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RELIABILITY-BASED STRUCTURAL OPTIMIZATION USING RESPONSE SURFACE APPROXIMATIONS AND PROBAB ILISTIC SUFFICIENCY FACTOR By XUEYONG QU 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 2004
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Copyright 2004 by Xueyong Qu
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This dissertation is dedicated to my lovely wife, Guiqin Wang.
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ACKNOWLEDGMENTS I want to thank Dr. Raphael T. Haftka for offering me the opportunity to complete my Ph.D. study under his exceptional guidance. He provided the necessary funding to complete my doctoral studies and support me to attend many academic conferences. Without his patience, guidance, knowledge, and constant encouragement, this work would not have been possible. Dr. Haftka made an immense contribution to this dissertation and my academic growth, as well as my professional and personal life. I would also like to thank the members of my supervisory committee: Dr. Peter G. Ifju, Dr. Theodore F. Johnson, Dr. Andr I. Khuri, and Dr. Bhavani V. Sankar. I am grateful for their willingness to review my research and provide constructive comments that helped me to complete this dissertation. Special thanks go to Dr. David Bushnell for his help with the PANDA2 program and stiffened panel analysis and design. Special thanks go to Dr. Vicente J. Romero for many helpful discussions and collaboration in writing papers. Financial support provided by grant NAG-1-2177, contract L-9889 and grant URETI from NASA is gratefully acknowledged. 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 Palaniappan Ramu, Thomas Singer, and Dr. Satchi Venkataraman for their collaboration in publishing papers. My parents deserve my deepest appreciation for their constant love and support. iv
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Lastly, I would like to thank my beautiful and loving wife, Guiqin Wang. Without her love, patience and support I would not have completed this dissertation. v
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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.........................................................................................................xiii ABSTRACT.......................................................................................................................xv CHAPTER 1 INTRODUCTION........................................................................................................1 Focus.............................................................................................................................2 Objectives and Scope....................................................................................................4 2 LITERATURE SURVEY: METHODS FOR RELIABILITY ANALYSIS AND RELIABILITY-BASED DESIGN OPTIMIZATION..................................................6 Review of Methods for Reliability Analysis................................................................7 Problem Definition................................................................................................7 Monte Carlo Simulation........................................................................................7 Monte Carlo Simulation Using Variance Reduction Techniques.........................8 Moment-Based Methods.......................................................................................9 Response Surface Approximations......................................................................10 Reliability-Based Design Optimization Frameworks.................................................12 Double Loop Approach.......................................................................................12 Inverse Reliability Approach...............................................................................14 Design potential approach............................................................................15 Partial safety factor approach (Partial SF)...................................................16 Summary.....................................................................................................................17 3 RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY-BASED DESIGN OPTIMIZATION........................................................................................19 Stochastic Response Surface (SRS) Approximation for Reliability Analysis............20 Analysis Response Surface (ARS) Approximation for Reliability-Based Design Optimization..........................................................................................................21 Design Response Surface (DRS) Approximation.......................................................23 vi
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Analysis and Design Response Surface Approach.....................................................24 Statistical Design of Experiments for Stochastic and Analysis Response Surfaces...25 4 DETERMINISTIC DESIGN OF COMPOSITE LAMINATES FOR CRYOGENIC ENVIRONMENTS.....................................................................................................28 Introduction.................................................................................................................28 Composite Laminates Analysis under Thermal and Mechanical Loading.................29 Properties of IM600/133 Composite Materials..........................................................30 Deterministic Design of Angle-Ply Laminates...........................................................34 Optimization Formulation...................................................................................35 Optimizations without Matrix Cracking..............................................................36 Optimizations Allowing Partial Matrix Cracking...............................................39 Optimizations with Reduced Axial Load Ny......................................................39 5 RELIABILITY-BASED DESIGN OF COMPOSITE LAMINATES FOR CRYOGENIC ENVIRONMENTS............................................................................41 Reliability-Based Design Optimization......................................................................41 Problem Formulation...........................................................................................41 Response Surface Approximation for Reliability-Based Optimization..............43 Analysis Response Surfaces................................................................................43 Design Response Surfaces...................................................................................45 Refining the Reliability-Based Design................................................................46 Quantifying Errors in Reliability Analysis..........................................................47 Effects of Quality Control on Laminate Design.........................................................48 Effects of Quality Control on Probability of Failure...........................................49 Effects of Quality Control on Optimal Laminate Thickness...............................50 Effects of Other Improvements in Material Properties...............................................51 Summary.....................................................................................................................54 6 PROBABILISTIC SUFFICIENCY FACTOR APPROACH FOR RELIABILITY-BASED DESIGN OPTIMIZATION..........................................................................56 Introduction.................................................................................................................56 Probabilistic Sufficiency Factor.................................................................................60 Using Probabilistic Sufficiency Factor to Estimate Additional Structural Weight to Satisfy the Reliability Constraint.................................................................62 Reliability Analysis Using Monte Carlo Simulation..................................................64 Calculation of Probabilistic Sufficiency Factor by Monte Carlo Simulation......66 Monte Carlo Simulation Using Response Surface Approximation.....................68 Beam Design Example...............................................................................................71 Design with Strength Constraint.........................................................................72 Design with Strength and Displacement Constraints..........................................75 Summary.....................................................................................................................79 vii
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7 RELIABILITY-BASED DESIGN OPTIMIZATION USING DETERMINISTIC OPTIMIZATION AND MULTI-FIDELITY TECHNIQUE.....................................80 Introduction.................................................................................................................80 Reliability-Based Design Optimization Using Sequential Deterministic Optimization with Probabilistic Sufficiency Factor.....................................................................81 Reliability-Based Design Optimization Using Multi-Fidelity Technique with Probabilistic Sufficiency Factor.............................................................................82 Beam Design Example...............................................................................................84 Reliability-Based Design Optimization Using Sequential Deterministic Optimization with Probabilistic Sufficiency Factor.....................................................................86 Reliability-Based Design Optimization Using Coarse MCS with Probabilistic Sufficiency Factor..................................................................................................87 Summary.....................................................................................................................88 8 RELIABILITY-BASED DESIGN OPTIMIZATION OF STIFFENED PANELS USING PROBABILISTIC SUFFICIENCY FACTOR..............................................90 Introduction.................................................................................................................90 Aluminum Isogrid Panel Design Example.................................................................92 Reliability-Based Design Problem Formulation.................................................92 Uncertainties........................................................................................................94 Analysis Response Surface Approximation........................................................95 Design Response Surfaces...................................................................................97 Optimum Panel Design........................................................................................98 Composite Isogrid Panel Design Example.................................................................98 Deterministic Design.........................................................................................100 Analysis Response Surface Approximation......................................................101 Reliability-Based Design Optimization Using Sequential Deterministic Optimization with Probabilistic Sufficiency Factor......................................103 Reliability-Based Design Optimization using DIRECT Optimization.....................106 DIRECT Global Optimization Algorithm.........................................................106 Reliability-Based Design Optimization Using Direct Optimization with Safety Factor Corrected by Probabilistic Sufficiency Factor....................................108 Summary...................................................................................................................108 APPENDIX A MATERIAL PROPERTIES OF IM600/133...........................................................110 B CONTOUR PLOTS OF THREE DESIGN RESPONSE SURFACE APPROXIMATIONS AND TEST POINTS ALONG THE CURVE OF TAR GET RELIABILITY.........................................................................................................113 LIST OF REFERENCES.................................................................................................115 BIOGRAPHICAL SKETCH...........................................................................................120 viii
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LIST OF TABLES Table page 4-1 Transverse strains calculated for conditions corresponding to the onset of matrix-cracking in the 90 plies of a quasi-isotropic (45/0/-45/90)2s in Aoki et al. (2000).33 4-2 Transverse strains of an angle-ply laminate ( 25)4S under the same loading condition as Table A1..............................................................................................34 4-3 Strain allowables for IM600/133 at F.............................................................34 4-4 Optimal laminates for different operational temperatures: 2u of 0.0110.................37 4-5 Optimal laminates for temperature dependent material properties with 2u of 0.0110 (optimized for 21 temperatures)...............................................................................38 4-6 Optimal laminate for temperature dependent material properties allowing partial matrix cracking: 2u of 0.011 for uncracked plies and 0.0154 for cracked plies......39 4-7 Optimal laminates for reduced axial load of1, 200 lb./inch by using load shunting cables (equivalent laminate thickness of 0.005 inch)...............................................40 5-1 Strain allowablesa for IM600/133 at F............................................................42 5-2 Coefficients of variation (CV) of the random variables...........................................42 5-3 Range of design variables for analysis response surfaces........................................44 5-4 Quadratic analysis response surfaces of strains (millistrain)....................................44 5-5 Design response surfaces for probability of failure (Probability calculated by Monte Carlo simulation with a sample size of 1,000,000)..................................................45 5-6 Comparison of reliability-based optimum with deterministic optima......................46 5-7 Refined reliability-based design []S (Monte Carlo simulation with a sample size of 10,000,000)..........................................................................................................47 5-8 Comparison of probability of failure from MCS based ARS and CLT....................47 5-9 Accuracy of MCS.....................................................................................................48 ix
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5-10 Effects of quality control of 2u on probability of failure for 0.12 inch-thick ()S laminates...................................................................................................................49 5-11 Effects of quality control of 1u, 1l, 2l, and 12 on probability of failure of 0.12 inch-thick ()S laminates........................................................................................50 5-12 Effects of quality control of E1, E2, G12, 12, Tzero, 1, and 2 on probability of failure of 0.12 inch-thick ()S laminates................................................................50 5-13 Effects of quality control of 2u on probability of failure for 0.1 inch-thick ()S laminates...................................................................................................................51 5-14 Effects of quality control of 2u on probability of failure for 0.08 inch-thick ()S laminates...................................................................................................................51 5-15 Sensitivity of failure probability to mean value of 2u (CV=0.09) for 0.12 inch-thick l()S aminates.........................................................................................................52 5-16 Sensitivity of failure probability to CV of 2u ( E(2u)=0.0154 ) for 0.12 inch-thick ()S laminates.........................................................................................................52 5-17 Maximum 2 (millistrain) induced by the change of material properties E1, E2, G12, 12, Tzero, 1, and 2 for 0.12 inch-thick [25]S laminate.......................................54 5-18 Probability of failure for 0.12 inch-thick [ 25]S laminate with improved average material properties (Monte Carlo simulation with a sample size of 10,000,000)....54 6-1 Random variables in the beam design problem........................................................62 6-2 Range of design variables for design response surface............................................72 6-3 Comparison of cubic design response surface approximations of probability of failure, safety index and probabilistic sufficiency factor for single strength failure mode (based on Monte Carlo simulation of 100,000 samples)................................73 6-4 Averaged errors in cubic design response surface approximations of probabilistic sufficiency factor, safety index and probability of failure at 11 points on the curves of tar get reliability....................................................................................................74 6-5 Comparisons of optimum designs based on cubic design response surface approximations of probabilistic sufficiency factor, safety index and probability of failure.......................................................................................................................75 6-6 Comparison of cubic design response surface approximations of the first design iteration for probability of failure, safety index and probabilistic sufficiency factor for system reliability (strength and displacement)...................................................76 x
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6-7 Averaged errors in cubic design response surface approximations of probabilistic sufficiency factor, safety index and probability of failure at 51 points on the curves of tar get reliability....................................................................................................76 6-8 Comparisons of optimum designs based on cubic design response surface approximations of the first design iteration for probabilistic sufficiency factor, safety index and probability of failure.....................................................................77 6-9 Range of design variables for design response surface approximations of the second design iteration.........................................................................................................78 6-10 Comparison of cubic design response surface approximations of the second design iteration for probability of failure, safety index and probabilistic sufficiency factor for system reliability (strength and displacement)...................................................78 6-11 Comparisons of optimum designs based on cubic design response surfaces of the second design iteration for probabilistic sufficiency factor, safety index and probability of failure.................................................................................................78 7-1 Random variables in the beam design problem........................................................85 7-2 Optimum designs for strength failure mode obtained from double loop RBDO.....86 7-3 Design history of RBDO based on sequential deterministic optimization with probabilistic sufficiency factor under strength constraint for tar get probability of failure of 0.00135.....................................................................................................86 7-4 Design history of RBDO based on sequential deterministic optimization with probabilistic sufficiency factor under strength constraint for tar get probability of failure of 0.0000135.................................................................................................87 7-5 RBDO using variable fidelity technique with probabilistic sufficiency factor under strength constraint....................................................................................................88 7-6 Range of design variables for design response surface............................................88 8-1 Amplitudes of geometric imperfection handled by PANDA2 software...................94 8-2 Uncertainties in material properties (Al 2219-T87) modeled as normal random variables...................................................................................................................94 8-3 Uncertainties in manufacturing process modeled as uniformly distributed random design variables around design (mean) value..........................................................94 8-4 Deterministic Optimum............................................................................................95 8-5 Range of analysis response surface approximations (inch)......................................96 xi
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8-6 Quadratic analysis response surface approximation to the most critical margins using Latin Hypercube sampling of 72 points.........................................................96 8-7 Probabilities of failure calculated by Monte Carlo simulation with 1106 samples97 8-8 Range of design response surface approximations (inch)........................................97 8-9 Cubic design response surface approximation to the probability of failure and probabilistic sufficiency factor (calculated by Monte Carlo sampling of 1106 samples)....................................................................................................................97 8-10 Optimum panel design..............................................................................................98 8-11 Probabilities of failure calculated by Monte Carlo simulation of 1106 samples....98 8-12 Uncertainties in material elastic properties (AS4) modeled as normal distribution with coefficient of variation of 0.03.......................................................................100 8-13 Uncertainties in material strength properties (AS4) modeled as normal distribution with coefficient of variation of 0.05.......................................................................100 8-14 Variation of the random design variables around nominal design value...............100 8-15 Safety factors used in deterministic design............................................................101 8-16 Deterministic Optimum (inch, degree, lb)..............................................................101 8-17 Quadratic analysis response surface approximation to the worst margins using Latin Hypercube sampling of 342 points.........................................................................102 8-18 Probabilities of failure calculated by Monte Carlo simulation of 106 samples (material and manufacturing uncertainties)............................................................102 8-19 Design history of RBDO based on sequential deterministic optimization using probabilistic sufficiency factor to correct safety factor directly by Equation (8-3)105 8-20 Design history of RBDO based on sequential deterministic optimization using probabilistic sufficiency factor to correct safety factor by actual safety margin using Equation (8-4)........................................................................................................105 8-21 Design history of RBDO based on DIRECT deterministic optimization with probabilistic sufficiency factor correct safety factor by actual safety margin using Equation (8-4)........................................................................................................108 xii
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LIST OF FIGURES Figure page 2-1 Double loop approach: reliability analysis coupled inside design optimization......12 2-2 Design potential approach: reliability constraints approximated at design potential point dpk; reliability analyses still coupled inside design optimization...................15 2-3 Partial safety factor approach: decouple reliability analysis and design optimization..............................................................................................................17 3-1 Analysis response surface and design response surface approach: decouple reliability analysis and design optimization.............................................................25 3-2 Latin Hypercube sampling to generate 5 samples from two random variables........27 4-1 Polynomials fit to elastic properties: E1, E2, G12, and 12....................................31 4-2 Polynomials fit to coefficients of thermal expansion: 1 and 2............................32 4-3 Geometry and loads for laminates............................................................................35 4-4 The change of optimal thickness (inch) with temperature for variable and constant material properties (2u of 0.0110)...........................................................................37 4-5 Strains in optimal laminate for temperature dependent material properties with 2u of 0.0110 (second design in Table 4-3)....................................................................38 5-1 Tradeoff plot of probability of failure, cost, and weight (laminate thickness) for [25]S........................................................................................................................53 6-1 Probability density of safety factor. The area under the curve left to s=1 measures the actual probability of failure, while the shaded area is equal to the tar get probability of failure indicating that probabilistic sufficiency factor = 0.8.............61 6-2 Cantilever beam subject to vertical and lateral beading...........................................62 6-3 Monte Carlo simulation of problem with two random variables..............................66 7-1 Cantilever beam subject to vertical and lateral beading...........................................84 xiii
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8-1 Isogrid-stiffened cylindrical shell with internal isogrid and external rings with isogrid pattern oriented along circumferential direction for increased bending stiffness in hoop direction........................................................................................93 8-2 Isogrid-stiffened cylindrical shell with internal isogrid and isogrid pattern oriented along circumferential direction for increased bending stiffness in hoop direction; the zero degree direction for the composite laminates in isogrid and skin panel are shown.......................................................................................................................99 8-3 1st iteration of DIRECT windowing for 2 dimensional example, Goldstein-Price (GP) function (Finkel, 2003), and (b) further iterations on GP function with potentially optimal boxes shaded and subsequently divided along longest dimension(s)...........................................................................................................107 A-1 Quadratic fit to 1 (1.0E-6/F)................................................................................110 A-2 Sixth-order fit to 2 (1.0E-4/F)..............................................................................110 A-3 Quadratic fit to E1 (Mpsi).......................................................................................111 A-4 Quartic fit to E2 (Mpsi)...........................................................................................111 A-5 Cubic fit to G12 (Mpsi)............................................................................................112 A-6 Quartic fit to 12......................................................................................................112 B-1 Contour plot of probabilistic safety factor design response surface approximation and test points along the curve of tar get reliability................................................113 B-2 Contour plot of probability of failure design response surface approximation and test points along the curve of tar get reliability. The negative values of probability of failure are due to the interpolation errors of the design response surface approximation.........................................................................................................114 B-3 Contour plot of safety index design response surface approximation and test points along the curve of tar get reliability........................................................................114 xiv
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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 RELIABILITY-BASED STRUCTURAL OPTIMIZATION USING RESPONSE SURFACE APPROXIMATIONS AND PROBABILISTIC SUFFICIENCY FACTOR By Xueyong Qu August 2004 Chair: Raphael T. Haftka Major Department: Mechanical and Aerospace Engineering Uncertainties exist practically everywhere from structural design to manufacturing, product life-time service, and maintenance. Uncertainties can be introduced by errors in modeling and simulation; by manufacturing imperfections (such as variability in material properties and structural geometric dimensions); and by variability in loading. Structural design by safety factors using nominal values without considering uncertainties may lead to designs that are either unsafe, or too conservative and thus not efficient. The focus of this dissertation is reliability-based design optimization (RBDO) of composite structures. Uncertainties are modeled by the probabilistic distributions of random variables. Structural reliability is evaluated in term of the probability of failure. RBDO minimizes cost such as structural weight subject to reliability constraints. Since engineering structures usually have multiple failure modes, Monte Carlo simulation (MCS) was used to calculate the system probability of failure. Response surface (RS) approximation techniques were used to solve the difficulties associated with xv
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MCS. The high computational cost of a large number of MCS samples was alleviated by analysis RS, and numerical noise in the results of MCS was filtered out by design RS. RBDO of composite laminates is investigated for use in hydrogen tanks in cryogenic environments. The major challenge is to reduce the large residual strains developed due to thermal mismatch between matrix and fibers while maintaining the load carrying capacity. RBDO is performed to provide laminate designs, quantify the effects of uncertainties on the optimum weight, and identify those parameters that have the largest influence on optimum design. Studies of weight and reliability tradeoffs indicate that the most cost-effective measure for reducing weight and increasing reliability is quality control. A probabilistic sufficiency factor (PSF) approach was developed to improve the computational efficiency of RBDO, to design for low probability of failure, and to estimate the additional resources required to satisfy the reliability requirement. The PSF is a safety factor needed to meet the reliability target. The methodology is applied to the RBDO of composite stiffened panels for the fuel tank design of reusable launch vehicles. Examples are used to demonstrate the advantages of the PSF over other RBDO techniques xvi
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CHAPTER 1 INTRODUCTION Aerospace structures are designed under stringent weight requirement. Structural optimization is usually employed to minimize the structural weight subjected to performance constraints such as strength and deflection. Deterministically optimized structures can be sensitive to uncertainties such as variability in material. Uncertainties exist practically everywhere from engineering design to product manufacturing, product life-time service condition and maintenance. Uncertainties can be introduced by manufacturing process such as variability in material properties and structural geometric dimensions; by errors in modeling and simulation; and by service conditions such as loading changes. Deterministic optimization can use large safety factors to accommodate uncertainties, but the safety and performance of the optimized structure under uncertainties (such as reliability) are not known, and the resulting structural design may be too conservative, and thus not efficient. To address this problem, reliability-based design optimization (RBDO) became popular in the last decade (Rackwitz 2000). The safety of the design is evaluated in terms of the probability of failure, with uncertainties modeled by probabilistic distribution of random variables. RBDO minimizes costs such as structural weight subject to reliability constraints, which are usually expressed as limits on the probability of failure of performance measures. 1
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2 Focus The focus of this dissertation is reliability-based structural optimization for use in the reusable launch vehicles (RLV). RLV is being developed as a safer and cheaper replacement of space shuttles, which suffer from high probability of failure and operating cost. For example, the probability of failure per mission launch is about 0.01 based on the shuttle launch history. The catastrophic failures both space shuttles, Challenger and Columbia, were all initiated from structural failure. The limited reuse of rocket boosters and fuel tanks also increase the operating cost of space shuttle. In order to reduce the operating cost, cryogenic fuel tank must be structurally integrated into the RLV, which motivates the use of composite materials. Composite materials are widely used in aerospace structures because of their high stiffness to weight ratio and the flexibility to tailor the design to the application. This extra flexibility can render deterministically optimized composite laminates very sensitive to uncertainties in material properties and load conditions (e.g., Grdal et al. 1999). For example, the ply angles of a composite laminate deterministically optimized under unidirectional loading are all aligned with the loading direction, which leads to poor design for even small loading transverse to the fiber direction. Design of composite structures for RLV poses a major challenge because the feasibility of these vehicles depends critically on structural weight. With traditional deterministic design based on safety factors, it is possible to achieve a safe design, but it may be too heavy for the RLV to take off. Therefore, reliability-based design optimization based is required for the structural design of RLV structures in order to satisfy both safety and weight constraints. The advantages of reliability-based design over deterministic design have been demonstrated (e.g., Ponslet et al., 1995). For designs with
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3 stringent weight requirements, it is also important to provide guidelines for controlling the magnitudes of uncertainties for the purpose of reducing structural weight. Deterministic structural optimization is computationally expensive due to the need to perform multiple structural analyses. However, reliability-based optimization adds an order of magnitude to the computational expense, because a single reliability analysis requires many structural analyses. Commonly used reliability analysis methods are based on either simulation techniques such as Monte Carlo simulation, or moment-based methods such as the first-order-reliability-method (e.g., Melchers, 1999). Monte Carlo simulation is easy to implement, robust, and accurate with sufficiently large samples, but it requires a large number of analyses to obtain a good estimate of low failure probability. Monte Carlo simulation also produces a noisy estimate of probability and hence is difficult to use with gradient-based optimization. Moment-based methods do not have these problems, but they are not well suited for problems with many competing critical failure modes. Response surface approximations solve the two problems of Monte Carlo simulation, namely simulation cost and noise from random sampling. Response surface approximations (Khuri and Cornell 1996) typically fit low order polynomials to a number of response simulations to approximate response. Response surface approximations usually fit the structural response such as stresses in terms of random variables for reliability analyses. The probability of failure can then be calculated inexpensively by Monte Carlo simulation using the fitted response surfaces. Response surface approximations can also be fitted to probability of failure in terms of design variables, which replace the reliability constraints in RBDO to filter out numerical noise in the
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4 probability of failure induced by Monte Carlo simulation and reduce the computational cost. Different ways of using response surface approximations for reliability analysis and reliability-based design optimization will be presented in subsequent chapters. Objectives and Scope The main purpose of this dissertation is to address the challenges associated with the reliability-based design of composite panels for reusable launch vehicles. The problems encountered include the high computational cost for calculating probabilities of failure and for performing reliability-based design optimization, and the control of structural weight penalty due to uncertainties. Therefore the main objectives are: 1. Investigate response surface approximation for use in reliability analysis and design optimization. Analysis and design response surface approximation are developed. 2. Develop methods that allow more efficient reliability-based design optimization when the probability of failure must be low. This motivates the development of a probabilistic sufficiency factor approach. 3. Explore the potential of uncertainty control for reducing structural weight for unstiffened and stiffened panels. 4. Provide reliability-based designs of selected composite panels A literature survey of methods for reliability analysis and reliability-based design optimization is presented in Chapter 2. Chapter 3 introduces the response surface approximation techniques developed for the efficient RBDO (objective 2). Chapter 4 presents deterministic design optimization for composite laminates in cryogenic environment. Chapter 5 demonstrates the reliability-based design the composite laminate for use in cryogenic environments, and tradeoffs of weight and reliability via the control of uncertainty (objective 1). Chapter 6 proposes a probabilistic sufficiency factor
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5 approach for more efficient reliability-based design optimization. Chapter 7 demonstrates the use of probabilistic sufficiency factor for RBDO. Chapter 8 provides reliability-based designs of selected composite stiffened panels for the fuel tank design of reusable launch vehicles.
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CHAPTER 2 LITERATURE SURVEY: METHODS FOR RELIABILITY ANALYSIS AND RELIABILITY-BASED DESIGN OPTIMIZATION The basic conceptual structure of the reliability-based design optimization (RBDO) problem, called RBDO framework, can be formulated as )d(minimizeFF P)x(Pthat suchtjjpnj,1, (2-1) where F is the objective function, d is a vector of design variables, Pj is the probability of failure of the jth failure mode, Pjt is the allowable probability of failure of the jth failure mode, np is the total number of failure modes, and x is a vector of random variables. To perform RBDO, reliability analyses must be performed to evaluate the probability of failure, which requires multiple evaluations of system performance (such as stresses in a structure). Depending on the specific reliability analysis method, the computational cost of a single reliability analysis is usually comparable to or higher than the cost of performing a deterministic local optimization. Furthermore, RBDO requires multiple reliability analyses, thus the computational cost of performing RBDO by directly coupling design optimization with reliability analysis is at least an order of magnitude higher than deterministic optimization. Efficient frameworks must be developed to overcome this computational burden. This chapter presents a literature review of state-of-the-art reliability analysis methods and RBDO frameworks, and concludes with the motivation to develop the methodologies in Chapters 3, 6, and 7 for solving the problems. 6
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7 Review of Methods for Reliability Analysis The most common techniques for reliability analysis are Monte Carlo simulation, approaches based on most probable point (MPP), and decoupled Monte Carlo sampling of a response surface approximation fit to samples from some experimental design. Different techniques are preferable under different circumstances. Problem Definition The limit state function of the reliability analysis problem is defined as G(x), where G(x) represents a performance criterion and x is a random variable vector. Failure occurs when G(x)<0. Thus the failure surface or limit state of interest can be described as G(x)=0. The probability of failure can be calculated as xxxXdfPGf0)()( (2-2) where fX(x) is the joint probability distribution function (JPDF). This integral is hard to evaluate because the integration domain defined by G(x)<0 is usually unknown and integration in high dimension is very difficult. Commonly used probabilistic analysis methods are based on either simulation techniques such as Monte Carlo simulation or moment-based methods such as the first-order-reliability-method (FORM) or second-order-reliability-method (SORM) (Melchers 1999). Monte Carlo Simulation Monte Carlo simulation (MCS) (e.g., Rubinstein 1981) generates a number of samples of the random variables x by using a random number generator. The number of samples required is usually determined by confidence interval analysis. Simulations (e.g., structural analyses) are then performed for each of these samples. Statistics such as mean,
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8 variance, and probability of failure can then be calculated from the results of simulations. This method is also called direct MCS or MCS with simple random sampling (SRS). Direct MCS is simple to implement; is robust; and is accurate with sufficiently large samples. But the usefulness of direct MCS in reliability analysis is quite limited because of its relatively low efficiency. For example, the probability of failure in engineering applications is usually very small, thus the number of limit state function evaluations required to obtain acceptable accuracy using direct MCS is very large (Chapter 5), which makes direct MCS very time-consuming. Direct MCS is usually used as a benchmark to verify the accuracy and compare the efficiency of other methods using approximation concepts. To improve the accuracy and efficiency of simple random sampling, various simulation methods using Variance Reduction Techniques (VRT) have been developed to reduce the variance of the output random variables. Monte Carlo Simulation Using Variance Reduction Techniques Rubinstein (1981) and in Melchers (1999) gave good overviews of VRT for general Monte Carlo sampling. The VRT can be classified into different categories, such as sampling method, correlation method, conditional expectation method, and specific method. Sampling methods reduce the variance of the output by constraining samples to be representative of (or distorting the samples to emphasize the important region of) the performance function. Commonly used sampling methods include importance sampling (Harbitz 1986), adaptive sampling, stratified sampling, Latin Hypercube sampling, and spherical sampling. Correlation methods use techniques to achieve correlation among random observations, functions, or different simulations to improve the accuracy of the estimators. Some commonly used techniques are antithetic variate, common random
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9 numbers, control variates, and rotation sampling. Conditional expectation methods utilize the independence of random variables to reduce the order of probabilistic integration to achieve higher efficiency. Some common techniques are conditional expectation, generalized conditional expectation, and adaptive conditional expectation. Specific methods include response surface method and internal control variables techniques. The VRTs can be combined further to increase the efficiency of simulation. A comparison of the accuracy and efficiency of several common VRT methods can be found in Kamal and Ayyub (2000). Latin hypercube sampling and response surface methods are studied in this dissertation. The VRT requires fewer limit state function evaluations to achieve the desired level of accuracy, but the simplicity of simulation is lost, and the computational complexity of each simulation cycle is increased. Moment-Based Methods Besides VRT, moment-based methods also reduce the computational cost drastically compared to MCS. The first-order-reliability method (FORM) and second order reliability-method (SORM) are well-established methods that can solve many practical applications (Rackwitz 2000). FORM and SORM methods first transform the random variables from the original space (X-space) to the uncorrelated standard normal space (U-space). An optimization problem is then solved to find the minimum distance point (most probable point, MPP) on the limit state surface (Z=0) to the origin of the U-space. The minimum distance, is called the safety index. The probability of failure is then calculated by using the normal cumulative distribution function )(1 fP in
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10 FORM (Rackwitz and Fiessler 1978), or by using second order correction in SORM (Breitung 1984). Thus the safety index can be used directly as a measure of reliability. One disadvantage of FORM and SORM methods is that there is no readily available error estimate. The accuracy of FORM and SORM must be verified by other methods, such as MCS. The errors of FORM and SORM may come from the errors associated with MPP search and the nonlinearity of the limit state. The search of MPP requires solving a nonlinear optimization problem, which is difficult to solve for some problems. Wrong MPP usually leads to poor probability estimates, which is common problem for MPP-based reliability analysis methods. FORM and SORM methods are also not well suited for problems with many competing critical failure modes (i.e., multiple MPPs). Due to the limitations of first-order and second-order approximations, FORM and SORM methods do not perform well when the limit state surface is highly nonlinear around MPP. This nonlinearity may come from the inherent nonlinearity of the problem or may be induced by the transformation from X-space to U-space (Thacker et al. 2001). For example, transforming a uniform random variable to a standard normal variable usually increases the nonlinearity of the problem. When FORM and SORM methods encounter difficulties, sampling methods with VRT such as Importance Sampling can be employed to obtain/improve results with a reasonable amount of computational cost compared to direct MCS. Response Surface Approximations Response surface approximations (RSA) (Khuri and Cornell 1996) can be used to obtain a closed-form approximation to the limit state function to facilitate reliability analysis. Response surface approximations usually fit low order polynomials to the structural response in terms of random variables. The probability of failure can then be
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11 calculated inexpensively by Monte Carlo simulation or FORM and SORM using the fitted polynomials. Therefore, RSA is particularly attractive for computationally expensive problems (such as those requiring complex finite element analyses). The design points where the response is evaluated are chosen by statistical design of experiments (DOE) so as to maximize the information that can be extracted from the resulting simulations. Response surface approximations can be applied in different ways. One approach is to construct local response surfaces around the MPP region that contributes most to the probability of failure of the structure. The DOE of this approach is iteratively adjusted to approach the MPP. Typical DOEs for this approach are Central Composite Design (CCD) and saturated design. For example, Bucher and Bourgund (1990), and Rajashekhar and Ellingwood (1993) constructed progressively refined local response surfaces around the MPP. This local RSA approach can produce good probability estimates given enough iterations. Another approach is to construct global RSA over the entire range of random variables (i.e., DOE around the mean values of the random variables). Fox (1993, 1994, and 1996) used Box-Behnken DOE to construct global RSA and summarized 12 criteria to evaluate the accuracy of response surfaces. Romero and Bankston (1998) used progressive lattice sampling, where the initial DOE is progressively supplemented by new design points, as the statistical design of experiments to construct global response surfaces. With the global approach, the accuracy of the RSA around the MPP is usually unknown, thus caution should be taken to avoid extrapolation around the MPP. Both the
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12 global and local approaches provide substantial savings in the number of total function evaluations. Reliability-Based Design Optimization Frameworks This section summarizes several popular RBDO frameworks. These frameworks are based on the concepts of design sensitivity analyses, approximated limit state function, approximated reliability constraints, and partial safety factor concept to convert reliability constraints to approximately equivalent deterministic constraints, and RSA. Double Loop Approach Outer Loop: design optimization Initial Design(deterministic optimum) Probabilistic Analyses:FORM/SORM Design Sensitivity /dfrom PA Reliability Converge? YesNoNo Design Converge?Yes Stop Inner Loop: iterative probabilistic analyses Approximate ReliabilityConstraints at DPdk Update Designusing Optimizer Outer Loop: design optimization Initial Design(deterministic optimum) Probabilistic Analyses:FORM/SORM Design Sensitivity /dfrom PA Reliability Converge? YesNoNo Design Converge?Yes Stop Inner Loop: iterative probabilistic analyses Approximate ReliabilityConstraints at DPdk Update Designusing Optimizer Figure 2-1. Double loop approach: reliability analysis coupled inside design optimization The traditional approach of RBDO is to perform a double loop optimization: outer loop for the design optimization (DO) and inner sub-optimization that performs reliability analyses using methods such as FORM or SORM. This nested approach is rigorous and
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13 popular, but it is computationally expensive and sometimes troubled by convergence problems, etc. (Tu et. al. 2000). The computational cost of RBDO with nested MPP may be reduced by sensitivity analysis. The sensitivity of the safety index to design variables can be obtained with little extra computation as by-products of reliability analysis (Kwak and Lee 1987). A simplified formula that ignores the higher order terms in the estimation equation was proposed by Sorensen (1987). Yang and Nikolaidis (1991) used this sensitivity analysis and optimized an aircraft wing with FORM subjected to system reliability constraint. Figure 2-1 shows the typical procedure for the double loop approach. With this approach, the reliability constraints are approximated at the current design point (DP) dk. For problems requiring expensive finite element analysis, this approach may still be computationally prohibitive; and FORM (e.g., classical FORM such as Hasofer-Lind method) may converge very slowly (Rackwitz 2000). Wang and Grandhi (1994) developed an efficient safety index calculation procedure for RBDO that expands limit state function in terms of intermediate design variables to obtain more accurate approximation. Reliability constraints can also be approximated to reduce the computational cost of RBDO. Wang and Grandhi (1994) approximate reliability constraints with multi-point splines within a double loop RBDO. Another way of improving the efficiency of multi-level optimization is to integrate the iterative procedures of reliability analysis and design optimization into one where the iterative reliability analysis stops before full convergence at each step of the optimization, as suggested by Haftka (1989). Maglaras and Nikolaidis (1990) proposed an integrated analysis and design approach for stochastic optimization, where reliability constraints are
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14 approximated to different levels of accuracy in optimization. Even combined with above approaches, nested MPP approach still suffers the problems of high computational cost and convergence. Several RBDO approaches are being developed to solve these problems. Inverse Reliability Approach Recently, there has been interest in using alternative measures of safety in RBDO. These measures are based on margin of safety or safety factors that are commonly used as measures of safety in deterministic design. The safety factor is generally expressed as the quotient of allowable over response, such as the commonly used central safety factor that is defined as the ratio of the mean value of allowable over the mean value of the response. The selection of safety factor for a given problem involves both objective knowledge (such as data on the scatter of material properties) and subjective knowledge (such as expert opinion). Given a safety factor, the reliability of the design is generally unknown, which may lead to unsafe or inefficient design. Therefore, using safety factor in reliability-based design optimization seems to be counter productive. Freudenthal (1962) showed that reliability can be expressed in terms of the probability distribution function of the safety factor. Elishakoff (2001) surveyed the relationship between safety factor and reliability, and showed that in some cases the safety factor can be expressed explicitly in terms of reliability. The standard safety factor is defined with respect to the response obtained with the mean values of the random variables. Thus a safety factor of 1.5 implies that with the mean values of the random variables, we have a 50% margin between the response (e.g., stress) and the capacity (e.g., failure stress). However, the value of the safety factor does not tell us what the reliability is. Birger (1970), as reported by Elishakoff (2001), introduced a factor, which
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15 we call here the Birgers safety factor that is more closely related to the target reliability. A Birgers safety factor of 1.0 implies that the reliability is equal to the target reliability; a Birgers safety factor larger than 1.0 means that the reliability exceeds the target reliability; and Birgers safety factor less than 1.0 means that the system is not as safe as we wish. Design potential approach Tu et al. (2000) used the probabilistic performance measure, which is closely related to Birgers safety factor, for RBDO using FORM. Figure 2-2 summarizes the design potential approach. Outer Loop: design optimization Initial design(deterministic optimum) Probabilistic Analyses:FORM/SORM Design Sensitivity /dfrom PA Reliability Converge? YesNoNo Design Converge?Yes Stop Inner Loop: iterative probabilistic analyses Approximate Reliabilityconstraints at DPPdkp Update DesignUsing Optimizer Outer Loop: design optimization Initial design(deterministic optimum) Probabilistic Analyses:FORM/SORM Design Sensitivity /dfrom PA Reliability Converge? YesNoNo Design Converge?Yes Stop Inner Loop: iterative probabilistic analyses Approximate Reliabilityconstraints at DPPdkp Update DesignUsing Optimizer Figure 2-2. Design potential approach: reliability constraints approximated at design potential point dpk; reliability analyses still coupled inside design optimization They showed that the search for the optimum design converged faster by driving the probabilistic performance measure to zero than by driving the probability of failure to
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16 its ta998 and 2001) developed a partial safety factor similar to Birgers h a series of deterministic optimizations by convesis. The shift of limit r, can be any MPP-based method) is decoupled rget value. Another major difference between the design potential approach and double loop approach is that the reliability constraints are approximated at the design potential point dpk (DPP), which defined as the design that renders the probabilistic constraint active, instead of the current design point. Since the DPP is located on the limit-state surface of the probabilistic constraint, the constraint approximation of DPM becomes exact at (DPP). Thus DPM provides a better constraint approximation without additional costly limit state function evaluation. Therefore, a faster rate of convergence can be achieved. Partial safety factor approach (Partial SF) Wu et al. (1 safety factor in order to replace the RBDO wit rting reliability constraints to equivalent deterministic constraints. After performing reliability analysis, the random variables x are replaced by safety-factor based values x*, which is the MPP of the previous reliability analy state function G needed to satisfy the reliability constraints is s, which satisfies P(G(x)+s)<0)=Pt. Both x* and s can be obtained as the byproducts of reliability analysis. Since in design optimization, the random variables x are replaced by x* (just as in the case of traditional deterministic design, where random variables are replaced by deterministic values after applying some safety factor), the method is called partial safety factor approach (Figure 2-3). The target reliability is achieved by adjusting the limit state function via design optimization. It is seen that the required shift s is similar to the target probabilistic performance measure g*. The significant difference between Partial SF and DPM or nest MPP is that reliability analysis (FORM in the pape
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17 from optimfailure, MCS was used to perform relysis. We developed an analysis RS appro and driven by the design optimization to improve the efficiency of RBDO. If n iterations are needed for convergence, the approach needs n deterministic optimizations and n probabilistic analyses. However, the convergence rate of subsequent probabilistic analyses is expected to increase after obtaining a reasonable MPP. Wu et al. (2001) demonstrated the efficiency of this approach by optimizing a beam subject to multiple reliability constraints. Deterministic Design Optimization Probabilistic Analyses:FORM/SORM Deterministic DesignOptimization with x*No Design and ReliabilityConverge?Yes Stop New design d Safety factor based x* Probabilistic analyses to replace random variable with deterministic values x*Deterministic design optimization with x* Deterministic Design Optimization Probabilistic Analyses:FORM/SORM Deterministic DesignOptimization with x*No Design and ReliabilityConverge?Yes Stop New design d Safety factor based x* Probabilistic analyses to replace random variable with deterministic values x*Deterministic design optimization with x* Figure 2-3. Partial safety factor approach: decouple reliability analysis and design ization Summary Since the reliability analyses involved in our study are for system probability of liability ana ach to reduce the high computational cost of MCS and design response surface approach to filter noise in RBDO (Chapter 3).
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18 The current RBDO frameworks mostly deal with the probability of failure of individual failure modes, an efficient framewor k must be developed to address RBDO for the sy stem probability of failure. Chapter 6 developed an inverse reliability measure, the probabilistic sufficiency factor, to improve the computational efficiency of RBDO, to design for low probability of failure, and to estimate the additional resources needed to satisfy the reliability requirement. Chapter 7 demonstrated the use of probabilistic sufficiency factor with multi-fidelity techniques for RBDO and converting RBDO to sequential deterministic optimization. The methodology is applied to the RBDO of stiffened panels in chapter 8.
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CHAPTER 3 RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY-BASED DESIGN OPTIMIZATION Response surface approximation (RSA) methods are used to construct an approximate relationship between a dependent variable f (the response) and a vector x of n independent variables (the predictor variables). The response is generally evaluated experimentally (these experiments may be numerical in nature), in which case f denotes the mean or expected response value. It is assumed that the true model of the response may be written as a linear combination of basis functions with some unknown coefficients in the form of )(~xZ xZT)( ~ Response surface model can be expressed as (3-1) bxxTZY)()( where Z(x) is the assumed basis function vector that usually consists of monomial functions, and b is the least square estimate of For example, if the a linear response surface model is employed to approximate the response in terms of two independent variables, x1 ans x2, the response surface approximation is ~ (3-2) 22110)(xbxbbYx The three major steps of response surface approximation as summarized by Khuri and Cornell (1996) are Selecting design points where responses must be evaluated. The points are chosen by statistical design of experiment (DOE), which is performed in such a way that the input parameters are varied in a structured pattern so as to maximize the information that can be extracted from the resulting simulations. Typical DOE for quadratic RSA is central composite design (CCD, Khuri and Cornell 1996). 19
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20 Determining a mathematical model that best fits the data generated from the design points of DOE by performing statistical test of hypotheses of the model parameters(Khuri and Cornell 1996 and Myers et al. 2002) Predicting response for given sets of experimental factors or variables by the constructed response surface approximation. Due to the close form nature of the approximation, RSA is particularly attractive for engineering problems that require a large number of computationally expensive analyses, such as structural optimization and reliability analysis. The accuracy of RSA is measured by error statistics such as the adjusted coefficient of multiple determination (R2adj), root mean square error predictor (RMSE), and coefficient of variation (COV =RMSE/Mean of Response). An R2adj close to one and a small COV close to zero usually indicate good accuracy. The RSAs in this dissertation were all constructed by JMP software (SAS Institute., 2000). The above error statistics are readily available from JMP after RSA construction. Khuri and Cornell (1996) presented a detailed discussion on response surface approximation. This chapter presents the response surface approach developed for reliability-based design optimization. Stochastic Response Surface (SRS) Approximation for Reliability Analysis Among the available methods to perform reliability analysis, moment-based methods (e.g., FORM/SORM) are not well suited for the composite structures in cryogenic environments because of the existence of multiple failure modes. Direct Monte Carlo simulation requires a relatively large number of analyses to calculate probability of failure, which is computationally expensive. Stochastic response surface approximation is employed here to solve the above problems. To apply RSA to a reliability analysis problem, the limit state function g(x) (usually the stress of displacement in the structures) is approximated by
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21 (3-3) bxxTZG)()( where x is the vector of input random variables. With the polynomial approximation )(xG the probability of failure can then be calculated inexpensively by Monte Carlo simulation or FORM/SORM. Since the RSA is constructed in random variable space, this approach is called stochastic response surface approach. Stochastic RSA can be applied in difference ways. One approach is to construct local RSA around the Most Probable Point (MPP), the region that contributes most to the probability of failure of the structure. The statistical design of experiment (DOE) of this approach is iteratively performed to approach the MPP. Another approach is to construct global response surfaces over the entire range of the random variables, where the mean value of the random variables is usually chosen as the center of DOE. The selection of RSA approach depends on the limit state function of the problem. Global RSA is simpler and efficient to use than local RSA for problems with limit state function that can be well approximated globally. Analysis Response Surface (ARS) Approximation for Reliability-Based Design Optimization In reliability-based design optimization (RBDO), the SRS approach needs to construct response surfaces to limit state functions at each point encountered in the optimization process, which requires a fairly large number of limit state function evaluation and RS construction. The local SRS approach is more computationally expensive than the global SRS approach due to multiple iterations involved in the RSA construction. This dissertation (see also Qu et al., 2000) developed an analysis response surface (ARS) approach in the unified system space (x, d) to reduce the cost of RBDO, where x
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22 is the vector of random variables and d is the vector of design variables. By including design variables in the response surface formulation, the efficiency of the RBDO is improved drastically for certain problems. The ARS is fitted to the response (limit state function) in terms of both design variables and random variables bdxZxTG),()( (3-4) The ARS approach combines probabilistic analyses with design optimization. Using the ARS, the probability of failure at every design point in the design optimization process can be calculated inexpensively by Monte Carlo simulation based on the fitted polynomials. The number of analyses required for ARS depends on the total number of random variables and design variables. Because the ARS fits an approximation in terms of both random variables and design variables it requires more analyses than SRS. For our applications, where the number of random variables is large (around 10) and the number of design variables is small (around four), this typically results in an ARS that is less than three times as expensive to construct as an SRS, which is due to the use of Latin Hypercube sampling that can generate an arbitrary number of design points for RSA construction (explained in last section of this chapter and demonstrated in chapter 5). This compares with a large number (order of 10 to 100) of SRS approximation required in the course of optimization. For a large number of variables (more than 20 to 30), the construction of ARS is hindered by the curse of dimensionality. SRS might be used to reduce the dimensionality of the problem. Besides the computational cost issue, the inclusion of design variables may increase the nonlinearity of the response surface approximation. It might be necessary to use RSA of order higher than quadratic, for
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23 which proper DOE must be employed. The DOE issues are discussed in the last section of this chapter. Design Response Surface (DRS) Approximation Direct Monte Carlo simulation introduces noise in computed probability of failure due to limited samples. The noise can be reduced by using a relatively large number of samples, which is computationally made possible by using response surface approximation. The noise can also be filtered out by using another response surface approximation, the design response surface (DRS). DRS fitted to probability of failure P as a function of design variables d can be shown as (3-5) bdZdTP)()( The use of DRS also reduces the computational cost of RBDO by approximating the reliability constraint by close-form function. The probability of failure is found to change by several orders of magnitude over narrow bands in design space, especially when the random variables have small coefficients of variation (Chapter 5). The steep variation of probability of failure requires DRS to use high-order polynomials for the approximation, such as quintic polynomials, increasing the required number of probability calculations (Qu et al. 2000). An additional problem arises when Monte Carlo simulations (MCS) are used for calculating probabilities. For a given number of simulations, the accuracy of the probability estimates deteriorates as the probability of failure decreases. The numerical problems associated with steep variation of probability of failure led to consideration of alternative measures of safety. The most common one is to use the
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24 safety index which replaces the probability by using the inverse standard normal transformation, ) (3-6) (1P Safety index is the distance measured as the number of standard deviations from the mean of a normal distribution that gives the same probability. Fitting DRS to safety index showed limited improvement of accuracy (Chapter 6), and it has the same problems of accuracy as the probability of failure when based on Monte Carlo simulations. Box-Cox transformation (Myers and Montgomery 1995) on the probability of failure was also tested, but showed very limited improvement. A probabilistic sufficiency factor approach is developed as an inverse reliability measure to improve the accuracy of DRS, estimate additional resources required to satisfy the reliability constraint, and convert RBDO to sequential deterministic optimization (Chapter 6 and 7). Analysis and Design Response Surface Approach Figure 3-1 summarizes the ARS/DRS-based RBDO approach. First the DOE of ARS is performed and ARS is constructed. Then DOE of DRS is performed, which should stay in the range of design variables of the DOE for the ARS, and DRS is constructed. Design optimization is then performed on the DRS. If the design does not converge, the DOE of the DRS can be moved toward the intermediate optimum and its range can be shrunk to improve the accuracy of DRS. If the intermediate optimum is near the boundary of the ARS, the DOE of the ARS needs to be moved to cover the potential optimum region better. The entire process is repeated until the optimization converges and the reliability of the optimum stabilizes.
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25 Stop Analyses & build ARS DOE toward optimum MCS & construct DRS Optimization Converge? YesNoNo DOE toward optimum Stop Analyses & build ARS DOE toward optimum MCS & construct DRS Optimization Converge? YesNoNo DOE toward optimum Analyses & build ARS DOE toward optimum MCS & construct DRS Optimization Converge? YesNoNo DOE toward optimum Figure 3-1. Analysis response surface and design response surface approach: decouple reliability analysis and design optimization Statistical Design of Experiments for Stochastic and Analysis Response Surfaces Statistical design of experiments selects design points for response surface approximation in such a manner that the required accuracy is achieved with a minimum number of design points. However, the exact functional form of the structural response to be approximated is rarely known, the errors in SRS and ARS usually include both variance and bias errors. Structural responses are usually computationally expensive to evaluate. Therefore, the selection of the DOE for ARS are primarily based on the following two considerations The number of design points in the DOE is flexible, since we want to reduce the number of analyses. The points in the design space have good space-filling distribution. The DOE is often used to provide a sampling of the problem space. Higher than quadratic polynomials may also be needed in order to provide good approximation of the response. Both desire space-filling DOE.
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26 ARS needs to include both the design and random variables, the number of variables is relatively large, often exceeding 15 variables. This excludes the use of many DOE, such as Central Composite Design (CCD). The CCD has 2n vertices, 2n axial points, and one center point, so the required number of design points is 2n+2n+1,where n is the number of variables involved. A polynomial of mth order in terms of n variables has L coefficients, where !))...(2)(1(mmnnnL (3-7) For n = 15, CCD requires 32799 analyses. On the other hand, a quadratic polynomial in 15-variable has 136 coefficients. From our experience, in order to estimate these coefficients, the number of analyses only needs to be about twice as large as the number of coefficients, which is less than one percent of the number of vertices for 15-variable space. Therefore, other DOEs such as CCD using fractional factorial design (Myers and Montgomery 1995) need to be used. The fractional factorial CCD is intended for the construction of quadratic RSA. Orthogonal arrays (Myers and Montgomery 1995) are used for the construction of higher order RS (Balabanov 1997 and Padmanabhan et al. 2000). Isukapalli (1999) employed orthogonal arrays to construct SRS. For problems where only very limited number of analyses is computationally affordable, Box-Behnken designs or saturated designs can be used (Khuri and Cornell 1996). In the paper of Qu et al. (2000), it is shown that Latin Hypercube sampling is more efficient and flexible than orthogonal arrays. The idea of Latin Hypercube sampling can be explained as follows: assume that we want n samples out of k random variables. First, the range of each random variable is divided into n nonoverlapping intervals on the basis of equal probability. Then one value is selected randomly from each interval. Finally, by
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27 randomly pairing values of different random variables, the n input vectors each of k dimension for Monte Carlo simulation are generated. Figure 3-2 illustrates a two-dimensional Latin Hypercube sampling. NormalUniform NormalUniform Figure 3-2. Latin Hypercube sampling to generate 5 samples from two random variables
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CHAPTER 4 DETERMINISTIC DESIGN OF COMPOSITE LAMINATES FOR CRYOGENIC ENVIRONMENTS This chapter presents deterministic designs of composite laminates for hydrogen tanks in cryogenic environments. The traditional way of deterministically designing the laminate with safety factors is employed in this chapter in order to investigate the design issues. Reliability-based design, explicitly taking account of uncertainties in material properties, is presented in chapter 5. Introduction The use of composite materials for the design of liquid hydrogen tanks at cryogenic temperatures has many challenges. The coefficient of thermal expansion (CTE) along the fiber direction is usually two orders of magnitude smaller than that transverse to fiber direction. In typical composite laminates, the ply angles are different in order to carry load efficiently, which results in a mismatch of the coefficients of thermal expansion. When the laminates are cooled down during manufacturing from stress-free temperature that is near the curing temperature, the mismatch of the coefficients of thermal expansion induces large thermal strains. Cooling to cryogenic temperatures substantially increases the thermal stresses. The residual thermal strains may result in matrix cracking leading to reduction in stiffness and strength of the laminate and possible initiation of delamination. A more detrimental effect of matrix cracking in hydrogen tanks is hydrogen leakage through the wall of the tank. Park and McManus (1996) proposed a micro-mechanical model based on fracture mechanics and verified the model by experiments. Kwon and 28
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29 Berner (1997) studied matrix damage of cross-ply laminate by combining a simplified micro-mechanics model with finite element analysis and showed that the prediction of damage is improved substantially with the incorporation of residual stresses. Aoki et al. (2000) modeled and successfully predicted the leakage through the matrix cracks. The present objective is to investigate options available to minimize the increase in thickness due to thermal residual strains for laminates designed subject to thermal and mechanical loads. Deterministic designs were performed to investigate the following effects: (i) temperature dependant material properties for strains analysis, (ii) laminates designed to allow partial ply failure (matrix cracking), and (iii) auxiliary stiffening solutions that reduce the axial mechanical load on the tank wall laminates. Composite Laminates Analysis under Thermal and Mechanical Loading Since the properties of composite materials, such as coefficients of thermal expansion and elastic moduli, change substantially with temperature, classical lamination theory (CLT) (e.g., Grdal et al. 1999) is modified to take account of temperature dependent material properties. The stress-free strain of a lamina is defined as where TF TF is the coefficient of thermal expansion (CTE). When is a function of temperature T, the stress free strain is given by the expression (4-1) TdTservicezeroTTF)( where Tzero is the stress-free temperature of the material and Tservice is the service temperature. From the equilibrium equation and vanishing of residual stress resultant, the equilibrium of a symmetric laminate subjected to pure thermal load with uniform temperature profile through the thickness can be expressed by
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30 )()()(220TdzTTNhhFNNQA (4-2) where is the non-mechanical strain induced by thermal load. The right hand side of Equation 4-2 is defined as the thermal load NN. From Equation 4-2, the non-mechanical strain induced by thermal load can be expressed by N0 ) (4-3) ()()(10TTTNNNA The residual thermal stress is given by the constitutive equation )()()()(0TTTTFNRQ (4-4) The mechanical strain is expressed by ) (4-5) ()()(1TTTMMNA Therefore, the mechanical stress is given by )()()(TTTMMQ (4-6) By the principle of superposition, the residual strain and total stress in the laminate are expressed by )()()()(0ResidualTTTTFNM (4-7) )()()(TTTMRTotal (4-8) Properties of IM600/133 Composite Materials The composite material used in the present study is the IM600/133 graphite-epoxy material system, which has a glass-transition temperature of 356F. Aoki et al. (2000) tested IM600/133 (material Aa in their paper) composite material system at various temperatures, ranging from 356F to .2F (180C to -269C), with mechanical tensile loads. The material properties of IM600/133 were taken from Aoki et al. (2000)
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31 and fitted with smooth polynomials as functions of temperatures in order to be used in calculation (Figures 4-1 and 4-2). The data points used in the fitting and the individual polynomials are shown in the Appendix A. 0.05.010.015.020.025.0-425.0-225.0-25.0175.0375.0Temperature (F)E1 Elastic Modulus in fiber direction-0.50.00.51.01.52.02.5E2-Elastic modulus in matrix direction, G12 Shear Modulus, nu12 Poissons Ratio E1 (Mpsi) E2 (Mpsi) G12 (Mpsi) nu12 Figure 4-1. Polynomials fit to elastic properties: E1, E2, G12, and 12 Aoki et al. (2000) showed that the fracture toughness of the material increased at lower temperatures; however, the increased strain energy due to the mismatch in the thermal expansion coefficients also increased the critical energy release rate. They also applied a micro-mechanics model proposed by Park and McManus (1996) for predicting micro-cracking and showed good correlation with experiments. Aoki et al. (2000) found that at cryogenic temperatures, quasi-isotropic laminates exhibited a large reduction in the transverse mechanical strain 2 that initiates micro-cracking (0.702% at room temperature to 0.325% at cryogenic temperatures). Experimental data from Aoki et al. (2000) were used to determine the strain allowables. They tested a 16-ply quasi-isotropic (45/0/-45/90)2s symmetric laminate in
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32 tension in the 0 direction at cryogenic temperatures. The nominal specimen thickness and width were 2.2 mm and 15 mm. The mechanical loads corresponding to matrix cracks (Table 4-1) were extracted from Figure 5 in Aoki et al. (2000). The strain transverse to the fiber direction, 2, is assumed to be the strain that induces matrix-cracking in the laminate. Based on the load condition and the configuration of the laminate, the transverse strain 2 in the 90 plies is the most critical strain in the laminate. -0.006-0.004-0.0020.0000.0020.0040.0060.0080.0100.012-500-400-300-200-1000100Temperature (F)Strains Epsilon1 Epsilon2 Gamma12 Figure 4-2. Polynomials fit to coefficients of thermal expansion: 1 and 2 Normally, strain allowables are calculated by loading laminates at room temperatures. However, for micro-cracking, the residual stresses are of primary importance, so all strains are calculated from the stress-free temperature, assumed to be 300 F. The calculations are made by integrating the thermal strains from the stress free temperature to the operational temperature as described in the next section.
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33 Table 4-1 shows the transverse strains 2 in the 90 plies corresponding to the loading at the onset of matrix cracking at selected temperatures. Aoki et al. (2000) found that the maximum mechanical strain before matrix cracking is reduced from 0.7% at room temperature to 0.325% at -452F. Older results (Aoki et al. 1999) (Table 4-1) indicated that the maximum mechanical strain at cryogenic temperature may be as low as 0.082%. However, the calculation indicates that the total strain (including the residual thermal strain) may vary anywhere from 1.5 to 1.9% depending on the temperature and the measurement. These values appear high, but this is because they include the residual strains that are usually not counted. For the quasi-isotropic laminate, these residual strains at room temperature are very high, at 0.86%, and are higher at lower temperatures Table 4-1. Transverse strains calculated for conditions corresponding to the onset of matrix-cracking in the 90 plies of a quasi-isotropic (45/0/-45/90)2s in Aoki et al. (2000) Room temperature (77 F or 25 C) LN2 temperature (-320 F or -196 C) LHe temperature (-452 F or 269 C) LHe temperature (-452 F or 269 C)a Mechanical load (Mpa) 390 330 200 50a Total 2 0.01564 0.01909 0.01760 0.01517a Thermal 2 0.00864 0.01365 0.01435 0.01435a Mechanical 2 0.00700 0.00544 0.00325 0.00082a a Older data obtained from Aoki et al. (1999) The importance of working with strains measured from the stress free temperature is demonstrated in Table 4-2, which shows the 2 in the angle-ply laminate (25)4S under the same loading condition as Table 4-1. At room temperature, the residual (thermal) strains are only about 0.4% compared to 0.86% for the quasi-isotropic laminate. An analysis based on strains measured from room temperature will not show the additional 0.46% strain that the (25)4S laminate can carry compared to a quasi-isotropic laminate.
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34 Based on the data from Table 4-1, we selected the allowable strain to be 1.54% for the probabilistic design and 1.1% (1.4 safety factor) for the deterministic design. Table 4-2. Transverse strains of an angle-ply laminate ( 25)4S under the same loading condition as Table A1 Room temperature (77 F or 25 C) LN2 temperature (-320 F or -196 C) LHe temperature (-452 F or C) LHe temperature (-452 F or C)a Mechanical load (Mpa) 390 330 200 50a Total 2 -0.00261 0.00360 0.00527 0.00656a Thermal 2 0.00393 0.00669 0.00699 0.00699a Mechanical 2 -0.00654 -0.00309 -0.00172 -0.00043a a Older data obtained from Aoki et al (1999) Table 4-3 shows the strain allowables for the lamina, where other strain allowables except 2u were provided to us by NASA. The strain allowables may appear to be high, but this is because they are applied to strain including residual strains that develop due to cooling from stress-free temperature of 300F. A quasi-isotropic laminate will use up its entire transverse strain allowable of 0.011, when cooled to -452F. Thus, this value is conservative in view of the experiments by Aoki et al. (2000) that indicated that the laminate can carry 0.325% mechanical strain at cryogenic temperature. Table 4-3. Strain allowables for IM600/133 at F Strain 1u 1l 2u 2l 12u Allowablesa 0.0103 -0.0109 0.0110 or 0.0154b -0.0130 0.0138 a Strains include residual strains calculated from the stress-free temperature of 300 F b The value 0.0110 is obtained from the extreme value 0.0154 divided by a safety factor of 1.4 Deterministic Design of Angle-Ply Laminates It is estimated that the minimum thickness needed to prevent hydrogen leakage is 0.04 inch, so it may be acceptable to permit matrix cracking if the undamaged part of the laminate has a minimum thickness of 0.04 inch. For the cracked part of the laminate, the
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35 elastic modulus transverse to fiber direction, E2, and the shear modulus, G12, are reduced by 20 percent and the transverse strain allowable, 2u, is increased. The rest of the laminate must not have matrix cracking and must provide at least 8 contiguous intact plies (0.04 inch) in order to prevent hydrogen leakage. Optimization Formulation Laminates with two ply angles, [ 1/ 2]S, see Figure 4-3, were optimized. The x direction here corresponds to the hoop direction on a cryogenic propellant tank, while the y direction corresponds to the axial direction. The laminates are made of IM600/133 graphite-epoxy material with ply thickness of 0.005 inch and subjected to mechanical load and an operating temperature of F. Nx is 4,800 lb./inch and Ny is 2,400 lb./inch. y xxy+ 12 Figure 4-3. Geometry and loads for laminates The design problem was formulated as (Thickness are in inches) 214tth minimize httuulul040.0,0.005such that211212222111 (4-9)
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36 where h is the laminate thickness, superscripts u and l denote upper and lower limits of associated quantities, and 1, 2, and 12 are the ply strains along fiber direction, transverse to fiber direction, and shear strain, respectively. The stack thickness of plies with ply-angle 1, which is allowed to have matrix cracking, is t1. The stack thickness of the plies with ply-angle 2, which are not allowed to crack and must provide in total a minimum intact thickness of 0.04 inch to prevent hydrogen leakage, is t2. The four design variables are the ply angles 1 and 2 and their stack thickness t1 and t2. The individual stack thickness from a continuous optimizer (SQP in MATLAB) is rounded up to the nearest multiple of 0.005 inch. Optimizations without Matrix Cracking In order to see the effect of mechanical and thermal loads, it is instructive to compare designs for different operational temperatures. Table 4-4 shows the optimum laminates at these temperatures. In the last row of Table 4-4, the numbers in the parentheses are the continuous thickness before rounding. Without thermal strains, a cross-ply laminate with thickness of 0.04 inch can easily (with 0.1% transverse strain as the margin of safety) carry the mechanical loads. When thermal strains are taken into account, the angle between the plies must decrease in order to reduce the thermal strains. The ply angles do not vary monotonically because both the residual strains and the stiffness of the laminate increase with the decrease of temperature. At cryogenic temperatures the angle decreases to 25.5, and at that angle the axial loads cannot be carried efficiently and the thickness increases to 0.1 inch. Figure 4-4 shows that the thickness of optimum laminates for temperature dependent and constant material properties at 77F changes substantially with the working temperature for a strain limit
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37 2u of 0.0110. Using temperature dependent material properties avoided a very conservative design with constant material properties. Table 4-4. Optimal laminates for different operational temperatures: 2u of 0.0110 Mechanical only Mechanical and Thermal load Temperature (F ) 77.00 77.00 -61.50 -242.00 -423.00 1 (degree) 90.00 34.82 38.13 33.57 25.50 2 (degree) 0.00 33.93 38.13 33.57 25.50 t1 (inch) 0.005 0.005 0.005 0.010 0.010 t2 (inch) 0.005 0.005 0.010 0.010 0.015 ha (inch) 0.040 (0.040) 0.040 (0.040) 0.060 (0.048) 0.080 (0.079) 0.100 (0.093) a Numbers in parentheses indicate unrounded thickness Figure 4-4. The change of optimal thickness (inch) with temperature for variable and constant material properties (2u of 0.0110) Desig temperatures, so for all designs discu2u is the active constraint at F for the second optimal design of Table 4-5. 00.020.040.060.080.10.120.140.160.180.2-500-400-300-200-1000100200Temperature (F)Optimal Thickness (inch) Variable material properties Constant material properties ns must be feasible for the entire range of ssed in the rest of the dissertation, strain constraints were applied at 21 temperatures, which were uniformly distributed from 77F to F. Table 4-5 shows that the design problem has multiple optima. Figure 4-5 shows that the tensile strain limit
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38 Table 4-5. Optimal laminates for temperature dependent material properties with 2u of 1 2 t1 t2 ha (inch) Probability of f 0.0110 (optimized for 21 temperatures) (degree) (degree) (inch) (inch) ailureb 0.00 28.16 0.005 0.020 0.100 (0.103) 0.019338 (0.014541) 27.04 27.04 0.010 0.015 0.100 (0.095) 0.000479 (0.001683) 94) 25.16 27.31 0.005 0.020 0.100 (0.0 0.000592 (0.001879) a insete ued thbrobabereted ethbed These optimal laminates have similar thickness but different ply angles. The failure proba Numbers parenthe s indica nround ickness odology descri The p ilities w calcula by the m in the chapter 5 -0.006-0.004-0.0020.0000.0020.0040.0060.008-500-400-300-200-1000100Temperature (F)Strains 0.012 0.010 Epsilon1 Epsilon2 Gamma12 Figure 4-5. Strains in optimal laminate for temperature dependent material properties with 2u of 0.0110 (second design in Table 4-3) probabilities of the continuous designs are shown in parentheses. The high failure bilities of the first design (continuous and discrete) clearly show a smaller safety margin than the other two. The second and third designs show that a slight rounding can change the failure probability significantly. Designs with two similar ply angles have much lower failure probabilities than designs with two substantially different ply angles. The failure probabilities of these laminates are too high (compared with 10-4 to 10-6), and this provides incentives to conduct reliability-based design.
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39 Optimizations Allowing Partial Matrix Cracking Plies of angle 1 are the plies allowed for matrix cracking for optimizations lies was increased to 0.0154, while the reopartial matrix cracking: 2 of 0.011 for uncracked plies and 0.0154 for tte allowing partial matrix cracking. The 2u of the 1 p st of the laminate still used 2u of 0.011. The lower limit of t2 was increased to 0.010 inch (total 2 thickness of 0.04 inch) to prevent hydrogen leakage. Table 4-6 shows the optimal design allowing partial matrix cracking. Its thickness is the same as that of the design withut partial matrix cracking (Table 4-5), and the ply angle of the cracked plies increased due to the increased strain limit, 2u. However, the failure probability is higher than the design that does not allow matrix cracking, which indicates that this option does not help. The active constraint is still the tensile strain limit 2u of 0.011 at cryogenic temperatures for the un-cracked plies. Table 4-6. Optimal laminate for temperature dependent material properties allowing ucracked plies 1(degree) 2(degree) 1 (inch) 2 (inch) ha (inch) Probability of failur 36.07 82) 25.24 0.015 0.010 0.100 (0.097) 0.003716 (0.0045 a Numbers in parentheses indicate unrounded thickness. OoexiaNy f the load is the axial load Ny, llant tank. A smaller axial load may be obtain ptimizati ns with R duced A l Load With small ply angles, the critical component o induced by pressure on the caps of the prope ed by using an auxiliary structure to carry part of this load, such as axial stiffeners or a cable connecting the caps. If the auxiliary structure does not directly connect to the wall of the hydrogen tank (such as attached to the caps of the tank), it will not be affected by the mismatch of the thermal expansion coefficients, i.e., the residual thermal strains. Here the possibility of reducing the axial load by half by carrying 1200 lb./inch of the
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40 axial load by a cable made of unidirectional material was explored. The required cross-sectional area of the composite cable is 5.05 inch2, which is equivalent to a laminate thickness of 0.005 inch for a tank with a 160-inch radius. Table 4-7 lists designs optimized with half of the axial load. The results indicate that reducing axial load is an effective way to reduce the laminate thickness. Higher probabilities of failure reflect rounding down of the thickness. Table 4-7. Optimal laminates for reduced axial load of1, 200 lb./inch by using load 1 2 t1 shunting cables (equivalent laminate thickness of 0.005 inch) (degree) (degree) (inch) t2 (inch) ha (inch) Probability of failure 0.00 29.48 0.005 0.005 0.040 (0.043) 0.010311 (0.00115 6) 2798 2 (0.473536) 42) .30.62 26.20 0.005 0.005 0.040 (0.043) 0.58573 11.31 0.005 0.005 0.040 (0.0 0.008501 (0.008363) a insate ded is st th of ly dth safety factors did not work well for this problem due to various uncertainties and the lamin Numb ers paren the es indic unroun thickness. It een tha e tradition al way deterministical esign the laminate wi ate cracking failure mode. Uncertainties in the material properties are introduced by the fabrication process, the temperature dependence of material properties, the cure reference temperature, and acceptable crack density for design. These uncertainties indicate a need to use reliability-based optimization to design laminates for use at cryogenic temperatures.
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CHAPTER 5 RELIABILITY-BASED DESIGN OF COMPOSITE LAMINATES FOR CRYOGENIC ENVIRONMENTS This chapter presents reliability-based designs of composite laminates for hydrogen tanks in cryogenic environments, comparison between deterministic and reliability-based design, identification of uncertainty parameters that have the largest influence on the optimum design, and quantification of the weight penalty associated with level of uncertainty in those parameters. The results indicate that the most effective measure for reducing thickness is quality control (refer also to Qu et al., 2001). The reliability-based optimization is carried out using response surface approximations combined with Monte Carlo simulation described in chapter three. Reliability-Based Design Optimization Problem Formulation The reliability-based optimization is formulated as 214tth minimize (5-1) 21005.0005.0such thatttPPu where h is the laminate thickness, t1 is the stack thickness of lamina with ply-angle 1 and has a lower limit of 0.005 inch on it, t2 is the stack thickness of lamina with ply-angle 2 and has a lower limit of 0.005 inch. The limits on t1 and t2 also ensure that the laminate has a minimum thickness of 0.04 inch to prevent hydrogen leakage. The reliability 41
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42 constraint is expressed as a limit Pu (i.e., Pu=10-4) on the probability of failure, P. The probability of failure is based on first-ply failure according to the maximum strain failure criterion. The four design variables are the ply angles 1 and 2 and their stack-thickness t1 and t2. Reliability-based optimization seeks the lightest structure satisfying the reliability constraint. The twelve random variables are four elastic properties (E1, E2, G12, and 12), two coefficients of thermal expansion (1 and 2), five ply strain allowables (1u, 1l, 2u, 2l, and 12u), and the stress-free temperature of the material (Tzero). The mean values of the strain limits are shown in Table 5-1 except for 2u, which is 0.0154. Table 5-2 shows the coefficients of variation (CV) of the random variables that are assumed to be normally distributed and uncorrelated. Those CVs are obtained based on limited test data provided to us by the manufactures, and are intended only for illustration. The mean value of the stress-free temperature is 300F. The mean values of other random variables change as function of temperature and are given in chapter four. Table 5-1. Strain allowablesa for IM600/133 at F Strain 1u 1l 2u 2l 12u Allowables 0.0103 -0.0109 0.0110 or 0.0154b -0.0130 0.0138 a Strains include residual strains calculated from the stress-free temperature of 300F b The value 0.0110 is obtained from the extreme value 0.0154 divided by a safety factor of 1.4 Table 5-2. Coefficients of variation (CV) of the random variables Random variables E1 E2 G12 12 1 2 Tzero 1u 1l 2u 2l 12u CV 0.035 0.035 0.030 0.06 0.09
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43 Response Surface Approximation for Reliability-Based Optimization For the present work, response surface approximation of two types was created. The first type is analysis response surface (ARS), which is fitted to the strains in the laminate in terms of both design variables and random variables. Using the ARS, the probability of failure at every design point can be calculated inexpensively by Monte Carlo simulation based on the fitted polynomials. The second type of response surface is design response surface (DRS) that is fitted to probability of failure as a function of design variables. The DRS is created in order to filter out noise induced by the Monte Carlo simulation and is used to calculate the reliability constraint in the design optimization. The details of the ARS/DRS approach are given in chapter three. Analysis Response Surfaces Besides the design and random variables described in the problem formulation, the service temperature was treated as a variable ranging from 77F to F in order to avoid constructing analysis response surfaces at each selected temperature. Therefore, the total number of variables was seventeen. However, the strains in the laminate do not depend on the five strain allowables, so the ARS were fitted to the strains in terms of twelve variables, which included four design variables, four elastic properties, two coefficients of thermal expansion, the stress-free temperature and the service temperature. The range of the design variables (Table 5-3) for the ARS was chosen based on the values of the optimal deterministic design. Ranges for random variables are automatically handled and explained below. Using the ARS and five strain allowables, probabilities of failure are calculated by Monte Carlo simulations, while the strain constraints were evaluated at 21 uniformly distributed service temperatures between 77F and F.
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44 Table 5-3. Range of design variables for analysis response surfaces Design variables 1 2 t1 t2 Range 20 to 30 20 to 30 0.0125 to 0.03 inch 0.0125 to 0.03 inch The accuracy of the ARS is evaluated by statistical measures provided by the JMP software (Anon. 2000), which include the adjusted coefficient of multiple determination (R2adj.) and the root mean square error (RMSE) predictor. To improve the accuracy of response surface approximation, polynomial coefficients that were not well characterized were eliminated from the response surface model by using a mixed stepwise regression (Myers and Montgomery 1995). The statistical design of experiment of ARS was Latin Hypercube sampling or Latin Hypercube design (LHS, e.g., Wyss and Jorgensen 1998), where design variables were treated as uniformly distributed variables in order to generate design points (presented in Chapter 3). Since the laminate has two ply angles and each ply has three strains, six ARS were needed in the optimization. A quadratic polynomial of twelve variables has 91 coefficients. The number of sampling points generated by LHS was selected to be twice the number of coefficients. Tables 4 shows that the quadratic response surfaces constructed from LHS with 182 points offer good accuracy. Table 5-4. Quadratic analysis response surfaces of strains (millistrain) Analysis response surfaces based on 182 LHS points Error Statistics 1 in 1 2 in 1 12 in 1 1 in 2 2 in 2 12 in 2 R2adj 0.9977 0.9956 0.9991 0.9978 0.9961 0.9990 RMSE Predictor 0.017 0.06 0.055 0.017 0.055 0.06 Mean of Response 1.114 8.322 -3.13 1.108 8.328 -3.14
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45 Design Response Surfaces The six quadratic ARS were used to calculate the probabilities of failure with Monte Carlo simulation. Because the fitting errors in design response surfaces (DRS) are generally larger than the random errors from finite sampling in probability calculation, Monte Carlo simulation needs only to be performed until relatively small errors estimated confidence intervals are achieved. Therefore, a sample size of 1,000,000 was employed. The design points of DRS combine Face Center Central Composite Design (FCCCD) and LHS. Table 5-5 compares the three DRS. Table 5-5. Design response surfaces for probability of failure (Probability calculated by Monte Carlo simulation with a sample size of 1,000,000) FCCCD 25 points LHS 252 points LHS 252 points + FCCCD 25 points Error Statistics quadratic 5th order 5th order R2adj 0.6855 0.9926 0.9982 RMSE Predictor 0.00053 0.000003 0.000012 Mean of Response 0.00032 0.000016 0.000044 The accuracy of the quadratic response surface approximation is unacceptable. The accuracy of fifth order response surface (with 126 unknown coefficient before stepwise regression) was improved by using a reciprocal transformation on the thickness t1 and t2, since the probability of failure, like most structural response, is inversely correlated with the stack thickness. We found that LHS might fail to sample points near some corners of the design space, leading to poor accuracy around these corners. We therefore combined LHS with FCCCD that includes all the vertices of the design space. The accuracy of DRS based on LHS combined with FCCCD is slightly worse than that of DRS based on LHS alone, because the probabilities at the corners of the design space are usually extremely low or high, presenting a greater fitting difficulty than without FCCCD. But the
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46 extrapolation problem was solved, and the side constraints are set as the range of the ARS shown in Table 5-3. The error of 0.000012 is much lower than the allowable failure probability of 0.0001. Table 5-6 compares the reliability-based optimum with the three deterministic optima from chapter 4 and their failure probabilities. The optimal thickness increased from 0.100 to 0.120, while the failure probability decreased by about one order of magnitude. Table 5-6. Comparison of reliability-based optimum with deterministic optima Optimal Design [1, 2, t1, t2] (degree and inch) Laminate thickness (inch) Failure probability from MCS of ARS 1,000,000 samples Allowable probability of failure [24.89, 25.16, 0.015, 0.015] 0.120 (0.120) 0.000055 0.0001 [0.00, 28.16, 0.005, 0.020] 0.100 (0.103) 0.019338a [27.04, 27.04, 0.010, 0.015] 0.100 (0.095) 0.000479 [25.16, 27.31, 0.005, 0.020] 0.100 (0.094) 0.000592 Deterministic optima a This deterministic optimum is out of the range of the analysis response surfaces; the probability of failure was calculated by Monte Carlo simulation based on another set of analysis response surfaces. Refining the Reliability-Based Design The reliability-based designs in Table 5-6 show that the ply angles close to 25 offer designs with low failure probability. Furthermore, good designs require only a single ply-angle allowing simplification of the configuration of the laminate from [1/2]S to [ ]S. Table 5-7 shows the failure probabilities of some chosen designs calculated with Monte Carlo simulation using ARS. The laminates with ply-angles of 24, 25, and 26 offer lower probabilities of failure than the rest. These three laminates will be further studied.
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47 Table 5-7. Refined reliability-based design []S (Monte Carlo simulation with a sample size of 10,000,000) (degree) h (inch) Probability of failure 21.00 0.120 0.0001832 22.00 0.120 0.0001083 23.00 0.120 0.0000718 24.00 0.120 0.0000605 25.00 0.120 0.0000565 26.00 0.120 0.0000607 27.00 0.120 0.0000792 Quantifying Errors in Reliability Analysis The reliability analysis has errors due to MCS with limited sample size and due to the approximation of CLT analysis by analysis response surfaces. To evaluate the amount of errors in reliability analysis, the probability of failure of the rounded design was evaluated by using MCS with the exact analysis (classical laminate theory, CLT), but only one million analyses were performed due to the cost of computation. Table 5-8 compares the results of MCS based on ARS and that based on CLT. The difference is about 1.2510-5. Table 5-8. Comparison of probability of failure from MCS based ARS and CLT Optimal Design [1, 2, t1, t2] (degree and inch) Laminate thickness (inch) Failure probability from MCS of ARS 1x107 samples Failure probability from MCS of CLT 1x106 samples [25, 25, 0.015, 0.015] 0.120 (0.120) 0.0000565 0.000069 By assuming each simulation as a Bernoulli trial and the N trails as Binomial distribution, the coefficient of variation (COV) of the probability (Pof) obtained by MCS can be estimated by PofNPofPofPofCOV)1()( (5-2)
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48 where N is the sample size of the MCS. The accuracy of MCS can also be expressed in terms of percentage error corresponding 95% confidence interval as %196)1(%TTPofNPof (5-3) where PofT is the true probability of failure. Table 5-9 shows the accuracy and error bounds for MCS. Together with Table 5-8 the error calculation indicates that the probability of failure of the rounded design is still below the target probability of failure of 0.0001. The errors can be reduced by more accurate approximations and advanced Monte Carlo simulations. Another reliability-based design cycle in a reduced size design region can be performed to obtain more accurate result. Table 5-9. Accuracy of MCS Coefficient of Variation (COV) Percentage errors (Absolute errors) for 95% CI MCS of 1x107 samples 4.2% 8.2% (4.66x10-6) MCS of 1x106 samples 12.05% 23.6%(1.63x10-5) Effects of Quality Control on Laminate Design Comparing deterministic designs to the reliability-based design, there is an increase of 20% in the thickness. In addition, the design failure probability of 10-4 is quite high. In order to improve the design the possibility of limiting the variability in material properties through quality control (QC) is considered. Here, quality control means that materials will be tested by the manufacturer and/or fabricator, and that extremely poor batches will not be accepted. Normal distributions assume the possibility (though very small) of unbounded variation. In practice, quality control truncates the low end of the distribution. That is, specimens with extremely poor properties are rejected. It is also
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49 assumed that specimens with exceptional properties are scarcer than those with poor properties. The normal distribution will be truncated on the high side at 3 (excluding 14 out of 10,000 specimens) and on the low side at different values corresponding to different levels of QC. The tradeoff between QC, failure probability and laminate thickness (weight) will be explored. Effects of Quality Control on Probability of Failure Since the primary failure mode of the laminate is micro-cracking, the tensile strain limit 2u is the first quantity to be improved by quality control. The normal distribution of 2u is truncated at 3 to exclude unrealistically strong specimens, and on the low side QC at -4, -3, and -2 was checked, which corresponds to rejecting 3 specimens out of 100,000, 14 specimens out of 10,000, and 23 specimens out of 1,000, respectively. Table 5-10 shows the change in the failure probability for selected reliability-based designs. Quality control on 2u is a very effective way to reduce the probability of failure. A relatively low cost QC of 2u at 3 will reduce the failure probability by more than two orders of magnitude. Table 5-10. Effects of quality control of 2u on probability of failure for 0.12 inch-thick ()S laminates Probability of failure from MCS 10,000,000 samples Un-truncated Normal Truncate at -4 (3/100,000) Truncate at -3 (14/10,000) Truncate at -2 (23/1,000) 24.0 60.5e-6 30.5e-6 0.0 0.0 25.0 56.5e-6 29.9e-6 0.1e-6 0.0 26.0 60.7e-6 31.0e-6 0.5e-6 0.0 Table 5-11 shows that truncating other strain limits even at -2 will not change the laminate failure probability substantially. This reveals the fact that the primary failure mode of the laminate is micro-cracking. Therefore, 2u is the critical parameter to study
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50 further. Table 5-12 shows that this conclusion applies also to the elastic moduli, coefficients of thermal expansion, and stress-free temperature. By comparing with Table 5-10, we see that truncating any of the other parameters at -2 does not change the failure probability as significantly as truncating 2u at -4 Note that some probabilities from truncated distributions are slightly larger than those from untruncated distributions, which is due to the sampling errors. Table 5-11. Effects of quality control of 1u, 1l, 2l, and 12 on probability of failure of 0.12 inch-thick ()S laminates Probability of failure from MCS 10,000,000 samples Un-truncated Normal Truncated 1u at -2 Truncated 1l at -2 Truncated 2l at -2 Truncated 12 at -2 24.0 60.5e-6 58.6e-6 61.5e-6 54.4e-6 55.4e-6 25.0 56.5e-6 53.0e-6 52.3e-6 54.0e-6 53.2e-6 26.0 60.7e-6 63.4e-6 62.0e-6 60.1e-6 61.0e-6 Table 5-12. Effects of quality control of E1, E2, G12, 12, Tzero, 1, and 2 on probability of failure of 0.12 inch-thick ()S laminates Probability of failure from MCS 10,000,000 samples by truncating at -2 E1 E2 G12 12 Tzero 1 2 24.0 62.2e-6 52.1e-6 57.8e-6 51.8e-6 54.6e-6 52.7e-6 58.2e-6 25.0 52.5e-6 48.1e-6 55.1e-6 49.7e-6 55.1e-6 56.8e-6 54.4e-6 26.0 54.5e-6 59.1e-6 60.4e-6 59.4e-6 59.8e-6 63.0e-6 60.4e-6 Effects of Quality Control on Optimal Laminate Thickness Quality control (QC) can be used to reduce the laminate thickness instead of the probability of failure. Table 5-13 shows that QC of 2u at -3 will allow 0.1 inch-thick laminates with failure probability below the required 0.0001.
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51 Table 5-13. Effects of quality control of 2u on probability of failure for 0.1 inch-thick ()S laminates Probability of failure from MCS 1,000,000 samples Un-truncated Normal Truncate at -4 (3/100,000) Truncate at -3 (14/10,000) Truncate at -2.5 (6/1,000) 24.0 0.002224 0.002163 0.001054 0.000071 25.0 0.001030 0.000992 0.000229 0.000007 26.0 0.000615 0.000629 0.000092 0.000003 Table 5-14 shows that QC of 2u at -1.6, which corresponds to rejecting 55 specimens out of 1000, will reduce the thickness to 0.08 inch with a failure probability below 0.0001. Therefore, the laminate thickness can be reduced to 0.08 inch if QC is able to find and reject 55 specimens out of 1000. Table 5-14. Effects of quality control of 2u on probability of failure for 0.08 inch-thick ()S laminates Probability of failure from MCS 1,000,000 samples Un-truncated Normal Truncate at -3 (14/10,000) Truncate at -2 (23/1,000) Truncate at -1.6 (55/1,000) 24.0 0.061204 0.060264 0.039804 0.015017 25.0 0.028289 0.027103 0.008820 0.001019 26.0 0.013595 0.012154 0.001243 0.000071 Effects of Other Improvements in Material Properties Instead of quality control, it is possible to improve the design by using a better material. Table 5-15 shows the effects of changing the mean value of 2u by 10 percent of the nominal value of 0.0154. Comparison with Table 5-10 shows that a 10% improvement has big influence on failure probability but is not as powerful as quality control at -3 level.
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52 Table 5-15. Sensitivity of failure probability to mean value of 2u (CV=0.09) for 0.12 inch-thick l()S aminates Probability of failure from MCS 10,000,000 samples E(2u)=0.0154 E(2u)=0.01694 E(2u)=0.01386 24.0 60.5e-6 2.5e-6 1082.3e-6 25.0 56.5e-6 3.4e-6 996.7e-6 26.0 60.7e-6 3.4e-6 1115.7e-6 The failure probability also depends on the coefficient of variation (CV) of 2u. The CV can be improved if the manufacturing could be more consistent. Table 5-16 shows that the failure probabilities are not as sensitive to changes of coefficient of variation as to changes in the mean value of 2u, but 10 percent reduction in the coefficient of variation can still reduce the failure probability by about a factor of five. Table 5-16. Sensitivity of failure probability to CV of 2u ( E(2u)=0.0154 ) for 0.12 inch-thick ()S laminates Probability of failure from MCS 10,000,000 samples CV=0.09 CV=0.099 CV=0.081 24.0 60.5e-6 209.5e-6 9.8e-6 25.0 56.5e-6 208.2e-6 10.8e-6 26.0 60.7e-6 224.2e-6 11.1e-6 Figure 5-1 combines several effects discussed earlier to show a tradeoff plot of probability of failure, cost (truncating and changing the distribution of 2u), and weight (thickness) for a laminate of [ 25]S. For probability of failure less than 1e-3, quality control at the -2 level is more effective for reducing the probability of failure than increasing the mean value by 10 percent or decreasing the coefficients of variation by 10 percent. The reason is that small failure probability is heavily affected by the tails of the
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53 distributions. For large failure probability, increasing the mean value of 2u is more effective. Increasing the mean value of 2u by 10 percent or truncating 2u at 2 can reduce the laminate thickness to 0.10 inch for a safety level of 1e-4. Combining all three measures together, the laminate thickness can be reduced to 0.08 inch with a safety level of 1e-7. Table 5-17 shows the changes of maximum 2 calculated by the laminate analyses. Ten percent changes of the mean values of E2, Tzero, and 2 (same CV) will lead to about 5% change in the maximum 2, which indicate that further study needs to focus on the three quantities. Table 5-18 shows the probabilities of failure are reduced by a factor of five by 10 percent change of the mean values of E2, Tzero, and 2 (same CVs). This reduction of probability shows the potential of further improvements via improvements in all three material properties. 1.0E-081.0E-071.0E-061.0E-051.0E-041.0E-031.0E-021.0E-011.0E+000.060.080.10.120.140.16Thickness (inch)Failure Probability Nominal Quality control to -2 Sigma 10% increase in allowable 10% reduction in variability All Figure 5-1. Tradeoff plot of probability of failure, cost, and weight (laminate thickness) for [25]S
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54 Table 5-17. Maximum 2 (millistrain) induced by the change of material properties E1, E2, G12, 12, Tzero, 1, and 2 for 0.12 inch-thick [25]S laminate Maximum 2 from deterministic analyses for 21 temperature Nominal maximum 2=9.859 E1 E2 G12 12 Tzero 1 2 0.9*Nominal 9.901 10.469 9.763 9.909 9.320 (5.47%) 9.857 9.399 (4.67%) 1.1*Nominal 9.824 9.313 (5.54%) 9.960 9.981 10.584 9.861 10.333 Table 5-18. Probability of failure for 0.12 inch-thick [ 25]S laminate with improved average material properties (Monte Carlo simulation with a sample size of 10,000,000) Nominal 1.1*E(E2) 0.9*E(Tzero) 0.9*E(2) All three measures Probability of failure 0.0000605 0.0000117 0.0000116 0.0000110 0.0000003 Summary The design of hydrogen tanks for cryogenic environments poses a challenge because of large thermal strains that can cause matrix-cracking, which may lead to hydrogen leakage. The laminate design must use ply angles that are not too far apart to reduce the thermal residual strains, compromising the ability of the laminate to carry loads in two directions. These small ply angles can cause the laminate thickness to more than double compared to what is needed to carry only the mechanical loads in the application study here. Satisfying reliability constraints increased the thickness further. Improving the probability of failure required increase of thickness. The most influential uncertainty was variability in the tensile strain allowable in the direction transverse to the fibers, 2u. Limiting this variability can reduce the required thickness. Of the different options studied in the chapter, quality control on the transverse tensile allowable, 2u, proved to be the most effective option. Quality control at the .6 level of 2u, corresponding to rejection of about 5.5% of the specimens, can reduce the required
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55 thickness by a third. Reductions in the coefficient of variation of 2u, or increase in its mean value also reduce the failure probability substantially. Increasing the transverse modulus E2, decreasing coefficient of thermal expansion 2, and reducing the stress free temperature Tzero can also help considerably.
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CHAPTER 6 PROBABILISTIC SUFFICIENCY FACTOR APPROACH FOR RELIABILITY-BASED DESIGN OPTIMIZATION A probabilistic sufficiency factor approach is proposed that combines safety factor and probability of failure for use in reliability-based design optimization. The probabilistic sufficiency factor approach represents a factor of safety relative to a target probability of failure. It provides a measure of safety that can be used more readily than probability of failure or safety index by designers to estimate the required weight increase to reach a target safety level. The probabilistic sufficiency factor can be calculated from the results of Monte Carlo simulation with little extra computation. The chapter presents the use of probabilistic sufficiency factor with a design response surface approximation, which fits it as function of design variables. It is shown that the design response surface approximation for the probabilistic sufficiency factor is more accurate than that for the probability of failure or for the safety index. The probabilistic sufficiency factor does not suffer like probability of failure or safety index from accuracy problems in regions of low probability of failure when calculated by Monte Carlo simulation. The use of probabilistic sufficiency factor accelerates the convergence of reliability-based design optimization Introduction Recently, there has been interest in using alternative measures of safety in reliability-based design optimization. These measures are based on margin of safety or safety factors that are commonly used as measures of safety in deterministic design. 56
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57 Safety factor is generally expressed as the quotient of allowable over response, such as the commonly used central safety factor that is defined as the ratio of the mean value of allowable over the mean value of the response. The selection of safety factor for a given problem involves both objective knowledge such as data on the scatter of material properties and subjective knowledge such as expert opinion. Given a safety factor, the reliability of the design is generally unknown, which may lead to unsafe or inefficient design. Therefore, the use of safety factor in reliability-based design optimization seems to be counter productive. Freudenthal (1962) showed that reliability can be expressed in term of the probability distribution function of the safety factor. Elishakoff (2001) surveyed the relationship between safety factor and reliability, and showed that in some cases the safety factor can be expressed explicitly in terms of reliability. The standard safety factor is defined with respect to the response obtained with the mean values of the random variables. Thus a safety factor of 1.5 implies that with the mean values of the random variables we have a 50% margin between the response (e.g., stress) and the capacity (e.g., failure stress). However, the value of the safety factor does not tell us what the reliability is. Therefore, Birger (1970), as reported by Elishakoff (2001), introduced a factor, which we call here the probabilistic sufficiency factor that is more closely related to the target reliability. A probabilistic sufficiency factor of 1.0 implies that the reliability is equal to the target reliability, a probabilistic sufficiency factor larger than one means that the reliability exceeds the target reliability, and probabilistic sufficiency factor less than one means that the system is not as safe as we wish. Specifically, a probabilistic sufficiency
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58 factor value of 0.9 means that we need to multiply the response by 0.9 or increase the capacity by 1/0.9 to achieve the target reliability. Tu et al. (2000) used probabilistic performance measure, which is closely related to Birgers safety factor, for RBDO using most probable point (MPP) methods (e.g., first order reliability method). They showed that the search for the optimum design converged faster by driving the safety margin to zero than by driving the probability of failure to its target value. Wu et al. (1998, 2001) used probabilistic sufficiency factors in order to replace the RBDO with a series of deterministic optimizations by converting reliability constraints to equivalent deterministic constraints. The use of the probabilistic sufficiency factor gives a designer more quantitative measure of the resources needed to satisfy the safety requirements. For example, if the requirement is that the probability of failure is below 10-6 and the designer finds that the actual probability is 10-4, he or she cannot tell how much change is required to satisfy the requirement. If instead the designer finds that a probability of 10-6 is achieved with a probabilistic sufficiency factor of 0.9, it is easier to estimate the required resources. For a stress-dominated linear problem, raising the probabilistic sufficiency factor from 0.9 to 1 typically requires a weight increase of about 10 percent of weight of the overstressed components. Reliability analysis of systems with multiple failure modes often employs Monte Carlo simulation, which generates numerical noise due to limited sample size. Noise in the probability of failure or safety index may cause reliability-based design optimization (RBDO) to converge to a spurious optimum. The accuracy of MCS with a given number of samples deteriorates with decreasing probability of failure. For RBDO problems with
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59 small target probability of failure, the accuracy of MCS around the optimum is not as good as in regions with high probability of failure. Furthermore, the probability of failure in some regions may be so low that it is calculated to be zero by MCS. This flat zero probability of failure does not provide gradient information to guide the optimization procedure. The probabilistic sufficiency factor is readily available from the results of MCS with little extra computational cost. The noise problems of MCS motivate the use of response surface approximation (RSA, e.g., Khuri and Cornell 1996). Response surface approximations typically employ low-order polynomials to approximate the probability of failure or safety index in terms of design variables in order to filter out noise and facilitate design optimization. These response surface approximations are called design response surface approximation (DRS) and are widely used in the RBDO (e.g., Sues et al. 1996). The probability of failure often changes by several orders of magnitude over narrow bands in design space, especially when the random variables have small coefficients of variation. The steep variation of probability of failure requires DRS to use high-order polynomials for the approximation, such as quintic polynomials (chapter 5), increasing the required number of probability calculations (Qu et al. 2000). An additional problem arises when Monte Carlo simulations (MCS) are used for calculating probabilities. For a given number of simulations, the accuracy of the probability estimates deteriorates as the probability of failure decreases. The numerical problems associated with steep variation of probability of failure led to consideration of alternative measures of safety. The most common one is to use the
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60 safety index, which replaces the probability by the distance, which is measured as the number of standard deviations from the mean of a normal distribution that gives the same probability. The safety index does not suffer from steep changes in magnitude, but it has the same problems of accuracy as the probability of failure when based on Monte Carlo simulations. However, the accuracy of probabilistic sufficiency factor is maintained in the region of low probability. The probabilistic sufficiency factor also exhibits less variation than probability of failure or safety index. Thus the probabilistic sufficiency factor can be used to improve design response surface approximations for RBDO. The next section introduces the probabilistic sufficiency factor, followed by the computation of the probabilistic sufficiency factor by Monte Carlo simulation. The methodology is demonstrated by the reliability-based beam design problem. Probabilistic Sufficiency Factor The deterministic equivalent of reliability constraint in RBDO can be formulated as ) ,(),(dxdxcrgg (6-1) where gr denotes a response quantity, gc represent a capacity (e.g., strength allowable), is usually the mean value vector of random variables, d is the design vector. The traditional safety factor is defined as x ),(),(),(dxdxdxrcggs (6-2) and the deterministic design problem requires (6-3) rss),(dx where sr is the required safety factor, which is usually 1.4 or 1.5 in aerospace applications. The reliability constraint can be formulated as a requirement on the safety factor
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61 rPs )1(Prob (6-4) where Pr is the required probability of failure. Birgers probabilistic sufficiency factor Psf is the s olution to Ps rP sf )(Probhows the probability density of the safety factor for a given design. The area under the curve left to s=1 represents the proresents the target proba (6-5) It is the safety factor that is violated with the required probability Pr. Figure 6-1 s bability that s<1, hence it is equal to actual probability of failure. The shaded area in the figure rep bility of failure, Pt. For this example, since it is the area left of the line s=0.8, Psf = 0.8. The value of 0.8 indicates that the target probability will be achieved if we reduced the response by 20 % or increased the capacity by 25% (1/0.8-1). For many problems this provides sufficient information for a designer to estimate the additional structural weight. For example, raising the safety factor from 0.8 to 1 of a stress-dominated linear problem typically requires a weight increase of about 20% of the weight of the overstressed components. Figure 6-1. Probability density of safety factor. The area under the curve left to s=1 measures the actual probability of failure, while the shaded area is equal to the target probability of failure indicating that probabilistic sufficiency factor = 0.8
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62 Using Probabilistic Sufficiency Factor to Estimate Additional Structural Weight to Satisfy the Reliability Constraint The following cantilever beam example (Figure 6-2) is taken from Wu et al. (2001) to demonstrate the use of probabilistic sufficiency factor. Figure 6-2. Cantilever beam subject to vertical and lateral beading s yielding, which is m There are two failure modes in the beam design problem. One failure mode i ost critical at the corner of the rectangular cross section at the fixed end of the beam )600600(),,,,(22XtwYwtRRtwYXRgS (6-6) where R is the yield strength, X and Y are the independent horizontal and vertical loads. Another failure mode is the tip deflection exceeding the allowable displacement, D0 2 2200wtEwtD 234),,,,(XYLDDDtwYXEg (6-7) whereRandom variables X Y R E E is the elastic modulus. The random variables are defined in Table 6-1. Table 6-1. Random variables in the beam design problem Distribution Normal (500, 100) lb Normal (1000, 100) lb Normal (40000,2000) Normal (29E6, 1.45E6) psi psi The cross sectional area is minimized subject to two reliability constraints, which e require the safety indices for strength and deflection constraints to be larger than thre
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63 (probability of failure less than 0.00135). The reliability-based design optimization problem, with the width w and thickness t of the beam as design variables that are deterministic, can be formulated as wtA minimize 05 0013 .0that (6-8) based on probability of failure, or p such wtA minimize 03 (6-9) based on safety index, where is the safety index, or thatsuch wtA minimize such 01sfPthat (6-10) based on the probabilistic sufficiency factor. Tthe above three forms, which are equivalently in terms of safety. The details of the beam design are given later in the paper. In order to demonstrate the utility of the P for estimating the required weight for correcting a safety deficiency, it is useful to see how the stresses and the displacements problem. If we have a given design with dimensions w0 and t0 and a Psf of Psf0, which is smaller than one, we can make the structure safer by scaling both w and t un he reliability constraints are formulated in sf depend on the weight (or cross sectional area) for this iformly by a constant c 00,cttcww (6-11) It is easy to check from (6-6) and (6-7) that the stress and the displacement will then change by a factor of c3, an d the area by a factor of c2. Since the Psf is inversely
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64 propo rtional to the most critical stress or displacement, it is easy to obtain the relationship 5.10)(APP (6-12) where A0=w0t0. This indicates that a one percent increase in area (corresponding to 0.5 percent increase in w and t) will improve the Psf by about 1.5 per 0Asfsfcent. Since non-uniform increases in the width and thickness may be mobe able to do better than 1.5 percent. Thus, if we have P=0.97, we can expect that we can make the structure safe with a weight increase under two percent. The probabilistic sufficiency factor gives a designer a measure of safety that can be used more readily than the probability of failure or the safety index to estimate the required weight increase to reach a target safety level.sf1302 (safety index re efficient than uniform scaling, we may sf The P of a beam design, presented in section 4 in details, is 0.9733 for a target probability of failure of 0.00135, (6-12) indicate that the deficiency in the Psf can be corrected by scaling up the area by a factor of 1.0182. Since the area A is equal to c2wt, the dimensions should be scaled by a factor c of 1.0091 (=1.01820.5) to w = 2.7123 and t = 3.5315. Thus the objective function of the scaled design is 9.5785. The probability of failure of the scaled design is 0.00 of 3.0110 and probabilistic sufficiency factor of 1.0011) evaluated by MCS with 1,000,000 samples. Such estimation is readily available using the probability of failure (0.00314) and the safety index (2.7328) of the design. Reliability Analysis Using Monte Carlo Simulation Let g(x) denote the limit state function of a performance criterion (such as strength allowable larger than stress), so that the failure event is defined as g(x) <0, where x is a random variable vector. The probability of failure of a system can be calculated as
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65 xxxXdfPgf0)()( (6-13) X where f(x) is the joint probability distribution function (JPDF). This integral is hard to evaluate, because the in and integration in high dimCarlo simulation (MCS) (e.g., Melchers 1999). Monte Carlo simulation is a good method to use for system reliability analysis with multiple failure modes. The present chapter achieve low relativccurate near the optimum. Furthermore, the probabilities of failure in some design regions may be so low that they are calculated as zero by MCS. tegration domain defined by g(x) < 0 is usually unknown, ension is difficult. Commonly used probabilistic analysis methods are either moment-based methods such as the first-order-reliability method (FORM) and the second-order-reliability method (SORM), or simulation techniques such as Monte focuses on the use of MCS with response surface approximation in RBDO. Monte Carlo simulation utilizes randomly generated samples according to the statistical distribution of the random variables, and the probability of failure is obtained by calculating the statistics of the sample simulation. Fig. 3 illustrated the Monte Carlo simulation of a problem with two random variables. The probability of failure of the problem is calculated as the ratio of the number of samples in the unsafe region over the total number of samples. A small probability requires a large number of samples for MCS to e error. Therefore, for fixed number of simulations, the accuracy of MCS deteriorates with the decrease of probability of failure. For example, with 106 simulations, a probability estimate of 10-3 has a relative error of a few percent, while a probability estimate of 10-6 has a relative error of the order of 100 percent. In RBDO, the required probability of failure is often very low, thus the probability (or safety index) calculated by MCS is ina
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66 This f lat zero probability of failure or infinite safety index cannot provide useful gradient information to the optimization. Figure 6-3. Monte Carlo simulation of problem with two random variables Calculation of Probabilistic Sufficiency Factor by Monte Carlo Simulation Here we propose the use of probabilistic sufficiency factor to solve the problems associated with probability calculation by MCS. Psf can be estimated by MCS as follows. Define the nth safety factor of MCS as )ix(1in )(sMimthnns (6-14) where M is the sample size of MCS, and the nth min means the nth smallest safety factor among actors M safety factors from MCS. Thus )(ns is the nth-order statistics of M safety f from MCS, which corresponds to a probability of n/M of )()(nss x. That is, w e seek to find the safr ety factor that is violated with the required probability P. The probabilistic sufficiency factor is then given as MPnsPrnsf for )( (6-15)
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67 For example, if the required probability Pr is 10-4 and the sample size of Monte Carlo simulation M is 106, Psf is equal to the highest safety factor among the 100 samples mpy factor ubility constraints, the most critical safety factor is calculated first for each Monte Carlo sample, (n=PrM) with the lowest safety factors. The calculation of Psf requires only sorting the lowest safety factors in the Monte Carlo sales. While the probability of failure changes by several orders of magnitude the probabilistic sufficiencsually varies by less than one order of magnitude in a given design space. For problems with k relia ickiggnmis1 )(ix (6-16) ththth riThen the sorting of the n minimum safety factor can be proceeded as in (6-14). When n is small, it may be more accurate to calculate Psf as the average between the n and (n+1) lowest safety factor in the Monte Carlo samples. The probabilistic sufficiency factor provides more information than probability of failure or safety index. Even in the regions where the probability of failure is so small that it cannot be estimated accurately by the MCS with given sample size M, the accuracy of Psf is maintained. Using the probabilistic sufficiency factor also gives designers useful insights on how to change the design to satisfy sa2.1. The estimate is not readily available from the probability of failure or the safety index. The probabilistic sufficiency factor is based on the ratio of allowable to response, which exhibits much less variation than the probability of failure or safety index. Therefore, approximating probabilistic sufficiency factor in design optimization is easier than approximating probability of failure or safety index as discussed in the next section. fety requirements as shown in section
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68 Monte Carlo Simulation Using Response Surface Approximation Monte Carlo simulation is easy to implement, robust, and accurate with sufficiently large samples, but it requires a large number of analyses to obtain a good estimate of small failure probabilities. Monte Carlo simulation also produces a noisy response and hence is difficult to use in optimization. Response surface approximations solve the two problems, namely simulation cost and noise from random sampling. Response surface approximations fit a closed-form approximation to the limit state function to facilitate reliability analysis. Therefore, response surface approximation is as those requiring comp coefficient vector estime Point (MPP) that contributes most to the probability of failure oent particularly attractive for computationally expensive problems such lex finite element analyses. Response surface approximations usually fit low-order polynomials to the structural response in terms of random variables bxxTZg)()( (6-17) where)(xgdenotes the approximation to the limit state function g(x), Z(x) is the basis function vector that usually consists of monomials, and b is the ated by least square regression. The probability of failure can then be calculated inexpensively by Monte Carlo simulation or moment-based methods using the fitted polynomials. Response surface approximations (RSA) can be used in different ways. One approach is to construct local RSA around the Most Probabl f the structure. The statistical design of experim (DOE) of this approach is iteratively performed to approach the MPP on the failure boundary. For example, Bucher and Bourgund (1990), and Sues (1996, 2000) constructed progressively refined local RSA around the MPP by an iterative method. This
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69 local RSA approach can produce satisfactory results given enough iterations. Another approach is to construct global RSA over the entire range of random variables, i.e., desigurface can be written as n of experiment around the mean values of the random variables. Fox (1993, 1994, 1996) used Box-Behnken design to construct global response surfaces and summarized 12 criteria to evaluate the accuracy of RSA. Romero and Bankston (1998) employed progressive lattice sampling as the design of experiments to construct global RSA. With this approach, the accuracy of response surface approximation around the MPP is unknown, and caution must be taken to avoid extrapolation near the MPP. Both approaches can be used to perform reliability analysis for computationally expensive problems. The selection of RSA approach depends on the limit state function of the problem. The global RSA is simpler and efficient to use than local response surface approximation for problems with limit state function that can be well approximated globally. However, the reliability analysis needs to be performed and hence the RSA needs to be constructed at every design point visited by the optimizer, which requires a fairly large number of response surface constructions and thus limit state evaluations. The local RSA approach is even more computationally expensive than the global approach in the design environment. Qu et al. (2000) developed a global analysis response surface (ARS) approach in unified space of design and random variables to reduce the number of RSA substantially and achieve higher efficiency than the previous approach. This analysis response s bdxdxTZg),(),( (6-18)
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70 x and d are the random variable and design variable vectors, respectively. They recommended Latin Hypercube sampling as the statistical design of experiments. The number of response surface approximations constructed in optimization process is reduced substantially by introducing design variables into the response surface approximation formulation. The selection of RSA approa ch depends on the limit state function of the problem and target probability of failure. The global RSiterative RSA is desirable for general reliability analysis probl is the cumulative distribution function of normal distribution, improves the accur A approach is more efficient than local RSA, but it is limited to problems with relatively high probability or limit state function that can be well approximated by regression analysis based on simple basis functions. To avoid the extrapolation problems, RSA generally needs to be constructed around important region or MPP to avoid large errors in the results of MCS induced by fitting errors in RS. Therefore, an em. Design response surface approximations (DRS) are fitted to probability of failure to filter out noise in MCS and facilitate optimization. Based on past experience, high-order DRS (such as quintic polynomials) are needed in order to obtain a reasonably accurate approximation of the probability of failure. Constructing highly accurate DRS is difficult because the probability of failure changes by several orders of magnitude over small distance in design space. Fitting to safety index =-1(p), where p is the probability of failure and acy of the DRS to a limited extent. The probabilistic sufficiency factor can be used to improve the accuracy of DRS approximation.
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71 Beam Design Example The details of the beam design problem mentioned in section 2 are presented here. Since the limit state of the problem is available in closed form as shown by (6-6) and (6-7), the direct Monte Carlo simulation with a sufficient large number of samples is used here (without analysis response surface) in order to in order to better demonstrate the advantage of probabilistic sufficiency factors ov er probability of failure or safety index better. By using the exact limit in the results of Monte Carlo simulFCCCD, Khuri and Cornell 1996)sampling can be used to construct higher order response surface (Qu et al. 2000). We state function, the errors ation are purely due to the convergence errors, which can be easily controlled by changing the sample size. In applications where analysis response surface approximation must be used, the errors introduced by approximation can be reduced by sequentially improving the approximation as the optimization progresses. The reliability constraints, shown by (6-8) to (6-10), are approximated by design response surface approximates that fit to probability of failure, safety index, and probabilistic sufficiency factor. The accuracy of the design response surface approximations is then compared. The design response surface approximations are in two design variables w and t. A quadratic polynomial in two variables has six coefficients to be estimated. Since Face Center Central Composite Design ( is often used to construct quadratic response surface approximation, a FCCCD with 9 points was employed here first with poor results. Based on our previous experience, higher-order design response surface approximations are needed to fit the probability of failure or the safety index, and the number of points of a typical design of experiments should be about twice the number of coefficients. A cubic polynomial in two variables has 10 coefficients that require about 20 design points. Latin Hypercube
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72 found that Latin Hypercube sampling might fail to sample points near some corners of the design space, leading to poor accuracy around these corners. To deal with this extrapolation problem, all four vertices of the design space were added to 16 Latin Hypercube sampling points for a total of 20 points. Mixed stepwise regression (Myers and Montgomery 1995) was employed to eliminate poorly characterized terms in the response surface models. Design with Strength Constraint The range for the design response surface, shown in Table 6-2, was selected based on the mean-based deterministic design, w = 1.9574" and t = 3.9149". The probability of failure was calculated by direct Monte Carlo simulation with 100,000 samples based on the exact stress in (6-6). Table 6-2. Range of design variables for design response surface System variables w t Range 1.5" to 3.0" 3.5" to 5.0" Cubic design response surfaces with 10 coefficients were constructed and their statistics are shown in Table 6-3. An R2 close to one and an average percentage error (defined as the ratio of root mean square error (RMSE) predictor and mean of response) close to zero indicate good accuracy of the response surfaces. It is seen that the design response surfaces for the probabilistic sufficiency factor has the highest R2adj and the srcentage er The standard error in probility calculated by Monte Carlo simulation can be estima adj mallest average pe ror. ba ted as M pp)1(where p is the probability of failure, and M is the sample size of the Monte Carlo p (6-19)
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73 simulation. If a probability of failure of 0.2844 is to be calculated by Monte Carlo simulation of 100,000 samples (the mean probability of failure in Table 6-3), the standard error due to the limited sampling is 0.00143. The RMSE error of the probability design response surface is of 0.1103. Thus t he error induced by the limited sampling (100,000) is much smaller than error of the response surface approximation to the probability of failure. Table 6-3. Comparison of cubic design response surface approximations of probability of failure, safety index and probabilistic sufficiency factor for single strength failure mode (based on Monte Carlo simulation of 100,000 samples) 16 Latin Hypercube sampling points + 4 vertices Error Statistics Probabilistic RS Probability RS Safety index RS sufficiency factor R2adj 0.9228 0.9891 0.9999 RMSE Predictor 0.1103 0.3027 0.002409 Mean of Response 0.2844 1.9377 1.0331 APE (Average ErroPredicResponse) (=RMSE Predictor of Pof/Mean of Pof) Percentage r=RMSE tor/Mean of 38.78% 15.62% 0.23% APE in Pof 38.78% 12.04% N/A The probabilistic sufficiency factor designse surface haerage error less than onhile the sx desise surface has an average error of about 15.6 percent. It must be noted, however, that the average percent errors of the three d surfatlyd, because oent error in probabilistic sufficiency factor does not correspond to one percent error in probability of faiindex design response surface were transformed to errorability as shown in Table 6-3. It is seen that safety index design respon s an av e percent, w afety inde gn respon esign response ce cannot be direc compare ne perc lure or safety index. Errors in safety s in terms of prob
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74 re sponse surface approximation is more accurate than the probability design response surface approximation. Besides the average errors over the design space, it is instructive to compare errors measured in probability of failure in the important region of the design space. For optimization problems, the important region is defined as the region containing the optimum. Here it is the curve of target reliability according to each design response surface, on which the reliability constraint is satisfied critically, and the probability of failure should be 0.00135 if design response surface approximation does not have errors. For each design response surface approximation, 11 test points were selected along a curve of target reliability and given in the Appendix. The average percentage errors at these test points, shown in Table 6-4, demonstrate the accuracy advantage of the probabilistic sufficiency factor approach. For the target reliability, the standard error due to Monte Carlo simulation of 100,000 samples is 8.6%, which is comparable to the response surface error for the Psf. For the other two response surfaces, the errors are apparently dominated by the modeling errors due to the cubic polynomial approximation. Table 6-4. Averaged errors in cubic design response surface approximations of points on the curves of target reliability Surface of failure sufficiency factor probabilistic sufficiency factor, safety index and probability of failure at 11 Design Response Probability of Safety Index (Pof) Probabilistic Average Percentage of Failure Error in Probability 213.86% 92.38% 10.32% The optima found by using the design response surface approximations of Table 6-3 are compared in Table 6-5. The probabilistic sufficiency factor design response surface clearly led to a better design, which has a safety index of 3.02 according to Monte Carlo simulation. It is seen that the design from probabilistic sufficiency factor design response
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75 surface approximation is very close to the exact optimum. Note that the valuesility based optimum and safety index ba of Psf for the probabsed optimum provide a good estimate to the ght increments. For examplehe design has a safety factor shortfall of 3.37 percent, uquire ore than 2.25 percent weight increment to remedy the problem. Indeed the optimum design is 2.08 percent heavier. required wei with a Psf=0.9663 t safety index based indicating that it sho ld not re m This would have been difficult to infer from a probability of failure of 0.00408, which is three times larger than the target probability of failure. Table 6-5. Comparisons of optimum designs based on cubic design response surface approximations of probabilistic sufficiency factor, safety index and probability of failure Minimize objective function F while 3 or 0.00135 pof Design response function F=wt from MCS of 100,000 samples surface of Optima Objective Pof/Safety index/Safety factor Probability 9.2225 0.00690/2.4624/0.9481 w=2.6350, t=3.5000 Safety index 9.3258 0.00408/2.6454/0.9663 9.5367 0.00128/3.0162/1.0021 9.5204 0.00135/3.00/1.00 w=2.6645, t=3.5000 Probabilistic sufficiency factor w=2.4526, t=3.8884 Exact optimum (Wu et al. 2001) w=2.4484, t=3.8884 Design with Strength and Displacement Constraints system reliability problem with strength and displacement constrain For ts, the pure is 0 sed on thetress lace allowable tip displacemchoe 2.25" impeting cu et al. 2hredesign restions in the range of design variables shown in Table 6-2 were constructed and their statistics are shown in Table 6-6. robability of failamples bas calculated by direct Monte Carlo simulation with 100,00 exact s and exact disp ement in (6-6) and (6-7). Th ent D0 is sen to b n order to have two co onstraints (W 001). The t e cubic ponse surface approxima
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76 Trisonigne surface e first teratbabiilure, safety index and probabilistic sufficiency factor for system reliability (strength and displacement) sampling points + 4 vertices able 6-6. Compadesign i of cubic desion for pro responslity of fa approximations of th 16 Latin Hypercube Error Statistics surface ty index surface Probabilistic response surface Probability Safe response response sufficiency factor R2adj 0.9231 0.9887 0.9996 RMSE Predictor 0.1234 0.3519 0.01055 APE (Average Percentage Predictor/Mean of Response) (=RMSE Predictor of 32.14% 10.51% N/A Mean of Response 0.3839 1.3221 0.9221 Error=RMSE 32.14% 26.62% 1.14% APE in Pof Pof/Mean of Pof) It is seen that the R2adj of probabilistic sufficiency factor response surface approximawhich implies probabilistic sufficiency factost a of averaged ee entire ce as sh. The critical errors of the three design response sre also ch dace approx 51 test pre selected along a curve of target reliability (probabof f.00135). ge perceors at these test points, shown in Table 6-7, demonstrate that the probabilistic sufficiency factor de approxiore accurate than the proband safety index response surface approximations. Table tion is the highest among the three response surface approximations, or design response surface approximation is the m ccurate in terms rrors in th design spaa own by Table 6-2. For eac urfaces ompared esign response surf imation, oints we ility ailure = 0 The avera ntage err esign response surfac mation is m ability of failure 6-7. Averaged errors in cubic design response surface approximations of probabilistic sufficiency factor, safety index and probability of failure at 51 points on the curves of target reliability Design Response Surface of Probability of failure Safety Index (Pof) Probabilistic sufficiency factor Average Percentage of Failure Error in Probability 334.78% 96.49% 39.11%
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77 The optima found by using the design response surface approximations of Table 6-6 are compared in Table 6-8. The probabilistic sufficiency factor design response surface led to a better design than the probability or safety index design response surface in terms of reliability. The probability of failure of the Psf design is 0.00314 evaluated by Monte Carlo simulation, which is higher than the target p robability of failure of 0.00135. The babilistic sufficiencymation. The probabilistic sufficiency factor to estimate tonal welia sc.7123 a315 was ob section 2.1. ctive functiMinimize objective function F while 3 or 0.00135 pof deficiency in reliability in the Psf design is induced by the errors in the pro factor design response surface approxi can be used he additi ight to satisfy the re bility constraint. A aled design of w = 2 nd t = 3.5 tained in The obje on of the scaled design is 9.5785. The probability of failure of the scaled design is 0.001302 (safety index of 3.0110 and probabilistic sufficiency factor of 1.0011) evaluated by MCS with 1,000,000 samples. Table 6-8. Comparisons of optimum designs based on cubic design response surface approximations of the first design iteration for probabilistic sufficiency factor, safety index and probability of failure Design response Optima Objective function F=wt Pof/Safety index/Safety factor from MCS of 100,000 samples surface of Probability t=3.5000 9.3069 0.00522/2.5609/0.9589 w=2.6591, Safety index t=3.5000 9.2654 0.00630/2.4949/0.9519 sufficiency factor t=3.500 9.4084 0.00314/2.7328/0.9733 w=2.6473, Probabilistic w=2.6881, nse surface by shrinking the design space around the current design. The reduced range of design response surface approximations is shown in Table 6-9 for the next design iteration. The design response surface approximations The design can be improved by performing another design iteration, which would reduce the errors in design respo
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78 constructed are compared in Table 6-10. It is observed again that the probabil factor response surface approximation istic sufficiencyis the most accurate. Tf designcond design System variables able 6-9. Range ose variables for design response surface approximations of the n iteratiow t Range 2.2" to 3.0" 3.2" to 4.0" Table 6-10. Comparisosigse surfacecond ilure, safety index and probabilistic ncy steility (stren16 Latin Hypercube sampling points + 4 vertices n of cubic de n respon approximations of the se design iteration for probability of fasufficie factor for sy m reliab gth and displacement) Erro r Statistics Probability response surface Safety index response surface Probabilistic sufficiency factor response surface R2adj 0.9569 0.9958 0.9998 RMSE Predictor 0.06378 0.1329 0.003183 Mean of Response 0.1752 2.2119 0.9548 APE (Average Percentage Error=RMSE Predictor/Mean of Response) 36.40% 6.01% 0.33% The optima based on design response surface approximations for the second design iteration shown in Table 6-10 are compared in Table 6-11. It is see n that the design e due to its superior accuracy over the probability of failure and safety index design sponse surfaces. Table 6-11.es of index and probability of failure Minimizenction or converges in two iterations with probabilistic sufficiency factor response design surfac re Comparisons of optimum designs based on cubic design response surfacthe second design iteration for probabilistic sufficiency factor, safety objective fu F while 3 0.00135 pof Design response surface of Optima e n F=wt fety in Objectivfunctio Pof/Safrom MCS of 100,000 samples dex/Safety factor Probability w=2.7923, t=3.3438 511/2.56 9.3368 0.00 83/0.9658 Safety index t=3.5278Probabilistic sufficiency w=2.6041, w =2.6878, 177/2.910 factor t=3.6746 9.5691 0.00130/3.0115/1.0009 9.4821 0.00 65/0.992
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79 Summary This chapter presented a probabilistic sufficiency factor as a measure of the safety level relative to a target safety level, which can be obtained from the results of Monte Carlo simulation with little extra computation. It was shown that a design response surface approximation can be m ore accurately fitted to the probabilistic sufficiency factor single or s was demonstrated that the design response stion based ond accelerates the convergence of relisign sufficiency factor a more information in regions of such low probability that t of ffety index cannot be estimated by Monte Carlo simulation wple size, which is h guiding thes shown that the probabilistic sufficiency factorloyed by the designer to estimate the requir than to the probability of failure or the safety index. Using the beam design example with ystem reliability constraints, it urface approxima probfficiency abilistic suability-based de factor has superior accuracy an optimization. The probabilistic lso providesailure or sa he probabilityith a given sam elpful in optimizer. Finally it wa can be emp ed additional weight to achieve a target safety level, which might be difficult with probability of failure or safety index.
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CHAPTER 7 RELIABILITY-BASED DESIGN OPTIMIZATION USING DETERMINISTIC OPTIMIZATION AND MULTI-FIDELITY TECHNIQUE Introduction The probabilistic sufficiency factor (PSF) developed in Chapter six is integrated to reliability-based design optimization (RBDO) framework in this chapter. The classical RBDO is performed in coupled double loop fashion, where the inner loop performs reliability analysis and the outer loop performs design optimization. RBDO using double loop framework requires many reliability analyses and is computationally expensive. Wu et al. (1998, 2001) developed a safety-factor based approach for performing RBDO in a decoupled single loop fashion, where the reliability constraints are converted to equivalent deterministic constraints by using the concept of safety factor. The similarity between Wu's approach and the probabilistic sufficiency factor approach indicates that it may be worthwhile to study the use of probabilistic sufficiency factor converting RBDO to sequential deterministic optimization. For many problems the required probability of failure is very low, so that good estimates require a very large MCS sample. In addition, the design response surface (DRS) must be extremely accurate in order to estimate well a very low probability of failure. Thus we may require an expensive MCS at a large number of design points in order to construct the DRS. A multi-fidelity technique using probabilistic sufficiency factor for RBDO is investigated to alleviate the computational cost. The two approaches of reducing computational cost of RBDO for low probability of failure are compared. 80
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81 Reliability-Based Design Optimization Using Sequential Deterministic Optimization with Probabilistic Sufficiency Factor Wu et al. (1998, 2001) proposed a decoupled approach using partial safety factor to replace reliability constraints by equivalent deterministic constraints. After performing reliability analsed values x*, which is the most probable point (MPous reliability analysis. The required shift een Wusblems where the design space is cha ysis, the random variables x are replaced by safety-factor ba P) of the previ of limit state function g in order to satisfy the reliability constraints is s, which satisfy P(g(x)+s)<0)=Pt. Both x* and s can be obtained as the byproducts of reliability analysis. The target reliability is achieved by adjusting the limit state function via design optimization. It is seen that the required shift s is similar to the probabilistic sufficiency factor (Qu and Haftka 2003) presented in Chapter six. The significant difference betw partial safety factor and coupled RBDO is that reliability analysis is decoupled from and driven by the design optimization to improve the efficiency of RBDO. Thus RBDO is performed in a deterministic fashion and corrected by reliability analysis after optimization. The PSF is employed in this chapter to convert RBDO to equivalent deterministic optimization. Converting RBDO to equivalent deterministic optimization enables further exploration of the design space for those pro racterized to have multiple local optima and only limited number of analyses are available due to its high computational cost, such as design of stiffened panels addressed in chapter eight. By starting from a mean value based design, where the deterministic safety factor is one, an initial design was found by deterministic optimization. Reliability analysis using Monte Carlo simulation shows the deficiency in probability of failure and probabilistic
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82 sufficiency factor. In the next design iteration, the safety factor of the next deterministic optimization is chosen to be )()1(),(),(kkPssdxdx (7-1) )(sfkwhich is used to reduce the yield strength of the material, R. The optimization problem is formulated as wtA minimize 0)d,x(1)(ksthatsuchThe process is repeated until the optimum converges and the reliability constraint is satisfied. Reliability-Based Design Optimization Using Multi-Fidelity Technique with Probabilistic Sufficiency Factor For problems with very low probability of failure, a good estimate of probability requires a very large MCS sample. In addition, the DRS must be extremely accurate in order to estimate well a very low probability of failure. Thus we may require an expensive MCS at a large number of design points in order to construct the DRS. The deterministic optimization may be used to reduce the computational cost associated with RBDO for very low probability of failure. However, since it does not use any derivative information for th R (7-2) e probabilities, it is not likely to converge to an optimum design when comphigher than the required probability Pr and can be estimated via a less expensive MCS eting failure modes are disparate in terms of the cost of improving their safety. A compromise between the deterministic optimization and the full probabilistic optimization is afforded by the Psf by using an intermediate target probability PI, which is
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83 and less accurate DRS. Then the Psf can re-calibrated by a single expensive MCS. This is a variable-fidelity techniqu e, with a large number of inexpensive MCS combined with a small number of expensive MCS. A compromise between the deterministic optimization and the full probabilistic rget probability PI, which is higher than the required probability Pr and ss accurate DRS. Then the Psf can re-calibrated by a single expensive MCS. This is a vnumber of inexpensive MCS combined with a small number of expensive MCS. of 0.0000135, and using as intermediate probability 0.00135, the value used as reum desiglude the h optimization is afforded by the probabilistic sufficiency factor Psf by using an intermediate ta can be estimated via a less expensive MCS and le ariable-fidelity technique, with a large For the beam example we illustrate the process by setting a low required probability quired probability for the previous examples. We start by finding an initial optim n with the intermediate probability as the required probability. This involves the generation of a response surface approximation of PSfI for the intermidiate probability as well as finding the optimum based on this response surface. We then perform an expensive MCS which is adequate for estimating the required probability. Here we use MCS with 107 samples. We now calculate the PSf from this accurate MCS, and denote it PSfA. At that design the PSfI predicted by the response surface approximation is about 1, because the initial optimization was performed with a lower limit of 1 on the PSf. In contrast, the accurate PSfA will in general be different for several reasons. These inc igher accuracy of the MCS, the response surface errors, and most important the lower probability requirements. For example, with 107 samples, at this initial design we may get PSfI=1.01 for the intermediate probability (based on the 13500 lowest safety
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84 factors) and PSfA=0.89 for the required probability (based on the 135 lowest safety factors). With a value of of PSfI and PSf A at the same point, we can define a scale factor f, as the ra tio of these two numbers I sfIAsfPPThis ratio can be used to correct the response surface approximation during the optimization process. Once an optimum design is found with a given f, a new accurate MCS can be calculated at the optimum, a new value of f can be calculated from E f (7-3) quation (7-3) at the new point, and the process repeated until convergence. As further refinement, we have also updated the response surface for the intermediate probability, centering it about the new optima. Beam Design Example The following cantilever beam example (Figure 7-1) is taken from Wu et al. (2001) to demonstrate the use of probabilistic sufficiency factor. Figure 7-1. Cantilever beam subject to vertical and lateral beading There are two failure modes in the beam design problem. One failure mode is yielding, which is most critical at the corner of the rectangular cross section at the fixed end of the beam
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85 )600600(),,,,(XYRRtwYXRg (7-4) 22twwtSwhere R is the yield strength, X and Y are the independent horizontal and vertical loceeding the allowable displacement, D0 ads. Another failure mode is the tip deflection ex 2 22wtwt 23004),,,,(XYELDDDtwYXEgD (7-5) where E is the elastic modulus. The random variables are defined in Table 7-1. Table 7-1. Random variables in the beam design problem variables Random X Y R E Distribution (40000,2000) (29E6, 1.45E6) Normal (500, 100) lb Normal (1000, 100) lb Normal psi Normal psi The cross sectional area is minimized subject to two reliability constraints, which require the safety indi ces for strength and deflection constraints to be larger than three (probability of failure less than 0.00135). Tty-based design optimization problem, with the width w and thickness t of the beam as design variables that are deterministic, can be formulated as he reliabili wtA minimize 000135.0pthatsuch (7-6) based on probability of failure, or wtA minimize (7-7) which are equivalently in terms of safety. The details of the beam design are presented in Chapter 6. thatsuch 01sfPbased on the probabilistic sufficiency factor. The reliability constraints are formulated in the above forms
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86 The optimum design for the case with strength failure musing probabilistic sufficiency factor and Wus exact optimum are shown in Table 7-2. The probability of failure is calculated by direct MCS with 100,000 samples based on the exact stress in Equation (7-1). It is seen that the design from probabilistic sufffactor DRS is very close to the exact optimum. Table 7-2. Optimum designs for strength failure mode obtained from double loop RBDO f ode obtained in Chapter 6 iciency Minimize objective function F while 0.00135 po DRS of Optima Objective function F=wt PofE /Safety factor from MCS of xact stress Probabilissufficiency tic factor w=2.4526, t=3.8884 9.5365 0.00128/1.0021 Exact optimum(Wu et al., 20 01) 9.5 w=2.4484 t=3.8884 204 0.00135/1.00 Reliability-Based Design Optimization Using Sequential Deterministic Optimization based on sequential deterministic optimization with probabilistic sufficiency factor under strength constraint for target probability of failure of 0.00135 Minimize objective fun with Probabilistic Sufficiency Factor To verify the validity of the proposed method, converting RBDO to sequential deterministic optimization using probabilistic sufficiency factor for target probability of failure of 0.00135 is performed (Table 7-3). Table 7-3. Design history of RBDO ction F while 3 or 0.00135 pof Probabilistic sufficiency factor Optima Objective function F=wt Pof/Safety index/Safety factor from MCS of 105 samples Initial design w=1.9574, (s1=1.0) t= 3.9149 7.6630 0.49883/0.00293/0.7178 s2=s1/0.7177 =1.3932 w=2.1862, t=4.3724 9.5589 0.00140/2.98889/0.9986 s3=s2/0.9986 =1.3951 w=.1872t=4.3744 9.5676 0.00130/3.01145/1.0006 2, It is seen that the final design has a slightly higher weight than the optimum in Table 7-2. The reason is that this method employs a safety factor based on the mean
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87 valuee to multiple DRS construction and attractive. The beam design with strraint with 0.0000135 is shown in Table 7-4. It is seen that the method provides an acceptable design w he total computational cost is two deterministic optimprobabilistic snt for target probability s of the random variables. Better prediction can be achieved by using a MPP based safety factor. When the target probability failure is very low, the RBDO using double loop is computationally prohibitive. RBDO using coarse MCS and multi-fidelity technique can also be computationally very expensive du corrections. Converting RBDO to deterministic optimization becomes computationally ength const a target probability of failure of ithin one desiations, each follo gn iteration. Twed by a reliability analysis. iz Table 7-4. Design history of RBDO based on sequential deterministic optimization with ufficiency factor under strength constrai of failure of 0.0000135 Minimize objective function F while 3 or 0.00135 pof Probabilistic function F=wt 107 samples sufficiency factor Optima Objective Pof/ Safety factor from MCS of Initial w=1.9574, design 7.6630 0.5001155/ 0.6352549 (s1=1.0) t= 3.9149 s2=s1/ 0.63525410.3698 0.0000130/1.0006877 9=w=2.2770, t=4.5541 1.5741712 ability-Bsign OptimCoac Sufficiency Factor design with a target probability of failure of 0.0000135 was repeated he ponse snd the optimere as initial design. The design process and ranges of DRS are shown in Table 7-5 and 7-6. It is seen Reli ased De ization Using rse MCS with Probabilisti The beam ere, so that th revious resp urface a um in Table 7-2 is used h that the target reliability is achieved in two design iterations of low fidelity DRS
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88 updated by two high fidelity reliability analysis. The total computational cost is three low fidelity design response surface construction and three high fidelity reliability evaluation. under strength constraint Converting RBDO to sequential deterministic optimization is computationally more efficient than multi-fidelity technique for problem with low probability of failures. However, it is seen in Table 7-4 and Table 7-5 that the design obtained by the latter approach is more reliable and lighter than the previous one. Table 7-5. RBDO using variable fidelity technique with probabilistic sufficiency factor Minimize objective function F while 4.1974 or 0.0000135 pof Probabilistic factor Optima Objective function F=wt Pof/Safety factor from MCS of 107 samples (PsfA) Sufficiency Initial design (f1=1.0) w=2.4526, t=3.8884 9.5367 0.0012350/0.891938 f2=0.891938 w=2.2522, t=4.6000 10.3600 0.0000160/0.996375 6375w=2.4071, t=4.3000 f3=f2*0.99=0.88704 10.3510 0.0000108/1.003632 Tnge of design variables for design response variablw t able 7-6. RaSystem surface es Range for f2 D2.0" to 3.0" 3.6" to 4.6" RS Range f or f3 D1.7" RS to 2.7" 4.3" to 5.3" he ufficiency factor represents a factor of safety relative to a targily than probability of failure or safeers or designers to estimate the required weight increase to reach a target Summary Since t probabilistic s et probability of failure. It provides a measure of safety that can be used more readty index by optimiz safety level. The RBDO is converted to equivalent deterministic optimization using probabilistic sufficiency factor. The probabilistic sufficiency factor also provides more information in the region of such low probability that probability of failure or safety index cannot be estimated by
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89 MCS given sample size, which is helpful in guiding the optimizer. In order to reduce the computational cost of RBDO, especially problems with very low probability of failure, a multiach is computationally more f oion in -fidelity based RBDO approach using probabilistic sufficiency factor was proposed, which can find satisfactory design with low probability of failure by using a correction factor to a response surface created by low-accuracy Monte Carlo Simulation. For problem with low probability of failure, the sequential appro efficient than the multi-fidelity approach, but the design is more efficient in term o bjective funct the multi-fidelity approach.
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CHAPTER 8 RELIABILITY-BASED DESIGN OPTIMIZATION OF STIFFENED PANELS USING PROBABILISTIC SUFFICIENCY FACTOR This chapter studies reliability-based designs of isogrid stiffened panels with random uncertainties in material properties and structural dimensions. The PANDA2 program used for the analysis provides information only on a critical subset of the multiple failure modes in the stiffened panel s. Therefore, we cannot follow each individual failure mode, and polynomial response surface approximation (RSA) are fit to the most critical safety margins. Probability of failure is calculated by Monte Carlo simulation (MCS) using these response surface approximation. A probabilistic sufficiency factor approach (Qu and Haftka, 2003) is employed to facilitate the design optimization. Introduction Stiffened panels are often used in aircraft and launch vehicle design to obtain lightweight structures with high bending stiffness. The design optimization of stiffened panels under buckling and strength constraints is characterized by a large number of local optima (Lamberti et al. 2003). Some of these designs are more sensitive to uncertainties than others (Singer et al., 2003). Therefore, it is important to provide designers with reliability-based optimum designs. Here, we consider uncertainties due to variability in material properties and geometric variations induced by the manufacturing process. Noor et al. (2001) investigated the variability in the nonlinear response of stiffened panels associated with 90
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91 variations in the material and geometric parameters. Fuzzy set analysis was employed to model the uncertainties in the response. Buckling loads of the thin walled stiffened panels are highly sensitive to geometric imperfectio1999) applied a convex conveculate pre-buckling, buckling and post-buckling responses with highly efficient analysis. PANDA2 also provterministic global optimization based on mulobal, local, inter-ring, and general ovalization of cylind ns. Elseifi et al. ( model to represent the worst-case geometric imperfection. The results obtained with this x model were compared to the traditional probabilistic models used to account for uncertainties in imperfections. This chapter presents reliability-based designs of isogrid stiffened panels. The problem is to minimize the weight of the stiffened panel subject to a reliability constraint. The reliability-constraint is evaluated by Monte Carlo simulation, which requires a large number panel analyses. The PANDA2 software (Bushnell, 1987) is employed to analyze the stiffened panels. PANDA2 uses a combination of approximate physical models, exact closed form (finite-strip analysis) models and 1-D discrete branched shell analysis models to cal ides limited de ltiple starting points strategy. Under compressive loads, the load carrying capacity of stiffened panels is greatly affected by geometric imperfections due to fabrication. The effects of geometric imperfections are taken into account in PANDA2 software directly by modifying the effective radius of a cylindrical panel and indirectly by redistributing the pre-buckling stress resultants over the various segments of the panel. Various geometric imperfections, such as g rical panels are considered, with a sophisticated methodology to identify the most detrimental imperfections. (Bushnell, 1996).
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92 Even with the low computational cost of PANDA2 analyses, they cannot be used directly in the MCS that requires a large number of analyses (thousands to millions). Instead, response surface approximations are employed. For the reliability-based design optimization, analysis response surface approximation (ARS) is fitted to the most critical margin in the isogrid panel in terms of both geometric design variables and the material properties, both of which are modeled by random variables. Using the ARS, the proba directions to the circumferential directions as shown in Figure 1. The tank barrel to be optimized is stiffened externally with J-shaped ring stiffeners, and internally with a blade-shaped isogrid oriented bility of failure at every design point can be calculated inexpensively by MCS based on the fitted polynomials. A design response surface approximation (DRS) is then fit to the probability of failure in order to filter out noise generated by MCS. The details of reliability analyses and design optimization using Monte Carlo simulation combined with response surface approximation (Qu et al., 2000) were introduced chapter three. Due to the high nonlinearity of probability of failure and safety index in the design space, a probabilistic sufficiency factor approach (Qu and Haftka, 2003) is used instead of the probability of failure in the design optimization, as shown in detail in chapter six. Example problems of the reliability-based design of isogrid stiffened panels are presented. Aluminum Isogrid Panel Design Example An isogrid panel design problem is taken from Lamberti et al. (2003) to demonstrate the reliability-based design methodology. Reliability-Based Design Problem Formulation Isogrid stiffened panel are cylindrical shells that have rectangular blade stiffeners positioned along the circumferential and
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93 circumpsi critical for buckling. Figure 8-1. Isogrid-stifd external rings with isogrid pattern oriented along circumferential direction for increased bending bject to the constraint that the ower than a given requirement. The reliab ferentially. The length, L1, of the tank barrel is 300 in, and the radius, r, is 160 in. A half cylindrical tank is considered due to symmetry. The design variables are the isogrid spacing, b, the isogrid blade height, h, and the thicknesses the skin, t1, and the thickness of the isogrid blades, t2. The geometry of the ring is fixed. Following Lamberti et al. (2003) the panel is designed subject to two load cases internal proof pressure of 35 psi critical for strength axial compressive load Nx = 1000 lb/in, with an internal (stabilizing) pressure of 5 fened cylindrical shell with internal isogrid an stiffness in hoop direction The weight W of the panel is minimized su probability of failure P of the panel must be l ility-based optimization of stiffened panels is formulated as ),,,(21tthbWW minimize u P P thatsuch (8-1)
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94 where b, h, t1 and t2 are the design variables; W is the weight of the panel; Pu=10-4 is the limit on the probability of failure, P. The four design variables are b, h, t1 and t2. Geometric imperfections such as global, local, inter-ring, and general ovalization of cylindrical panels are taken into account in PANDA2 software directly by specifying the geometric imperfection amplitudes as shown in Table 81. ftware OBuckling modal general Table 8-1. Amplitudes of geometric imperfection handled by PANDA2 so ut of roundness (in) amplitude (in) Initial local amplitude (in) 0.8 0.8 0.008 Uncertainties The material used in the study is Al 2219-T87, which has a density of 0.1 lb/in3. Two for elastic properties (Youngs modulus E and Poissons Ratio ) and the strength allowable a are assumed to be normally distributed and uncorrelated random variables with mean values and coefficients of variation shown in Table 8-2. Table 8-2. Uncertainties in material properties (Al 2219-T87) modeled as normal random variables Youngs Modulus (E) Poisson Ratio () Stress Allowable (a) Mean value 0.107108 psi 0.34 0.58105 psi Coefficient of variation 0.03 0.03 0.05 Table 8-3. ing process modeled as uniformly distributed random design variables around design (mean) value 12 Uncertainties in manufactur b h t t Percentage 1% 2% 4% 4% variation The four design variables (b, h, t1 and t2) are also random around design (mean) value due to manufacturing uncertainties. They are assumed to be uniformly distributed with percent variation shown in Table 8-3. These data are intended only for illustration.
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95 Analysis Response Surface Approximation Reliability analysis of systems with multiple failure modes often employs Monte Carloently, it is impossible to for each of thxtractealysirations are fitted to therst safety margins in tf random bility of failures for the panel system and each load cases are calculated by pe simulation, combined with approximation to the failure evaluations. PANDA2 provides the margins of those failure modes that are critical for the stiffened panel, and these may change during the design optimization. Consequ follow each individual failure mode, and instead the most critical safety margins e two load cases are e d from the PANDA2 an s report. Analysis response surface app oxim wo erms o variables. Proba rforming Monte Carlo simulation using the approximation. The deterministic optimum obtained by PANDA2 global optimization in Lamberti et al. (2003) is shown in Table 8-4 (Lamberti et al. 2003). Table 8-4. Deterministic Optimum b (in) h (in) t1 (in) t2 (in) Weight (lb) 101 091 .7 2.049 0. 86 0.126 2598 sis respsuations (ARS) were fitted to thar two load cases in terms of seven variables, which included four design nt is Latin Hypercube sampling (LHS, e.g., Wysnd Jorgensen, 1), where designdom variables were treated as uniformly distributed variables ove5. Ranges for normal random variables are automatically handled by Latin Hypercube sampling. Using the ARS, probabilities of failure are calculated by Monte Carlo simulations Two analygins of the onse rfaces approxim e critical m random variables, and three material random variables. Statistical design of experime s a 998 n ra r the range shown in Table 8
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96 Table 8-5. Range of analysis response surface approximations (inch) b h t1 t2 9.5-10.8 1.94652.1515 0.09367-0.1035 0.1198-0.1324 e number of sampling pointsonstructed from LHS with 72 Table 8-6. Quadratic analysis response surfation to the most critical msing Latin Hbe sampling of 72 points Critical mf load case 1 cal margiad case 2 The accuracy of the ARS is evaluated by statistical measures provided by the JMP software (Anon. 2000), which include the adjusted coefficient of multiple determination (R2adj.), and the root mean square error (RMSE) predictor. To improve the accuracy of response surface approximation, polynomial coefficients that were not well characterized were eliminated from the response surface model by using a mixed stepwise regression ( Myers and Montgomery 1995). A quadratic polynomial of seven has 36 coefficients. Th generated by LHS was selected to be twice the number of coefficients. Table 8-6 shows that the quadratic response surface approximations c points offer good accuracy. ace approxim argins u ypercu argins o Criti ns of lo Rsquare adj. 0.9413 0.9986 RMSE predictor 0.0203 0.00273 Mean of response 0.4770 0.1986 The deterministic design is then evaluated under material and manufacturing uncertainties. The probability of failure of the deterministic optimum under uncertainties in material properties is shown in Table 8-7. The dominant failure mode is local skin triangular buckling.
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97 Table 8-7. Probabilities of failure calculated by Monte Carlo simulatio n with 1106 mples Probability of failure of se 1 Probability of failure of d caseSystem probability of sa load ca loa 2 failure 0 69310-6 69310-6 Design Response Surfaces onlinear, the reliability constraint of Equation (8-1) is replac avoid extrapolation in probability calculation. The error probability ofilure is poor, whe enoTable 8esign response surface approximations (inch) b h t1 t In order to filter the noise in the results of MCS, design response surface approximation (DRS) are constructed to approximate the reliability constraints. Using the ARS constructed in previous section, the probability of failure at each design point of the design of experiment of DRS can be evaluated inexpensively by MCS. Since the probability of failure is highly n ed by an equivalent form of constraint in terms of probabilistic sufficiency factor introduced in chapter six. The range of the cubic DRS is shown in Table 8-8, which is a subset of the range of ARS shown in Table 8-5 to statistics are summarized in Table 8-9. It is seen that the accuracy of the DRS to fa ile DRS to PSF is accurat ugh for RBDO. -8. Range of d 2 9.6-10.7 1.9854 2.1084 0.0974-0.0986 0.1246-0.1271 able 8-9. Cubic design response surface approximation to the probability of failure and samples) factor T probabilistic sufficiency factor (calculated by Monte Carlo sampling of 1106 Probability of failure Probabilistic sufficiency Rsquare adj. 0.6452 0.9990 RMSE predicto r 0.000211 0.00176 Mean of response 0.0000580 1.079
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98 Optimum Panel Design Using the DRS of td. The optim he PSty-baseis performeum design is shown in ility evaluated by MCS sing ARS, whose error statistics are shown in Table 8-5, with 106 samples is shown in Table 8-11. It is seen that the reliability-based design is slightly lighter and safer than the deterministic design. The probability of failure of reliability-based design is slightly higher than the target reliability, which is due to the errors in the ARS approximation. The design can be refined by another design iteration that use a smaller design domain centered around it. Table 8-10. Optimum panel design F in Table 8-9, reliabili d design optimization Table 8-10. Its reliab u b h t1 t2 Weight (lb) 10.70 1.98 54 0.1004 0.1246 2581 Table 8-11. Probabilities of failure calculated by Monte Carlo simulation of 110 Probability of failure of Probability of failure of System probability of 6samples load case 1 load case 2 failure 0 12310 12310 -6-6 Next we consider a composite isogrid panel. The panel and the zero degree irection for the composite angle ply laminates in isogrid and skin are shown in Figure 2. is a good design for cryogenic environments due to reduced residual strains. Two additional design variables are the ply angles of angle ply laminates iogrid and skins,tively. Reliabilittimization seeks the lightest structures Composite Isogrid Panel Design Example d It is shown in chapter four that angle ply laminates with plies oriented close to each other n the is 1 and 2, respec y-based op
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99 satisfying the reliability co nstraints, which is expressed as the probability of failure of the panel Figusogrid-stiflindrical sh internal id isogrid riented alomferential for increding stiff hoop direction; the zero degree direction for the composite laminates in m st52F re ( -423Fred together withg two load cases roof pressure of 35 psi critical for strength; 2. axial compressivizing) pressure of 5 psi critical fo thermal expansion (CTE) along and transverse to fiber direction, 1 and 2, respectively. It is shown in chapter three that 1 is two orders of magnitude smaller than 2, 1 is ignored in the analysis to reduce the number of variables in response surface P less than Pu=10-4. re 8-2. Io fened cyng circu ell with direction sogrid anased ben patternness in isogrid and skin panel are shown Thermal loading fro ) are conside ress free temperature ( 3 the followin ) to service temperatu 1. internal p e load Nx = 1000 lb/in, with an internal (stabilr buckling. The amount of geometric imperfections is the same as those shown in Table 8-1. The material used in the study is AS4, which has a density of 0.057 lb/in3. For composite panel, the residual stress induced from cooling for cure temperature to service temperature must be considered. The residual stress calculation needs the coefficients of
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100 approximation. The uncertainties in the material properties are represented by 11 random variables, six for elastic p roperties (E1, E2, G12, G23 and 2) and five for strength allowables (1t, 1c, 2t, 2c and max), which are assumed to be normally distributed and uncorrelated. The mean values and coefficients of variation of the uncertainties in material elastic properties and strength allowables are shown in Table 8-12 and Table 8-13 respectively. The six design variables (b, h, t1, 1, t2 and 2) are assumed to be uniformly distributed with percent variation given in Table 8-14. These data are intended only for illustration. Table 8-12. Uncertainties in material elastic properties (AS4) modeled as normal distribution with coefficient of variation of 0.03 Mean value Youngs modulus along fiber direction (E1) 20.6106 psi Youngs direction modulus transverse to fiber (E2) 1.49106 psi In plan shear modulus (G12) 1.04106 psi Minor Poisson ratio () 1.9510 -2 Transverse shear modulus (G23) 0.75 106 psi CTE across fiber direction (2) 0.1510-4 8-13. Uncertainties in mater Tableial strength properties (AS4) modeled as normal distribution with coefficient of variation of 0.05 1122 t c t c max Mean value 331000 psi 208900 psi 8300 psi 33100 psi 10300 psi Table 8-14. Variation of the random design variables around nominal design value b h t1 1 t2 2 Variation 1% 2% 4% 1 4% 1 The design problem is formulated as Deterministic Design
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101 ),,,,,(2211 tthbWW minimize Ni csaii (8-2) ththTable 8-15. Safety factors used in deterministic design General instability i,1,01thatsuchwhere N is the total number of failure modes, ai and ci is the allowable and response for the i failure modes, and si is the safety factor for the i failure mode. Deterministic optimization use safety factors to account for uncertainties. A set of safety factor typically used in the design of stiffened panel aerospace application (Lamberti et al., 2003) is shown in Table 8-15. Local buckling Stiffener buckling Strength failure 1.4 1.2 1.2 1.2 The deterministic optimum found by using the global oppability in PANDA2 software is shown in Table 8-16. The composite design is heavier than the Aon, because composite bstantially p shown in chapter four. Tptimum (inch, degree, lb) 1 t2 Weight timization ca luminum designs in previous secti design is su enalized by thermal loading as able 8-16. Deterministic O B h t1 2 9.999 1.583 0.0222 9 87.57 0.03950 3386 14.84 Analysis Response Surface Approximation The m ost criticaafety margins of the two locases are exted from PANDA2 nase S) d to ma ables, and 11 random variables. Statistical design of experimeis Latin Hercube sam(LHS). The range of the design random vas for thS was c as 5%ed l s ad trac report. Two a lysis respon surfaces (AR were fitte the critical rgins of the two load cases in terms of sixteen variables, which included six design random vari nt yp pling riable e AR hosen bas
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102 on the values of the opt imal deterministic design. Using the ARS, probabilities of failure are caA quadr 17 variables has 171 coefficients (=18x19/2). The number of sampling points generated by LHS was secoefficients. Table 8-17 shows that the quadratic response surfaces constructed from LHS with 342 points offer good accuracy. Quadratic ARS over as % design perturbation is employed to perform reliability analysis and design optimization. Table 8-17. Quadratic analysis response surface approximation to the worst margins Critical margins of load Critical margins of load case 2 lculated by Monte Carlo simulations atic polynomial of lected to be twice the number of using Latin Hypercube sampling of 342 points case 1 Rsquare adj. 0.9686 0.9772 RMSE predictor 0.01615 0.010 80 Mean of response 0.4181726 6 0. The deterministic design is then evaluated under material and manufacturing uncertainties. The probability of failure of the deterministic optimum under uncertainties is shown in Table 8-18. The dominant failure mode is in-plane shear and transverse tensile strength failure for load case 1) of internal pro of pressure of 35 psi. The high safety factor or reliability-based design optimization. Table 8-18. Probabilities of lculaMonimul 106 s (material and manufacturing uncertainties) failure of lo1 (strSystem probability System Probabilistic failure probability can be reduced by using increased failure ca ted by te Carlo s ation of amples Probability of Probability of ad case ength) failure of load case 2 (buckling) of failure Sufficiency Factor for 10-4 probability 89.7810-4 0 89.7810-4 0.837178 As shown by the aluminum panel design example, RBDO using a design surface approximation around deterministic optimization yields a reliability-based design near
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103 deterministic optimization. The approximation domain of design response surface approximation and thus the search domain of the optimization is rather limited compared to the whole design space. In order to fu lly explore the potential of reliability-based desigtion with ProbaFor the deterministic dhere the deterministicfactors are given in Table 8-15, reliability analysis using Monte Carlo simulation shows the critical failure mailure model a probabilistic sufficiencyr. The deficiency safety factor of strength failure mode can be corrected by using the probabilistic sufficiency factor, which is the factor to be applied to the response to meet the target probability of failure requirement. In the next design iteration, the safety factors of those failure modes that are not active in the last design iteration remains the same. The safety factor of the critical failure modes are chosen to be n optimization, RBDO and design response surface construction must be performed repeatedly to cover the entire design space, which is computationally prohibitive. To address this problem, RBDO is converted to equivalent deterministic optimization in order to reduce the computational cost and explore the entire design space as shown in Chapter 7. Reliability-Based Design Optimization Using Sequential Deterministic Optimizabilistic Sufficiency Factor esign, w safety ode is the strength f nd the facto in )()1(ksfikiPs (8-3) )(ks duoriluPANDA2 the safety margins in optimization and it usually rrors on the conservative side in approximation, the actual safety margins from which are used to reSince ce the allowables ai f approximates the corresponding fa re modes. e
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104 PAND A2 analysis that doesnt employ those approximation used in optimization might be slightly higher than zero due to approximation errors. The result is that the PANDA2 optimum design maybe corresponds to a slight higher safety factor than the one used in optimization. Another correction formula based on the actual safety margin from PANDA2 analysis is proposed as the following )( )1()(1sfikPkms ki (8-4) The correction is also only done for those safety factors that correspond to active failure modes. The o ptimization problem is formulated as ),,,,,(2211 tthbWW minimize Nigsaki,1,01)1( (8-5) iitThe process is repeated until the optimum converges and the reliability constraint is satisfied. Two design iterations are carried out using Equation. (8-3) and (8-4) using PANDA2 optimization, respectively. The results are shown in Table 8-19 and Table 8-20. It is seen that the second design (S2=1.437514) in Table 8-19 has much lower probability of failure than the desired probability of failure predicted by using the safety factor calibrated by PSF. It must be noted that the critical failure mode switches from in plane shear failure of the isogrids as of the first design (S1=1.2) to transverse tensile failure of the tank skin a of the second desigconvergence. thasuch sn, which causes the oscillation in design
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105 Tableusing probabilistic sufficiency factor to correct safety factor directly by Minimize objective function W while pof 0.0001 8-19. Design history of RBDO based on sequential deterministic optimization Equation (8-3) Safety fastrength fa6 ctor for ilure mode [b, h, t1, 1, t2, 2] Objective function: weight (lb) Pof/PSF from MCS of 10 samples* Critical strength failure mode (load case 1) Initial d(s1shear esign =1.2) [9.999, 1.583, 0.02229, 87.57, 0.0395, 14.84] 3386 .008978/0.837178 Isogrid in-plane Ss1=Skin transverse-tensile 2= /0.8371781.437514 [9.997, 0.9701, 0.01828, 89.54, 0.07011, 0.1484] 3192 0.00000/1.179381 S3= s2/1.179381=1.218872 [9.997, 0.8254, 0.005205, 90.00, 0.07087, 0.9826] 1878 0.001875/0.944237 Skin transverse-tensile pof = probability of failure, PSF = probabilistic sufficiency factor Table 8-20. Design history of RBDO based on sequential deterministic optimization using probabilistic sufficiency factor to correct safety factor by actual safety margin using Equation (8-4) ction W while pof 0.0001 Minimize objective fun Safety factor for strength failure mode [bObjective function: weight (lb) Pof/PSF from MCS of 106 samples Critical strength failure mode (load case 1) h, t1, 1, t2, 2] Initial design (s1=1.2) [9.999, 1.583, 0.02229, 87.57, 0.0395, 14.84] 3386 0.00shear 8978/0.837178 Isogrid in-plane S2= s=1.437514 [9.997, 0.9701, 0.07011, 0.1484] 1/0.8371780.01828, 89.54, 3192 0.00000/1.179381 Skin transverse-tensile Ss2/1.179=1.218[9.997, 1.137, Skin transverse3= 381872 0.006349, 90.00, 0.05445, 0.001495] 2058 0.000080/1.002482 tensile pof = probability of failure, PSF = probabilistic sufficiency factor It is seen that the third designs in Table 8-19 and quite different from that in Table 8-20, because the different correction formulation used. The designs in Table 8-19 use direct correction to the safety factors employed in the previous design iteration shown by Equation (8-3), while designs in Table 8-20 employed correction to the actual safety margins of the previous design iteration shown by Equation (8-4). Comparing the third
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106 designs in Table 8-19 and 8-20, it is seen that the correction with actual safety margins is more accurate than that using the safety factor, because PANDA2 Superopt employs fety margin in in. In the second design, the actual safety margin of PANDA2 design (S2= 1.437514) is ich sif PANioin ondte differerd s in Ta0. Th, the secola accuone, Equation (8-4). eliign Oization using DIREimdi fac actual sabation optimizatiog global optimizationmed afeter PANns. CT Gltion Algorithm IRECd Lipschitzian optimization algorithm (Jones et al. 1993). es,es algorithm works. It works in a space normalized to 0 xi 1 for i = [1, 2, N], s f pace. DIRECT begins by evaluating the ei for i = [1, 2, N] (the 0 approximation in the optimization process, which might results in slightly positive sats optimum desig superoptdid not employ approxim 0.0532, whptimization a hould be zero results in h DA2 Superopt atn nce in the thi designore ble 8-19 and 8-2 erefore nd correction formu Equation (8-3) is mR rate than the first ability-Based Des ptim CT Opt ization The using an o screpancy of safetyimizer such as DIR tor and fety margin can also e eliminated by ptECT (Jones et al. 1993) that doesnt use approxim a nd use only PANDA2 analysis. However, DIRECT is a sampling-based global n algorithm that is computationally more expensive than the multiple startin pointsseveral in PANDA2, DIRECT is chosen to be perfor DA2 design iteratio DIRED obal OptimizaT is a modifie The name is an acronym for DIvide RECTangl which is a simple d cription of how thewhere N i the dimensionality o the design s function to be minimized, f, at the center of the design space, c1 = 0.5 th iteration). It then evaluates the function at the points c1 1/3 ei (the 1st iteration). Each dimension then has 2 function evaluations, and DIRECT defines
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107 11 11min, for 1, 2, ... N33wfcfci iii ee (6) Figure 8-3. 1st iteration of DIRECT windowing for 2 dimensional example, Goldstein-with potentially optimal boxes shaded and subsequently divided along longest Price (GP) function (Finkel, 2003), and (b) further iterations on GP function dimension(s). The design space is divided into thirds along the dimension with the minimum wi. The center box is then divided again into thirds along the dimension with the next smallest wi, and so on. Thus, the minimum function value for the iteration is in the largest subdivision of the original box in the iteration (Figure 8-3 (a)). DIRECT then determines potentially optimal boxes, using both the function value at the center of the box and the size of the box. DIRECT divides potential optimum boxes along their longest dimension(s), using the same procedure at the initial division. Thus, if, as in the initial division, the box is a hypercube in n = N dimensions with the side length of the cube greater than the other side lengths, it is divided along those n dimensions. Multiple
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108 iterations of DIRECT for a two dimensional space might look like Figure 8-3 (b); given infinite iterations, DIRECT will evaluate every point in the design space A more detailed in Jones et al. (1993) and Finkel (2003). Reliability-Based Design Optimization Using Direct Optimization with Safety actor Corrected by Probabilistic Sufficiency Factor DIRECT optimizer is employed to avoid the problem that PANDA2 Superopt may not drive the actual safety margin to zero. DIRECT optimization starts from the last design in Table 8-20. The optimum is shown in Table 8-21. It is seen that design indeed satisfies the reliability constraint. Table 8-21. Design history of RBDO based on DIRECT deterministic optimization with probabilistic sufficiency factor correct safety factor by actual safety margin using Equation (8-4) Minimize objective function W while pof 0.0001 explanation of this procedure is given F Safety factor for strength failure mode [b, h, t1, 1, t2, 2] Objective function: weight (lb) Pof/PSF from MCS of 106 samples Critical strength failure mode (load case 1) S3= 1.514/1.179381=1.2837[5.895, 0.9167, 0.0005, 89.81, 0.0400, 1.67] 1952.4 0.000100/0.999641 Skin transverse-tensile 25 pof = probability of failure, PSF = probabilistic sufficiency factor Summary Reliability-based design optimization of aluminum and composite isogrid stiffened panels is investigated. The uncertainties in the panels including both material properties and manufacturing process variation are modeled by random variables. The reliability-based design optimization is carried out using Monte Carlo simulation and response surface approximation. Due to the high nonlinearity of probability of failure, probabilistic sufficiency factor approach is employed to construct design response surface approximation.
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109 RBDO using a design surface approximation is computationally prohibitive to explore the entire design space, since it requires a large number of design response surface approximation. In order to full explore the potential of reliability-based design optimization, probabilistic sufficiency factor is employed to convert reliability-based design optimization to equivalent sequential deterministic optimization, which enable the that is safer and lighter than the deterministic design using PANDA2 global optimization. The reliability-based panel design is lighter and safer than the deterministic optimum obtained by global optimization. use of global optimizer DIRECT. DIRECT optimization found optimum design
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APPENDIX A MATERIAL PROPERTIES OF IM600/133 The material properties of IM600/133 measured by Aoki et al. (200) are the polynomial curve fittings are shown in the following pictures. Figure A-1. Quadratic fit to 1 (1.0E-6/F) Figure A-2. Sixth-order fit to 2 (1.0E-4/F) -0.4 -0.1 -0.0 0.1 -0.3 -0.2 -600 -400 -200 0 200 400Temperature (F) 0.0 0.2 0.4 0.6 0.8 -500 -250 0 250 500Temperature (F) 110
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111 Figure A-3. Quadratic fit to E1 (Mpsi) Figure A-4. Quartic fit to E2 (Mpsi) 21.1 21.2 21.4 21.5 21.6 21.3 21.7 21.8 -500 -250 0 250 500Temperature (F) 0.0 0.5 1.0 1.5 2.0 2.5 -500 -250 0 250 500Temperature (F)
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112 1.25 Figure A-5. Cubic fit to G12 (Mpsi) Figure A-6. Quartic fit to 12 0.25 0.50 0.75 1.00 -600 -400 -200 0 200 400Temperature (F) 0.375 0.345 0.350 0.355 0.360 0.365 0.370 -500 -250 0 250 500 Temperature (F)
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APPENDIX B CONTOUR PLOTS OF THREE DESIGN RESPONSE SURFACE APPROXIMATIONS AND TEST POINTS ALONG THE CURVE OF TARGET RELIABILITY To compare critical errors of the design response surface approximations, 11 test points were selected along a curve of target reliability (probability of failure=0.00135) for each design response surface approximployed in error calculation for the probabilistic sufficiency factor, probability of failure and safety index response surface approximations are shown by the following figures. The average percentage errors at these test points are shown in Table 6-4. Figure B-1. Contour plot of probabilistic safety factor design response surface approximation and test points along the curve of target reliability ation. The contour plots and test points em 1.5 2 2.5 3 3.5 4 4.5 5 w (inch)t (inch)0.576020.75180.927571.27911.45491.63071.8064 1.1033 113
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114 Figure B-2. Contour plot of probability of failure design response surface approximation ative values of probability of failure are due to the interpolation errors of the design response Figure B-3. Contour plot of safety index design response surface approximation and test points along the curve of target reliability and test points along the curve of target reliability. The neg surface approximation 1.5 2 2.5 3 3.5 4 4.5 5 .018562 w (inch) t (inch) 1.0 8690.994740.902620.81050.626260.534150.442030.34991 0.257790.165670.073556 0.71838 1.5 2 2.5 3 3.5 4 4.5 5 t (inch).8403.0325.2247.416870.390951.19882.00662.81443.62224.434 .43 w (inch)
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LIST OF REFERENCES Aoki, T., T. Ishikawa, H. Kumazawa, and Y. Morino (2000), Mechanical Performance of CF/Polymer Composite Laminates Under Cryogenic Conditions. 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, Georgia, AIAA Paper 2000-1605. Aoki, T., T. Ishikawa, H. Kumazawa, and Y. Morino (1999), Mechanical Behavior of CF/Polymer Composite Laminates under Cryogenic Environment. 12th International Conference on Composite Materials (ICCM-12), Paris, France, Paper No.172. Balabanov, V.O. (1997), Development of Approximations for HSCT Wing Bending Material Weight using Response Surface Methodology. Ph.D. dissertation, Virginia 2004) Birger, I.Ars and Diagnostics. Problems of Mechanics of Solid Bodies, 71-82. Leningrad: Sudostroenve (in Russian) Breitung, K. (1984), Asymptotic Approximation for Multinormal Integrals. Journal of Engineering Mechanics, ASCE, 110 (3): pp. 357-366. Bucher, C. G. and Bourgund, U.(1990), A Fast and Efficient Response Surface Approach for Structural Reliability Problems. Structural Safety, 7, pp. 57-66. Elishakoff, I. (2001), Interrelation between Safety Factors and Reliability. NASA/CR-2001-211309 Fox, E. P. (1996), Issues in Utilizing Response Surface Methodologies for Accurate Probabilistic Design. Proceedings of 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper No. AIAA-1996-1496. Fox, E. P. (1994), The Pratt & Whitney Probabilistic Design System. Proceedings of 35th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper No. AIAA-1994-1442. Fox, E. P. (1993), Methods of Integrating Probabilistic Design within an Organizations AIAA/uctural Dynamics, and Materials Conference, Paper No. AIAA-1993-1380. Polytechnic Institute and State University, Blacksburg, VA, 1997. (Available athttp://scholar.lib.vt.edu/theses/available/etd-82597-124631/, last checked March. (1970), Safety Facto Design System Using Box-Behnken Matrices. Proceedings of 34th ASME/ASCE/AHS /ASC Structures, Str 115
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BIOGRAPHICAL SKETCH helor of Science in Aircraft Design from Nanjing University of Aeronautics and Astronautics in July 1996, and a Master of Science in Aircraft Design from the same university in April 1999. His interest in conducting research motivated him to join the Structural and Multidisciplinary Optimization Group of Professor Haftka at the University of Florida in June 1999, to pursue his Ph.D. degree in Aerospace Engineering. Xueyong Qu was born Changchun, China in 1974. Mr. Qu received his Bac 120