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An inverse reliability method and its applications in engineering design Li, Hong 1999

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A N I N V E R S E R E L I A B I L I T Y M E T H O D A N D ITS APPLICATIONS IN ENGINEERING  DESIGN  BY HONG LI B. Sc., Hohai University, 1984 M.Sc, Hohai University, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering  We accept this thesis as conforming to thejiequired standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1999 ©Hong Li, 1999  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  available for  copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and study. scholarly  or for  her  Department The University of British Vancouver, Canada  DE-6 (2/88)  Columbia  purposes  gain  shall  requirements that  agree  may  representatives.  financial  permission.  I further  the  It not  be is  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall make for  the that  without  it  extensive  head  of  my  copying  or  my  written  Abstract The concepts and methods of reliability theory have been developed to take into account intervening engineering uncertainties in modern engineering design. To achieve a certain performance an engineer must find design parameters which meet corresponding pre-set target reliability levels for the different limit states considered. Traditionally, this inverse reliability problem is implemented by means of "trial and error," using a forward reliability procedure and interpolating the design parameters at the desired reliability. This approach is inefficient and involves difficulties resulting from repetitive forward reliability analysis. Thus, it is desired to develop an efficient and more direct approach to determine the design parameters for specified target reliabilities.  A general inverse reliability methodology is proposed in the present work, which allows the direct determination of the design parameters when the corresponding target reliabilities are given. The design parameters can also be associated with random variables, such as their mean and standard deviation. As an extension of the inverse reliability method, an inverse reliability-based optimization procedure is also developed.  In contrast to traditional  reliability-based optimization, this method permits the separation of the ordinary optimization method and the inverse reliability analysis. For cases requiring an intensive computational effort or having a non-smooth limit state function, a response surface method for inverse analysis is proposed to implement and speed up the calculations.  II  The efficiency and applicability of the proposed method are verified with several practical engineering  applications, ranging from offshore  and earthquake engineering to  manufacturing of carbon fiber reinforced composite components.  iii  Table of Contents ABSTRACT  H  T A B L E OF CONTENTS  IV  LIST OF TABLES  IX  LIST OF FIGURES  XI  ACKNOWLEDGEMENTS  XH  CHAPTER 1 INTRODUCTION  1  1.1 General  1  1.2 Review of previous work  3  1.3 Research objectives and thesis scope  7  1.4 Thesis outline  8  CHAPTER 2 INVERSE RELIABILITY - THEORETICAL DEVELOPMENT AND CASE STUDD2S  10  2.1 Introduction  10  2.2 Review of forward reliability method  10  2.2.1  Basic concepts  10  2.2.2  FORM procedure  14  2.2.3  Bounds for a series system  20  2.3 Inverse reliability with a single design parameter  21  2.3.1  Statement of the problem  21  2.3.2  Proposed algorithm  23  2.3.3  Case studies  26  2.3.3.1 Deterministic design parameter  26  2.3.3.2 Mean value as the design parameter  27  2.3.3.3 Standard deviation as the design parameter  30  IV  2.4 Inverse reliability with multiple design parameters 2.4.1  Statement of the problem  30 30  2.4.2 Proposed algorithm  32  2.4.2.1 Individual target reliability levels are specified  32  2.4.2.2 Target reliability level of a series system is specified  34  2.4.3  Case studies  36  2.5 Inverse response surface method 2.5.1  41  Review of the response surface method  41  2.5.2 Inverse response surface method  44  2.5.3  45  Case studies  2.6 Inverse problem with multiple solutions for the design parameters 2.6.1  The existence of the problem  48 48  2.6.2 Equivalent deterministic method  49  2.6.3  50  Hybrid of the Newton-Raphson method and bisection method  2.6.4 Examples  53  2.7 Summary  55  CHAPTER 3 INVERSE RELIABILITY-BASED OPTIMIZATION AND CASE 57  STUDIES 3.1 Introduction  57  3.2 Procedures  58  3.3 Case studies  62  3.4 Summary  72  CHAPTER 4 APPLICATIONS IN ENGINEERING DESIGN  74  4.1 Introduction  74  4.2 Reliability-based codified design  75  4.2.1  Basic concepts  76  4.2.2 Determining the load factors, an example  79  4.2.2.1 Problem description  79  4.2.2.2 Limit state functions  80  V  4.2.2.3 Intervening variables  80  4.2.2.4 Determination of factors an and CCQ  82  4.3 Application in offshore engineering  84  4.3.1  Background and description of the problem  84  4.3.2  Loads on the structure  86  4.3.3  Intervening Variables  88  4.3.4  Determination of exceedence loads and the structural weight  88  4.3.4.1 Target reliability levels  88  4.3.4.2 Limit state functions  90  4.3.4.3 Results  91  :  4.3.5  Simultaneous determination of mean and standard deviation of the structure  weight  94  4.4 Applications to earthquake engineering 4.4.1  96  Linear response  96  4.4.1.1 Determination of the design peak acceleration  97  4.4.1.2 Dynamic analysis  99  4.4.1.3 Random variables  100  4.4.1.4 Limit-state functions  102  4.4.1.5 Results of basic case  103  4.4.1.6 Sensitivity analysis  103  4.4.2 Nonlinear response  107  4.4.2.1 Random variables  Ill  4.4.2.2 Limit-state function and target reliability  112  4.4.2.3 Seeking the design parameter, mass of the pilecap  112  4.5 Summary  :  115  CHAPTER 5 APPLICATIONS IN T H E MANUFACTURE OF A LAMINATED COMPOSITE COMPONENT  116  5.1 Introduction  116  5.2 Manufacturing process modeling  121  VI  5.2.1  A typical manufacturing process  121  5.2.2  A brief review of the process model COMPRO  122  5.3 Description of the composite laminate used for case studies 5.3.1  Laminate and cure cycle  125  5.3.2 Materials used in the case studies  128  5.4 Basis for probabilistic-based process modeling 5.4.1  125  128  Source of uncertainties  128  5.4.2 Random variables  129  5.4.3  131  Failure mode  5.5 Case study 1: Variability prediction in the spring-in  131  5.5.1  Variability prediction for the basic case  132  5.5.2  Sensitivity analysis  134  5.5.3  Conclusionsfromthe sensitivity analysis  138  5.6 Case study 2: Seeking design parameter - an inverse approach fraction  139  5.6.1  Mean of the fiber volume  140  5.6.2  Standard Deviation of VF  141  5.6.3  Target temperature  142  5.7 Case study 3: Seeking multiple design parameters - Inverse Reliability-based optimization 5.7.1  144  Minimizing the initial cost under reliability constraint  145  5.7.1.1 Objective function and constraints  145  5.7.1.2 Response surface.....  147  5.7.1.3 Optimization results  149  5.8 Case study 4: Seeking multiple design parameters - Reliability-based optimization 152 5.8.1  The problems of interest  152  5.8.2  Approach  157  5.8.3 Numerical examples  159  5.9 Summary  170  VII  CHAPTER 6 SUMMARY, CONTRIBUTIONS AND FUTURE W O R K  172  6.1 Summary and contributions  172  6.2 Future work  176  REFERENCES  177  viii  List of Tables Table 2.1: Example 1 (Casel) Results Table 2.2: Statistics Table 2.3: Design Mean Value H for Different Cases (m) Table 2.4: Design Standard Deviation Table 2.5: Case 1: Statistics Table 2.6: Case 1: Solution Table 2.7: Case 2: Statistics Table 2.8: Case 2: Results Table 2.9: Statistics Table 2.10: Results and Iteration Details Table 2.11: Results and Iteration Details (initial design variable d is assumed as 7.0) Table 2.12: Sensitivity of Design Variable with Respect to Initial Values Table 2.13: Statistics of the Random Variables Table 2.14: Summary of the Results Table 3.1: Example 1 (Case 1) Statistics Table 3.2: Example 1 (Case 1) Optimal Solutions Table 3.3: Example 1 (Case 2) Statistics Table 3.4: Example 1 (Case 2) Optimal Solutions Table 3.5: Example 2 Statistics Table 3.6: Example 2 Optimal Solutions Table 3.7: Statistics Table 3.8: Optimum Results Table 3.9: Sensitivity to COV(fi= 3.0) Table 4.1: Occupancy Loads Table 4.2: Statistical Parameters for the Roof Load Ratio r Table 4.3: Statistics of the Variable g for three Cities Table 4.4: Reliabilities for Different Load Situations Table 4.5: Summary of Specified Variables and their Statistics Table 4.6: Loads at Specified Annual Exceedence Probabilities Table 4.7: Wave Loads (MN) at Specified Annual Exceedence Probabilities Table 4.8: Iceberg Force (MN) at Specified Annual Exceedence Probabilities Table 4.9: Mean Structure Weight for Sliding with a 1QT Annual Failure Probability Table 4.10: Design Parameters for Different Target Reliabilities, Offshore Platform Table 4.11: Random Variable Statistics Table 4.12: Design Mean Column Stiffness for the Basic Case Table 4.13: Results for Various C W of the Stiffnesses Table 4.14: Results for Various COV of the Masses Table 4.15: Results for Different Spatial Distributions Table 4.16: Results for Different Damping Ratios Table 4.17: The Effect of the Peak Acceleration Table 4.18: Statistics of the Random Variables 4  IX  27 29 29 30 37 37 38 38 40 41 46 47 53 55 64 65 66 66 68 68 70 70 71 81 81 82 83 89 92 93 93 94 95 102 103 104 105 105 106 107 Ill  Table 4.19: Summary of the Results (mass in tonne) Table 5.1: Summary of Potential Random Variables Table 5.2: Probability of G > 0 for Various Tolerances Table 5.3: Statistics and Error for Different Distribution Models Table 5.4: The Required Mean Value ofVF Table 5.5: The Required Standard Deviation ofVF Table 5.6: The Desired Mean of Target Temperature T Table 5.8: Response Surface Points (J3 = 1.5) Table 5.9: Detailed Comparison Results Table 5.10: Optimization Results (aj = 50.0) Table 5.11: Statistical Parameters for Different Model Considered Table 5.12: Optimal Results Table 5.13: Optimal Results.... Table 5.14: Design Parameters and Associated Costs  X  114 130 133 133 140 142 143 148 149 150 161 161 166 169  List of Figures Figure 2.1: Geometric Representation of Reliability Calculation Figure 2.2: Sketch of the Cantilever Beam Figure 2.3: Sketch of a Simply Supported Beam Figure 2.4: Graphic Interpretation of IRSM Figure 2.5: Relationship Between /? and Design Parameter d Figure 2.6: Relationship Between D and M Figure 3.1: Flow Chart of the Inverse Reliability-Based Optimization Figure 3.2: Sketch of a Simply Supported Box Beam Figure 4.1: An Example of P -</> Relationship Figure 4.2: Definition Sketch of Iceberg - Structure Geometry Figure 4.3: Contours of fi, and fi = 2.0 Figure 4.4: Multi-story Frame Under Earthquake Excitation Figure 4.5: Ground Acceleration of Event in Landers, California (Joshua Station) Figure 4.6: Pile in an Earthquake Excitation Figure 4.7: Relationship Between Mass and Maximum Deflection Figure 5.1: Flow Chart of the Procedure Figure 5.2: A Typical Autoclave Cure Cycle Figure 5.3: Sketch of the C-shaped Part and Tool Figure 5.4: Process-induced Spring-in 6 Figure 5.5: Cure Cycle Figure 5.6: Distribution of the Spring-in Figure 5.7: Sensitivity to the Mean of VF at Different Tolerance Level Figure 5.8: Sensitivity to the COV at Different Tolerance Level Figure 5.9: Model Prediction for Different Means Figure 5.10: Model Prediction for Different COV. Figure 5.11: Mean of VF against Target Reliability Figure 5.12: Optimal Costs Figure 5.13: Optimal Design Parameters .* Figure 5.14: The Sketches of the Parts and Tool Figure 5.15: Graphic Illustration of the Possible Events in Structural Assembly Figure 5.16: Probabilities of the Events Figure 5.17: Optimal Costs for Different Quality Ratings Figure 5.18: Optimal Probabilities for Different Quality Ratings Figure 5.19: Costs vs. #/for Different Quality Ratings Figure 5.20: Optimal Costs Figure 5.21: Optimal Probabilities for Different C o max  2  q  xi  15 29 40 47 49 54 60 69 79 86 96 97 101 108 113 119 122 126 127 127 134 136 136 137 137 141 150 151 153 154 156 161 162 163 166 167  Acknowledgements I would like to express my most sincere and extreme gratitude to Dr. Ricardo Foschi for his guidance, financial support, constant encouragement and help throughout this work.  I  would also like to very much thank Dr. Reza Vaziri for his guidance and help in my work done with the UBC Composites Group.  Many thanks to Dr. Anoush Poursartip and Dr. Goran Fernlund for sharing their expertise and offering their valuable time to discuss various problems on composite manufacturing process with me. Many thanks also go to Dr. Robert Sexsmith and Dr. Carlos Ventura for their guidance and proof reading of my thesis. The assistance offered by Mr. Robert Courdji in helping me understand and run COMPRO is highly appreciated.  My friends, classmates and the fantastic members of the UBC Composite Group have played an important role through these four years, their friendship, enthusiasm and encouragement made this part of my life especially memorable.  Finally I would like to express my appreciation to my wife Jianshan Wang and my daughter Mengdi Li for their support, patience and understanding throughout these years.  xn  Chapter 1 Introduction  Chapter 1 Introduction 1.1  General  The variability in the performance of a structural system results from the uncertainties involved in the materials properties, load conditions and the accuracy in construction or manufacturing. When performing either safety assessment or modern engineering design, it is essential to take these uncertainties into account through probabilistic analysis. In combination with an analytical model for the system, safety assessment requires forward reliability methods for estimating the reliability or the probability that it will perform as intended. On the other hand, engineering design requires an inverse reliability approach of determining the design parameters to achieve pre-specified target reliabilities.  While forward reliability methods have been applied widely and successfully in various engineering fields, inverse reliability approaches have not received the same degree of application, although they are particularly useful due to their important role in engineering design. Inverse reliability problems appear when; for example: 1) a target reliability is specified in the design, be it for ultimate capacity or serviceability criteria. In this case, design parameters must be determined to achieve the given reliability level.  For example, in designing the piers of Prince Edward Island's  Confederation bridge for the possibility of sliding or overturning under the action of ice loads, a target reliability index (j3 = 4.25 using maximum loads in a 100 years window)  1  Chapter 1 Introduction  was specified in the contract (Lester and Tadros, 1995), requiring calculation of the appropriate pier mean weight and its variance. 2) a reliability-based design code is being calibrated. Code design procedures usually include performance and load factors, which are used to account for the uncertainties and to produce a design with the desired reliability. To achieve this objective, the performance and load factors may be calculated using the inverse reliability approach, taking note, however, that full code calibration may involve achieving a target reliability in a number of design situations which exceed the number of factors being sought. 3) a target quality is specified for a manufactured product. Several design parameters in the manufacturing process, rangingfrommaterial properties to process implementation, may have to be obtained in order to ensure that the processed product meets a prespecified quality or tolerances with a desired reliability.  In general, an inverse problem involves finding either a single design parameter to achieve a given single reliability constraint or multiple design parameters to meet specified multiple reliability and/or geometric constraints. The design parameters may also be associated with random variables, for example, their mean values and standard deviations. More generally, when an objective function related to cost or the weight of the structure is introduced into the inverse reliability problem, the task is to find the design parameters to minimize the objective function and, at the same time, satisfy the equality reliability constraints. Such a problem can be classified as reliability-based optimization. It is desirable to include  2  C h a p t e r 1 Introduction  optimization in order to generalize the inverse reliability problem and to provide a general, complete approach.  1.2 Review of previous work The fundamental theory and methods of structural reliability have experienced several breakthroughs during the last three decades. To replace tedious and complex mathematical calculations, application-oriented methods have been developed to provide an efficient and accurate estimate of structural reliability. These include: 1) FORM (First Order Reliability Method) (Ffasofer and Lind, 1974, Rackwitz and Fiessler, 1978), and SORM (Second Order Reliability Method) (Tvedt, 1983, Breitung, 1984 and Der Kiureghian et al. 1987) for calculating the reliability in a single performance or limit state mode. These are approximate methods that have also been extended to multiple modes through estimation of probability bounds (Ditlevsen, 1979). 2) Efficient simulation methods, replacing direct Monte Carlo simulation, such as the importance sampling technique (Ffarbitz, 1986, Schueller and Stix, 1987) and the adaptive sampling method (Bucher, 1988, Schueller, et al. 1989), which provide an efficient and precise alternative for evaluating small probabilities of non-performance.  These well-established methods, classified as forward reliability methods, allow various practical approaches to determine the reliability of a system when the statistics and parameters of the intervening variables are given, and the system behavior is quantified by a performance model.  3  Chapter 1 Introduction  The inverse reliability problem can be solved by trial and error by repeating forward reliability analyses such as FORM/SORM, interpolating the design parameters being sought. This is inefficient and time consuming, especially when multiple design parameters and multiple reliability constraints are involved.  To approach the inverse reliability  problem directly, some efficient methods have been proposed under the name of " inverse reliability procedures?'' A reliability contour method has been described by Winterstein et al. (1994) and applied to problems in offshore environmental loads in the context of limitstate functions of the form G(x, y ) =y cap  cap  - R{x)  (1.1)  where JC is the vector of random variables associated with the response R(x) and y  cap  is a  given deterministic threshold. Given information on the environmental variables, a contour with a prescribed reliability level is constructed first, and the maximum R^ ofR(x) is then found along that contour. The vector JC associated with i?  max  provides the variable  combination corresponding to the design point, and the companion y  cap  (1.1) using G = 0. In this case, y  cap  is obtainedfromEq.  is treated as a deterministic design variable. Another  contour method was proposed based on linearization of an arbitrary response function (Huyse and Maes 1996, Maes and Huyse 1997). In this method, the contour is constructed directly in the original variable domain without performing the inverse transformation. This contour-based approach allows the uncoupling between the environmental or external loading variables and the structural response.  4  Chapter 1 Introduction To extend the method to general limit states functions, Der Kiureghian et al. (1994) proposed an iterative algorithm based on the modified Hasofer-Lind-Rackwitz-Fiessler scheme (Hasofer and Lind, 1974, Rackwitz and Fiessler, 1978). The limit-state function G was considered to be dependent on a vector u of known intervening random variables and a single, deterministic design parameter 9. The iterative algorithm was applied in the standard manner using the vector (u, 9). At the end of each iteration, however, the next components of the vector were chosen at a point that was obtained by means of a line search to minimize the error in the solution and to make the iteration procedure more efficient, particularly for the case when the limit-state function has a strong outward curvature.  Reliability-based optimization has been studied extensively for the last three decades. Using the forward reliability method (FORM/SORM) for formulating nonlinear constraints, various methods have been proposed to approach the problem effectively and efficiently (see, for example, Mau, 1971, Moses, 1977 and Frangopol 1985). Due to the complexity of the problem and the requirement of an intensive computational effort, this is an area still under development. In most cases a two-level algorithm is used, in which the first is used to iterate the design parameters to minimize the cost or weight and the second, which addresses the reliability constraints, is called by first level, resulting in a nested iteration scheme.  5  Chapter 1 Introduction  To avoid the two-level approach, recent developments have been proposed and have already shown success in practical applications. In order to avoid the calculation of secondderivatives of the limit-state function and the repeated computation of the first order reliability index, Kirjner, Polak and Der Kiureghian (1995) formulated an "outer approximations" algorithm for solving the reliability-based optimization problem. The convergence properties of the method have been proven. Based on the modified quasiNewton algorithm, Pederson and Thoft-Christensen (1994,1996) proposed a prototype for an iterative optimization system, which allows the designer to adjust simple bounds, fix or relax design variables and include or exclude constraints.  Considering the fact that modem reliability methods like FORM are formulated as an optimization problem, Kuschel and Rackwitz (1997, 1998) proposed and formulated a onelevel approach method to reliability-based optimization. Two types of problems were analyzed: 1) minimization of the total cost, including initial cost and expected cost of failure, subjected to reliability and other constraints; and 2) maximizing the reliability of a structural system subjected to cost and other constraints. Formulated as a one-level optimization problem, the reliability-based optimization can be solved by using any standard optimization routine. Apparently, this is superior to the traditional two-level formulation, both in computational effort and robustness. However, the calculation of second derivatives of the limit state function is required.  From the previous review, it is noted that the inverse problem has been studied with only a single and deterministic design parameter. However, in practical applications, multiple  6  Chapter 1 Introduction  design parameters are sometimes needed if multiple reliability and/or geometric constraints are specified. For example, when designing a multi-story building under a random earthquake base excitation, it may be required to determine the mean value of the lateral stiffness for each floor to reach target reliability levels for not exceeding allowable interstory drifts. In addition, as discussed before, if an objective function is introduced, the reliability-based optimization can be treated as a special case of the inverse reliability problem. Thus, for completeness, it is necessary to take into account optimization in a general inverse reliability method. In this sense, an inverse reliability-based optimization method would offer an alternative approach to the other reliability-based optimization procedures that have been studied extensively.  1.3 Research objectives and thesis scope This thesis will focus on: 1) the development of a general inverse reliability method and corresponding software; 2) the development of an inverse reliability-based optimization method and corresponding software.  In order to demonstrate the efficiency and applicability of the methods to various engineering fields, several applications will be presented as well, ranging from: 1) reliability-based codified design; 2) offshore engineering; 3) earthquake engineering; and 4) control of manufacturing processes of carbon fiber reinforced composite components.  7  Chapter 1 Introduction  1.4 Thesis outline The research presented in the thesis is organized as follows: Chapter 1 - Introduction: The background and the objective of the research are described and previous work on inverse reliability and reliability-based optimization are reviewed. Chapter 2 - Inverse reliability-theoretical development and case studies: The development of the inverse reliability method is proposed with corresponding numerical examples. Included are 1) the theoretical development of the method for a single and for multiple design parameters; 2) implementation of the response surface in the inverse reliability problem; 3) strategy for problems with multiple solutions.  Chapter 3 - Inverse reliability-based optimization and case studies: The development of the inverse reliability-based optimization method is outlined.  Included are the proposed  procedures and cases studies.  Chapter 4 - Applications in engineering design: Several applications are presented, including: 1) reliability-based codified design; 2) the determination of the design wave and iceberg load, and the calculation of a platform weight in offshore engineering; 3) the determination of design lateral stiffnesses of a three-story building under earthquake excitation; and 4) the design weight applicable to a pile at its cap, under the earthquake load, when considering the nonlinear interaction between pile and the surrounding soil.  8  Chapter 1 Introduction  Chapter 5 - Applications in the manufacture of a laminated composite component: The variability prediction of a process-induced component deformation is considered first. Given a target reliability, the manufacturing design parameters, including the properties of raw composite material and the autoclave settings, are determined by the inverse procedure. Using inverse reliability-based optimization, an application of determining multiple design parameters to minimize the initial cost in the manufacture and to meet a target reliability is presented. Finally, for completeness of the application, the total expected manufacturing cost was minimized through traditional reliability-based optimization.  Chapter 6 - Summary, contributions andfuture work: A summary of the work is provided and it is followed by highlights of the thesis' contributions. Recommendations for future work are also presented.  9  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Chapter 2 Inverse Reliability - Theoretical Development and Case Studies  2.1 Introduction In this Chapter, a general method is developed to approach the inverse reliability problem for either single or multiple design variables. In general, these are also regarded as random variables, with corresponding mean values and standard deviations. The problem may include the determination of the means alone when the coefficients of variation are specified, the standard deviations when the means are given, or both means and standard deviations. The objective of the inverse procedure is to find, directly, the design vector d so that the target reliability level corresponding to each specified limit state will be satisfied. For cases requiring an intensive computational effort or having a non-smooth limit state function, a response surface method for inverse analysis is proposed to implement and speed up the calculations. In order to describe the inverse methodology clearly and systematically, the basic reliability concepts and forward reliability method are first briefly reviewed.  2.2 Review of forward reliability method 2.2.1  Basic concepts  The reliability of a general system is the probability that it will perform as required in the given conditions within a specified period of time. For example, the reliability level may  10  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  express the probability that the maximum deflection of a tower will not exceed a given threshold in next 50 years under the load conditions at a given site. In addition to the design parameters, the performance of the system is normally controlled and influenced by several intervening variables, some representing material, mechanical and geometric properties and others characterizing the external effects such as the load demands. From the probabilistic point of view, these variables are regarded as random and have to be described in probabilistic terms, giving their mean values, standard deviations and distributions. However, those with a small degree of uncertainty or more accurately known can be treated as deterministic parameters represented by a nominal value.  The probabilistic description of a random variable for reliability analysis may be achieved by 1) laboratory tests or surveys, providing statistics of the random variables and an estimate of the corresponding distributions; and 2) engineering experience and judgement when such statistical information is lacking. Furthermore, to save computational effort, a pre-sensitivity analysis can be used to infer the number of variables that should be specified as random.  The implementation of the reliability analysis is based on a description of the limit state of interest by a performance function G(x) as follows: G(x) = G(x ,x ,...,x ) 1  where the JC = (x x , lt  2  2  (2.1)  n  x„) is an n-dimensional vector of intervening random variables. T  Some of these may affect the demand on the system, denoted by D, while the others may  11  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  influence the system capacity C to withstand the demand. By convention, the performance function (2.1) may be written as G = C-D  (2.2)  Thus, the system will not perform as intended (failure) if the combination of the intervening random variables results in G < 0. The corresponding probability of such event (Prob( G < 0 )) is then called the probability of failure. Conversely, the combination of the intervening random variables resulting in G > 0 will make the system perform as required (survival) and the corresponding probability (Prob( G > 0 )) is termed the reliability. The situation G = 0 is a limit-state between failure and survival and the corresponding surface, G(x) = 0, is called the limit-state failure surface. To calculate the performance function we require a computational model describing the problem of interest. For example, let G be given as G=D -D(x ,x ,...,x ) 0  1  2  (2.3)  n  where Do is an allowable deflection of a wood floor and D is the maximum deflection of such floor system, with uncertain material and geometric properties, under a random load. The calculation of such maximum deflection may be implemented by running a separate computer program based on a finite element mathematical model. In the reliability analysis, this kind of program must be able to provide the output of interest and be integrated with the reliability procedure.  The probability of failure can be obtained by calculating the probability of the event G< 0. Since, in general, there could be a number of random variables involved in G, the exact  12  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  calculation requires the joint probability density function of all random variables and an integration over the failure region G < 0, (2.4) G<0  However, this exact approach can hardly be applied since the joint probability function/is unknown and very difficult to find. An alternative method is the straightforward, standard computer simulation (Monte Carlo method) which is simple to implement and can converge to the exact solution. However, it could be very computationally demanding, especially when dealing with a low probability of failure. For example, if the probability of failure is 10", the performance function must be evaluated 100000 times in order to observe, on 5  average, one outcome G< 0. Particularly, if the G function must be obtained by running a separate computer program (for example, a nonlinear dynamic analysis with time integration), the standard simulation approach will be very demanding, even using the latest computers.  A second alternative is the use of approximate methods that have been developed during the last three decades, such as the FORM/SORM procedures (First Order or Second Order Reliability Methods), and which are based on the calculation of the reliability index (3. From this index, the probability of failure Pf and the reliability P can be estimated r  approximately as shown in Eqs. (2.5) and (2.6), by use of the Standard Normal probability distribution function <D(.). (2.5)  13  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  and  P =l-®(-/J) r  (2.6)  = ®(/J)  These procedures have proven effective and relatively accurate. The following section discusses the FORM procedure in detail.  2.2.2  F O R M procedure  In order to avoid computationally intensive simulation, an efficient procedure called First Order Reliability Method (FORM) (Hasofer and Lind, 1974, Rackwitz and Fiessler, 1978) was developed in the early 1970's and has been modified and extended since then. For the last two decades, these computer-based methods have been widely used in day-to-day engineering design and in the calibration of modern reliability-based design codes.  The forward reliability problem can be stated as follows: given statistical information on the random variables x, (/ = J, 2, .. ri) and the limit-state function G = G( xj, x , .... x„), 2  the reliability index B or probability of failure Pf is to be sought. This goal can be achieved by using the following FORM algorithm. First, assume that the random variables x, are normally distributed and uncorrelated (independent). A new set of random variables w, are defined in the Standard Normal space and are obtained through the mapping  (i=l,2,...ri)  u. =  where JC,. is the mean value of Xj and  cr  Xi  its standard deviation. The origin u -  mapping of the mean vector of JC.  14  (2.7)  0 is  the  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  The reliability index  is the minimum distance between the origin « = 0 and the limit state  surface G(u) = 0, as illustrated in Fig. 2.1 for the case of two variables uj and u . Taking 2  advantage of this geometric interpretation of /?, several iterative computer algorithms have been proposed and are well established to find this minimum distance (Madsen, et al. 1986). The point u* on the limit-state surface which is the closest point to the origin is called the design point representing the most likely combination of random variables at failure. For Standard Normal, uncorrelated variables and a linear function G(u) = 0, it can be proven that Eq. (2.5) or (2.6) are exact. In the general case this use of the index /? gives an approximate estimate of Pf.  Figure 2.1: Geometric Representation of Reliability Calculation  The reliability index is then obtained by minimizing the distance between the origin and the failure surface, G(u) = 0. Thus, letting P = u u + AG(u)  x  T  (2.8)  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  where X is a Lagrange multiplier, the optimum solution can be reached if the vector u meets the following conditions V ¥(K,X) = 0  (2.9)  U  and  ^  dX  = G(u) = 0  (2.10)  where V (.) denotes the gradient with respect to u. U  FromEqs. (2.9) and (2.8),  u = ~V G  (2.11)  u  Multiplying both sides of Eq.(2.11) by V G , R  U  X  VGu  2  V G V„G  T  =  (2.12)  u  R  U  and substituting Eq. (2.12) into Eq. (2.11)  u=  "f V G V..G V..G " V  R  V.G'V.G  therefore,  and  (2.13)  U  /? =  »  "  (V.G'V.G)  .  (2.15)  (V G'V G); U  B  16  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  If i* satisfies Eq. (2.10) and Eq. (2.13), the reliability index  can then be calculated from  Eq. (2.15).  To seek the solution vector u, an iteration scheme is used (Ffasofer and Lind 1974).  This is based on a Taylor series expansion of G at a point u up to and including linear terms, G thereby being replaced by the tangent hyperplane at u .  G(u) = G(u) + V G ( « * ) ( « - « * )  (2.16)  T  u  By setting  G(u*) + V„G ( « * ) ( « - " * ) = 0  one obtains  V „ G ( « > = V G (u)u  (  r  r  -G(u)  T  u  V G («*)V G(«*) r  B  u  . "  Combining Eqs. (2.18) and (2.19),  -G(Q y.G>y V.G'COV.GCO " +  Similarly, forEq. (2.15), we have G(Q-V,,G»*  (V.G'COV.GCO)  1  17  )  (2.18)  Applying now Eq. (2.13) to the tangent hyperplane, V G > >  2 1 7  .  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Eqs. (2.20) and (2.21) provide the answer directly if the failure surface G were, in fact, the tangent hyperplane at u*. Since, in general, this is just an approximation, the calculated u is used as the new u and the procedure is repeated until convergence is achieved. The convergence of this quasi-Newton iteration scheme is not assured, however, as it is sensitive to the initial vector ***. The reliability index  is finally calculatedfromEq. (2.21)  or Eq. (2.15).  In general, when the random variables JC are correlated and non-Normal distributed, transformations are required before the procedure is implemented.  Suppose that the vector of the random variables in the original, basic space, is denoted as x = (x x ,... ,x„f ]t  (2.22)  2  The limit-state equation is given by G(x) = g(x ,x ,...,x„) 1  2  =0  (2.23)  A mapping of u into JC is achieved through the transformation x =F-'(<l>(« )) (  (j=l,2...,n)  (  (2.24)  where F (.) is the cumulative distribution function (CDF) for the variable x,, and <I>(.) is the Standard Normal function. When the basic variables are correlated with a given correlation matrix R, an intermediate vector y is introduced before the transformation Eq.(2.24) is applied. The variables y are a set of Standard Normal random variables obeying  18  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  *i=^~ (*G',))  (i=l,2...,n)  1  (2.25)  and having a correlation matrix Ro , which can be obtained from R. (Der Kiureghian and Liu, 1986)  Finally, the vector u is related to y through (2.26)  y = Lu  where the matrix I, resultsfromthe Cholesky decomposition of Ro into a lower and upper triangular form: (2.27)  R = LV 0  It can be shown that the resulting u are a set of Standard Normal, uncorrelated variables.  Based on the V G(JC), the gradient of G with respect to JC, V„G(M) can be formulated as: X  (2.28)  V„G(«) = IcrVG(jc) r  x  where rj is a matrix defined as follows:  (2.29)  and  cr, =  9iy )  (2.30)  t  19  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  (p(.) is the density function of Standard Normal distribution andy() the density function of random variable x,.  The reliability analysis (FORM) for a general case (Non-Normal correlated variables) can then be implemented in the following steps:  1) An initial vector u is taken in the Standard Normal, uncorrelated space. 2) Since the performance function is specified in the basic space defined as G(x), the function G(u*) is calculated by converting the uncorrelated, Standard Normal vector « to the correlated, Standard Normal variables y using Eq.(2.26), and then, the basic variables x are obtainedfromEq.(2.25). 3) The gradientV G(«') can be obtained by using Eq.(2.28) depending on the gradient U  V G(jc*)that can be calculated either by taking the derivative directly for an explicit x  function or numerically through finite differences centered at JC* for an implicit function. 4) The new vector u is calculated using Eq.(2.20) and the procedure is continued until convergence is reached. 5) The reliability index ft is then calculatedfromEq. (2.21) or Eq.(2.15)  2.2.3  Bounds for a series system  When studying a system with multiple failure modes, for example a statically determinate truss with multiple-components, the failure of any component may result in the failure of  20  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  the whole system. Such system behaves as a series or weakest-link system and its probability of failure will be the probability of the union of the several failure events (modes). The performance function is written for each failure mode and then studied individually by FORM, obtaining the corresponding reliability index p.  The results for  each failure mode can then be used to estimate the overall probability of failure of the system. While it is difficult to calculate the exact probability of failure for a system having more than two failure modes, a method (Ditlevsen 1979) permits the calculation of a low bound and an upper bound for the system reliability. If these bounds are sufficiently narrow, their average can be accepted as the estimation of the overall reliability.  The procedures described above have been implemented in several commercially available computer packages (PROBAN, STRUREL, CALREL, ISPUD, etc.). For this thesis, the package developed at the Civil Engineering Department of University of British Columbia was used. This is called RELAN, (Foschi, et al. 1996) and carries out a variety of reliability calculations for both single and multiple failure modes.  2.3 Inverse reliability with a single design parameter 2.3.1 Statement of the problem Suppose that the limit-state function G in the Standard Normal, uncorrelated space is G(u) = g(x,d)  (2.31)  21  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  where « = ( u , uj ) is the random, Standard Normal vector with components u and uj T  x  x  corresponding, respectively, to JC and d in the original, basic space. Here JC is the vector of basic intervening random variables and d is the design variable. We also treat d (with mean value d and standard deviation a) as a single random variable with a specified distribution. This will allow us to treat either the mean d or the standard deviation a as the actual design parameters being sought by the inverse procedure.  For a target reliability index J3, the inverse problem can then be stated as: Given /?, Find: d (mean value of d) or o (standard deviation of d)  (2.32)  Subject to: min (u u) = B and G(u) = g(x,d) = Q T  2  The mapping of « and uj into, respectively, x and d is achieved through the transformation x  xi=^ (*(««))  zW,2...,«  _1  d = F; (*(u,))  (2.33)  (2.34)  1  where F (.) and Fd(.) are cumulative distribution functions for the variable JC, and d, respectively, and <D(.) is the Standard Normal function.  As already discussed for the  forward procedure when the basic variables are correlated, an intermediate vector of Standard Normal variables y is obtained first by using y=Lu  (2.35)  22  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  In the above equation, the matrix L satisfies  (2.36)  R =LL  T  0  where Ro is the correlation matrix of the variables y, which can itself be obtainedfromthe correlation matrix R of the basic variables, as already discussed. Eqs.(2.33) and (2.34) are then used to obtain the basic variablesfromthe vector v.  2.3.2  Proposed algorithm for a single design parameter  A direct approach to find the design parameter associated with the design variable is now described.  As shown in the development of the forward reliability procedure FORM, the vector u at the design point must satisfy V„G M r  « = (•V „ G  r  (2.37)  V„G  where V G denotes the gradient of G with respect to u. M  The reliability index /? is then given by -V G « r  P =( V „ G  (2.38)  t t  r  V G ) 1/2 u  Combining (2.37) and (2.38), one obtains  23  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  u —  -PV G  (2.39)  U  ( V „ G V G ) 1/2 r  t t  Equation (2.39) together with the constraint G = 0, is now used as the basis for the proposed inverse reliability algorithm. Given u in the Standard Normal space, the value of the function G could be obtained by calculatingfirstx and d from, respectively, u and uj_. x  Since the statistics for d influence the value of G through the mapping of uj into d, G can be regarded as dependent on both u and d (or a). Let us assume that the desired design parameter is d with an initial value d = d .By using a truncated Taylor expansion of G on 0  d, at d and conditional on u = u , n  0  G(u ,d) = G(u ,d ) 0  0  0  dG(u ,d) 0  dd  (2.40)  (d-d ) = 0 0  From which G(u ,d ) g  d=d 0  (2.41)  0  dG(u^,d) 3d  For an initial pair (« , d ), and corresponding gradient V G(«,<i)| 0  0  u  U|)  ^ , Eq.(2.39) is used  once to obtain a new vector u, which will then satisfy u u = fi . Conditional on this u, T  (  dG(^,d). dd  )U  2  is calculated numerically and Eq. (2.41) is iterated to find d that satisfies  the constraint G = 0. With this new d, Eq.(2.39) is used again to obtain an upgrade of u. The process is repeated until convergence for both d and « is achieved. Convergence is obtained when, for example, the change in the Euclidean norm of u is less than a specified  24  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  small fraction of the norm for the previous u. In the case that the standard deviation of d is the design parameter, the same procedure is used by replacing, in the above relationships, tf and d with o and a, respectively. The gradient W G(u,d) 0  0  u  or numerically by using Eq. (2.28) based on V G(x,c/)|  ^ can be obtained directly  . Since the limit-state function  1  x  u  G is given in the basic space, the uncorrelated, Standard Normal variables u are, in general, converted first to correlated, standard Normal variables y using Eq.(2.35), and then the basic variables JC and the design variable d are obtained by the transformations in Eqs. (2.33) and (2.34).  Three special cases can be noted: 1) If the limit-state function G is of the form G( u, y  ) = g(x, y  cap  with y  cap  cap  ) =y  - r(x) = y  cap  cap  - R(u)  (2.42)  a deterministic design parameter, and r(x) or R(u) some given functions,  Eq.(2.39) becomes  ^ " n  (2-43)  (v R VuRY T  u  This equation can now be used to iterate for the vector u. When convergence is achieved, y  cap  is directly obtainedfromy  = cap  R(u).  2) If the design parameter d is deterministic, u will only contain the u components x  corresponding to the vector JC. Accordingly, the V G will only include the gradients U  with respect to u . With only this modification, the procedure based on Eqs.(2.39) and x  25  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  (2.41) can now be equally applied to finding the value of d. An alternative is to use the general procedure, regarding d as a random variable, but setting for it a very small coefficient of variation and searching for the mean value. 3) If the target reliability is B = 0, then it = 0 and the design parameter will be directly obtained by iterating Eq.(2.41). This case may be encountered, for example, in some serviceability limit states when only near average performance is required in design.  2.3.3 Case studies To demonstrate the efficiency and accuracy of the proposed method the following case studies were carried out by using the software IRELAN (Inverse RELiability Analysis) developed as part of this thesis.  2.3.3.1 Deterministic design parameter A limit state function with a single design parameter, as shown in the work of Der Kiureghian et al. (1994) is used first to show the applicability of the proposed method and to offer a comparison with results in the literature. The function is G = exp[ - 0( it! + 2 u + 3 u )] - u + 1.5 2  3  (2.44)  4  where the vector of random variables u = (uj, u , us, U4 ) is in the Standard Normal space T  2  and uncorrelated. The specified reliability index is j3 = 2.0.  Here we treat 6 as a deterministic design variable. The initial values for the variables were u = (0.2, 0.2, 0.2, 0.2) and 6 = 0.15. The tolerance for convergence was 10". Table 2.1 r  0  4  0  26  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  shows the results and iteration details using the described procedure. After 7 iterations the design parameter was obtained as 0.367183. To check its accuracy, using 9 = 0.367183 in Eq.(2.44) results, by forward reliability FORM, infi=1.999964. The value of 9 agrees with that given in the Der Kiureghian et al. (1994) example.  Table 2.1: Example 1 (Casel) Results Iteration  u  9  1  0.200000, 0.200000, 0.200000, 0.200000  0.150000  2  0.226887, 0.453774, 0.680661, 1.810890  0.367813  3  0.210263, 0.420527, 0.630790, 1.838760  0.367722  4  0.218043, 0.436086, 0.654128, 1.825760  0.367428  5  0.218261, 0.436522, 0.654782, 1.825530  0.367288  6  0.218272, 0.436545, 0.654816, 1.825580  0.367218  7  0.218275, 0.436550, 0.654823, 1.825610  0.367183  2.3.3.2 Mean value as the design parameter A rectangular cantilever beam with width B and depth H is subjected to two point loads: one at the free end of the beam, and another at a distance Lj from the fixed support as shown in Fig.2.2. Failure is considered to occur when the deflection at the free end of the beam exceeds a maximum. This deflection capacity at thefreeend is given by  (2.45)  where L is the span and K the limiting deflection factor. The deflection at thefreeend due to the applied loads is the demand on the system and can be calculated from simple beam  27  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  theory as  A  =  PA  3EI\  1  +  3^  ,  A ( A + 4 )  21,  3  (2.46)  3EI  the limit-state function can thus be written;  (2.47)  G =A  max  The intervening random variables were assumed to be the vertical loads Pj and P , the 2  modulus of elasticity E, the position Lj of the load Pj, the width of the beam B, and the depth H. L and K were treated as deterministic variables and assumed as L = 2m and K = 100. The target reliability level was specified as /3 =3.0 and the design parameter was the mean value H , of the random variable  All variables were first assumed uncorrelated and then, to show the effect of correlation, the variables Pi and P were assumed to have the correlation coefficients p = 0.5 or p = 0.9. 2  Also, upper bounds for Pj and P were considered. Table 2.2 gives the statistics of the 2  random variables and Table 2.3 presents the results when considering the different correlation coefficients and the upper bounds for the loads Pi and P . In Table 2.3, Pi and 2  P  2u  u  indicate respectively, those upper bounds.  The results show that when the correlation of the loads Pi and P increases the required 2  design parameter H increases as well. Also, the closer the bounds for the loads are to the mean values, the lower the required mean depth.  28  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  /  Pi  P  U  2  A  Figure 2.2: Sketch of the Cantilever Beam  Table 2.2: Statistics Variable  Means  Standard Deviation  Distribution  Xj:  Applied load Pj  100 K N  20 K N  Lognormal  X:  Applied load P  200 K N  50 K N  Lognormal  2  2  X3: Modulus: E  100xl0 KN/m 6  5  Width of the beam, B  X6~- Depth of the beam, H  15xl0 KN/m 6  2  Lognormal Uniform  0.5 m <L < 1.5 m  X4: Location of load Pi, Lj X:  2  }  0.10 m  0.005m  Normal  ?  COV = 0.05  Normal  Table 2.3: Design Mean Value H for Different Cases (m) Upper Bounds of Pi and P (KN) 2  ]u  Pi  u  Pi = u  = 220, P = 2u  = \60,P  2u  140, P  2u  Pi =\20,P = u  2u  =0.0  0  =0.5  0  =0.9  0.455589  0.459738  0.463377  500  0.455551  0.459698  0.463333  = 350  0.451924  0.455806  0.458912  = 300  0.444962  0.448246  0.450556  250  0.431632  0.433858  0.435251  Plu = <*>, P2u - °° P  0  29  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  2.3.3.3 Standard deviation as the design parameter For the same problem described above, the mean value H is now fixed at 0.4m and its standard deviation is the design parameter. The calculations were carried out for different target reliability indices and correlation coefficients.  No bounds for the loads were  considered. The results are given in Table 2.4.  Table 2.4: Design Standard Deviation Target fi  p  =0.0  p  =0.5  p  =0.9  2.0  0.077971  0.077413  0.076953  2.5  0.058984  0.058167  0.057466  3.0  0.045107  0.043912  0.042828  These results show, as expected, that increasing the target reliability level results in a decrease in the design standard deviation, reflecting the linkage between reliability and low manufacturing tolerances.  2.4 Inverse reliability with multiple design parameters 2.4.1  Statement of the problem  When multiple design parameters are considered, three distinct cases need to be addressed: 1) The number of the design parameters is the same as the number of constraints, which might be reliability-related or geometric, and 2) The number of design parameters is greater than the number of constraints.  30  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  3) The number of design parameters is less than the number of constraints. This thesis will address the first two cases, while the third will be left as a topic of future research.  For the first case, a unique solution may be obtained since enough non-linear constraint equations are provided. For the second case, a unique solution can only be obtained if the problem is coupled with the optimization of a relevant objective function, as will be discussed in Chapter 3.  For the first case, when the number of design parameters is the same as the number of constraints, we assumefirstthat the constraints are n limit state functions G, with associated target reliabilities /?,. Suppose that d = (d d , ... d ) is a vector of n design parameters r  ]t  2  n  which might include mean values or standard deviations for some design variables, regarded as random. According to the given target reliability level, two sub-problems are now stated as follows:  •  When all the individual target reliability levels are specified, Given:  fi  (i=J,2,3,...n)  Find:  di  Subject to:  mm(u u) = /?, and G,(«) = g,{x,d) = 0  (2.48)  (i=l,2,3,...ri) (design parameters) T  2  31  (i =12,'A...,n)  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  •  When an overall target reliability level for a series system is specified, Given:  /3 t ov  Find: d , and /?, (/'=1,2,3, ...ri) (design parameters and target reliability t  mm(u u) = fi, and G,(«) = g,{x, d) = 0  Subjectto:  T  2  vector) (2.49)  (/ =1,2,3,...,n)  P = Prob(Gj<0 u G <0 ... vG <0 ) = 0(-/&v,)  And  f  n  2  In this case, the probability of failure will be given by Prob(G7<0 KJ G <0 . ..^JG„<0 ). This 2  probability can be bounded using narrow bounds (Ditlevsen,  1979). Given their  narrowness, the inverse problem can be implemented to achieve the target f3ovt of the average of the bounds.  2.4.2  Proposed algorithm for multiple design parameters  2.4.2.1 Individual target reliability levels are specified In this section a procedure is proposed to approach the problem described in Section 2.4.1 (Eq.(2.48)) (Li and Foschi, 1998). Applying the solution strategy previously outlined for a single design parameter, any design parameter d could be obtained conditional on all others t  remaining fixed for a specified reliability level /?,-. Implicitly, one could then write d in the t  form d =f {d ,d i  i  1  2  d .i,d j,...d„,{3 ) i  i+  i  (i =1,2,3,. ..,ri)  (2.50)  where /?, is the target reliability corresponding to the constraint G,. Relationships in the form of Eq.(2.50) can be similarly developed for each of the reliability constraints. The final determination of the parameters d involves the solution of this resulting system of t  32  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  nonlinear equations, a task which can be accomplished, for example, through a NewtonRaphson iterative algorithm. To this end, a vector F of functions F can be written from t  Eq.(2.50), ( i = l,2,..,n)  Fi = di -f ( d d ,..., di. d ,...,d„,Bi) u  2  u  i+I  (2.51)  with the Newton-Raphson iterations applied to arrive at F = 0. That is, startingfroman t  initial combination J*= ( d*, d *,... d„*) , the function.F, can be expanded in Taylor series T  2  up to linear terms  or  F  t  =d;  -/ (<,^..x,,<.,X,A)+Z^rlr(^ - O i  (' = ' >-rt ]  2  (2.53) Finally by setting F, = 0,(i = 1,2,...,«), we obtain n linear equations  Z^rlrVj-d])=fi(d;,di-.-,dU,d; ,..  (»= ^ - > « )  +l  (2.54)  Thus, the solutions can be obtained by iterating  d = d* -M\d*  where  M.. "  -/)  *  M,=1.0  (2.55)  (i = l,2,...,n), i= /  33  (j = l,2,...,n)  (2.56)  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  and  f=(fuf2,f3,...,fn)  (2.57)  T  It is noted that the matrix M has a unit diagonal and the off-diagonal terms can be obtained numerically by finite differences centered at d*, requiring repeated inverse reliability calculations for a single design parameter. Starting from an initial vector of design parameters, Eq. (2.55) is used as an iterative scheme until convergence is reached. When this is achieved, the design parameters obtained satisfy simultaneously the multiple reliability constraints. A general convergence criterion is not given here and probably does exist, as convergence is sensitive to the choice of the initial vector d. In practice, if convergence is difficult or it cannot be achieved, a new initial d could be tried.  In the case that geometric constraints are also included, they can be added to the reliabilityrelated constraints in the nonlinear system of equations (2.50), with the rest of the procedure remaining the same. It should be noted that a solution may not be achieved in all cases, since the set of contours represented by Eqs.(2.50) at fixed /?, may not have an intersection in the feasible domain.  2.4.2.2 Target reliability level of a series system is specified For the problem (b), when a given overall target reliability p t is specified for a series ov  system, the procedure described in the previous section can also be applied. However, in this case, one has to determine the target P= (j3i, 02,-, Pnf vector first, before proceeding with the inverse analysis in which PJ, P2,-, Pn will be treated as target reliability levels for  34  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  limit-state functions Gj, G2, ... G„, respectively. In this problem, the unknowns are n design parameters and n target reliability indeices. The n unknown design parameters can be obtained using the proposed method in 2A.2.1, as long as the target reliability indices are given. The calculated design parameters are then used to upgrade the target /? vector. This coupled problem can be carried out as described below:  1) Given an initial target reliability vector p , 0  2) Find the design parameters d based on fi  0  using the inverse reliability method  previously described in 2.4.2.1, 3) Based on the obtained design parameters, calculate the corresponding overall reliability Ps by taking the average of the system bounds, 4) Compare the p with the overall target reliability level p : if the difference between s  ovt  them is sufficiently small, the calculation is finished; otherwise, upgrade the target P vector by using an interpolation or Newton-Raphson method, returning to step 2), until convergence is reached.  It is noted that there is an infinity of combinations of /?, which can result in the same overall target reliability being achieved, if there are not enough pre-specified constraints among the components of the vector p. For simplicity, in this thesis, these constraints are considered only as the ratios among the components of the P vector,  p = k Pi; 2  2  p = hp 3  i;  p„ = k„Pi  35  (2.58)  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  where k , k , ...£„ are specified constant ratios and fij is the free component to be determined 2  3  by the Newton-Raphson method to achieve the target overall reliability fl . ovt  That is, if  r = 0M)-fio«  (- ) 2  59  then  ^ /dB > P^P; x  After obtaining /3j, the other components of the /? vector can then be calculated by Eqs.(2.58). 2.4.3  Case studies  Example 1 In this example, the proposed method is applied to a multiple design variable problem when individual target reliability levels are given for each reliability constraint. A set of three limit state functions G are given involving four random variables, x x , x and xs. J}  Gi = xi  2  3  - 4x - 2x3X4 2  G = 2xix -x x 2  G3  2  4  — x 1X2X4  2  -  (2.61)  3  2x$  with given target reliability indices fii = 3.0, /3 = 3.5 and fi - 4.0. 2  36  3  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Case 1: the mean values of three of these variables are taken as the design parameters. All coefficients of variation, the mean value of the fourth variable, and the assumed distributions are shown in Table 2.5. All variables are assumed uncorrelated.  Table 2.5: Case 1: Statistics Variable  Mean Value  Coefficient of Variation  Distribution Type  Xl  ?  0.01  Normal  x  ?  0.2  Lognormal  x  ?  0.1  Lognormal  x  1.0  0.1  Gumbel  2  3  4  For an initial value of 5.0 for each of the design parameters, thefinalvalues were obtained after a few iterations. Table 2.6 gives the solution and iteration details, with corresponding means of4.364, 2.162 and 1.783.  Table 2.6: Case 1: Solution Iterations  Mean Value of xj  Mean Value of x  Mean Value of X3  0  5.000  5.000  5.000  1  4.290  1.508  0.754  2  3.748  1.631  1.168  3  4.307  2.086  1.650  4  4.359  2.157  1.777  5  4.364  2.162  1.783  6  4.364  2.162  1.783  Forward B  Pi  Check  3.000  2  Ps  3.500  37  4.000  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Case 2: In this case the standard deviation of variable xj is chosen as one of the design parameters, together with the means of x and x . The statistics are shown in Table 2.7, with 2  3  the corresponding results in Table 2.8.  It can be verified that, in either Case 1 or Case 2, a forward reliability calculation does validate the target reliability indeices, as shown in the tables.  Table 2.7: Case 2: Statistics Variable  Mean Value  Coefficient of Variation  Type  X]  6.0  ?  Normal  X2  ?  0.2  Lognormal  X3  ?  0.1  Lognormal  1.0  0.1  Gumbel  x  4  Table 2.8: Case 2: Results Iterations  Standard Deviation of xj  Mean Value of x  Mean Value of x  0  0.600  3.000  3.000  1  0.769  2.130  2.330  2  0.773  2.182  2.059  3  0.768  2.196  2.079  4  0.768  2.196  2.079  Forward /?  Pi  P2  Ps  Check  3.000  3.500  4.000  2  38  3  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Example 2 This example shows an application when the overall target reliability level B  ovt  is given for  a series system. The problem is the design of the cross section of a simple supported beam with depth H and width B. (Fig.2.3) Two failure modes corresponding to the maximum normal stress and the maximum deflection, respectively, were considered below.  PL  Here a  y  3  is the yield stress, P the applied point load, D  cap  the allowable deflection assumed  30mm and L the beam span assumed 5000mm.  The objective was to find the design variables so that a target overall reliability level B  ovt  can be reached. As described in section 2.4.2.2, the initial B vector was given with a fixed ratio between two individual indices Bj and B corresponding to, respectively, Gj and G . 2  2  Let H and B have normal distributions with coefficient of variation 0.01 and 0.01, and assume that all the variables are uncorrelated and with the statistics and distribution types given in Table 2.9.  39  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  P  Figure 2.3: Sketch of a Simply Supported Beam  Table 2.9: Statistics Coeff. Of Variation  Distribution  0.1  Lognormal  0.1  Lognormal  50000 N  0.1  Lognormal  B  ?  0.01  Normal  H  ?  0.01  Normal  Variable  Mean  °>  350 N/mm  E  350000 N/mm  P  2  2  Given the target overall reliability fc = 3.0, the inverse reliability analyses were carried vt  out for different ratios between fc and fc to show the effect of ratios on the design parameters.  Table 2.10 shows iteration details and the final results(in bold characters). The inverse analysis gives the desired design parameters, so that the individual reliabilities fc and fc, in the chosen ratio, allow the overall target reliability to be reached. In this example the ratios between fc and fcwere assumed as fc/fc = 1.2, 1.5 or 2.0. As expected, changes in this have a significant influence on the final results.  40  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Table 2.10: Results and Iteration Details  Ratio=1.2  Ratio=1.5  Ratio=2.0  2.5 2.5.1  Pi  2.00  3.005  3.028  3.029  P2  2.40  3.606  3.634  3.635  Pov  1.897  2.975  2.999  3.000  H  147.68  152.30  152.41  152.41  B  65.376  70.951  71.088  71.091  Pi  2.00  2.982  3.000  3.000  P2  3.00  4.474  4.501  4.501  Pov  1.984  2.982  3.000  3.000  H  161.06  173.22  173.45  173.45  B  54.961  54.673  54.668  54.668  Pi  2.00  2.999  3.000  3.000  P2  4.00  5.999  6.000  6.000  Pov  2.00  2.999  3.000  3.000  H  186.12  215.45  215.4.7  215.47  B  41.157  35.426  35.423  35.423  I n v e r s e response surface m e t h o d Review of the response surface method  Reliability analysis requiring the evaluation of the performance of a large or complex system is usually carried out using other computer programs, which can sometimes be computational-intensive. For instance, the nonlinear dynamic behavior of a structure under earthquake loading can be obtained from a complete time-history dynamic analysis which may demand a substantial computational effort. In reliability analysis (FORM, SORM), such an analysis must be performed many times in the calculation of the limit-state function  41  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  as well as its gradient, which may result in an unacceptable computational cost. The response surface method (RSM) was proposed as an approximate procedure to reduce the computational effort to an acceptable level. A response surface provides a relationship between system response and values of the input variables. Some contributions and applications have been made in enhancing the efficiency and accuracy of this method. Bucher and Bourgund (1990) presented an adaptive scheme to reach the real response surface in the range of the design point. Yao and Wen (1996) proposed a methodology in conjunction with the fast integration technique suggested by Wen and Chen (1987) to provide a limit-state formulation which is computationally simple for applications to timevariant reliability analysis. In the method presented by Kim and Na (1997), by using a gradient projection, the response surface is constructed based on linear type response functions rather than quadratic functions. All of these methods are based on having information about the structural system and all the intervening variables, which of course are only available for the forward reliability analysis.  The response surface allows for an easy and simple computation of the structural system behavior without losing account of its essential properties. Based on observations from the original system, an alternative function representing the relationship between input parameters (loading and system features) and output parameters (response in terms of deflections, stresses, etc.) is constructed to make the reliability analysis computationally feasible.  42  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  The RSM that will be considered in this thesis is that proposed by Bucher and Bourgund (1990), in which an incomplete two-degree polynomial type function gn(x) was suggested to represent the original function g(x).  (2.64) i=l  where x , t  ( /' = J,...,  ri) are the basic variables and the parameters a,  c,, ( i=l,...,n  b  u  ) have  to be found by collocation. It is noted that 2n+l observations are required to obtain the 2n+l unknown coefficients. The suggested procedure to obtain observation points and construct a response surface in the reliability analysis was as follows:  1) Choosing the mean values of the intervening random variables as the anchor point, the actual gix) is calculated at the mean and, in turn for each variable, at the mean plus three standard deviations and the mean minus three standard deviations. 2) Based on the 2n+J calculated values of g(x), the parameters  a, b  h  c,, (  i=l,...,n  ) are  obtained by solving the set of linear equations for the collocation. 3) Reliability analysis (FORM, SORM) is then implemented to determine estimates of the reliability index and design point using the response surface gR. Since the real surface will not be covered sufficiently only by one interpolation, an upgrade of the response surface is then required. 4) Once the initial design point XD is found, gixo) is evaluated and the new anchor point XM for next interpolation is given by linear interpolation (Bucher and Bourgund, 1990) between the mean value and design point XD, i.e.,  43  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  X  » * =  +  (  X  °-  X  \W-1(*  D  )  ( 2  '  6 5 )  5) Based onxM, step (1) to step (4) are repeated, an updated response surface is obtained, as well as a new design point. This process is repeated until convergence is reached for B or the design point.  2.5.2  Inverse response surface method  The response surface is an alternative to the real limit-state function. However, in contrast to the forward approach, the function values that are used to construct the response surface are not available until the desired design variables are determined.  Based on the adaptive response surface method (Bucher and Bourgund, 1990) and the previously proposed inverse reliability algorithm, an inverse response surface method (DR.SM) is now presented. Numerical examples will be given below to demonstrate the efficiency and accuracy of this method.  In the inverse reliability analysis, some system parameters representing mean values, standard deviations or deterministic parameters are to be found for a given reliability level. An iterative scheme to upgrade the response surface and, at the same time, to accomplish the inverse reliability analysis is proposed as follows:  1) With initial values for the design parameters, the initial response surface is constructed using the foregoing response surface method. Based on this approximate response  44  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  surface the inverse reliability analysis is carried out and a new estimate of design parameter is obtained as well as the design point xz>.  2) Take xo as anchor point together with the previously obtained design parameter and update the response surface. Based on this updated response surface the inverse reliability is carried out again to seek the new design parameter and design point XQ.  3) This process is repeated until the convergence is achieved at the design parameter with acceptable tolerance. Again, no convergence theorem is developed here but convergence has been observed in all cases tried during extensive testing of the scheme.  2.5.3  Case studies  An explicit nonlinear limit-state function used by Kim and Na (1997) is selected here to demonstrate the validity and accuracy of the proposed procedure. g(x) = exp ( 0.4 (xj + 2 ) + 6.2 ) - exp ( 0.3 x + d) - 200 2  (2.66)  where d is a deterministic parameter to be found and the target reliability index is assumed P = 2.710. xj and x are standard normal variables. In the inverse analysis, the parameter d 2  is treated as a random variable with very small standard deviation. The software IRELAN (Inverse RELiability ANalysis) developed based on above proposed procedure was used to solve this problem. Assume that an incomplete two-degree polynomial is chosen as response surface in the form of: g = Co + d xj + C x + C xi + C x 2  R  2  2  3  45  4  2 2  (2.67)  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  where  C = B +B d 0  0  ]  +Bd  (2.68)  2  2  Since d is treated as a random variable, Bo, Bj and B are needed for interpolation. After 2  each iteration the constant Co can be obtained by Eq. (2.68). Table 2.11 shows the final results and iteration details.  From Table 2.11, we can see that after 6 iterations the converged value d = 5.0005 was obtained against the d = 5.0 obtained by JRELAN using the true G(x)fromEq.(2.66), the error is 0.01%. It was also determined that the sensitivity of the design variable with respect to the initial value is quite small. The only difference is the required number of iterations. Table 2.12 gives the sensitivity analysis results.  Table 2.11: Results and Iteration Details (initial design variable d is assumed as 7.0) Iterations  1  2  3  4  5  6  Design variable d  7.000  4.863  4.750  5.026  5.003  5.0005  C  -199.97  854.97  818.56  780.86  790.69  790.55  Q c  551.78  487.72  376.01  374.19  379.46  378.99  -375.24  -29.101  -39.000  -51.766  -49.974  -49.864  C  98.777  60.987  34.816  34.503  35.414  35.332  -52.771  -12.897  -6.9266  -8.925  -9.327  -9.2475  0  2  3  c  4  From a sensitivity analysis it is known that the choice of initial design variable will not significantly affect the final result. For some complex systems the response surface method is the only way to approach both inverse analysis and forward analysis since there is no other method which can give the solution with an acceptable level of computational effort.  46  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Table 2.12: Sensitivity of Design Variable with Respect to Initial Values Initial Value of Design Variable  Design Variable  Number of Required Iteration  1.0  4.9999  7  3.0  5.0020  6  6.0  5.0018  5  7.0  5.0005  6  10.0  5.0010  9  In order to give the graphic interpretation of the proposed iteration scheme, we rewrite the foregoing limit-state function (2.66) as g(x) = exp ( 0.4 ( xj + 2.0 ) + 6.2) - exp ( d) - 200.0  (2.69)  Using the same statistics of random variable and target reliability as above, the design variable d is found as d= 5.142. Figure 2.4 shows how response surfaces approach the real surface graphically.  1500 G  i  1000 -  /  Original function  |3rd u p d a t e d R S I 2nd updated R S "V  /  / I  -4  L^^^^^^  -3  I  |  ^ ^ ^ ^  J^"^  [Initial R S |  ^^^a00 -  I  0  -1  I /  2.710  /  [1st u p d a t e d R S | 1  ^-500 •  Figure 2.4: Graphic Interpretation of IRSM  47  2  3  x,  4  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  2.6 Inverse problem with multiple solutions for the design parameters 2.6.1 The existence of the problem It was observed that in some inverse reliability problems, the search for the value of the design parameter might offer difficulties, as several solutions may be available for the same design requirement. Also, the Newton-Raphson method may fail to find the solution due to the existence of the local extrema. Figure 2.5 gives an illustration of such an issue. Assume that a curve representing the relationship between B and the design parameter d was built by repeating the forward reliability analysis. At the target reliability level Bj= Bj a unique d would be found, but at B = B , four values of d would be found, all of which meet the T  2  reliability requirement. For example, in dynamic analysis, different masses could result in the same dynamic response under a ground excitation due to the occurrence of resonance. The change of mass will change the natural frequency of the system, thus resulting in a variation of the mass at which resonance will occur. In such a case, the original direct inverse reliability method, which is based on the Newton-Raphson method, may fail to locate all the solutions or even fail to achieve convergence if the initial design parameter is not chosen carefully. There remain two problems in 1) how to locate all the solutions in the feasible domain and 2) how to choose a solution among the multiple solutions. The only robust method is to consider all the feasible values of the design parameters, in small increments, repeating the forward reliability analysis, and choosing the solution corresponding to the target reliability level. This problem is inevitable when dealing with the dynamic response of a structure in an earthquake-related inverse problem. Two methods to approach the problem are discussed in the following sections.  48  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Figure 2.5: Relationship Between B and Design Parameter d  2.6.2  Equivalent deterministic method  A typical problem in earthquake engineering is used to show the proposed method. When conducting inverse reliability analysis, if the acceleration spectrum is fixed, in general the most important random variable is the peak ground acceleration, denoted as aa. By neglecting the randomness of the other intervening variables, there is only one random variable in the problem. Therefore, the given target reliability index /3T is equal in magnitude to the normalized value of the peak ground acceleration. Based on this fact, the value of ao at the design point can be determined according to its statistical distribution, thus fixing the ground excitation. The dynamic analysis can then be carried out for the design parameter varying from the lower bound to the upper bound of the feasible domain in a reasonable increment. The relationship of maximum specified earthquake response, for example the maximum deflection, or the maximum stress, against the design parameter can  49  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  be obtained, providing a constant reliability contour at PT. This curve permits an easy and fast way to locate all the possible design parameters approximately. This method permits a pre-selection of the design parameters and the bounds used in the following direct inverse reliability method to estimate the effect of the other random variables on the design parameters.  2.6.3  Hybrid of the Newton-Raphson method and bisection method  By hybridizing the Newton-Raphson method with the bisection method (Press, et al., 1986), all the solutions can be located, one by one, as long as one can isolate intervals for them. Let DL be the lower bound and Du the upper bound of a potential solution, and G(DjJ the limit-state function at the lower bound and G(Du) the limit-state function at the upper bound. The modified procedure used in the inverse reliability method is described as follows: 1) Given a lower bound DL and a upper bound Du of a potential solution so that the product G(DL)G(DU) is negative, inferring a solution exists between DL and Du. Any value inside the interval can be chosen as the initial design parameter. 2) With an initial design point u and the do, the iterations start using Newton-Raphson method and an upgraded design parameter d is obtained using Eq. (2.41). Once d is out of the given interval, instead of the Newton-Raphson method, the bisection method is used to obtain the upgraded d by linear interpolation or taking average of the bounds, which ensures the solution being in the interval. In the process of the iteration, the  50  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  interval is narrowed using the newly obtained d. Finally the solution can be found conditional on the u vector. 3) Conditional on the newly obtained design parameter d, the u vector is upgraded using Eq.(2.36). Step (2) is then repeated to seek the new design parameter. The iterations repeat until the convergence is reached. 4) The next solution can be obtained by providing another set of bounds, and proceeding throughout the entire feasible domain.  The advantage of this hybrid method is that it will never fail to find the solution with a reasonable convergence speed, since the bisection method will only be used when the solution is outside the bounds. It should also be noted that the drawback of this approach is that one has to provide the initial bounds ahead of the analysis, however as mentioned before, the equivalent deterministic analysis can provide such bounds approximately. Furthermore, in the implementation, due to the upgrading of « vector, the inequality G(DL)G(DU)  <  0 cannot be always satisfied. If the product G(Dj)G(Du) is positive, a slight  adjustment for the bounds will be used to change the bound which gives the lower absolute value of the G. After obtaining all the possible solutions, the most suitable solution, from the safety point of view, is the one which has the lowest sensitivity of the reliability index B with respect to the design parameter. The following gives the derivation of the formula for calculating such sensitivity factor.  Let p be a parameter in the limit-state function G.  51  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  From the definition of B we have B =uu 2  (2.70)  T  where u is the vector of random variable defined in the standard Normal space. Take the derivative for both sides of Eq. (2.70) with respect to p,  2 / ' j " = i > , t dp t{ dp  dl^u du L  dp  since  L  i==1  =  (2.71)  rdu  n  B dp  (  2  )  (2.73)  u  r  ?  dp  V G(n) n=- _ ^" „„ (V G VG)' r  2  1 / 2  u  V =  dp  U  G » ^  f  (2-74)  ^17T  (VG («)VG(«)) r  ,  1/2  taking the derivative for the limit state function with respect to the parameter p, we have  ^ =- V G » — dp dp  (2.75)  Combining Eq.(2.74) with Eq.(2.75), one obtains the formula for calculation of the sensitivity dG dp _ dp  dp (V G V„G) T  (2.76)  1/2  U  52  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  2.6.4 Examples Numerical examples are shown below to illustrate these two approaches. To simulate a dynamics problem, let the limit-state function be: G = D - Dmax  (2.77)  Do = 0.2 m  (2.78)  0  With  D  max  Where D  max  = xj sin(2M)-x cos(2.5M)+xj Msin(3M)+x Mcos(0.0JM) 2  2  (2.79)  simulates the maximum deflection of a single degree offreedomstructure, say  a pile, M the mass carried by it, xj the peak acceleration and x the other intervening 2  random variable. Do is assumed as the allowable maximum deflection. The objective is to find the mass M so that a given target reliability level B = 2.0 could be reached. Table 2.13 provides the statistics of the two random variables.  Table 2.13: Statistics of the Random Variables Variables  *(2)  Mean Value  COV  Type  0.05  0.6  Normal  0.025  0.4  Normal  Based on forgoing discussion, two methods will be used to approach this problem.  (1) using equivalent deterministic method  53  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  Given a value of M, a forward reliability analysis indicated that the most important random variable is xj. Therefore, the x was assumed deterministic and fixed at its mean value. 2  Since the normalized xj at the design point must equal 2.0, the target B, one can obtain the value of the x* at the design point in the basic space using the following simple relationshipfromthe Normal distribution: x * = Mean + B xStandard deviation = 0.05 + 2.0(0.6 x 0.05) = 0.11 }  Substituting x* into Eq. (2.79) and calculating the D  max  (2.80)  for various M, changingfrom0.0 to  8.0 in increments of 0.5, a contour of B = 2.0 was obtained as shown in Fig. 2.6. It is seen that three possible solutions could be found to meet the target reliability, if the limit is Do = 0.2, and a unique solution could be located if the limit is Do = 0.4. For the former the three solutions areM= 0.8922,M= 1.640, andM= 2.502, respectively.  1  -0.2  Figure 2.6: Relationship Between D  max  54  and M  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  (2) using the hybrid method The original direct inverse reliability method, combined with the hybrid method described above, was used to seek the solutions. Two random variables are now considered. From Fig. 2.6 we can easily determine approximately the location of the solutions as well as their bounds required in the inverse analysis. Inverse reliability analyses were carried out to find the required values of M and their sensitivities to the reliability index B. The summary of the results is given in Table 2.14. The sensitivity results indicate that the most suitable solution is M = 1.670, which gives the lowest sensitivity of 2.245. Comparing the results with those obtained by the previous approach we can see there are some minor differences resultingfromthe consideration of the random variable x . 2  Table 2.14: Summary of the Results Cases  Lower Bound  Upper Bound  Mass  Sensitivity  Solution 1  0.5  1.0  0.876  -5.616  Solution 2  1.0  2.0  1.670  2.245  Solution 3  2.0  3.0  2.483  -4.750  2.7 Summary In this chapter, the theoretical background for solving various inverse reliability problems has been presented. Each strategy has been illustrated with several examples to show the efficiency and applicability of the procedures. These cover the cases of single or multiple design variables, which can be treated as deterministic or random. In the latter case, the method allows for the determination of the mean and standard deviations of the  55  Chapter 2 Inverse Reliability - Theoretical Background and Case Studies  corresponding design variable. Imposing a number of reliability-related or geometric constraints equal to the number of design parameters is a necessary, but not sufficient, condition for a unique solution. In the multiple-design variable problem, the strategy calls for using the single variable algorithm in conjunction with reliability contours developed for each limit state. Furthermore, an inverse response surface method (IRSM) was proposed to approach complicated and time-consuming problems. A hybrid of the Newton-Raphson technique with a bisection method has been described for inverse reliability problems when there are multiple solutions for the design parameter or difficulties in convergence due to local extrema. The inverse reliability procedures presented here are efficient methods to estimate directly design parameters corresponding to given target reliabilities. In some applications, particularly in multiple design variable problems, the procedure provides a cost-saving alternative to applying and interpolating results from a standard forward method.  56  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  3.1 Introduction As discussed in Chapter 2, in order to find a unique solution to an inverse reliability problem with multiple design parameters, their number has to be equal to the total number of reliability or other type of constraints. In practice, however, the number of design parameters may be larger than the number of all intervening constraints. For example, in designing the cross section of a box beam, for only a serviceability limit-state with a given target reliability level, there could be three design parameters to be found: the depth, th$ width and the thickness of the beam walls. Conditional on any two of them, the third one can be obtained by the inverse reliability method. In such a way, one could find an infinity of solutions satisfying the reliability constraint. A unique optimum solution could be obtained by introducing an optimization with an objective function related, for example, to the area of the cross section. The problem thus becomes one of reliability-based optimization, as reviewed in Chapter 1, which has been studied extensively in the last two decades. Conventionally, it can be approached by taking the reliability requirements as nonlinear constraints in the optimization, requiring repeated forward reliability analysis.  As an alternative approach and extension of the inverse reliability method, an inverse reliability-based optimization procedure is now proposed. In contrast to traditional reliability-based optimization, this method permits the separation of the ordinary  57  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  optimization method and the inverse reliability analysis. The latter is performed to provide the equality constraints, as will be discussed in the following section, which can be eliminated by substituting them into the objective function. Any general non-linear programming method, for example, the quasi-Newton method, quadratic approximation method, penalty function method and generalized reduced gradient method, simplex and complex method etc., can be used to obtain the independent (free) design parameters by performing the optimization. The optimum results will meet both equality reliability constraints and the minimization requirement. Numerical examples demonstrate the validity and applicability of this alternative approach as well as the comparisons with other work in the literature.  3.2 Procedures The reliability-based optimization problem to be solved is stated as follows: Find the design parameters:  d = (dj, d , ... d )  (3.1)  T  2  n  Minimize the objective function: F = F(dj, d ,... d„)  (3.2)  2  Subject to the reliability constraints associated with the limit-state functions: Gt (xj, x ,... xi) = 0 2  with given /?, (i =1,2, ...m)  ^  (3.3)  where: d is a vector of n design parameters which might include mean values or/and standard deviations of the design variables regarded as random; JC = (xj, x , ... x/) is a T  2  vector of the intervening random variables associated with the limit-state functions G, (  58  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  i=l,2,...m)  and F is assumed as a deterministic objective function involving the design  parameters d. It is assumed that the problem has only deterministic constraints (simple geometric constraints) and m reliability constraints associated with the limit-state functions G, (z = 1,2, ...,m),  with corresponding target reliability levels /?,(/' = l,2,...,m). Taking advantage of  the inverse reliability method outlined in Chapter 2, the reliability constraints can be eliminated in the manner described as follows: 1) A pre-specified combination of m design parameters, for example, the first m of the n, could be implicitly expressed by using the inverse reliability method conditional on all others remainingfixed.Therefore, one could implicitly write di  (i = 1,2,...,m) in  the  form dt =ft (d i, d dn,Bi) m+  2)  ( i = 1,2, ...,m)  m+2  Substitute the calculated d (/ = 1, t  2, ...,m)  (3.4)  into the objective function F = F(dj,  d, 2  dr), thus causing the number offreedesign parameters in the objective function to be reduced to n-m. An upgraded objective function is then represented by F = F (d +j, m  d +2 m  ,...,d„)  with n-mfreeparameters. Any nonlinear programming  technique can then be employed to implement the optimization and thefinaldesign parameters can be obtained which will meet all the reliability constraints as well as the geometric constraints.  59  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  The procedure is illustrated by a simple flow chart as shown in Fig. 3.1:  Start optimization  Ask for objective function F with upgraded d , d d „  Determine the free design parameters:  m+J  m  +  2  dm+l, d +2 d„ m  Main optimization engine  Inverse reliability di =fi (d ,  d  m+]  m + 2  d^Bj)  (i=l,2, ...m) End of operation with solutiond (i = 1,2,...,m) t  Calculate objective function F = F(dj,  d , ...,d ,d i, 2  m  m+  d  ,  m+2  ...,d„)  Figure 3.1: Flow Chart of the Inverse Reliability-Based Optimization  Based on above flow chart, the inverse reliability-based optimization can be implemented as follows: 1) According to the number of reliability constraints, determine the free and dependent design parameters. 2) Carry out the nonlinear optimization on the free design parameters. When requiring the evaluation of the objective function, an inverse reliability analysis is used to calculate  60  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  the non-free design parameters conditional on the upgraded free design parameters using Eq. (3.4), thus permitting the calculation of the objective function F. 3) The iteration is repeated until convergence at thefreedesign parameters is reached.  As a general procedure, the design variables are treated as random variables in the proposed scheme. Within this general procedure, a deterministic design variable can also be included, regarding it as random, searching for its mean value and setting a very small coefficient of variation. It is noted that, by eliminating the reliability constraints using the inverse reliability technique, the optimization can be carried out in an ordinary manner.  If the limit-state functions associated with the reliability constraints need intensive computational efforts, the explicit expressions of the dependent design parameters d ( / = t  1,2, ...,m) conditional on free design parameters d +j, d +2, ...,d„ can be constructed in the m  m  form of Eq.(3.4) by running IRELAN repeatedly in order to interpolate a response surface as described in Chapter 2. These expressions (response surfaces) are constructed ahead of the optimization based on the reliability-related constraints and the simple geometric constraints in which the observation points are obtained by inverse reliability analysis. These explicit expressions will remain the same in the later optimization thus no more limit-state function will be calculated, resulting in a drastic decrease of the computational effort. By doing so, the approximate solution, which is acceptable from the engineering point of view, could be obtained efficiently with reasonable computational efforts. A practical example of using the response surface to explicitly represent the reliability constraint will be shown in Chapter 5.  61  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  3.3 Case studies The reliability-based optimization procedure described above has been implemented in a computer program, called IRELBO (Inverse RELiability-Based Optimization), which is the combination of IRELAN and a non-linear optimization software (Xu, 1992) coded based on the Complex method, a constrained Simplex method, proposed by Box (1965). The Complex method permits one to find the optimal solution through direct searches by forming a configuration with 2n vertices without calculating the gradient of the objective function with respect to the design parameters. The iteration repeats to upgrade the configuration by replacing the worst vertex with a new one until convergence is reached.  It should be indicated that the program IRELBO was written in such a manner that any nonlinear programming routine could be easily introduced to perform the reliability-based optimization.  Example 1: This example illustrates two cases of reliability-based optimization to show the validity and applicability of the proposed method. There are four intervening random variables in the problems and three of them are treated as design variables denoted as xi, x and x . One 2  3  reliability constraint was considered with a given limit state function and associated target B. In the first case, the mean values of design variables were treated as design parameters, and were required to minimize a given objective function. In the second case, a combination of both mean values and standard deviation of the design variables were  62  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  considered as design parameters. The effects of correlation were investigated by considering the correlation between x and x with a correlation coefficient p = 0.8 in both 2  3  cases.  Case 1: Find the design parameters: (3.5)  d = (dj, d , d )  T  2  3  where di : the mean value of random variable xi d : the mean value of random variable x 2  2  d : the mean value of random variable x 3  3  with the geometric constraints: 0.0< di < 2.0 (3.6)  d >0.0 2  d >0.0 3  Minimizing the objective function: (3.7)  F = 2d? +3d +5d] 2  2  Subject to the reliability-related constraint G = x +x x -x  with a given target B  2  2  3  4  63  (3.8)  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  The statistics of intervening random variables are assumed in Table 3.1. The inverse reliability based optimization was carried out for different target B and the optimum solutions are shown in Table 3.2.  Table 3.2 shows that the required design parameters and the optimum function F increase with the increase of the target B, and that the correlation between xj and x does have a 2  significant effect on the magnitude of the design parameters. It is also noted that the mean value of xj is always chosen as its upper limit, no matter what the target B, resulting from its contribution to the limit state function and objective function, while the mean of xj has a tendency to exceed its upper bound. The forward reliability calculation does validate the target reliability being achieved.  Table 3.1: Example 1 (Case 1) Statistics Variable  Mean Value  Coefficient of Variation  Distribution Type  xi  ?  0.2  Normal  x  ?  0.1  Lognormal  x  ?  0.1  Lognormal  100.0  0.1  Lognormal  2  3  x  4  64  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Table 3.2: Example 1 (Case 1) Optimal Solutions p  Mean Value of x  Mean Value of xj p = 0.0  ,0  2  = 0.8  Mean Value of x  3  Optimum Value, F  p = 0.0  p = 0.&  p = 0.0  p = 0.&  p = 0.0  p = 0.8  1.0  2.000  2.000  12.16  12.43  9.46  9.63  898.4  935.36  2.0  2.000  2.000  13.30  13.86  10.30  10.72  1068.8  1158.9  3.0  2.000  2.000  14.53  15.44  11.23  11.94  1271.6  1436.1  4.0  2.000  2.000  15.84  17.19  12.27  13.31  1513.0  1779.9  Case 2: Find the design parameters: d = {dj,d ,d f 2  (-) 3  3  9  where di : the mean value of random variable xj d : the standard deviation of random variable x 2  2  d : the mean value of random variable x 3  3  with geometric constraints: dj >0.0 .  0.0<J <3.5  (3.10)  2  d >0.0 3  Minimize the objective function: F = 2rf, +3(4-rf ) +4rf 2  J  2  2  (3.11)  2 3  65  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Subject to the reliability constraint: G = x? + x x - x 2  3  with a given target B  4  (3.12)  The statistics of the random variables are assumed in Table 3.3, and the optimum solutions for different target B are given in Table 3.4.  Table 3.3: Example 1 (Case 2) Statistics Variable  Mean Value  Coefficient of Variation  Distribution Type  Xl  ?  0.1  Normal  x  40.0  ?  Lognormal  x  ?  0.2  Lognormal  x  100.0  0.1  Lognormal  2  3  4  Table 3.4: Example 1 (Case 2) Optimal Solutions p  Mean Value of xj  Std. Dev. of x  2  Mean Value of x  3  Optimum Function F  p = 0.0  p = 0.i  p = 0.0  p = 0.i  p = 0.0  /? = 0.8  p = 0.0  p = 0.S  1.0  0.3977  0.4439  3.5000  3.5000  3.2071  3.3771  42.21  46.77  2.0  0.6139  0.7324  3.5000  2.8746  4.0551  4.3731  67.28  81.37  3.0  0.9090  1.0410  3.1041  1.7007  5.0653  5.3398  106.69  132.08  4.0  1.3442  1.3051  2.5099  0.2993  6.2447  6.1565  166.26  196.10  In this case two mean values and one standard deviation were treated as design parameters. The reliability constraint was assumed the same as those in the first case. It can be seen that 1) the correlation has a significant effect on the second design parameter, especially, when the target /?is larger, and 2) at P= LO and P=2.0, the standard deviation of x is limited by 2  66  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  the geometric constraints and 3) the higher the target reliability level B the smaller the required standard deviation of x . 2  Example 2: This example is given to demonstrate the performance of the proposed method when multiple reliability constraints are considered at the same time.  Find the design parameters: d = (d d ,d ,d )  (3.13)  T  h  2  3  4  Where the d], d , d and d are the mean values of random variables X;, x , x and x^ 2  3  4  2  3  respectively. Minimize an objective function assumed as F = d +d + d + (6-d ) 2  2  2  2  (3.14)  2  4  Subject to two reliability constraints: G, = 4 X j X + 2x - xl  with a given target Bj  (3.15)  G = x,x x - 2x  with a given target B  (3.16)  2  3  2  2  3  4  2  The statistics of the random variables are given in Table 3.5, and the optimum solutions are shown in Table 3.6 for different pairs of target B.  67  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Table 3.5: Example 2 Statistics Variable  Mean Value  Coefficient of Variation  Distribution Type  Xl  ?  0.2  Normal  x  ?  0.1  Lognormal  x  ?  0.1  Lognormal  x  ?  0.1  Normal  2  3  4  Table 3.6: Example 2 Optimal Solutions Pi and B  dj  d  d  d  Optimal Value F  2.0,2.5  2.678  2.703  2.695  4.407  24.278  2.5,2.5  2.653  2.542  2.623  3.997  24.395  3.0,2.5  2.573  2.404  2.562  3.579  24.822  3.0,2.0  2.375  2.160  2.389  3.341  23.084  3.0,1.5  2.279  1.935  2.225  3.159  21.956  3.0,1.0  2.094  1.788  2.123  2.981  21.204  2  2  3  4  It can been seen that the multiple reliability constraints-related optimization can also be approached easily by introducing the inverse reliability procedure. However more computational effort is needed for obtaining the multiple design parameters in the inverse reliability analysis.  Example 3: To illustrate a practical case and to offer a comparison with the results in the literature, the problem of designing a simply supported beam, as shown in Wu and Wang (1996), was chosen.  68  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Fig 3.2 shows a simply supported box beam subjected to a concentrated load at the center. The span of the beam was assumed 3.048m (120 inch), treated as a deterministic variable, and the others, modulus of elasticity E, load P and cross-section dimensions t, B, H were taken as random variables. Their statistics are given in Table 3.7. The goal is to seek the dimensions of the cross section of the beam t, B and H to ensure that the mid-span deflection exceeds 0.00102m (0.04inch) only with a probability of 10' , which corresponds 3  to a reliability index B = 3.09. Furthermore, it is required that the cross-sectional area be minimized. This reliability-based optimization problem can be stated mathematically as:  Find:  the mean values of t, B and H, denoted by t , B and H . m  Minimize: area of the cross section C = B H m  m  m  m  - (B - 2t )(H - 2t ) m  m  m  m  Subject to reliability constraint: G = 0.04-PL /48EI with target B= 3.09 corresponding to 3  the probability of failure P = 0.001, f  and the simple geometric constraints:  0.00254m <t < 0.0127m, m  B > 0.127m and m  H < 0.381m m  L/2  L/2 wr  w\  H  A  7>77  B  Figure 3.2: Sketch of a Simply Supported Box Beam  69  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  The inverse reliability-based optimization was carried out for specified target reliability level. As an extension of the original problem, the sensitivity of the solution to target reliability levels and to the coefficient of variation of the design variables was investigated. Table 3.8 presents the optimum solution for different target reliability level and Table 3.9 shows the sensitivity of the solution to the coefficient of variations of the design variables, when all the design variables were assumed to have the same coefficient of variation.  Table 3.7: Statistics Coeff. of Variation  Distribution  1.99955xl0 kN/m  0.1  Normal  P  4.448 kN  0.1  Normal  t  ?  0.1  Normal  B  ?  0.1  Normal  H  ?  0.1  Normal  Variable  Mean Values  E  8  2  Table 3.8: Optimum Results Target B  t (m)  B (m)  Hm(m)  Minimum Area C (m )  0.0  0.00254  0.127  0.228  0.00178  1.0  0.00254  0.127  0.260  0.00193  2.0  0.00254  0.127  0.298  0.00213  3.09*  0.00254*  0.127*  0.351*  0.00240*  4.0  0.00254  0.165  0.381  0.00275  5.0  0.00254  0.298  0.381  0.00342  m  m  70  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  Table 3.9: Sensitivity to COV(B = 3.0)  t  cov  m  (m)  H  B (m)  H {m)  Minimum Area C (m )  m  m  0.10  0.00254  0.127  0.346  0.00237  0.12  0.00254  0.127  0.376  0.00253  0.14  0.00254  0.173  0.381  0.00279  0.16  0.00254  0.243  0.381  0.00308  0.18  0.00254  0.337  0.381  0.00363  0.20  0.00310  0.381  0.381  0.00468  From Table 3.8, we can see that the solution marked with an asterisk (*) agrees closely with the work done by Wu and Wang (1996) which gave t = 0.1 inch (0.00254m), B = m  m  5.0 inch (0.127m) and H = 13.78 inch (0.350m) with minimum area C = 3.72 inch  2  m  (0.00240m ) and which used finite elements to perform the structural analysis. It is also 2  seen that when the target B varies from 1.0 to 3.0 the desired mean values of t and B are constrained geometrically by their lower bounds and for B = 4.0 and B = 5.0 the mean of the H is constrained by its upper bound. A higher target reliability reflects a larger demand on the cross-section. It shows that the thickness t is insensitive to the target reliability level m  as it is always limited to its lower bound. Looking at the results of the sensitivity to the coefficient of variations, these were varied, simultaneously, from 0.1 to 0.2 with increment of 0.02. As expected, significant changes were observed for different COV. The required minimum area of the cross-section is almost doubled when the COV changes from 0.1 to 0.2. It is noted that, when increasing the COV up to 0.18, since the mean of the crosssection H is constrained by 0.381m, the mean of B has to be sufficiently larger to meet the reliability requirement. And B will keep increasing as the COV increases, until it reaches m  71  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  0.381m, which is assumed, in addition, as the upper bound of the mean width. When the other two dimensions are limited by their upper bounds, the mean thickness t  m  has to  increase to meet the reliability constraints, as shown in the last row of Table 3.9. It is expected that a larger COV in this problem may result in no solution since the t is limited m  at 0.0127m. This sensitivity analysis highlights the important role that quality control has in engineering design.  3.4 Summary An inverse reliability-based optimization method has been presented offering an alternative approach to standard reliability-based optimization. In addition, along with the methods proposed in Chapter 2, this approach completes the theoretical development of the general inverse reliability methodology presented in this thesis. The problem of inverse reliabilitybased optimization can be treated as a special case of a general inverse reliability problem, in which an objective function is introduced to obtain the optimum solution. Some advantages of using this approach are observed as: 1) through inverse reliability analysis, the reliability constraints can be converted to geometric constraints which maintain certain relationships among design parameters, allowing a separation between reliability analysis and programming; 2) it is easy to handle the problem with multiple reliability constraints and 3) for the problem which needs intensive computational effort, an explicit response surface representing each reliability constraint can be constructed prior to the optimization, offering tremendous saving in computations. Three numerical examples have shown the applicability and validity of the proposed method. The optimum solution can be reached  72  Chapter 3 Inverse Reliability-Based Optimization and Case Studies  with the reliability constraints being satisfied simultaneously. There is no special requirement for choosing a nonlinear optimization routine to be combined with IRELAN. In this Chapter, the optimization subroutine used is based on the Complex method (Box, 1965) where no calculation for the gradient of the objective function with respect to design parameter is required. For all the examples the required inverse reliability evaluations are below thirty. It should be indicated that the performance of the optimization is in general sensitive to the initial points, which itself is the common problem of all optimization procedures. A good initial point may be obtained based on engineering experience. The efficiency of the reliability-based optimization heavily depends on the optimization procedure being utilized (Wu and Wang, 1994).  73  Chapter 4 Applications in Engineering Design  Chapter 4 Applications in Engineering Design  4.1 Introduction The main objective in designing an engineering system is to ensure that it fulfills its function satisfactorily and safely during its service life. The performance of a designed system is dependent not only on the intervening variables, either related to load demands or strength characteristics, but also on the design parameters which are determined by the designer to meet given safety and serviceability criteria. Normally most design variables are recognized as random due to uncertainties, which may arise from variations in loads, material properties, dimensions of the components, insufficient knowledge in modeling the system response, and human error or lack of care in the design and construction. For example, the yield stress in a column and the shear demand will be part of uncertain design variables. On the other hand, the dimensions of the column will be part of the design parameters. The design parameters can be determined by taking into account these uncertainties, using the principles of probability and statistics theories, through reliabilitybased design. The proposed general inverse reliability method outlined in Chapter 2 provides a direct and efficient way to implement reliability-based design. In this Chapter, the method will be applied to practical engineering problems as follows: 1) calibration of an existing code or codification of a new code in which load factors or performance factors are determined on a  74  Chapter 4 Applications in Engineering Design  probabilistic basis; and 2) design of complex or special systems when there is no specified code to follow.  A general procedure of reliability-based codified design is reviewed and studied first to show how load factors and performance factors are determined traditionally and how the inverse reliability method accomplishes the purpose more efficiently. Several applications, focusing on practical engineering design, are then given to demonstrate the efficiency and applicability of the inverse reliability method, including design examples in offshore and earthquake engineering. For the former, the design maximum wave load and ice load for given annual exceedence probabilities are calculated. For the earthquake application, both a linear and a nonlinear example are discussed. In the linear case, the design mean values of the column stiffnesses in a multi-story elastic building are calculated to achieve specified target reliabilities for individual inter-story drifts. In the nonlinear case, the design mass carried by a pile foundation was calculated for a specified target reliability associated with pilecap displacement.  4.2 Reliability-based codified design A typical reliability-based design code procedure contains a design equation with loads and resistance factors applied to nominal loads and capacities. This example considers the use of inverse reliability to calibrate the factors in order to achieve target reliabilities. First, the traditional method of calibration using forward reliability is reviewed.  75  Chapter 4 Applications in Engineering Design  4.2.1  Basic concepts  Consider, for the above purpose, the elastic bending limit-state of a simply supported beam with rectangular cross-section. From static structural analysis, the maximum bending stress is given by  6 X 2  °"  WH  (D + Q)  (4.1)  and the performance function G is then written as:  G =R - a ^ = R - ^ ( D  +  Q)  (4.2)  where R is the bending strength; D is a uniformly distributed dead (or permanent) load; Q is a uniformly distributed live load, maximum throughout the service life of the structure; L, B, and H are the beam span, width and depth, respectively. As described in Chapter 2, the situations of G < 0, G > 0 and G = 0 correspond, respectively, to the failure, survival and limit-state of the system. The objective of reliability-based design (or limit-state design) is to find the design parameters B, H so that the probability of G > 0 will meet a specified target reliability level, formulated as Prob ( G > 0) = ®(J3), where 0(.) is the standard Normal cumulative function and 8 is the reliability index.  In order to provide a familiar design format for the designer and code user, Eq. (4.1) can be rewritten in the form of a design equation in which nominal loads and strengths will be used with associated adjusting factors (load factors and performance factor). The design equation is then written as  76  Chapter 4 Applications in Engineering Design  6 L  2  (a  D  D  + a  n  Q  Q  ) — — — =  n  <t>R  (4.3)  0  where D„ is the nominal (or design ) dead load and Q„ is the nominal live load. D„ is usually considered as average weights for materials where Q „ is usually a load with a low probability of being exceeded during the service life of the system, ao and CXQ are load factors associated with the dead load and live load, respectively; Ro is a characteristic bending strength, usually a lower percentilefromthe distribution of the variable R; and ^ is a resistance (or performance) factor associated with R.  The reliability-based calibration includes the determination of the load factors and performance factor, so that a target reliability level can be reached. Letting D Y =—  (4.4)  Qn  Q „ can be obtained by assuming that the design equation (4.3) is satisfied, which gives  ^_ _^_ %BH  </>R,  6Lr  ay)+ a  2  Qn=  2  r2  D  _  (4  Q  5)  Combining the Eq. (4.5) with Eq. (4.2), the G function can be expressed as  G  = R - - ^  T  Q  n  ( — r + ^ - )  which, from Eq. (4.5), gives:  77  (4-6)  Chapter 4 Applications in Engineering Design  G = R-—^—(3d (a y + a ) D  + q)  (4.7)  e  Where, d and q are the dead and live loads, respectively, normalized with respect to their corresponding design values:  rf-^. D.  , = -2.  (4.8)  Q.  Given the statistics of the random variables R, d and q, the performance function of Eq.(4.7) can be used to study the probability of the failure associated with the design equation for different combination of the load and performance factors. Conversely, if the target reliability level is given one of these factors can be obtained by the inverse method while others remain fixed. In this design case, if the load factors do and OQ arefixed,the unknown factor ^ can be found to meet a target reliability. This calculation can also be done by "trial and error" or interpolation by repeating forward reliability calculations. Figure 4.1 gives a graphic explanation of such approach, in which a curve representing the relationship between B and factor 0 is builtfirst,then the design factor ^ * can be found at the intersection of the curve with an horizontal at the target reliability level B . target  It is noted that the result will change if the statistics of the random variables R, d, q, or the ratio y are changed. A universal performance factor could be found to provide adequate reliability over a range of situations by carrying out a large number of such inverse calculations. Furthermore multiple factors also could be found simultaneously if the number of the limit-state functions is equal to the number of unknown factors. The  78  Chapter 4 Applications in Engineering Design  following shows an example of determining two design load factors by using inverse reliability method.  fitarget  Performance factor <j>  Figure 4.1: An Example of  4.2.2  -<j> Relationship  Determining the load factors, an example  4.2.2.1 Problem description This example considers the limit states of bending strength of a single lumber member under either dead load alone or dead load plus occupancy load or dead load plus snow load in combination. The species is Douglas fir, grade Select Structural and size 2 x 10 (38mmx235mm). To determine the dead load factor ar, and the live load factor CCQ simultaneously, an inverse reliability analysis was carried out for the performance factor <> / = 0.9. Two reliability constraints were considered, giving a unique solution.  Chapter 4 Applications in Engineering Design  4.2.2.2 Limit state functions In the inverse analysis, two limit states were considered as follows:  G,  G  2  0.05  =R-  =R-  a  (4.9)  d  D  D  (4.10)  iyd + q)  ya +a  Q  where Gj is the limit state of bending strength for dead load only and G corresponds to 2  dead load and occupancy or snow load in combination. R is the bending strength and Ro.os is the characteristic value, taken at the 5 - percentile of the variable R. yis the ratio of the th  design dead load to the design live load. When considering the snow loads, the live loads q in Eq.(4.10) are rewritten as (4.11)  q = rg  where g, Gumbel distributed, expresses the variability in the total ground snow and r, lognormal distributed, presents the variability between actual roof load and that at the ground. A detailed discussion of this point is given by Foschi et al. (1989).  4.2.2.3 Intervening variables From the literature (Foschi, et al. 1989) the intervening variables and their statistics are listed below: •  R: Bending strength which obeys 2-P Weibull distribution with the parameters m = 44.55MPaandK=4.10.  •  Ro.os'- Deterministic value Ro.os 21.60MPa =  80  Chapter 4 Applications in Engineering Design  •  d. Dead load. It can be written as d=\.0 + 0.\R„  (4.12)  where R„ is a Standard Normal variable. •  q : Live load (occupancy load or snow load)  The occupancy live load is an Extreme Type I (Gumbel) distributed variable. Table 4.1 gives the maximum statistics of the occupancy load in a window of 30 years. For the snow load, Table 4.2 presents the statistics of the variable r, which is treated as lognormal and Table 4.3 shows the statistical parameters of the variable g which is also Extreme type I distributed.  Table 4.1: Occupancy Loads Maxima 30 Years  Occupancy Type Mean  COV  A  B  Residential  0.812  0.272  5.8060  0.71258  Office  0.925  0.236  5.8752  0.82675  Table 4.2: Statistical Parameters for the Roof Load Ratio r Location and Roof Type  Mean Value  COV  All locations or sloping roofs  0.6  0.45  Vancouver or sheltered flat roofs  0.8  0.0  81  Chapter 4 Applications in Engineering Design  Table 4.3: Statistics of the Variable g for three Cities Maxima 30 Years  Location Mean  COV  A  B  Vancouver  1.1530  0.28713  3.8741  1.004  Ottawa  1.1053  0.20558  5.6446  1.003  Quebec City  1.0670  0.13529  8.8850  1.002  4.2.2.4 Determination of factors ao and CCQ Using the inverse reliability method, the design load factors CCD = 1.43 and CCQ = 1.60 were obtained for the two reliability constraints as given in Eqs (4.9) and (4.10), when dead load and office occupancy load (office) were considered and the corresponding target reliabilities were 0j = 2.4 and 02 ~ 2.6. The reliability indices corresponding to residential occupancy and to snow loads from three Canadian cities were carried out based on the obtained load factors {OCD = 1,43 and OCQ = 1.60). The results are presented in Table 4.4. For comparison, the existing load factors (CCD = 1.25 and CCQ = 1.5) in the Canadian Code, with performance factors either <f> = 0.8 or <j> = 0.9, were also used to calculate the reliability levels for all the load situations considered. The results are also included in Table 4.4.  82  Chapter 4 Applications in Engineering Design  Table 4.4: Reliabilities for Different Load Situations Reliability Load Situations  OD=  1.43  ao= 1.25  CCQ =  1.60  OQ=  1.50  CCD=  OQ=  1.25  1.50  0 = 0.9  ^=0.8  ^ = 0.9  Dead load  2.400  2.373  2.191  Dead load and office load  2.600  2.657  2.491  Dead load and residential load  2.722  2.776  2.617  Dead load and snow load (Vancouver)  2.571  2.628  2.462  Dead load and snow load (Ottawa)  2.784  2.836  2.683  Dead load and snow load( Quebec City)  2.844  2.896  2.744  It is impossible to introduce design recommendations that would result in the same reliability level across all situations. The objective is to find design factors so that the variability of B for all the situations is not substantial. It should be indicated that the calibration using B = 2.4 and B=2.6 agrees with the load factors recommended in the U.S. (do = 14, CCQ = 1.6 and <j) = 0.9) but not with those suggested by the Canadian Code (cfo = 1.25, CCQ = 1.5 and <f> = 0.9). The reliability for the case of dead load only is too low (j3 = 2.191). However, more compatible results are obtained at <f> = 0.8, inferring that <j> = 0.8 would be better than <j> = 0.9. Without changing to <j> = 0.9, the load factors CCD and CCQ may be changed in Canada to respectively, 1.4 and 1.6. It is understood here that full code calibration would require the consideration of many more design situation or load cases.  83  Chapter 4 Applications in Engineering Design  4.3  A p p l i c a t i o n i n offshore e n g i n e e r i n g  Reliability theory has been found very useful in offshore engineering where the structures are subjected to extreme environmental conditions. In this section, inverse reliability will be used to seek directly the design parameters, including the design loads and the design weight of a platform when target reliabilities are specified.  4.3.1  Background and description of the problem  One of the most important steps in the design process is the determination of the design loads for the structure. Due to their high uncertainties, loads such as those produced by wind or earthquake should be determined on a probabilistic basis. A design load, denoted by Fo, can be defined so that the probability of an annual maximum load exceeding Fo equals a given exceedence target. For example, the 50-year load is defined as a level that has an annual exceedence risk of 0.02.  In offshore engineering, it is of critical importance to determine the proper design loads when offshore structures being designed are intended for operation in extreme environments. Such design loads may include those due to waves, wind, earthquakes, ice, and iceberg collisions. To satisfy a certain safety level in gravity-based platforms, the required weight of the structure must be calculated against the risk of overturning or the sliding on the seabed under the random environmental loads. The Canadian CSA Offshore Structures Code CAN/CSA-S471-92 (Canadian Standards Association, 1992) has been subjected to a comprehensive verification process, and Foschi et al. (1994,1996,1998) have  84  Chapter 4 Applications in Engineering Design  developed a methodology to calculate the load combination factors for wave and iceberg collisions. Design loads due to wave alone, an iceberg collision alone, and an iceberg and waves in combination, have been calculated for a range of iceberg and wave parameters, with the results applied to a first-order reliability analysis to study force levels corresponding to specified annual exceedence probabilities. To determine the combination factors in the code, the exceedence forces were obtained in a traditional manner: 1) repeat the forward reliability calculation for different force levels, and 2) find the force corresponding to a given target reliability (or annual risk) by interpolation. The combination factor yis used in the code as follows: E = E + yE r  (4.13)  f  where E is the combined load (an iceberg with waves) with an annual exceedence probability of 10", E is the iceberg-alone load with an annual exceedence probability of 4  r  IO", and Ef is the wave load with an annual exceedence probability of 10' . 4  2  The objective of this example is to use the inverse reliability procedure described in Chapter 2 as a direct alternative and verification of the design loads and structural weight without the interpolation required by the forward procedure used by Foschi, et al. (1996). Three cases were studied: 1) determination of the exceedence loads due to wave load alone and iceberg load alone with the given annual exceedence probabilities; 2) determination of the design mean value of the structural weight with a given coefficient of variation when a reliability constraint associated with a limit-state of seabed sliding is given; and 3) determination of the mean value and the standard deviation of the structural weight  85  Chapter 4 Applications in Engineering Design  simultaneously to meet two reliability constraints which are associated with the limit-states of sliding and overturning. As a case study, conditions similar to that of the Hibernia platform located in the Grand Banks off the Newfoundland coast were used. This is a gravity-based reinforced concrete structure and is protected from iceberg impact by a concrete cylindrical wall. Fig.4.2 shows the assumed shape of the iceberg with a circular plan and ellipsoidal elevation and the profde of the cylindrical wall, of radius a, in water with a depth d. For the iceberg, the waterline length is D, the draft is h and the maximum radius is R.  Still water level  Figure 4.2: Definition Sketch of Iceberg - Structure Geometry  4.3.2  Loads on the structure  Based on the environmental conditions, with associated wave and iceberg statistics, the calculation of the random loads on the platform structure due to wave alone, ice alone and wave and ice in combination have been studied and described in detail by Foschi et al.  86  Chapter 4 Applications in Engineering Design  (1996) and Foschi (1994). The following gives a brief description of how wave load and iceberg load were calculated. To determine the wave load without ice collision, a three-dimensional linear wave diffraction theory is used in combination with a boundary element method, (Sarpkaya and Isaacson, 1981). For regular waves of height H and period T, the force on the structure varies sinusoidally with period T. The numerical results were used to develop a response surface for wave periods J ranging from 10 to 20 sec. The loads due to random waves can then be obtained as an extension of the regular waves by taking the wave height as random variable. To determine the iceberg load in the absence of waves, the force is considered dependent on the distance x that the structure penetrates into the ice mass during the collision. The distance x determines the area of contact which, multiplied by the ice crushing pressure, gives the load F. The crushing pressure itself is area-dependent, as shown by testing of ice in compression. The distance x is determined through an energy balance, i.e. to stop the iceberg, its initial kinetic energy must be completely spent in the work done to crush the ice an amount x. The randomness of the ice load can be considered by taking several intervening random variables, including iceberg shape and dimensions, the iceberg mass, ice-crushing strength, and the iceberg's initial impact velocity.  In this case study, the mechanical model and the corresponding computer subroutines (WUSER and ICEUSER, Foschi, et al. 1996) of calculating the wave load and ice load were employed directly and linked with IRELAN to implement the inverse reliability calculation.  87  Chapter 4 Applications in Engineering Design  4.3.3 Intervening Variables According to the environmental conditions and statistics of the site, the dimensions of the iceberg, current velocity (iceberg impact velocity in calm water), significant wave height, sand friction angle, minimum ice-crushing pressure and the weight of the structure were treated as random variables. The uncertainties in the model, collision eccentricity and minimum ice-crushing pressure were also considered through three associated random variables. Water depth and the radius of the structure were treated as deterministic. The random variables distributions and statistics are given in detail in Table 4.5. (Foschi, et al., 1996, 1998)  4.3.4 Determination of exceedence loads and the structural weight 4.3.4.1 Target reliability levels In the case of an iceberg impact, the target annual exceedence probability or the annual failure probability (sliding failure) denoted by p is first given as the target reliability level. a  The corresponding force exceedence probability or failure probability during an event, denoted by p , conditional on the occurrence of an iceberg impact, is then calculated. It is e  assumed that the events follow a Poisson pulse process with a given mean rate of annual occurrence (events per year), denoted p. Thus, the relationship between annual risk p and a  conditional exceedence probability of the event, p , is obtained by e  p = 1- exp (-ppe)  (4.14)  a  88  Chapter 4 Applications in Engineering Design  In the case of wave alone, the annual force exceedence risk or the annual failure probability are already calculated for an annual window, and therefore Eq.(4.14) is not used in this case.  Table 4.5: Summary of Specified Variables and their Statistics Variable  Distribution  Characteristics  Iceberg length, L  Gamma  H = 107.19 m, a = 35.60 m  Iceberg draft, h  Beta  |x = 70.89m, a = 15.41m min= 0.00m, max = 100.00m  Current velocity, U  Lognormal  \x = 0.28 m/s, a = 0.2 m/s  Significant wave height  Extreme type I  \x= 14.175m, a = 1.695m  Normal  p: = 4.0 Mpa,  (annual maximum) Minimum crushing pressure, Po <j> sand friction angle R„i, associated with model  a = 1.0 Mpa Normal Normal  uncertainty Rn2, associated with ice-crushing  u = 35.0°, rj = 7.0° u. = 1.0, a = 0.10 (wave load) o = 0.15 (ice load)  Normal  p = 0.0, a = 1.0  Normal  u. = 0.0, a= 1.0  Uniform  Min = 0.0  pressure p Rn3, associated with Rayleigh distribution for wave height H Rn4, associated with collision  Max =1.0  eccentricity W, weight of the structure  Normal  Depth d  Deterministic  100 m  Structure radius a  Deterministic  50 m  u = ?,  G =  ?  The question marks in Table 4.5 indicate values that will )e the design parameters.  89  Chapter 4 Applications in Engineering Design  4.3.4.2 Limit-state functions In order to calculate the exceedence loads due to wave and iceberg and the weight of the structure, two limit-state function G(x) are considered: 1) the exceedence of a force level Fo, with a corresponding annual exceedence probability po, and 2) the sliding of the platform on the seabed.  In thefirstcase, the limit state function is (4.15)  G(x)=F -FM(x)R j 0  n  where, in both cases, FM(X) is the maximum force developed on the structure due to waves, or iceberg impact, x denotes a set of specified random variables characterizing the structure, the iceberg, and the wave conditions. R„j is a random variable quantifying the model error in the calculation of FM- The probability of the event G < 0 corresponds to the probability that the maximum load FM exceeds the load level Fo.  In the second case, the limit state function for sliding is defined as G(x) =  (4.16)  Wtan(0)-F (x)R j M  n  Where <f> is the sand friction angle and IT is the weight of the platform. In this case, we want to find the design parameter, the mean value of W, which is required to meet a prescribed reliability level as a function of the coefficient of variation of W, reflecting the accuracy with which a target IF can be achieved in practice.  90  Chapter 4 Applications in Engineering Design  4.3.4.3 Results In conjunction with the program WUSER (wave load calculation) and ICEUSER (ice load calculation), respectively, based on limit-state function (4.15) the inverse reliability program IRELAN was first run to obtain the forces for waves alone and iceberg load alone at the target annual exceedence probabilities 10" and 10", respectively, based on the 2  4  statistics provided in Table 4.5. Table 4.6 gives the corresponding results together with those obtained by interpolation using the forward approach. The results show that there is agreement between the two methods. However, the inverse procedure offers advantages as to efficiency and execution time. For example, in the first case, the IO" iceberg load of 4  3394.89MN required 6.04 sec. in a Pentium-based 166Mhz computer when the forward method was utilized. In this case, 50 calculations were done at different Fo levels, increasing this value from 1000MN to 3500 M N in steps of 50MN. These results then permitted the interpolation at the desired 10" exceedence level. For the inverse procedure, 4  the same computer required only 0.49 sec. and produced the desired Fo level directly. Apart from the gain in computer time, an advantage of the inverse procedure is that no a priori guessing of a range for Fo is required.  To investigate the influence of the significant wave height in the maximum wave load, and the effect of the current velocity and the iceberg arrival rate in the iceberg force, sensitivity analyses were carried out. The results are given in Table 4.7 and Table 4.8 for wave load and iceberg force, respectively. Table 4.7 indicates that the wave loads are strongly dependent on the significant wave height, which corresponds to the sea state, reflecting the  91  Chapter 4 Applications in Engineering Design  importance of the selection of a sea state in the design. Table 4.8 shows that both arrival rate and the current velocity have significant effect on the iceberg force. For example, the IO" force changesfrom846 M N at the rate of 0.04 and the current velocity of 0.14 m/s to 4  5074 M N at the arrival rate of 1.0 and a current velocity of 0.42 m/s.  Table 4.6: Loads at Specified Annual Exceedence Probabilities Load Case  Method  Annual Risk Forward  Direct Inverse  FORM  Algorithm  Wave load  10"  1431.79  1430.33  (MN)  4  10"  2230.63  2228.80  Iceberg load  10"  1164.67  1163.82  (MN)  io-  3394.89  3395.28  4  (Iceberg arrival rate =1.0 collision/year)  To determine the structure weight W for the sliding limit state, the target annual failure probability is set at 10". In the case of iceberg loading, the iceberg arrival is assumed as 1.0 4  collision /year. The inverse reliability analysis was run to seek the mean value of the weight IT for various coefficients of variation. The results are given in Table 4.9. This Table shows that the coefficient of variation does have some effect on the desired weight but not significantly, since the random variable IT is not dominant in the reliability analysis in comparison to the other intervening random variables.  92  Chapter 4 Applications in Engineering Design  Table 4.7: Wave Loads (MN) at Specified Annual Exceedence Probabilities Annual  Mean of Significant  Exceedence  Wave Height and its  Probability  Std. Dev.(m)  10-  2  10  Maximum Wave Load F (MN) m  15.0 (a = 1.5)  1493.49  17.0(a= 1.7)  1660.92  20.0(a = 2.0)  1794.31  15.0 (a = 1.5)  2280.88  17.0(a= 1.7)  2478.00  20.0(a = 2.0)  2632.91  Table 4.8: Iceberg Force (MN) at Specified Annual Exceedence Probabilities Annual  Iceberg  Exceedence  Arrival  Probability  Rate n  10 "  2  10 -*  Maximum Iceberg Force FM (MN) 0.14 m/s  0.28m/s  0.42 m/s  (cj=0.1m/s)  (o=0.2m/s)  (o=0.3m/s)  0.04  158.82  317.38  475.70  0.08  231.85  463.25  694.20  0.20  342.39  683.92  1024.61  0.40  438.36  875.46  1311.31  1.00  582.69  1163.37  1742.06  0.04  846.58  1689.45  2528.60  0.08  999.92  1994.93  2985.05  0.20  1227.40  2447.90  3661.51  0.40  1419.95  2831.12  4233.50  1.00  1703.49  3395.08  5074.72  93  Chapter 4 Applications in Engineering Design  Table 4.9: Mean Structure Weight for Sliding with a IO" Annual Failure Probability 4  Coeff. of Variation for W  Load Case  Mean Value of the Weight W (MN)  0.0  5654.86  0.04  5680.60  0.08  5764.48  0.0  5997.76  0.04  6018.86  0.08  6086.38  Wave load  Iceberg load  (Iceberg arrival rate = 1.0 collision/year)  4.3.5  Simultaneous determination of mean and standard deviation of the structure weight  In this case, both the mean value and the standard deviation of the structural weight were treated as design parameters and determined simultaneously. As reliability constraints two limit-states were considered: 1) the sliding of the platform on the seabed as shown in Eq.(4.16) and 2) the overturning of the platform considering an allowable kern area of diameter b = a/2.  For the first constraint, the limit state function Gi is given by Eq. (4.16). For the second constraint, the limit state function G is given by 2  G =W-2-F (x)R a 2  M  (4.17)  n  94  Chapter 4 Applications in Engineering Design  Wis assumed to be Normally distributed. The two reliability-related constraints permit the calculation of the two design parameters at specified indices B B . Table 4.10 presents the lt  2  results corresponding to different target reliabilities. They show that when B\ is fixed, for example, increasing the specified B results in an increase in the mean value of W as well as 2  its standard deviation and coefficient of variation. This implies that when increasing the mean weight, the quality control or variability in W can be relaxed while still maintaining the target reliabilities. If pi equals B there is no solution in a practical range to satisfy the 2>  two constraints. Figure 4.3 shows, for example, that the contours for Bi = 2.0 and B = 2.0 2  have an intersection at very large values of W, outside the practical range. Such results are also obtained whenever B is specified greater than Bi, a consequence of the statistics used 2  for the friction angle and the ratio aVa adopted for this example.  Table 4.10: Design Parameters for Different Target Reliabilities, Offshore Platform Standard Deviation, Weight JF(MN)  C.O.V  4036.97  1573.30  0.39  1.5  7127.67  3169.71  0.45  2.0  2.0  No feasible solution  No feasible solution  -  3.0  1.5  4682.61  1118.57  0.24  3.0  2.0  6797.57  1948.10  0.29  3.0  2.5  11568.8  3583.14  0.31  Pi  P2 Mean Value, Weight  2.0  1.0  2.0  (MN)  95  Chapter 4 Applications in Engineering Design  7000 -rZ  6000 -  ~  5000 -  •A  2CO 4000 £  3000 -  g  2000 -  S  — • — s l i d i n g mode - - - overturn mode  1000 -  o  0  500  1000  1500  2000  Standard deviation (MN)  Figure 4.3: Contours of Bj and B - 2.0 2  4.4  Applications to earthquake engineering  Due to high variance inherent in the earthquake loading, reliability theory, in general, is very important to earthquake engineering for either risk assessment of existing structures or the limit-state design of new ones. A proper seismic design must take into account the uncertainties of both ground motion and material properties. In this section, inverse reliability will be applied to determination of the design parameters so that the pre-specified reliabilities can be achieved. Both linear and nonlinear dynamic behaviors are considered.  4.4.1  Linear response  The following example is a case of a serviceability limit-state design in a three-story building excited by ground motion. The building, with lumped masses in each floor, as shown in Figure 4.4, is assumed to respond in an elastic manner. This is only an assumption  96  Chapter 4 Applications in Engineering Design  for the purpose of illustrating the approach, as a nonlinear response could be equally considered. The main objective is to find the design mean values for the lateral stiffness of the columns, denoted by ki, k2 and k , simultaneously, such that target reliability levels are 3  achieved for not exceeding allowable inter-story drifts (and the associated damage) under a random earthquake base excitation. Thus, three reliability constraints are involved. The excitation was considered to have a random peak acceleration with a fixed frequency spectrum as recorded at Joshua Tree Station during the1992 California Landers earthquake.  mi  3x3m  «o(t)  Figure 4.4: Multi-story Frame Under Earthquake Excitation  4.4.1.1 Determination of the design peak acceleration Usually the design peak ground acceleration <ZG is assumed to follow a lognormal distribution and therefore, obeys a relationship of the form:  97  Chapter 4 Applications in Engineering Design  a=  " VI  where a  M  e^^^  M  G  +v  (4.18)  2  is the mean value of the design peak acceleration OG, V is its coefficient of  variation and RN is a Standard Normal variable. From typical attenuation relationships, the standard deviation of ln(aG), expressed as -^\n(\+V ), is assumed to be 0.55, to which a 2  value of V = 0.6 is associated. This relatively high value is consistent with many soil attenuation relationships widely used in earthquake engineering. In accordance with the Code criteria, the mean value of the design acceleration is calculated so that the design acceleration, corresponding to an annual exceedence probability of 1/475 is OQ = 0.25g. Such annual risk is consistent with an exceedence risk of 0.1 in a 50-year window. The following gives the detailed calculation of the mean value of peak ground acceleration.  It is also assumed that the earthquakes under consideration occur once in every 10 years. The probability PE that the peak acceleration «G exceeds the design acceleration OD if the earthquake happens is derivedfromPoisson process  — = \-e475  *  (4.19)  0AP  from which  P  E  -ln(l——) = ^ - = 0.021075 0.1  (4.20)  From the definition of the probability for a variable lognormal distributed, the ps can be calculated by using OD= 0.25g and  98  Chapter 4 Applications in Engineering Design  ln(a )-ln(^ VI+ V !-<£(D  p  E  =PROB(a >a ) a  =  D  ) =  Vln(l + F )  l-<D(i<„)  (4.21)  2  Since (4.22)  i?v= O" (l-/te) = 2.032 1  The mean value  is obtained as O.lg which, together with V= 0.6, will be used as the  statistics for the random variable CIG in the following inverse reliability analysis.  Assume now that the failure probability of theframe,in each of the three limit states and over the next 30 years, is required to be 0.001. According to a Poisson arrival, with a rate v = 0.1, process, the conditional target failure probability p if the earthquake does happen is e  given by  0.0003335  (4.23)  Thus, the associated, conditional, target reliability index B is B = Or (1 - p ) = O (1 - 0.0003335) = 3.4 1  - 1  e  (4.24)  4.4.1.2 Dynamic analysis To proceed with the inverse reliability analysis, a linear dynamic analysis of the structure is needed in the calculation of the limit-state function. It can be performed by using the time step integration method (constant average acceleration method) of the equation of motion  99  Chapter 4 Applications in Engineering Design  (4.25)  M x+ C x+ K x = -MI a (t) s  where JC is the vector of degree of freedom ( horizontal mass displacement), M and K are respectively, the mass and the stiffness matrices. J is a unit vector given as  (4.26)  The damping matrix C is taken as (4.27)  C=aM+0K  - i - i  CO,  = 2.0x  where IAJ  CD,  1 CO,  (4.28) CO*  Here coj, CO3 are the first and third natural frequencies of the structure and based on current structural properties.  and  are damping ratio corresponding to the first and the third  mode. Both of them are assumed to have a mean value of 5%. {/} is a unit vector and a (t) s  is the historic spectrum of the event in Landers, California 1992, as shown in Fig. 4.5.  4.4.1.3 Random variables A total of nine (9) random variables were considered in this example. They were: 1) the earthquake peak ground acceleration aa, with statistics as previously discussed; 2) the damping ratio & and £3, which were assumed Normally distributed with 10% coefficient of variation; 3) the three lumped floor masses  mj, rri2, ni3  100  assumed Normal with 1% coefficient  Chapter 4 Applications in Engineering Design  of variation; and 4) the column stiffnesses kj , k , k , assumed Normal with 5% coefficient 2  3  of variation. The correlation between column stiffnesses was taken into account, and the coefficient of correlation of 0.8 was assumed, i.e. pkj,k2=0.S,  pki.ki  = 0.8 and p\2,k3=0.8. All  statistics are summarized in Table 4.11 and will be referred as the basic case in the following discussions. Other variables could be used if the problem would require consideration of, for example, wave directionality. However, the procedure illustrated here would be equally applied.  -0.3 10  20  30  40 50 time (sec)  60  70  80  90  Figure 4.5: Ground Acceleration of Event in Landers, California (Joshua Station)  101  Chapter 4 Applications in Engineering Design  Table 4.11: Random Variable Statistics Variable  Mean values  Coefficient of variation  Distribution type  a ntj  O.lg  0.60  Lognormal  10.0 tonne  0.01  Normal  m  15.0 tonne  0.01  Normal  m ki  20.0 tonne  0.01  Normal  ?  0.05  Normal  k k  ?  0.05  Normal  ?  0.05  Normal  0.05  0.1  Normal  0.05  0.1  Normal  G  2  3  2  3  6  4.4.1.4 Limit-state functions Three limit-states functions are used:  G, = D . -—lx. n  -x,\  '  2  lmax  "o  "o  G, —D 3  , —— Ix, I max 3  |  3|  m a x  "0  where ao is the maximum peak ground acceleration of the historic event under consideration (0.278g). The frequency spectrum of such event was used in this example to obtain, through a dynamic analysis, the maximum displacements for each floor, which are,  102  Chapter 4 Applications in Engineering Design  respectively, xj, X2andx .  The allowable inter-story drifts are given as 0.5% of the column  height, thus A n a x i  = Anax3  3  =  D  max2  = 3m / 200 = 0.015m.  4.4.1.5 Results of basic case The basic case was studied with the statistics in Table 4.11. The mean values of the floor masses varied from floor to floor with a pyramidal distribution. By running IRELAN in conjunction with the dynamic analysis program, the desired mean values of the column stiffnesses for eachfloorwere obtained simultaneously and are given in Table 4.12.  Table 4.12: Design Mean Column Stiffness for the Basic Case kj (kN/m)  £ (kN/m)  k (kN/m)  12588.2  24776.3  35895.2  2  3  It can be verified, through a forward FORM reliability analysis, that indeed the target reliability index B = 3.4 is achieved for each of the three limit state functions.  4.4.1.6 Sensitivity analysis Sensitivity analyses are seen as one of the most important potential applications of the inverse reliability method in codified design, to deal with variations in the statistics of the materials, loads and their combinations. Also these studies can be used to identify the major factors influencing the results. For the current problem, the effect of the statistics of the intervening random variables on the mean stiffness for each column is examined, including the effects of the coefficient of variation of the masses and stiffnesses, spatial distribution  103  Chapter 4 Applications in Engineering Design  of the masses, mean value of the damping ratio and mean values of the peak acceleration, respectively. For all the sensitivity analyses the target reliability level for each limit state is 3.4.  •  Effect of coefficient of variation on the design parameters  Four cases were examined for the coefficient of variation (COV) of the stiffness of each column varying from 0.01 to 0.07 with increment of 0.02, while the other statistics remaining the same as given in Table 4.11. Similarly, the inverse analyses were carried out to examine the effect of the COV of the masses on the design parameters. The results are summarized following in Table 4.13, and Table 4.14, respectively, for stiffness and mass. The results show that the required mean stiffness for each column increases with increasing the COV of the stiffnesses. A little change in the required mean stiffness is observed with an increase of the COV of the masses. For example,, in Table 4.14, comparing the results at  COV= 0.02 with those at COV = 0.08, the maximum change in the design parameters is only about 2%, which validates that the assumption of using small COV for masses in the basic case.  Table 4.13: Results for Various COV of the Stiffnesses  COV  £j(kN/m)  £ (kN/m)  £i(kN/m)  0.01  12318.0  24410.6  35589.3  0.03  12381.2  24471.2  35676.9  0.05  12588.2  24776.3  35895.2  0.07  12823.4  25142.2  36222.7  2  ;  104  Chapter 4 Applications in Engineering Design  Table 4.14: Results for Various COV of the Masses  •  COV  £ (kN/m)  £ (kN/m)  ^(kN/m)  0.02  12611.1  24806.4  35840.4  0.04  12664.1  24860.2  35969.9  0.06  12787.9  25002.0  35909.2  0.08  12902.1  25148.0  35747.6  2  7  Effect of the mass spatial distribution on the design parameters  The inverse calculations were implemented for different spatial distribution of the masses and the results are presented in Table 4.15, while other statistics remainingfixedas per the basic case. From Table 4.15, it is seen that the distribution will significantly influence the required column stiffnesses at the same reliability level. It is also noted that combination of bigger masses may result in obtaining smaller stiffnesses, for example the results at mj = 15.0 tonne, m = m = 20.0 tonne are smaller than those at mj = m = m =15.0 tonne, 2  3  2  3  indicating the influence of different resonance characteristics.  Table 4.15: Results for Different Spatial Distributions mi (tonne)  m (torme) mjftonne) 2  */(kN/m)  fe(kN/m)  fc (kN/m) 3  10.0,  10.0,  10.0  11201.4  18160.6  21847.4  10.0,  15.0,  15.0  11606.6  23523.9  29930.5  15.0,  15.0,  15.0  16802.2  27241.1  32771.6  15.0,  20.0,  20.0  10518.9  20374.7  29210.2  20.0,  20.0,  20.0  13025.9  23624.2  33684.6  ,  105  Chapter 4 Applications in Engineering Design  •  Effect of damping ratio on the design parameters  In the basic case the mean damping ratios were assumed as 5% for first and third mode. However, in practice, these ratios may change from 1% to 10% reflecting the different connections and different types of structures. It is difficult to determine accurately the damping ratios for a structure. To show their effect, sensitivity studies were carried out when the mean value of both  and £ was taken as, respectively, 3% and 6% with 10%  coefficient of variation, with other statistics remaining as in the basic case. Table 4.16 presents the required mean value of the stiffnesses for each column. As expected, the damping ratios significantly affect the results.  Table 4.16: Results for Different Damping Ratios  •  6 = 6  fo(fcN/m)  /b(kN/m)  fe(kN/m)  0.03  13979.9  27034.7  38629.0  0.06  11552.8  22308.2  29426.2  Effect of the peak acceleration on the design parameters  Finally, a sensitivity study was conducted to examine the effect of the peak ground acceleration, reflecting reliability-based design in different seismic zones. While the other random variables remained as in the basic case, the mean value of the peak acceleration was changed from 0.12g to 0.20g with coefficient of variation of 0.6. The obtained design parameters are given as shown in Table 4.17. As expected, the bigger the peak acceleration, the larger the required stiffnesses of columns.  106  Chapter 4 Applications in Engineering Design  Table 4.17: The Effect of the Peak Acceleration Mean of Peak  Mean Value of  Mean Value of  Acceleration(g)  Ar(kN/m)  Jb(kN/m)  fo(kN/m)  0.12  13732.6  26332.9  39342.4  0.14  16652.5  31635.7  44259.2  0.16  17445.6  34786.4  48480.3  0.18  20559.4  42800.8  62418.8  0.20  25062.2  48511.9  67578.2  4.4.2  ;  Mean Value of  Nonlinear response  This example is also a design problem in earthquake engineering, but demanding a nonlinear dynamic analysis for the structural response. A pile, as shown in Fig. 4.6, assumed elasto-plastic, is driven into a soil, which is modeled as a nonlinear foundation. The pile carries a mass M at its head and is excited by the free-field motion of the soil during an earthquake. The objective of this application is to calculate the mass M which can be supported while meeting a prescribed reliability in a serviceability limit-state. The pile is a 30.0m - long steel tube with an outside diameter of 356mm and a wall thickness of 10 mm.  Under the time-varying excitation, the relationship between the horizontal displacement A of the pile-head and the shear force V is typically in the form of a hysteresis loop, a representation of the nonlinear response resulting from the elasto-plastic properties of the pile and the nonlinear interaction between the surrounding soil and the pile. During the  107  Chapter 4 Applications in Engineering Design  shaking, the gaps developed between these two elements contributes to a "pinched" characteristic for the hysteretic loop, due to the absent of tensile stresses at the interface soil/pile. It is known that such hysteretic properties play an important role in the analysis of the dynamic response of the structure to an earthquake ground excitation. Therefore, it is essential to employ a nonlinear model with a reasonably accurate representation of hysteresis in the dynamic analysis, which is needed in the calculation of the limit-state function. Two approaches are briefly reviewed in the following.  <—  V r  M \  yy  /  a (t) G  Figure 4.6: Pile in an Earthquake Excitation  •  Empirical Model  In modeling this nonlinear behavior, empirical models are commonly used by defining a set of rules for loading and unloading paths. The intervening model parameters can be calibrated to an observed experiment history. This approach ranges from coarser piecewise  108  Chapter 4 Applications in Engineering Design  linear models to sophisticated and versatile techniques. For example, a model proposed by Baber, Noori and Wen (1981,1985) and modified by Foliente (1995), is capable of modeling complex hysteresis loops with pinched behavior, strength and/or stiffness degradation. In this model, a first order differential equation for the hysteretic force is integrated as a function of the displacement. A total of 13 parameters are involved in the equation and must be calibrated first from a test.  In these mathematical modeling  techniques, the question remains as to whether the same set of parameters would provide a proper representation of loops for displacement histories other than the one used for the calibration. This issue was raised and discussed in detail by Foschi (1998 a).  •  Finite element model  As an alternative, another approach in modeling the non-linearity is to calculate the hysteresis loop at each time step, based on the finite element method and using the basic stress/strain information on the structural members and the nonlinear behavior of the surrounding medium. Such a finite element model has been developed. (Foschi, 1998 b) The main advantage of this formulation is that, starting from the basic properties, it will automatically adjust to any displacement history. In addition, the input information has clear physical meaning, like modulus of elasticity, yield strength, etc. In contrast, it is difficult to assign a physical meaning to some of the calibrated parameters in an empirical model. Furthermore, when conducting a reliability analysis it is very hard or even impossible to provide a proper statistical representation for a calibrated parameter in an empirical model. Since the statistics for the basic material properties are easier to obtain, the finite element approach is more reliable.  109  Chapter 4 Applications in Engineering Design  Based on the above discussion, the finite element model proposed by Foschi (1998 b) was used in conjunction with the dynamic analysis program to implement the calculation of the limit-state function in the application.  The pile length is divided into beam elements. The lateral displacement w and the axial displacement u are defined as fifth degree and cubic polynomials in x, respectively. These displacements are referred to the center of gravity of the pile cross-section. Considering the p-A effect, the strain s of a point at the distance^fromthe center of gravity is expressed as du dw 1 dw ^ =-T--J ^X +-(T-) dx dx 2 dx  ,.\  2  ;  2  (4.30)  The stress a(e) in the element is assumed to obey an elasto-plastic constitutive relation, with either strain hardening or softening. The pressure p(w) from the soil medium, per unit length, is assumed as a function of displacement w, and acts only in compression.  The equation of motion of this nonlinear system can be formulated and integrated in the time domain, using a constant average acceleration routine. Within each time step, iterations are carried out by the Newton-Raphson technique, until dynamic equilibrium is achieved. The detailed description of how this model has been developed is shown elsewhere (Foschi, 1998 b).  no  Chapter 4 Applications in Engineering Design  4.4.2.1 Random variables Although many uncertainties are involved in this problem, associated with the earthquake as well as with the pile and the soil characteristics, it is assumed here that the only random variables are the peak ground acceleration eta, the yield stress 05, of the pile and the soil relative density D . The earthquake excitation was assumed the same as that in the forgoing r  linear problem, the historical Landers event. The random variable CIG was assumed to be lognormal with mean value of 1.0 m/sec and a coefficient of variation of 0.6, which is 2  consistent with a design acceleration of 0.23g when a return rate of 0.05 was assumed for the earthquakes. Let J? be a Normal distributed random variable associated with the uncertainty in the yield stress of the pile. The summary of the statistics of the random variables is given in Table 4.18. The design mass Mis assumed Normal distributed with the coefficient of variation of 0.01. All variables are assumed uncorrelated.  Table 4.18: Statistics of the Random Variables Variables  Mean Value  COV  Type  a  1.0m/sec  0.6  Lognormal  R, Associated with the yield  1.0  0.1  Normal  M  ?  0.01  Normal  D  75.0  0.2  Normal  2  G  stress (250.0MPa)  r  111  Chapter 4 Applications in Engineering Design  4.4.2.2 Limit-state function and target reliability The serviceability limit state function is defined as G = D - DraJiao, 0  (4.31)  Oy,M,D ,...) r  where the maximum pile-cap displacement Z W is determined by the nonlinear dynamic analysis, as a function of two intervening random variables and other deterministic variables. Do is the maximum allowable pile-cap displacement. The reliability index against exceeding A  max  during the earthquake event is a target B = 2.5, corresponding to an  exceedence probability of approximately 6.2 x 10". 3  4.4.2.3 Seeking the design parameter, mass of the pilecap In this problem, as discussed in Chapter 2, there might be multiple solutions for the design parameter, the mass M. An inverse approach was first used to give an approximate estimate of M. Thus, only one random variable, CXG, was considered as it tends to have the most significant influence on the reliability. Its normalized value, in the Standard Normal uncorrelated space, must be equal to the specified target reliability index. Thus, the value of QG at its design point is obtained by 1.0  Vl + 0.6  2  112  0.35g  (4.32)  Chapter 4 Applications in Engineering Design  The nonlinear dynamic analysis was then run for this peak acceleration, calculating the maximum displacement A for different masses ranging from 5.0 tonne to 200.0 tonne at the increments of 1.0 tonne. The results are shown in Fig. 4.7, which provides the relationship M-A for a constant reliability contour at B = 2.5.  1.80E-01 T  0.00E+00  1  5.00E + 01  1.00E+02  1.50E + 02  2.00E+02  2.50E + 02  mass  Figure 4.7: Relationship Between Mass and Maximum Deflection  The required mass is found at the intersection of the curve with a horizontal at the required limit of 0.1m, which gives these possible values f o r M = 45.94, 50.01 and 55.52 tonne, indicating that the solution is not unique at this allowable displacement. When only the peak acceleration is taken into account, while ignoring the uncertainties of other intervening variables, these solutions provide the approximate design mass that the pile can carry to meet the desired reliability level under the excitation.  113  Chapter 4 Applications in Engineering Design  To investigate the influence of the other uncertainties involved (yield stress, soil relative density and mass), the direct inverse reliability technique, in conjunction with the hybrid of the Newton-Raphson with the bisection method (as discussed in Chapter 2), was carried out as an alternative method. To save the computational effort, the response surface was first fitted locally in the specified interval with a given anchor point (initial M), which was taken from the previously obtained possible values of the mass (one random variable case). After a few iterations, the approximate results were obtained as shown in Table 4.19, which gives all the possible design masses with corresponding sensitivities and the required iteration times to achieve the convergent response surface. It is noted that the bounds required in the analysis and the initial mass can be obtainedfromFig. 4.7.  Table 4.19: Summary of the Results (mass in tonne) Lower  Upper  Initial  No. of  Design  Bound  Bound  Mass  Iterations  Mass  Solution 1  40.0  48.0  45.0  2  45.79  -0.071823  Solution 2  48.0  52.0  50.0  3  50.54  0.064959  Solution 3  52.0  60.0  55.0  2  55.04  -0.051307  Cases  Sensitivity  It is seen that the effect of considering the uncertainty in the yield stress has, in this case, a very slight influence on the desired mass. The solution M = 55.04 tonne, with lower sensitivity, is chosen as the proper solution.  From Fig.4.7, one can see that unique  solutions are obtained if Do is approximately less than 0.9 or greater than 0.14.  114  Chapter 4 Applications in Engineering Design  4.5 Summary Several practical design examples, drawnfromcode calibration, offshore and earthquake engineering, have demonstrated the applicability and the efficiency of the proposed inverse reliability methodology. Compared with the traditional approach, the direct inverse reliability method not only shows a satisfied saving in computational efforts, but also provides a useful design tool to the designer. For the applications in earthquake engineering, two problems were used to demonstrate a reliability-based design procedure. The linear problem was used to show how multiple design parameters are sought simultaneously to meet specified reliability constraints. In the nonlinear problem, multiple solutions of a design parameter were found to satisfy a serviceability reliability criteria, while a nonlinear model was used to evaluate the limit-state function.  115  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Chapter 5 Applications in the Manufacture of a Laminated Composite Component 5.1 Introduction The control of process-induced deformation has been one of the most critical challenges in the manufacture of structural components using composite materials. This deformation may result in final component dimensions being seriously beyond design requirements, resulting in severe mismatch during the assembly of the components in a final structure like an aircraft. The residual stresses developed during the manufacturing process are responsible for the deformation which may range from small in-plane deformations and local spring-in of angular sections to large-scale warpage in larger composite structures. One of the major objectives during the manufacture is to maintain the residual deformations within a pre-set tolerance limit. From a probabilistic point of view, the process must have a very small probability of exceeding the tolerance limit. To meet this goal, a traditional approach (still popular in today's industry practice) is to modify the material properties, tool geometry and processing cycle until an acceptable component geometry is obtained. This trial-and-error iterative procedure is, of course, expensive, time-consuming and not very efficient, especially when applied to large components.  A probabilistic model of the complete manufacturing process is thus needed to replace the usual empirical technique. During the last two decades, many theoretical methods of  116  Chapter 5 Applications in Manufacture of a Laminated Composite Component  modeling the composite manufacturing process have been developed. These have different focuses: 1) empirical formulae (Nelson and Cairns, 1989; Huang and Yang 1997); 2) analytical and numerical models ranging from one-dimensional analyses (Loos and Springer, 1983, White and Hahn, 1990) and two-dimensional analyses (Bogetti and Gillespie, 1991, 1992) to three-dimensional analysis (Chen and Ramkumar, 1988). These process-modeling methods include the analysis of thermochemical behavior, resin flowcompaction, process-induced deformation and so forth. However, relatively little work has been done towards modeling the industrial autoclave process for structures of practical size and complexity. This remains an ongoing research area with significant theoretical and practical value.  Since 1992, a comprehensive computer software for modeling the entire composite manufacturing process, named COMPRO, has been developed by the Composites Group at The University of British Columbia, with the collaboration of The Boeing Company, Seattle. COMPRO allows the complete modeling of the manufacturing process for large and complex shape composite structures (Hubert, 1996, Johnston, 1997). Several characteristics developed in the process, for example, degree of resin cure and the processinduced deformation, can be predicted efficiently. The accuracy and the applicability of the model have been verified through comparison with experimental results and actual field data (Hubert, et al. 1996, Johnston, 1997 and Fernlund, et al., 1998).  However, since many uncertainties (random variables) are involved in the manufacturing process including, among others, material properties, tooling, ply lay-up and autoclave  117  Chapter 5 Applications in Manufacture of a Laminated Composite Component  loading, the process-induced deformation is also a random variable. From an engineering point of view, a high or low variability reflects, respectively, poor or high quality of the processed component.  The variability of the individual components will affect the  variability of the completed mechanical assembly, both in geometry and functional performance. Some studies called 'tolerance analysis' have been done on the variability of a completed assembly.  In these studies, based on the variability of the individual  components, the propagation of the variability in the assembly can be determined by using 2D and 3D tolerance analysis (Chase, et al., 1996). When applying the method of tolerance analysis to an assembly of composite components, for example, the wing of an aircraft, one has to know a priori the variability of the individual components.  Therefore, there is a need to model the composite manufacturing process on a probabilistic basis by considering the uncertainties involved in the raw materials and the process implementation. This probability-based modeling will provide the required variable quantities for a processed component, which can then be used in the calculation of the variability of the assembly using 'tolerance analysis.' Fig. 5.1 shows a flow chart with 4 steps to illustrate the procedure.  To achieve a pre-specified quality for a composite assembly, a probability-based design is needed to determine the relevant design parameters. This requires an inverse analysis.  118  Chapter 5 Applications in Manufacture of a Laminated Composite Component  1: Determination of the variations in the raw material and process implementations  2: Determination of the variability of the individual processed components by using probability-based process modeling  3: Determination of the variability of a complete mechanical assembly by using 'tolerance analysis'  4: Evaluation of the quality of the assembly  Figure 5.1: Flow Chart of the Procedure  The inverse reliability approach presented in this thesis can be used to determine the design parameters with associated tolerance ranges. To do this, the parameters are treated as random variables with mean values and standard deviations, acting as the actual design parameters.  The objective of this application is to carry out reliability-based design for several design situations in the composite manufacturing process. The computer code COMPRO, and the forward and inverse reliability methods proposed in Chapter 2 and Chapter 3, were used to formulate the probability-based design process. This allows, for example, the variability  119  Chapter 5 Applications in Manufacture of a Laminated Composite Component  prediction of the process-induced deformation and the determination of a quality index for the composite product. This is associated with the probability that the process-induced deformation will exceed a pre-specified tolerance limit. In addition, to meet a given probability-based criterion for a composite component or a structure, the design parameters associated with material behavior, tooling, ply lay-up and bagging implementation and the cure cycle are desired. This can be done through the reliability-based design by using the inverse reliability method. Furthermore, either initial cost (cost of making a component or structure) or the total expected cost (the expected cost of failure plus the initial cost) can be minimized by selecting the proper design parameters via either an inverse reliability or forward reliability based-optimization technique. Integrated methodologies for approaching these problems do not appear to exist in the field of composite manufacturing process. Among the barriers to the implementation of such methodologies are the complexity of analysis models like COMPRO and that of the reliability evaluation itself. With emphasis on illustrating the proposed approach rather than solving a real problem, four case studies will be discussed here: 1) predicting the variability of the process-induced deformation; 2) determining a design parameter to meet the given quality index (target probability of failure for a component); 3) seeking the design parameters required to minimize the initial cost, and 4)findingthe design parameters to minimize the total expected cost. Some assumptions were made to supplement the shortage on the statistics  120  Chapter 5 Applications in Manufacture of a Laminated Composite Component  for material properties, tooling, ply lay-up and autoclave loading (cure cycle) and other relevant variables.  5.2  Manufacturing process modeling  5.2.1 A typical manufacturing process Among the number of techniques that may be used to manufacture fiber reinforced composites, the autoclave process (simulated here by COMPRO) is usually used in highperformance structures.  In this procedure, thin layers of prepreg (a material of high  stiffness fiber impregnated with a partially cured resin) are laid up to form a laminate against a tool having the desired shape. The fibers can be oriented in any direction to achieve the desired mechanical properties. The shaped, raw component is then covered by several absorbent layers of cloth to absorb excess resin (bleeder cloth) and to provide a path for removal of air and volatile gases from the part (breather cloth) during the cure. The complete assembly, including laminate, tool, bleed and breather is sealed inside a vacuum bag. The next step is the "cooking " of the laminate. The whole tool-composite assembly is placed in an autoclave and subjected to high temperature and pressure controlled by a predetermined cure cycle as shown in Fig. 5.2. In the autoclave, the temperature is used to trigger the polymerization reaction of the resin and the pressure is used to keep the laminate fully against the tool surface and to prevent any voids from developing during the cure. After completion of the cycle, the cured part is cooled, removed from the tool and made ready for further process or assembly.  121  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Temperature(°F), Pressure(psi)  Cure cycle time (hours) Autoclave Temperature  Autoclave pressure  Figure 5.2: A Typical Autoclave Cure Cycle  5.2.2 A brief review of the process model COMPRO To model the manufacturing process for a laminated composite, a two-dimensional planestrain finite element model (COMPRO) has been developed using bilinear quadrilateral isoparametric elements (Hubert, 1996 and Johnston, 1997). COMPRO incorporates the effects of heat transfer, resin flow-compaction, resin cure kinetics, resin cure shrinkage and stress-deformation development.  A wide range of problems and configurations can be  modeled, including multidirectional laminates, fabrics, tool-part interfaces, honeycomb, tool, inserts and autoclave types. The model predicts the temperature of the curing parts, resin degree of cure, resin viscosity, fiber volume fraction, resin pressure, part thickness  Chapter 5 Applications in Manufacture of a Laminated Composite Component  variation and process-induced deformation. The overall structure of the process model is an integration of a series of'sub-models' or 'modules' that are analyzed independently and in sequence over time. The following is a brief background to the three major modules used in COMPRO.  •  Thermochemical Module  The thermochemical module is used to calculate the distribution of component temperature T and the degree of resin chemical advancement a (degree of cure). Two sub-models, heat transfer and resin reaction kinetics, are considered in the analysis, permitting the prediction of current temperature and degree of cure, two fundamental 'state' variables used to upgrade the properties of the composite material during the entire process. The governing equation of the thermochemical module is expressed as follows (Johnston, 1997):  ~(pC T) dt p  - —(*„ —) +—(k —) + — ( * „ —) + — ( * „ — ) + dx dx dz dz dx dz dz dx !2  fi  (5.1)  where p and C are the composite density and the composite specific heat, respectively and p  ley are the composite thermal conductivities. The rate of resin heat generation Q is formulated as:  Q = ^-V )PrH f  (5.2)  R  where p and V/ are the resin density and fiber volume fraction, respectively and HR is the r  resin heat of reaction, defined as the total amount of heat generated during a complete resin reaction. Based on a finite element formulation and time integration, the two-coupled  123  Chapter 5 Applications in Manufacture of a Laminated Composite Component  equations can be solved to provide the current temperature and degree of cure used by the other modules throughout the process.  •  Flow-Compaction Module  The flow-compaction module allows to upgrade the resin viscosity, the resin pressure, composite deformation and local composite fiber volume fraction during the process. Two governing equations, the stress equilibrium and the resin continuity equations, are involved in the module. The stress equilibrium is given as: da.  da BP _ . , da, da„ 8P „ . - + —+ + 7^=0 and -+ —+ +F =0 dx dz dx dz dx dz  _ (5.3) x  Z  and the resin continuity is governed by: •  '  d X  x  dP.  dx p dx where a , x  a  t  and  the resin pressure; e  x  d ,K  Z  dz  dP.  p dz  a„ are the effective stresses; F and F are the body forces; P is  and s  x  z  z  are the composite strain rates and K and K are thefiberbed x  z  permeabilities, p is the resin viscosity defined as a function of its temperature and degree of cure and can be obtained from several empirical models. For a given set of boundary conditions the coupled equations can be solved using the finite element method together with time-step integration.  124  Chapter 5 Applications in Manufacture of a Laminated Composite Component  •  Stress-deformation Module  The stress-deformation module is responsible for modeling the internal stresses and stressinduced deformation developed within a component during the entire processing. Through integration of this module with analyses of component temperature and resin degree of cure (thermochemical module), the model can examine all five major sources of process-induced stress and deformation identified in the literature (Johnston, 1997). These are:  1) thermal expansion 2) resin cure shrinkage 3) gradients in component temperature and resin degree of cure 4) resin pressure gradients (resulting in resin flow) 5) mechanical constraints due to tooling  The fundamental equations, and the associated system of algebraic equations for the discretized problem, are obtained from the principle of stationary potential energy. By solving these equations, the nodal displacements for the complete component or structure can be determined, allowing the calculation of the stress-induced deformation.  5.3  5.3.1  D e s c r i p t i o n o f the composite l a m i n a t e used f o r case studies  Laminate and cure cycle  A C-shaped composite laminate, as shown in Fig.5.3, was chosen for all case studies. As one of the simulated results of COMPRO, the process-induced spring-in, as illustrated in  125  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Fig. 5.4, is of particular interest. The prepreg was cut and laid up in 16 layers in the orientations of [0/45/-45/90/0/45/-45/90/90/-45/45/0/90/-45/45/0], against a pure aluminum tool. A single-hold cure cycle used for autoclave loading was assumed as shown in Fig. 5.5. The autoclave temperature was increased to a target level of 350 °F, at the heating rate of 5 °F per minute, held for two hours, and then cooled down to 151°F at the rate of -6°F per minute. The autoclave was pressurized to 65 psi and then kept constant throughout the entire cycle.  Figure 5.3: Sketch of the C-shaped Part and Tool  126  Chapter 5 Applications in Manufacture of a Laminated Composite Component  TandP  350°F v  65psi Temperature Pressure  \  151°F  Time (minutes) 120  Figure 5.5: Cure Cycle  127  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.3.2  Materials used in the case studies  The composite material used in all the case studies in this Chapter is Hercules AS4/8552, a unidirectional carbon-fiber epoxy prepreg. Its relevant properties mainly include fiber volume fraction, density, specific heat capacity, conductivity, thermal expansion, resin cure kinetics, resin cure shrinkage and elasticity of the fiber. The tool material is chosen as pure aluminum with specified properties. A detailed description of these materials is provided elsewhere (Johnston. 1997).  5.4 Basis for probabilistic-based process modeling 5.4.1  Source of uncertainties  In the probability-based process modeling, the sources of uncertainty should first be determined in order to estimate the related probability models. Those uncertainties may be associated with material behavior, the implementation of the tooling, ply lay-up and autoclave loading as well as the analytical process model. The uncertainties in material properties are usually caused by: 1) poor material characterization due to difficulties and high cost of performing tests; 2) materials with poor quality produced by a low-level manufacturing (for example, lack of knowledge of what to control), resulting in high variation against the nominal design value. In laying up the prepreg to form a shaped laminate prior to curing, human error or lack of care and experience may cause uncertainties in the orientations and the bond condition between the tool and the laminate. In the autoclave, the internal airflow can cause some variation in the temperature and  128  Chapter 5 Applications in Manufacture of a Laminated Composite Component  pressure, so that the pre-specified cure cycle may not be achieved as designed. Also, although the accuracy of COMPRO has been confirmed by comparison with extensive experimental results, a complete probability-based modeling must include a random variable accounting for model error due to many empirical formulas used and the complexity of the problem. This random variable can be established by comparing the experimental results with the numerical model predictions.  5.4.2  Random variables  There are many parameters involved in COMPRO modeling, ranging from material properties and tooling to autoclave schedule, and it may not be necessary to take into account the variability in all these parameters. Only those exhibiting significant effects should be taken as the random variables. Due to either a lack of statistical data or the complexity of a sensitivity analysis, some simplifying assumptions are needed to facilitate the solution. For example, if a high-quality raw composite material is used in the process, one could neglect the variability in the material properties by treating the relevant parameters as deterministic variables.  To simplify the analysis in this Chapter, the following assumptions are made: 1) the variability in composite materials and in autoclave loading are taken into account; 2) highlevel performance in the tooling, ply lay-up and bagging is assumed, implying that the associated uncertainties can be neglected and 3) model error is considered by introducing a random variable with a mean value of 1.0 and standard deviation of 0.1. Based on these  129  Chapter 5 Applications in Manufacture of a Laminated Composite Component  assumptions and previous sensitivity analyses (Johnston 1997), five parameters, as shown in Table 5.1, are chosen as the random variables.  Table 5.1: Summary of Potential Random Variables Variables  Mean Value  COV  Type  RCS (Associated with resin cure shrinkage)  0.1  0.1  Normal  VF (Volume fraction of fiber)  0.57  0.1  Normal  CTE3 (Coef. of thermal expansion) / °C  2.86E-5  0.1  Normal  T (Target temperature) °F  350.0  0.01  Normal  R„ (Model error)  1.0  0.1  Normal  As shown in Table 5.1, the five variables are assumed to be normally distributed. In the absence of more detailed statistical data for these random variables, the nominal values of the material properties are used as the mean values for RCS, VF, and CTE3 with an associated 10% coefficient of variation. The mean value of the target temperature T is taken as the 350 °F, based on the pre-designed cure cycle, as shown in Fig. 5.5. From experimental experience, the real loading temperature may be +/- 10 °F offfromthe 350 ° F, thus, a 0.01 coefficient of variation is obtained approximately by equating the range of 20 °F to six standard deviations. The variable model error R„ is used as a factor applied to COMPRO predictions. As assumed, it has a mean of 1.0 and coefficient of variation of 10%.  130  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.4.3  Failure mode  As already indicated, the process-induced spring-in angle 0 is assumed as the only deformation of interest. Therefore, a typical performance function G can be defined as:  G=e -R O 0  (5.5)  n  where 6 is the spring-in obtainedfromCOMPRO, which is a function of random variables and other deterministic parameters, and 6o is the given tolerance limit, which depends on the actual requirements for the assembly. R„ is the random variable associated with model inaccuracy.  The probability of the event G < 0 is then the probability that the actual  spring-in will exceed 6Q.  5.5  Case study 1: Variability prediction in the spring-in  The objective of this case study is to predict the variability of the process-induced deformation (spring-in) during the manufacturing process and also to investigate the sensitivities of the results to the mean value and coefficient of variation of the material properties. The results of the sensitivity analysis will be used to facilitate the following inverse reliability analysis (case study 2), inverse reliability-based optimization (case study 3) and reliability-based optimization (case study 4). The variability prediction can be done by carrying out a forward reliability analysis, which permits an estimate of the probability that the spring-in will exceed a given tolerance limit. To perform this calculation, COMPRO was combined with RELAN, a general RELiability-ANalysis software package (Foschi, et al,  1998)  that implements first-and second-order reliability analysis  131  Chapter 5 Applications in Manufacture of a Laminated Composite Component  (FORM/SORM).  To save computational efforts,  the reliability calculations were  implemented using a response surface technique (RSM), a user option in RELAN, since COMPRO, a time-dependent model, requires substantial time to perform a single run.  5.5.1 Variability prediction for the basic case The variability of process-induced spring-in 9 was first investigated by calculating the probability of G > 0, or Prob (R„0 < 9o), for various values of the tolerance limit Oo. The statistics given in Table 5.1 are referred to as the basic case. The analytical results are presented in Table 5.2. To give a complete description of the variability of the spring-in, the analytical results were fitted by a probabilistic model using the least-squares method. To obtain the best fit, four probabilistic models, Normal, Lognormal, Weibull and Gumbel, were fitted to the results in Table 5.2.  Table 5.3 shows the obtained statistics and the  cumulative error for different fitted probabilistic models. It is seen that the Normal distribution with a mean value of 1.498° and a standard deviation of 0.168°, provides the best representation of the variability of the process-induced spring-in. Figure 5.6 shows both the analytical results and the fitted Normal distribution function. This predicted probabilistic model is essential to evaluate not only the quality of the composite product, but also to carry out the tolerance analysis of an assembly system composed of processed components (Chase et al. 1996).  132  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Table 5.2: Probability of G > 0 for Various Tolerances 0o  P  Prob(G>0)  0o  P  Prob(G>0)  1.0  -3.028  0.00122911  1.50  0.056  0.52244085  1.2  -1.772  0.03818301  1.55  0.348  0.63598665  1.25  -1.461  0.07206547  1.60  0.637  0.73784753  1.30  -1.151  0.12489526  1.65  0.922  0.82167772  1.35  -0.843  0.19949003  1.70  1.208  0.88655745  1.40  -0.539  0.29498303  1.80  1.746  0.95962975  1.45  -0.241  0.40495483  2.00  2.781  0.99728775  Table 5.3: Statistics and Error for Different Distribution Models Distribution Type  Distribution Parameters  Error  Mean(°)  Standard Deviation(°)  Normal  1.49772  0.168171  0.00059  Lognormal  1.48677  0.175425  0.00285  Weibull  Loc = 0.896756, Seal e=0.665323, Shape=3.82307  0.00148  Gumbel  A=7.21916,B=1.38014  0.03267  133  Chapter 5 Applications in Manufacture of a Laminated Composite Component  1.00 0.90  c  o 3 <fl  0.80  Mean = 1.49772  0.70  Std.Dev. = 0.16817  0.60  a t  0.50  E  0.30  Ia 3  u  0.40 '  — N o r m a l Distribution  0.20  • Analysis Results  0.10 0.00 0.80  1.00  1.20  1.40  1.60  Spring-in 0  1.80  2.00  2.20  O  Figure 5.6: Distribution of the Spring-in  5.5.2  Sensitivity analysis  To identify the major factors influencing the results of the process model, a sensitivity analysis was performed by investigating the effect of the statistics of the random variables on the probability of failure Pf, that is, the probability of the process-induced deformation exceeding a given tolerance limit Oo. In this section, the effect on the results of the mean values and coefficient of variations (COV) are studied.  134  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The sensitivity to the mean value of the fiber volume fraction VF was conducted by obtaining results for mean values of 0.5 and 0.6, at different threshold Oo rangingfrom1.4° to 1.8°. The statistics of the other random variables remained as in Table 5.1, which is the basic case. The results are shown in Fig. 5.7.  To investigate the effects of the variability in the random variables, the COV of the three variables associated with material properties were changed simultaneously from 5% to 20%. The COV of the target temperature and the COV of the model error remained the same as in the basic case. Figure 5.8 shows the corresponding probabilities of failure at different Oo. For both cases, the analytical results were fitted, respectively, by Normal distributed functions, as shown in Fig.5.9 and Fig 5.10.  135  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Figure  5.8:  Sensitivity to the  COVaX  136  Different Tolerance Level  Chapter 5 Applications in Manufacture of a Laminated Composite Component  1.00  0.90 0.80 -\  M e a n of V F = 0 . 5  in  § 0.70  /  I  :§ 0.60 •K =5 0.50 a>  M e a n of V F = 0.6  /  M e a n = 1.46 Std.dve = 0.174  /  >  J  i  O  0.40 0.30  ^  M e a n = 1.57 Std.dve = 0.176  0.20 0.10 0.00 0.75  1.00  1.25  1.50  1.75  2.00  2.25  2.50  S p r i n g - i n B„  Figure 5.9: Model Prediction for Different Means  1.00 0.90 0.80  .1 0 . 7 0 3  H  in  a V  0.60  M e a n = 1.493 Std.dev = 0.2318  COV=0.05  H  0.50  -COV=0.20  _>  5  3 E 3  0.40  M e a n = 1.491 Std.dev = 0.1578  0.30  Q 0.20 0.00 0.10  0.80  1.00  1.20  1.40  1.60  Spring-in 0  1.80  0  Figure 5.10: Model Prediction for Different COV  137  2.00  2.20  Chapter 5 Applications in Manufacture of a Laminated Composite Component  From Fig.5.7 and Fig.5.9, it is seen that increasing the mean value of VF from 0.5 to 0.6 will shift the probability distribution of the predicted spring-in leftward, without varying the curve shape, resulting in the change of the mean value of the spring-in 0from 1.57° to 1.46°, and a very slight change in the standard deviation (from 0.176° to 0.174°).  Figure 5.8 and Fig.5.10 show that the decrease of the COV from 20% to 5% will decrease overall variability of the model without shifting it. The mean value of the predictions remains almost the same (from 1:491° to 1.493°) while the standard deviation varies significantly from 0.2318° to 0.1578°. It is also seen that the influence of the COV on the Pf depends highly on the location of the tolerance limit Oo, or the probability of failure being calculated. For example, at Bo =1.7°, the probability of failure Pf changes from 0.186189 at COV = 20% to 0.092465 at COV = 5%.  However, at 0 =1.5°, which O  corresponds to Pf= 0.5, no significant change of Pf is observed between COV = 20% and COV= 5%. The opposite trend happens when, for example, do =1.3°.  5.5.3  Conclusions from the sensitivity analysis  The sensitivity analysis leads to the following conclusions: 1) If the threshold 9o is close to the nominal spring-in 0 predicted by COMPRO, using the mean values of the random variables, the coefficient of variation of the intervening random variables (COV) has very little influence on the probability of failure. A  138  Chapter 5 Applications in Manufacture of a Laminated Composite Component  decrease in the COV or, in other words, increasing the quality of the materials or the process will not improve the quality of the processed component significantly. 2) When the nominal 0is different from the threshold 60, the influence of the COV could be substantial. In this case, the manufacturing engineer should pay attention to the quality of the materials and implementation of the process rather than modifying the mean design values.  5.6 Case study 2: Seeking design parameter - an inverse approach This case study will demonstrate the application of the inverse reliability method to the composite manufacturing process. The objective is to find the proper design parameters involved in the composite material or manufacturing process so that the pre-specified quality level for a composite product can be satisfied. In contrast to the previous studies, this is an inverse approach with emphasis on the engineering design in the manufacturing process. To perform the inverse calculation, COMPRO was combined with ERELAN. The design parameters may be chosen from the raw material properties or autoclave loading in the cure cycle, depending on the problem at hand. In this section, three cases will be discussed for a given target reliability: 1) seeking the mean value of the fiber volume fraction VF, when its COV is specified; 2) finding the standard deviation of fiber volume fraction P F when the mean value is given; 3) designing the mean of target temperature Tin the cure cycle. For all three cases, the reliability constraint is associated with the limit-state function as shown in Eq.5.5 with a given target reliability level. The following provides the analysis of results for each case in detail.  139  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.6.1  Mean of the fiber volume fraction  In this case the design parameter is the mean value of the VF, one of the most important parameters controlling the behavior of the composite material. The other parameters involved, either in the material, ply lay-up, cooling and autoclave loading are assumed to be either constants or random variables with given statistics and distributions. The statistics of the random variables remains the same as those given in Table 5.1, except for VF. VF is assumed Normally distributed with a given coefficient of variation, implying an allowable tolerance for the quality of the material. Inverse analyses were carried out for various coefficients of variation of VF ranging from 0.05 to 0.15 and target reliability levels ranging from 1.0 to 1.5. The threshold of the spring-in Oo is assumed 1.8°. The inverse results are given in Table 5.4 and graphically illustrated in Fig. 5.11.  Table 5.4: The Required Mean Value of VF Target Reliability Level B  COV =0.05  COV= 0.1  COV = 0.15  1.0  0.4501  0.4553  0.4660  1.5  0.5238  0.5346  0.5520  2.0  0.5896  0.6117  0.6457  140  Chapter 5 Applications in Manufacture of a Laminated Composite Component  1  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2  P  Figure 5.11: Mean of VF against Target Reliability  It can be seen from Fig.5.11 that the required mean value of VF increases with increasing target reliability for a specified coefficient of variation COV. And at B = 2.0 the changes in the mean value against COV are significantly greater than those at B = 1.0, which is consistent with the conclusions addressed in the previous section.  5.6.2  Standard Deviation of VF  In this case the design parameter is chosen as the standard deviation of the fiber volume fraction while its mean value is given. This is the problem of determining the quality level of the materials when their nominal design values (mean values) are given. The inverse reliability analyses were carried out for target reliability indices of 1.0 and 1.5 respectively, for different given mean values. Table 5.5 presents the results.  141  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Table 5.5: The Required Standard Deviation ofVF Target Reliability Level B  Mean Value=0.55 Mean Value=0.5 7 Mean Value=0.60  1.0  0.19350  0.2181  0.2530  1.5  0.08127  0.1030  0.1336  From Table 5.5, it is seen that the desired standard deviation must be significantly reduced to about 50%, when the target reliability increases from 1.0 to 1.5. This case study demonstrates that when the nominal design values of the material are given, high reliability can also be achieved by specifying a low variation, or higher accuracy in the quality of the material being used.  5.6.3  Target temperature  When the properties of the composite material are given, the design of a proper cure cycle is of importance for the manufacturing engineer to achieve a target reliability associated with the process-induced deformation. Many parameters affect the process including the autoclave pressure, temperature, heating rate, cooling rate and the number of hold stages. So far, the only feasible method to fix all these parameters is based on engineering experience and an empirical approach through 'trial and error'. With the development of the numerical process modeling method, the determination the cure cycle can be still done using 'trial and error' through computer simulation rather than through experiments. In this case study, a simple case of determining a single design parameter in the cure cycle, on a probabilistic basis, using the inverse procedure, was performed. The cure cycle is assumed the same as in Fig. 5.1, but with an unknown mean value for the target temperature T,  142  Chapter 5 Applications in Manufacture of a Laminated Composite Component  which is treated as the design parameter. Its coefficient of variation is still assumed 0.01. All the other statistics of intervening variables remained the same as in Table 5.1. The threshold Oo is, still, Oo = 1.8°. The analyses were carried out for various target reliability levels B = 1.5, 1.75 and 2.0. The results are given in Table 5.6.  Table 5.6: The Desired Mean of Target Temperature T MeanofT(°)  Target Reliability Level B 1.5  362.5  1.75  349.6  2.00  337.9  From Table 5.6, it is seen that the design parameter, the mean value of the temperature, decreases with increasing target reliability, indicating that high temperature may cause larger residual deformations. However, the temperature and hold time have to be sufficient to ensure that the composite part is fully cured, i.e. the degree of cure should be near 1.0. On the other hand, if the temperature is too high the laminate can burn out. Thus, the obtained temperature should be checked against some other conditions, for example, when the hold time is set to two hours, the temperature must be in the domain of 300°F < T < 400° F.  143  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.7 Case study 3: Seeking multiple design parameters - Inverse Reliability-based optimization This case study presents a determination of multiple design parameters to minimize the initial cost (or material cost) in the manufacturing process. Based on previous assumptions, the design parameters are considered associated only with the properties of the composite material.  A quality requirement of a structural assembly implies a certain quality level (reliability) of the individual composite component. For example, in order to assemble an important structure that is the combination of such components, it may be required that the probability of exceeding a certain level for a process-induced deformation be less than 10' . 2  This  quality goal can be achieved at a higher cost, such like selecting higher quality raw material and stricter control in the tooling and autoclave loading, etc. However, the manufacturer is more likely interested in how to minimize the manufacturing cost C / (called initial cost) and, at the same time, meet the quality requirement. This demands an optimization on the reliability basis, in which the initial cost Q is taken as the objective function and the quality criteria is considered as a reliability constraint. The objective function is dependent on the design parameters, and so does the reliability constraint. Based on the above discussions, the following problem will be discussed.  144  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Minimizing the initial cost under reliability constraint  5.7.1  This problem can be solved by the inverse reliability-based optimization proposed in Chapter 3. The objective function, associated with the manufacturing cost (initial cost), is defined in terms of design parameters. The process-induced spring-in is considered in the reliability constraint as discussed in the last section.  5.7.1.1 Objective function and constraints The evaluation of the initial cost should take into account the cost of the materials, the cost of design, the cost of implementation of the process (including ply lay-up, tooling and autoclave loading) as well as the cost of labour. To simplify the analysis and to keep consistency with the previous studies, two assumptions are made: 1) the initial cost C/ is associated only with the raw composite material properties, and 2) the initial cost Q is, in general, proportional to the mean value of the material properties and the sum of the inverse of their coefficient of variations (COV).  Based on the previous sensitivity analysis, it has been seen that both the mean value and the coefficient of variation are important to the probability of the process-induced spring-in exceeding a given tolerance. Thus, four design parameters are considered: 1) Myf = The mean value of thefibervolume fraction VF 2) Vj = The COV of the resin cure shrinkage 3) V = The COP of thefibervolume fraction VF 2  4) V = The COV of the coefficient of thermal expansion 3  145  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Thus the objective function (initial cost) d can then be expressed as: (5.6)  Ci = ajMyf + a (II Vj+ 1/ V + 1IV ) 2  2  3  Where aj is the cost per unit Myf and a is the coefficient related to the COVs. In Eq.(5.6) 2  the first term is the cost associated with the mean of the fiber volume fraction and the second term is the cost related to the coefficient of variations (quality control). These terms are denoted, respectively, Cyf and Cco»  For the reliability constraint, the associated limit-state function is the same as shown in Eq. (5.5), with threshold Oo = 1.8°. The target reliability is B = 1.5, corresponding to an exceedence probability of 0.066807.  The optimization problem can then be stated as: Minimize the objective function: Ci = F(Mvf, V VT, V ) = aiMyf + u  3  a (lf 2  Vi+ 1/V + 1IV ) 2  3  (5.7)  Subject to: Reliability constraint: Prob ( G < 1.8°-R„ •  0) = 0.066807, equivalent to B= 1.5  Other constraints: OA < Myf <0.7  (5.8a)  0.0 < Vj < 0.3  (5.8b)  0.0 <V < 0.3  (5.8c)  2  146  Chapter 5 Applications in Manufacture of a Laminated Composite Component  0.0 < V < 0.3  (5.8d)  3  For each iteration required by the optimization algorithm, the inverse reliability calculation is performed to represent implicitly one of the design parameters as a function of the others, which forms a contour of the specified target reliability B = 1.5. The reliability constraint is then eliminated by substituting the expression for the contour into the objective function, thus resulting in the number of free design parameters reducedfromfour to three. However, due to the complexity of the COMPRO calculation, an intensive computational effort is needed to calculate the limit-state function. Therefore, the efficiency of the optimization is affected. To solve this problem, a response surface is used here to form a contour of B = 1.5 in terms offreedesign parameters. The design parameter My/ is expressed approximately as a function of the others through this response surface. By doing so, the inverse reliability calculation in the optimization can be replaced by a calculation of an explicit function. To interpolate the response surface, the observation points are obtained by repeating the inverse reliability analyses for selected values of the free design parameters in the feasible domain. This strategy allows the separation of the optimization and the inverse reliability analysis, reducing dramatically the computational efforts within a reasonable engineering error.  5.7.1.2 Response surface The response surface, Myf= F(Vi, V2, V ), 3  was constructed by repeating the inverse  reliability analysis for target reliability B =1.5 and various combinations of the selected  147  Chapter 5 Applications in Manufacture of a Laminated Composite Component  design parameters. The results were then fitted by a second-degree polynomial expression using the least-squares method.  Using the statistics shown in Table 5.1, the inverse reliability analyses results are shown in Table 5.8. The fitted quadratic was then obtained as:  M,  -0.50739 + 0.006284IF, -0.0991 \2V + 0.022593F, + 0.32233F  2  2  yf  i  2  + 2.13284F +0.66683F" 2  2  3  i  (  5  9  )  2  3  It should be noted that the Eq.(5.9) is a contour surface for the reliability constraint imposed (/?= 1.5). Substituting Eq.(5.9) into Eq.(5.7), the optimization problem can then be easily solved.  Table 5.8: Response Surface Points (6= 1.5) Vj  v  V  Myf  0.15  0.15  0.15  0.5671  0.01  0.15  0.15  0.5590  0.3  0.15  0.15  0.5898  0.15  0.01  0.15  0.5332  0.15  0.3  0.15  0.6962  0.15  0.15  0.01  0.5490  0.15  0.15  0.3  0.6155  2  3  148  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The accuracy of the response surface was verified by comparing the predicted values Myf from response surface Eq.(5.9) with those directly calculated by inverse reliability analysis (IRELAN). From the comparison, an average relative error of 0.34% was obtained, which is sufficiently small to be accepted. Table 5.9 shows the details of the comparison.  Table 5.9: Detailed Comparison Results 2  V  M^(RS)  M/IRELAN)  Error(%)  0.2  0.2  0.2  0.618  0.615  0.57  0.01  0.01  0.01  0.507  0.507  0.0  0.01  0.1  0.1  0.528  0.530  0.47  0.1  0.01  0.2  0.542  0.541  0.05  0.2  0.1  0.2  0.564  0.565  0.12  0.15  0.1  0.1  0.536  0.540  0.74  0.05  0.2  0.3  0.641  0.642  0.22  0.3  0.1  0.1  0.559  0.559  0.23  0.1  0.1  0.1  0.532  0.535  0.57  0.1  0.05  0.1  0.521  0.524  0.63  Vi  v  3  5.7.1.3 Optimization results The reliability constraint is then eliminated by replacing M^with the response surface Eq.(5.9) in the objective function Eq (5.7). Thus, Q = aj (0.50739 + 0.00628407Vi- 0.0991117V + 0.0225928V + 0.322332V/ + 2  3  2.13284V + 0.666831 Vs ) + a (1/ Vj+ 1/ V + 1/V ) 2  2  2  2  2  3  (5.10)  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The problem, with the modified objective function Eq.(5.10) and with the additional constraints in Eq.(5.8), can now be solved easily by any nonlinear programming technique for given coefficients aj and a . 2  In this case study, the Complex method was used to  perform the optimization. In order to assess the effect of the ratio between aj and a on the 2  optimization, the optimal calculations were carried out for a = 0.5 and 1.0 for a given aj. 2  The minimized initial cost and corresponding obtained design parameters are shown in Table 5.10. Figures 5.12 and 5.13 show graphically the optimal costs and design parameters.  Table 5.10: Optimization Results (ctj= 50.0) Cost on Myf Cost on COV  Vj  a  Ci  0.5  31'.47  58.54  29.27  8.2  0.25  0.14  0.19  1.0  44.72  63.08  31.54  13.2  0.30  0.18  0.24  2  M J%) V  v  2  44.7  Cl  Cvf  Figure 5.12: Optimal Costs  150  Ccov  v  3  Chapter 5 Applications in Manufacture of a Laminated Composite Component  0.631  Mvf  V1  V2  V3  Figure 5.13: Optimal Design Parameters  From Figs. 5.12 and 5.13, it can be seen that increasing the constant a (the coefficient 2  associated with COV oi materials) the optimal initial cost C/ increases along with all of the design parameters. This reflects the fact that, if the cost of increasing the quality of materials is high, cost minimization and the reliability constraint can be achieved by increasing the amount of the requiredfiberand decreasing the material quality.  151  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.8 Case study 4: Seeking multiple design parameters - Reliability-based optimization For completeness of the application, a total cost-oriented optimization problem in composite manufacturing is approached in this case study. It takes into account the total expected cost involved in both the manufacturing process and the structural assembly. As discussed previously, the quality of the raw material and the process implementation may significantly influence the quality of composite product. The manufacturer has the choice of either selecting the raw material to be used in the process, which may exhibit different qualities, or improving the quality of the process. For example, having a material with low quality, the initial saving on the material is compensated by paying more during structural assembly through 'shim to fit' or even failure.  On the other hand, choosing initially a  material with high quality and high cost, the penalties during assembly may be negligible. Obviously, there is an optimal choice between these two extremes, at which the total expected cost, including the initial cost, the expected cost of shimming and the expected cost of failure is at its minimum. This problem can be classified as another reliability-based optimization problem of minimizing the total expected cost (Moses, 1969, Mau, 1971, and Frangopol, 1985), which is expressed as the function of design parameters, thus no reliability constraint is directly imposed.  5.8.1 The problems of interest As shown in Fig. 5.14, a carbon fiber composite part A is required to fit another existing part B . To this end, part A is shaped using a tool during manufacturing,  152  cti is the  Chapter 5 Applications in Manufacture of a Laminated Composite Component  interference angle of the part B and  a i too  is the tool angle. Due to the inevitable process-  induced spring-in angle 9, the final angle of cured part A is expressed as: cc/=  ckooi - 9  Part A  (511)  Part B  Tool  Figure 5.14: The Sketches of the Parts and Tool  Because of the uncertainties involved in the raw materials and the process, the spring-in 6 is, in fact, a random variable and its variability depends on the quality of the raw material and the process. As illustrated in Fig. 5.15, four possible consequences (events) will happen in the structural assembly. In order to express the formulas in term of 0, define 9j and 02 small angles representing, respectively, the tolerances corresponding to acceptable and fixable limits. A working angle 0j is defined as: 9d =  (5.12)  atooi-cci  153  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The part fails (a,f < ai)  The part isfixablethrough shamming  The part is acceptable (a/ < a/< aj+ 0j)  The part fails (a/ > aj+ 62)  (ai+dj <af<ai+ 6 ) 2  Figure 5.15: Graphic Illustration of the Possible Events in Structural Assembly  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The above four events can be summarized as: •  When Of < otiipr 0> Od), the c^is too small to fit the part B, the part fails and the cost of failure must be paid.  •  When ai<Of<ai+0i  (or 0d-9i<0< Od), the composite part is acceptable to be used  without extra costs. •  When cci +0] < af< aj + 02 (or Od-02 < 0< Od-Oi), the part still can be used but extra cost are incurred to fix the mismatch in the assembly, that is, shimming is required.  •  When a/ > ai+ 0 (or Od < Od-O2) the 2  is too large to be shimmed. The part also fails  and the cost of failure must be paid.  According to the above events, the corresponding probabilities are defined as: •  Pf] = Prob (0> Od), the probability of making a bad part (failure).  •  P - Prob (0d-0j <0< Od), the probability of making an acceptable part.  •  P = Prob (Od-02 < 0  •  Pf2 = Prob (0 < Od-02), the probability of making a bad part (failure),  a  <  s  Od-Oi), the probability of requiring shimming.  and they must satisfy: Pf] + P + P + Pf2 = LO a  (5.13)  s  A graphic illustration of above probabilities is shown in Fig. 5.16.  155  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Shim  Survival  Fail  Fail /  \  P / 2  Pa  Ps  0D-02  Ny  dj-Oj  d  d  Figure 5.16: Probabilities of the Events  For a given material and process setting, the statistical model of the process-induced springin 0can be predicted by performing reliability analyses, as discussed in Case Study 1. The mean and variability of 0 are determined by the design nominal values and the quality of raw material or process. Since dd influences the probability of failure and, in turn, c w influences dd, manufacturers sometimes prefer to adjust the tool angle a i rather than too  modify the statistics of # through changes in the material properties.  A manufacturer is interested in minimizing the total expected cost through a proper design. Considering the possibilities of failure and shimming, the total expected cost E includes the t  manufacturing and material cost Ci (initial cost), the expected cost of failure ECF and the expected cost of shimming Tics, E =Q+Ecs t  +EF  (5.14)  C  156  Chapter 5 Applications in Manufacture of a Laminated Composite Component  The goal of this case study is to design the tool angle  Otooi  and the quality level of the  process and the raw material to minimize the total expected cost E . t  To show the progress of the approach, two problems will be studied. 1) Find an appropriate tool angle o w to minimize the total expected cost when the statistical model of the spring-in 0is given. 2) Determine both tool angle  a i too  and os, the standard deviation of the spring-in 9  (representing the quality level of material and the process), to minimize the total expected cost.  5.8.2 Approach If C is the cost of shimming and Cp and Cp are the costs of failure, the expected shimming s  cost and the expected costs of failure are defined, respectively as: •  Ecs = CsPs(9 cr )  •  ECFI  = CpPfl(9 ,ag)  (5.15b)  •  E 2 = CpPp(9 ,o-g}  (5.15c)  A  CF  (5.15a)  6  d  d  Where dd and a$ are the two design parameters associated with, respectively, the tool angle and the variability of process-induced spring-in 0.  Using the reliability-based optimization technique, the above two problems can be, in general, achieved by minimizing an objective function:  157  Chapter 5 Applications in Manufacture of a Laminated Composite Component  E,= CM  + C P/0d,o-e) + C Pfl(9 ,o-<) +C P (d ,a^ s  fl  d  f2  f2  d  (5.16)  The initial cost Cj is associated with the cost of manufacturing and materials. It can be formulated as (5.17)  Ci(ae) =Co+C (a-Q} q  where Co is a constant, considered as the basic cost, and QfoV is that related to the quality of process and material, which may be expressed as inversely proportional to <JQ (implying that the lower the o& the higher the cost). The cost of shimming results from fixing the mismatch between parts in the assembly, and includes the cost of shimming materials as well as labour. Although the cost of shimming depends on the amount of the mismatch, to simplify the study it is assumed that this cost C , is a constant and can be taken as the average of the costs corresponding to different gaps. The estimate of the cost of failure Q? and Cf2 must consider the costs associated with the wasted time and materials, impacts on the progress of the assembly and loss of reputation.  With the total expected cost E expressed as a function of the design parameters, the t  minimization of the total cost can be performed through a nonlinear programming technique. With a given feasible range of dd and OQ, the optimization problem is simply stated as:  Minimize:  E (dd, t  (5.18a)  CTQ}  Subject to:  (5.18b)  158  Chapter 5 Applications in Manufacture of a Laminated Composite Component  (5.18c)  <J <CT0<OU L  where 9t and Ou are lower and upper bound of 0* , and ai and ou are those of standard deviation ae.  Once the cost of failure, the cost of shimming and the initial cost are given, the minimization could be implemented with optimal results obtained. The optimal 0* can then be used to determine an appropriate tool angle and the optimal ae can be taken as the design quality level of spring-in 9.  For simplicity, ae is chosen in this case study as the quality-related design parameter. However, for a general case, the variability of the spring-in is controlled by that of intervening random variables rangingfrommaterial properties to process implementation. As a result, other design parameters will be involved in the optimization instead of a$ , which requires a forward reliability analysis for evaluating the probabilities conditional on design parameters as previously discussed in Case Study 1.  5.8.3  Numerical examples  Example 1: It is assumed that the raw material has been chosen and the process has been designed. From the variability prediction as outlined in Case Study 1, the mean value, the standard deviation and the distribution of the spring-in 9 are thus obtained. One design parameter 9d is considered.  159  Chapter 5 Applications in Manufacture of a Laminated Composite Component  FromEq.(5.15), the problem is stated as Minimize:  E,= C +C P (0 ) + C P (0 )  Subject to:  0.0 < 9 < 3.0  q  s  s  d  p  p  d  (5.19a)  + CpPrfOd)  (5.19b)  d  Based on discussion in Case Study 1, the statistical distribution of spring-in 0, Fe, is assumed Normal distributed with given mean Me and standard deviation are. In reality, this distribution function can be determined by performing reliability analysis as previously described in Case Study 1. Thus, Eq.(5.19) can be rewritten as: E,= C + q  C fFe(0 -0j) s  d  -Fe(0 -02)1 + C (Jd  FL  Fg(0aj) + CpFg(0 -02) d  (5.20)  where C is a constant conditional on the ere being considered. Assuming that the shimming q  cost is Cs = $4 and the costs of failure are Cji= Cp= $40. In the following description the unit of the cost is omitted for clearness.  To investigate the effect of different variability of spring-in 0 on total expected cost, the optimization analyses were carried out for three different models for Fg. They are refereed as Model A, B and C, which represent, respectively, "very good" (COV=4%), "good" (COV=10%) and "bad" (COV=20%) quality of manufacture.  Table 5.11 shows their  statistical parameters, distributions and the corresponding cost C . Table 5.12 presents the q  optimal results. Figures 5.17, 5.18 and 5.19 illustrate graphically the optimal probabilities, optimal costs and the relationship between the costs and 0 , which provides graphic d  explanation of the optimization.  160  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Table 5.11: Statistical Parameters for Different Model Considered Model  Rating  <0  COV(%)  Q  Distribution  A  Very good  1.5  0.06  4.0  5.0  Normal  B  Good  1.5  0.15  10.0  2.0  Normal  C  Bad  1.5  0.3  20.0  1.0  Normal  Md°)  Table 5.12: Optimal Results Model  E  0d  t  Shimming  Failure 1  Failure 2  Ecs  Ps  ECFJ  Pfl  EcF2  Pfl  A  6.03  1.640  0.66  0.1647  0.37  0.009  0.0  0.0  B  6.64  1.780  2.72  0.6797  1.30  0.0326  0.62  0.0155  C  15.58  1.784  1.86  0.4637  6.89  0.1723  5.83  0.1457  ^COV=4%  15.58  @COV=10% • COV=20% 6.89  Figure 5.17: Optimal Costs for Different Quality Ratings  6.64  Chapter 5 Applications in Manufacture of a Laminated Composite Component  COV=4% ("Very g o o d " )  0.8261  0.1647  0  |  0.009  •Rllll  Pf2  i  Ps  Pf1  Pr (a)  COV=10% ("Good")  0.6797  |  0.0155  0.2722  ffiMHl  hmmmm.  Pf2  Ps  0.0326  Pr  Pf1  0.2181  0.1723  (b)  COV=20%("Bad")  0.4637  I  0.1457 I  Pf2  I  Ps  I  Pr  Pf1  (c) Figure 5.18: Optimal Probabilities for Different Quality Ratings  162  Chapter 5 Applications in Manufacture of a Laminated Composite Component  Figure 5.19: Costs vs. &for Different Quality Ratings  163  Chapter 5 Applications in Manufacture of a Laminated Composite Component  From Figs.5.17 and 5.18, it can be that the optimal expected costs Ep, Es and E/i are significantly controlled by the quality of manufacture (material and process). Having been optimized, the probabilities of failure, the probability of shimming and the probability of survival are redistributed by adjusting the 9 so that the total expected cost reaches its d  minimum. From Fig. 5.18, it is also seen that increasing the quality may reduce or even eliminate the probability of failure, resulting in drastic reduction on the expected cost. The effects of the quality on the optimal total expected cost E, depends on not only the qualityrelated cost C but also the quality level cr,? (or, COV). For example, decreasing COV from q  10% to 4%, the total expected cost is only reduced from 6.64 to 6.03, while decreasing COV from 20% to 10%, a significant saving on total expected cost is observed (from 15.58 to 6.64). Thus, it is necessary to include standard deviation  or COV as design parameter  so that the optimal Oj and the oe can be determined simultaneously to minimize the total expected cost. This will be discussed in Example 2.  From Fig. 5.19, it is seen that the contribution of the expected costs of failure to the total expected cost varies with changing quality ratings throughout all the feasible range. For the model A (COV - 4%), the total expected cost reaches it minimum at 6d = 1.64°. However, this design angle may not be used in reality, since the total expected cost is very sensitive to the left of 9d. To reduce the risk of paying a high cost due to the uncertainties in the implementation and accuracy of statistics, it is recommended not to choose 0a near the sensitive range even if it may be an optimal solution. In this problem, choosing 0j =1.7° or 6d = 1.8° may increase slightly the expected shimming cost but will stabilize the total  164  Chapter 5 Applications in Manufacture of a Laminated Composite Component  expected cost. The calculation of this sensitivity can be done by taking derivative of the E  t  with respect to design parameter. For the model B (COV = 10%), the optimal design parameter 0 = 1.78° was obtained. It shows that the total expected cost is stable in the d  neighborhood of 0 = 1.78°. For the model C (COV = 20%), the probabilities of failure d  control the total expected cost that is quite insensitive to the optimal design parameter within the neighborhood of optimal 0 . d  Example 2: This example demonstrates the determination of the tool angle and the standard deviation of the design spring-in 0. It is assumed that the quality of raw material and the process can be improved by increasing the cost. The quality related cost is assumed inversely proportional to the standard deviation. Two design parameters, 0 and og are determined to d  minimize the total expected cost. The problem is stated as: Minimize:  E = C (og) + C P (0 , og) + C P (0 , t  s  q  s  d  fl  fl  d  og)  +C P (9 , p  p  d  Subject to: 0.0< &<3.0  (5.21a) (5.21b)  ;  and  og)  0.0K og< 0.5  (5.21c)  The statistical model of spring-in 0, Fg, and its mean Mg are assumed the same as in Example 1. Thus, Eq.(5.21a) can be rewritten as Et = C  q0  /cTg  +  C (lfl  + C [Fg((0 -0j), S  d  Fg(0 , d  Og))  dg)  - Fg((0 -02), Og)] d  + Cf2Fg((0 -02),  165  d  Og)  (5.22)  Chapter 5 Applications in Manufacture of a Laminated Composite Component  where C o is a constant coefficient associated with the cost of improving the quality. To q  investigate the effect of C o on total expected cost, the optimization analyses were carried q  out for C o = 0.1, C o - 0.3 and C o = 0.6, which are refereed as, respectively, "low cost", q  q  q  "high cost" and "very high cost." Table 5.13 shows the optimal results and Figs 5.20, 5.21 illustrate graphically the optimal probabilities and optimal costs.  Table 5.13: Optimal Results Cost  e  ET  d  Shimming  oe  CqO  Failure 2  Failure 1  Ecs  Ps  ECFI  Pfl  EcF2  Pfl  0.1  2.5  1.63  0.05  0.25  0.063  0.17  0.004  0.0  0.0  0.3  5.8  1.67  0.08  1.47  0.37  0.60  0.015  0.0  0.0  0.6  8.4  1.76  0.13  2.74  0.69  0.83  0.02  0.19  0.005  8.39 E C q = 0.1  I  B C q = 0.3  5.83  0.00 0.00 °-  Ef2  :  Es  Eft  Cq  Figure 5.20: Optimal Costs  166  Et  '  Chapter 5 Applications in Manufacture of a Laminated Composite Component  C  = 0.1 (low cost)  q 0  0 . 9 3 3 i  l  l  mm mm  mm WmSfflm  0 . 0 6 3  vmmmm  0  Ps  Pf2  Pr  0 . 0 0 4  Pf1  (a) C  q 0  = 0.3(high cost) 0.616  0.369  0.015  0  Ps  Pf2  Pr  Pf1  (b) C  q 0  = 0.6(very high cost)  0 . 6 8 5  mm PiP mm  0 . 2 8 9  Wmmm, WW*,W&<  0 . 0 0 5  Pf2  lllll  0 . 0 2 1  msssmm. Ps  Pr  Pf1  (c)  Figure 5.21: Optimal Probabilities for Different C o q  167  Chapter 5 Applications in Manufacture of a Laminated Composite Component  It can be seenfromFigs. 5.20 and 5.21 that •  For the case of low cost where C o = 0.1, there are almost no probabilities of failure and q  the probability of shimming is very small as shown in Fig 5.21(a). The total expected cost is dominated by the initial cost C . In this case, having high quality (reliability P ) q  a  apparently is the best way to minimize the total expected cost. •  For the case of high cost where C o = 0.3, the probabilities of failure still appear quite q  small but the probability of shimming is larger as shown in Fig 5.21(b). The total expected cost is mostly contributed by both expected cost of shimming and the initial cost. In this case, the minimization of total expected cost can be achieved by allowing a certain probability of shimming. •  For the case of very high cost where C o = 0.6, the probabilities are still quite small but q  the probabilities of shimming becomes dominant as shown in Fig. 5.21 (c). Due to the high cost of increasing the quality, it is not cost-effective to have high reliability Pa. Permitting a large probability of shimming will do a better job on saving total expected cost.  Example 3: As an extension of Example 2, this example focuses on the determination of the design parameters dd and oo when target probabilities of failure Pfi and Pp are specified, due to, for example, extreme high failure cost (such as Cp = Cp = 4000.0). Based on previous discussion, two limit states are considered:  168  Chapter 5 Applications in Manufacture of a Laminated Composite Component  G> = 0d-O  (5.23)  G = 0-(0 -O.6)  (5.24)  2  d  The mean value of spring-in 0 is kept as Me =1.5° and the spring-in 0 is assumed normal distributed. For several set of target probabilities of failure Pp (Prob(Gj<0)) and Pp (Prob(G2<0j), inverse reliability analyses were carried out to obtain design parameters dd and ere simultaneously. Table 5.14 presents the results along with the associated expected costs of failure Ep and Ep and quality related cost C . q  Table 5.14: Design Parameters and Associated Costs E =4000Pp Ep=4000Pp  Pfl  Pfl  fl  C =0.3/ag q  IO"  10"*  1.842  0.111  4.0  40  2.7  IO"  4  io-  3  1.828  0.088  0.4  4.0  3.4  IO"  If/  4  1.821  0.075  0.04  0.4  4.0  3  5  The design parameters in Table 5.14 will ensure that the two target probabilities of failure are satisfied simultaneously. It can be seen that increasing the target reliability will decrease the expected costs of failure and increase the quality-related cost.  169  Chapter 5 Applications in Manufacture of a Laminated Composite Component  5.9  Summary  Reliability theory has been introduced to study different problems in a composite manufacturing process: •  Prediction of the variability of process-induced deformation  To determine the variability of process-induced deformation, forward reliability analyses were carried out. The results of variability analysis can be used not only for predicting and evaluating the quality of processed composite component, but also for selecting the design parameters in a reliability-based design or optimization. •  Determination of one design parameter to meet given target exceedence probability  Inverse reliability analyses were performed to determine directly the design parameters when a given target exceedence reliability is given. A single design parameter was found to satisfy a reliability constraint, the specified probability of process-induced spring-in not exceeding a threshold. Both mean value and standard deviation were considered as the design parameters. •  Determination of multiple design parameters through inverse reliability-based optimization  Inverse reliability-based optimization was carried out to determine the design parameters for minimizing the initial cost and satisfying a reliability constraint. In order to reduce the required computational efforts, the response surface technique was used.  170  Chapter 5 Applications in Manufacture of a Laminated Composite Component  •  Determination of multiple design parameters through reliability-based optimization  A practical problem of minimizing the total expected cost in the composite manufacturing and structural assembly was proposed and approached. Considering the possibility of the shimming and failure, the total expected cost includes the initial quality related-cost, the expected cost of failure and the expected cost of shimming. As a special case, inverse reliability method was used again to determine the multiple design parameters when the probabilities of failure were specified.  Due to the shortage of the statistics of the random variables and the complexity of the problem, some assumptions were made to simplify the study. This case study was only intended to provide a practical methodology to different problems in the composite manufacturing when the uncertainties of the intervening variables have to be considered. With sufficient information on materials, processing implementation and analysis model, real problems can be approached practically.  171  Chapter 6 Summary and Future Work  Chapter 6 Summary, Contributions and Future Work 6.1 Summary and contributions As the major contribution of this thesis, a general inverse reliability methodology has been proposed. It allows the direct and efficient determination of design parameters when target reliabilities for corresponding limit states are specified. The accuracy, efficiency and robustness of the procedure were verified by performing a variety of numerical case studies, including comparisons with the work done by others. An inverse reliability method for a single deterministic design parameter, previously proposed by others, is significantly extended to solve problems with either single or multiple design parameters. When seeking multiple design parameters, either individual target reliability indices for each reliability constraint or an overall target reliability index of a series system can be achieved directly by means of the proposed inverse approach. In addition, the design parameters can be treated as deterministic or they can be associated with the mean value or the standard deviation of a random variable. To obtain a unique solution, the number of reliability constraints or/and geometric constraints must equal the number of design parameters.  When the solution of the inverse reliability problem requires an intensive computational effort for the evaluation of the limit state function, an inverse response surface method is  172  Chapter 6 Summary and Future Work  proposed. The accuracy and efficiency in using a response surface approach were also evaluated through case studies.  Since there may be multiple solutions for a design parameter or, in some cases, difficulties in convergence due to local extrema, a practical inverse reliability-based design procedure is proposed by combining Newton-Raphson's with a bisection method. This has been shown to be very effective in an example involving nonlinear dynamic analysis.  As a special case of a general inverse reliability procedure, an inverse reliability-based optimization method is also proposed and developed in this thesis. This approach is useful when the number of design parameters exceeds the number of reliability and/or geometric constraints. As an alternative approach to standard reliability-based optimization, the inverse reliability-based optimization attempts to offer a unique solution by minimizing an objective function subject to single or multiple equality reliability constraints. It has been shown in the case studies presented that the inverse reliability-based optimization is superior to the other approaches, particularly, when multiple equality reliability constraints are considered.  In the proposed inverse reliability-based optimization, the reliability constraints are converted to implicit geometric constraints for design parameters, by performing inverse reliability analyses. When the problem needs intensive calculations, it is proposed that the reliability constraints be expressed as explicit geometric constraints by using response surfaces. These can be fitted to observation points by performing several inverse reliability  I 173  Chapter 6 Summary and Future Work  analyses. In this way, the optimization is separated from the reliability analysis. Large computational savings were then observed in a case study.  Based on the proposed methods, two computer programs were developed. IRELAN (Inverse RELability Analysis) can be used to approach the problem with multiple design parameters and IRELBO (Inverse RELiability-Based Optimization) is aimed at finding multiple design parameters to optimize an objective function and to meet reliability constraints.  The efficiency and applicability of the proposed methods have been verified by considering several practical engineering problems, as follows: •  An application in offshore engineering addressed issues in reliability-based codified design.  Design wave and iceberg loads were obtained directly to satisfy specified  annual exceedence risks. The corresponding code load factors were then determined. The calculation of the mean weight of an exploration platform and its standard deviation showed an example of simultaneously finding two parameters associated with a random variable, the platform weigh, when two design criteria were specified. •  Two problems in earthquake engineering were used to demonstrate a reliability-based design procedure which is not covered by codes of practice. In a linear response problem, the example showed how multiple design parameters could be obtained simultaneously to meet specified reliability constraints under the ground excitation. In a  174  Chapter 6 Summary and Future Work  nonlinear response case, using a response surface method, multiple solutions of a design parameter were studied to satisfy again a serviceability criterion. •  To show the wide usefulness of the inverse reliability method and to illustrate the potential benefits in manufacturing processes, another significant contribution is made in this thesis through several applications to the fabrication of carbon fiber reinforced composite components. In conjunction with the numerical process model COMPRO, the variability of a process-induced deformation and its sensitivity to mean value and coefficient  of variation of the intervening process random variables was first  investigated. The inverse reliability method was then used to determine a single design parameter when a target reliability (target quality level) is specified for the processinduced deformation. Integrating the response surface technique with inverse reliabilitybased optimization, multiple design parameters were found to minimize the initial cost in manufacture and to meet a target reliability. As an extension and comparison, traditional reliability-based optimization was also used to minimize the total expected cost, including initial cost, expected cost of shimming and expected cost of failure.  It is concluded that inverse reliability methods provide design engineers with a useful and efficient tool to optimize and customize the design, satisfying reliability standards. The methods can be applied widely to structural, geotechnical, coastal and ocean engineering, and in industrial manufacturing.  175  Chapter 6 Summary and Future W o r k  6.2 Future work Several issues arose during the course of this research, indicating needs for future developments in both theory and applications.  •  In some special cases, when a highly nonlinear limit state function is involved, the reliability index B cannot accurately represent the actual reliability. A corresponding inverse reliability procedure, based on other probability estimates, should be developed to deal with this situation.  •  For the problem when the number of design parameters is less than the number of reliability and/or geometric constraints, a method of choosing the design parameters should be found to, for example, minimize the errors in reliability achieved at each limit state.  •  In earthquake engineering, especially in nonlinear problems, multiple solutions for a design parameter satisfying the same target reliability may exist. Using a special procedure, the problem was solved for the case of single design parameter. A more general procedure for the case of multiple design parameters should be developed.  ©  In this study, reliability concepts were first introduced in the manufacturing of a carbon fiber reinforced composite component. It was not intended to solve a specified problem but to provide an approach. Since the numerical model of the process is still being developed, more refined versions, as they become available, should be incorporated into the inverse reliability assessment, to provide a reliability-based design tool.  176  References Baber, T and Noori, M.N. (1985)" Random vibration of degrading, pinching systems" J. of Engineering Mechanics, ASCE, 111(8): 1010-1026. Barber, T. and Wen, Y.K. (1981) "Random vibration of hysteretic degrading systems." J. of Engineering Mechanics, ASCE, 107(6): 1069-1089. Breitung, K. (1984) "Asymptotic approximations for multinormal integrals." J. of Engineering Mechanics Division, ASCE, 110: 357-366. Box, M. J. (1965) "A new method of constrained optimization and a comparison with other methods." Computer Journal, 8, 42-52. Bogetti, T. A. and Gillespie Jr., J. W. (1991) "Two-dimensional cure simulation of thick thermosetting composites." Journal of Composite Materials 25 (3), 239-273. Bogetti, T. A. and Gillespie Jr., J. W. (1992) "Process-induced stress and deformation in thick-section thermoset composite laminates, Journal of Composite Materials 26(5), 626660. Bucher, C.G. and Bourgund, U. (1990) "A fast and efficient response surface approach for structural reliability problems." Structural Safety, 7(1) 57-66. Bucher, C. G. (1988) " Adaptive sampling - an iterative fast Monte Carlo procedure." Structural Safety, 5(2), 119-126 Chase, K. W., Gao, J., Magleby, S. P. and Sorensen, C. D. (1996) " Including geometric feature variation in tolerance analysis of mechanical assemblies." HE Transactions (Institute of industrial engineers), 28(10), 795-807.  177  Chen, P.C. and Ramkumar, R. L. (1988) "RAMPC - An integrated three-dimensional design tool for processing composites." 33rd International SAMPE Symposium, March 710, 1697-1708. Ditlevsen, O. (1979) "Narrow reliability bounds for structural systems." J. of Structural Mechanics, 7(4) 453-472. Der Kiureghian, A., Zhang, Y. and Li, C C . (1994) "Inverse reliability problem." J. of EngineeringMechanics, ASCE, 120(5), 1154-1159. Der Kiureghian, A. and Liu, P. L. (1986) "Structural reliability under incomplete probability information." J. ofEngineering Mechanics, ASCE, 112(1), 85-104. Der  Kiureghian, A., Lin, H. and Hwang,  S. (1987). "Second-order  reliability  approximations." J. ofEngineering Mechanics, ASCE, 113(8), 1208-1225. Fernlund, G., Abdel-rahman, N., Courdji, R , Poursartip, A. Willden, K. and Nelson, K. (1998) " Effects of cure cycle and lay-up on the dimensional fidelity of autoclave-processed composite parts." To be submitted.  ;  Foschi, R. O., Folz, B., Yao, F. and Li, H. (1997) "RELAN: RELiability Analysis." User's Manual, Department of Civil Engineering, University of British Columbia, Vancouver, B. C. Canada. Foschi, R. O., Isaacson, M . , Allyn, N. and Yee, S. (1996) "Combined wave-iceberg loading on offshore-structures." Canadian Journal of Civil Engineering, 1996, 23(5), 10991110. Foschi, R.O., Folz, B and Yao, F. (1989) " Reliability-based design of wood structures." Structural Research Series Report No. 34, Department of Civil Engineering, University of British Columbia, Vancouver, B. C. Canada  178  Foschi, R. O., Isaacson, M., Allyn, N. and Saudy, I. (1998) "Assessment of the waveiceberg load combination factor." International Journal of Offshore and Polar Engineering, 8(1), 1-8. Foschi, R.O. (1998 a) "Reliability-Based Design in Earthquake Engineering and Hysteresis Modelling", Third International Conference on Computational Stochastic Mechanics, Santorini, Greece. Foschi, R.O. (1998b) "Modeling hysteretic response for reliability-based design in earthquake engineering" J. of Engineering Mechanics, ASCE, (Submitted). Foschi, R.O. (1994) "Reliability applications in iceberg collision forces with offshore structures." Proceedings of the Symposium on Risk Analysis, University of Michigan, Ann Arbor, Michigan, 55-62. Foliente, G. (1995) "Hysteresis modeling of wood joints and structural systems." J. of structural engineering, ASCE, 121(6): 1013-1022 Frangopol, D. M . (1985). "Structural Optimization Using Reliability Concepts", Journal of Engineering Mechanics, ASCE, 111(11), 2288-2301 Gasser, M . and Schueller, G. I. (1998) "Reliability-based optimization utilizing approximate method." Proc. of 7  Th  International conference on structural safety and  reliability." Kyoto, Japan. Hasofer, A. M., Lind, N.C. (1974)" An exact and invariant first order reliability format." J. of Engineering Mechanics Div., ASCE, 100(1), 111-121. Hubert, P., Vaziri, R. and Poursartip, A. " A two dimensional flow model for the process simulation of complex shape composite laminates" Int. J. for Num. Meth. In Eng., in press.  179  Harbitz, A. (1986) "An efficient sampling method for probability of failure calculation." Structural Safety, 3(2), 109-115. Huyse, L. and Maes, M. A. (1996) " Methods for Response-based design criteria." Proc. of the 7  th  JPTP WG 7.5 conference, Reliability and Optimization of Structural Systems,  Boulder, CO, Chapman and Hall, pp.181-188. Hubert, P. (1996) "Aspects of flow and compaction of laminated composite shapes during cure." Ph.D. Thesis, The University of British Columbia, Vancouver, B.C., May. Hubert, P., Johnston, A., Vaziri, A. and Poursartip, A. (1995) "A two-dimensional finite element processing model for FRP composite components." Proc. of ICCM-10, Whistler, B. C . , Canada, August. Johnston, A. (1997) "An integrated model of the development of process-induced deformation in autoclave processing of composite structures." Ph.D. Thesis, The University of British Columbia, Vancouver, B.C. May. Kirjner-Neto, C , Polak, E. and Der Kiureghian, A. (1995) " Algorithms for reliability based optimization design." Proc. of the 6 IFIP WG 7.5 conference, Reliability and th  Optimization of Structural Systems, Assisi, Italy, 145-152. Kuschel, N. and Rackwize, R. (1997) "Two basic problems in reliability-based structural optimization." Mathematical methods of operations research, 46(3), 309-333. Kuschel, N. and Rackwize, R. (1997) "Design for optimal reliability." Proc. of 7  th  International conference on structural safety and reliability." Kyoto, Japan, (in press) Kim, S. H. and Na, S. W. (1997) " Response surface method using vector projected sampling points." Structural Safety, 19(1), 3-19.  180  Loos, A. C. and Springer, G. S. (1983) "Curing of epoxy matrix composites." Journal of Composite Materials 17 (2), 135-169. Li, H. and Foschi, R.O. (1998) " An Inverse Reliability Method and Its Application." Structure Safety, Elsevier, 20(3), 257-270. Lester, B. and Tadros, G. (1995) "Design Requirements." Nova Scotia Business Journal, November. Mau, S. T. (1971) "Optimum design of structures with a minimum expected cost criterion." Report No.340, Department of Structural Engineering School of Civil Engineering Cornell University. Madsen, H. O., Krenk, S. and Lind, N.C. (1986) Methods of structural safety, PrenticeHall Inc., Englewood Cliffs, NJ. Maes, M . A. and Huyse, L. (1997) "Developing structural design criteria with specifies response reliability." Canadian Journal of Civil Engineering, 1997, 24(2), 201-210. Moses, F. (1969) "Approaches to structural reliability and optimization" In: An introduction to structural optimization, M . Z. Cohn, Ed., Solid Mechanics Division, University of Waterloo, Study No. 1, 81-120. Nelder, J. A. and Mead, R. (1965) "A simplex method for function minimization." Computer Journal, 7, 308-313. Nelson, R. H. and Cairns, D. S. (1989) "Prediction of dimensional changes in composite laminates during cure." 34th International SAMPE Symposium, May 8-11, 2397-2410. Pedersen, C. and Thoft-Christensen, P. (1995) " Interactive structural optimization with quasi-Newton-algorithm." Proc. of the 6  th  JTTP WG 7.5 conference, Reliability and  Optimization of Structural Systems, Assisi, Italy, 225-232.  181  Press, W. H., Flannery, B. P., Teukolsky, S.A. and Vetterling, W.T. (1986) Numerical Recipes, Cambridge University Press, Cambridge. Rackwitz, R. Fiessler, B. (1978) "Structural reliability under combined random load sequences." Comp.& Struct. 9, 484-494 Sarpkaya, T. and Isaacson, M. (1981) "Mechanics of wave forces on offshore structures." Van Nostrand Reinhold, New York. Schueller, G. I, Bucher, C. G , Bourgund, U. and Ouypornprasert, W. (1989) " On efficient computational  schemes to calculate  structural failure probabilities." Probabilistic  Engineering Mechanics, 4(1), 10-18 Schueller, G. I. and Stix, R. (1987) " A critical appraisal of methods to determine failure probabilities." Structural Safety, 4(4), 293-309 Tvedt, L. (1983) " Two second-order approximations to the failure probability." Veritas Report RDIV/20-004-83, Det norske Veritas, Oslo, Norway Winterstein, S. R , Ude, T. C , Cornell, C. A., Bjerager, P. and Haver, S. (1994) "Environmental parameters for extreme response: Inverse form with omission factors." Structural safety & reliability, Schueller, Shinozuka & Yao (eds), Balkema, Rotterdam, 551-557. White, S. R. and Hahn, H. T. (1990) "Mechanical property and residual stress development during cure of a Graphite/BMI composite." Polymer Engineering and Science 30(22), 1465-1473. Wu, Y. T. and Wang, W. (1996) " A new method for efficient reliability-based design optimization" Proc. of the seventh specialty conference, ASCE, Probabilistic Mechanics & Structural Reliability, Worcester, Massachusetts, USA, 274-277.  182  Xu, S. L. (1992) Numerical recipe in Fortran, Tsinghua University Press, Beijing. Yao, T. H. J. and Wen, Y. K. (1996) " Response surface method for time-variant reliability analysis."./ of Structural Engineering, ASCE, 122(2), 193-201.  183  

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