45 I4 9472, 11001, 14575, 17130, 21277, 24231, 28099, 31225, 34907, 38178 31.11 I1, I3, I3, I3, I3, I3, I3 (Optimum schedule) 9472, 14383, 18120, 22941, 26770, 31495, 35433 29.84 Comparing Table 4-9 to the fi rst row of Tables 4-6 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 4-9, the fuel cost is 66%, the manuf acturing cost 8% and inspections are 26% of the total cost. It shoul d be noted; however fatigue is not the only structural design driver, so th at at lower thicknesses, othe r structural constraints may dominate. Summary A methodology for developing optimum in spection type sequences, time, and structural thickness was develope d for fuselage panels. Uncertai nty in material properties, crack sizes and loads were considered. Th e 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 simula tion. Inspections a nd structural sizes were optimized so that a given threshold probability of failure was never exceeded. PAGE 87 65 CHAPTER 5 EFFECT OF SAFETY MEASURES ON RELIABILITY OF AIRCRAFT STRUCTURES SUBJECTED TO FATIGUE DAMAGE GROWTH Introduction Safety of aircraft structures is largel y maintained by using conservative design practices to compensate for the uncertaintie s in the design proce ss and service usage. Typically, conservative initial defect specification, A-basis material properties, safety factor in fatigue life and conservative loads ar e used to design struct ures for fatigue crack growth. This is further augm ented by quality control measur es like certification testing and inspections. The main reason for using se veral layers of safety measures is the uncertainties involved (e.g., loading, materi al properties, accidental impact etc.). It has been shown that the life of a stru cture cannot be accurately determined even in carefully controlled cond itions because of variability in material properties, manufacturing defects etc. (Sin clair and Pierie, 1990). Because of uncertainty in damage initiation and growth, a damage tolerance appr oach 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 duri ng 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 a ll structures designed for damage tolerance PAGE 88 66 should be shown to avoid failure due to fa tigue, manufacturing de fects and accidental damage. As an example, we study here the design of fuselage structures for fatigue failure due to pressure loading and compute the serv ice failure probability and the effect of safety measures and certification tests. De sign of fuselage structures for fatigue is described in detail in Niu (1990). Ahmed et al. (2002) studied th e initiation and distribution of cracks in fu selage panel by performing fa tigue testing. Structural optimization and inspection scheduling of fusela ge 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 fa tigue crack growth and found that damage growth and residual strength were two main design drivers. Widespread fatigue damage in fuselage structure has been investig ated by testing panels according to FAA regulations (Gruber et al., 1996, FAA/AR-95/47). The combined effects of various safety meas ures used to design structures for static strength were studied by Acar et al. (2005). They studied the in teraction of uncertainties, safety factors and certification tests on safety of aircraft st ructures designed for static strength. As in earlier studies, the effect of variability in geometry, loads, and material properties is readily incorpor ated here by the appropriate random variables. However, there is also uncertainty due to lack of knowledge (epistemic uncertainty), such as modeling errors in the analysis. To simulate these epistemic uncertainties, we transform the error into a random variable by consideri ng the design of multiple aircraft models. As a consequence, for each model the structure is di fferent. It is as if we pretend that there are hundreds of companies (Airbus, Boeing, Bombardier, Embraer etc.) each designing PAGE 89 67 essentially the same airplane, but each having di fferent errors in their structural analysis. The objective of this chapter is to study the e ffect of individual safety measures used to design structures for fatigue damage growth. The effectiveness of cer tification testing as a function of magnitude of safety fact ors 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 5-1 distinguishes be tween (1) uncertainties that apply equally to the entire fleet of an aircraft model and (2) uncertainties that vary for individual aircraft. The distin ction is important because safe ty 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 gi ven aircraft model (e.g., Boeing 737-400) 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, erro rs in load calculations, or use of materials and tooling in construction that are differe nt from those specified by the designer. Systemic errors affect all the copies of th e structural components made and are therefore fleet-level uncertainties. They can reflect differences in analysis, manufacturing and operation of the aircraft from an ideal. The id eal aircraft is designe d 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 specificati ons, and that the operating environment agrees on average PAGE 90 68 with the design specifications. The other type of uncertainty reflects variability in material properties, geometry, or loading betw een different copies of the same structure. Table 5-1: Uncertainty classification Type of uncertainty Spread Cause Remedies Systemic error (Modeling errors) Entire fleet of components designed using the model Errors in predicting structural failure. Testing and simulation to improve math model. Variability Individual component level Variability in tooling, manufacturing process, and flying environment. Improve tooling and construction. Quality control. In this chapter, we focus on design of fuselage structures for fatigue damage growth due to pressure loading. To simula te the effect of these modeling errors we pretend that there are several aircraft co mpanies designing the same airplane but each having some different stress calculations and crack growth model leading to different designs. Because these models are only approxi mate, 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 la rgely using code-based design rather than probabilistic approaches. Safety is improve d through conservative de sign practices that include use of safety factors and conservative material properties. It is also improved by certification tests that ca n 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 PAGE 91 69 airplanes and in the Department of Defe nse, Joint Services Specification Guide-2006 (JSSG). Safety Margin on Load. Aircraft structures should be demonstrated to withstand 1.5 times the limit load without failure. For damaged fuselage stru cture, it should be demonstrated by tests that th e structure has enou gh 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 c onservative material properties. This is determined by testing a specified number of coupons selected at random from a batch of material. The A-basis property is defined as the value of a material property exceeded by 99% of the population with 95% confidence, an d the B-basis property is that exceeded by 90% of the population with 95% confidence. For structures without redundancy, A-basis properties are used and for st ructures with redundancy, B-ba sis properties are used. The conservative material properties considered here are A-Basis values of crack growth exponent (leading to rapid crack propagation), yield stress Y and fracture toughness KIC. Appendix H describes the methodology for obtai ning these properties from coupon tests. Conservative Initial Crack. The FAA requires all damage to lerant structures to be designed assuming initial flaw of maximum probable size that could exist because of manufacturing or fabrication. The JSSG-2006 damage tolerance guide lines specify this value as the B-basis crack size. This is the value that can be detected by an eddy current inspection with a 90% probabi lity and 95% confidence. Safety Factor on Life. Typically, transport aircraft are designed with a safety factor of two on the fatigue life. Fatigue te sts are conducted to validate the structural PAGE 92 70 design and the FAA requires that no aircraft be operated for more than half the number of cycles accumulated on a full-scale fatigue test. Inspections. Inspections provide protection agai nst structural failure by detecting damage. The FAA requires that inspection schedule should be in place to detect and repair damage before it grows to unacceptabl e size causing structural failure. The JSSG2006 damage tolerance guidelines specify that fail-safe multiple load path structures should be designed for depot-level inspecti ons every one quarter of service life. Component and Certification Tests. Component tests and certification tests of major structural components reduce stress a nd material uncertaintie s due to inadequate structural models. These te sts are conducted in a buildi ng block procedure. First, individual coupons are tested, and then a subassembly is tested followed by a full-scale test of the entire structure. Since these tests cannot apply to every load condition, they leave uncertainties with respect to some lo ad 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. Ce rtification testing may be c onducted in two ways. A panel may be randomly chosen out of a batch and s ubjected 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 B-basis value of crack size, which is also used to design the panel. A summ ary of fatigue testing of fuselage panels is documented in FAA/AR-95/47 (Gruber et al., 1996). Simulation Procedure for Calculation of Probability We simulate the effect of these safety m easures by assuming statistical distributions of uncertainties and incorpor ating them in Monte Carlo simulation. For variability, the simulation is straightforward. However, while systemic errors are uncertain at the time of PAGE 93 71 the design, they will not vary for a single st ructural 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 aircra ft. This creates a two-level Monte Carlo simulation, with different aircraft models being considered at the upper level, and different instances of the same aircraft at the lower level. Figure 5-1: 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 fi xed for each aircraft manufacturing company. We have N different aircraft companies designing essentially the same airplane but with different erro rs in their calculations. During the design process, the only random quantities are the conservative material properties and the PAGE 94 72 model errors k and e. In the outer loop of the Monte Ca rlo simulation (Figure 5-1, 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 di fferent because of manufacturing variability. We simulate the effect of this variability by producing M instances of the design obtained in step A (Figure 5-1) from a ssumed statistical distributions for material properties and thickness distribution. Next we simulate certification testing in step B of Figure 5-1 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 5-1) a nd counting the number of failed structures. Damage Growth Model The typical structural desi gn process involves determini ng structural sizes that would be sufficient to meet given strengt h 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 to a minimum value of zero in one flight. One cycle of fatigue lo ading consists of one flight. To model fatigue, we assume that crack growth follows the Paris model. Equation 5-1 represents the rate of crack growth da/dN in terms of stress intensity. PAGE 95 73 m calcK D dN da (5-1) where a is the crack size in meters, N is the number of cycles (flights), dN dais the crack growth rate incycle meters /,and the stress intensity factor range K is in meters MPa. For 7075-T651 aluminum alloy, D and m are material parameters related by Equation 5-2 obtained from Sinclair and Pierie (1990). D has units in m MPa m meters 2 1 47 12 2 3 me D (5-2) We use the subscript calc to note that relations, (such as Equation 5-1), that we use in the analysis, provide only approximations to true values. For a cen ter cracked infinite panel with far field stress and one cycle of fatigue lo ading per flight we have a Kcalc (5-3) Equation 5-1 is integrated to obtain estimated fatigue lifecalc fN m calc m i m c calc fm D a a N 2 12 1 2 1 (5-4) where the computed stresscalc (as obtained from finite element analysis) is different from the actual stresstrue in the structure. Here ai is the initial crack size and the critical crack length ac is the crack length at which crack gr owth becomes unstable. The critical crack length can be obtained as a function of fracture toughness KIC and from Equation 5-5. 2 IC cK a (5-5) PAGE 96 74 Equation 5-1 represents an approximate value of crack growth rate because it is obtained by fitting empirical model to observed crack growth data and calculated stresscalc The true crack growth rate is di fferent from that estimated by Equation 5-1. We include and error factor e in analysis and then Equa tions 5-1 and 5-4 become. m trueda eDK dN (5-6a) m true m i m c true fm D a a e N 2 1 12 1 2 1 (5-6b) The actual stresstrue in the structure due to applied loading is different from the calculated stress calc used to design the structure. Equati on 5-7 represents the error in the calculated stress, through an error parameter k. calc truek 1 (5-7) 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. PAGE 97 75 Table 5-2: Distributions of errors, design and material parameters for 7075-T6 aluminum Variables Distribution Mean Coefficient of variation, (standard deviation/ mean) ai, initial crack Lognormal 0.2 mm 35 % e, error in crack growth Lognormal 1.0 Variable k, error in stress Uniform 0.0 Variable KIC, fracture toughness Lognormal 30.5 MPa-m.5 10% m, paris exponent Lognormal 2.97 17% Nl, service life Deterministic40,000 flights -Ns, design life Deterministic10,000 flights -p, pressure load Lognormal pd 8.3 psi (0.057 MPa) 2.5 % r, fuselage radius Deterministic3.25 m -SFL safety factor on life Deterministic2 -SF, safety factor on load Deterministic1.5 -tact, actual thickness Lognormal tdesign mm 3% Y, yield stress Lognormal 495 MPa 5% (Source: Sinclair and Pierie, 1990 and Niu, 1990. A lognormal distribution for 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 5-2 lists uncertainties in form of e rrors and variability in the life prediction and structural design model assumed here fo r 7075-T6-aluminum alloy. Typical service life of 40,000 flights is obtained from Back man (2001). In this chapter we demonstrate the effect of safety factors on two design crite ria, (i) safe-life: structure is designed for safe crack growth for the entire servi ce life of 40,000 flights; no inspections are performed (ii) fail-safe: structure is designed for safe crack growth until the next inspection (10,000 flights). The typical inspection in terval of 10,000 flights was obtained from JSSG-2006. PAGE 98 76 Calculating Design Thickness This section determines the design thickne ss calculation in step A of Figure 5-1. The calculated stresscalc on the structure is found from Equation 5-8 representing hoop stress due to pressure loading. t pCcalc (5-8) where C is a function of the geometry, p is applied pressure differential and t is structural thickness. In subsequent calcu lations for stress in fuselage components in hoop direction, the parameter C was approximated for convenience by the value for a cylindrical pressure vessel.*** Cr (5-9) where r is the radius of fuselage. Combining E quations 5-7 and 5-8, the stress in the structure is calculated as t rp ktrue 1 (5-10) To design a panel for fixed life we first obtain the stress level required to grow the initial crack ai to critical crack ac during the design life. This is obtained by solving Equation 56b with all the safety measures. The safety measures considered are using mA, the ABasis value of m aiB the B-basis value of ai, the conservative value of critical crack acA obtained using A-basis value of KIC in Equation 5-4, A-Basis 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. PAGE 99 77 0 2 1 1 12 1 2 A A Am iB m fatigue design A IC A s FL m fatigue designa K m eD N S (5-11) Solving Equation 5-11 for fatigue designgives the stress that should not be exceeded so that the fatigue life does not fall below twice the se rvice life. Further, 1.5 times the maximum load in any component should not exceed the yiel d stress of the material to prevent static failure. We assume that the A-Basis property of yield stress is used to design the structure for static strength. yield Basis A yield design (5-12) Equations 5-11 and 5-12 give two different va lues of allowable design stress. Also the FAA requires that the damaged structure shou ld have sufficient residual strength to withstand 1.15 times the limit load without fail ure. 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 th e three criteria are satisfied. For the undamaged structure, a safety factor SF = 1.5 on design load is also specif ied, then structural thickness is designed with a design load pd multiplied by a safety factor SF, hence the design thickness of the structural is ca lculated from Equation 5-13 as yield design d yield design d fatigue design d mean designr p k r p k r p k t 80 0 15 1 1 5 1 1 1 max (5-13) PAGE 100 78 For the example considered here, the second and thir d components of Equation 513 are less critical than the first. The thickness obtained from Equation 5-13 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 (t hat is, design thickness determined from Equation 5-13), we simulate certification test ing for the aircraft. We assume that the structure will not be built with complete fi delity 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 5-1 (i) randomly selected structur al 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 Fatigue certification test: 0 2 12 1 2 1 m act d m i m c s FLt r p m D a a N S (5-14a) Static certification test: 0 f act d F ft r p S (5-14b) where the actual thickness tact is uniformly distributed with mean equal to tdesign and 3% bound. If the structural design fa ils either test than that de sign is rejected. Here the thickness t initial crack ai, fractures toughness KIC, yield stress f and Paris model constant m are random variables (see Table 5-2). This procedure of design and testing is PAGE 101 79 repeated (steps A-B, Figure 5-1) for N different aircraft mode ls. For each new model, different random error factor k and e are picked for the design, and different allowable properties are generated from coupon test ing (Appendix I). If the design passes the certification tests, then M instance of the design are generated to simulate variability in the aircraft fleet. Service Simulation To simulate failure in service we assume that the structure is required to withstand service stress for the specified design life Ns. The inner loop in Figure 5-1 (steps C-E) represents the simula tion of a population of M fuselage designs that all have the same design. However, each structure is different due to variability in geometry (thickness t ), material properties ( KIC, ai, m f) and loading, p (step D, Figure 5-1). We subject the structure in each airplane to actual random maximum (over a lifetime) service loads (step E, Figure 5-1) and calculate whet her it fails using Equation 5-15. 0 0 2 12 1 2 1 f ac t f m act m i m c st pr t pr m D a a N (5-15) where the actual thickness tact is uniformly distributed with mean equal to tdesign and 3% bound. If the structural design fa ils either test than that de sign is rejected. Here the thickness t initial crack ai, fractures toughness KIC, yield stress f and Paris model constant m are random variables (see Table 5-2). Th is procedure of design and testing is repeated (steps A-B, Figure 5-1) for N different aircraft mode ls. For each new model, different random error factor k and e are picked for the design, and different allowable PAGE 102 80 properties are generated from coupon testing (Appendix H). If the design passes the certification tests, then M instance of the design are generated to simulate variability in the aircraft fleet. We add up the number of structures failed for each airplane, and calculate the failure probability by dividing the number of failure s by the number of airplane models that passed certification times M The values of N and M used here for calculation of failure probability are 2000 a nd 5000, respectively. The following symbols are used to describe the results Table 5-3: Nomenclature of symbols used to calculate failure probability and describe the effect of certification testing Symbol Description Bk Bounds on error k Pc Probability of failure with certification Pnc Probability of failure without certification R Percentage of designs rejected by certification SFeff Ratio of mean design thickness with safety measures to that without any safety measure tcert Structural thickness of certified structures tdesign Thickness of designed structures In this chapter we demonstrate the effect of safety factors on two design criteria (i) safe-life: structure is designed for safe crack growth (fatigue life) for the entire service life of 40,000 flights; no inspecti ons are performed (ii) fail-saf e: structure is designed for safe crack growth until the next inspection (10, 000 flights). Typical inspection interval of 10,000 flights was obtained from JSSG-2006. Results Effect of Errors and Testing on Structural Safety Effects of All Safety Measures. We calculate first the failure probability of structures designed using all the safety measures. The sa fety measures considered are the A-Basis m the B-basis ai, the A-basis KIC, the A-Basis yield stress, SF of 1.5, the safety factor on PAGE 103 81 static load and SFL of 2, safety factor on fatigue life. Calculations are done for two levels of e and three levels of k For small probability of failure, simulations performed using M = 10,000 can be inaccurate because some batche s may have no failure at all. This can produce large scatter in fa ilure probability among de signs. We also performed calculations with N = 500 designs and M = 100,000 instances of each design. These are not necessarily more accurate, but they give an idea of the MCS errors for these low probabilities. Tables 5-4 and 5-5 show the fa il-safe structures de signed for fatigue life until the next inspection (10,000 flights) and Tables 5-6 and 5-7 demonstrate the calculation for safe-life structure designed fo r the entire service life (40,000 flights). Table 5-4: Probability of failure for 10 % COV in e and different bounds on error k using all safety measures for fail-safe desi gn for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 <10-8 <10-8 0.0 2.08, 0.14 2.08, 0.14 -0.3 6-6 (6.7) 6-6 (6.7)0.4 2.08, 0.38 2.08, 0.38 1.0 0.5 3-3 (5.7) 7-3 (3.9)6.0 2.14, 0.57 2.08, 0.60 0.42 *Using N = 500 designs and M = 100,000. For Bk = 0.3, we have Pc = 510-6 (5.9) and Pnc = 510-6 (5.6). The coefficient of variation among batches reduces b ecause failure probability of each batch is computed more accurately because of larger M Since calculation of failure probability is based on Monte Carlo simulation (MCS), there will be error because of finite sample size. The failure probability before and after certification tests presented in each row is the average failure probability of all the designs used in service. The st andard error (estimate of standa rd deviation) in this value due to MCS is reduced by a factor of or 70.7 from the standard deviation of the probability. For example, in Table 5-4 the probability of failure is given as 6-6 with a coefficient of variation of 6.7. The coeffici ent of variation of th e mean probability is PAGE 104 82 6.7/70.7 or approximately 10%. So we only present the number of significant digits consistent with the size of MCS. Accuracy can be improved by using larger sample sizes. Table 5-5: Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for fail-safe desi gn for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 8-5 (8.2) 9-5 (7.4) 1 2.05, 0.39 2.04, 0.40 0.88 0.3 8-4 (7.5) 2-3 (8.7) 3 2.07, 0.50 2.04, 0.51 0.4 0.5 3-3 (6.9) 2 -2 (3.8)12 2.17, 0.64 2.04, 0.70 0.15 Using N = 500 designs and M = 100,000. For Bk = 0.1, we have Pc = 410-5 (8.3) and Pnc = 610-5 (7.5) and for Bk = 0.3, we have Pc = 810-4 (6.9) and Pnc = 310-3 (7.7) Table 5-6: Probability of failure for 10 % COV in e for different bounds on error k using all safety measures for safe-life desi gn of 40,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc RB %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 <10-8 <10-8 0 3.58, 0.24 3.58, 0.24 1.0 0.3 3-5 (5.2) 4-5 (4.8)0.8 3.60, 0.64 3.59, 0.64 0.75 0.5 3-3 (4.3) 5-3 (3.6)4 3.68, 0.98 3.61, 1.02 0.6 Using N = 500 designs and M = 100,000. For Bk = 0.1, we have Pc = 110-7 (8.1) and Pnc = 110-7 (8.1) and For Bk = 0.3, we have Pc = 410-5 (5.1) and Pnc = 410-5 (4.9) Table 5-7: Probability of failure for 50 % COV in e for different bounds on error k using all safety factors for safe-life design of 40,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design) Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 3-4 (6.2) 3-4 (5.7)1 3.53, 0.74 3.52, 0.74 1.0 0.3 3-3 (6.7) 4-3 (5.7)4 3.49, 0.87 3.44, 0.88 0.75 0.5 4-3 (4.8) 2-2 (3.6)10 3.68, 1.17 3.49, 1.24 0.2 Using N = 500 designs and M = 100,000. For Bk = 0.1, we have Pc = 3-4 (7.2) and Pnc = 510-4 (6.9) PAGE 105 83 The second and third columns give the failu re probabilities and the fourth column shows the percentage of designs rejected by th e certification test. This number increases with error magnitude because large errors may produce less conservative designs which will be rejected by testing. The coefficient of variation in the proba bilities is given in parentheses and represents the variation in fa ilure probability between structures designed by different aircraft companies. The fifth column gives the mean and standard deviation in thickness of certified designs and sixt h column gives the corresponding design thicknesses. The variability in design thickness is due to uncertainty in model errors. Certification testing increases the mean va lues and reduces the standard deviations because extremely thin designs will fail certific ation. We see that probability of failure is high when errors are high and it decreases w ith errors. The effectiv eness of certification tests, measured by the ratio of probability improvement Pc /Pnc, is high when errors are high, indicating that testing is more important for large errors. Large errors produce some super-strong and some super-weak designs. Th e super-weak designs are mostly caught by the certification tests, leaving the super-strong designs to reduce the pr obability of failure. This is also indicated by large ba tch rejection rate for high errors. Comparing Tables 5-4 and 5-5 with Tables 5-6 and 5-7 we see that there is a large difference in design thickne ss (about 40%) between fail-safe and safe-life design. The reason for this is that the sa fe-life structure is designed for entire service life (40,000 flights) which requires greater thickness fo r maintaining stable crack growth over a longer period of time. In contrast the fail-sa fe structure is designe d for 10,000 flights only which require lower structural thickness. From the last columns, comparing the ratio Pc PAGE 106 84 / Pnc we see that certification tests are more eff ective for fail-safe structures than safe-life structures even though the batch rejection rates fo r the latter are higher. Next, we perform simulations for structure designed without any safety measure for structure designed usi ng fail-safe criteria. Table 5-8: Probability of failure for different bounds on error k for 10 % COV in e without any safety measures for fail-safe design for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 0.22 (1.1) 0.52 (0.2)53 0.90, 0.05 0.88, 0.05 0.42 0.3 0.14 (1.5) 0.52 (0.6)52 0.99, 0.11 0.88, 0.15 0.27 0.5 9-2 (2.0) 0.51 (0.7)51 1.09, 0.16 0.88, 0.25 0.17 Table 5-9: Probability of failure for different bounds on error k for 50 % COV in e without any safety measures for fail-safe design for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 0.16 (1.5) 0.56 (0.4)56 0.95, 0.13 0.86, 0.14 0.28 0.3 0.13 (1.7) 0.57 (0.5)56 1.01, 0.16 0.86, 0.19 0.22 0.5 9-2 (2.1) 0.54 (0.7)54 1.10, 0.20 0.87, 0.28 0.16 Tables 5-8 and 5-9 shows that the probability of failure without testing is about 50%. This is expected because the structure is designed without any safety factors so that about 50% of the designs will fail below design life because of unconservative designs. Also we see that the probability of failure af ter certification decreases with increase in errors. When the errors are large, the average thickness of the designs that pass certification is high which leads to decrease in failure probability. Thus for this case we PAGE 107 85 get counterintuitive results th at large errors produce safer designs. This phenomenon manifests itself when safety factors are low. We further explore the effectiveness of testing by testing components with artificially machined cracks in next section. Effect of Certification Te sting With Machined Crack From Tables 5-4 and 5-5 we see that certification testing is not very effective in reducing the probability of failure. The prim ary reason for this is because of the nonlinear damage growth model which is very sensitive to variability in material properties leading to huge scatter in fatigue life. Becau se of this, the structure passing certification tests will not closely represent entire fleet. As a result, copies of th e certified structures will still fail in actual service because of vari ability in material properties. Testing only reduces model errors; variability in material properties of tested panels reduces the effectiveness of testing. The effectivene ss of certification testing may be improved by machining a large crack in the structure. Such tests are documented in FAA/AR-95/47 (Gruber et al. 1996) and are primarily used to determine material properties and understand component failure. Here we simulate this proced ure by introducing a B-basis crack and checking if fatigue life is less than twice the design life. The use of machine cracks removes the uncertainty in certification testing associated with the distribution of initial crack sizes. Table 5-10 show the results for fail-safe structure designed for 10,000 flights and Table 5-11 show the calculations for design of safe-life structure for the service life of 40,000 flights. PAGE 108 86 Table 5-10: Probability of fa ilure for different bounds on k and 10 % COV in e for structures designed with all safety meas ures for fail-safe for 10,000 flights and tested using a machine cracked pane l. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 <10-8 <10-8 1 2.08, 0.14 2.08, 0.14 -0.3 1-6 (11.6) 4-6 (6.1)10 2.13, 0.34 2.07, 0.36 0.25 0.5 7-6 (11.8) 6-3 (4.2)26 2.35, 0.47 2.09, 0.61 1.16-3 Using N = 500 designs and M = 100,000. For Bk = 0.3, we have Pc = 2-6 (7.1) and Pnc = 610-6 (4.9) For Bk = 0.5, we have Pc = 1-5 (11.7) and Pnc = 610-3 (3.9) Table 5-11: Probability of fa ilure for different bounds on k and 10 % COV in e for structures designed with all safety meas ures for safe-life of 40,000 flights and tested using a machine cracked pane l. Simulations are performed with N = 5000 designs and M = 10000 instances of each design). Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 ---3.59, 0.25 3.59, 0.25 -0.3 3-5 (6.5) 7-5 (7.0)7 3.65, 0.63 3.59, 0.65 0.42 0.5 6-4 (8.9) 6-3 (3.3)20 3.91, 0.90 3.57, 1.06 0.1 Using N = 500 designs and M = 100,000, For Bk = 0.1, we have Pc = 2-7 (8.6) and Pnc = 310-7 (8.1) For Bk = 0.3, we have Pc = 210-5 (5.5) and Pnc = 4-5 (4.6). For Bk = 0.5, we have Pc = 310-4 (6.2) and Pnc = 610-3 (3.4) Comparing Tables 5-4 and 56, to Tables 5-10 and 5-11 we note large improvement in effectiveness of certification and reduction probability of failure due to certification testing. Effectiveness of certification testing is largely affected by variability in material properties and using machined cracked panels reduces the variability in crack sizes, thereby improving effectiveness of tests. We see that failure probability can be reduced by a marginal amount (less than an order of magnitude) using certification for safe-life structures. For fail-safe structures certificati on is very effective and a large improvement PAGE 109 87 in failure probability can be achieved. We c onclude that certificati on is more effective when structures are designed and certified for short fatigue life. Effect of Variability in Material Propertie s on Structure Designed With all Safety Measures Tables 5-4 through 5-11 used 17% COV in material parameter m The COV in m was obtained from test data (reported by seve ral different experimentalists) presented in Sinclair and Pierie (1990). In practice, aircraft compan ies employ stringent quality control on materials so that the variability in material properties is reduced. To demonstrate the effect of variability in material properties on fatigue life we obtain failure probabilities for 8.5% COV in m and compare it to the results for 17% COV in m Table 5-12: Probability of fa ilure for different bounds on k and 10 % COV in e for structure designed with all safety meas ures for fail-safe for 10,000 flights and COV in material property m reduced to 8.5%. Simulations are performed with N = 5000 designs and M = 10000 instances of each design). Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 <10-8 <10-8 0 1.83, 0.12 1.83, 0.12 -0.3 1-6 (11.7) 1-6 (11.7) 0 1.82, 0.32 1.82, 0.32 1.0 0.5 2-3 (9.4) 1.2-2 (4.2)10 1.92, 0.48 1.83, 0.53 0.16 Using N = 500 designs and M = 100,000. For Bk = 0.3, we have Pc = 1-6 (7.6) and Pnc = 110-6 (7.4) Table 5-13: Probability of fa ilure for different bounds on k and 50 % COV in e for structures designed with all safety me asures for fail-safe criteria for 10,000 flights and COV in material property m reduced to 8.5%. Simulations are performed with N = 5000 designs and M = 10,000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 3-5 (13.8) 4-5 (11.7)0.4 1.80, 0.31 1.80, 0.32 0.75 0.3 5 10-4 (12.3) 3 10-3 (8.7)4 1.85, 0.41 1.82, 0.42 0.16 0.5 1-3 (9.6) 3 10-2 (3.8)12 1.92, 0.53 1.80, 0.58 0.03 Using N = 500 designs and M = 100,000. For Bk = 0.1, we have Pc = 2-6 (8.3) and Pnc = 210-5 (8.6). For Bk = 0.3, we have Pc = 7 10-4 (11.1) and Pnc = 2 10-3 (7.3) PAGE 110 88 Comparing Tables 5-4 and 55 to Tables 5-12 and 5-13, we see that certification testing becomes more effective when variability is low. This is reasonable because large variability increases the chan ce of large difference between the tested design and the actual structures used in serv ice. Also we see that failure probability reduces when variability is lesser. This is because of the use of conservative value of m in design. Effect of Conservative Material Properties. Failure probability is calculated next for structures designed using onl y conservative material spec ification. In Table 5-14 we calculate the failure probability for stru cture designed using ABasis property for m only. Tables 5-15 and 5-16 present the results using A-Basis property for fracture toughness Kth and crack growth parameter m (17% COV), B-basis value of crack size and A-Basis value of yield stress. All other safety measures have a value of 1.0. Table 5-14: Probability of fa ilure for different bounds on k and 10 % COV in e for structures designed using only A-Basis m for fail-safe criteria for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10,000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 2-2 (1.3) 2-2 (1.2) 2 1.23, 0.08 1.23, 0.08 1.0 0.3 6 10-2 (1.8) 1 10-1 (1.5)10 1.26, 0.20 1.23, 0.22 0.6 0.5 6-2 (2.6) 2 10-1 (1.4)23 1.37, 0.29 1.23, 0.36 0.3 Compared to Table 5-9 where structure is designed without any safety measures, we see from Table 514 that using A-basis m improves the failure probability by an order of magnitude. However, comparing Table 5-14 to Table 5-13 we s ee that reducing the variability in material property m by 50% is more effective than using conservative material property for m PAGE 111 89 Table 5-15: Probability of fa ilure for different bounds on k, 10 % COV in e for structure designed using conservative properties for fail-safe design for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design). Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 3-5 (3.1) 3-5 (3.1)0 1.61, 0.10 1.61, 0.10 1.0 0.3 3-3 (3.3) 3-3 (3.2)0.7 1.64, 0.28 1.64, 0.28 1.0 0.5 3-2 (3.0) 7-2 (2.3)6 1.67, 0.44 1.61, 0.47 0.42 Using N = 500 designs and M = 100,000. For Bk = 0.1, we have Pc = 310-5 (2.8) and Pnc = 310-5(2.8) Table 5-16: Probability of fa ilure for different bounds on k 50 % COV in e for structures designed using conservative properties fo r fail-safe criteria for 10,000 flights. Simulations are performed with N = 5000 designs and M = 10000 instances of each design Bk Pc Pnc R %certt (mean, standard deviation) mm designt (mean, standard deviation) mm Pc /Pnc 0.1 4-3 (4.8) 6-3 (4.8) 0.6 1.59, 0.28 1.58, 0.28 0.66 0.3 1.6-2 (3.4) 2-2 (3.2) 2 1.59, 0.33 1.56, 0.34 0.8 0.5 4-2 (3.0) 0.10 (2.2) 10 1.65,0.48 1.57, 0.51 0.4 Comparing Tables 5-4 and 5-5 with Tables 5-15 and 5-16 it can be seen that failure probability is very sensitive to structural de sign. When all safety measures are used, the effective safety factor (ratio of design thic kness with safety measures to that without any safety measures) is about 2.36, while when onl y conservative material properties are used the effective safety factor is about 1.8. For th is 20% decrease in effective safety factor the failure probability increases by more than an order on magnitude when errors are high and about four orders of magnitude when errors are low. To illustrate the interaction between safe ty measures, errors and variability we obtain the effective safety factor in Table 5-17 for two levels of error bounds. The ratio PAGE 112 90 PSF/ PNSF indicates the effectiveness of safety f actors in reducing failure probability. The other ratio Pc/Pnc is the ratio of probability of failure with certification test to that without any test and measures the effectiv eness of certifica tion testing only. Table 5-17: Effective safety factor and measur es of probability improvement in terms of individual safety measures and erro r bounds for structure designed using failsafe criteria of 10,000 flights. A = cons ervative properties, C = certification, M = machined crack certification, S = design safety factors, SFeff = ratio of structural thickness with all safety meas ure to that without any safety measure, PSF/ PNSF = ratio of probability of failure of structure design ed using safety measure to that designed without any safety measure. Safety measure Error bounds ( k = 10%, e = 10%)Error bound ( k = 50%, e = 50%) SFeff PSF/ PNSF Pc /Pnc SFeff PSF/ PNSF Pc /Pnc S+A+M 2.36 2 -8 1.0 2.69 4 10-5 7 -4 S+A+C 2.36 2 -8 1.0 2.50 6 -3 0.4 S+A 2.36 2 -8 -2.34 4 -2 -C+A 1.83 6 -5 1.0 1.90 0.08 0.4 A 1.83 6 -5 -1.80 0.2 -C 1.02 0.4 0.4 1.26 0.16 0.16 We can see that certificati on tests are not very useful when errors are low. The effective safety factor is almost the same with or without certification for low error bounds ( k = 10%, e = 10%). For high errors certificati on tests become more useful as indicated by column seven of Table 5-17. All th e tables indicate that the when the errors increase, ( Pc/Pnc) decreases revealing that the efficien cy of testing increases. Another way of looking at this effect is to note that when there are no errors, ther e is no point to the tests. The ratio PSF/ PNSF in Table 5-17 shows that safety measures are most effective when errors are lower and greater improvement in failure probability can be achieved as compared to that at high errors. From PSF/ PNSF we see that among all the safety measure the conservative material prope rties (A) contribute the most to probability improvement when errors are low, for high errors, testing is more effective. This is expected because fatigue is very sensitive to material properties and using c onservative material properties PAGE 113 91 reduces the effect of variability. The other safety measures provide less protection. We can see that certification testing is most effective when safety factors are low, errors are high and variability is low and effectivene ss of tests can be improved by using panels with machined cracks. The combined effect of all safety measures can reduce the failure probability to about 10-7. PAGE 114 92 CHAPTER 6 A PROBABILISTIC MODEL FOR INTERNAL CORROSION OF GAS PIPELINES Introduction Internal corrosion in a pipeline leads to reduction of pipe wall thickness and ultimately to leak or burst failure. The presence of electrolytes such as water and species such as CO2, H2S, and O2 are some of the prominent fact ors causing internal corrosion. Consequently, periodic inspections and repa irs are necessary to maintain pipeline integrity and prevent failure. Despite r ecent advances in inline inspection (ILI) technology, not all portions of a pipeline can typically be inspected due to various geometrical and economic constraints. The surest way to determine the location and extent of internal corrosion is to excavate and examine the pipe. However, typical oil and gas pipelines are hundreds of miles long and extensive excavation is impractical given the negative economic consequence of shutting the pipeline down to perform the excavation. Recently, an internal corros ion direct assessment (ICDA) methodology has been developed for essentially dry gas lines to prioritize locations for excavation and detailed examination by Burwell et al. (2004). This chapter addr esses the uncertainties entailed in the ICDA method us ing a probabilistic framework. Bich and Eng (2002) and Nguyen and Heaver (2003) conducted corrosion measurements at a few discrete locations al ong a pipeline. Predicted corrosion rates were compared with observed corrosion rates and factors related to corrosion (CO2, H2S, O2, Sulfides, etc.) were identified and matc hed to morphology and location of observed damage. Preventive measures like thermal regul ation, increased inhi bitor use, inspection PAGE 115 93 pig runs, and reduction of corrosion enhanci ng species were recommended to reduce corrosion damage. Philipchuk (1998) develope d a regression equation to forecast the number of leaks per year in terms of annual leak histories and other measured variables driving corrosion growth. He also determin ed the most significant variables affecting leak. An important limitation of these approaches is that they are local in nature because the analysis is made on few discrete locatio ns. However, since these corrosion models are not coupled to models that id entify locations along a pipeline where corrosive electrolyte could accumulate, they provide no informa tion about the location where excavation should be performed to perform inspections. W ithout incorporating this information into a predictive model, it is uncle ar how what is learned in one area can be applied to another. For a specific area, it is not possible for a pipeline operator to identify all other areas that are similar in si ze, age and conditions and ex amine the corrosion depth. Previous work on dry gas pipelines conduc ted at Southwest Research Institute (SwRI) used deterministic flow analysis to iden tify locations in a pi peline where water is likely to accumulate first. Internal corrosion is considered to be present at these locations and is most severe where the inclination angle of the pipe is great er than the critical flow angle. Excavations can be performed at thes e locations, and other locations can be inspected if substantial corro sion is detected at these locations. Although this method gives a reasonable idea about the presence of corrosion, no account of the inherent variations in flow informa tion (pressure, temperature, and species concentration), The inclination angle is the angle between horizontal and the current section of pipe Water is assumed to accumulate if the inclinat ion angle is greater than the critical angle. PAGE 116 94 elevation data, and modeling error are take n into account. Furthermore, the simplified flow modeling is not ade quate for wet gas lines. Probabilistic analysis can be used to incorporate uncertainty in data and obtain the relative likelihood of failu re in a pipeline subjected to co rrosion growth. Here the model parameters are represented as random variable s, each with a probabi lity distribution. The state of failure is represented by the proba bility that the corrosion depth will exceed a critical amount at a specif ied location and time of operation of the pipeline. The combination of physics-based modeling, associated parameter and model uncertainties, and inspection-based model updating provide s a more rational framework for making inspect/repair/replace de cisions that does traditional deterministic analysis. An example application of probabilistic analysis to the prediction of life governed by mechanical failure in gas distribution pi pelines is given in Thacker et al (1992). Muhlbauer (1996) developed a risk indexi ng system that relies on establishing subjective weighing factors derived from judg ment of the corrosivity of the product and presence/absence of mitigation methods (monitoring coupons, ILI, inhibitor injection, gas treatment, and internal coating). Other risk assessment methods use a fault-tree/event-tree approach but assign subjective probabilities to vari ous processes leading to a leak. These approaches are limited by the fact these indices or probabilities are fixed, subjective, and do not allow updated pipeline information. Ahammed and Melchers (1995) used the Fi rst Order Reliability Method (FORM) to predict the pipe leak probability at a si ngle location subjected to pitting corrosion. Ahammed and Melchers (1997) performed reliab ility analysis to incorporate uncertainty in data and obtain failure proba bility of a single s ection of pipe subjected to widespread PAGE 117 95 corrosion growth. Ahammed and Melchers (1997) and Caleyo et al. (2002) assumed that the number and location of defective sites we re known and used a probabilistic approach to compute the reliability of a pipeline segment subjected to corrosion growth in the presence of multiple defects. Vinod et al. (2003) used Markov chains and FORM to estimate inspection time for a pipe segment for maintaining a specified probability of failure. They used the erosion-corrosion growth model to estimate the time required for corrosion depth to exceed a critical depth. Hong (1999) also used Markov chains to develop a method for obtaining optimum inspec tion time for a pipeline subjected to corrosion growth including gene ration of new defects during th e service life of a pipeline. All of these methods are based on calculating th e reliability estimate of a single section of pipe, which is assumed to govern the overall structural integrity of the pipeline. Consequently, they provide no framework to id entify the critical location in the pipeline itself. Gartland et al (2003) developed a model to pred ict the corrosion profile throughout the length of a pipeline. The model combines pipeline profile and flow information into multiphase flow modeling software to obtain wa ter wetting factors at different locations along pipe length. This is combined with a point corrosion model and inhibitor effect to estimate CO2 corrosion along the pipe length. They also developed a framework to combine the model predictions with inspecti on and monitoring data to obtain updated estimates. However they did not account for un certainty in pipeline pr ofile information in calculating water wetting factor s. Also their results are c onditioned upon predictions of a single model, which may not be suitabl e for all conditions in pipelines. PAGE 118 96 The proposed approach is aimed at deve loping a probabilistic model for assessing the extent of internal corros ion along the length of a pipe line. The probabilistic model can incorporate inspection data, so the model as we ll as the results can reflect observational data. The probabilistic model uses either Monte Carlo simulation or an approximate FORM solution to perform the probability inte gration. A Bayesian approach is used to update the model prediction with field data. Because the true corrosion rate model is unknown, three candidate corrosion rate models are used to obtain the probability estimate. The corrosion rate models are combined as a weighted average, where the weight factors are updated us ing the corrosion depth measured from inspection data. Proposed Methodology Corrosion Rate Model The nature of corrosion growth largely depe nds on the presence of electrolyte such as water, concentration of species such as CO2, O2, H2S, pH, and flow parameters such as temperature, pressure, and velocity. Here we focus on the internal region of a pipeline where the presence of widespread corrosion is prominent. Various empirical equations are available to represent the corrosion rate as a function of aforementioned parameters. Three candidate models were selected, M1 represents the de-Waard Milliams model (deWaard and Milliams, 1975), M2 represents the de-Waard Lotz model, (de-Waard and Lotz, 1993) and M3 represents the SwRI model, (Sridhar et al ., 2001) M1: 2 10log 67 0 1710 8 510pCo T IC k dt da mm/ years (6-1) M2: 2 10log 67 0 1710 8 510pCo T ICF C k dt da mm/ years (6-2) PAGE 119 97 M3: pH O O S pH O pCo S pH pCo pH O O C k dt daI 2 3 2 2 3 2 2 5 2 2 2 2 2 7 2 310 11 1 10 37 2 10 82 4 10 93 4 31 1 10 48 1 10 86 9 7 8 0254 0 mm/ years (6-3) Where the temperature correction factor for M2 is given as 0 1 log 61 0 7 6 2400 .. .. log 61 0 2400 7 6 102 10 2 10 CF else T pCo when pCo T CF (6-4) These three models are refe rred to as DM, DL and SwRI respectively. In the Equations 6-1 to 6-4, a is the corrosion depth in mm t is time in seconds T is temperature in degree K, pCO2 partial pressure of CO2 in the mixture in Pascals pH2S is partial pressure of H2S in the mixture in Pascals O2 is the concentration of O2 in parts per million, k is the modeling error, CI is inhibitor corre ction factor, and CF is the temperature correction factor give n by the following Eq. (6-4) (Bert et al ., 2002). Although a complete discussion of the applicabil ity of these models to pipeline systems is beyond the scope of this chapte r it should be noted that th e DM and DL models have been derived from experiments simulating production systems wher eas the SwRI model simulates gathering lines. Inhibitor Correction Model Corrosion inhibitors can be a dded to the inlet of a pipe line to reduce the corrosion rate. Since the effect of inhibitor will diminish as a function of distance from the injection point, an exponential model is assumed to represent the redu ction in corrosion rate with distance along pipe length. Th e inhibitor correction factor is represented by the following equation PAGE 120 98 01L L A Ie C (6-5) where A is the model parameter, L is the distance along the pipe length, and L0 is the characteristic length to describe the effect of inhibitor. The effect of continuous inhibitor injection on corrosion in multiphase fl ow system was examined by Erickson et al. (1993). Their modeling showed that th e inhibitor effectiveness in a condensate pipeline is a complex function of gas and liquid flow rate and pipeline elevati on profile. However, they made a general observation that increas ing condensation occurs as a function of distance away from the inlet end. Because the inhibitor does not partition to the condensed phase, the concentra tion of inhibitor decr eases as the liquid flow increases due to condensation. The result is that the inhib itor effectiveness decrea ses as a function of distance away from the inlet end. The decay distance is a function of condensation and gas flow rates. Water Accumulation Flow in wet gas pipelines can be classified into four major types: (i) bubble flow occurs in the form of gas bubbles in a c ontinuous liquid phase wh en the gas volume is low, (ii) annular flow occurs at high gas velo cities in the form of a liquid layer contacting pipe wall and a gas core, (iii) slug flow occurs in the form of pockets of liquid followed by pockets of gas, and (iv) st ratified flow occurs in the form of a liquid layer at the bottom of the pipe with oil or gas flowing above it. In the case of stratified flow, the shear stress between the liquid-vapor interfaces prov ides the mechanism for water to flow from one location to another along the pipe length. Ho wever, if the pipeline inclination angle is greater than the critical a ngle then water is assumed to accumulate. Water formation PAGE 121 99 occurs at all angles greater than critical angle because the shear stress is no longer sufficient to overcome gravity (Burwell et al. 2004). The critical angle is given by: i g l g ggD V F 2 1sin (6-6) where 24i P gD H V 273 101325 P Z T S Hp p Z T R MW Pg where F is the Froude number, g is the gas density in 3/ kgm Vg is the gas velocity in m/s l is water density in 3/ kgm g is acceleration due to gravity in m/s2, Di is internal diameter of pipe in mm Hp is high-pressure flow rate in m3/s Sp is flow rate at standard temperature and pressure 2/ms, Z is compressibility factor, MW is molecular weight of gas in /kgmoleand R is universal gas constant in // JKmole. Typical values of these constants are given in Table 6-1. Table 6-1: Typical wet gas pipeline flow parameters Constant Value l 1000 3/ kgm G 9.812/ ms Di 0.559 m Sp 136.111 2/ms Z 0.83 MW 0.015 / kgmole R 8.314 // JKmole Froude number, F 0.56 (>2 deg), 0.35 ( <2 deg) 0.14 +0.28 (0.5<<2 deg) Pipe thickness 8.33 mm PAGE 122 100 Probabilistic Model Corrosion Damage The probability of corrosion damage at a specific location is the probability of corrosion depth exceeding a critical value times the probability that water is present at that location. The likelihood of water formation at a location is given by P Pw (6-7) where is inclination at a particular location. Fi gure 6-1 illustrates that there will always be some probability that the in clination angle will exceed the critical angle. The uncertain inclination angle arises from uncertainties in mapping measurements, cover depth and axial location (discussed later in the chapter). The uncertain critical angle arises from uncertainties in the flow velocity, pressure temperature and pipe diameter. The corrosion probability at a location is calculate d by using three candidate models 3 3 2 2 1 1W a a P W a a P W a a P Pc M c M c M cr (6-8) Where W1 + W2 + W3 = 1 and ac is the critical corrosion depth, aMi is the corrosion depth predicted by ith model, Wi is the weight factor for the ith model, and P ( aMi ac) is the probability of exceeding the crit ical corrosion depth for the ith model. For oil and gas lines the critical corrosion depth has been assumed to be 80% of wall thickness (Caleyo et al. 2002; Vinod et al., 2003). The total corrosion probabi lity at a location given that water is present at that location is given by cr P w P tot P (6-9) PAGE 123 101 Distance 2 Figure 6-1: Uncertainty in inclination and critical angle Input Uncertainties Safety measures such as inspections and repairs are scheduled to reduce the chance of leaks and structural fail ure in face of uncertainties. These uncertainties include parameters affecting, for example, corrosion growth, and water flow and elevation data. These uncertainties should be accounted for by assessing the extent of corrosion damage at given location along pipe le ngth and scheduling excavation and repairs. Probabilistic analysis can be used to account for randomn ess in these parameters. Table 6-2 presents random variables and their associated proba bility distributions for a demonstrative pipeline scenario. Table 6-2: Typical wet gas pipe line corrosion growth parameters Random variable (units) Distribution typeMean Standard deviation T (degree K) Normal 289 28.9 % CO2 (mole) Lognormal 5 1 O2 (ppm) Lognormal 5000 1500 pH Lognormal 6 1 % H2S (mole) Lognormal 0.05 0.005 P (Pascal) Lognormal 4080000808000 k Corrosion Model errorLognormal 1.0 0.5 A, Inhibitor factor Lognormal 1.0 0.5 PAGE 124 102 Mapping Uncertainty The uncertainty in pipeline inclination data occurs because of inaccuracies in elevation data present in digital maps and uncertainties in pipeline burial depth. The mapping inaccuracies are location specific and there is substantial evidence to suggest that it is positively correlated to th e ruggedness of terrain (e.g., Holmes et al. 2000; Riley et al ., 1999; Sakude et al ., 1988 and Tang et al ., 2000). A linear equation between the terrain ruggedness index (TR I) and the accuracy in elevation can be obtained by a regression analysis on data from Tang et al. (2000) and Weng (2002) used the USGS database to obtain the maximum error of 11 me ters and minimum error of three meters in the elevation data obtained from digital maps. The elevation data is used to calculate the terrain ruggedness index (this is the root mean square erro r between the elevation at a location and eight neighboring locations) fo r each location along the pipe. The regression equation developed above was used to obtain an estimate of elevat ion error at each location. The following equations are used 82 4 4 i j i j j iy y TRI (6-10) 2 1C TRI Cy (6-11) where TRI is terrain ruggedness index, y is the error in elevation, and C1 and C2 are regression constants, yi is the location at which TRI is to be determined and yjs are adjacent locations around yi. The error yis used to calculate maximum and minimum inclination angles at these lo cations. We further assume that the inclination angle follows PAGE 125 103 a normal distribution with a 6 range between the maximu m and minimum inclination angles (this was chosen to capture the majority of the data). Inspection Updating There will be uncertainty in our initial prediction because of lack of accurate information on model weights, physics consid ered (or neglected) in competing models, and the assumed distribution of random variables. Data collected from inspections can be used to update the reliability estimate. Baye sian updating provides a systematic method for incorporating measured data with prior information to estimate future outcome (e.g., Rajasankar et al ., 2003; Simola and Pulkkinen, 1998; and Zhang and Mahadevan, 2001). The underlying assumption in the approach taken is that the correct form of the corrosion rate model is unknown. Consequently, and as a demonstration of the methodology, three candidate models are considered. Based on inspection data, the weight factors are adjusted to reflect this additional information in the next prediction. We have used the reliability-updating model developed by Zhang and Mahadevan (2001). We further assume that there is no uncertainty in the det ection process so that the detected damage is the actual damage at a location. The event of damage detected with size ad is expressed as d M Aa a Di (6-12) where i Ma is the corrosion depth predicted by ith model at the most probable point (MPP). The updated model weight and reliabilit y in the event of a detected corrosion depth ad can be expressed as 3 10 | | 0 |i Md i i i M d i iiMd iaa M iA iMd iaa i MPaa W a WD Paa W a (6-13) PAGE 126 104 3 1| 0 | 0i d a i M a i M d i M i d a i M a i M d i M c i M i Updateda a a P a a a a a P P (6-12) Equations 6-13 and 6-14 are solved numerically Note that the updating only affects the component weighting of the component mode ls. No adjustment to the probability distribution of the component models is performed. Example 1: Determination of Critic al Location Prior to Inspection A typical gas transmission pipeline was chosen for demonstrating the proposed methodology. The pipeline elevation data at 1000 locations was used to calculate the inclination angles and the a ssociated uncertainties in them using Equations 6-10 and 611. These are actual elevation data from wh ich company-specific geographic information has been removed. The probability of water formation is obtained from Equation 6-7 and the probability of corrosion damage is obtaine d from Equation 6-8 using inputs in Tables 6-1 and 6-2 after a time period of 10 years. As a first illustration, we demonstrate the methodology by calculating the corrosion probability at each of the 1,000 locations. Since no information about pipeline co rrosion is available in the be ginning, we assume that all corrosion models represent corrosion growth in the pipeline with equa l probability so that the model weights of each corrosion model in Equation 6-8 are 1/3. Figure 6-2 shows that the probability of water formation is at a ma ximum at location 971. Figure 6-3 shows that the probability of corrosion depth exceeding cr itical depth increases monotonically with pipe length. This is because the corrosion inhibitor reduces the corrosion rate in the beginning and its effectiveness di minishes with pipe length. PAGE 127 105 Figure 6-2: Probability of water formation along pipe le ngth with highest probability observed at location 971 Figure 6-3: Probability of corrosion depth exceeding criti cal depth along pipe length assuming water is present at all locations PAGE 128 106 Figure 6-4: Total probability of co rrosion exceeding critical depth along pipe length As we can see from Figure 6-4, the proba bility of corrosion is maximum at location 971 (highest probability of water formation and far from the corrosion inhibitor injection). Consequently ex cavation and inspection is r ecommended at this location. Example 2: Updating Corrosion Modeling with Inspection Data In Example 1, three different corrosion mode ls with equal model weights were used because there is no information regarding the accuracy of individual model prediction in a typical pipeline setting. The DM and DL models (derived for production systems) estimate corrosion rates that are higher than those obtained from th e SwRI model. Also each model is based on different uncertain para meters. Inspections can be done to repair the damaged part of a pipeline with the data collected subsequently utilized with Bayesian analysis technique to improve reli ability estimates. As more information from inspections becomes available, accuracy s hould improve as the most appropriate model or weighting of the three models is modified from the initial prediction. An example of the updating process is provided in Table 6-3 to show how se veral observations affect the model weights. PAGE 129 107 Table 6-3: Updating of model weights gi ven assumed observations corresponding to input component models Observed corrosion depth, mm Observation derived from Model weight W1 (DM) Model weight W2 (DL) Model weight W3 (SwRI) 0.05 SwRI 0.104 0.104 0.792 0.07 SwRI 0.019 0.019 0.962 0.11 SwRI 0.003 0.003 0.994 0.13 SwRI 0.0003 0.0003 0.9994 5.12 DM 0.500 0.500 0.000 The observations were made so that they corresponded to either the SwRI model (first four samples) or the DM model (last sa mple) at a given location. The analysis began with an equal weighting of 0.3333 for all three models. For the first four observations derived from the SwRI corrosion model, it is clear that the model weights rapidly approach 0.0 for both the DM and DL models while the SwRI goes to 1.0. The reason for this is the large disparity in the predictions between the competing models resulting from the differing intended applications for each model. The last observation corresponds to the DM model and there is no gradual transition in the weight factors. The reason for this is the probability associated with the SwRI model generating a 5.12 mm crack are exceptionally small compared to DM a nd DL. As such, the Bayesian updating immediately removes the SwRI model from ac tive consideration in order to reflect the latest observation. Since the DM and DL models are equivalent for the conditions being considered, their corresponding weights each go to 0.50. Typically pipelines are hundreds of miles long and excavating the entire length is impractical and uneconomical. Excavations can be scheduled at locations where the probability of corrosion is highest. Data obtai ned from each excavation can be used to update the reliability along the pipe length a nd to predict the next excavation location. This can continue until a specified level of re liability at each location on the pipeline is PAGE 130 108 obtained. Here we assume that if corrosion is detected at a location, it is repaired or replaced such that the location becomes defect free. Table 6-4 shows the results of a series of inspections and model updates. Only a few locations are illustrated in the table. Additionally, th e method outlined below is one approach for updating. Depending on the insp ection methods and procedures, other types of updating can be performed. In the illust rative example shown in Table 6-4, the following sequences of steps are performed (1) Before any inspections are performed, location 971 is predicted to have the highest probability with ot her locations up and downstream from it having a lower probability of corrosion exceeding certain depth. (2) An inspection at location 971 is performe d and the depth of corrosion is found to be 8.2 mm. Based on the predicted and de tected corrosion depth at location 971, the model is updated and the next location of maximum probability is predicted to be location 923. (3) An inspection is then performed at loca tion 923, a defect depth of 7.5 mm is measured. This is then compared to model prediction and the model updated again. This updating then modifies the probabilities of corrosion downstream and upstream from this location. (4) This process is repeated at the next highest probability location until all the highest probability locations upstream and downstream from 739 are inspected. Note that as the model is updated based on inspections, the previously inspected location probabilities will change. Howe ver, since they would have been PAGE 131 109 inspected already and mitigation measures adopted, the purpose of this assessment is considered to be fulfilled. Table 6-4: Inspection locations along pipeline Location of inspection Detected corrosion depth ( mm ) Maximum updated probability Location of maximum updated probability 971 8.2 0.7247 923 923 7.5 0.7100 739 739 5.1 0.6662 580 Note that updating procedure may be modi fied to suite individual pipeline needs. Furthermore, modeling and updating may be a dopted such that a certain proportion of pipeline is inspected in de tail and downstream from these inspections, the updated model is used for locating further examinations. Summary A preliminary methodology to predict the mo st probable corrosion damage location along a pipeline and update this prediction using inspection data has been developed. The approach computes the probability of critical corrosion damage as a function of location along the pipeline using physical models for flow, corrosion rate, and inspection information as well as uncertainties in el evation data, pipeline geometry and flow characteristics. The probability of corrosion damage is computed as the probability that the corrosion depth exceeds a critical depth given the presence of electrolytes such as water. Water is assumed to be present at locations where the pipeline inclination angle is greater than the critical angl e. Three candidate corrosion ra te models were employed to reduce the chance of selecting the incorrect model. Monte Carlo simulation and the firstorder reliability method (FORM) implemented in a spreadsheet model was used to perform the probability integration. Bayesian updating was used to incorporate inspection PAGE 132 110 information (e.g., in-line, excavation, etc.) and update the predic tion of most probable damage location. This provides a systematic method for focusing costly inspections on only those locations with a high probability of damage and incorporating the results of the inspection in a manner that improves confidence in future predictions. PAGE 133 111 CHAPTER 7 CONCLUSIONS The primary objective of this dissertati on was to demonstrate the advantage of simultaneous design optimization of struct ure and inspection sc hedule. There is uncertainty in structural failure because of randomness in geometry, material properties, and loading. These uncertaintie s are readily incorporated in to the design process by using statistical distributions a nd reliability methods. Here a methodology for cost optimal reliability-based structural design and inspecti on planning of aircraft structures subjected to fatigue damage growth was developed. An optimization problem was formulated to minimize the expected lifetime cost while maintaining a minimum acceptable reliability level. The effect of the st ructural design and the inspect ion schedule on the operational cost and reliability was explored. Calculat ing structural reliability in presence of inspection is computationally challenging because distribution of some of the parameters has to be updated after each inspection to simulate replacement. Exact evaluation using Monte Carlo simulation is time consuming because large sample size is required for estimating low probability of failure accurate ly. In this dissertation an approximate method using a combination of Monte Carlo si mulations (MCS) and first-order reliability method (FORM) to expedite reliability calcula tions was presented. This method was used to perform combined optimization of structur al design and inspection schedule of aircraft structures. The study led to the following conclusions (1) The multiple safety measures used to design structure for stable fatigue crack growth lead to heavy design for structures designed using safe -life criterion. PAGE 134 112 (2) Large variability associated with estima ting fatigue life reduces the effectiveness of safety measures. Safety measures are also more effective for fail-safe design than safe-life design. For the same reason certification tests ar e not very effective in improving reliability. Certification tests can be made more effective by reducing variability in crack sizes. (3) Even if only their effect on fatigue is cons idered, inspections ar e cost effective in maintaining reliability levels through damage detection and replacement. From our examples, there was 25 % in lifetime cost due to inspections over a design without any inspections. Their advantages for detecting other types of damage, such as that due to corros ion, tool drops, and accidenta l impact, only add to their usefulness. As fuel cost rise, additi onal inspections become profitable and improvement in inspection effectiveness also makes them more attractive. (4) Combined optimization of structural we ight and inspection reduces the lifetime cost penalty associated with single ty pe of inspection, thus allowing possible simplification of the inspection regime. Wh en inspections are scheduled after the structure has been designed, mixture of inspection types may lead to additional cost savings. (5) Designing structures for multiple load transf er capability (that is stiffened panels) can be much more cost effective and failure resistant than single load path when structure is designed without any inspectio ns, however with inspections there is much lower gain from stiffeners. (6) The variability associated with locati on and severity of corrosion damage in pipelines makes probabilistic design suita ble for predicting location of maximum PAGE 135 113 damage in pipeline. Inspections are scheduled at locations of maximum probability of failure and results from in spections are used to update corrosion probabilities. The methodology presented in this disse rtation eases the co mputational burden associated with calculation of low failure probabilities. However this method suffers from the fact that probability dist ribution in the tail region is poorly approximated because only the mean and standard deviations are estimated. An alternative method is to use extreme value distributions (EVDs) to estima te the tail region of probability distribution and use it directly in reliability analysis The may lead to more accurate reliability computations at slightly higher computati onal expense. The use of importance sampling to reduce the sample size of MCS may be he lpful in reducing the computational expense. The through the thickness crack model assu med in this work represents the worst case scenario and was chosen to demonstrate that inspections are cost effective for the worst case. Actual cracks present at th e time of initiation may have different configurations but will finally evolve into a through the thickness crack. The methodology presented in this dissertation ca n be applied to different crack growth models. Furthermore, a simple structural design with constant amplitude loading is considered (fuselage) to facilitate the dem onstration of the proposed reliability method. For structures with variable amplitude loading like wing panels, detailed load history effects can be included by using cycle c ounting techniques in combination with crack growth acceleration and retardation models. PAGE 136 114 APPENDIX A DISPLACEMENT COMPATIBILITY ANALYSIS FOR CALCULATION OF STRESS INTENSITY OF STIFFENED PANEL Introduction This appendix presents detailed explana tion and clarification of the displacement compatibility method developed by Swift (1984) to facilitate easy implementation and also serves as errata for printing errors in th e original paper. These errors were corrected from private communication with the author. Th e figures and equations presented in this chapter have been extracted with permissi on from special techni cal publication STP 842, Damage Tolerance of Metallic Structures, An alysis Methods and A pplications, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA, 19428 As the crack propagates in a stiffened panel, load is transferred from the skin to the intact stiffeners by means of fasteners. The st ress intensity factor at the crack tip can be obtained by displacement compatibility analysis In this method the displacement in the cracked sheet at fastener location are made equal to the stiffener plus fastener displacement. The effect of stiffening is measured by the geometric factor which is the ratio of stress intensity factor with stiffening to that without stiffening. To demonstrate the application of the displacement compatibility analysis we consider a center cracked s tiffened panel as shown in Figure A-1 with two intact stiffeners placed symmetrically across from crack center line and a broken stiffener along the crack centerline ( y -axis). This is a typical exampl e of a two bay crack with centre broken stiffener used to certify aircraft for damage tolerance. PAGE 137 115 Figure A-1: Half-geometry of a center crac ked stiffened panel with a central broken stiffener and two intact stiffeners placed symmetrically across from crack Displacement Compatibility Method The stress intensity factor at the crack tip of a stiffened panel is lower than that on an unstiffened panel because of the reduced st resses at the crack tip as shown in Figure A-1. The panel is assumed to be in a state of plane stress and the stiffeners are assumed to be one dimensional rods placed symmetrically across the crack with one broken stiffener along the crack centerline. The displacements in the panel at fastener locations are obtained by superposition of five cases shown in Figure A-2. (1) V1, the displacement anywhere in the crack ed sheet caused by the applied gross stress. (2) V2, the displacement in the uncracked sh eet resulting from fastener loads, F (3) V3, the displacement in the uncracked sheet resulting from broken fastener loads, P The highlighted text in this appendix is different from original paper. The additional details were obtained from private communication with the author of the paper. PAGE 138 116 (4) V4, the displacement in the cracked sheet resulting from stress applied to the crack face equal and opposite to th e stresses caused by rivet loads. (5) Stiffener displacement at location yi resulting from direct fastener load. The total number of fasteners on a single stiffener is 2n equally distributed on either side of crack. Using symmetry we need to solve for fastener forces only in a quarter of the panel. For solving the matrix equation for fastener forces we number the index of fastener on central broken stiffener from 1 through n and those on intact stiffener from n+1 through 2 n Figure A-2: Description of a pplied stress and resulting fasten er forces and induced stress on stiffened panel PAGE 139 117 Figure A-3: Description of pos ition coordinate of forces and displacement location with respect to crack centerline as y axis where the variables are defined in nomencla ture. Figure A-2 shows the displacement due to gross stress V1, displacements due to concentrated fastener forces V2 and V3 and displacement due to stress at the crack tip. Displacement V1 The displacement at any location ( x, y ) in the cracked sheet resulting from overall gross stress can be determined using Westre gaards stress function (Westergaard, 1939) below y r r yr v r r E V 2 1 2 1 2 1 2 1 12 2 cos 1 2 sin 2 (A-1) The variables as shown in Figures A-2 a nd A-3 are measured from the coordinate ( xi, yi) of the point at which displacement is measured. PAGE 140 118 Displacement V2 and V3 The stress distribution anywhere in and in finite plate resulting from a concentrated force F can be determined from work of Love (1944). The displacement at any location ( x y ) due to a concentrated force can be obtained as C y x x y x tE F VF 2 2 2 2 21 log 2 3 4 1 (A-2) where x, y is measured from point of application of force and C is a constant. Superposing the displacement due to four ri vet load placed symmetrically opposite of the crack centerline the resulting displacement can be obtained as follows (the constant C cancels out) 1 2 tan 1 2 tan 1 2 tan 1 2 tan 3 1 4 1 1 log 1 1 1 log 1 1 1 log 1 1 1 log 1 16 3 1 , ,2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 A B B B B B B B B A A A A A A A B B A B B B B A B B B A A A A B A A A A j j i iX Y Y Y X Y Y Y X Y Y Y X Y Y Y Y X Y X X Y X Y X X Y X Y X X Y X Y X X Et F y x y x V (A3-1) for the central broken stiffener, super position of the two rivet forces placed symmetrically opposite of th e crack centerline we get 1 2 tan 1 2 tan 3 1 4 1 1 log 1 1 1 log 1 16 3 1 , ,2 2 1 2 2 1 2 2 2 2 2 2 2 2 3 A B B B A A A A B A A A A B A A A A j j i iX Y Y Y X Y Y Y Y X Y X X Y X Y X X Et P y x y x V (A3-2) PAGE 141 119 where j i B j i A j i B j i Ay y d Y y y d Y x x d X x x d X 2 2 2 2 the ith term is the point at which displacement is required and the jth term represents coordinates of the forces. Displacement V4 The displacements due to concentrated poi nt forces at the rivets induce stresses along the crack length. The disp lacement in the cracked shee t resulting from this stress distribution is given as a i i j j j a i i j j j jdb b y x b y x P db b y x b y x F Et y V0 0 2 4, , , , 2 1 (A-4) Figure A-4: Description of pos ition coordinate of forces a nd induced stress distribution along the crack length 2 2 2 2 2 2 2 2 2 2 2 22 2 1 1 1 3 ,j j j j j j j j j j j jy x b x b y x b x b y x b y x b b y x (A-5) PAGE 142 120 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 2 3 2 1 5 0 2 2 2 1 1 1 1 1 5 0 2 2 2 2 2 1 1 1 1 1 5 0 2 2 2 22 1 log , D B C A y D A C B x y x D A C B y D B C A x y b x r r r r b a y r r D A C B b a b a r r D A C B b a b a a b y xi i i i i i i i i j i (A-6) and 5 0 2 1 5 0 2 1 5 0 1 1 5 0 1 1, ,a x r D a x r C a x r B a x r Ai i i i the highlighted text in this appendix is di fferent from original paper. The additional details were obtained from private communication with th e author of the paper. Intact Stiffener Displacement The outer stiffener is assumed to be supported on three frames running normal to the stiffeners. The center frame is on the skin crack centerline. Stiffener extension at the fastener shear face is determined because of axial loads and bending from fastener loads and direct loads resulting from axial stress es. Stiffener bending is induced since the fastener shear faces are offset from the stiffener neutral axis. The average bending moment between each fastener, obtained through the use of the three moment equation is given by 2 2 2 31 2 1 2 3 2i i n j n j j j j n j i j j Ay y L y Ly F L C CF Mi (A-7) stiffener displacement caused by bending from fastener loads is given by i i n i i i A My y M E I Ci i1 1 (A-8) stiffener displacement caused by direct fastener loads is given by i j n j n j i j j i j j DF AE y y F AEi1 2 11 (A-9) PAGE 143 121 stiffener displacement resulting from gross stress is given by E yi Gi (A-10) Broken Stiffener Displacement The broken stiffener is supported at the fr ames that pass across the center of the panel along the crack length and at the edge of the stiffeners. Since the fasteners are not located at the neutral axis of the stiffene rs there will be displacement due to bending moment also. The average bending moment between each fastener is given by 2 4 3 4 51 1 2 2 1 i i n j j j j i j j j Ay y L y P L C CP Mi (A-11) stiffener displacement resulting from be nding from fastener loads is given by 1 1 n i i i i i A My y M E I Ci i (A-12) stiffener displacement resulting fr om direct load is given by i j j n j i j j n j j i n Dy y P AE P y y AEi1 1 11 1 (A-13) Fastener Displacement The elastic displacement in shear can be represented by the empirical relation 2 1 2 1t D t D H H ED FR (A-14) where the symbols are defined in nomenclat ure and the parameters for aluminum and steel are given below. H1: 5.0 for aluminum and 1.666 for steel fasteners H2: 0.8 for aluminum and 0.86 for steel fasteners Compatibility of Displacements For displacement compatibility the total sk in displacement at any fastener location should be equal to the stiffener plus fastener displacement at that location. This gives 2n PAGE 144 122 by 2n matrix equation to be solved for unknow n fastener forces. For central broken stiffener, the center of the stiffener undergoes displacement, so we fix the nth fastener on this stiffener as datum and subtract the displacement at the nth location from displacements at all the other fasteners on this stiffener. The compatibility equation is given by n n i i n j j a i i j j j j j i i j n j n j a i i j j j j j i i j i i i n n j j a i i j j j j j n n j n j n j a i i j j j j j n n jy x V y x V db b y x b y x Et y y x y x v P db b y x b y x Et y y x y x v F R M D R db b y x b y x Et y y x y x v P db b y x b y x Et y y x y x v F, , , 2 1 , , , 2 1 , , , 2 1 , , , 2 1 , ,1 1 1 0 2 3 2 1 0 2 2 1 0 2 3 2 1 0 2 2 (A-15) Additional equation is obtained from the equi librium of broken stiffener. The summation of fastener forces on the broken stiffener s hould be equal to the gross stress times the broken stiffener area. n i s iA P1 (A-16) for outer intact stiffeners the datum is fixe d at the center of sti ffener since it does not undergo displacement. i i i n j j a i i j j j j j i i j n j n j a i i j j j j j i i j i i iG y x V db b y x b y x Et y y x y x v P db b y x b y x Et y y x y x v F M D R , , 2 1 , , , 2 1 , ,1 1 0 2 3 2 1 0 2 2 (A-17) PAGE 145 123 using Equations A-15 and A-16 for calculating displacement at n fasteners on the central broken stiffener and Equation A-17 for n fastener on outer intact stiffener 2n fastener forces are determined from 2n equations. The stress intensity factor due to equal and opposite pair of fastener fo rces placed symmetrically across the crack centerline (x-axis) is given as 2 11 2 3 2 I I t a Fy K (A-18) and the total stress intensity due to fastener forces is n n i n i FI I t a Fy I I t a Py K2 1 2 1 1 2 11 2 3 2 1 2 3 2 (A-19) where 5 0 2 2 2 2 2 2 2 2 2 5 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 14 2 1 4 2 4 y x x a y x a y y x x a y y x a y y x x a y x a I y x x a y y I (A-20) The total stress intensity is obtained by supe rposition of stress intensity due to far field stress on an unstiffened plate, plus that due to each set of fasteners (paying attention to load direction. If a stiffener is broken then the fastener forces on that stiffener will be tensile otherwise for intact stiffener they are compressive). a K KF total (A-21) The geometric factor is the ratio of this stress intensity factor with stiffeners to that of an unstiffened panel. A complete description of procedure can be found in Swift (1984). PAGE 146 124 Effectiveness of Stiffeners in Re ducing Crack Tip Stress Intensity The effectiveness of stiffeners in redu cing crack growth rate increases with stiffener area and decreases with stiffener sp acing. If stiffener area is very small or stiffener spacing is very large compared to critical crack length, the effectiveness of stiffeners in reducing crack growth will be compromised. Stiffeners will however continue to be very effectiv e in arresting cracks, besides they provide protection against buckling and pillowing effect due to bending loads in fuselage. To demonstrate this, the stress intensity a K is plotted in Figure A-5 for unstiffened and stiffened panel made of 7075-T6 Aluminum alloy. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 half crack length, mStress Intensity, MPa a K:Stiffened K:Unstiffened Figure A-5: Comparison of stre ss intensity factor for a panel with skin thickness = 2.34 mm and stiffener area of 2.30 10-3 meter2 PAGE 147 125 From Figure A-5 it can be seen that stiffene rs have arrested the crack. The critical stress intensity factor for 7075-T6 aluminum is 36.58 meter MPa. To demonstrate a case where stiffening will not be very effec tive we plot the stress intensity for another design in Figure A-6. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 50 100 150 half crack length, mStress Intensity, MPa a K:Stiffened K:Unstiffened Figure A-6: Comparison of stre ss intensity factor for a panel with skin thickness = 1.81 mm and stiffener area of 7.30 10-4 meter2 PAGE 148 126 APPENDIX B CALCULATING CRACK GROWTH FOR STIFFENED PANELS USING NUMERICAL INTEGRATION AND RESPONSE SURFACE The main computational expense associat ed with reliability based design for fatigue cracking is that for most structur al design problems there is no analytical expression to track crack size as a function of applied load cycles. Crack growth can only be determined using computa tionally intensive methods. Fo r the stiffened panel design the number of fatigue cycles accumulated in growing a crack from initial size ai to final size aN can be obtained by integrating a crack growth rate equation f, between the initial crack ai and final crack aN. Alternatively, the final crack size aN after N fatigue cycles can be determined by solving Equation B-1. N ia am K f da N, (B-1) where a is crack length f represents the crack growth rate function m represents one or more material pa rameters depending on crack growth model K is the cyclic stress intensity range is the effect of structural geometry Accurate numerical integration will require us to determine the stress intensity factor K at large number of integration points usi ng detailed finite element analysis or displacement compatibility method (Appe ndix A), which is cost prohibitive. PAGE 149 127 To reduce the computational burden associat ed with evaluating the stress intensity factor we develop response surface approxima tions (RSAs) for the geometric factor due to stiffening ( ). The geometric factor accounts for the effect of stiffening and depends on crack length a, panel dimension, stiffener dimension and stiffener spacing. Stiffeners can break during crack growth if stiffener strength is exceeded thereby increasing the stress intensity fact or. For accurate computation of at each integration point (crack size), the state of the stiffeners (broken or intact ) was also incorporated into the estimation of by having four RSAs for each stiffener states. Using symmetry, for a given crack le ngth and stress, only the following are considered (a) all stiffeners intact (b) two i nner stiffener broken (c ) four inner stiffener broken (d) all stiffeners broken. Typical response curves for geometric factor and stiffened panel dimensions are shown in Figur e B-1. Calculating crack growth during the reliability based optimization can be extremely time consuming. To reduce the computational cost we use cubic respons e surface approximation (RSA) to estimate at crack tip and maximum force on stiffener as a function of plate thickness, stiffener area and crack length. A Latin hypercube design of experime nts in these three variables with 600**** sampling points is used (200 for each of the th ree regimes of crack length). To take into account the state of stiffener (b roken or intact) in computing fatigue life, we first obtain the maximum number of stiffeners that coul d have been broken at the given state of **** To improve the accuracy of RSAs, three different approximations were obtained for each case above (a, b, c and d) for crack lengths between (1) center of plate to first sti ffener (2) first stiffener to s econd stiffener (3) second stiffene r to third stiffener. In all we have 12 RSAs for and stiffener force. The original reason for the large number of points was to improve the accuracy of single cubic RSA that was fitted to the geomet ric factor curve in Figure B-1 in our previous paper (Kale et al., 2005). With three individual RS As accurate results can be ac hieved with fewer points. For example, with 150 sample points the error in th e reliability index is of the order of 3% PAGE 150 128 structure (structural design, crack length). The maximum stre ss on each stiffener is then calculated using the three RSAs for maximum stress on each stiffenermax ener FirstStiffF,max fener SecondStifF,max ener ThirdStiffF. If this stress exceeds yield stress of material than the stiffener is broken. Depe nding on state of stiffener the appropriate RSA for is used, e.g., if none of the stiffeners are broken than = 0_broken is used to estimate the stress intensity f actor at that crack length. Effect of stiffeners on the geometric factor 0 0.5 1 1.5 2 2.5 0.00E +00 2.00E01 4.00E01 6.00E01 8.00E01 1.00E +00 half crack length (m)Geometric factor All stiffeners intact 2 inner stiffeners broken 4 inner stiffeners broken all stiffeners broken Figure B-1: Typical response curves for e ffect of stiffening on geometric factor for a stiffener area of 1.5 mm2 and skin thickness of 1.5 mm PAGE 151 129 APPENDIX C ACCURACY ESTIMATES OF RESPO NSE SURFACE APPROXIMATIONS Response Surface Approximations for Geometric Factor The response surface approximation (RSA) for geometric factor due to stiffening is obtained by first computing it for some selected design points in the domain using the displacement compatibility analysis The design of experiments for fitting the RSA for was obtained by constructing Latin hyp ercube design in three variables namely plate thickness, stiffener area a nd crack length with 600 sampling points and calculating and stiffener forces at each point A cubic equation is then fitted by minimizing the least square error betw een predicted and actual value. To achieve accurate results we partition the design domain and construct several RSAs depending on structural design con cept. For safe-life design without any inspections, the stiffener area and skin thickne ss will be higher than the inspection based design. Table C-1 shows the bounds on design va riables used to construct the design of experiment for estimating for safe-life design calculation. The bounds on stiffener area and skin thickness were obtained by successi ve windowing. Table C-2 show the error estimate for the RSAs used to approximate and stiffener stress for safe-life design. Table C-1: Bounds on design va riables used to evaluate response surface approximation for safe life design (this design domain is used for calculating results in Table 3-10) Design variableLower bound Upper bound Skin thickness 2.04 mm 4.08 mm Stiffener area 1.5-3 meter2 3.0-3 meter2 Crack length 0.86 meters 0.1 mm PAGE 152 130 Table C-2: Error estimate of analysis respons e surfaces used to obtain safe-life stiffened panel design (these RSAs are used in crack growth calculations for safe-life design of stiffened panels in Table 3-10), -RSA and F-RSA Error estimates Normalized values Typical value eav erms R2 R2 adj 0_broken 0.5 1.0 0.004,0.007, 0.006 0.006, 0.01, 0.009 0.95 0.95, 0.96 0.94, 0.94,0.96 1_broken >1 0.006, 0.008, 0.007 0.009, 0.012, 0.01 0.98, 0.98, 0.91 0.97, 0.98, 0.90 2_broken >1 0.006, 0.009, 0.01 0.009, 0.01, 0.014 0.98, 0.98, 0.97 0.97, 0.97, 0.98 3_broken >1 0.006, 0.01, 0.01 0.009, 0.013, 0.012 0.98, 0.98, 0.97 0.98, 0.97, 0.97 max ener FirstStiffF >1 0.009, 0.02,0.04 0.012, 0.03,0.05 0.99, 0.99,0.99 0.98, 0.99,0.99 max fener SecondStifF >1 0.002, 0.02, 0.04 0.004, 0.03, 0.05 0.99, 0.98, 0.99 0.99, 0.97, 0.99 max ener ThirdStiffF >1 0.006,0.003, 0.04 0.008, 0.001, 0.05 0.99, 0.99, 0.97 0.99,0.99,0 .96 Since addition of inspections leads to reduc tion in structural si zes, we use closer bounds on the design of experiments to constr uct RSA for inspection based design. These bounds are shown in Table C-3. The upper bound on thickness is based on maximum possible plate thickness for unstiffened panel optimum with inspection (2.56 mm + thickness equivalent to cost of 3 inspectio ns); bounds on stiffener area were reduced based on successive windowing. Table C-4 shows the error estimate of the RSAs. Table C-3: Bounds on design va riables used to evaluate response surface approximation for inspection based design. (This design domain is used to calculate results for inspection based design show n in Table 3-12 through 3-14) Design variableLower boundUpper bound Skin thickness 1.0 mm 3.0 mm Stiffener area 3.0 10-4 m2 2.0 10-3 m2 Crack length 0.86 m 0.1 mm PAGE 153 131 Table C-4: Error estimate of analysis response surfaces used to obtain inspection based stiffened panel design (these RSAs are used in crack growth calculations for inspection based design of stiffened panels in Table 3-12 through 3-14), RSA and F-RSA Error estimates Response Nominal value eav erms R2 R2 adj 0_broken 0.5 1.0 0.004, 0.009, 0.008 0.007, 0.013, 0.011 0.96,0.95, 0.97 0.94, 0.94, 0.96 1_broken >1 0.009, 0.011, 0.008 0.014, 0.016, 0.012 0.98, 0.98,0.93 0.98,0.98, 0.92 2_broken >1 0.009, 0.015, 0.011 0.014, 0.019, 0.016 0.99, 0.98, 0.98 0.98, 0.97, 0.98 3_broken >1 0.01, 0.014, 0.012 0.015, 0.02, 0.017 0.99, 0.98, 0.98 0.98,0.97, 0.98 max ener FirstStiffF >1 0.005, 0.024, 0.027 0.008, 0.032, 0.036 0.99, 0.99, 0.99 0.99,0.99,0. 99 max fener SecondStifF >1 0.0023, 0.017, 0.033 0.0051, 0.022, 0.043 0.99, 0.98, 0.99 0.99, 0.98, 0.99 max ener ThirdStiffF >1 0.0014, 0.0018, 0.04 0.002, 0.0024, 0.05 0.99, 0.99, 0.97 0.99, 0.99, 0.96 Response Surface Approximation for Reliability Index (Beta) This section presents the accuracy estima tes of design response surface to estimate the reliability index as a function of design variables. These RSAs are constructed by fitting a cubic equation to data at sampled locations. Table C-5 presents the RSA for reliability index for the stiffened panel safe-life design. The bounds on the design variables are presented in Table C-1 and RS A is calculated by fitting data to 100 sample points. Table C-5: Error estimate of design response surf aces (this is used to estimate reliability index as a function of design variable s for safe-life design calculations in Table 3-10), d-RSA Response, reliability inde xNominal target valueeav erms R2 R2 adj Beta 5.0 6.0 0.180.35 0.95 0.93 PAGE 154 132 Computational cost of reliability analys is with inspection schedule is reduced by using a cubic response surface approximation to predict the crack size mean and standard deviation after an inspection as a function of skin thickness, stiffener area, standard deviation in applied stress, initial crack m ean, initial crack standard deviation and crack growth time. For accurate results we estimate the crack size mean and standard deviation after the first inspection using the RSA in Table C-7 as a function of skin thickness, stiffener area, inspection interval and standard deviation in stress (the initial crack size distribution is fixed for first inspection). The reliability inde x for first inspection is also approximated by a cubic RSA in same variable s as shown in Table C-9. A minimax LHS design of experiment with 200 level of each de sign variable is used. For all inspections following the first one we estimate the crack size mean and standard deviation after inspections using the RSA in Table C-8 as a function of skin thickness, stiffener area, inspection interval and standard deviation in stress, initial crack m ean and initial crack standard deviation. The reliability index is approximated by a cubic RSA in same variables is shown in Table C-10. A minimax LH design of experiments with 400 level of each design variable is used. The range of each variable is shown in Table C-6. These RSAs are used for calculations in Tables 3-12 through 3-14. Table C-6: Bounds on design vari ables used to evaluate resp onse surface for crack sizes parameters after inspection and reliability index Design variable Lower bound Upper bound Skin thickness 1.0 mm 3.0 mm Stiffener area 3.0 10-4 meter22.0 10-3 meter2 Initial crack mean ai 0.2 mm 1.0 mm Initial crack standard deviation ai0.2 mm 1.0 mm Crack growth time 5000 flights 20000 flights Standard deviation in Stress 2 MPa 10 MPa PAGE 155 133 Table C-7: Error estimate of crack size respons e surfaces used to estimate the crack size distribution parameters after the first inspection (response surface is constructed by normalizing variables, stiffener ar ea, plate thickness and inspection time and standard deviation in stress), ai-RSA, ai-RSA Response Nominal target value mm eav, erm R2 R2 adj Mean ai after inspection 0.2 1.0 0.015 0.013 0.990.99 Standard deviation of ai after inspection 0.2 1.0 0.027 0.036, 0.960.95 Table C-8: Error estimate of crack size respons e surfaces used to estimate the distribution after inspection (response surface is constructed by normalizing variables), ai-RSA Response Nominal target value mm eav, erm R2 R2 adj Mean ai after inspection 0.2 1.0 0.006 0.008 0.990.99 Standard deviation of ai after inspection 0.2 1.0 0.016 0.023, 0.980.98 Table C-9: Error estimate of reliability inde x response surfaces us ed to schedule first inspection (design variables are stiffene r width, plate thickness and inspection time), d-RSA Response, reliability inde xNominal target valueeav erms R2 R2 adj Beta 5.0 0.170.24 0.99 0.98 Table C-10: Error estimate of reli ability index response surfaces, d-RSA Response, reliability inde xNominal target valueeav erms R2 R2 adj Beta 5.0 0.180.28 0.98 0.97 PAGE 156 134 APPENDIX D COST OF STRUCTURAL WEIGHT Fuel cost and material manufacturing cost may account for more than 80% of the total life cycle cost. We calculated the fuel cost based on data obtained from Venter (1998). The fuel cost is assumed to be $0. 89 a gallon and it is assumed that a pound of structural weight will cost 0. 1 pounds of fuel in a flight. Th is leads to a fuel cost of $0.0134 per pound per flight (assuming a gallon of jet fuel weighs 6.7 pounds). In this dissertation a slightly higher value of $0.015 per pound per flight is used for convenience. The material manufacturing cost was obtaine d by scaling down the material cost of the composite ($250) from Venter (1998) to $150 per pound for aluminum from previous paper (Kale et al., 2003). The scaling down is base d on a rough estimate only to demonstrate the methodology rather than perfor ming true cost calculation. The rough cost estimate of about $110 per pound was obtained from Petit et al. (2000). They obtained the cost estimate for stiffened metallic fusela ge panels for the B777 series aircraft. The structure used in Petit et al. (2000) consists of a 3.04 3.04 m2 structural component with 14 stringers, 7 frames and 7 fail safe straps bonded to each frame. The fuselage length is l = 62 m, the radius is r = 3.2 m and the skin thickness is 1.6 mm. The total manufacturing cost for the fuselage structure is $2.78 million. The computation of the total volume of a single structural component for weight calculation is give n in Table D-1. PAGE 157 135 Table D-1: Area of st ructural dimensions for cost calculation Structural component Area, m2 Number of components Total area m2 Total volume, m3 Total area Width Stringer 2 10-4 14 28 10-4 85.64 10-4 Frame 3 10-4 7 21 10-4 63.84 10-4 Tear strap 0.96 -47 6.77 10-4 20.48 10-4 (Source: Petit et al., 2000) Single panel skin area = 9.24 m2. The total fuselage surface area is 21246 2m rl This corresponds to 135 panels. The skin pa nel volume is, skin area times thickness = 9.24.0016 m3. Assuming aluminum density, = 2670 kg/m3 Total fuselage weight = number of panels (stringer volume+ frame volume + strap volume + skin volume) = 11512 kg Cost per unit weight = 241 $ 512 11 000 780 2 per kg or $110 per pound PAGE 158 136 APPENDIX E PSEUDO CODE FOR COMBINED OP TIMIZATION OF STRUCTURE AND INSPECTION SCHEDULE Introduction Combined optimization of structure and inspection schedule can be done by performing a one dimensional search on struct ural size for fixed number of inspection. When only one inspection type is used to ge nerate optimum inspect ion schedule this is straightforward since the cost function va ries monotonically with thickness for fixed number of inspections. The use of multiple in spection types leads to several local minima for fixed number of inspection depending on the inspection type sequence used for generating the inspection schedule. For exampl e, to obtain minimum cost for a schedule consisting of two inspections generated using four inspection types; it will require structural optimization for 16 inspection sche dules (4). Since the number of inspection that will lead to minimum cost is unknown, several iterations on structural sizes and number of inspections are required to obt ain optimum design and inspection schedule. This is computationally expensive. To solv e this problem we perform a one dimensional minimization on structural thickness in small steps. At each step we reduce thickness by an amount equivalent to cost of cheapest in spection. For reducing to tal lifecycle cost, at most one inspection can be added for this st ructure after iteration. This expedites the optimization. The pseudo code is shown below. (1) Start the one dimensional minimizati on using safe life de sign as initial guess i guesst = tsafe life, initialize thickness PAGE 159 137 0 i, initialize iteration counter Nallowable = 0, set the allowable num ber of inspections to zero (safe life design does not require any inspection (2) Reduce the thickness by an amount equiva lent to cost of cheapest inspection and obtain optimum inspection schedul e such that the number of inspections in the schedule is at most equal to one additional inspection from optimum of previous step. 1 i i, update iteration counter t t ti guess i guess 1, decrease thickness by fixed amount 1min allowable iN N Obtain cost of inspection schedule iCmin Obtain total minimum cost at ith iterationi TotalC (3) Find all the insp ection type sequences with co st bounded between the costs of inspection at current iteration and that at previous iteration. The constraints for generating inspection sequence is given below. i new i i new iN N N C C Cmin min 1 min min min 1 min (4) Obtain the minimum thickness required to ma intain the threshold reliability level for each inspection type sequence obtained in st ep 3. This gives the minimum cost at ith iteration. Obtain optimum structural designi optt, number of inspectionsi optN, minimum cost i TotalC and cost of inspections,i optC. PAGE 160 138 (5) If the structural thickness ha s reached a minimum allowable value, find the structural design and inspection schedule with minimum total cost from optimum of each iteration else update the variables as shown below and go to step 2. Update variables at ith iteration i opt allowable i opt i i opt i i opt i guessN N C C N N t t min min Optimization of Inspection Types Optimization of an inspection schedule with different inspection types is computationally time-consuming because the inspection time and type of each subsequent inspection depends on the inspecti on time and type of previous inspections. As an illustration, if four inspection types ar e to be chosen to schedule four inspections for minimum cost, this will require a reliabil ity analysis on 4 4 = 256 different type sequence. However, in a typical optimization problem, the number of inspections required to satisfy reliability constraint is unknown and is obtained directly from reliability analysis (optimization of in spection times for fixed reliability). The computational cost may be prohibitive if a brut e force approach is used. We seek a mix of inspection types and reduce the number of se quence by eliminating im possible or clearly sub-optimal inspection types. The terminology used in the algorithm presented in Chapter 4 is described below. For aircraft structures the minimu m structural thickness is about 1.0 mm Typically, if the cost starts increasing at successive thickness reduction, the iterations can be terminated PAGE 161 139 (1) Baseline inspection schedule (Nkb, Ckb). Optimum number of inspections of kth type and Nkb and the corresponding cost Ckb if only the kth inspection type is used to optimize the inspection schedule for a given reliability constraint. (2) Combination sequence. Number of inspection of each type that can be used to generate the inspection sche dule (time of inspection). Fo r example, a combination represented by [N1 =1, N2 = 2, N3 = 1, N4 = 2] means that there is one inspection of type 1, two inspections of type 2, one inspection of type 3 and two inspections of type 4 in an inspection schedule. In all on them there ar e 6 inspections available to generate the inspection schedule. (3) Inspection type sequence. The order in which the various inspections are done. For example, a combination sequence represented by [N1 =0, N2 =1, N3 = 1, N4 = 1] can have six different orders in which an inspec tion of each type can be conducted. These are (1) I2, I3, I4 (2) I2, I4, I3 (3) I3, I2, I4 (4) I3, I4, I2 (5) I4, I2, I3 (6) I4, I3, I2. The order of the inspection type is important in gene rating the optimum inspection schedule for minimum cost, because the reduction in the proba bility of failure of one sequence can be different from others, probability of detection Pd being a function of crack size and crack size being a function of time. (4) Constraint 1. ck kI C Nmin where [] is a rounded up integer. This constraint essentially means that the maximum number of inspections of the kth type that can occur in a combination sequence should be such that the cost due to inspection of the kth type is less than or equal to the minimum cost Cmin. PAGE 162 140 (5) Constraint 2: 4 1 4 1 k b k bN N NThe reason for this constraint is that N1b is the optimum number of inspections obtained using the most effective inspection type. A total number of inspections in a combination sequence 4 1 k kN less than N1b is a direct violation of the relia bility constraint. N4b is the optimum number of inspections obtained using the least effective inspection type. The total number of inspection in a combination sequence 4 1k kN greater than N4b is a direct violation of the cost constraint. PAGE 163 141 APPENDIX F EFFECT OF CRACK SIZE PROBABILITY DISTRIBUTION ON FAILURE PROBABILITY AND INSPECTION INTERVAL Intuitively it appears that when we increase the standard deviation of a distribution, we increase the probability of extreme valu es of the random variable. This appendix shows that this is not always true for the l ognormal distribution used here to model initial crack size. Consequently, we can run into si tuations as in Table F-1 where increase in mean and standard deviation of the initial cr ack sizes can lead to increase in inspection interval. In Table F-1 we generated insp ection times for a structural size of 2.48 mm for a required reliability level of 10-7. Table F-1: Inspection sche dule and crack size distribution after inspection (ah = 1.27 mm) for an unstiffened plate thickness of 2.48 mm and a threshold probability of 10-7 Number of inspections, N Inspection time (flights) Inspection interval (flights) 1 n nS S Crack size distribution after inspection (Mean,mm, cov) 0 Initial crack distribution (0.200 mm, 0.35) 1 14569 14569 (0.33,1.39) 2 18991 4422 (0.29,1.06) 3 25952 6961 (0.32,1.31) 4 30128 4176 (0.36,1.16) 5 38167 8037 Table F-1 shows the inspection times in colu mn 2 and inspection interval in column 3. The crack size mean and standard deviati on after the inspection is shown in column 4 (these values are different for each row becau se the crack growth time is different for each inspection interval and because the de tected cracks are repl aced by new component with much smaller cracks after an inspection). From Table F-1, third inspection interval PAGE 164 142 6961 flights is smaller than fi fth inspection interval 8037 f lights even though the crack size distribution parameter before the third interval (mean = 0.29 mm, cov = 1.06) is benign than the parameters before fifth interval (mean = 0.36 mm, cov = 1.16). That is, by just looking at the mean a nd standard deviation from Tabl e F-1 it appears that larger cracks are growing slower which is counter intuitive because crack grows faster with crack length (e.g., Paris law). This counterintuitive nature of the lognor mal distribution is illustrated by keeping the mean of the distribution at 1.0, varyi ng the standard deviati on and calculating the probability that the variable is greater than 2.0. Figure F-1: Probability of exceeding 2.0 for a lognormally distributed random variable with a mean of 1.0. Note that large standard deviation decreases probability PAGE 165 143 Figure F-2: Comparison of failu re probability (1CDF) of two probability distributions with mean 10-5 and standard deviation of 2 and 10 units Figure F-2 illustrates the cumulative density f unction of two distributions with standard deviation of 1.8 and 10.0 units. It can be seen that the failure probability of the distribution with higher standard deviation is lesser than th at of the distribution with lower standard deviation. PAGE 166 144 APPENDIX G WHY ARE AIRPLANES SO SAFE STRU CTURALLY? EFFECT OF VARIOUS SAFETY MEASURES ON STRUCTURAL SAFETY This chapter is close to Acar et al. (2005). A preliminary methodology to investigate the interaction of error, variability and testing in structural design was developed by me using M onte Carlo simulation method. This work was done in collaboration with Erdem Acar who impr oved the work by developing analytical approximation for calculating failure probability and investigated the effectiveness of certification testing and its eff ect on distribution of errors. Erdems contribution to this work in fully acknowledged. Introduction In the past few years, there has been growing interest in applying probability methods to aircraft structural design (e .g., Lincoln, 1980; Wirsching, 1992; Aerospace Information Report 5080 (Society of Auto motive Engineers, 1997) and, Long and Narciso, 1999). However, many engineers are skeptical of our ability to calculate the probability of failure of structural designs for the following reasons. First, data on statistical variability in material properties geometry and loading distributions are not always available in full (e.g., joint distribu tions), and it has been shown that insufficient information may lead to large errors in probability calculations (e.g., Ben-Haim and Elishakoff, 1990 and Neal et al., 1992). Second, the magnitude of errors in calculating loads and predicting structural response is not known precisely, and there is no consensus on how to model these errors in a probabilistic setting. As a result of these concerns, it is PAGE 167 145 possible that transition to probability based desi gn will be gradual. In such circumstances it is important to understand the impact of existing design practices on safety. This chapter is a first attempt to explore the eff ects of various safety measures taken during aircraft structural design using the deterministic design approach based on FAA regulations. The safety measures that we include here are (1) the use of safety factors, (2) the use of conservative material properties (A-b asis), and (3) the use of final certification tests. These safety measures are representative rather than all inclus ive. For example, the use of A-basis properties is a representative measure for the use of conservative material properties. We do not include in this discussion the additiona l safety due to structural redundancy and due to conserva tive design load specifica tion. The use of A-Basis property rather than B-basis is due to the f act that we did not include redundancy. FAA suggests that (FAR 25.613) when there is a si ngle failure path, A-Basis properties should be employed, but in case of multiple failure paths, B-Basis properties are to be used. The effect of the three individual safety measur es and their combined effect on the probability of structural failure of the aircraft are de monstrated. We use Monte Carlo simulations to calculate the effect of these safety measures on the probability of fa ilure of a structural component. We start with a structural design employi ng all considered safety measures. The effects of variability in geometry, loads, a nd material properties ar e readily incorporated by the appropriate random variables. However, there is also uncertainty due to various errors such as modeling errors in the anal ysis. These errors are fixed but unknown for a given airplane. To simulate these epistemic uncertainties, we transform the error into a PAGE 168 146 random variable by considering the design of mu ltiple 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, etc.) each designi ng essentially the same airplane, but each having different errors in their structural an alysis. This assumption is only a device to model lack of knowledge or er rors in probabilistic setting. However, pretending that the distribution represents a large number of ai rcraft companies helps to motivate the probabilistic setting. For each model we simulate certification tes ting. If the airplane passes the test, then an entire fleet of airplanes with the same design is assumed to be built with different members of the fleet having di fferent geometry, loads, and material properties based on assumed models for variability in these pr operties. That is, the uncertainty due to variability is simulated by considering multiple realizations of the same design, and the uncertainty due to errors is simulated by desi gning different structures to carry the same loads. We consider only stress failure due to extreme loads, which can be simulated by an unstiffened panel designed under uniaxial load s. No testing of components prior to certification is analyzed fo r this simple example. Structural 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. We propose in Table G-1 a clas sification that distinguishes between (1) uncertainties that apply equally to the en tire fleet of an aircraft model and (2) uncertainties that vary for the individual ai rcraft. The distinction is important because safety measures usually target one or the other. While type 2 ar e random uncertainties PAGE 169 147 that can be readily modeled probabilistically, type 1 are fixed for a given aircraft model (e.g., Boeing 737-400) but th ey are largely unknown. That is, the uncertainty in the failure of a structural member can also be divided into two types: systemic errors and variability. Systemic errors reflect inaccurate modeling of physical phenomena, errors in struct ural analysis, errors in load calculations, or use of materials and tooli ng in construction that are different from those specified by the designer. Systemic errors affect all the copies of the structural components made and are therefore fleet-level un certainties. They can reflect differences in analysis, manufacturing and operation of the aircraft from an ideal. The id eal aircraft is an aircraft designed assuming that it is possi ble to perfectly predict stru ctural 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 de sign specifications, and that there exists an operating environment that on average agrees with the design speci fications. The other type of uncertainty reflects variability in material properties, geometry, or loading between different copies of th e same structure and is called here individual uncertainty. Table G-1: Uncertainty classification Type of uncertainty Spread Cause Remedies Systemic error (modeling errors) Entire fleet of components designed using the model Errors in predicting structural failure and differences between properties used in design and average fleet properties. Testing and simulation to improve math model and the solution. Variability Individual component level Variability in tooling, manufacturing process, and flying environments. Improve tooling and construction. Quality control. PAGE 170 148 Safety Measures Aircraft structural design is still done, by and large, us ing code-based design rather than probabilistic approaches. Safety is im proved through conservative design practices that include use of safety fact ors and conservative material pr operties. It is also improved by tests of components and certifi cation tests that can reveal in adequacies in analysis or construction. In the following we detail some of these safety measures. Safety Margin. Traditionally all aircraft structures are designed with a safety factor to withstand 1.5 times the limit loads without failure. A-Basis Properties. In order to account for uncertain ty in material properties, the Federal Aviation Administration (FAA) recomm ends the use of conservative material properties. This is determin ed by testing a specified numb er of coupons selected at random from a batch of material. The A-basi s property is determined by calculating the value of a material property exceeded by 99% of the population with 95% confidence. Component and Certification Tests. Component tests and certification tests of major structural components reduce stress a nd material uncertainties for given extreme loads due to inadequate structural models. These tests are conducte d in a building block procedure. First, individual coupons are tested, and then a sub assembly is tested followed by a full-scale test of the entire structure. Since these tests cannot apply every load condition to the structure, they leave uncertainties with resp ect to some loading conditions. It is possible to re duce the probability of failure by performing more tests to reduce uncertainty or by extra structural weig ht to reduce stresses. If certification tests were designed together with the structure, it is possible that additional tests would become cost effective because they w ould allow reduced structural weight. PAGE 171 149 We simulate the effect of these three sa fety measures by assuming the statistical distribution of the uncertainti es and incorporating them in approximate probability calculations and Monte Carlo simulation. For variability the simulation is straightforward. However, while systemic erro rs are uncertain at the time of the design, they will not vary for a single structural com ponent on a particular ai rcraft. Therefore, to simulate the uncertainty, we assume that we have a large number of nominally identical aircraft being designed (e.g., by Airbus, Boeing, Bombardier, etc.), with the errors being fixed for each aircraft. This creates a two-le vel Monte Carlo simula tion, with different aircraft models being consider ed at the upper level, and diffe rent instances of the same aircraft at the lower level. To illustrate the procedure we consider a simple example of an unstiffened panel designed for strength under uniaxial tensile loads. This will still simulate reasonably well more complex configurations, such as stiffe ned panels subject to stress constraints. Aircraft structures have more complex failure modes, such as fatigue and fracture, which require substantially different treatment and the consideration of the effects of inspections (see Kale et al., 2003). However, this simple example serves to furthe r our understanding of the interaction be tween various safety measures. The procedure is summarized in Figure G-1, which is described in detail in the next section. Panel Example Definition Design and Certification Testing We assume that we have N different aircraft models, i.e., we have N different companies producing a model with systemic errors. We consider a generic panel to represent the entire aircraft structure. The true stress (true ) is found from the equation PAGE 172 150 wt Ptrue (G-1) where P is the applied load on the panel of width w and thickness t. In a more general situation, Eq. (G-1) may apply to a small element in a more complex component. When errors are included in the analysis, the true stress in the panel is different from the calculated stress. We include th e errors by introduci ng an error factor e while computing the stress as true calce 1 (G-2) Positive values of e yield conservative estimates of the true stress and negative values yield unconservative stress estimation. The other random variables account for variability. Combining Eqs. (G-1) and (G-2), the stress in the panel is calculated as wt P ecalc 1 (G-3) The design thickness is determined so that the calculated stress in the panel is equal to material allowable stress for a design load Pd multiplied by a safety factor SF, hence the design thickness of the panel is calculated from Eq. (G-3) as a d F designw P S e t) 1 ( (G-4) PAGE 173 151 Figure G-1: Flowchart for Monte Carlo simulation of panel design and failure where the panel width, w, is taken here to be 1.0 meter, and a is the material stress allowable obtained from testing a batch of coupons according to procedures that depend on design practices. Here, we assume that A-basis properties ar e used (Appendix H). During the design process, th e only random quantities are a and e. The thickness obtained from Eq. (G-4) (step A in Fig. (G-1)) is the nominal thickness for a given aircraft model. The actual thickness will vary due to individuallevel manufacturing uncertainties. After the panel has been de signed (that is, thickness determined) from Eq. (G-4), we simulate certification testing for the aircraft. Here we assume that the panel will not be built with complete fidelity to the desi gn due to variability in geometry (width and thickness). The panel is then loaded with the design axial force of (SF times Pd), and the stress in the panel is record ed. If this stress exceeds th e failure stress (itself a random PAGE 174 152 variable, see Table G-2.) then the design is rejected, otherwise it is certified for use. That is, the airplane is certified (step B in Fig. G-1) if the following inequality is satisfied 0Fd ffSP wt (G-5) and we can build multiple copies of the airpla ne. We subject the panel in each airplane to actual random maximum (over a lif etime) service loads (step D) and decide whether it fails using Eq. (G-6). f PRtw (G-6) Here, P is the applied load, and R is the resistance or load ca pacity of the structure in terms of the random width w, thickness t and failure stress f. A summary of the distributions for the random va riables used in design and cert ification is listed in Table G2. Table G-2: Distribution of random variables used for panel design and certification Variables DistributionMeanScatter Plate width (w) Uniform 1.0 (1%) bounds Plate thickness (t) Uniform tdesign (3%) bounds Failure stress (f) Lognormal 150.08 % coefficient of variation Service load (P) Lognormal 100.010 % coefficient of variation Error factor (e) Uniform 0.0 10% to 50% This procedure of design and tes ting is repeated (steps A-B) for N different aircraft models. For each new model, a different random error factor e is picked for the design, and different allowable properties are genera ted from coupon testing (Appendix H). Then in the testing, different thic knesses and widths, and different failure stresses are generated at random from their distributions. PAGE 175 153 Effect of Certification Tests on Distribution of Error Factor e One can argue that the way cer tification tests reduce the probability of failure is by changing the distribution of the error factor e. Without certification testing, we assume symmetric distribution of this error factor However, designs ba sed on unconservative models are more likely to fail certifi cation, and so the distribution of e becomes conservative for structures that pass certificat ion. In order to quan tify this effect, we calculated the updated distribu tion of the error factor e. The updated distribution is calculated analytically by Bayesian updating by making some approximations, and Monte Carlo simulations are conducted to ch eck the validity of those approximations. Bayesian updating is a commonly used t echnique to obtain up dated (or posterior) distribution of a random va riable upon obtaining new in formation about the random variable. The new information here is that the panel has passed the certification test. Using Bayes Theorem, the upda ted (posterior) distribution ) (Uf of a random variable is obtained from the in itial (prior) distribution ) (If based on new information as d f P f P fI I U) ( ) | ( ) ( ) | ( ) ( (G-8) where P(|) is the conditional probability of observing the experimental data given that the value of the random variable is For our case, the po sterior distribution ) ( e fUof the error factor e is given as b b I I Ude e f e C P e f e C P e f ) ( ) | ( ) ( ) | ( ) ( (G-9) PAGE 176 154 where C is the event of passing certification, and P(C|e) is the probability of passing certification for a given e Initially, e is assumed to be uniformly distributed. The procedure of calculation of P(C|e) is described in Appendix J, where we approximate the distribution of the ge ometrical variables, t and w as lognormal, taking advantage of the fact that their coefficient of variation is small compared to that of the failure stress (see Table G-2). We illustrate the effect of certification te sts for the panels designed with A-Basis material properties. An initial and upda ted distribution plot of error factor e with 50 % bound is shown in Fig. G-2 Monte Carlo simula tion with 50,000 aircraft models is also shown. Figure G-2 shows that the certification tests greatly reduce the probability of negative error, hence eliminating most unconser vative designs. As seen from the figure, the approximate distribution calculated by the analytical approach matches well the distribution obtained from Monte Carlo simulations. Probability of Failure Calculation by Analytical Approximation The stress analysis represented by Eq. (G-1 ) is trivial, so that the computational cost of Monte Carlo simulation of the probabi lity of failure is not high. However, it is desirable to obtain also analytical probabiliti es that may be used for more complex stress analysis and to check the Monte Carlo simu lations. In order to take advantage of simplifying approximations of the distribu tion of the geometry parameters, it is convenient to perform the pr obability calculation in two stages, co rresponding to the inner and outer loops of Fig. G-1. That is, we first obtain expressions for the probability of failure of a single aircraft model (that is, given e and allowable stress). We then calculate the probability of failu re over all aircraft models. PAGE 177 155 Initial and Updated Distribution of Error Factor e and Comparison of Analytical Approx. with Monte Carlo Simulations0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 error factor, eInitial and Updated Distributions Monte Carlo, initial Monte Carlo, updated Analytical Approx., initial Analytical Approx., updated Figure G-2: Initial and upda ted probability distribution functions of error factor e Error bound is 50% and Monte Carlo simulation done with a sample of 50,000. The mean value of the probability of failure over all aircraft models is calculated as design design design f fdt t f t P P ) ( ) ( (G-10) where tdesign is the non-deterministic di stribution parameter, and f(tdesign) is the probability density function of parameter tdesign. It is important to have a measure of variab ility in this probability from one aircraft model to another. The standard deviation of failure probability gives a measure of this variability. In addition, it provides inform ation on how accurate is the probability of failure obtained from Monte Carlo simulations. The standard deviation can be calculated from 2 / 1 2) ( ) ( design design f design f f Pdt t f P t P (G-11) PAGE 178 156 Probability of Failure Calculatio n by Monte Carlo Simulations The inner loop in Fig. G-1 (steps C-E) re presents the simulation of a population of M airplanes (hence panels) that all have th e same design. However, each panel is different due to variability in geometry, failu re stress, and loading (step D). We subject the panel in each airplane to actual random maximum (over a lifetime) service loads (step E) and calculate whether it fails using Eq. (G-6). For airplane model that pass certification, we count the number of panels failed. The failure probability is calculated by dividi ng the number of failures by the number of airplane models that passed certification, times the number of copies of each model. The analytical approximation for the probability of failure suffers due to the approximations used, while the Monte Carlo simulation is subject to sampling errors, especially for low probabilitie s of failure. Using large samples, though, can reduce the latter. Therefore, we compared the two me thods for a relatively large sample of 10,000 aircraft models with 100,000 instances of each model. In addition, the comparison is performed for the case where mean material properties (rather than A-basis properties) are used for the design, so that the probabil ity of failure is high enough for the Monte Carlo simulation to capture it accurately. Ta ble G-3 shows the results for this case. Table G-3: Comparison of probability of failures ( Pfs) for panels designed using safety factor of 1.5, mean value for a llowable stress and error bound of 50% Value Analytical approximation Monte Carlo simulation* % error Average Value of Pf without certification ( Pnt)1.715-1 1.726-1 0.6 Standard Deviation of Pnt 3.058-1 3.068-1 0.3 Average Value of Pf with certification ( Pt) 3.166-4 3.071-4 3.1 Standard Deviation of Pt 2.285-3 2.322-3 1.6 Average Value of Initial error factor ( ei) 0.0000 -0.00024 --Standard Deviation of ei 0.2887 0.2905 0.6 Average Value of Updated error factor ( eup) 0.2468 0.2491 0.9 Standard Deviation of eup 0.1536 0.1542 0.4 N = 10,000 and M = 100,000 is used in the Monte Carlo Simulations PAGE 179 157 The last column of Table G-3 shows the percent error of the analytical approximation compared to Monte Carlo simu lations. It is seen that the analytical approximation is in good agreement with the values obtained through Monte Carlo simulations. It is remarkable that the standa rd deviation of the probability of failure is almost twice the average value of the probability (the ratio, the coefficient of variation, is about 178%) before certification, and about seven times larger afte r certification. This indicates huge variability in the probability of failure for different aircraft models, and this is due to the large error bound, e=50% With 10,000 different aircraft models ( N ), the standard deviation in the Monte Carlo estimat es is about 1%, and the differences between the Monte Carlo simulation and the analyt ical approximation are of that order. Effect of Three Safety Measures on Probability of Failure We next investigate the effect of other safety measures on failure probability of the panels using Monte Carlo simulations. We performed the simulation for a range of variability in error factor e for 5000 airplane models ( N samples in outer loop) and 20,000 copies of each airplane model ( M samples in inner loop). Here, we compare the probability of failure of a structure designed with three safety measures (safety factor, conservative material property and certificati on testing) to that of a structure designed without safety measures. PAGE 180 158 Table G-4: Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and A-basis prop erty for allowable stress. Numbers in parenthesis denote the coefficient of va riation of the quantity. Average design thickness without cert ification is 1.271. Error Bound e Average design thickness after certificati on* Certificat ion failure rate % Probability of failure after certification ( Pt) x10-4 Probability of failure without certification ( Pnt) x10-4 Probability ratio ( Pt/Pnt) Probability difference ( PntPt) 50% 1.453 (0.19) 29.3 1.664 (7.86) 449.0 (2.74) 3.706-3 4.473-2 40% 1.389 (0.17) 24.3 1.586 (6.92) 89.77 (3.22) 1.767-2 8.818-3 30% 1.329 (0.15) 16.3 1.343 (5.28) 9.086 (3.46) 1.479-1 7.742-4 20% 1.283 (0.12) 6.2 0.304 (4.81) 0.477 (3.51) 6.377-1 1.727-5 10% 1.272 (0.07) 1.2 0.027 (4.71) 0.029 (4.59) 9.147-1 2.490-7 *Average over N=5000 models Table G-4 presents the results when all safety measures are used for different bounds on the error. The second column shows the mean and standard deviation of design thicknesses generated for panels that passed certification. These panels correspond to the outer loop of Fig. G-1. The variability in design thickness is due to the randomness in the error e and in the stress allowable. The average thickness before certification was 1.269, so that the column s hows the conservative effe ct of certification testing. When the error bound is 10% then 98.8% of the panels pass certification (third column in Table G-4), and the average thickness is increased by only 0.24% due to the certification process. On the other hand, when the error bound is 50%, 29% of the panels do not pass certification, and this raises the av erage thickness to 1.453. Thus, the increase in error bound has two opposite effects. With out certification testing, increasing the error bound greatly increases the probability of failure. For example, when the error bound changes from 30% to 50%, the probability of failure without certification changes from PAGE 181 159 0.00091 to 0.0449 or by a factor of 49. On the ot her hand, with the increased average thickness, after certification the probability increases only from 1.343x10-4 to 1.664x10-4. The effectiveness of the certification test s can be expressed by two measures of probability improvement. The first measure is th e ratio of the probability of failure with the test, Pt, to the probability of failure without tests, Pnt. The second measure is the difference of these probabilities. The ratio is a more useful indicator for low probabilities of failure, while the difference is more mean ingful for high probabilities of failure. However, when Pt is high, the ratio can mislead. That is, an improvement from a probability of failure of 0.5 to 0.1 is more substantial than an improvement in probability of failure of 0.1 to 0.01, because it saves more airplanes. However, the ratio is more useful when the probabilities are small, a nd the difference is not very informative. Table G-4 shows that certif ication testing is more important for large error bounds e For these higher values the number of pane ls that did not pass certification is higher, thereby reducing the failure probability for those that passed certification. While the effect of component tests (building block tests) is not simulated, their main effect is to reduce the error magnitude e This is primarily due to th e usefulness of component tests in improving analytical models and revealin g unmodeled failure modes. With that in mind, we note that the failure probability for the 50% error range is 1.70-4, and it reduces to 2.7-6 for the 10% error range-that is, by a factor of 63. The actual failure probability of aircraft pane ls is expected to be of the order of 10-8 per flight, much lower than the best number in the fourth column of Table G-4. However, the number in Table G-4 is for a lifetime for a single structural component. Assuming about 10,000 flights in the life of a panel and 100 independent structural components, this PAGE 182 160 10-5 failure probability for a panel will translate to a per flight probability of failure of 10-7 per airplane. This factor of 10 discrepancy is exacerbated by other failure modes like fatigue that have not been considered. Ho wever, other safety measures, such as conservative load specifications ma y account for this discrepancy. Table G-5: Probability of failure for different bounds on error e for panels designed using safety factor of 1.5 and mean value for allowable stress. Numbers in parenthesis denote the coefficient of va riation of the quantity. Average design thickness without cert ification is 1.000. Error bound e Average design thickness after certification Certific ation failure rate+ % Probability of Failure after certificatio n ( Pt) x 10-4 Probability of failure without certificatio n ( Pnt) x10-4 Probability ratio ( Pt/Pnt) Probability difference ( Pnt-Pt) 50% 1.243 (0.13) 50.1 3.420 (5.82) 1681 (1.81) 2.035-3 1.677-1 40% 1.191 (0.11) 50.1 4.086 (6.78) 969.0 (1.99) 4.217-3 9.649-2 30% 1.139 (0.09) 50.8 5.616 (5.45) 376.6 (2.00) 1.495-2 3.700-2 20% 1.086 (0.07) 50.7 6.253 (3.19) 92.67 (1.83) 6.748-2 8.642-3 10% 1.029 (0.05) 51.0 9.209 (1.70) 19.63 (1.25) 4.690-1 1.043-3 *Average over N=5000 models +With only 5000 models, the standard deviation in the certification failure rate is about 0.71%. Thus, all the number in this column are about 50% as may be expected when mean ma terial properties are used. Table G-5 shows results when average rather than conservative material properties are used. It can be seen from Table G-5 that the average thickness determined using the mean value of allowable stress is lower than that determined using the A-basis value of allowable stress (Table G-4). This is equivalent to adding an additional safety factor over an already existing safety factor of 1.5. For the distribution (lognormal with 8% coefficient of variation) and nu mber of batch tests (40 tests) considered in this chapter, a typical value of the safety factor due to A-Basis property is around 1.27. PAGE 183 161 Without the A-basis properties, the stress in the certification test is approximately equal to the average ultimate service stress, so that about 50% of the panels fail certification. When the errors are large, this raises substantially the average thickness of the panels that pass certificatio n, so that for an error bound of 50% the certification test is equivalent to a safety factor of 1.243. Large errors produce some super-strong and some super-weak panels (see Fig. G-3). The su per-weak panels are mostly caught by the certification tests, leaving the super-strong panels to reduce the pr obability of failure. Another way of looking at this effect is to note that when there are no errors, there is no point to the tests. Indeed, it can be seen that the probability of failure without certification tests improves with reduced error bound e but that the reduced effect of the certification tests reverses the trend. Thus for this case we obtain the counter-intuitive results that larger errors produce safer designs. Comparin g the first row of Table G-5 to Table G-3 we see the effect of the smaller sample fo r the Monte Carlo simulations. Table G-3 was obtained with 10,000 models and 100,000 copies per model, while Table G-5 was obtained with 5000 models, and 20,000 c opies per model. The difference in the probability of failure after ce rtification between the two tables is about 11 percent. However, the two values straddle the analytical approximation. The effects of building block type of te sts that are conducted before certification are not included in this study. These test s reduce the errors in analytical models. For instance, if there is 50% error in the analytical model the buil ding block type of tests may reduce this error to lower values. Hence, th e difference between the rows of Table G-4 may be viewed as indicating the benefits of reducing the error by building block tests. PAGE 184 162 Figure G-3: Design thickne ss variation with low and high error bounds. (Note that after certification testing only the designs above the minimum thickness are built and flown. Those on the right have a much higher average design thickness than those on the left). Table G-6: Probability of failure for different bounds on error e for safety factor of 1.0 and A-basis property for allowable stre ss. Numbers in parenthesis denote the coefficient of variation of the quant ity. Average design thickness without certification is 0.847. Error bound e Average design thickness after certification Certificat ion failure rate % Failure probability after certificatio n ( Pt) x10-2 Failure probability with no certificatio n ( Pnt) x10-2 Probability ratio ( Pt/Pnt) Probability difference ( Pnt-Pt) 50% 0.969 (0.19) 29.4 6.978 (2.12) 29.49 (1.31) 2.366-1 2.251-1 40% 0.929 (0.17) 25.0 7.543 (1.98) 24.56 (1.38) 3.071-1 1.702-1 30% 0.886 (0.15) 16.6 8.923 (1.73) 17.11 (1.43) 5.216-1 8.184-2 20% 0.855 (0.11) 5.7 8.171 (1.40) 9.665 (1.34) 8.454-1 1.494-2 10% 0.847 (0.06) 1.3 4.879 (0.97) 4.996 (0.97) 9.767-1 1.163-3 *Average over N=5000 models Table G-6 shows the effect of not using a safety factor. Although certification tests improve the reliability, again in a general tre nd of high improvement with high error, the lack of safety factor of 1.5 limits the impr ovement. Comparing Tabl es G-4 and G-6 it can be seen that the safety factor reduces the pr obability of failure by two to four orders of magnitudes. It is interesting to note that the effect of the error bo und on the probability of PAGE 185 163 failure after certification is not monotoni c, and this phenomenon is discussed in Appendix I. Table G-7: Probability of failure for different error bounds for panels designed using safety factor of 1.0 and mean value for allowable stress. Average design thickness without cert ification is 0.667. Error bound e Average design thickness after certification Certific ation failure rate % Probability of Failure after certificatio n ( Pt) Probability of failure without certificatio n ( Pnt) Probability ratio ( Pt/Pnt) Probability difference ( Pnt-Pt) 50% 0.830 (0.12) 50.1 0.125 (1.39) 0.505 (0.83) 2.463-1 3.808-1 40% 0.796 (0.11) 50.2 0.158 (1.20) 0.504 (0.79) 3.140-1 3.459-1 30% 0.761 (0.09) 50.4 0.205 (0.92) 0.503 (0.72) 4.075-1 2.981-1 20% 0.727 (0.08) 50.9 0.285 (0.64) 0.503 (0.58) 5.653-1 2.189-1 10% 0.686 (0.05) 50.7 0.412 (0.34) 0.500 (0.34) 8.228-1 8.869-2 *Average over N=5000 models Table G-7, shows results when the only sa fety measure is certification testing. Certification tests can reduce the probability of failure of panels by 38%, the highest improvement corresponds to the highest error. As can be expected, without certification tests and safety measures, the probability of failure is near 50%. Tables G-4 through G-7 illustrates the probability of failure for a fixed 8 % coefficient of variation in failure stress. Th e general conclusion that can be drawn from these results is that the error bound e is one of the main parameters affecting the efficacy of certification tests to improve reliability of panels. Next, we will explore how another parameter, variability, influences the efficacy of tests. This is accomplished by changing the coefficient of variation of failure stress f between 0 16% and keeping the error bound constant. PAGE 186 164 Table G-8: Probability of failu re for uncertainty in failure stress for panels designed using safety factor of 1.5, 50% error bounds e and A-basis property for allowable stress. Numbers in parenthesis denote the coefficient of variation of the quantity. cov of f Averag e design thickne ss without certific ation Averag e design thickne ss after certific ation Certific ation failure rate % Probabili ty of failure after certificati on ( Pt) x10-4 Probabili ty of failure without certificati on ( Pnt) x 10-4 Probability ratio ( Pt/Pnt) Probability difference ( PntPt) 0 % 0.998 (0.29) 1.250 (0.11) 50.2 0.017 (6.85) 1699 (1.87) 1.004-5 1.698-1 4% 1.127 (0.29) 1.347 (0.15) 38.9 0.087 (7.20) 970.4 (2.35) 8.973-5 9.703-2 8 % 1.269 (0.29) 1.453 (0.19) 29.3 1.664 (7.86) 449.0 (2.74) 3.706-3 4.473-2 12 % 1.431 (0.29) 1.574 (0.22) 20.9 13.33 (7.71) 206.1 (3.08) 6.469-2 1.927-2 16% 1.616 (0.30) 1.723 (0.25) 14.1 22.52 (5.54) 107.3 (3.24) 2.100-1 8.476-3 *Average over N=5000 models Table G-9: Probability of failu re for uncertainty in failure stress for panels designed using safety factor of 1.5, 30% error bound e and A-basis properties cov of f Averag e design thickne ss without certific ation Averag e design thickne ss after certific ation Certific ation failure rate % Probabili ty of failure after certificati on ( Pt) x10-4 Probabili ty of failure without certificati on ( Pnt) x 10-4 Probability ratio ( Pt/Pnt) Probability difference ( PntPt) 0 % 1.001 (0.17) 1.148 (0.08) 50.1 0.026 (4.79) 223.8 (2.50) 1.163-4 2.238-2 4 % 1.126 (0.17) 1.232 (0.11) 31.6 0.146 (6.03) 35.25 (2.97) 4.149-3 3.511-3 8 % 1.269 (0.17) 1.329 (0.15) 16.3 1.343 (5.28) 9.086 (3.46) 1.479-1 7.742-4 12 % 1.431 (0.18) 1.459 (0.17) 7.2 2.404 (3.87) 4.314 (3.45) 5.572-1 1.911-4 16% 1.617 (0.18) 1.630 (0.18) 3.3 2.513 (3.73) 3.102 (3.54) 8.099-1 5.896-5 PAGE 187 165 Table G-10: Probability of failure for uncerta inty in failure stress for panels designed using safety factor of 1.5, 10% error bounds e and A-basis properties cov of f Averag e design thickne ss without certific ation Averag e design thickne ss after certific ation Certific ation failure rate % Probabili ty of failure after certificati on ( Pt) x10-4 Probabili ty of failure without certificati on ( Pnt) x 10-4 Probability ratio ( Pt/Pnt) Probability difference ( PntPt) 0 % 1.000 (0.06) 1.048 (0.03) 50.3 0.075 (2.91) 1.745 (1.78) 4.304-2 1.669-4 4 % 1.126 (0.06) 1.131 (0.06) 5.9 0.053 (3.85) 0.070 (3.56) 7.548-1 1.716-6 8% 1.269 (0.06) 1.272 (0.07) 1.2 0.027 (4.71) 0.029 (4.59) 9.147-1 2.490-7 12 % 1.431 (0.07) 1.432 (0.07) 0.8 0.049 (4.30) 0.051 (4.23) 9.623-1 1.926-7 16% 1.623 (0.08) 1.624 (0.08) 0.5 0.085 (3.50) 0.083 (3.55) 9.781-1 1.853-7 *Average over N=5000 models. See Tables G-8 through G-10 ro ws 3, 4, 5 with 8 16% standard deviation in failure stress. The increase in the variability in failure stress has a large effect on the allowable stress because A-basis properties specify an allowable that is below 99% of the sample. Increased variability reduces the allowabl e stress and therefore increases the design thickness. It is seen from Tables G-8 through G-10 that when the variability increases from 0% to 16%, the design thickness incr eases by more than 60%. This greatly reduces the probability of failure without certification. However, the probability of failure with certification still deteriorates. That is, the use of A-basis properties fails to fully compensate for the variability in material properties. This opposite behavior of the probability of failure before and after certification is discussed in more detail in Appendix I. The variability in failure stress greatly changes the effect of certification tests. Although the average design thicknesses of the panels increase with the increase in PAGE 188 166 variability, we see that when the variability is large, the value of the tests is reduced because the tested aircraft can be greatly diffe rent from the airplanes in actual service. We indeed see from the Tables G-8 through G10 that the effect of certification tests is reduced as the variability in the failure st ress increases. Recall that the effect of certification tests is also reduced when the error e decreases. Indeed, Table G-8 shows a much smaller effect of the tests than Ta ble G-10. Comparing the second and third columns of Tables G-8 through G-10 we s ee that as the bound of error decreases, the change in the average value of design thicknesses of the panels become less which is an indication of loss in the effi cacy of certification tests. Up to now, both the probability difference ( PntPt) and the probability ratio ( Pt/ Pnt) seem to be good indicators of efficacy of test s. To allow easy visualization, we combined the errors and the variability in a single ratio (Bound of e ) / VR( / f) ratio (ratio of error bound e to the coefficient of variation of the stress ratio). The denominator accounts for the major contributors to the variability. The value in the denominator is a function of four variables; service load P width w thickness t and failure stress f. Here, P and f have lognormal distributions but w and t are uniformly distributed. Since the coefficient of variations of w and t is very small, they can also be treated as lognormally distributed to make calculation of the denominator easy while plotting the graphs. Since the standard deviations of the variables are small, the denominator is now the square root of the sum of the squares of coefficient of variations of the four variables mentioned above, that is ) ( ) ( ) ( ) ( ) (2 2 2 2 f R R R R f RV t V w V P V V (11) PAGE 189 167 The effective safety factor is the ratio of the design thickness of the component when safety measures (such as usage of A-basis values for material properties and safety factor) are applied to the thickness of the co mponent when no safety measures are taken. Figure G-4: Influence of effective safety fact or, error, and variabil ity on the probability ratio (3-D view) Figure G-5: Influence of effective safety f actor, error and variability on the probability ratio (2-D contour plot) Figures G-4 and G-5, present the Pt/Pnt ratio in visual formats. It can be seen that as expected, the ratio decreases as the (Bounds on e )/ VR( / f) ratio increases. However, PAGE 190 168 these two figures do not give a clear indicati on of how certification tests are influenced by the effective safety factor. Figures G-6 and G-7 show the probability difference, PntPt. In these cases, the dependence on the effective safety factor is monotonic. As exp ected, it is seen that as the effective safety factor increases, the improv ement in the safety of component decreases; meaning that the certification tests become le ss useful. The probability difference is more descriptive as it is proportional to the nu mber of aircraft failures prevented by certification testing. The probability ratio lack s such clear physical interpretation, even though it is a more attractive measure when the probability of failure is very small. Considering the results presented by Figure s G-4 through G-7, the probability difference ( PntPt) is the more appropriate choice for ex pressing the effectiveness of tests. Figure G-6: Influence of effective safety factor, error and variability on the probability difference (3-D view) PAGE 191 169 Figure G-7: Influence of effective safety f actor, error and variability on the probability difference (2-D contour plot) Concluding Remarks We have used a simple example of point stress design for yield to illustrate the effects of several safety measures taken in aircraft design: safety factors, conservative material properties, and certification tests. Analytical calculati ons and Monte Carlo simulation were performed to account for both fl eet-level uncertainties (such as errors in analytical models) and individual uncertainties (such as variability in material properties). It was seen that an increase of the systemic errors in the analysis causes an increase in the probability of failure. We found that the systemic errors can be reduced by the use of certification tests, thereby reducing the probability of failure. Also we found that design thicknesses of the panels increased as the bounds of systemic errors increased. We found that the effect of certification tests is most important when errors in analytical models are high and when the vari ability between airplanes is low. This leads to the surprising result that in some situations larger error variability in analytical models reduces the probability of failure if certifi cation tests are conducted. For the simple PAGE 192 170 example analyzed here, the use of conservative (A-basis) properties was equivalent to a safety factor of up to 1.6, depending on the scatter in failure stresses. The effectiveness of the certification te sts is expressed by two measures of probability improvement. The ratio of th e probability of failure with the test, Pt, to the probability of failure without tests, Pnt, is useful when Pt is small. The difference is more meaningful when the probability is high. Us ing these measures we have shown that the effectiveness of certification tests increases when the ratio of error to variability is large and when the effective safety factor is small. The effect of building-block type tests th at are conducted before certification was not assessed here. However, these tests reduce the errors in the analytical models, and on that basis we determined that they can redu ce the probability of failure by one or two orders of magnitude. The calculated probabilities of failure w ith all the considered safety margins explain why passenger aircraft are so safe st ructurally. They were still somewhat high about 10-7compared to the probability of failure of actual aircraft structural componentsabout 10-8. This may be due to additional safety measures, such as conservative design loads or to the effect of design against additional failure modes. PAGE 193 171 APPENDIX H CALCULATION OF CONSERVATIVE MATERIAL PROPERTIES The conservative material properties are specified by the probability level and confidence level of the probability distribution of material property. For example the Abasis value is the value exceeded by 99% of the population with 95% confidence. This is given by A-basis = k1 (H-1) where is the mean, is the standard deviation and k1 is the tolerance coefficient for normal distribution given by Equation H-2. a ab z z kp p 2 1 1 1 (H-2) N z z b N z ap2 1 2 1 2 1; ) 1 ( 2 1 (H-3) where, N is the sample size and z1-p is the critical value of normal distribution that is exceeded with a probability of 1p The tolerance coefficient k1 for a lognormal distribution is obtained by first transformi ng the lognormally distributed variable to a normally distributed variable. Equation H1 and H-2 can be used to obtain an intermediate value. This value is then co nverted back to the lo gnormally distributed variable using inverse transformation. In order to obtain the A-basis values, 40 panels are randomly selected from a batch. Here the uncertainty in material property is due to allowable stress. The mean and standard deviation of 40 random values of allowable stress is calculated and used in PAGE 194 172 determining the A-basis value of allowable stre ss. For instance, when the failure stress is lognormal with 8% coefficient of variation a nd 40 tests are performed, the coefficient of variation of A-basis valu e is about 3 percent. PAGE 195 173 APPENDIX I CONFLICTING EFFECTS OF ERROR A ND VARIABILITY ON PROBABILITY OF FAILURE As explained in the discussion of Table G-5 and Figure G-3, large errors coupled with certification tests can improve the average (over all companies) safety of an aircraft model. This was most apparent when mean ma terial properties are used for design (Table G-5) because for this case airplanes would be tested at their average failure load, so that fifty percent will fail certification. A large e rror bound means a wide variation in design thicknesses. Certification testing fails most of the airplane models with unconservative designs and passes a group of airplane models with high average thickness (that is, overdesigned planes). When the additional safety factor of conservative material properties is used, as in Table G-6, the picture is more complex. Certification is still done at the same loads, but the test airplane is designed for higher loads because of the conservative material properties. For high errors, many airplanes will still fail certification, but small errors will be masked by the conservative properties. Thus in Table G-6, the certification failure rate varies from 29.4% for the largest errors to 1.3% for the smallest errors. At the highest error bound (50%), the certifi cation process increases the aver age thickness from 0.847 to 0.969, and this drops to 0.866 for 30% erro r bound. This substantial drop in average certified model thicknesses increases the probability of failure. Below an error bound of 30%, the change in thickness is small, and then reducing errors reduces the probability of PAGE 196 174 failure. This is because small negative errors are not caught by certification, but they still reduce the effective safety factor. A similar phenomenon is observed when th e variability is changed in Tables G-8 through G-10. When the coefficient of variatio n in failure stress is increased from 0% to 16%, the average design thickness before certif ication increases by about 60% and so the probability of failure without certification is reduced by factors of 16-70. Note that for the smallest error bound (Table G-10), the dr op occurs from zero to 8% coefficient of variation. At the higher coefficients of variation the probability of failure before certification increases again as the increas ed design thickness does not suffice to compensate for the large variation between airp lanes .Once certification is included in the process, variability is mostly detrimental. Certification doe s not amount to much for large variability, because the certified airplane can be very different from the production aircraft. For large error bounds (Table G-8) there are large errors that can be masked during certification by the high material sa fety factor. Thus in Table G-8, while the probability of failure without certification is reduced by a factor of 16, the probability of failure with certification is incr eased by a factor of 1320 as the coefficient of variation in the failure stressed is increased from 0 to 16%. For small errors (Table G-10) the picture is more mixed as the non-monotonic behavi or without certifica tion is mirrored with certification. PAGE 197 175 APPENDIX J CALCULATIONS OF P(C|E), THE PROBABILITY OF PASSING CERTIFICATION TEST An analytical method to obtain failure probability following certification test is presented here. This method was developed by Erdem Acar. This method was used to validate the results from Mont e Carlo Simulation and update the distribution of errors following certification tests. ) ( ) ( ) | (S R P P S t w P t w P S P P e C Pd F f d F f f (J-1) where w t Rf and S = SF Pd. (J-2) S is a deterministic value, and since the coefficient of variations of t and w is small compared to the coefficient of variation of f, we assume t and w can be assumed lognormal, so R can be treated as lognormal. In orde r to take advantage of the properties of lognormal distribution fo r calculating the distribution parameters, we can take S as a lognormally distributed random variable with ze ro coefficient of variation. Hence, both R and S are lognormally distributed random va riables with distribution parameters R, R, S and S. Then, ) ln(d F SP S and 0 S (J-3) w t Re ef ) ( ) ( and 2 2 2 2w t Rf (J-4) PAGE 198 176 where 2 25 0 ) 1 ( ln 5 0 )) ( ln( ) (t a d F t design tw P S e e t e (J-5) Then, P(C|e) can be calculated as ) ( 2 2 22 exp 2 1 ) ( ) ( ) ( ) | (e S R S Rdx x e e S R P e C P (J-6) Mean and Standard Deviation of Probability. Failure is predicted to occur when the resistance of the structure ( R ) of the problem is less than the load ( P ), see Equation J7. Then, the probability of failure is given as: P R Pf Pr (J-7) The load P is lognormally distributed, and as expl ained in above in this appendix, the distribution of R can also be approximated by a lognor mal distribution, which allows us to immediately obtain the probabi lity of failure of a single ai rcraft model. To calculate the probability of failure over all aircraft mode ls, we take into account the fact that that tdesign is a random variable. Then, the expected va lue of probability of failure is given as: design design design f fdt t f t P P ) ( ) ( (J-8) where tdesign is the non-deterministic distribution parameter, and f(tdesign) is the probability density function of parameter tdesign. The standard deviation of failure probability can be calculated from 2 / 1 2) ( f f f f PdP P f P Pf (J-9) PAGE 199 177 where ) (design f ft P P f design design fdP dt t f P f ) ( ) ( (J-10) design f design fdt dP dt dP1 Hence, Equation J-9 can be re-written as 2 / 1 2) ( ) ( design design f design f Pdt t f P t Pf (J-11) As seen from Equations J-8 and J-11, th e mean and standard deviation of the probability of failure can be expressed in terms of the probability density function (pdf) f of the design thickness, tdesign. 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Wu, J., Shin, Y., Probabilistic Function Evaluation System (ProFES) for Maintenance Optimization, 46th AIAA/ASME/ASCE/AHS/ASC St ructures, Structural Dynamics and Materials Conference 13th AIAA/ASM E/AHS Adaptive Structures Conference AIAA-2005-2214, Austin, 2005. Yu, J. C., Stress Intensity Factor for an Edge Crack in a Stiffened Sheet, American Society for Testing of Materials, ASTM STP 945 1988, pp. 247-258. PAGE 208 186 Zhang, R., Mahadevan, S., Model Uncertain ty and Bayesian Updating in Reliability Based Inspection, Structural Safety Vol. 22, No. 2, 2000, pp. 145-160. Zhang, R., Mahadevan, S., Reliability Based Reassessment of Corrosion Fatigue Life, Structural Safety Vol. 23, No. 1, 2001, pp. 77-91. PAGE 209 187 BIOGRAPHICAL SKETCH Amit Anand Kale was born on 25th of October 1978 at Bhopal in India. He received a Bachelor of Technology degree in aerospace engineering in May 2000 from the Indian Institute of Technology, Kharagpur. He then joined Deneb Robotics Inc, India, as a software programmer. After working for a year he joined the Department of Mechanical and Aerospace Engineering at the University of Florida to pursu e doctoral studies in August 2001. While at the University of Florid a he completed a four month internship at Southwest Research Institute. |