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- Permanent Link:
- https://ufdc.ufl.edu/UFE0024834/00001
## Material Information- Title:
- Measuring, Using, and Reducing Experimental and Computational Uncertainty in Reliability Analysis of Composite Laminates
- Creator:
- Smarslok, Benjamin
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2009
- Language:
- english
- Physical Description:
- 1 online resource (148 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Mechanical Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- Haftka, Raphael T.
- Committee Co-Chair:
- Ifju, Peter
- Committee Members:
- Sankar, Bhavani V.
Kim, Nam Ho Uryasev, Stanislav - Graduation Date:
- 8/8/2009
## Subjects- Subjects / Keywords:
- Composite materials ( jstor )
Estimate reliability ( jstor ) Estimation methods ( jstor ) Fiber volume fraction ( jstor ) Laminates ( jstor ) Material properties ( jstor ) Random variables ( jstor ) Simulations ( jstor ) Standard deviation ( jstor ) Statistical discrepancies ( jstor ) Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF carlo, composites, correlation, error, failure, material, measurement, monte, probabilistic, probability, properties, reliability, sampling, separable, uncertainty, variability, variance - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Mechanical Engineering thesis, Ph.D.
## Notes- Abstract:
- The failure of the composite hydrogen tanks on the X-33 Reusable Launch Vehicle (RLV) from combined thermal and mechanical failure modes created a situation where the design weight was highly sensitive to uncertainties. Through previous research of sensitivity and reliability analysis on this problem, three areas of potential uncertainty reduction were recognized and became the focal points for this dissertation. The transverse elastic modulus and coefficient of thermal expansion were cited as being particularly sensitive input parameters with respect to weight. Measurement uncertainty analysis was performed on transverse modulus experiments, where the intermediate thickness measurements proved to be the greatest contributor to uncertainty. Data regarding correlations in the material properties of composite laminates is not always available, however the significance of correlated properties on probability of failure was detected. Therefore, a model was developed for correlations in composite properties based on micromechanics, specifically fiber volume fraction. The correlations from fiber volume fraction were combined with experimental data to give an estimate of the complete uncertainty, including material variability and measurement error. The probability of failure was compared for correlated material properties and independent random variables in an example pressure vessel problem. Including the correlations had a significant effect on the failure probability, however being unsafe or inefficient can depend on the material system. Reliability-based design simulations often use the traditional, crude Monte Carlo method as a sampling procedure for predicting failure. The combination of designing for very small failure probabilities and using computational expensive finite element models, makes traditional Monte Carlo very costly. The separable Monte Carlo method, which is an extension of conditional expectation, takes advantage of statistical independence of the limit state random variables of the response and capacity for improved accuracy in reliability calculations. The separation of response and capacity sampling enables flexible sample sizes, permitting low samples of the more expensive component (usually the response). In turn, this motivates the beneficial reallocation of uncertainty by reformulating the limit state. The variance estimator was derived for separable Monte Carlo and three example problems were used to compare the Monte Carlo methods. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2009.
- Local:
- Adviser: Haftka, Raphael T.
- Local:
- Co-adviser: Ifju, Peter.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28
- Statement of Responsibility:
- by Benjamin Smarslok.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Smarslok, Benjamin. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 2/28/2010
- Resource Identifier:
- 489240979 ( OCLC )
- Classification:
- LD1780 2009 ( lcc )
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PAGE 1 MEASURI NG, USING, AND REDUCING EXPERIMENTAL AND COMPUTATIONAL UNCERTAINTY IN RELIABILITY ANALYSIS OF COMPOSITE LAMINATES By BENJAMIN P. SMARSLOK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 PAGE 2 2009 Benjam in P. Smarslok 2 PAGE 3 To m y parents, Paul and Diane, and sister, Brianne 3 PAGE 4 ACKNOWL EDGMENTS I would first like to send my thanks and apprec iation to my advisors Dr. Raphael T. Haftka and Dr. Peter Ifju. Prof. Haftkas patience, gu idance, and innovative id eas have taught me not only about how to conduct research, but all aspects of life. Ill always cherish our interesting conversations and carry his advice throughout my professional career. In addition to Prof. Haftkas reliability and optimization expertise, I also had knowledgeable in sight from Prof. Ifju on composite materials and experimental techniqu es. Im extremely fortunate to have had the support of two great professors. Many thanks are extended to the other profe ssors and colleagues for their technical help and friendship. Dr. Laurent Carrar o, Dr. David Ginsbourger, and Dr Rodolphe Le Riche for their brilliant mathematical assistance and hospitality while I visited cole Nationale Suprieure des Mines de Saint-tienne, France. Christian Gogu fo r his contributions from our collaborations on spatial variation and correlated material properties. Dr. Lucian Speriatu and William Schulz for their experimental data and thoughtful discussi ons. Bharani Ravishankar for her work with separable Monte Carlo and our co llaborative paper. Dr. Nam Ho Kim, Dr. Bhavani Sankar, Dr. Stanislav Uryasev for their willi ngness to serve on my Ph.D. committee. Other thanks goes to Dr. Erdem Acar, Dr. Theodore F. Johnson, and Dylan Alexander for their assistance and consultation. Financial support provided by the NASA Constellation University Institute Program (CUIP) is gratefully acknowledged. Finally, this dissertation could not have been completed without the endless loving support from my family and friends. Their encouragemen t and inspiration have carried me through my doctoral studies and I am so blessed and grateful to have them in my life. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.......................................................................................................................10 LIST OF ABBREVIATIONS........................................................................................................12 NOMENCLATURE......................................................................................................................13 ABSTRACT...................................................................................................................................16 CHAPTER 1 INTRODUCTION................................................................................................................. .18 Motivation and Background...................................................................................................18 Objectives and Outline......................................................................................................... ..24 2 LITERATURE REVIEW.......................................................................................................26 Probabilistic Analysis of Aerospace Structures......................................................................26 Measurement Error and Material Variability in Composites.................................................28 Uncertainty Classification...............................................................................................28 Measurement Uncertainty Analysis................................................................................29 Modeling Uncertainty in Composite Materials...............................................................30 Efficient Simulation Techniques for Estimating the Probability of Failure...........................32 3 EXPERIMENTAL UNCERTAINTY IN TEMPERATURE DEPENDENT TRANSVERSE ELASTIC MODULUS................................................................................34 Experimental Procedure......................................................................................................... .35 Classification of Measurement Uncertainties.........................................................................37 Propagation of Uncertainty..................................................................................................... 40 Results of Transverse Modulus Experiments.........................................................................41 Component Uncertainty Analysis...........................................................................................42 Uniaxial Load Uncertainty..............................................................................................42 Transverse Strain Uncertainty.........................................................................................44 Thickness and Width Uncertainty...................................................................................45 Transverse Modulus Uncertainty............................................................................................48 4 SPATIAL VARIATION IN COEFFI CIENT OF THERMAL EXPANSION.......................50 5 PAGE 6 Analytic al Model of Axisymmetric Plate with Spike in Coefficient of Thermal Expansion............................................................................................................................50 Example Problem for Displacements and Strain s from a Spike in Coefficient of Thermal Expansion............................................................................................................................54 Strain Predictions for Vari ous Engineering Materials............................................................55 5 CORRELATION MODEL FOR COMP OSITE MATERIAL PROPERTIES......................57 Investigating the Effect of Variation in Fiber Volume Fraction on Thermal Strain..............58 Combining Correlated Material Vari ability and Measurement Error.....................................66 Uncertainty Model with Correlation...............................................................................66 Correlated material variability from fiber volume fraction......................................69 Correlated measurement error from experimental data............................................73 Illustrative Results for Graphite /Epoxy and Glass/Epoxy Laminates.............................75 Effect on Failure Probability for a Pressure Vessel.........................................................78 6 SEPARABLE MONTE CARL O SIMULATION METHOD................................................81 Methods and Expectations of Proba bility of Failure Estimators............................................81 Crude Monte Carlo Method (CMC)................................................................................82 Conditional Expectation Method (CE)............................................................................84 Separable Monte Carlo Method (SMC)..........................................................................87 Conditional Expectation w ith an Analytical Estimate of a Random CDF......................95 7 IMPROVING ACCURACY OF FAILURE PROBABILITY ESTIMATES WITH SEPARABLE MONTE CARLO............................................................................................98 Basic Example with Normal Distributions for Response and Capacity.................................99 Bending in a Composite Plate Example...............................................................................101 Comparison of Monte Carlo Methods...........................................................................103 Reformulation of Limit State Func tion as Variance Reduction Technique..................107 Separable Sampling of a Non-separable Limit State............................................................113 Regrouping a General Limit State for Separable Sampling..........................................113 Application to Composite Materials for the Tsai-Wu Failure Criterion.......................115 8 CONCLUSIONS.................................................................................................................. 123 APPENDIX A RADIAL DISPLACEMENT AND STRA IN DERIVATION FOR SPIKE IN COEFFICIENT OF THERMAL EXPANSION IN AN AXISYMMETRIC, CIRCULAR PLATE.............................................................................................................127 Infinite Circular Plate...........................................................................................................130 Finite Circular Plate..............................................................................................................132 B DERIVATION OF COVARIANCE TER M IN SEPARABLE MONTE CARLO VARIANCE ESTIMATOR..................................................................................................134 6 PAGE 7 C SYMMETRY OF RESPONSE AND CAPACITY RANDOM VARIABLES IN SEPARABLE MONTE CARLO..........................................................................................136 D MONTE CARLO EFFICIENCY COMPAR ISON VIA ANALYTICAL EXAMPLE.......138 LIST OF REFERENCES.............................................................................................................142 BIOGRAPHICAL SKETCH.......................................................................................................148 7 PAGE 8 LIST OF TABLES Table page 3-1 NIST component measurement uncertainty table for transverse modulus experiments....44 3-2 Three-way analysis of variance of thic kness data with subtracted specimen means.........46 3-3 Transverse modulus uncertainty from NIST analysis and experimental results................48 3-4 Transverse modulus com ponent uncertainty summary......................................................49 4-1 Analytical displacement and stra in results for a circular plate..........................................55 5-1 Linear coefficients for fiber volume fraction of graphite/epoxy (Vf0 = 0.6)......................59 5-2 Laminate material properties of IM7/977-2 carbon fiber/epoxy at Vf...............................59 5-3 Corresponding standard deviation and coefficient of variation for Vf: N(0.6, 0.05).........63 5-4 Component properties of S-glass fibe r and epoxy matrix (Gibson 1994, Sect. 3.3).........70 5-5 Linear coefficients for fiber volume fraction of glass/epoxy and graphite/epoxy.............72 5-6 Correlation and covariance measurement da ta from vibration testing on glass/epoxy.....73 5-7 Mean, standard deviation and coeffi cient of variation for S-glass/epoxy.........................76 5-8 Combined covariance correlation matrix ( total) for glass/epoxy...................................76 5-9 Mean, standard deviation, and coefficient of variation for IM7/977-2 graphite/epoxy.....77 5-10 Combined covariance correlation matrix ( total) for graphite/epoxy properties..............77 5-11 Propagation of independent and correlat ed random composite properties to strain..........79 6-1 Summary of crude and separabl e Monte Carlo simulation methods.................................96 7-1 Comparison of experiment al results for crude and se parable Monte Carlo methods......100 7-2 Distribution parameters of random variables for bending...............................................103 7-3 Comparison of empirical and estimated results for CMC and SMC (original)...............104 7-4 Mean and standard deviation of er ror ratios for CMC and SMC (original)....................104 7-5 Mean and standard deviation of ex pectation components for SMC (original)................105 7-6 Comparison of empirical and estimated results for CMC and SMC (reformulated).......108 8 PAGE 9 7-7 Mean and standard deviation of erro r ratios for CMC and SMC (reformulated)............109 7-8 Empirical and estimated variance from SMC with the reformulated limit state.............109 7-9 Mean and standard deviation of exp ectation components for SMC (reformulated)........110 7-10 Extrapolated variance for SMC with the reformulated limit state...................................110 7-11 Mean and coefficient of random materi al properties, pressure, and strengths................116 7-12 Comparison between CMC and SMC for the Tsai-Wu limit state..................................118 7-13 Comparison of the scatter of the orig inal and regrouped limit states for SMC...............120 9 PAGE 10 LIST OF FI GURES Figure page 1-1 Simulated view of NASAs X-33 Reusable Launch Vehicle (RLV)................................18 3-1 Location of thickness and width measurements on composite laminate specimen...........36 3-2 Example data set distinguishing system atic and random uncertainty distributions...........38 3-3 Transverse modulus as a function of temperature.............................................................41 4-1 Circular coordinates for analytical model..........................................................................51 4-2 Distribution of coefficient of thermal expa nsion on the axisymmetric, circular plate......51 4-3 Analytical radial displacement and stra in for example axisymmetric plate problem........54 5-1 Relationships of carbon fi ber/epoxy composite properties and fiber volume fraction......58 5-2 Range of material coordina te strains from CLT for a symmetric laminate..................62 5-3 Mean and coefficient of variation (CV) of the material coordinate mechanical strains....65 5-4 Example uncertainty distributions for m easurement error and material variability..........68 5-5 Relationships of glass/epoxy composite properties with fiber volume fraction (Vf).........71 6-1 Illustration of Monte Carlo method sa mple comparisons A) CMC B) SMC..................88 6-2 Separable MC method us ing random data points of R and the empirical CDF of C .........89 7-1 Composite laminate of dimensions L x L under transverse loading, q( x, y )....................101 7-2 SMC extrapolated CV contour pl ot for the original limit state.......................................106 7-3 Probability density functions of R and C for A) Original B) Reformulated..................108 7-4 SMC extrapolated CV contour plot for the reformulated limit state...............................111 7-5 Estimated coefficient of variation of SM C and CMC for the reformulated limit state...112 7-6 Illustration of separable sampling A) Or iginal limit state B) Regrouped limit state.....114 7-7 Standard Deviation of CMC, SMC, and regrouped limit state SMC...............................119 7-8 Probability density functions of the A) Original limit st ate B) Regrouped limit state..120 A-1 Example CTE impulse distribution..................................................................................128 10 PAGE 11 A-2 FEA displacement curve for a CTE impulse...................................................................129 D-1 General uniform probability density functions for response and capacity......................138 D-2 Sample size of crude and separa ble MC for uniform distributions.................................140 D-3 Actual and estimated ratio of CE to CMC sample sizes for uniform distributions.........141 11 PAGE 12 LIST OF ABBRE VIATIONS ASTM American Society fo r Testing and Materials CDF Cumulative distribution function CE Conditional Expectation method CMC Crude Monte Carlo Method CLT Classical Lamination Theory cov Covariance CTE Coefficient of thermal expansion CV Coefficient of variation FEM Finite element modeling i.i.d. Independent and identically-distributed ISO International Organiza tion for Standardization LPU Law of propagation of uncertainties MCS Monte Carlo Simulation NASA National Aeronautics and Space Administration NIST National Institute of Standards and Technology PDF Probability density function RBDO Reliability-based design optimization RLV Reusable launch vehicle SMC Separable Monte Carlo method stdev Standard deviation TPS Thermal protection system var Variance 12 PAGE 13 NOMENCL ATURE A = laminate extensional stiffness matrix b = systematic uncertainty C = random variable for capacity of a system c = random sample value of the capacity D = laminate bending stiffness matrix D* = laminate bending stiffness com ponent flexural response solution E = expected value of random variable E1 = lamina longitudinal modulus E2 = lamina transverse modulus exp Xe = measurement error in material property, X 112266,1212,,,, FFFFFF = Tsai-Wu strength coefficients FC = actual cumulative distribution function of the capacity C F = random estimate of cumulative distribution function of the capacity FR = actual cumulative distribution function of the response RF = random estimate of cumulative distribution function of the capacity fC = probability density function of the capacity fG = probability density function of the limit state function fR = probability density function of the response G = limit state function for probability of failure G12 = lamina shear modulus h = lamina thickness I = indicator function 13 PAGE 14 i,j = vector or matrix indices K =constant in circular plate k = slope of linear approximation wi th respect to fiber volume fraction M = number of capacity random samples m = total number of standard deviation estima tes in calculating pooled standard deviation N = size of probability of failure simulati ons or number of response random samples n = total number of specimen P = random variable for internal pressure load P = mean value of pressure load pf = actual probability of failure ce p = estimate of probability of failure using conditional expectation cmc p = estimate of probability of failure using crude Monte Carlo f it p = estimate of failure probability using condi tional expectation with random, fitted CDF s mc p = estimate of probability of failure using separable Monte Carlo q0 = transverse pressure loading on composite laminate plate R = random variable of response of a system R1, R2 = independent random variables of law R r = radial coordinate of axisymmetric, circular, thin plate S = vector of random variables of strength of the composite S = vector of mean values of strength of the composite T = temperature t = laminate or specimen thickness u = radial displacement of axis ymmetric, circular, thin plate 14 PAGE 15 Vf = fiber volume fraction of composite laminate w = out-of-plane displacement of laminate wall = allowable out-of-plane displacement of laminate X = generic random variable or material property 1,XX2 = mutually independent random variable vectors = coefficient of thermal expansion = systematic error in measurements 12 = shear strain in the 1-2 plane of a composite lamina 1 = normal strain in the 1-direction of a composite lamina 2 = normal strain in the 2-direction of a composite lamina = random error in measurements = lamina fiber orientation v12 = lamina Poissons ratio = covariance matrix 1 = normal stress in the 1-direction of a composite lamina 2 = normal stress in the 2-direction of a composite lamina u = stress per unit load 12 = shear stress in the 1-2 plane of a composite lamina Vf X = material variability in material property, X = independent random variable for uncertainty C = probability distribution of the capacity 15 PAGE 16 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MEASURING, USING, AND REDUCING EXPERIMENTAL AND COMPUTATIONAL UNCERTAINTY IN RELIABILITY ANALYSIS OF COMPOSITE LAMINATES By Benjamin P. Smarslok August 2009 Chair: Raphael T. Haftka Cochair: Peter G. Ifju Major: Mechanical Engineering The failure of the composite hydrogen tanks on the X-33 Reusable Launch Vehicle (RLV) from combined thermal and mechanical failure modes created a situation where the design weight was highly sensitive to uncertainties. Through previous research of sensitivity and reliability analysis on this problem, three areas of potential uncertainty reduction were recognized and became the focal points for this dissertation. The transverse elastic modulus and coefficien t of thermal expansion were cited as being particularly sensitive input para meters with respect to weight. Measurement uncertainty analysis was performed on transverse modulus experi ments, where the intermediate thickness measurements proved to be the gr eatest contributor to uncertainty. Data regarding correlations in the material properties of composite laminates is not always available, however the significance of correla ted properties on probability of failure was detected. Therefore, a model was developed fo r correlations in compos ite properties based on micromechanics, specifically fiber volume fraction. The correlat ions from fiber volume fraction were combined with experimental data to give an estimate of the complete uncertainty, including material variability and measurement error. The probability of failure was compared for 16 PAGE 17 17 correlated material properties a nd independent random variables in an example pressure vessel problem. Including the correlations had a significant effect on the failure probability, however being unsafe or inefficient can depend on the material system. Reliability-based design simulations often use the traditional, crude Monte Carlo method as a sampling procedure for predicting failure. The combination of designing for very small failure probabilities and (~10-8 10-6) and using computational expensive finite element models, makes traditional Monte Carlo very costly. Th e separable Monte Carlo method, which is an extension of conditional expecta tion, takes advantage of statistical independence of the limit state random variables of the response and capacity for improved accuracy in reliability calculations. The separation of response and capacity sampling enables flexible sample sizes, permitting low samples of the more expensive com ponent (usually the response). In turn, this motivates the beneficial realloca tion of uncertainty by reformulati ng the limit state. The variance estimator was derived for separable Monte Carl o and three example problems were used to compare the Monte Carlo methods. PAGE 18 CHAP TER 1 INTRODUCTION Motivation and Background In the design of aerospace structures, weight is directly related to performance, efficiency, and operating costs (Acar et al 2005). Substantial research has focused on reducing the weight of aerospace structures by selecting optimal materials and reducing uncertainties. Composites have emerged as the preferred material for aeros pace structures, because of their high strengthto-weight ratio and versatility over metals. However, composites present more uncertainties than isotropic materials, and there are special situatio ns where the weight of composite structures is highly sensitive to these uncertainties. One such case was analyzed by Qu et al (2003) regarding the hydrogen tank of NASAs X-33 Reusable Launch Vehicle (RLV) which had both thermal and mechanical failure loads (Final Report of the X-33 Liquid Hydrogen Ta nk Test Investigation Team 2000). A simulated view of the X-33 RLV is shown in Fig. 1-1 (Final Report of the X-33 Liquid Hydrogen Tank Test Investigation Team 2000). Figure 1-1. Simulated view of NASAs X-33 Reusable Launch Vehicle (RLV) The X-33s hydrogen tank was a pressure vessel composed of carbon fiber/epoxy composite laminate. The hydrogen tank was subjected to cryogenic temper atures that created 18 PAGE 19 large therm al strains, which caused matrix crack ing. To reduce the thermal strains, the fiber angle between plies was restricted However this compromised the ability to carry loads in two directions, which are present in pressure vessels. The sensitivity and reliability analysis by Qu et al (2003) of this problem produced three observa tions that are furthe r addressed in this dissertation: 1. The weight of the composite hydrogen tank was found to be very sensitive to uncertainty in transverse elastic modulus E2, and transverse coefficient of thermal expansion 2. 2. Increasing the transverse modulus E2, and decreasing the transverse coefficient of thermal expansion 2, would reduce the probability of failure. 3. When calculating the small failure pr obabilities of safe designs (~10-7) with traditional Monte Carlo simulations, there was a co mputational resource problem. Based on the observations listed above, this dissertation focuses on improving analysis of uncertainties significant in composite cryogenic tanks and on exploring ways to reduce them. The main topics of this analysis include id entifying the main sources of input (response) uncertainty for this problem, looking at the co rrelation between mate rial properties, and developing an efficient Monte Carl o simulation technique. Consider the roles of these topics in the propagation of uncertainty shown in Fig. 1-1. The input parameters are used to calculate response R (e.g. stress or strain via finite element an alysis), which is compared to the capacity C in the limit state function to estimate the probability of failure pf. 19 PAGE 20 Figure 1-2. Propagation of uncertainty to probability of failure calculation The boxes in Fig. 1-2 emphasize the input uncertainty and probability of failure calculation, which encompasses the previously discussed observat ions made by Qu et al. (2003) regarding composite property uncertainty and accurate pf estimates. Note that uncertainty exists in every step of probabilistic analysis, but Fig. 1-1 is a visual guide of the areas of interest in this dissertation. First, it is necessary to clar ify terminology and describe all of the types of uncertainty considered in this study. The total uncertain ty contains computati onal and experimental uncertainty, as well as material variability. Accord ing to the definition used for this dissertation, computational uncertainty includes all numerical errors, randomness and bias from sampling statistical distributions, and m odeling error from either a finite element model (FEM) or inaccurate assumption (such as, a linear elasti c material using Hookes law). In addition, experimental uncertainty includes random and systematic effects in measurements, which correspond to scatter and bias, respectively. Fi nally, there is material variability between specimens, which may be due to random manuf acturing imperfections (e.g. fiber volume fraction). Further defining the terminology, sour ces of uncertainty can be either aleatory (inherent, e.g. variability) or ep istemic (lack of knowledge, e.g. sparse data). As a preview, the 20 PAGE 21 m ain topics of uncertainty covere d in this dissertation are: anal ysis of random and systematic effects in measurement techniques, accurately modeling correlations in composite properties, and variance in estimates of the probability of failu re. The remainder of Chapter 1 provides further motivation and descriptions of th ese topics, and then Chapter 2 summarizes existing literature on the subjects of interest. Efficiency and safety of composite aerospace structures are directly connected to the amount of information known about the materi al system. For example, the X-33s liquid hydrogen tank failed due to errors in the analysis of the composite s performance when subjected to the extreme environment. Therefore, reducing epistemic uncertainty by using more accurate models for material properties (e.g. temperatur e dependence) or performing more tests to reduce epistemic uncertainty (lack of knowledge) are bo th methods of uncertainty reduction that can lead safer structures or si gnificant weight savings (Sc hulz et al. 2005b, Acar et al 2005). Measurement uncertainty and material variabilit y are inherent in experimental observations; however, differentiating real sp ecimen variability from that which is manifested by the measurement process is a difficult task. By breaking down experimental components of the desired measurement and accounting for all of th e identifiable sources of uncertainty (both scatter and bias), then one can determine wher e improved accuracy is desirable. Weight was found to be sensitive to two of the i nput components identified by Qu et al (2003), namely transverse elastic modulus E2, and transverse coefficient of thermal expansion 2. Chapter 3 analyzes the uncertainty in measuring E2 and identifies the main c ontributor of the observed uncertainty in E2. Then, Chapter 4 focuses on an analytic al model for spatial variation in the coefficient of thermal expansion 2, to determine its significance. 21 PAGE 22 Furtherm ore, independent random variables for material properties are frequently used in reliability-based design of composite structures, due to lack of information about correlations. However, it is unlikely that material propertie s vary independently. For composites, the most common mechanisms for changes in material proper ties are variations in fiber volume fraction, fiber orientation, and fiber packing (Conceicao Antonio and Hoffbauer 2007, Gusev et al. 2000). The correlations that may exist from these comm on physical effects are no t properly represented in the statistical distributions of the random compos ite properties. This research presents a simple way to estimate correlations in elastic properties from estimates of variability in the underlying physical effects. In Chapter 5, a correlation mo del for the material variability was proposed based on simplified micromechanics for fiber volume fraction. The correlation model for fiber volume fraction is used in two ways. The effect of correlated trends in composite properties on thermal strain was first analyzed. Then the correlation model for material variability was combined with statistical data (covariance-varian ce matrix) for measurement error. An example pressure vessel problem was used to compare the propagation of mech anical properties as independent and correlated random variables to strain and failure probability. Finally, the uncertainties and correlations in material pr operties were analyzed to deduce whether an inefficient or unsafe design would result from neglecting presence of variability from fiber volume fraction. The last observation by Qu et al. (2003) a ddressed in this diss ertation was regarding computational uncertainty in the probabilistic an alysis from propagating th e material variability and measurement error to the probability of failure calculation. Reliability-based design of structures has advantages over safety factor-based deterministic approaches in that it can reduce the probability of failure more efficiently, and consequently decrease weight. However, common 22 PAGE 23 sam pling procedures used in reliability-based me thods, such as traditio nal, crude Monte Carlo simulations (CMC), are inaccurate for small probabilities of failure (~10-8 10-6), given a computational budget for performing expensive structural response si mulations (Qu et al 2003). As a result, numerous efforts have been made to increase the efficiency of probabilistic simulations, but this dissertation focuses on a technique named separabl e Monte Carlo (SMC). This method uses the same amount of informati on used in traditional Monte Carlo, but takes advantage of the statistical properties of indepe ndence in the limit state. Separable Monte Carlo is an extension of the conditional expecta tion method (CE) (Ayyub and Chao-Yi 1992), since it also uses the conditional probability of failure ba sed on a control variable in the limit state of a statistically independent response and capacity. However, separable MC is more general, in that it is based on comparing combinations of sample s, so the actual distribution of the control variable is not required. The improvement in accu racy over traditional methods is not apparent without extensive, empirical resu lts or comparison of variance estimators. The latter of the two approaches provides more convi ncing and robust conclusions, even though a variance estimator for separable Monte Carlo does not currently ex ist. Therefore, the variance estimate for calculating the probability of failure using the separable Mo nte Carlo was derived using conditional calculus to compensate for the ra ndomness in both the response and capacity. In addition to the inherent accuracy improve ments, separable Monte Carlo has two other characteristics that can be used to reduce the un certainty in the probability of failure estimate. First, given the nature of the SMC calculation me thod, the number of samples of response and capacity do not have to be equal, as in cr ude Monte Carlo. Thus, more samples of the inexpensive component (usually the capacity), wh ich increases the accuracy of the estimate at almost no cost. With this in mind, reform ulating the limit state by moving uncertainty 23 PAGE 24 (independent of the expensive re sponse calculation) to the cheap capacity enhances the previously mentioned advantages of separable M onte Carlo. Chapter 8 demonstrates the benefits of separable MC through several example problems and the accuracy of the probability of failure estimates is compared with crude Monte Carlo and conditional expectation methods. Combining all of the knowledge from experiment al uncertainty analysis, material property correlations, and efficient Monte Carlo simula tions reduces the overall uncertainty in the probability of failure prediction. The next se ction provides a detailed description of the objectives of this work. Objectives and Outline The objectives of this disserta tion are centered to the three observations described in the previous section by Qu et al (2003) regarding the uncertainty in the cryogenic, composite tank. First, Chapter 2 presents a literature review on measurement uncertainty, composite material variability, and efficiency of probability of failure calculations. Then, Chapters 3 and 4 address the first observation regarding sensitivity to uncertainty in E2 and transverse CTE, respectively. Chapter 3 covers the un certainty analysis of E2 measurements and identifies the greatest contributor to experimental unc ertainty and which intermediate measurements would benefit from reducing epistemic error. Next, Chapter 4 looks at the impact of spa tial variation of CTE in a circular plate on strain. The second observation by Qu et al. (2003) was that independent random variables were used for the material properties, even though so me desirable trends between E2 and 2 were noticed. Chapter 5 considers a correlation model based for composite material properties based on fiber volume fracti on, which can be used in the absence or to supplement other correlation data from material variability and measuremen t error. The uncertainty from independent and correlated material properties was propagated through 24 PAGE 25 25 Classical Lamination Theory (CLT) to strain and compared for an example pressure vessel problem. Finally, the third observation was regardi ng the issues of computational resources when calculating very small probability of failures. The separable Monte Carlo method was described in Chapter 6 and variance estimators were also derived. Chapter 7 applies and compares separable MC to crude MC and conditional exp ectation for several example problems, including a basic limit state, the flexural response of a co mposite plate, separable sampling of the Tsai-Wu failure criterion, and an analyt ical example using uniform distri butions. Finally, the conclusions of this dissertation are in Chapters 8. PAGE 26 CHAP TER 2 LITERATURE REVIEW The literature review of this dissertation is divided into three sections. First, a discussion of probabilistic analysis used for reliability-based design of composite aerospace structures. Next is a review of uncertainty and corr elations in composite materials and how they are analyzed and used. The last section contains a survey of efficient ways to calculate the probability of failure with focus on the Monte Carlo method and simulation-based techniques. Probabilistic Analysis of Aerospace Structures One of the most common and important uses of composite materials is in aerospace structures. Furthermore, reliability-based desi gn optimization (RBDO) has been emerging as a useful tool, especially in compos ite structures, due to its more efficient risk allocation between sources of failure (Wirsching 1992, Long and Narcis o 1999). In other words, significant weightsavings can be achieved by using RBDO in lieu of a single safety factor when the structure has multiple failure modes (Acar 2006b, Yang 1989). Acar et al. (2006a) showed that other errors can be avoided by using probabilistic methods in structural design versus deterministic. Other examples of probabilistic analys is of composite structures in clude the following. Lin (2000) looked at different probabilistic methods for cal culating reliability pred ictions of laminated composite plates with random system parameters Cederbaum et al (1990) studied the failure probability of laminated plates subjected to in-plane loads. Probabilistic analysis of a composite aircraft structure based on first ply failure was conducted by Murthy and Chamis (1995). Pai (1990) performed probabilistic structural analys is of a truss for a typical space station. There are certain situations when the composite structure is subjected to multiple failure modes (thermal and mechanical), the sensitivity to uncertainty re duces the potential weight gains over aluminum. Therefore, substantial research ha s been devoted to reducing the uncertainty in 26 PAGE 27 com posite materials, which directly translates to weight reduction in aerospace structures (Acar et al. 2005, Long and Narciso 1999, Neal et al. 1992). Even with reducing the weight as a top priority, safety is of the utmost importance. In order to design a safe structure, one must have extensive knowledge of the material system which may not always be the case due to insufficient statistical data. Inaccurately mode ling the uncertainty in material properties, geometry, and loading may result in large errors in probability of failure estimates (Neal et al. 1992). There were also errors in the analysis of the X-33 Reus able Launch Vehicle (RLV) which failed due to microcracking from residual stresses in the hydro gen tanks while subjected to cryogenic temperatures (Final Report of the X-33 Liquid Hydrogen Tank Test Investigation Team 2000). Failure could have been prevented by reducing epistemic uncertainty (lack of knowledge) by using the temperat ure dependent material prop erties and compensating for chemical shrinkage in the composite laminate (If ju et al. 2006, Schulz et al. 2005a). The failure of the hydrogen tanks on the RLV spurred numerous research projects that focused on analyzing all of the uncertainty that was not initially considered. Qu et al. (2003) focused on the reliabili ty-based optimization of a graphite-epoxy composite laminate used in hydrogen tanks in a cryogenic environment. There were several key findings in the paper that are expoun ded on in this dissertation. First, they determined that weight was very sensitive to the magnitude of uncerta inty. Another observation during the reliabilitybased optimization was that reduci ng the uncertainty of the material properties can dramatically lower weight. In addition to uncertainty reduction, Qu et al. (2003) also looked at how improving the material properties could re duce the failure probability. However, as discovered in the study by Qu et al. (2003) when designing a structure for a very small probability of failure, the cost 27 PAGE 28 burden of simulations becom es great. The conclusions made by Qu et al. (2003) regarding the uncertainty in material propertie s and failure probability became fo cal points of this dissertation. Measurement Error and Material Variability in Composites Uncertainty Classification There are several different standards on meas urement uncertainty and numerous ways of statistically representing the data, depending on the design of experiments. When performing experiments, there is uncertainty associated with each measurement, caused by both systematic (bias) and random (scatter) effect s (Dieck 2002, Sect. 2-2). Uncerta inties are also inherent in each specimen in the form of material variabi lity (covered in Chapters 4 and 5); however, this chapter will only discuss meas urement uncertainty. The terminology used to describe measurement effects varies depending on the standa rd being used or discipline of interest. The standard considered for this research is by the National Institute of Standards and Technology (NIST), which is one of the more comprehens ive and descriptive standards for presenting uncertainty statistics (Taylor and Kuyatt 1994). The NIST sta ndard categorizes measurement uncertainty by random or systematic effect, as well as whether the uncertainty is based on statistical calculations or other methods. The American Society for Testing and Materials (ASTM) has similar terms for random and systema tic effects on uncertainty, which they refer to as precision and bias, respectively. However, ASTM does not distinguish the method used to characterize the uncertainty (ASTM E 177 90a 1990). Conversely, the International Organization for Standardization (ISO) classifies the uncertainties only by whether they were evaluated by statistical analysis or by other methods, coined Type A and Type B evaluations, respectively (ISO Guide to Expression of Uncertainty in Measurement 1993). Another distinction between random and systematic uncertainty is provided in a modeling context by Oberkampf et al (2002), where uncertainty is divided into aleatory (v ariability) and epistemic 28 PAGE 29 (lack of knowledge) components, which roughly corresponds to random and systematic, respectively. Measurement Uncertainty Analysis The statistical method used to measure the signi ficance of a component in an experiment is dictated by the design of experiments used. In general, there are tw o types of design of experiments: nested and factorial. In a ne sted design, essentially each combination of components in the measurement process is uni que (Box et al. 1978), such as different users measuring each specimen. For a nested design of experiments, variance component analysis is performed, as done for acoustic reciprocal measurements by Gu et al. (1998). However, in the work for this dissertation, the same users repeat ed the measurements of all of the specimens, so this follows a factorial design. More concisely, for the thickness measurements in Chapter 3, the variables in the design of experi ments were crossed and random, which were used in a three-way analysis of variance (ANOVA) (Montgomery 2004, Chap. 9). When uncertainty for each component of the experiment is determined, then those uncertainties are propagated to the output. Cord ero et al. (2004b) evalua ted the uncertainty of electronic speckle-pattern interferometry by using the law of propagation of uncertainties (LPU). Similarly, Cordero and Lira (2004 a) looked at uncertainty in speck le-pattern interferometry, but focused on phase-shifting. Other observations f ound in this paper were that environmental perturbations and optical noise followed a Gaussian trend, which is also an assumption made for the random uncertainty variable s used in this paper. Interlaboratory studies are performed to measure the random uncertainty between laboratories of a particular measurement technique or estimate the epistemic uncertainty existing in an individual lab (ASTM E 691-92 1992). Wilson et al. (1990) performed an interlaboratory study on V-notch beam shear tests for composites, in compliance with ASTM standards. He 29 PAGE 30 showed that while the repeatab ility within a lab may appear sma ll, the variation between labs could be substantially large. Therefore, the inte rpretation of what nominal value should be used for optimization is unclear and introduces substa ntial epistemic errors. Bayesian techniques have also been used to extract measurement uncer tainty information by estimating a prior error distribution and performing Baye sian updating with the experimental data to estimate the epistemic error distribution (Acar et al. 2009, Gogu et al. 2009, Mahadevan and Rebba 2005). With respect to the material used for the X-33s hydrogen tank, Schulz et al (2005b) performed precise measurements on the temperat ure dependence of IM7/977-2 transverse elastic modulus E2, and transverse coefficient of thermal expansion (CTE), to re move some of the epistemic error. Based on these experiments, Chapter 3 of this dissertation analyzes the measurement uncertainty. Modeling Uncertainty in Composite Materials The combination of high specific strength and customizable properties from fiber orientation make composites a desirable material especially for building light-weight, safe aircrafts. However, composites also pose a unique set of challenges, such as the discrepancy between macro and micro material property ch aracterization and using an accurate failure criterion for an orthotropic mate rial (Grdal et al. 1999, Sect. 1.3). As in the nature of their properties and the manufacturing process, comp osite materials often have greater spatial variability than isotropic mate rials. Thus, there has been a great deal of work done on determining what experimental methods are good for composites and how much variation exists. Burr et al. (1995) looked at the a ppropriate strain gage size for an anisotropic material. It was determined that there was significa nt variation in measurement read ings for different strain gage sizes, and that the larger the gage, the smaller the variance. The conclusion one can draw from this is that since such large variability exists with small strain gages, then there is perhaps 30 PAGE 31 signif icant spatial variation in the material properties of the composite. For the experimenter to physically observe the spatial variation on the surface of the specimen, full-field measurement methods are often used. Grediac (2004) performed a survey of full-field techniques used with composite materials, in which he addressed thei r uses and limitations. However, the origins of the apparent spatial variation can be quite unclear whether it is from measurement error or material variability. Gogu et al. (2007a) showed that spatial variation from material variability in thermal protection system (TPS) was particularly detrimental to the thermal buckling load. In fact, Gogu et al. (2007c) went on to perform antioptimization to determine the worst-case distribution of coefficient of th ermal expansion and the other el astic properties in a composite plate. Chapter 4 of this disserta tion investigates an analytical model for understand ing the spatial variation in displacement and strain due to a change in material properties. Often measurement error and material vari ability are treated as independent random variables in reliability analysis, due to lack of knowledge. However, researchers have found large errors when neglecting correlations, such as Annis et al. (2004). Conceicao Antonio and Hoffbauer (2007) acknowledge that considering the joint (or co rrelated) effects of input parameters of the response (such as, material properties) is important, but were neglected in their work of analyzing the sensitivity of independent parameters. Research in reliability-based design optimization (RBDO) has been conducted to mode l correlations in material properties using copulas (Choi et al. 2006 and 2007, Noh et al. 2 009). Qu et al. (2003) al so treated material properties as independent random variables; however it was observed that increasing the transverse modulus E2, and decreasing the transverse coefficient of thermal expansion 2, would reduce the probability of failure. A potential rela tionship between properties is likely connected by some physical phenomena. For composites, the most common mechan isms of material 31 PAGE 32 variability in com posites is from fiber volume fraction, fiber orientat ion, or fiber packing (Conceicao Antonio and Hoffbauer 2007, Gusev et al. 2000). Caruso and Chamis (1986) looked at the variation in the mechanical properties of typical graphite/epoxy composites and validated that the simplified composites micromechanics m odel with three-dimensional finite element analysis. A similar study was done by Rosen a nd Dow (1987), however they focused on the thermal properties of composites (longitudinal a nd transverse coefficient of thermal expansion). Therefore, a correlation model for composite properties based on fiber volume fraction was proposed in Chapter 5. This relationship of the properties with fiber volume fraction represents an estimate of the material variability that is often unavailable. Measurement error can also be correlated, such as from expe riments conducted in the same laboratory or identification of multiple elastic properties from a single experiment. Bayesian statistics can be used to obtain correlated data from the analysis of identification experiments such as from vibration testing or an open-hole tensile test (G ogu et al. 2009, Pedersen and Frederiksen 1992, Molimard et al. 2005). Efficient Simulation Techniques for Estimating the Probability of Failure In addition to reducing input uncertainties (such as, material properties), the uncertainty can also be reduced by improving the accuracy of the propagation to the output. When designing safe structures using reliability-based methods, the output is the probability of failure. One method used for calculating the probability of failure is Monte Carlo simulations (Melchers 1999, Chap. 3, Hurtado and Barbat 1998). However, for small failure probabilities (~10-7) the simulations can be very comput ationally expensive (Qu et al 2003). As a result, much research has been done regarding varian ce reduction techniques to help re duce the uncertainty (Rubinstein 1981, Chap. 5, Ayyub and McCuen 1995). Response surface approximations is a method often used in reliability analysis to reduce co mputation time (Kale et al. 2005, Qu et al 2000 and 2004, 32 PAGE 33 33 Zheng and Das 2004). Kim et al. (2006) used a technique called tail modeling to accurately predict the extreme tail of the limit state function using generalized Pareto distributions. Harbitz (1986) developed a method called importance samp ling that restricts the sampling domain near the failure region. Adaptive importance sampling uses the concept of importance sampling, but also uses the information from each simulation to refine the sampling domain (Mahadevan and Ragothamachar 2000, Wu 1994). Conditional expecta tion method uses a control variable with a known distribution to calculate the conditional pr obability of failure for each random response sample (Ayyub and Chao-Yi 1992, Padamanabh an et al. 2006). Ge neralized conditional expectation method simply a generalized form of conditional expectation method that permits multiple control variables (Ayyub and Chao-Yi 1992, Momin et al. 2000). All of the afore mentioned methods are eff ective at reducing computation time; however the technique discussed in this dissertation is applicable in co mbination with these methods to further capitalize on the increased efficiency. The technique of interest for this dissertation is called separable Monte Carlo (SMC) (Smarsl ok et al. 2006a and 2008b). The separable Monte Carlo method is an extension of the conditional expectation method. Ayyub and Chao-Yi (1992) performed studies to measure the variance reduction and improved efficiency from conditional expectation. This dissertation will expand on th eir research by explaining the various ways of using separable Monte Carlo and deriving their vari ance estimators. In addition to the examples of separable Monte Carlo presented in Chapter 7, the method has been implemented in other reliability problems; namely stress failure in a te n-bar truss, thermal and buckling constraints in the probabilistic design optimization of an integrated thermal protection system, and joint work with Ravishankar on separable sampling of the Ts ai-Wu failure criterion limit state (Acar et al. 2007b, Kumar et al. 2008, Ra vishankar et al. 2009). PAGE 34 CHAP TER 3 EXPERIMENTAL UNCERTAINTY IN TEMPERATURE DEPENDENT TRANSVERSE ELASTIC MODULUS As described in the Chapter 1, Qu et al (2003) observed that the probability of failure was particularly sensitive to transverse elastic modulus E2, in the X-33 hydrogen tank problem. The other property was transverse co efficient of thermal expansion 2, which is covered in Chapter 4. The failure of the X-33 next generation shuttle, which occurred because of insufficient knowledge of the temperature effects on the materi al system used in the hydrogen tanks (Final Report of the X-33 Liquid Hydrogen Tank Test Investigation Team 2000). As a result, experiments were conducted to determine the re sidual stress and material properties of a carbon fiber/epoxy composite panel (IM7/977-2) over a temperature range from cryogenic to near cure conditions, approximately -165C to 150C (Schulz et al 2005b). The focus of this chapter is on the uncertainty analysis of these experiments. The observed uncertainty from an experiment is a combination of errors actually from the measuremen ts (e.g. user error), as well as variability in the specimen (e.g. manufacturi ng variability). This study brea ks down the elements of the E2 experiments and identifies the dominant sources of systematic uncertainty (bias) and random uncertainty (scatter). Distinguish ing the types of uncertainty pe rmits us to differentiate how much variability is real and how much is manifested from measurement techniques. Transverse elastic modulus E2, measurements were selected for analysis due to the sensitivity of this material property with respec t to the combined thermal and mechanical failure modes from the cryogenic environmen t of the X-33 hydrogen tank (Qu et al 2003). This study on E2 experiments included an extensive evaluati on of uncertainty components, using NIST guidelines (Taylor and Kuyatt 1994). The uncer tainty classification method used by NIST distinguishes between random and systematic effects on uncertainty, as well as Type A and B, 34 PAGE 35 which will b e discussed more later. The a ppropriate component uncertainties were then propagated through to E2 uncertainty and compared to experimental results. Experimental Procedure The experimental method described by Schulz et al (2005b) for determining temperature dependent transverse modulus was used. These te sts were specially designed to accommodate a wide temperature range, as well as reduce meas urement uncertainty. The experiments centered around performing a tensile test on an 18 laye r, unidirectional IM7/977-2 specimen with a nominal cross-section of 0.09x1in. The thicknes s and width measurements were taken after specimen preparation, but prior to the applicat ion of the strain gages. Then two custom unidirectional strain gages were applied to each side of the specimens and the readings were averaged between them. The WK-13-250BG-350 strain gages were from the Vishay Micromeasurements Group and capable of measuring in a temperature range of C. The thermal environment was varied from -165C to 150C, while taking 14 measurements per specimen at approximately 25C intervals. Once the desired temperature was reached according to the thermocouple inside the testing chamber, the specimen was allowed to soak for 23 minutes to reach a uniform, equilibrium temperature, as per ASTM standard. After the thermal soak period, the strain gages were zeroed and the coupon was loaded using an Interface SM series 1000 lb load cell to 200 lb and unloaded to 0 lb to obtain the load vs. strain data. This loading cycle was repeated five times at each temperat ure. The experiments followed a plane stress condition, so the modulus was determined using a simple manipulation of Hookes Law, shown in Eq. (3-1) .1 1 The actual experimental calculations for transverse modulus were done by averaging the linear fits of the slopes from the load vs. strain data for the five repetitions, then dividing by the cross-sectional area. 35 PAGE 36 2 2P E wt (3-1) Where the subscript refers to the principal material coordinates in the transverse direction, P is the uniaxial load in the 2-direction, width (w) and thickness (t) compose the crosssectional area of the specimen (A=wt), and 2 is the strain in the 2-di rection (Gibson 1994, Chap. 2). The four components in Eq. (3-1) were the sources analyzed in uncertainty propagation. Note that application of Hookes Law is a potential source of m odeling error if the material does not exhibit linear elastic properties; however, that is neglected here. Though a seemingly basic procedure, the potential impact of uncertainty in measuring the thickness and width on the uncertainty in E2 was found to be substantial; therefore, a detailed description of the measurement procedure is di scussed. After machining the specimen to the desired dimensions, the surface across the width was prepared for strain gage application by smoothing with fine grade sandpaper. Then at this point, prior to applying the strain gages, the width and thickness were measured with a micr ometer of .00005in precision, as shown in Figure 3-1. Figure 3-1. Location of thic kness and width measurements on 18-layer transverse composite laminate specimen The light region in the middl e of the specimen represents the smoothed area where the strain gage was eventually applied. Before appl ication, the thickness was measured at the three numbered points shown above and averaged. The wi dth was measured only once, in-line with the 36 PAGE 37 thickne ss measurements. Finally, the strain gage s were applied and the specimen was ready for the tests. Classification of Measurement Uncertainties Measurement error is due to effects that ge nerate a random variation in measurements (random error or scatter) and systematic departure from the true value (systematic error or bias). For this discussion for experimental uncertainty, error refers to the difference between the true value and the measured value. Equation (3-2) represents the idea of measurement errors by considering an individual m easurement xi, that differs from the true value Xtrue, by a random error X, and a systematic errorX, where X and X are random variables. itrueXXxX (3-2) Since the true value is never known, then the measurement error is never known. Therefore, uncertainty is defined as our estimate of the actual error. Since the true value is never known, then the experimental mean X is used as the estimate. Then uncertainty analysis is used to estimate the error with certain level of confiden ce, such as in the form of standard deviation. Total measurement uncertainty is an estimat e of all identifiable sources of random and systematic uncertainty. More precisely, this uncertainty classification identifies whether an uncertainty component came from a random or syst ematic effect in the measurement process, according to NIST standards (Taylor and Kuyatt 1994). To further explain the difference between systematic and random error, an example set of measurement data is graphically represented in Figure 3-2. 37 PAGE 38 Figure 3-2. Example data set distinguishing sy stematic and random uncertainty distributions The mean of one set of measurements (sample) is X which has an experimental population mean that is different from the true value Xtrue. The sample mean refers to measurements taken by one user or one specimen, for example. The population mean hypothetically refers to the average of a very la rge number of experiments in the same laboratory or same conditions (many samples). The true va lue is the actual value of the property being measured of a specimen, which is never exac tly known, so the sample mean is used for calculations. The bias of a sample is the diff erence between the experimental sample mean X and true value Xtrue, which is the systematic error X. The data points are scattered about the sample mean, which is the measurement scatter X. Even though the systematic effect X, is constant through the duration of a measurement, it may vary between experiments within the range 95% level of a normal distribution. The scatter about the experimental mean X is also represented with a normal distribution. Note that normal distributions are a ssumed, but the actual shape is unknown. Since the experimental mean X is used for both X and X, then the random variables in Eq. (3-2) are centered about zero. Thus, the goal of this research is to estimate the standard deviations of X and X through uncertainty analysis. 38 PAGE 39 Now that th e two types of uncertainties are identified, it is important to understand how they exist in the measurement process. Systematic uncertainty (or bias) exists when each measurement in a sample set has the potential to be off by the same amount. In other words, this includes calibration errors, incorrect assumptions, or other factors that woul d cause an equivalent departure from the true value for each reading. Systematic uncertainty is also the part of uncertainty that is often not dir ectly quantifiable through statistical analysis, so estimates of the maximum range of measurements are used. Meas urement scatter (or random uncertainty) on the other hand, is inherent varia tion in the experimental system, which causes scatter. Random uncertainty is often created by cap abilities of the instruments precision or the repeatability of a measurement. Confidence levels of the uncertainty analys is are used in the results based on the assumption of a standard normal distribution. The table of sources of uncer tainty in the results section classifies the component uncertainties as random or systematic, and distinguishes whether they were calculated from statistical analysis (Type A) or by other means (Type B). Type A uncertainties were calculated from data and are reported as the standard deviation of the average x s, as shown later in Eq. (3-9) In this work, the standard deviation was determined by pooling sev eral data sets, which is further ex plained in the thickness and width uncertainty analysis section. The other way to identify stan dard uncertainties is called a Type B evaluation, which means the uncertainty value was determined by methods other than statistical analysis. The range of systematic uncertainty in the meas urement process is assumed to correspond to two standard deviations or the 95% level of a normal distribution. The systematic uncertainty b, corresponds to one standard deviation or the 68%. 39 PAGE 40 Taking into consideration the proposed m ethod to estimate the random ( x s) and systematic (b) uncertainties, in addition to using the sample mean X as our best estimate of the true value; then the random variables X and X in Eq. (3-2) are represented as the normal distributions: X = N(0, x s) and X = N(0, b). Propagation of Uncertainty Since E2 is a parameter that wa s not directly measured, propagation of component uncertainties must be conducted. Similar to the studies by Cordero et al. (2004b), we will consider the law of propagation of uncertainties (LPU). The uncer tainty was divided into random and systematic classifications a nd the uncertainty propagation of E2 (Eq. (3-1) )is presented in Eqs. (3-3) and (3-4) respectively. 22 22 2222 2222 2 2()()()()xE xT xT xT xTEEEE ssPssts Pt 2 2 w w (3-3) 22 222 22222 222222 2 2()()()()2 ()()ETTTT TEEEEEE bbPbbtbwbtb Ptwt T w w(3-4) The component uncertainty terms have an additional subscript T which designates that those uncertainty values are the totals of all the various factors within that component. When performing uncertainty propagation, one must also consider if com ponents are correlated. In this case, only thickness and width have similar measur ements, thus having correlated uncertainties. In this case, only systematic uncertainties associated with w and t were correlated, so a covariance term was added to the end of Eq. (3-4) for thickness and width (Dieck 2002, Sect. 54). Finally, the overall unce rtainty in E2 is quantified as standa rd deviation. Equation (3-5) expresse s the total uncertainty of E2 with 68% confidence, which corresponds to 1. To find the 40 PAGE 41 uncertainty at dif ferent levels of confidence, one must multiply the result by the appropriate Students t distribution value. 222 68()()Ex EEUsb 22 (3-5) Results of Transverse Modulus Experiments The scatter in the empirical results of E2 will be used for comparison to the results of the uncertainty analysis. Figure 3-3 summarizes the experimental results pr esented by Schulz et al. (2005b). Figure 3-3. Transverse modulus as a function of temperature from 10 experiments, measured at approximately the 14 temperat ures shown in the table The transverse modulus averages are based on 10 experiments performed on different specimens. The graph displays a smoothed plot of the E2 data for the 10 specimens at each of the 14 temperatures shown. The uncertainty propa gation used in the following analysis was evaluated at the nominal values for room temperature (22.9 C). Note that undulations in the curves could be a real physical phenomena or ju st additional epistemic error; therefore, future work could involve investiga ting this effect. This would involve performing identical 41 PAGE 42 experim ents in other labs as a part of an in terlaboratory study, as done by Wilson (2004) with Iosipescu Shear Tests. Component Uncertainty Analysis Uniaxial Load Uncertainty The uncertainty associated with the applied load is dependent on the capabilities of the load cell and the unaccounted thermal loads. For this experiment, an Interface SM series 1000 lb load cell was used. The load cell precision, resolution, nonlinearity, hysteresis, and nonrepeatability were all calculated from the manufacturers specifica tions, therefore making each of these a Type B evaluation of uncertainty The load cell capabilities were defined as random uncertainty since the each parameter is assumed to vary continuously throughout the duration of the test, thus generating scatter. The other uncertainty from the load component is the effect of temperature drift. Once the desired temperature was reached and permitted to soak for 23 minutes, the specimen was incrementally loaded with 200 lb, five times to obtain an average E2. Even though the strain gage was zeroed at the desired temperature, a maxi mum of 0.1C temperature drift was observed during the loading process. Equation (3-6) displays the mechanical and thermal loading com ponents applied to the specimen. totalmechthermNNN (3-6) Where, 200 0 0mech therm k kNl bNT Q d z (3-7) The total load {N total}, has scatter in the mechanical component from the variability in the load cell, as well as systematic effects from the thermal component due to temperature drift. 42 PAGE 43 From the equation for {N therm}, T is the temperature drift, [ Q] is the transformed lamina stiffness matrix for the kth lamina, and {} is the coefficient of thermal expansion (CTE) for the kth lamina (Gibson 1994, Sect. 7.6). The material properties used in this calculation were taken from their nominal values at 150C, where the greatest thermal loading occurred. At 150 C, a 0.1C temperature change corresponded to a thermal load of up to 0.29 lb. Since the temperature drift would cause a systematic departure from the mean load, this is a Type B, bias uncertainty. Table 3-1 shows a summary of the uniaxial load uncertainty components, as well as the other component uncertainties that are yet to be covered. 43 PAGE 44 Table 3-1. NIST com ponent measurement uncerta inty table for transverse modulus experiments Standard uncertainties from random effects in the measurement process Standard uncertainties from systematic effects in the measurement process Component Source of uncertainty Type A Type B Type A Type B instrument precision 0.2 resolution 0.011 nonlinearity 0.15 hysteresis 0.02 nonrepeatability 0.05 temperature drift 0.145 Load (lb) RSS 0.256 0.145 strain gage precision 2.5 strain gage sensitivity factor 1.7 Strain () RSS 2.5 1.7 surface variation 0.000375 user bias* 0.00025 measurement repeatability 0.000025 pooled random uncertainty 0.000072 RSS 0.000076 0.00045 Thickness (in) *correlated uncertainty 0.00025 surface variation 0.00005 user bias* 0.00025 measurement repeatability 0.000025 pooled random uncertainty 0.000081 RSS 0.000085 0.00025 Width (in) *correlated uncertainty 0.00025 Transverse Strain Uncertainty The strain measurement uncertainty depends on tw o main factors: strain gage precision and strain gage sensitivity factor. The strain ga ge was estimated to be accurate to within from experience with the equipment. Similar to the load cell precision, the strain gage precision is listed in Table 3-1 as a Type B, random effect si nce precision creates sca tter during the test. The strain gage sensitivity factor uncertainty is th e level of confidence in converting the resistance 44 PAGE 45 into strain, which was provided by the m anufactur er as .2%. Since the same resistance would generate a bias effect, the strain gage sensitivity factor is listed as a Type B, systematic effect on uncertainty. Other sources of error that were not considered here may include strain gage alignment and strain gage reinforcement. Strain is one factor that is particularly in fluenced by specimen variability, which exists in the form of ply angle variati on and fiber volume fraction, for example. However, specimen variability from ply angle was determined to have ve ry small role in the overall strain uncertainty since a unidirectional laminate under transverse loading is not particularly sensitive to variability in fiber orientation. Furthermore, since this wo rk strictly deals with measurement uncertainty, then material variability analysis is not included here. Thickness and Width Uncertainty The cross-sectional ar ea calculation was broken down into its basic components: thickness t, and width w. Due to the nature of their measurements thickness and width have similar types of uncertainty. During the initial measurements of cross-sectional area, it appeared that thickness uncertainties had a significant e ffect on the total uncertainty of E2. Even though the magnitudes of uncertainty of thickness and width were comp arable, thickness was more sensitive, having a nominal value of 0.09in versus 1in for width. Ther efore, the coefficient of variation for thickness was about an order of magnitude greater than width for equivalent uncertainties (standard deviation). Referring to the experi mental procedure and Figure 31, a set of experiments were designed to analyze four factor s of uncertainty: specimen, surface variation, user bias, and measurement repeatability. Four specimens were measured at the three locations across the width, which was performed by four different user s, repeating the measurements three times at each location (144 total measurements). Given the na ture of this design of experiments, a threeway analysis of variance (ANOVA) of crossed, random variables was conducted (Montgomery 45 PAGE 46 2004, Chap. 9). Since the cross-sectio nal area wa s measured for each specimen, a nominal value was not required between specimens. For example, one specimen may have an average thickness of 0.089in and a second specimen could have 0. 091in, both were acceptable for calculating E2, as long as the mean of each specimen was known. Ther efore, to remove the specimen effect from the analysis, the specimen means were subtracted from data used in the analysis of variance. Table 3-2 summarizes the conditions and results of the ANOVA for the thickness measurements described in Figure 3-1. Table 3-2. Three-way analysis of variance of thickness data with subtracted specimen means Factors Description Levels 2var(in) t stdev(in) tp-value A specimen 4 ~0 ~0 ~1 B position 3 1.36E-07 0.00037 0.006 C user 4 4.15E-09 0.00006 0.3253 AB 2.26E-08 0.00015 0.0024 AC 2.01E-08 0.00014 0.0107 BC ~0 ~0 0.5036 ABC 1.81E-08 0.00013 0.0016 n repetitions 3 3.44E-08 0.00019 The three main factors and th eir descriptions are shown in Table 3-1, along with a restricted maximum likelihood estimation of variance. The ANOVA table values were not directly used in the uncertainty analysis; however, stdev(t) does estimate the magnitudes of uncertainty for surface variation, user bias, and measurement repeatability, as well as identify which factors are significant. The position factor, or surface variation effect, was the most significant with a p-value of 0. 006, which corresponds to observations during experiments. The estimate of standard deviation of 0.00037in was used to validate the Type B evaluation of surface variation in Table 3-1. The ANOVA estimate for user bias is an estimate of a random user effect, however only one user actually m easured the specimens dimensions during the E2 experiment, so a different systematic effect is reported in Table 3-1. 46 PAGE 47 As m entioned before, the thickness and width ha ve the same uncertainty factors, but with different magnitudes. The surface variation, user bias, and measurement repeatability were all Type B evaluations since the numbers are ba sed on measurement ranges from experience and converted into a standard deviation. The surface variation is a systema tic effect caused using only three measurement points to average the th ickness, which may have a complicated surface contour due to sanding during strain gage preparation. The user bias is the maximum amount of uncertainty created by the users technique with the micrometer. The user bias is also the only correlated uncertainty in the E2 measurement process, since it is the only equivalent systematic uncertainty that occurs in two different components of E2. Measurement repeatability is defined by the precision of the micrometer used, which was .00005in. The data collected during the ANOVA experi ments was used to calculate the pooled random uncertainty for the thickness and width. P ooling is used when multiple estimates of the same standard deviation exists. In this case, th e standard deviation for thickness and width were pooled across the specimens, since the measur ements for each specimen have the same random uncertainty effect, but with diffe rent specimen means. In other words, this method negated the effect of specimen variability. The calculation of the pooled standard devi ation and the standard deviation of the average are shown in Eqs. (3-8) and (3-9) respectively (Dieck 2002, Sect. 2-2). 1/2 2 1 () 11 1m xi i i xpooled m i ins s n (3-8) () x pooled xs s n (3-9) where, m are the number of standard deviat ion estimates from the specimen ( m = 4) and n is the total number of measurements per specimen ( n = 3*3*4 = 36). 47 PAGE 48 Transverse Modulus Uncertainty The total random and systematic uncertainty for each component wa s calculated by taking the square root of the sum of squares, shown in the rows labeled RSS in Table 3-1. The next step was to apply Eqs. (3-3) and (3-4) to propagate the total uncertainties of each component, for the random ( 2 x Es ) and systematic ( ) uncertainties of the transverse modulus m easurement process. The uncertainty propagation result s are presented with the experimental E2 results in Table 3-3. 2 Eb Table 3-3. Transverse modulus uncertainty from NIST unc ertainty analysis and experimental results at room temperature Method Nominal (GPa) 2 x Es (GPa) % of U68E 2bE 2 (GPa) % of U68E 2 U68E 2 (GPa) CV (%) NIST 9.01 0.0192 13.9 0.0477 86.1 0.051 0.57 Experiments 8.99 0.093 1.03 The results show that the systematic eff ects composed a dominan t 86.1% of the total E2 uncertainty, with the remaining 13.9% due to random effects. Note that the percent effects are calculated in terms of variance, not standard deviation, since you cannot add standard deviations. The uncertainty in the transverse elastic m odulus measurement process was 0.051GPa, with 68% confidence. The standard deviati on of the empirical transverse m odulus at room temperature was 0.093GPa. Comparing the coefficients of variation, the measurement uncertainties listed in Figure 3-3 accounted for about 30% of the total e xperimental scatter. This means the remainder is likely from material variability, which is discussed further in Chapter 5.. Finally, Table 3-4 summarizes the components random and sy stematic uncertainties, as well as the percent effect of each component on the random and systematic E2 uncertainty. 48 PAGE 49 49 Table 3-4. Transverse modulus component uncertainty summary with percent effect of uncertainty Random Systematic Parameter Nominal x Ts % of 2 x Es bT % of 2Eb Load (lb) 200 0.256 36.24 0.145 1.88 Strain ( ) 1700 2.5 47.83 1.7 3.57 Thickness (in) 0.09 0.000076 15.77 0.00045 89.36 Width (in) 1 0.000085 0.16 0.00025 0.23 cov( w,t ) 0.00025 4.96 The components are listed with their corres ponding nominal values at room temperature. The random ( x Ts ) and systematic ( bT) uncertainty columns are from the RSS values in Table 3-3. Referring to Table 3-4, the systematic uncer tainties were more significant in the E2 uncertainty, with 89.36% originating from thickness measurem ents. The random uncertainties were more evenly distributed, where 36.24%, 4 7.83%, and 15.77% of the scatter in 2 x Es came from load, strain, and thickness measurements, respectively. There were three important findings as a result of this study. First, the apparent measurement uncertainty of the E2 experiments was particularly lo w at around 1% coefficient of variation at room temperature, as shown in Figure 3-3. However, the undulations in the E2 curve with respect to temperature leads one to believe that there is bias error from this laboratory that can only be approximated by an interlaboratory study. Next, from uncertainty analysis it was determined that only about 30% of the apparent measurement uncertainty was accounted for in the component analysis. This means that the rest of the uncertainty is likely to come from material variability, which is discussed in Ch apters 4 and 5. Finally, the relatively mundane thickness measurements were the source of the majority of the identifiable experimental uncertainty. Thus, greater attention should be paid to the relatively simple task of measuring the specimens thickness. PAGE 50 CHAP TER 4 SPATIAL VARIATION IN COEFFI CIENT OF THERMAL EXPANSION Nonuniformity in the material properties of a st ructure is naturally i nherent, but being able to measure and model the nonuniformity accurately results in a safer, more robust design. The shape of nonuniformity distributions of interest fo r this study is for large variation over a small area (or spike) which may go unnotic ed experimentally. This spike in material property degrades the capabilities of the structur e, especially for the coeffici ent of thermal expansion, which substantially reduces the buckling eigenvalu e of a thermally loaded plate (Gogu et al. 2007b). In addition, Qu et al (2003) cited that the weight of the X-33 hydrogen tank was particularly sensitive to uncertainty in the coefficient of thermal expansion 2. Therefore, an analytical model was derived to easily predict (without finite elem ent modeling) the change in displacement and strain from a spike in CTE in a large, axisymmetric, circular, thin plate with isotropic properties. Analytical Model of Axisymmetric Plate with Spike in Coefficient of Thermal Expansion The objective of this study was to analyze the effect of nonuniformity in CTE over a very small area by using an analytical model, so several simplifying assumptions were made. The large variation (spike) in CTE over a very small ar ea of the plate lends itself to using a large, axisymmetric, circular, thin plate of radius rR for the model. The idea of a variation over a very small area could logically be modeled for an infin ite plate, but to keep this model practical and capable of being compared with a finite element model, a finite, axisymmetric plate with fixed boundary conditions at the edge was used. The expressions for in-plane displacement and strain are derived in this chapter for a finite plate, however a more deta iled derivation for the finite and infinite plate is in Appendix A. 50 PAGE 51 For the plane stress condition, the axisymmetric model ha s the benefit of CTE and displacem ent only being a functi on of the radial coordinate r ; as opposed to x and y for the rectangular case. Figure 4-1 shows the coordinate system used for this problem. Figure 4-1. Circular coordi nates for analytical model The spatial variation of CTE (( r )) considered in this proble m is a step impulse (spike) which acts on a small radius r0, about the center of the plate ( r = 0). The region from 0 to r0 will have a constant coefficient of thermal expansion, which will be greater than that of the nominal, outer region ( r0 to rR) called nominal. In fact, since only the difference in CTE is of interest for observing the variation in displa cement, then for calculations nominal was set to zero and the absolute variation was used. Figure 4-2 illustrates the step impulse of with radius r0 on an axisymmetric plate of radius rR. Figure 4-2. Distribution of coefficient of thermal expansion on the axisymmetric, circular plate of radius rR 51 PAGE 52 The constitu tive relations for axisymmetric, plane stress are shown in Eqs. (4-1) and (4-2) and 0zzrzrz 211rrErT Eduu v vdrrv (4-1) 211 ErT Eduu v vdrrv (4-2) The simplified equilibrium equation is shown in Eq. (4-3) 0rr rrrr (4-3) After applying the appropriate assumptions to the constitutive relations and equilibrium equation Eqs. (4-1) and (4-2) are combined with Eq. (4-3) to get the differential equation for CTE as a function of r, as written in Eq. (4-4) 2 2211 1 dr duduu vTdrrdrrdr (4-4) The approach to this problem was to impose a CTE distribution and observe the change in displacement and strain. In this case, a step impulse was used fo r CTE, which is a discontinuous function applied to a continuous plate. Therefore, it was necessa ry to have a condition that equated the displacements at r0. Integrating Eq. (4-4) and using the condition u ( r0) = u0, the general form for the radial di splacement is shown in Eq. (4-5) 2 000111 22urKrurKr r (4-5) Where, K is a constant and u0 is the displacement at r0. Since the function of coefficient of ther mal expansion was discontinuous, then the displacement function was broken up into inner and outer regions (re ferring to inside and outside r0). The first of two physical boundary cond itions used for solving the differential 52 PAGE 53 equation is a t the origin, which has a radial displacement of zero for an axisymmetric plate (u(0) = 0). The other is a fixed boundary conditi on at the edge of the finite plate (u(rR) = 0). The radial displacement equations (Eqs. (4-6) and (4-7) ) for inside and outside r0 were solved using the corresponding boundary conditions for each. 0 0inu urr r (4-6) 2 001outururrr r 0 (4-7) Where, was used to simplify a recurring fraction, as defined in Eq. (4-8). 0 22 0 Rr rr (4-8) The balance of forces was used to find u0. The stress at r0 has to be equal with respect to the inside and outside (rr(r0) = 0 in = 0 out). The equations for radial stress at r0 are shown in Eqs. (4-9) and (4-10) with the CTE of the inner region equal to and the outer region equal to zero. 00 0 0 2 00 0111inuu u EETEE v vrrvvrv 1 T (4-9) 2 000000 22 01 1 1outv E uvurur vr (4-10) Finally, equating Eqs. (4-9) and (4-10) the inner and outer radi al d isplacements become 0 0 0 2 000 01 0 21 1 1 21RTv rr r ur Tvr rrrrrr rr r (4-11) Via differentiation, the radial strains are 53 PAGE 54 0 0 0 2 000 2 01 0 21 1 1 21rr RTv rr r r Tvr rrrrr rr (4-12) Example Problem for Displacements and Strain s from a Spike in Coefficient of Thermal Expansion The material properties used in this example fo r the axisymmetric, thin plate are for Nextel 720/aluminosilicate, which is from the simila r work of Gogu et al. (2007b). The properties assumed to be constant through the plate were elastic modulus and Poissons ratio with E = 130 GPa and = 0.23, respectively. In accordance with th e model derived in the previous subsection, the nominal CTE (nominal = 6105.75C) was set to zero and the step impulse for 0 to r0 () was +10% of nominal, so the difference was 6100.575C. The radius of the spike in CTE was r0 = 2mm in a 5cm radius plate which was e xposed to a temperature increase of T = 100C. The analytical results for this sample problem are shown in Figure 4-3. Figure 4-3. Analytical radial displacement and strain for exam ple axisymmetric plate problem using Nextel 720/aluminosilicate with E = 130 GPa, = 0.23, nominal = 6105.75C, and = 61010%C nominal 54 PAGE 55 Analyzing the shape of the displacem ent and strain curves, the two regions of CTE are obvious. The inner region has a linear displacement and constant strain, which would be the case for a plate of constant CTE. The curves in Figur e 4-3 for the outer region exhibit a hyperbolic shape, which decrease at a rate that is dependent on r0 and rR. The maximum displacement and maximum magnitude of strain for this problem were umax = 0.0706m rr max = -35.42, respectively. A finite element model was constructe d to validate the analytical results, where the displacement results were identical to four significant digits, thus confirming that the assumptions used in th e derivation were valid. Strain Predictions for Various Engineering Materials A variety of materials were anal yzed for a realistic geometry to predict if a spike of 10% in CTE is detectable through measurement. The results Table 4-1 are for a fixed, finite, circular plate, with the same dimensions as the previ ous example. The CTE variation was again a +10% spike over a region r0, of 2mm with T = 100C. Table 4-1. Analytical displacement and strain re sults for a 20cm circular plate exposed to 10% increase in CTE with r0 = 2mm and T = 100C Material E (GPa) nominal( 610C)umax (m) rr max () Nextel 720 / Aluminosilicate (FEM) 130 0.23 5.75 0.071 -35.4 Carbon fiber/Epoxy IM7/977-2 (0) 150 0.34 0.45 0.006 -3.0 Carbon fiber/Epoxy IM7/977-2 (90) 9.0 0.02 23.0 0.235 -117.3 Steel 203 0.29 12.4 0.160 -80.0 Brass 110 0.33 19.4 0.258 -129.0 Aluminum Alloy 7075 T6 72 0.29 23.6 0.304 -152.2 Titanium Ti 6Al 4V110 0.31 9.50 0.124 -62.2 Alumina 370 0.22 8.2 0.100 -50.0 55 PAGE 56 56 The materials considered were ceramic, unidi rectional carbon fiber/epoxy composite, steel, brass, aluminum, titanium, and alumina. The ceramic material (Nextel 720/aluminosilicate) had the same properties used in Fig. 4-3. Since this is an estimate using isotro pic material properties, the composite was analyzed for when the fibers were oriented in the radial direction (0) and also when they were perpendicular (90), using the longitudinal (1) and transverse (2) coefficient of thermal expansion, respectively. Even though a relatively large spike of 10% was used, the re sults in Table 4-1 did not display a significant enough effect on strain to raise concern in de sign. However, this was just a study to analyze the effects of a nonuniformity over a small area on a plate; this does not mean that spatial variation in material prope rties is not important in some cases. PAGE 57 CHAP TER 5 CORRELATION MODEL FOR COMPOS ITE MATERIAL PROPERTIES In the previous chapter, a circular plate m odel was used to estimate the displacement and strain from variation in CTE al one; however, it is likely that cha nges in material properties are the result of common physical phenomena, thus correlated. Even though independent random variables are often used in probabilistic de sign or uncertainty pr opagation, it does not appropriately represent potential correlations in the mate rial properties (Conceicao and Hoffbauer 2007). Furthermore, Qu et al. (2003) ob served that the probability of failure changed in a particular way with E2 and 2, thus indicating a possible significance of correlated properties. When considering a laminate, correlation stems from several factors, including fiber misalignment, fiber packing, and fiber volume fraction. The mechanism assumed to be the most significant was fiber volume fraction. This chapter has two main sections. First, the relationship of composite properties with fiber volume fraction is analyzed. From that stud y, it was determined that fiber volume fraction can cause significant changes in strain. Therefore, a follow-up study on using the correlations from fiber volume fraction for material variab ility. Even though correlations in material variability are expected to exist and are significan t, the correlation data is not often available. So this work focuses on using a physical effect to derive the existing co rrelation in material variability. Namely, fiber volume fraction was used as the basis for this correlation model. Then the material variability was combined with measur ement error from experimental data to get the total uncertainty. Finally, the combined material property correlations were propagated to strain in a pressure vessel exampl e, similar to that analyzed by Qu et al. (2003). 57 PAGE 58 Investigating the Effect of Variation in Fiber Volume Fraction on Thermal Strain The first part of this study was to determine the significance of correlations in material variability generated from fiber volume fraction. Caruso et al.(2006) and Rosen et al. (1987) have studied the trends of graphite/epoxy prope rties to fiber volume fraction using simplified micromechanics models and rule of mixtures. Th e material used in this study was IM7/977-2, which has a nominal fiber volume fraction of Vf0 = 0.6. Linear approximations of the published trends were based on the relative change in pr operties for the realistic range of fiber volume fraction from 0.5 to 0.7. Figure 5-1 shows the published plots of the dependence of graphite/epoxy laminates on fiber volume frac tion (Caruso and Chamis 2006, Rosen and Dow 1987). Figure 5-1. Relationships of carbon fiber/epoxy composite properties and fiber volume fraction (originally published by Caruso a nd Chamis 1986, Rosen and Dow 1987) Linear approximations were made for th e composite properties with respect to Vf. Let f V be a random variable with a normal distributi on, where the range {-1,1} corresponds to two standard deviations or approxima tely 95% confidence interval. Therefore, the normal parameters 58 PAGE 59 for f V are N(0, 0.5). The linear relationships for of the material propert ies to normalized fiber volume fraction are given in Eq. (5-1). *1fV Xf X kVX (5-1) Where, X is a component of the av erage material properties T 12121212,,,,,EEvG at the nominal fiber volume fraction of 0.6, and kX is the corresponding linear coefficients for fiber volume fraction, shown in Table 5-1. Table 5-1. Linear coefficients for fiber volume fraction of graphite/epoxy (Vf0 = 0.6) Property 1Ek 2Ek 12vk 12Gk 1k 2k Graphite/Epoxy 0.017 0.011 -0.06 0.21 1 -0.12 The coefficient of thermal expansion in the longitudinal direction (1) was estimated as a constant since its magnitude is sma ll and it is not very sensitive to Vf over the range considered. Also note that Poissons ratio (v12) and transverse CTE (2) are negatively correlated with Vf. The nominal values were based on previous experimental results of IM7/977-2 (Schulz et al. 2005b). Table 5-2 shows the nominal composite prop erties, in addition to the value and percent change of the material properties for a fiber volume fraction change to 0.5. Notice that the percent change corresponds to the coefficient in the linear model (kX); however in this case the signs are opposite since a negative change in fiber volume change was considered. Table 5-2. Laminate material properties of IM7/977-2 car bon fiber/epoxy at nominal (Vf0: = 0.6) and low (Vf: = 0.5) fiber volume fraction Property Nominal Vf0 = 0.6 Low Vf = 0.5 Percent change E1 (GPa) 150 124.7 -17% E2 (GPa) 9.0 8.0 -11% v12 0.34 0.36 +6% G12 (GPa) 4.6 3.6 -21% (10-6/C) 0.45 0.45 0% (10-6/C) 23 25.8 +12% 59 PAGE 60 Looking at the m aximum variation in properties gives an idea of the correlations and the ranges of the properties. It is also interesting to observe how these changes propagate to stress and strain, via classical lamination theory. From basic composite mechanics, the constitutive equations for a symmetric, orthotropic laminate subjected to thermal loading are shown in Eq. (5-2) (Gibson 1994, Sect. 7.3). 0totalNA (5-2) Where, A is the laminate in-plane stiffness matrix, {0} are the midplane strains, and {Ntotal} is the total force per unit length of the mechanical and thermal components, as defined in Eq. (5-3) totalmechthermNNN (5-3) Since this is a thermal problem, the interactio n of composite properties in the thermal load explains how variation Vf propagates to stress or strain. The thermal load is a function of material properties and fiber orientation, and a conven ient way to analyze the material property interaction is by separating these effects (Grdal et al. 1999, Chap. 3). Equation (5-4) presents the the rmal load as a function of K1 and K2, which are constants based on the material properties. 12 1 12 1 2 1cos2 1 cos2 2 sin2N kk k N therm kk k N kk kKhKt NKhKt Kt T (5-4) The m aterial constants in the thermal load are given in Eqs. (5-5) and (5-6) 112121122 1 2 2 12 11 1 EE K E E (5-5) 60 PAGE 61 112121122 2 2 2 12 11 1 EE K E E (5-6) Equation (5-4) separates the part of the thermal load which depends on ply angles ( K2) from the part that depends only on thickness ( K1). Breaking down the numerator of Eqs. (5-5) and (5-6) 1 is constant and the effect from 12 is small, therefore K1 and K2 are essentially driven by the products E11 and E22. Since 1 is constant, then variability in the first term is dictated by variability in E1. The second term however, is the product of two properties that are sensitive to Vf, but are negatively correla ted. Thus, changes in E2 and 2 can somewhat cancel each other out, depending on their magnitudes. An example was considered using CLT to anal yze the effect of fi ber volume fraction on a laminate subjected to thermal loading by anal yzing the thermal strains in the material coordinates. Consider a composite laminate with nominal material propertie s as those listed in Table 5-2 subjected to T = 100C. To prevent bending and permit analysis for various lay-up orientations, a symmetric, four layer laminate ([ ]s) was used with a lamina thickness of t = 125 m and laminate thickness of h. A free-free laminate was considered (laminate free to expand) and the uncertainty was pr opagated to material direction mechanical strains. In other words, Eq. (5-2) was used with only thermal loading a nd the m echanical strains were then calculated from Eq. (5-7) 111 222 12120mechtotal mechtotal mechtotalT (5-7) 61 PAGE 62 W here total strain is calculated from Eq. (5-2) and converted into material coordinates and the second term is the thermal strain component. The mechanical strain s are more releva nt since they are more associated with failure. The mechanical strains were plotted against ply angle at high and low Vf of 0.5 and 0.7, respectively, to observe the fiber orientations that crea te a large range of strain. Figure 5-2. Range of mate rial coordinate strains (mechanical) from CLT for a symmetric laminate, thermally loaded (T =100C) with carbon fiber/epoxy composite material properties adjusted for low Vf = 0.5 and high Vf = 0.7 A) longitudinal strain, B) transverse strain and C) shear strain 62 PAGE 63 The shape o f the three strain curves is elem entary, yet informative with respect to the mechanics of the thermal loading and fiber orienta tion. At 0 ply-angle, there is no mechanical strain since the laminae are free to expand in the material coordinates since the laminae all have the same orientation. Conversely, when = 45, the + and layers are perpendicular to each other, so the fiber and matrix have the maximu m mechanical strain on eac h other. With respect to fiber volume fraction, the plots display a maximum change in all of the strains from low to high Vf of almost 30%. The next objective was to compare the diffe rence in standard deviation of output mechanical strain when considering the material properties as independe nt random variables and when the material properties were governed by their relationship to random variable, Vf from Eq. (5-1) A normal distribution was assumed for fibe r volume fraction with the range of .1 considered b efore was assumed to be at the 95% level (or standard deviations), therefore the normal distribution for Vf was N(0.6,0.05). The standard devia tions of each material property are presented in Table 5-3. Table 5-3. Corresponding standard deviation and coefficient of variation of the composite material for a normal random fiber volume fraction Vf: N(0.6, 0.05) Material property Nominal value Standard deviation from Vf range Coefficient of variation E1 (GPa) 150 12.71 8.5% E2 (GPa) 9 0.49 5.5% v12 0.34 0.01 3.0% G12 (GPa) 4.6 0.48 10.5% 1 (10-6/C) 0.45 0 0.0% 2 (10-6/C) 23 1.38 6.0% 63 PAGE 64 The Vf dependent material properties are perf ectly correlated either positively or negatively, because a linear mode l was used for the effect of Vf. The trends are perfect in this case; however they will not be when other uncertainties are present from measurement error. Using the standard deviations for each materi al property in Table 7-2, random variables were sampled by Monte Carlo simulation with 105 samples in CLT thermally loaded at 100C. Figure 5-3 displays mean and coefficient of varia tion of the material mechanical strains for fiber orientations between 0 to 45 (since it was shown in Figure 5-2 that the strains were symmetric about 45). The coefficient of variation of longitudinal (1) and transverse (2) mechanical strain display similar trends in coefficient of variati on, but with slightly different magnitudes. The trend for the shear strain is a bit more di fficult to distinguish be tween independent and Vf dependent properties. 64 PAGE 65 Figure 5-3. Mean and coefficient of variation (CV) of the material coordinate mechanical strains from 105 samples propagated in CLT for independent and Vf dependent material properties of a symmetric IM7/977-2 composite laminate, thermally loaded (T =100C) To better quantify the significance of difference in coefficient of variation, consider the longitudinal strain since 1 mech is generally design against failure of the structure. For example a laminate with = 25 has a decrease in CV(1 mech) of 17.5% to 9.7% for independent and Vf 65 PAGE 66 dependent random variables, respectively. Which tran slates to a decrease in probability of failure from 0.10 to 0.005 for maximum strain failure (assuming 1max = 100 ). From the previous study, it was deduced that material variability from fiber volume fraction can have significant e ffects on strain. Therefore, furt her research was conducted on how to use this correlated material variability in relia bility analysis and how can it be combined with measurement error, which is c overed in the next section. Combining Correlated Material Va riability and Measurement Error In this section, an uncertainty model is proposed that in cludes the a definition of the overall uncertainty in composite properties, the derivation of the material variability correlation model based on fiber volume fraction, and interp reting statistical data on measurement error. Then the proposed model for material variability was combined with existing data on measurement error for glass/epoxy and graphi te/epoxy composites. The effects of including material variability and using correlated proper ties instead of independe nt are shown through an example pressure vessel problem. Uncertainty Model with Correlation The uncertainty in composite material elastic constants has an aleatory (or random) part, i.e. variability in material properties that is due to manufacturing, and an epistemic part (lack of knowledge) primarily due to errors from measuring the properties experimentally. The variability in elastic properties is mostly caused by variabilit y in fiber orientation, fiber packing, and fiber volume fraction (Conceicao Antonio Hoffbauer 2007, Caruso and Chamis 1986, Gusev et al. 2000). Even though we expect material variability to exist from these sour ces, there is often no data on the correlations. Therefore, we aim to deri ve a correlation model fo r material variability that can be used, even if expe rimental data is unavailable. We assume that the fiber volume fraction is the dominant effect and use relationshi ps for the elastic constants in terms of volume 66 PAGE 67 fraction from si mplified micromechanics mode ls (Caruso and Chamis 1986). However, the resulting covariance matrix is presented in ge neral terms which can a ccommodate other sources of uncertainty and correlation, such as those from fiber orientation and waviness. The other uncertainty component is measurement error, whic h depends on the accuracy of the experimental technique. The laminate properties used in th is study include longitudinal modulus E1, transverse modulus E2, Poissons ratio 12, and shear modulus G12. Let X be the vector of elastic properties, as shown in Eq. (5-8) T 121212,,, EEvG X (5-8) Consider an experiment to measure one com ponent of the vector of material properties X of a composite laminate. The expe riments yield a measured average exp X however due to epistemic, measurement error the true averag e exp Xe X differs from the experimental results. That is, the true mean of the material propert y is never specifically known, so the measured average (or published value) exp X is most commonly used in design. Equation (5-9) gives the express ion for uncertainty from measurement for error the material properties in Eq. (5-8) and their corresponding errors, exp Xe 1expexp X X eX (5-9) Where, is the corresponding measurement errors for each property X. Note that if the material properties are measured from a limite d number of experiments, or even a single specimen, then there is some additional error that is not represented in the measured data. In that case, then there is an error from the test and also an error from specimen variability exp Xetest Xe s pecimen X. Equation (5-10) shows the complete measurement error term. 67 PAGE 68 (5-10) exptestspecimen XXXeeFurtherm ore, when the composite material is used in a structure there is uncertainty in X due to material variability f V X namely from fiber volume fraction Vf, in this case. Thus, the material property for a particular specimen is X. Equation (5-11) shows the material variability com ponent of uncertainty in composite properties. 1fV X X X (5-11) Where, f V X is the corresponding material variability for each property X and X is the true average from Eq. (5-9) Figure 5-4 illustrates the role of m easurement error and material variability in material property X. Figure 5-4. Example uncertainty di stributions for measurement error and material variab ility exp XeVf X ,of property X The objective of this uncertainty model is to have a basis for generating random samples of the true material property X, from a combination of measuremen t error and material variability. Therefore, substituting Eq. (5-9) into Eq. (5-11) the true value of property X is based on the uncertainty model represented in Eq. (5-12) 111fV expexptotalexp XXX X eXeX (5-12) 68 PAGE 69 W here, the total uncertainty in the material properties X, is approximately equal to the sum of the measurement error and material variability, as written in Eq. totale(5-13) f V totalexp XXee X (5-13) Furthermore, if correlations are considered for the measurement error and material variability of all of the composite properties in Eq. (5-8) then the variance-covariance matrices of X can be calculated and will be represented as f V and exp respectively. Using the approximation in Eq. (5-13) then the total variance-covariance is approximately the sum of the m aterial variability and meas urement error, as in Eq. (5-14) ftotalVexp (5-14) The correlation ri, j. between properties is calculated from the covariance as in Eq. (5-15) ,cov(,) varvarij ij ijXX r X X (5-15) The following two subsections sections will ex plore the correlated nature of material variability and measurement error, and the cons truction of their respec tive variance-covariance matrices. The discussion on uncertainty in compos ite properties is for two material systems, which are considered in the example in the following section: S-glas s/epoxy and IM7/977-2 carbon fiber/epoxy. Correlated material variability from fiber volume fraction Since correlated data is not often available for material variability, then a correlation model was derived from the dependence of the la mina properties on fiber volume fraction Vf, based on simplified micromechanics, shown in Eq. (5-16) (Caruso and Chamis 1986). 69 PAGE 70 11 2 21 211 1111fffm fffm m mm ff m f fEVEVEvVvVv E EG EG VV EG G (5-16) Where, the f and m subscripts refer to the fiber and matrix component properties, respectively. The component properties for S-gl ass/epoxy are given in Table 5-4 (Gibson 1994, Sect. 3.3). Since the nominal fiber volume fraction was not provide d for the glass/epoxy laminate, it was deduced to be approximately 0.7 given the experimental values from Gogu et al. (2009). Using the functions of Vf in Eq. (5-16) linear approximations were used near the nom inal fiber volume fraction (Vf0 = 0.7). Figure 5-5 shows the re lationships of the glass/epoxy lamina properties with respect to fiber volume fraction, as well as the linear approximation. A similar approach was used for the IM7/977-2 carbon fiber/epoxy composite, but only linear approximations were used at the nominal fiber volume fraction of Vf0 = 0.6 based on typical graphite/epoxy-Vf relationships published by Caruso and Chamis (1986), which was also mentioned in the previous section (Smarslok et al. 2007). Table 5-4. Component propertie s of S-glass fiber and epoxy matrix (Gibson 1994, Sect. 3.3) S-glass fiber Component value Epoxy matrix Component value Ef (GPa) 86 Em (GPa) 4.5 vf 0.22 vm 0.4 Gf (GPa) 35 Gm (GPa) 1.6 70 PAGE 71 Figure 5-5. Relationships of glass/epoxy composite propertie s with fiber volume fraction (Vf) The linear approximation was used for the trends with respect to fiber volume fraction to make this a more general case for any S-glas s/epoxy lamina properties, by normalizing with respect to the nominal (or measured) values. Th erefore, this correlation model for material variability from fiber volume fraction can be extende d to different experime ntal results with only slight modifications and without requiring the component proper ties. Alternately, Monte Carlo simulations could have been directly performed on the si mplified micromechanics (Eq. (5-16) ) with Vf as a normal random variable, then the correlation of the composite properties calculated empirically. The correlations from the linear ap proximations was checked with direct Monte Carlo method since E2 and G12 would not be perfectly Gaussian after propagating Vf, due to the nonlinearity in the rela tionship. Considering the glass/e poxy composite, the normal parameters for the random variable Vf were N(0.7, 0.025). For the linea r approximation case, the material variability is perfectly correlated, which means ri, j (from Eq. (5-15) )is equal to 1 or -1, depending 71 PAGE 72 on the slop e of the line in Fig. 5-4. Using 105 Monte Carlo samples of Vf, the lowest magnitude observed in the empirical corr elation matrix was 0.996, thus confirming that the linear approximations is a reasonable approach. A reasonable variation in fiber vol ume fraction was assumed to be + 0.05 about the nominal value of Vf0 = 0.7 for glass/epoxy, which is emphasized w ith bold lines on the plots in Fig. 5-5. The fiber volume fraction model is the same as in Eq. (5-1) however the range of fiber volume f raction is half 1 0.05 0 ffVVV f, so the coefficients are half of Table 5-1. The linear coefficients for fiber volume fraction kX, are shown in Table 5-5. Table 5-5. Linear coefficients fo r fiber volume fraction of glass/epoxy (Vf0 = 0.7) and graphite/epoxy (Vf0 = 0.6) Property 1Ek 2Ek 12vk 12Gk Glass/Epoxy 0.066 0.14 -0.034 0.14 Graphite/Epoxy 0.085 0.055 -0.03 0.105 Since the variability in properties is due to f V then the material variability from Eq. (5-11) can be written as *fV Xf X kV (5-17) With the linear relationship between the prope rties and the volume fraction, then the covariance matrix of material variability Vf, is perfectly correlat ed and given in Eq. (5-18) in general covariance term s, as well as the pr oduct of normal standard deviations in Eq. (5-19) cov,ffVV VfXX (5-18) ,, f fifjVijVXVX (5-19) Where, i,j = 1 to 4 to account for the four compos ite properties and make the 4 x 4 covariance matrix, which is perfectly correlated from the linear estimates with respect to fiber volume 72 PAGE 73 fraction. Th e covariance and correlation data from fiber volume fracti on are discussed in the results for glass/epoxy and graphite/epoxy. Next, th e measurement error is discussed, which can be used in combination with the pro posed material variability covariance. Correlated measurement error from experimental data The other source of uncertainty in Eq. (5-14) is measurement error exp. The information available on measurement error depends on the natu re of the experiment. Correlated data may be directly available if multiple properties were measured from a single test, such as in identification from a single openhole tensile test or vibration testing (Molimard et al. 2005, Pedersen and Frederiksen 1992). Ot herwise, independent estimates of the error or statistical analysis of the test procedure can yield covariance (e.g. Bayesian analysis). Ten natural frequencies were obtained from a vibration testing experiment on a laminated glass/epoxy plate (Pedersen and Fred eriksen 1992). From this data, Bayesian statistics were used to identify the joint probability distributions and variance-covariance data for the elastic properties, which can be used in the proposed uncertainty model. The published correlation and covariance data is shown in Table 5-6 (Gogu et al. 2009). To eliminate the redundancy of the symmetric correlation and covariance matrices, a combined format was used for correlation data. The matrix in Table 5-6 (and subsequent correlat ion data) will be presented with variance along the diagonal, covariance in the lower triangular matrix, and co rrelation in the upper triangular matrix. Table 5-6. Correlation (above diagona l) and covariance measurement data exp, from vibration testing on glass/epoxy laminate E1 E2 12 G12 E1 3.45e18 -0.141 -0.378 -0.626 E2 -3.05e17 1.36e18 -0.593 -0.358 12 -2.31e7 -2.28e7 1.10e-3 0.768 G12 -6.85e17 -2.46e17 1.49e7 3.47e17 73 PAGE 74 Note that sin ce the data show n in Table 5-6 is from a singl e specimen, then the error is actually so we are approximating that To account for the specimen variability, then m ore tests would need to be performed, or we can approximate test Xeexptest XXee 2 f V specimen XX since we are assuming that fiber volume fraction is the primary source of speci men variability. This additional consideration of specimen variab ility will be address in future work. Based on the vibration testing re sults, the data in Table 5-6 shows us that all of the correlations between elastic properties are negative except G12 v12. Furthermore, the correlation between E1 and E2 is relatively small, however G12 E1 and G12 v12 have stronger correlations. The variance covariance data shown in Table 5-6 can be combined with the glass/epoxy material variability f V, as in Eq. (5-14) The next section will disc uss the correlation data further, as w ell as use this data in an example problem. When experimental correlation data is not directly availabl e from testing, measurement error can still be estimated from individual ma terial property experiments. The magnitude of measurement errors are largely dependent on th e nature of the experiment and laboratory Smarslok et al. 2006b, Wilson 1990) In a related study on IM7/977-2, a coefficient of variation (CV) of 1% was observed in E2 measurements and CV = 1.7% in G12 measurements (Smarslok et al. 2006b, Schulz et al. 2005b). Th erefore, we assume that all the elastic properties can be measured within 3% CV, but E1 is slightly more accurate at 1% CV. Since no correlation data is available, then we assume that the measuremen t errors for graphite/epoxy are uncorrelated. A normal distribution was used for the experimental error component with parameters where is the measurement uncertainty coefficient of variation for each property. S ince measurement errors ( ) were assumed to be uncorrelated in this case, then the 0,exp XNCVexp XCVexp Xe 74 PAGE 75 covariance exp, is a diagonal matrix with the squares of the standard deviations (variance) on the diagonal. Equations (5-20) and (5-21) give the experimental covariance, where exp,Xi are the normal standard deviations and ij is Kronecker delta. cov,expexp expXX (5-20) 2 ,iexpijexpXij (5-21) The next section uses the uncertainty model that includes measurement error and material variability in an example pressure vessel problem for both glass/epoxy and graphite/epoxy laminates. Illustrative Results for Graphi te/Epoxy and Glass/Epoxy Laminates This section will present the mean material properties for the glass/epoxy and graphite/epoxy laminates, as well as the combin ed uncertainty from fiber volume fraction and measurement error. First, consider the S-glass/epoxy laminate with mean property values exp X shown in Table 5-7. To get a better sense of the magnitude of uncertainties, the coefficients of variation (CV) and standard deviatio ns from material variability (Eq. (5-19) ) and measurement error (Eq. (5-21) ) are summarized in Table 5-7. 75 PAGE 76 Table 5-7. Mean, stand ard deviation and coefficient of variation for material variability, measurement uncertainty, and combin ed uncertainty for S-glass/epoxy Material property exp X Vf CVVf exp CVexp 22fVex p CVtotal E1 (GPa) 61 2.01 3.3% 1.86 3.05% 2.73 4.48% E2 (GPa) 21 1.47 7.0% 1.15 5.46% 1.86 8.86% 12 0.27 0.005 1.7% 0.033 12.2% 0.033 12.3% G12 (GPa) 9.9 0.693 7.0% 0.590 5.96% 0.91 9.23% The total uncertainty in the material properties CVtotal, is the same order of magnitude, but the uncertainty in E1 is about half that of the other prope rties. Furthermore, if the material variability from fiber volume fraction was not included, then the relatively large uncertainty in E2 and G12 (7.0% for both) would be missed. As suming the uncertainties are normally distributed, Table 5-8 shows the combined cova riance and correlation matrix for glass/epoxy total, using material variability in Eq. (5-19) and measurement error in 0 Table 5-8. Com bined cova riance correlation matrix (total) for glass/epoxy E1 E2 12 G12 E1 7.47e18 0.523 -0.345 0.289 E2 2.66e18 3.46e18 -0.461 0.461 12 -3.21e7 -2.84e7 1.10e-3 0.394 G12 7.22e17 7.82e17 1.19e6 8.34e17 Note that after combing the material va riability and measurement error that the E1 E2 correlation and E2 G12 correlation got much stronger from the fiber volume fraction relationship. Table 5-9 summarizes all of the graphite/epoxy composite prope rties with their respective component standard deviation and co efficient of variation from Eqs. (5-19) and (5-21) 76 PAGE 77 Table 5-9. Mean, standard deviation, and coeffi cient of variation of m aterial variability and measurement error for IM7/977-2 graphite/epoxy Material property exp X Vf CVVf exp CVexp 22fVex p CVtotal E1 (GPa) 150 6.4 4.25% 3.8 1% 6.54 4.36% E2 (GPa) 9.0 0.25 2.75% 0.23 3% 0.365 4.06% 12 0.34 0.005 1.5% 0.009 3% 0.011 3.36% G12 (GPa) 4.6 0.24 5.25% 0.12 3% 0.278 6.04% In Table 5-9, the uncertainties are of the same order of magnitude, but variability in Vf has different effects on uncertainty for each property, most notably with E1. If we were to only consider experiments, then we would exp ect the uncertainty to be the least in E1. However, due to its strong dependence on Vf, then the total uncertainty in E1 actually becomes larger than E2 with 4.4% and 4.1%, respectively. The signi ficance of this increase of uncertainty in E1 will be more obvious in the pressure vessel problem in the next section. Next, the two covariance matrices were added as in Eq. (5-14) and are presented in 0 Table 5-10. Combined cova riance correlation matrix (total) for graphite/epoxy properties E1 E2 12 G12 E1 4.27e19 0.659 -0.436 0.844 E2 1.57e18 1.34e17 -0.303 0.586 12 -3.25e7 -1.27e6 1.30e-4 -0.390 G12 1.53e18 5.95e16 -1.24e6 7.72e16 From Table 5-10, strong co rrelations are observed for E2 and G12 with E1. This is a result of the strong correlations from fiber volume fr action combined with independent measurement uncertainty. Again, these correlations from material variability would not ha ve been captured if the properties were treated as independent random variables, or if onl y the measurement error was considered. 77 PAGE 78 The com bined covariance matrices for glas s/epoxy and graphite/e poxy properties can now be used with their respective mean values to generate corre lated random values for probabilistic analysis. The subsequent section compares th e propagation of independent and correlated random elastic properties for both laminates. Effect on Failure Probability for a Pressure Vessel Consider an example problem of a cylindric al pressure vessel, based on the hydrogen tank of the X-33 Reusable Launch Vehicle (Qu et al. 2003). A symmetric, four layer laminate ([2525]s) was used with a lamina thickness of h = 125 m and diameter of 1m, all which were deterministic. The same laminate desi gn was considered for both the glass/epoxy and graphite/epoxy cases. However, the internal pressure load of the glass/epoxy and graphite/epoxy was 100 kPa and 50 kPa, respectively, to obtain co mparable strains. The strain was calculated with Classical Lamination Theory as shown in Eq. (5-22) (Gibson 1994, Sect. 7.3). 0 1 0 00 x Hoop y xyN N A x i a l A (5-22) The midplane strains in Eq. (5-22) were transformed to get the m aterial coordinate strains {}12 for analysis. Monte Carlo simu lations were performed to calculate the mean and coefficient of variation (CV) of strain from 105 random samples of the material properties X for independent and correlated variables. The corre lated random variables were sampled from their respective combined covariance matrices. The proba bility of failure was also calculated for the transverse strain for each case base d on maximum strain criterion, where 2max was deterministic and equal to 1700 and for the glass/epoxy and graphite/epoxy, respectively. Transverse strain was proven to be the main failure mode of the X-33 vehicle since it lead to fuel leakage, hence why it is used in this example. 78 PAGE 79 The stra in results from Eq. (5-22) for independent and correlated random material properties of glass/epoxy and graph ite /epoxy are shown in Table 5-11. Table 5-11. Propagation of inde pendent and correlated random co mposite properties to strain and probability of failure for maximum transverse strain Independent Correlated mean() CV() (%) pf mean() CV() (%) pf 1 1640 4.30 1640 4.32 2 1399 8.50 0.010 1402 6.96 0.005 Glass/Epoxy 12 -286.5 41.1 -283.8 27.0 1 389.3 4.26 389.2 4.04 2 1245 3.20 0.0069 1246 4.07 0.026 Graphite/Epoxy 12 1020 4.81 1021 4.45 We see that the independent and correlated ma terial properties result in nearly the same mean strain values. Therefore, the difference be tween the two cases is in the coefficient of variation of the strain CV(), and thus the probability of failure pf. Furthermore, for this problem the longitudinal strain 1, and shear strain 12, show only slight differences in CV() between the independent and correlated cases. Note that the large coefficient of variation in 12 for glass/epoxy is an artifact of the small magnitude of the shear stra in in this problem. Transverse strain 2, is most affected from the different uncerta inty propagation, therefore we will focus most of the analysis on those values. The uncertainty in 2 decreased from 8.5 to 6.96 for the glass/epoxy with independent a nd correlated properties, respec tively. However, the opposite occurred for graphite/epoxy, where CV() increased from 3.20 to 4.07. In terms of failure probability, the correlated properties reduced pf by half for glass/epoxy (0.010 vs. 0.005), but pf was more than three times larg er if correlated properties were used for graphite/epoxy (0.0069 vs. 0.026). Therefore, not considering the correlat ions would result in an inefficient design with 79 PAGE 80 80 glass/epoxy, but an unsafe design for graphite/ep oxy. The opposite trends are mainly a result of the relative magnitude of nomi nal composite properties. Since E1 for graphite/epoxy is large (150 GPa) by comparison to the other properties and has strong correla tion from fiber volume fraction, then it dominates the variability in the stiffness matrix [ A ] and increases the uncertainty in 2. Whereas, the nominal properties of glass/ep oxy (Table 5-7) are comparable, then the interactions in the stiffness matrix actually reduces the uncertainty in 2 for the correlated case. Thus, not considering correlated properties can result in an undesirable design, however being unsafe or inefficient can depend on the material sy stem. Also recall that th e correlations for the graphite/epoxy is solely due to material variabilit y, therefore not including the relationship of the properties with fiber volume fraction can sign ificantly affect the probability of failure. PAGE 81 CHAP TER 6 SEPARABLE MONTE CARLO SIMULATION METHOD When considering the small failure probabilities of safe structures (~10-7), there is often an unmanageable computational cost associated w ith performing simulations to get an accurate estimate of the probability of failure. This probl em was encountered by Qu et al. (2003) when analyzing relationship of probability of failu re and thickness under uncertainty for the X-33 hydrogen tank. Therefore, this chapter develops an existing variance redu ction technique, called the conditional expectation method, into a more ge neral form that can be applied even when the actual distribution of the control variable is not known, referred to as the separable Monte Carlo method. Using separable Monte Carlo as an al ternative to traditional, crude Monte Carlo simulations can significantly improve the accuracy of the probability of failure estimate. This chapter presents three simulation-base d methods for calculating the probability of failure: crude Monte Carlo, conditional expectatio n, and separable Monte Carlo. In addition, further accuracy improvements with separable M onte Carlo are discussed, which include sample size allocation and reformulation of the limit stat e. Finally, a variance estimate for separable Monte Carlo was derived using conditional calculus. Methods and Expectations of Probability of Failure Estimators The probability of failure of a system is de fined by the failure condition, or limit state function. Consider a limit state G that is a function of random variables capacity C and response R Furthermore, assume that uncertainty in the response is due to one set of random variables X1, and the uncertainty in the capacity is due to a second set of random variables X2, and that X1 and X2 are statistically independent. The general lim it state function for pr obability of failure calculations is shown in Eq. (6-1) where R and C are vectors of random va riables and failure in the system occurs when G < 0 and it is safe when G 0. 81 PAGE 82 1, GGCR XX2 (6-1) The probability of failure for the limit state function G is shown in the integral form in Eq. (6-2) 00f gpPGfgdgG (6-2) Where fG(g) is the probability density function (PDF) of the limit state function. Since the exact probability of failure pf, is not known, then this chapter will focus on three different simulationbased ways of estimating the failure probability based on Eq. (6-2) including crude Monte Carlo (CMC), condition al expectation (CE), a nd separable Monte Carlo (SMC) methods. Crude Monte Carlo Method (CMC) The crude Monte Carlo is a basi c, yet widely used approach that involves assigning 0 or 1 for each run in the simulation, correspondi ng to pass or fail, respectively. Let Ci and Ri be independent and identically-distributed (i.i.d.) replicates of the capacity C and response R respectively, where 1,, i cmcN. Each of the N pairs of response and capacity are tested for failure and result in a 0 or 1. Based on Eq. (6-2) the crude Monte Carlo sim ulation estimate of probability of failure ( p ) is shown in Eq. (6-3) 11 (,)0N cmcii ipIGCR N (6-3) Where, I is the indicator function, which equals 1 if the condition is true and 0 if the condition is false. For the crude Monte Carlo method, follows a binom ial distribution of length N and probability pf, as a sum of independent Bernoulli random variables Thus, from definition the expectation and variance for crude Monte Carlo are shown in Eqs. cmcNp (6-4) and (6-5) respectively. Ecmcf p p (6-4) 82 PAGE 83 (1) var f f cmc p p p N (6-5) A convenient way to view the accuracy in cmc p is by considering the square root of the variance divided by the probability of failure, which is the coefficient of variation, as shown in Eq. (6-6) 1 var 1 CVf cmc cmc ffp p p f p pNpN (6-6) From Eq. (6-6) for a given probability of failure the accuracy is res tricted by the computational budget N For example, considering the approximation for small pf CV1cmc f p pN then for a probability of failure of one in a million, 100 million simulations are needed for 10% accuracy (using th e one standard deviation confidence level). In practice, the variance of crude MC in Eq. (6-5) is based on the estimate of the probability in Eq. (6-3) Equation (6-5) shows that for small probabilities the relative erro r in the estimate of the variance is the same as the relative error in the estimate of the probability. Equation (6-3) convenien tly provides an estimate of the probability of failure and accuracy from a single simulation of N samples (from Eq. (6-6) ). With that in mind, simulation estimates for the com ponents of the variance for conditional expect ation and separable MC are also included in the following two sections, respectively. In this dissertation, the focus is on the case wh ere the limit state itself is separable, because the failure criterion is when the response (e.g ., stress) exceeds the capacity (e.g., strength). Therefore, this limit state is expressed as the difference between capacity and response, shown in Eq. (6-7) GCR (6-7) 83 PAGE 84 Rewriting th e general form of CMC for the limit state in Eq. (6-3) yields Eq. (6-8) 11N cmcii iN p ICR (6-8) Conditional Expectation Method (CE) W hen the conditional probability of failure fo r a given value of the response is known, then obviously fCR p rfrdr (6-9) Where, C is the conditional expectation of the control random variable C for random samples of R Using N independent and identically -distributed (i.i.d.) samp les of the response, the probability of failure is estimated as the c onditional expectation method (CE), shown in Eq. (6-10) (Ayyub and Chao-Yi 1992). 11 N ceCi i p R N (6-10) Equation (6-10) has a similar form as that of CMC (Eq. (6-3) ), but recall that the N in crude Monte Carlo refers to the sam ple size of response and capacity, whereas the conditional expectation method samples only one random variable, R The steps for deriving the expected value and variance of Eq. (6-10) are shown next. The expected v alue of the CE estimate in Eq. (6-10) is shown in Eq. (6-11) 11 EEN ceCi i p R N (6-11) The expectation can be brought inside the summation, as shown in Eq. (6-12) 11 EEN ceCi i p R N (6-12) As described in Eq. (6-9), the expectation can be simplified as, 84 PAGE 85 11 EN cef i p p N (6-13) Expanding the summation and simplifying, 1 Ece ff p Npp N (6-14) In summary, the expected value of the condi tional expectation method is presented in Eq. (6-15) E=Ece C f p Rp (6-15) Similarly, we follow the same steps to obtai n the variance expression. From the conditional expectation estimate in Eq. (6-10) the variance is written as shown in Eq. (6-16) 11 varvarN ceCi i p R N (6-16) As done for the expectation in Eq. (6-12) the variance is brought inside the summation, as shown in Eq. (6-17) 2 11 varvarN ceCi i p R N (6-17) Expanding the summation and simplifying, 1 varvarce C p R N (6-18) Writing the variance in general form, 2 21 varEEce C CpR NR (6-19) Rewriting the second expect ation according to Eq. (6-15) we get the variance expression for the conditional expectation m ethod in Eq. (6-20) 2 21 varEce C f p Rp N (6-20) 85 PAGE 86 For the sim ple case of Eq. (6-7) C is equal to the cumulativ e distribution function (CDF) FC of the capacity (Melchers 1999, Sect. 3.3). Therefore, the moments from Eqs. (6-15) and (6-20) now include the CDF of the capacity FC, as shown in Eqs. (6-21) and (6-22) 11 ()N ceCi i p FR N (6-21) 221 varE()ce C f p FRp N (6-22) The remaining expectation in Eq. (6-22) can also be estimated during a simulation by using Eq. (6-23) 2 2 11 EN C iFRFR N ( )C i (6-23) Note that Eq. (6-23) and the other simulation estimates are simply used as estimates, and the squared term s actually have a bias, but it is negligible for large sample sizes. An alternate form of Eq. (6-21) is where the analytical CDF of the response FR, is known, which would yield, 11 1()N ceRi i p FC N (6-24) Equation (6-24) has the same expected value (or pf) as Eq. (6-21) however the accuracy between the two form s is dependent on the relative uncertainty in the response and capacity. Using probabilistic definitions, the variance reduction of separable Monte Carlo with respect to CMC is apparent when comparing Eqs. (6-5) and (6-22) Consider the probability of the squared term in the remaining expected value in Eq. (6-22) 22 1212()P()P,Pmax,CFrCrCrCrCCr (6-25) Where, the random variables C1 and C2 are independent copies of C Thus, the squared expectation from Eq. (6-22) becomes, 86 PAGE 87 2 121 varPmax,ce f p CCRp N (6-26) Comparing this result back to the CMC variance in Eq. (6-5) one can see that the probability of the m aximum of two random variables C1 and C2 being less than R is naturally less than or equal to just one random variable C1 being less than R 12 1Pmax,P f CCRCRp (6-27) In Eq. (6-27) the former probability corresponds appears in SMC variance, while th e latter probability is a term in CMC; which brings the conclusion of Eq. (6-28) varvarce cmcpp (6-28) The magnitude of the accuracy improvement de pends on the characteristics and types of distributions used for the response and capacity. Ch apter 7 contains several example studies that reinforce the previously mentioned conclusion. Separable Monte Carlo Method (SMC) When the capacity and response are stochas tically independent random variables, then C (X1) and R (X2) can be sampled independently. So, instead of comparing N sets of response and capacity samples as in Eq. (6-3) we can evaluate all possible combinations of C and R for f ailure. Equation (6-29) gives the estimate of the probability with M rand om capacities Cj, where 1,, j M and N random responses Ri, where 1,,iN 111 (,)0NM smcji ijpI G C R MN (6-29) For the case of the limit stat e in the form of response exceeding capacity from Eq. (6-7) this yields Eq. (6-30) 111 NM s mc ji ij p ICR MN (6-30) 87 PAGE 88 Figure 6-1 illustrates the eval uation m ethod of the limit state for a simulation of crude and separable MC, respectively. A B Figure 6-1. Illustration of Mont e Carlo method sample comparisons A) Crude Monte Carlo B) Separable Monte Carlo In Fig. 6-1a, the direct one-to-one comparisons of crude MC are shown for N samples. Whereas, Fig. 6-1b shows that sepa rable MC looks at all of the possible combinations of random samples, which makes it inherently more accura te than CMC. The advantages of SMC over crude Monte Carlo are obvious. First, with separable MC we do not need to have the same sample size for capacity and res ponse. Typically sampling capacity is cheaper computationally, so more samples can be obtained. This advantage may be enhanced further if we can reformulate the limit state so as to transfer some random va riables from the expensiv e response component to the cheaper capacity component. Consider for example, a case where the response may be written as R = R0. Where, is a random variable and a separate source of uncertainty (e.g. modeling error (Acar et al. 2007, Kumar et al. 2008) or random force amplitude) that is independent of the other random variables. Thus, the limit state can be reformulated to reduce the uncertainty in the comput ationally intensive response R as shown in Eq. (6-31) 0(,) C GCRR (6-31) 88 PAGE 89 The bending in a composite plate example in Chapter 7 uses the reformulation concept presented in Eq. (6-31) The second advantage o f SMC is that we make maximum use of each response sample by comparing it to all available cap acity samples. It is intuitivel y clear that the accuracy of Eq. (6-30) will be better than the accuracy obtained with CMC using N sam ples (similar cost), but not as good as CMC with MN samples (much higher cost). The variance estimates in the next section confirm this intuitive observation. When the random capacity samples from Eq. (6-30) are sorted to construct an experimental CDF (eCDF), then the c onnection between sepa rable Monte Carlo and conditional expectation method becomes apparent. Figure 6-2 show s the evaluation of a response sample Ri, in the experimental CDF of the capacity obtained from M random capacity samples. Figure 6-2. Separable MC met hod using random data points of R and the empirical CDF of the capacity The experimental CDF C F from separable Monte Carlo simulations is comparable to the conditional probability C from Eq. (6-20) All of the random capacity values below the 89 PAGE 90 random response sample Ri, represent failure and are summed and divided by the total amount of capacity data M This value gives an empirical estim ate of the value capacitys CDF at Ri. To account for randomness from both the response and capacity, conditional calculus must be used to determine the e xpectation and variance of s mc p The general expected value expression for the separable Monte Carlo estimate for the simple case of Eq. (6-7) is shown in Eqs. (6-32) Note that subscripts are writt en with expecta tions in Eq. (6-32) to clarify the corresponding random variable, such as if the expectation is taken with respect to the capacity, then it has the subscript C EEEE| s mc C RCC f p FRFRRp (6-32) The variance will be discussed in greater detail, as done in Eqs. (6-16) through (6-20) Since we are dealing with multiple random variable s and there is correlation in the random sets of response and capacity used to calculate s mc p then Eq. (6-33) has both variance and covariance com ponents. 2 1111 varvar2cov,NN N smc Ci CiCj ii j ipF RF R F N R (6-33) As done previously, expanding the summations gives Eq. (6-34) The random variables R1 and R2 are independent copies of R introduced to account for the covariance in random sets of R for a unique empirical CDF of the capacity. 12 221 1 varvarcov, 2smc C C CNN pNFRFRFR N (6-34) Finally, after simplifying, the general form for the variance of SMC is shown in Eq. (6-35) 11 1 varvarcov,smc C C CN2 p FR FRFR NN (6-35) 90 PAGE 91 The task of sim plifying Eq. (6-35) requires conditional calculus and substantial explanation, therefore the remainde r of this section is devoted to deriving a useful form of the variance estimator for separable MC. For the variance of the separable Monte Carlo estimate of pf, we must account for the randomness from both the response and capacity. Since deriving the variance for SMC is not trivial, consider a simpler case with a fixed point c in the empirical CDF of the capacity. Therefore, the expectation at c is (6-36) ECCCFcFc To transition from the fixed point c to the random variable R consider the intermediate case with two fixed points c1 and c2. The covariance of those two poi nts in the empirical CDF of the capacity is shown in Eq. (6-37) 12 1122 cov,EEECCCCCCCCCCFcFcFcFcFcFc (6-37) Simplifying the expectations in Eq. (6-37) we obtain Eq. (6-38) 12 12121 cov,min,CCC C CCFcFcFccFcFc M (6-38) Appendix B shows the steps for getting from Eq. (6-37) to Eq. (6-38) Equation (6-38) will be particularly useful for writing the covariance for random variable s, rather than fixed points. Recall the general variance expression for s mc p is written in Eq. (6-35) Similar to fixed points c1 and c2, random variables R1 and R2 in the covariance term are independent copies of R introduced to account for the co variance in random sets of R for a unique empirical CDF of the capacity. In order to evalua te this expression that contains the randomness of C F and R one must use conditional calculus. To provide order to the explanation, the right side of Eq. (6-35) 91 PAGE 92 will be an alyzed in two parts. First consider the variance part, which is written in terms of conditional expectation in Eq. (6-39) var()Evar|varE()|CRCCRCCFRFRRFRR (6-39) The varC term can be related to our finding in Eq. (6-38) ; however here we only have one random variable R so the covariance from Eq. (6-38) becomes a variance, illustrated in Eq. (6-40) 1 var|cov|,|CC CCC CCCFRRFRRFRRFRFRFR M (6-40) The varR term in Eq. (6-39) is a straightforward definition of variance. Rewriting Eq. (6-39) 2 221 var()E()()E()E()CRCCRCRCFRFRFRFRFR M (6-41) Sim plifying expectations in Eq. (6-41) the variance part of Eq. (6-35) is shown in Eq. (6-42) 22 21 var()EECfCCFRpFRFRp M f (6-42) Moving onto the covariance part in Eq. (6-35) the random variables in the covariance expression for this scenario includes R1, R2, and Consider the general expression of covariance for conditional expectation in Eq. C(6-43) (Rice 1995, Sect. 4.4, Whittle 1992, Chap. 2). 12 1 212 ,12 1 2cov,Ecov,|covE|,E|YXX YXX X X XXY XYXY (6-43) Matching terms to our scenario of response and capacity, we obtain: 11221 ,,CC 2 X FRXFRYRR (6-44) Therefore, the covariance part of Eq. (6-43) written using conditional calculus is shown in Eq. (6-45) Notice that the second part of Eq. (6-43) is zero in this case, since R1 and R2 are independent. 92 PAGE 93 12 12 12,1212 cov,Ecov ,|,CCCCRR CC FRFRFRFR FRFRRR (6-45) Again, we can relate Eq. (6-45) back to Eq. (6-38) which gives us Eq. (6-46) 12 12121212 ,1 cov,|,min,CCCC C CC FRFRFRFRRRFRRFRFR M (6-46) Substituting and simplifying Eq. (6-46) 2 12121 cov,Emin,CC C fFRFRFRRp M (6-47) Finally, Eqs. (6-42) and (6-47) are substituted back into Eq. (6-35) to give Eq. (6-48) 22 22 121 111 varEEEmin,smc fC C f C fN ppFRFRpFRRp NMNM (6-48) Consider a few observations from Eq. (6-48) First, recall the vari ance for the conditional expectation m ethod with known, analytical CDF in Eq. (6-22) Equation (6-48) should approach this exp ression when an infinite number of capacity samples ( M ) are used in the empirical CDF. Also, for the extreme case when only one random sample of the response and capacity are compared, then Eq. (6-48) should be equivalent to the crude Monte Carlo m ethod (Eq. (6-5) ). By setting N = 1 and M = 1 in Eq. (6-48) the variance of s mc p does in fact equal Furtherm ore, even though the derivation of the SMC variance estimator was derived using the empirical CDF of the capacity, the variance estima te is the same if the empirical CDF of the response was used instead. Appendix C demons trates this symmetry between response and capacity random variables for SMC variance. varcmcpLooking further at Eq. (6-48) the expression can be viewed as having three groups of expectations, which do not include the sam ple sizes M and N Let the three expectation components be defined as RR and 12, RR as shown in Eq. (6-49) 93 PAGE 94 122 2 2 2 ,12E E Emin,Cf RRfC RRCfFRp pFR FRR p (6-49) Rewriting Eq. (6-48) in terms of the expectation components, 12,1 111 varsmc RR RRN p NMNM, (6-50) Equation (6-50) is interesting becau se if the expectation components in E q. (6-49) are known (or estim ated) then the variance of s mc p can be extrapolated for any combination of response and capacity samples. This means, Eq. (6-50) is a valuable tool for the user to determ ine how many samples to use for a desired leve l of accuracy. In fact, if a cost function is used for N and M samples, then the cost can be minimized by using Eq. (6-50) As done with the variance estim ates, the remaining expectations in Eq. (6-48) can be approxim ated concurrently with the probability of failure by using the simulation estimates of the expectation. The variance estimator consists of three expectations: E[ FC( R )] (or pf), E[ FC( R )2], and E[ FC(min( R1,R2))]. The first two estimated expect ations terms are essentially the same as those conditional expectation, however he re the values from the empirical CDF must be used, as shown in Eqs. (6-51) and (6-52) 111 ENM ji RCf ijICR FRp NM (6-51) 2 2 111 ENM ji RC ijICR FR NM (6-52) 94 PAGE 95 The term E[FC(min(R1,R2))] can be approximated by taking the minimum of pairs of the response for evaluation in the limit state. The disc rete form of the estimate for the third term is shown in Eq. (6-53) 2 212 12 11min, 2 Emin,N M ji RC ijICRR FRR NM i (6-53) Therefore, the simulation estimate of the variance in s mc p uses existing information during the simulation to give a prediction of the accuracy in the probability of failure calculation. The component estimates in Eq. (6-49) are then available to project the accuracy in for oth er sample size combinations. Conditional Expectation with an Anal ytical Estimate of a Random CDF A common approach to perfor m simulations when the CDF of neither the response nor capacity is known, without using the empirical CD F, is when the CDF is fitted with a smooth function based on M random samples. Which in essence, is a combination of separable Monte Carlo and conditional expectation methods. This is a valid approach, however, this does introduce a bias from the fit. The pr ocess described here is two stages: i estimate the CDF of the capacity, and ii. use random responses in the CDF estim ate to approximate the probability of failure. The predictor of the proba bility of failure for an analyti cal estimate of a random CDF is shown in Eq. (6-54) 11 ()N fit f it Ci i p FR N (6-54) This draws obvious similarities to Eq. but here the expectation is not the exact pf. Since a random estimate of the CDF is used in this cas e, then it will provide an estimate of the probability of failure f p thus a bias error. 95 PAGE 96 EEEE|fit fit f it C RCC f p FRFRRp (6-55) Since bias error exists in this estimate along w ith the randomness, then deriving the variance of f it p is not straightforward and would seemingly not produce as meaningful of a result as the SMC method. Therefore, deriving f it p variance will not be cove red in this dissertation. To review the Monte Carlo methods, Table 61 lists all of the cas es and summarizes the simulation processes, sources of variance, and bias. Table 6-1. Summary of crude and separable Monte Carlo simulation methods Estimate Simulation Source of variance Bias cmc p 11N ii iN I CR Binomial law No ce p 11 ()N Ci iFR N Random response (R) No s mc p 111NM ji ij I CR MN Random response Random capacity (R, C) No f it p 11 ()N fit Ci iFR N Random response Random capacity (R, C) Yes In summary, three methods for calculating th e probability of failure were presented, including crude Monte Carlo (C MC), conditional expectation (C E), and separable Monte Carlo (SMC). Crude Monte Carlo estimates yield the great est uncertainty in the probability of failure, however it is applicable to the most gene ral case where response and capacity are not independent. When the response and capacity are i ndependent, and the probability of failure for given a response is available analytically (FC) or can be very accurately fit from sampling ( f it CF), then the conditional expectation should be used When the probability of failure for given response is not available, SMC should be used, and the accuracy estimates derived above may help in choosing the number of samples of resp onse and capacity needed for desired accuracy. 96 PAGE 97 97 Furthermore, accuracy improvements with SMC can be enhanced by reformulating the limit state, which is demonstrated in Chapter 7. Conditional expectation and separable Monte Carlo can be used in combination with moment-b ased distribution estim ates, response surface approximations, tail modeling, importance sa mpling, quasi-Monte Carlo sampling, and other variance reduction techniques. In particular, quasi-Monte Carlo sa mpling may be more beneficial for SMC because of the smaller number of samples compared to CMC. The next chapter uses example problems to co mpare separable Monte Carlo with the other methods, both experimentally and analytically as well as validate th e derived SMC variance estimate. PAGE 98 CHAP TER 7 IMPROVING ACCURACY OF FAILURE PROBABILITY ESTIMATES WITH SEPARABLE MONTE CARLO Three probability of failure example problems are presented to compare crude Monte Carlo conditional expectation, and se parable Monte Carlo. Additional discussion was provided to explain why separating the limit state was appropriate in each case. The goal of this chapter is to use relatively simple problems to illustrate the implementation of separable MC, as well as show the accuracy improvements over traditional Monte Carlo. These concepts on separable Monte Carlo have been realized in wo rk by Acar et al. (2007) and Kuma r et al. (2008) for stress failure in a ten-bar truss and probabilistic design optim ization of an integrat ed thermal protection system, respectively. Therefore, separable MC would also be an option for reducing computational burden (or improved accuracy) in the probability of failure calculation from Qu et al. (2003). The first example problem in this chapter is basic with both distribut ions of the response and capacity being known; however, s mc p and f it p were still calculated for demonstration purposes. Next, bending in a composite plate ex ample is used to not only compare the Monte Carlo methods for the original limit state, but also to exhibit effect s of reformulating the separable limit state with SMC. Furthermore, th e derived variance estimator for separable MC was validated with the simulation estimates. The third example is joint work with Ravishankar (Ravishankar et al. 2009), which covers the more general concept of separable sampling (shown with the Tsai-Wu failure criter ion), where the response and cap acity are non-separable in the limit state. Note, Appendix D also shows an analytical example using uniform distributions to compare the required sample size to obtain a given level of accuracy using CE and CMC. 98 PAGE 99 H owever, since that work this work doesnt de al with SMC, then it was omitted from the main body of the dissertation. Basic Example with Normal Distribut ions for Response and Capacity The first example considered is a straightfo rward problem with th e following independent normal distributions for th e response and capacity: R = N(10,1.25) and C = N(13, 1.5) (Melchers 1999, Sect. 3.4). The actual probability of failure for this problem is pf = 0.06221. All of the simulation methods covered in the previous sec tion were used in this experiment. For the s mc p and f it p methods, the given nor mal distributions for R and C were ignored and estimates of the CDFs were based on random samples. The method of moments based on M samples of the capacity was used to estimate the normal parameters of the CDF for f it p The experimental results from this study are presente d in Table 7-1 for three cases of N and M equal to different combinations of 10 and 100. Note that cmc p and ce p only refer to the sample size N, since just one sample size is required for each, as per Eqs. (6-8) and (6-21) Also, the variance for the crude Monte C arlo predictor was calculated directly from Eq. (6-5) The samples sizes were relatively sm all, but this was to emphasize the difference in variance and bias. The bias is simply the difference of the mean probability of failure( an me f it p ) and the actual probability of failure pf. Since f it p uses an estimate of the CDF, then it has a bias in the pf estimate. The other methods are all completely random, so only variance is present. Each simulation was repeated 104 times to obtain the expect ed values and standard deviations. In the next subsection, the large numbe r of repetitions to obtain the expectations of the variance predictors will be replaced with a single simulation estimate based on Eqs. (6-22) and (6-48) The results of this simulation experim ent are shown in Table 7-1. 99 PAGE 100 Table 7-1. Com parison of experimental results for M onte Carlo methods with 104 repetitions for R = N(10,1.25) and C = N(13, 1.5) (*CMC variance was evaluated at pf = 0.06221) N = 100 M = 100 Mean Standard deviation Bias cmc p 0.0622* 0.0242* ce p 0.0622 0.0097 s mc p 0.0623 0.0162 f it p 0.0627 0.0157 0.0005 N = 10 M = 100 Mean Standard deviation Bias cmc p 0.0622* 0.0764* ce p 0.0622 0.0313 s mc p 0.0622 0.0344 f it p 0.0627 0.0338 0.0005 N = 100 M = 10 Mean Standard deviation Bias cmc p 0.0622* 0.0242* ce p 0.0622 0.0097 s mc p 0.0623 0.0421 f it p 0.0672 0.0405 0.0050 Analyzing Table 7-1, the c onditional expectation method ( ce p ) was consistently the most accurate, which makes sense given the derivation of variance and since the exact CDF of the capacity was used. Consider ing the first case with N = M = 100, it is obvious that separable MC is more accurate that CMC for the same number of response and capacity samples. The variance in s mc p and f it p were similar, however f it p uses an estimate of the CDF, thus it has inherent bias. The magnitude of the bias in f it p changes with the sample size M, since that corresponds to the fit of the capacitys CDF. This leads one to conclude that with f it p the magnitude of M affects the variance (standard deviation) and bias of the pf estimate, regardless of N. In contrast, the results show that the magnitude of N is only associated with the variance, not bias. 100 PAGE 101 Bending in a Composite Plate Example A comparison of the accuracy of crude M onte Carlo, separabl e Monte Carlo, and conditional expectation methods wa s illustrated through a composite plate example. The variance (presented as standard deviati on and coefficient of variation) was estimated through repeated simulations, as well as from simulation estimates (via Eqs. (6-5) (6-22) and (6-48) ). The limit state of the exam ple problem is for allowable de flection of a composite plate. The calculations involving composite mechanics were done usi ng Classical Lamination Theory (CLT). The example was chosen to illustrate the ability of SMC to take advant age of the reformulation of the limit state, which is the s econd part of the results. A common engineeri ng situation where the limit state is separable occurs when the loading conditions (response) are independent of the stiffness, stre ngth, or allowable limits of the structure (capacity). One such failu re condition that lends itself to this separation is failure due to excessive out-of-plane displacement w of a composite laminate. Consider a simply supported square laminate plate of dimensions L x L under transverse loading as shown in Fig. 7-1 (Di Sciuva and Lomario, 2003). Figure 7-1. Composite la minate of dimensions L x L under transverse loading, q(x, y) The conventional way to formulate the limit stat e function for flexural response in a plate is shown in Eq. (7-1) where the allowable deflection wall, is the capacity and the deflection from loading w, is the response. (,)all allGwwww (7-1) Let the loading conditions be sinusoidally varying pressure, as defined in Eq. (7-2) 101 PAGE 102 0,sinsin x y qxyq ab (7-2) The solution of the governing equations for out-o f-plane displacement at the center of the plate for a balanced laminate is shown in Eq. (7-3) (Grdal et al. 1999, Chap. 8). 0*q w D (7-3) Where D* is composed of terms of the laminate bending stiffness matrix. 4 11126622 4*22DDDDD L (7-4) The laminate bending stiffness (D*) is a function of plate le ngth L, lamina thickness t, fiber orientation and lamina material properties (G ibson 1994, Section 7.9). Therefore, D* is a function of the random va riables in Table 7-2 (E1, E2, v12, G12, and L). Substituting Eq. (7-3) in the lim it state (Eq. (7-1) ) lends itself to three main random components: allowable out-of-plane displacem ent wall, pressure loading q0, and laminate bending stiffness D*. The three groups of random variables are independent of each other, which means that condi tional expectation and separable MC methods are applicable. Under pressure loading, the maximum deflection occurs at the center of the plate. A [90, 45, -45]s symmetric laminate with lamina thickness of 125 m will be considered. Deterministic thickness and fiber orientati on are used. The normal random variables for IM7/977-2 carbon fiber/epoxy and th e corresponding coefficients of variation are presented in Table 7-2, as well as length (L) and transverse loading (q0), and the allowable deflection (wall) as lognormal (Smarslok et al. 2007). 102 PAGE 103 Table 7-2. Distribution param eters of random variables for bending Random variable Mean Coefficient of variation Distribution E1 (GPa) 150 5% Normal E2 (GPa) 9 5% Normal v12 0.34 5% Normal G12 (GPa) 4.6 5% Normal L (mm) 75 2% Normal wall (mm) 8 3% Lognormalq0 (kPa) 130 15% Normal Comparison of Monte Carlo Methods The Monte Carlo simulations had a sample size of 103 and were repeated 104 times in order to obtain reliable statistics. The empirical variance of each pf estimator is calculated from the 104 simulations. On the other hand, the varian ce estimates are also calculated for each individual simulation, and their average and standard deviation provide accuracy measures of these estimates. Recall, the expressions for vari ance of CMC, CE, and SMC were shown in Eqs. (6-5) (6-22) and (6-48) respectively; along with the corr esponding probability of failure estim ates (Eqs. (6-8) (6-22) and (6-48) ). Given the limit state arrangement in Eq. (7-1) the distribution of the capaci ty random variable (wall) is known, thus making the conditional expectation method in Eq. (6-22) applicable. The nominal failure probability of 33.9810 was determined from 105 repetitions of the CE method. Table 7-3 shows the empirical and estimated standard deviations (stdev, square root of the variances) and coefficients of variation (CV) for the similar cases (with respect to response samples) of Ncmc = 103, Nce = 103, and Nsmc = Msmc = 103, all repeated 104 times. 103 PAGE 104 Table 7-3. Com parison of empi rical and estimated results for crude and separable Monte Carlo methods for laminate bending problem with pf = 3.98x10-3 for the limit state in Eq. (7-1) Empirical standard deviation v alues were calculated for 104 repetitions of MCS, and estimated variance values were averaged (standard deviation) and found the coefficient of variation ( CV(var()) p ) over the same 104 repetitions Empirical variance Estimated variance Method Sample size stdev p CV p stdev p CV p CVvar p cmc p Ncmc = 1000 0.00198 49.3% 0.00199 49.6% 49.6% ce p Nce = 1000 0.00160 40.0% 0.00161 40.1% 51.2% s mc p Nsmc = Msmc = 1000 0.00161 40.0% 0.00161 40.2% 51.4% These results serve two purposes: a) comparison of the accuracy of the variance estimates, b) comparison of the accuracy of the probability of failure estimates. First, consider the accuracy of the variance estimates. The results in Tabl e 7-3 shows that the SMC variance estimate is unbiased, by comparing empirical and estimated CV p Furthermore, the coefficient of variation of the variance estimate ( CV(var()) p ) was similar for all methods. To check that the SMC estimated variance has a similar level of accuracy as CMC, consider the error ratio: ()/stdevf p p p. Table 7-4 contains the m ean and standard deviation of the error ratio based on 104 repeated simulations for 103 samples (corresponding to the case in Table 7-3). Table 7-4. Mean and standard deviation of error ratios ()/stdevf p p p for CMC and SMC failure probability and variance estimates for the limit state in Eq. (6-50) calculated fro m 104 repetitions Error ratio mean stdev cmc p 0.84 0.77 s mc p 0.90 0.80 Since the mean of error ratios are both near 1, then the simulation estimates of the standard deviation are close to the actual error in the pf estimate. In addition, the standard deviations show 104 PAGE 105 that there is large v ariability in the variance estimates, however error estimates are only expected to give a rough idea of the errors, and in addition, the SM C variance estimates perform at about the same accuracy as the commonly used CMC estimates. Moving on to comparing the accuracy in the f p estimates from Table 7-3, the coefficient of variation for ce p and s mc p were nearly identical at 40%, but only about a 20% improvement over CMC. Both observations are a result of th e coefficient of variat ion relationship of the response and capacity. The re sponse random variable (q0/ D*) has coefficient of variation of 17%, whereas the capacity ( wall) is nearly deterministic at 3%. So, 103 samples of the capacity (Msmc) produced an empirical CDF which was nearly identical to the actual CDF of wall. To get a better understanding of the relationship of Nsmc and Msmc to the accuracy of s mc p the component expectations in Eq. (6-49) were estimated for limit state in Eq. (7-1) with Nsmc = Msmc = 103, as shown in Table 7-5. Table 7-5. Mean and standard deviation of ex pectation components from separable Monte Carlo variance estimator (Eq. (6-49) ) calculated from simulation estimates with Nsmc = Msmc = 103 repeated 104 times Componentmean CV 0.00258 51.0% ,RR 0.00140 34.5% 12,RR 5.7e-6 2380% The component 12,RR has a nearly insignificant contri bution to the variance, which also explains the huge coefficient of variation. The other two components had significant mean values and reasonable accuracy. The expectation components in Table 7-5 were used in Eq. (6-50) to extrapo late the variance (shown as coefficient of variation) for other combinations of Nsmc and Msmc in Fig. 7-2. 105 PAGE 106 Figure 7-2. Separable Monte Carlo coefficient of variation contour plot for the limit state in Eq. (7-1) using extrapolated values from Eq. (6-50) with expectation components calcu lated with Nsmc = Msmc = 103 In Fig. 7-2, notice that th ere are only very small improvements in the accuracy with increased samples of the capacity ( Msmc). This indicates that the SMC accuracy converges to the accuracy of CE for a relatively small number of cap acity samples. This corre sponds to the data in Table 7-3 where CV for conditional expectation and separable MC methods are nearly equal. Conversely, increasing the number of response samples ( Nce and Nsmc) would still reduce the failure probability variance significantly. However, assuming that the response calculations required for D* are computationally expensive (because for more general loading and boundary conditions they will require finite element analysis) and only 103 samples are manageable. The dashed line in Fig. 7-2 references the N = 103 level. Note that the assumption was made that the cost of generating D* samples is much greater than any ot her random variable and the evaluation of the limit state to determine failure is trivial. The extrapolation variance data in the next ex ample is more interesting with respect to Msmc, so more analysis is included in the following section. A way to improve the accuracy with 106 PAGE 107 separab le MC is by reformulating the limit stat e to reduce the uncertainty in the expensive response term, as shown in the next section. Reformulation of Limit State Functi on as Variance Reduction Technique When one (or more) of the random components in the limit state function are independent and more expensive to sample than the other random variables, then it is beneficial to reformulate the limit state in a way that reduces the uncertainty in the term associated with the expensive component, as explained in Eq. (6-31) The bending example was repeated by rearranging the lim it state compone nts to illustrate the variance reduction feature of separable Monte Carlo. The transverse load q0 has a large coefficient of variation at 15%, but is inexpensive to generate. However, in Eq. (7-1) q0 is paired with the D* which makes the response term expensive. Therefor e, it would be beneficial to reformulate the limit state as shown in Eq. (7-1) by moving q0 to the capacity and consequen tly reducing the uncertainty in the expensive response component. 0 01 ,*,all allw GqDw qD (7-5) Equation (7-5) is an equally valid limit state for this problem and has the same nominal probability of failure. Now the response (1 /D*) has coefficient of varia tion of 7.5% and that of the cheap capacity ( wall/ q0) is now 16.5%. Therefore, the uncertainty associated with the expensive response term was reduced from 17% to 7.5%. Figure 7-3 visually shows the shift in uncertainty in the response and capacity between the original and reformulated limit state. 107 PAGE 108 A B Figure 7-3. Probability density functions and coefficients of variation in the response R and capacity C for the A) Original limit state in Eq. (7-1) B) Reformulated limit state in Eq. (7-5) Table 7-6 shows the results for th e reformulated limit state in a similar manner to Table 73. Note that in this reformulation the exact analytical CDF required for conditional expectation method is lost, so that method is excluded from this example. Table 7-6. Comparison of empi rical and estimated results for crude and separable Monte Carlo methods for laminate bending problem with pf = 3.98x10-3 for the reformulated limit state in Eq. (7-5) Empirical standard deviation values were calculated for 104 repetitions of MCS, and estimated variance values were averaged (standard deviation) and found the coefficient of variation ( CV(var()) p ) over the same 104 repetitions Empirical variance Estimated variance Method Sample size stdev p CV p stdev p CV p CVvar p cmc p Ncmc = 1000 0.00197 49.4% 0.00199 50.0% 49.1% s mc p Nsmc = Msmc = 1000 0.00097 24.3% 0.00098 24.5% 47.8% The results in Table 7-6 show the same coef ficient of variation for CMC as Table 7-3, since the pf is the same (Eq. (6-5) ). However, the SMC method was about twice as accurate than the prev ious results with a coefficient of vari ation of 24.3%. Therefore, even using the same number of samples with the reformulated limit state improved the variance in the pf estimate. Also, as done for the previous limit state, the erro r ratio of the estimates is shown in Table 7-7, 108 PAGE 109 using 103 samples for the estimates and repeated 104 times (corresponding th e case in to Table 76). Table 7-7. Mean and standard deviation of error ratios ()/stdevf p p p for CMC and SMC failure probability and variance estimates for the reformulated limit state in Eq. (7-5) calcu lated from 104 repetitions Error ratio mean stdev cmc p 0.86 0.78 s mc p 1.02 1.44 As with the error ratios with the original limit state in Table 7-4, the mean of the SMC and CMC error ratios were both near 1, but this time variability in the error was greater for SMC at 1.44. Recall that separable MC permits different sample sizes of R and C So keeping the expensive response term fixed at Nsmc = 103, we are still able to sample more from the capacity ( Msmc) at a small computational cost. Table 7-8 shows the separable Mo nte Carlo results for increasing number of samples of the capacity, from 250 to 105. Table 7-8. Empirical and estimated vari ance from SMC for laminate bending of the reformulated limit state in Eq. (7-5) ( pf = 3.98x10-3) Sample size Empirical variance Estimated variance Nsmc Msmc stdev s mcp CV s mcp stdev s mcp CV s mcp CVvarsmcp 1000 250 0.00188 47.2% 0.00188 47.2% 99.3% 1000 500 0.00134 33.7% 0.00134 33.7% 69.8% 1000 1000 0.00097 24.3% 0.00098 24.5% 47.8% 1000 5000 0.00052 13.0% 0.00052 13.1% 21.3% 1000 10,000 0.00043 10.8% 0.00043 10.9% 20.0% 1000 50,000 0.00035 8.7% 0.00035 8.7% 26.2% 1000 100,000 0.00033 8.3% 0.00033 8.3% 27.7% The results in Table 7-8 show that when the number of samples of the response is fixed at Nsmc = 1000, but the number of samples of the capacity is increased, the co efficient of variation of the failure probability reduces from 47.2% to 8.3%. In addition, changing the number of 109 PAGE 110 capacity samples ( Msmc) used in the estimates of variance components in Eqs. (6.28-30) also improves the accuracy of the estimates of the variance ( ).The expectation com ponents in Eq. CV (var())smcp (6-49) were also estimated for Nsmc = Msmc = 103 repeated 104 times, as shown in Table 7-9. Table 7-9. Mean and standard deviation of ex pectation components from separable Monte Carlo variance estimator (Eq. (7-5) ) calculated from simulation estimates with N = M = 103 repeated 104 times Componentmean CV 0.000105 37.0% RR 0.00387 24.5% 12,RR 0.000852 52.8% The values for RR and 12, RR in Table 7-9 were more similar in magnitude that those in Table 7-5 and the same can be said for their coefficients of variati on. As done for the first limit state, the values in Ta ble 7-9 were used in Eq. (6-50) to extrapolate the coefficients of variation for different combinations of Nsmc and Msmc. Figure 7-4 shows the contour plots for the coefficients of variation using the expectation compone nts for the reformulated limit state, and Table 7-10 shows the corresponding data for the cases presented in Table 7-8. Table 7-10. Extrapolat ed variance from Eq. (6-50) for SMC for laminate bending of the reform ulated limit state in Eq. (7-5) ( pf = 3.98x10-3) Sample size Extrapolated variance estimates Nsmc Msmc stdev s mcp CV s mcp CVvarsmcp 1000 250 0.00188 47.2% 51.6% 1000 500 0.00135 33.8% 50.4% 1000 1000 0.00098 24.5% 47.8% 1000 5000 0.00053 13.2% 38.4% 1000 10,000 0.00044 11.0% 34.5% 1000 50,000 0.00035 8.8% 34.2% 1000 100,000 0.00034 8.5% 35.3% 110 PAGE 111 Figure 7-4. Separable Monte Carlo coefficient of variation contour plot for the limit state in Eq. (7-5) using extrapolated values from Eq. (6-50) with expectation components calcu lated with Nsmc = Msmc = 103 The results in Table 7-10 should be compared to the empirical and estimated variance in Table 7-8. The variance data matches very closely to the empirical results, even though this data was extrapolated from just the case where Nsmc = Msmc = 103. Recall that the values for RR and 12,RR in Table 7-9 were averaged over 104 repetitions, but uncertainty in these components from a single simulation will propagate to th e variance estimate, which is quantified as in Table 7-10. Observe that for Nsmc = Msmc = 103 has the same value of 47.8% as in Table 7-8, since that wa s the number of response an d capacity samples used to estimate ())smcp CV (var CV (var())smcp ,RR and 12,RR The extrapolated variance performed reasonably well over the range of capacity samples. In fact, the extrapolat ion estimates were more accurate for values of Msmc less than 103, but slightly worse for Msmc greater than 103. Table 7-10 indicates that once we have performed SMC, we can then estim ate by how much we need to increase Msmc and Nsmc to 111 PAGE 112 im prove the accuracy to a desired level. Then a particular combination of Msmc and Nsmc may be chosen based on the relative cost of response and capacity evaluations. Figure 7-5 shows the SMC data from Table 7-8 with Nsmc = 1000 fixed and varying Msmc, as well as the corresponding crude Monte Carlo values ( Ncmc = 1000). The plot for coefficient of variation in SMC in Fig. 7-5 corresponds a slice (dashed line) in Fig. 7-4 at Nsmc = 103 Figure 7-5. Estimated coefficient of variation of SMC for varying Msmc with Nsmc = 1000 for laminate bending limit state in Eq. (7-5) with 1 standard deviation bounds on the varian ce estimate (pf = 3.98x10-3) Figure 7-5 shows that about Msmc = 250 and Nsmc = 1000 would give nearly the same accuracy as CMC with 1000 samples. Using a very large number of cheap capacity samples in Eq. (7-5) can bring the coefficien t of variation in the pf estimate to under 10%. The one standard deviation bounds (dashed lines in Fig. 7-5) on the estimated CV() s mcp (Eq. (7-6) ) demonstrate t im proved accuracy of the variance estimate as well. he 112 PAGE 113 1 varstdevvar CV p p bp p (7-6) Therefore, not only is the accuracy of th e failure probability estimate improved as a function of Msmc, but also a more accurate simulation estimat e of the variance. In conclusion, this type of problem where one of the random va riables (response or cap acity) is relatively inexpensive to generate and also has large uncertainty, then reformulating the limit state and sampling more from the cheap capacity can result in a more accurate probability of failure estimate. Separable Sampling of a No n-separable Limit State The previous two examples of separable Mont e Carlo only explored simple limit states, expressed as a difference between a single response and a single cap acity. This section looks at a more general limit state function that has several components of response and capacity combined such that they are non-separable. The objective is to use separable sampling and reformulation to shift uncertainty away from the computationally expensive com ponents to improve accuracy in the probability of failure estimate. The example considered here is the failure of composite pressure vessel subjected to in ternal pressure. For composite materials, failure is generally predicted using the Tsai-Wu failure criterion, which serves as the limit state (Whitney 1987). The Tsai-Wu limit state is non-separabl e, but is composed of independent random components, thus making regrouping with SMC possible. Regrouping a General Limit State for Separable Sampling In most structural problems, failure of the system depends on the strength of the material S and stresses the structure sustains Consider a general limit state G(,S), where the stress are considered the response and the strength is the cap acity. As in the previous example, the stress is often a computationally expensive calculation, whereas the strength may be defined as a 113 PAGE 114 statistical distribu tion. Since we are limited with the number of samples of the response it would be helpful to reduce the uncertainty associ ated with it. Therefor e, a method for regrouping random variables in the limit state is pr esented to be used with separable MC. In linear problems, stresses are a linear function of the load P as in Eq. (7-7) (7-7) uP W here, u are stresses per unit load. The randomness in the load P is often independent of the random variables that affect u (geometry and materi al properties), but P adds large uncertainty to the computationally expensive stress calculati on. Therefore, since the stress per unit load, load, and strength are independent components, then the random variables can be separated, as shown in Eq. (7-8) ,,uuGGP S,S (7-8) Now there are different possi ble groupings of the random variables for sampling with SMC. Figure 7-6 illustrates the original and regrouped limit state in Eq. (7-8) for separable Monte Carlo sim ulations. A B Figure 7-6. Illustration of sepa rable sampling of the separable M onte Carlo method A) Original limit state B) Regrouped limit state 114 PAGE 115 Figure 7-6 is a m ore specific arrangement than its counterpart in Fig. 6-1.The separable Monte Carlo formula that corresponds to Fig. 7-6b is shown in Eq. (7-9) A similar form of Eq. (7-9) could be written for Fig. 7-6a but with different indices. 1111 [,,MN uuu smcijj jipI GP MNS 0] (7-9) The next section looks the failure probability estimate of the case of the Tsai-Wu failure criterion where the independent random variab les in the non-separable limit state can be regrouped and separately sampled. The accuracy (presented as standard deviation) of the pf estimate using SMC and CMC is compared for the tw o groupings of the limit st ate as in Fig. 7-6. Note that statistical expression for standard deviation in SMC is only presently available for the simple limit state in Eq. (6-7) not for the more general, non-separable limit state. Thus, the em pirical standard deviations of the pf estimates were calculated by performing repeated simulations. Application to Composite Materials for the Tsai-Wu Failure Criterion The problem involves prediction of failure of composite pressure vessel according to the Tsai-Wu failure criterion A pressure vessel of d = 1 m diameter made of composite laminate is subjected to an internal pressure of 100 kPa. Th e composite laminate is a graphite/epoxy with the lamina sequence of [+25/-25]s and each layer being 125 m thick. The material properties of the composite are shown in Table 7-11 (Stamblewski et al. 2007). In general, the stress calculat ion is computationally expensiv e when it is calculated through finite element analysis, or in this case Cla ssical Lamination Theory (CLT) (Gibson 1994, Section 7.3). The stresses 1 and 2 (normal) and 12 (shear) acting in each pl y of the laminate are calculated as in Eq. (7-10) 115 PAGE 116 1 2 121 11 2 12/2 1/2 [][]/4[][]1/4 00u u uPd TQAPdTQAPP (7-10) Where, the stresses acting in each ply of the la minate are a function of the in-plane stiffness matrix of the laminate [ A ], reduced stiffness matrix of each lamina [] Q transformation matrix of each lamina [ T ], the pressure load P, and the diameter ( d = 1 m) of the pressure vessel. Observe that the CLT stress calculation in Eq. (7-10) was separated into u and P as described in Eq. (7-7) The m ost widely used criterion for composites is the Tsai-Wu criterion (Whitney 1987). The criterion is a function of the strength S and the stresses in the fiber and transverse direction (e.g. 1 and 2 direct ion, respectively). The parame ters of the random composite properties, pressure, and strengths used in th e Tsai-Wu criterion are presented in Table 7-11. Table 7-11. Mean and coefficient of random material properties, pressure, and strengths Properties Mean CV% Strength Mean CV% E1(GPa) 159.1 5% S1T (MPa) 2312 10% E2(GPa) 8.3 5% S1C (MPa) 1809 10% G12(GPa) 3.3 5% S2T (MPa) 39.2 10% 12 0.253 5% S2C (MPa) 97.2 10% P (kPa) 100 15% S12 (MPa) 33.2 10% According to the Tsai-Wu criterion, a layer of the laminate is assumed to have failed when the limit state in Eq. (7-11) is greater than or equal to zero. (7-11) 222 111222661211221212(,) 1 GFFFFFFS 116 PAGE 117 W here, 11 1 1111 22 2 22 22 1122 66 12 2 1211 11 1 2TCTC TCTCFF SSSS FF SSSS FF FF S 1 1 (7-12) The in-plane norm al and shear stresses, in each layer of the laminate is obtained from Eq. 1212,,T(7-10) and the Tsai-Wu coefficients 22661212,,,,,11 F FFFFF are functions of the tensile and compressive strengths S. The uncertainty in the stresses is due to randomness in material properties (or u) and pressure load P and the uncertainty in the Tsai-Wu coefficients are from the randomness in the strengths S. For unit pressure load (P =1), stresses are equal to u. Therefore, the original li mit state function in Eq. (7-11) ( G (,S)) can be reorganized so that it is composed three sets of statistica lly independent random variables: u, S and P Thus, the regrouped limit state becomes Gu(u, P ,S). Now, the large uncertainty from the load P of 15% is separated from the expensive st ress per unit load calculations u. The inexpensive loads can be grouped with the inexpensive strengths S, so there is not a restriction to the number of samples capable of being generated (within reason). Th ese ideas regarding the regrouping of the limit state with separable MC will be further discussed next. The probability of failure of a composite pressure vessel was calculated using crude Monte Carlo and separable Monte Carlo with the original and regrouped limit state. It was assumed that our computational budget only perm itted 500 stress calculations (). Therefore, crude Monte Carlo used an equal number of random response () and capacity (S) random variables ( N = 500) were sampled and compared. In the case of separable Monte Carlo, the response samples () 117 PAGE 118 were lim ited (N = 500) and the capacity samples (S) were varied for M = 500 to 50,000 samples. The actual probability of failure the original and regrouped limit state is 0.0121. In order to assess the accuracy of the met hods, the empirical standard devi ation was obtained by performing 104 repeated simulations shown in Ta ble 7-12. Comparing the results of the original limit state, it is evident that separable MC predicts probabi lity of failure more accurately than crude MC. Increasing the number of capacity samples in SM C causes a reduction in the standard deviation of the estimate. When N =500 and M = 50,000 samples for the original limit state ( G( ,S)), the uncertainty in the probability of failure estimate with SMC CV()smcp drops down to 15.6 %, compared to the 40.0% with CMC. Table 7-12. Comparison between CMC and SMC for increasing samples of M for a fixed response sample (N =500) repeated 104 times Crude Monte Carlo Separable MC with original limit state Separable MC with regrouped limit state M stdevcmcp CVcmcp stdev s mcp CV s mcp stdevu s mcp CVu s mcp 500 0.00488 40.0% 0.00249 20.6% 0.00437 36.3% 1000 0.00222 18.4% 0.00313 26.0% 5000 0.00194 16.2% 0.00141 11.7% 7000 0.00193 16.0% 0.00119 9.9% 10000 0.00192 16.0% 9.90E-04 8.2% 25000 0.0019 15.7% 6.52E-04 5.4% 50000 0.00187 15.6% 4.77E-04 4.0% In the original limit state ( G (,S)), the expensive stress calculation contains the large uncertainty from the load P Therefore, rearranging the calcula tion to obtain stress per unit load and load permits the separation of these random variables. This arrangement will enable separable sampling of th e regrouped limit state G (,P ,S), similar to that shown in Eq. (7-9) This is an eq uivalent limit state to the original case, however this formulation shifts the large uncertainty in the load away from the expensive calculation (). By reallocating more samples uu u 118 PAGE 119 to the load a nd strengths ( P S), the accuracy in the probability of failure estimate is significantly improved. The empirical standard devi ation of the regr ouped limit state ( stdev()u s mcp ) is shown in Table 7-12. The corresponding data is also presented in Fig. 7-7. Figure 7-7. Standard Deviation of CM C, SMC and regrouped limit state SMC where N =500 (fixed) and M is varying for 10,000 repetitions Figure 7-7 clearly illustrates the effect of regrouping of the inexpensive random variables of the limit state. Crude Monte Carlo only has a single data point since we are limited to only 500 u samples. Whereas, the two separable Monte Ca rlo methods can use different sample sizes ( M ) for the other random variables. Observe that the standard deviation from the original limit state of SMC levels off to a nearly constant value of 0.0019 for M samples greater than 5000. On the other hand, the standard devi ation for the regrouped limit state continually decr eases with the number of M samples all the way to stdevu s mcp = 0.0005 (or 4.0% CV). In other words, CMC can only estimate the failure probability to 40 % accuracy, but SMC can achieve 15.6% with the original limit state or 4% with the regrouped limit state. That is, for nearly the same 119 PAGE 120 com putational cost, separable MC with regroupin g can estimate the failure probability 85% more accurately than crude Monte Carlo. To understand the reason for the accuracy im provement with SMC for the regrouped limit state, consider the comparison of the probabi lity density functions (PDF) of the effective response and capacity for the two cas es in Fig. 7-7. Table 7-13 s hows the numerical values that are represented graphically in Fig. 7-8. Table 7-13. Comparison of the scatter of th e original and regrouped limit states for SMC Mean Standard deviation Coefficient of variation SMC G (, S) 0.616 0.132 21.5% G (S, ) 0.594 0.0899 15.1% SMC regrouped Gu(u S, P ) 0.651 9.68e-3 1.5% Gu( P,S, u ) 0.592 0.146 24.7% A B Figure 7-8. Probability density functions of the A) Original limit state B) Regrouped limit state In the Fig. 7-8a, the PDF labeled G (, S) is the original limit state with randomness in the stresses at the mean strength values S and the PDF labeled G (S, ) is the original limit state with randomness in the strength at the mean stress values Likewise, Fig. 7-8b is the same as 120 PAGE 121 Fig. 7-8a, but for the regrouped lim it state. That is, Fig. 7-8 shows the PDF of the limit state with random stress per unit load Gu(u S, P ), and the PDF of the regrouped limit state with random load and strengths Gu( P ,S, u ). Keep in mind that the sample sizes for the expensive stress random variables ( or ) and cheap strength and load random variables (S and P ) are N and M respectively. Comparing the probability distri bution functions of the compone nts of the limit state in Fig 7-8, one can see the reduction in uncertainty associated with th e expensive calculation, that is G (, S) to Gu(u S, P ), or 21.5% to 1.5%. Therefore, fewe r samples are required to accurately represent the distribution Gu(u S P ). So for the reformulated limit state, N = 500 samples of u was sufficient. However, note that the uncertainty was shifted to the inexpensive calculation, thus making the PDF for Gu( P,S, u ) much wider. This is the desired scenario since many more samples of the cheap random variables (e.g. M = 50,000) are easily available. However, if a small number of samples were still used for the load and strength (e.g. M = 500), then distribution is not estimated very accurately. This explains why CVu s mcp was greater than CV s mcp M = 500, with values of 36.3% and 20.6%, respectively. However, with increasing the samples of u s mc p to M = 50,000, then the accuracy of the estimate improved drastically to 4%, as shown in Table 7-12. Also recall that the accuracy in s mc p leveled off at about M = 5,000. This indicates that 5,000 samples are all that is requi red to sufficiently estimate the distribution of G (S, ). Therefore, it is advantageous to regroup th e random variables in th e limit state to shift uncertainty away from the expensive random variable. In conclusion, separable Monte Carlo was su ccessfully extended to a general limit state that is non-separable, but sti ll consists of statistically independent random components. 121 PAGE 122 122 Separable sampling inherently improves the accur acy in the probability of failure estimate over crude Monte Carlo; especially when the limit state random va riables are regrouped to shift uncertainty away from expensive calculations. PAGE 123 CHAP TER 8 CONCLUSIONS The failure of the hydrogen tanks on NAS As X-33 Reusable Launch Vehicle (RLV) motivated much research to investigate the erro neous analysis of the design. This problem was interesting with respect to proba bilistic analysis since the design weight was highly sensitive to uncertainties for a given failure probability. In previous research by Qu et al. (2003) on sensitivity and reliability analysis for this probl em, three areas of potential uncertainty reduction were recognized and addressed in this dissertation. First, weight was found to be particularly sensitive to the transv erse elastic modulus E2, and transverse coefficient of thermal expansion 2. Measurement uncertainty analysis was performed in compliance with NIST standards on experiments to determine the transverse elastic modulus of a composite laminate over a wide te mperature range. The uncertainties associated with load, strain, thickness, and width measurements were propaga ted to estimate the uncertainty in E2 measurements. Uncertainties from systematic effects in the E2 measurement process were more apparent than random effects. The syst ematic uncertainty composed of 86.1% of the E2 uncertainty, whereas only 13.9% from random eff ects. The thickness component of systematic uncertainty contributed 89.4% of the total sy stematic uncertainty in transverse modulus measurements. An analysis of variance identified surface variation as the most significant factor in uncertainty due to thickness measurements. Random uncertainties had a smaller impact on E2 uncertainty. Spatial variation in the transverse co efficient of thermal expansion was investigated with an analytical model of an axisymmetric, circular plate. It was determined that observed changes in spatial variation in CTE were not significantly detrimental. Based on the second observation by Qu et al. (2003) that correlations in composite properties can significantly affect the failure probability, a corre lation model was developed that 123 PAGE 124 is capab le of combining material variability with measurement error. Often correlation data is unavailable for material variability, however correl ations can be derived from the relationship of elastic properties to a common phys ical effect, such as fiber volume fraction. The correlation model from fiber volume fraction was based on simplified micromechanics for graphite/epoxy and glass/epoxy laminates. Assuming a reasonable uncertainty in fiber volume fraction, then the covariance and correlation in composite proper ties could be generated. Next, the material variability from fiber volume fraction was combined with measurement error data for both laminates. The glass/epoxy laminate had correlated data available from vibration testing on a single specimen, whereas the measurement error in graphite/epoxy was estimated from several moduli experiments. The composite properties were treated as corre lated and independent random variables and propagated to strain in an example pressure vessel problem. The coefficient of variation in strain and probabi lity of failure were compared correlated and independent random variables for both materials. Compared to the independent case, correlated properties were observed as having varying effects on the coefficient of variation of transverse strain and failure probability. Therefore, if corr elated material properties were not considered, then it could result in an inefficient or unsaf e design. Furthermore, th e significant increase in probability of failure between the correlated a nd independent material properties for the graphite/epoxy case was a direct result of including the material variability (fiber volume fraction), since the measurement error was assumed independent for that case. Thus, considering material variability from the correlations thr ough fiber volume fraction can play a large role in the reliability-based design of composite structures. Finally, the third area of potential uncertainty reduction was in estimating failure probabilities. Since probability of failure calc ulations using traditi onal Monte Carlo can be 124 PAGE 125 com putationally intractable for accurate estimates of safe structures, the separable Monte Carlo method (SMC) was developed to improve the fa ilure probability accuracy. Separable Monte Carlo is a form of Monte Carlo sampling that is us ed to calculate the probability of failure when response and capacity are independent random vari ables. It allows sampling the response and capacity separately and using all possible combin ations of random response and random capacity samples. Accuracy estimates of SMC were de rived for the case where the limit state has the simple form of capacity minus response. The variance estimator for separable MC was derived using conditional calculus and va lidated using simulation estimates. The simulation estimates of variance were shown to be of comparable accura cy to those obtained for crude MC. SMC may be viewed as an extension of the conditional expectation method, which also needs independent response and capacity. The variance estimates essent ially provide the additional errors associated with having to sample capacity instead of using a given conditi onal probability (or cumulative distribution function (CDF)) for the capacity. Se parable Monte Carlo also allows different sample sizes of the response and capacity, de pending on the computational budget. To improve the accuracy of the failure probability estimate, SMC can be used in combination with a reformulated limit state that shifts uncertainty from the expensive response to the cheap capacity. It was demonstrated that the variability estimate can help us choose the number of samples of capacity and response needed for given accuracy. Separable MC, crude MC and conditional exp ectation were applied to three example problems, including an example with a basic limit state, out-of-plane displacement of a composite laminate, and separable sampling of the Tsai-Wu limit state. Conditional expectation was observed to be the most accurate method in a ll of the example problems, however either the response or capacity CDF must be known. The concept of reformulating (o r regrouping) the limit 125 PAGE 126 126 state was applied to the composite plate and Ts ai-Wu examples. Reformul ating of the limit state of independent random variables led to improved accuracy. A desirable situation is one where an inexpensively sampled random variab le (usually the capacity), that also has a large coefficient of variation and can be isolated in the limit state from the expensive calculations (usually the response). The example of out-of-plane displaceme nt of a composite laminate met the previous description. The uncertainty in the probability of failure estimat e was significantly decreased by sampling more from the wider response distribu tion, while fixing the number capacity samples, which contained the more expensive Cla ssical Lamination Theory calculations. PAGE 127 APPENDIX A RADIAL DISPLAC EMENT AND STRAIN DE RIVATION FOR SPIKE IN COEFFICIENT OF THERMAL EXPANSION IN AN AXISYMMETRIC, CIRCULAR PLATE Cylindrical coordinates were nominated as a good candidate for analyzing stress distributions from variation since the number of signi ficant coordinates for plane stress condition is reduced from two in rectangular coordinates ( x y ) to just one in cylindrical ( r ). The idealized problem setup is consid ering an impulse function acting over a very small area at zero on an infinite, axisymmetric, thin plate. This is a good assumption for large, localized variations in CTE. In general, the displacement field has the terms: ,,,,,, urzvrzwrz (A-1) However, for this problem, we are considering a thin, axisymmetric, thin plate, therefore the displacement terms become ,0 urzvwrz (A-2) Further simplifying, we note that for all intensive purposes of this study, we are only interested in the distributions in the plane of the plate, so the displacements will not be a function of z When taking all of these assumptions into consideration, the cylindrical strains are 0rr zzrzrzduu drr (A-3) Continuing from the differential equation for CTE in Eq.(4-4) .There are two physical boundary conditions, one at the origin and th e other at the edge of the plate. The radial for an axisymmetric plate, the radial displa cement at the origin ( r = 0) is zero. The othe r boundary condition depends on the problem definition. We are going to consider a fixed boundary condition at the edge for an infinite plate and a finite plate of radius rR. Therefore, the boundary conditions that are used to solve Eq. (4-4) are 127 PAGE 128 00,00Ruuorur (A-4) The variation in CTE that we are going to cons ider in the problem is a step impulse which acts on a radius r0 from the center of the plate. The region from 0 to r0 will have a spike in coefficient of thermal expansion of which will be greater than the outer region from r0 to L (2), which will be set to zero. Figure A-2 shows an arbitrary example of the type CTE distribution we are interested in. Figure A-1. Example C TE impulse distribution Finite element analysis (FEA) was used for the CTE distribution described above to help us anticipate the solution to the problem. Fi gure A-3 displays some example results for displacement, u. The data shown corresponds to the following properties: 6610 10 12 2 010.525.75 1300.2350 0.06CCEGPavTC rm 128 PAGE 129 Figure A-2. FEA displacement curve for a CTE impulse Several characteristics from this curve will be us eful for the initial conditions used in the calculations below. First, we see that there is a linear relationship from zero to a point we will call r0, where the displacement is a maximum, u0. Also, the displacement decays to zero as r goes to infinity. The differential equation (Eq.(4-4)) can be integrated to get Eq. (A-5). 11 1 duu C vTdrr (A-5) As mentioned before, we are goi ng to be breaking the displacement curve into an inner and outer region (for inside and outside r0). The inner region will be equal to a constant a1, whereas the outer region will be set to zero. Therefore, from Eq. (A-5), for to be equal to a constant (or zero) then the differentia l terms must be equal to a constant, K duu K drr Multiply both sides by r. du ruK dr r We observe that the left side can be condensed by the Product Rule. 129 PAGE 130 d urkr dr By integrating, we finally obtain, 21 2 C ukr r (A-6) Analyzing Eq. (A-6) we can see that it has the two components observed in Fig. A-2, the linear term and the hyperbolic term. Since we are going to be writing two separate expressions for the two parts of the displacement curve, it is n ecessary to have a condition that equate the displacement at r0, because the plate is continuous. Ther efore, we can rewrite the constant C2 as, 2 2 002001 22 C ukruruCurkr r 01 2 000111 22 ukrurkr r (A-7) At this point of the derivation, we are able to go in two directions: infinite or finite plate. The next two sections will explain the details of each scenario. Infinite Circular Plate For the infinite plate, we are going to use Eq. (A-7) and the boundary equations listed below to solve for K and u0. 00,0 uu (A-8) Each boundary conditio n will enable us to solve for the displacement expression for the inner and outer region. Referring to Eq. (A-6), we see that C2 is forced to be zero when r = 0. This gives us the linear equati on for the inner region. 01 for0 2inuKrr r (A-9) 130 PAGE 131 Again considering Eq. (A-6) and when r goes to infinity, then K m ust be zero. Since this boundary condition occurs at the edge of the plat e, then this is the expression for the outer region. 0 00foroutr uurr r (A-10) Equations (A-9) and (A-10) have two unknowns ( K and u0), thus we need two more relationships to solve for them. First, we know that the disp lacement equation for the inside and the outside have to be equal at r0. By setting the two previous equations equal, then we can get u0 in terms of K 0 00 001 2 2 r kruk rr 0u (A-11) Equation (A -9) becomes 0 0 inu u r r (A-12) The second relationship that we can take advantage of to solve for the constants is the inner and outer stress relationship. Si nce the CTE is equal to zero on the outside and the outer boundary is fixed, then we know that the stress at r0 has to be equal with respect inside and outside. 000 rrinoutr (A-13) Using constitutive equations and the displacement equations, the stress terms for the inner and outer region are: 00 0 0 2 00 0111inuu u EETEE v vrrvvrv 1 T (A-14) 00 0 2 0011outuuu E v vrrvr 0 0E (A-15) 131 PAGE 132 Setting Eqs. (A-14) and (A15) equal, we find that u0, and therefore K are 001 11 2 urTvKTv Finally, we can substitute these constants back into the expressions for the inner and outer radial displacements. Differentia ting these terms gives the radial strain component. Below is a summary of the findings for an infinite circular plate. 011 11 22in rrinurTvrrTvrr 0 (A-16) 22 00 211 11 22out rrouturrTvrrTvrr rr 0 (A-17) Finite Circular Plate In this section, we will consider a finite circular plate of radius rR, with a fixed boundary condition at K Again, assume we have the same impulse distribution, where = constant and nominal = 0. 000Ruur (A-18) Let us begin again by analyzi ng Eq.(A-7). As with the infi nite plate, the constant C2 must equal zero when r = 0. However, when looking at the outer boundary condition, we notice that since we are dealing with a finite length, then K does not have to equal ze ro as with the infinite plate. These observations mean that the displace ment function for the outer region will contain a hyperbolic and linear term. This was mentioned now because we are going to distinguish the two coefficients of the linear parts for the inside and outside as Kin and Kout, respectively. First, use the boundary conditi on at the origin to find Kin. 2 0 2000 02 1 000 2in inu uCurKrK r (A-19) Therefore, inner term remains the same as before. 132 PAGE 133 133 0 01 2ininu uKr r r (A-20) Next, we can use the oute r boundary condition to find Kout. 2 00 000 22 02 111 00 22R outR out out R Rur urKrurKrK r rr (A-21) To simplify fraction in Eq.(A-21), let 0 22 0 Rr rr (A-22) The outer displacement function becomes, 2 001outuurrr r 0 (A-23) As done previously for the infinite plate, we must take advantage of the relationship in Eq. (A-13) to find the constant u0. 00 0 0 2 00 0111inuu u EETEE v vrrvvrv 1 T (A-24) 2 000000 22 01 1 1outv E uvurur vr (A-25) Setting Eqs. (A-24) and (A-25) equal, we find 0 0 01 21 Tvr u r (A-26) Therefore, the inside and outside 00 rr 0 Rrrr radial displacements and strains become the expressions in Eqs. (4-11) and (4-12). PAGE 134 APPENDIX B DERIVATION OF COVARIANC E TERM IN SEPARABLE MONTE CARLO VARIANCE ESTIMATOR The derivation of the simplifie d covariance term for the sepa rable Monte Carlo estimator in Eq. (6-38) merits further explanation. Star ting with Eq. (6-37) and rewriting the expectation gives Eq.(B-1). 11 1jM CCccC jFcEFc Fc M (B-1) Furthermore, the covariance is rewritten as 1212 1 2 2 111 cov,11jkMM CCCccCccC jkFcFc EFcFc M (B-2) Note that the sum is centered, that is th e expectation is zero. For simplicity, let, 1111jkjccC kccCuFcvF22c (B-3) As m entioned before, the expectation of the diffe rences in Eq. (B-2) ar e centered, thus zero. 0andjkEu Ev 0 When j = k the terms are not necessar ily independent since if c1 < c2, then c < c1 also implies c < c2. The opposite is true for j k therefore the r.v.s uj and vk are independent 0jk jkEuvEuEv For covariance we are only concerned with the te rms that are not independent, therefore we can use only one subscript. Replacing the differences in Eq. (B-2) with the definitions from Eq.(B-3), 12 2 11 cov,M CCCjj jFcFcEuv M (B-4) Looking at the expectation and simplifying, 21 12121 min,111jj jjj CccCccCC ccc 2 E uvE FcFcFcFc 134 PAGE 135 135 12122112min,jjC CCCCCCEuvFccFcFcFcFcFcFc 1212min,jjC CC E uvFccFcFc (B-5) Substituting back into Eq. (B-4), 12 121 2 11 cov,min,M CCCCCC jFcFcFccFcFc M 2 12 1212 21 cov,min,CCC C CCFcFcMFccFcFc M Which finally simplifies to Eq. (6-38). PAGE 136 APPENDIX C SYMMETR Y OF RESPONSE AND CAPACITY RANDOM VARIABLES IN SEPARABLE MONTE CARLO This proof shows that the deri ved variance estimator for SMC in Eq. (6-48) is the same, regardless of the empirical CDF being consider ed. Begin with finding the general component expression for s mc p variance. Recall the exp ectation component form of the variance estimator for separable Monte Carlo from Eq.(6-50). To demonstrate th e symmetry of the response and capacity samples, consider the expectation components for a unifo rm distribution from Eq. (D-1) for convenience. 21 3RRfCR p pp (C-1) 21 3CRf 2 p pp (C-2) 122 ,1 3RRCRf 2 p pp (C-3) where, 1 2 f CR p pp (C-4) The problem of interest is for when pC = pR, then values of M and N should be interchangeable to keep symmetry. Thus, consider pC = pR = p and simplify the components above. 211 23RR 3 p p (C-5) 311 344 p p (C-6) 123 ,11 34RR 4 p p (C-7) At this point it is apparent that and 12, RR are equal. To simplify the expression further, reduce the three components to two constants. 1 RRC (C-8) 12,RRC2 (C-9) 136 PAGE 137 137 This permits us to rewrite the variance equa tion in a more telling form with respect to M and N 1 2111 varsmcC pC NMNMNM (C-10) Now, it is more obvious that N and M are equivalent for the case of pC = pR since the function is only made up of sums and products of N and M Thus, the variance of separable Monte Carlo using a random empirical CDF is symmetric when the response and capacity have the same distribution shape and same standard deviation. PAGE 138 APPENDIX D MONTE C ARLO EFFICIENCY COMPARISON VIA ANALYTICAL EXAMPLE To explore the efficiency of the two methods, expressions for the probability of failure and required sample size were derived for uniform di stributions. For simplicity, assume that the random determinants of response and capacity produced a uniform distribution for each. Uniform distributions were chosen for conveni ence of calculations. Figure D-1 shows a general scenario of probability of failure for two uniform distributions. Figure D-1. General uniform probability density functions for response and capacity To reduce the number or parameters used to describe the relationshi p of the response and capacity, the overlap ratios of each distribution we re used. These parameters shown in Eq. (D-1) represent the probability of each of the random variables being in the failure region. RC RC R RR CCba ba p ba ba Cp (D-1) The upper and lower bounds of the uniform distribution are a and b, respectively. The parameters defined in Eq. (D-1) permit a convenien t representation of the probability of failure and variance. Consider the integral form of calcu lating the probability of failure in Eq. (6-9). Substituting the cumulative distribution function and probability density function in terms of pR and pC yields a discrete expression for pf. 211 2()()2R Rb CR C f RC RRCCRRCC aab ra p dr pp babababa (D-2) 138 PAGE 139 First, consider the variance of the crude Mont e C arlo method from Eq. (6-5). As a way to express the cost of a simulation, Eq. (6-5) was rewritten in terms of the number of samples required, Ncmc, to achieve a specified level of acc uracy or coefficient of variation. 22(1) 12 1f cmc fppRCp N pCVCVpp (D-3) This form better displays the inverse nature of sample size to probability of failure and the overlap ratios. Analyzing Eq .(D-3), it is apparent that for very small probability of failures, or very small pR and pC, a very large sample size is required. Similarly, the variance from the conditional expe ctation method (Eq. (6-22)) is rewritten in Eq. (D-4). 2 var()()uni cecfr Rr p Frpfrdr (D-4) 211 varR Rb uni C cef RRCC ara p pd Nbaba r (D-5) Integrating and substituting in the parameters in Eq. (D-1). 24 var1 3f uni ce Rp p Np (D-6) As mentioned before, the parameters associ ated with the random response are the only contributors to the variance in ce p From this result, give n uniform distributions (or approximations) of the response and capacity one can analytically predic t variance (or required number of samples) of a probability of failure simulation. 214 1 3ce pRN CVp (D-7) 139 PAGE 140 Considering small probabilities, pR and pC are typically small, so the second term in Eqs. (D-3) and (D-7) may be neglected. Then afte r some manipulation of the equations, another descriptive formula for efficiency analysis is shown in Eq. (D-8). 22 3ce R f cmcCN p p Np (D-8) Dissecting the components of the equation s hows when SMC is more efficient than standard Monte Carlo. First, the smaller the probability of failure gets, the smaller the number of samples are required for SMC with respect to cr ude MC. Also, the method is advantageous when more of the uncertainty comes from th e capacity rather than the response ( pR> pC). For example, when the response is deterministic, pR = 1. To graphically illustrate the effect of pf on both simulation methods, a plot of sample size vs. pf was generated in Fig. D-2. Figure D-2. Sample size of crude and se parable MC for uniform distributions with CVp = 1% and pR= pC For a better representation of the advantage of SMC at small probability of failures, the ratio of sample sizes were compared in Fi g. D-3 versus the tw o terms of interest, pf and pR/ pC. 140 PAGE 141 141 The actual plots use the ratio of Eqs. (D-3) and (D-7), whereas the estimated plots are from Eq. (D-8). Figure D-3. 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Literatu re review on mechanical reliab ility and probabilistic design. probabilistic structural analysis methods fo r select space propulsion system components (PSAM). NASA Contractor Report 3, 189-159. Wu, Y. T. (1994). "Computational methods for efficient structural reliability and reliability sensitivity analysis." AIAA J ., 32(8). 1717-1723. Yang, J.S. (1989). System reliability optimization of aircraft wings. PhD Dissertation, Virginia Polytechnic Institute, Blacksburg, Va. Zheng, Y., and Das, P.K. (2000). Improved re sponse surface method and its application to stiffened plate reliability analysis. Eng. Struct., 22, 544-551. PAGE 148 BIOGR APHICAL SKETCH Benjamin P. Smarslok was born in Pittsburgh, Pennsylvania in 1982. In 2004, he received his Bachelor of Science degree in mechanical engineering and a minor in engineering mechanics. While studying at Penn State, he also worked as a research assi stant at the Center for Innovative Sintered Products (CISP). The exposure to resear ch motivated him to attend graduate school to obtain his Ph.D. Ben was admitted into the direct Ph.D. program at the University of Florida in August 2004 under the guidance of Prof. Raphael Haft ka and Prof. Peter Ifju. During his Ph.D. studies, he also had the opportunity to perform re search at Ecole Nationale Suprieure des Mines in Saint-Etienne, France in May 2007, and do an internship at Savannah River National Laboratory in Aiken, South Carolina in the summer of 2008. His research interests include statistical analys is of uncertainty in composite properties and probabilistic techniques. |