reliability-based structural optimization using response surface approximations and probabilistic

136 Pages·2004·1.24 MB·English

RELIABILITY-BASED STRUCTURAL OPTIMIZATION USING RESPONSE
SURFACE APPROXIMATIONS AND PROBABILISTIC SUFFICIENCY FACTOR

By

XUEYONG QU

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2004
Copyright 2004

by

Xueyong Qu
This dissertation is dedicated to my lovely wife, Guiqin Wang.
ACKNOWLEDGMENTS
I want to thank Dr. Raphael T. Haftka for offering me the opportunity to complete
my Ph.D. study under his exceptional guidance. He provided the necessary funding to
complete my doctoral studies and support me to attend many academic conferences.
Without his patience, guidance, knowledge, and constant encouragement, this work
would not have been possible. Dr. Haftka made an immense contribution to this
dissertation and my academic growth, as well as my professional and personal life.
I would also like to thank the members of my supervisory committee: Dr. Peter G.
Ifju, Dr. Theodore F. Johnson, Dr. André I. Khuri, and Dr. Bhavani V. Sankar. I am
grateful for their willingness to review my research and provide constructive comments
that helped me to complete this dissertation. Special thanks go to Dr. David Bushnell for
his help with the PANDA2 program and stiffened panel analysis and design. Special
thanks go to Dr. Vicente J. Romero for many helpful discussions and collaboration in
writing papers.
Financial support provided by grant NAG-1-2177, contract L-9889 and grant
URETI from NASA is gratefully acknowledged.
My colleagues in the Structural and Multidisciplinary Optimization Research
Group at the University of Florida also deserve thanks for their help and many fruitful
discussions. Special thanks go to Palaniappan Ramu, Thomas Singer, and Dr. Satchi
Venkataraman for their collaboration in publishing papers.
My parents deserve my deepest appreciation for their constant love and support.
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Lastly, I would like to thank my beautiful and loving wife, Guiqin Wang. Without
her love, patience and support I would not have completed this dissertation.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS.................................................................................................iv
LIST OF TABLES.............................................................................................................ix
LIST OF FIGURES.........................................................................................................xiii
ABSTRACT.......................................................................................................................xv
CHAPTER

1 INTRODUCTION........................................................................................................1
Focus.............................................................................................................................2
Objectives and Scope....................................................................................................4

2 LITERATURE SURVEY: METHODS FOR RELIABILITY ANALYSIS AND
RELIABILITY-BASED DESIGN OPTIMIZATION..................................................6
Review of Methods for Reliability Analysis................................................................7
Problem Definition................................................................................................7
Monte Carlo Simulation........................................................................................7
Monte Carlo Simulation Using Variance Reduction Techniques.........................8
Moment-Based Methods.......................................................................................9
Response Surface Approximations......................................................................10
Reliability-Based Design Optimization Frameworks.................................................12
Double Loop Approach.......................................................................................12
Inverse Reliability Approach...............................................................................14
Design potential approach............................................................................15
Partial safety factor approach (Partial SF)...................................................16
Summary.....................................................................................................................17

3 RESPONSE SURFACE APPROXIMATIONS FOR RELIABILITY-BASED
DESIGN OPTIMIZATION........................................................................................19
Stochastic Response Surface (SRS) Approximation for Reliability Analysis............20
Analysis Response Surface (ARS) Approximation for Reliability-Based Design
Optimization..........................................................................................................21
Design Response Surface (DRS) Approximation.......................................................23

vi
Analysis and Design Response Surface Approach.....................................................24
Statistical Design of Experiments for Stochastic and Analysis Response Surfaces...25

4 DETERMINISTIC DESIGN OF COMPOSITE LAMINATES FOR CRYOGENIC
ENVIRONMENTS.....................................................................................................28
Introduction.................................................................................................................28
Composite Laminates Analysis under Thermal and Mechanical Loading.................29
Properties of IM600/133 Composite Materials..........................................................30
Deterministic Design of Angle-Ply Laminates...........................................................34
Optimization Formulation...................................................................................35
Optimizations without Matrix Cracking..............................................................36
Optimizations Allowing Partial Matrix Cracking...............................................39
Optimizations with Reduced Axial Load Ny......................................................39

5 RELIABILITY-BASED DESIGN OF COMPOSITE LAMINATES FOR
CRYOGENIC ENVIRONMENTS............................................................................41
Reliability-Based Design Optimization......................................................................41
Problem Formulation...........................................................................................41
Response Surface Approximation for Reliability-Based Optimization..............43
Analysis Response Surfaces................................................................................43
Design Response Surfaces...................................................................................45
Refining the Reliability-Based Design................................................................46
Quantifying Errors in Reliability Analysis..........................................................47
Effects of Quality Control on Laminate Design.........................................................48
Effects of Quality Control on Probability of Failure...........................................49
Effects of Quality Control on Optimal Laminate Thickness...............................50
Effects of Other Improvements in Material Properties...............................................51
Summary.....................................................................................................................54

6 PROBABILISTIC SUFFICIENCY FACTOR APPROACH FOR RELIABILITY-
BASED DESIGN OPTIMIZATION..........................................................................56
Introduction.................................................................................................................56
Probabilistic Sufficiency Factor.................................................................................60
Using Probabilistic Sufficiency Factor to Estimate Additional Structural Weight
to Satisfy the Reliability Constraint.................................................................62
Reliability Analysis Using Monte Carlo Simulation..................................................64
Calculation of Probabilistic Sufficiency Factor by Monte Carlo Simulation......66
Monte Carlo Simulation Using Response Surface Approximation.....................68
Beam Design Example...............................................................................................71
Design with Strength Constraint.........................................................................72
Design with Strength and Displacement Constraints..........................................75
Summary.....................................................................................................................79

vii
7 RELIABILITY-BASED DESIGN OPTIMIZATION USING DETERMINISTIC
OPTIMIZATION AND MULTI-FIDELITY TECHNIQUE.....................................80
Introduction.................................................................................................................80
Reliability-Based Design Optimization Using Sequential Deterministic Optimization
with Probabilistic Sufficiency Factor.....................................................................81
Reliability-Based Design Optimization Using Multi-Fidelity Technique with
Probabilistic Sufficiency Factor.............................................................................82
Beam Design Example...............................................................................................84
Reliability-Based Design Optimization Using Sequential Deterministic Optimization
with Probabilistic Sufficiency Factor.....................................................................86
Reliability-Based Design Optimization Using Coarse MCS with Probabilistic
Sufficiency Factor..................................................................................................87
Summary.....................................................................................................................88

8 RELIABILITY-BASED DESIGN OPTIMIZATION OF STIFFENED PANELS
USING PROBABILISTIC SUFFICIENCY FACTOR..............................................90
Introduction.................................................................................................................90
Aluminum Isogrid Panel Design Example.................................................................92
Reliability-Based Design Problem Formulation.................................................92
Uncertainties........................................................................................................94
Analysis Response Surface Approximation........................................................95
Design Response Surfaces...................................................................................97
Optimum Panel Design........................................................................................98
Composite Isogrid Panel Design Example.................................................................98
Deterministic Design.........................................................................................100
Analysis Response Surface Approximation......................................................101
Reliability-Based Design Optimization Using Sequential Deterministic
Optimization with Probabilistic Sufficiency Factor......................................103
Reliability-Based Design Optimization using DIRECT Optimization.....................106
DIRECT Global Optimization Algorithm.........................................................106
Reliability-Based Design Optimization Using Direct Optimization with Safety
Factor Corrected by Probabilistic Sufficiency Factor....................................108
Summary...................................................................................................................108

APPENDIX

A MATERIAL PROPERTIES OF IM600/133...........................................................110
B CONTOUR PLOTS OF THREE DESIGN RESPONSE SURFACE
APPROXIMATIONS AND TEST POINTS ALONG THE CURVE OF TARGET
RELIABILITY.........................................................................................................113
LIST OF REFERENCES.................................................................................................115
BIOGRAPHICAL SKETCH...........................................................................................120
viii
LIST OF TABLES
Table page

4-1 Transverse strains calculated for conditions corresponding to the onset of matrix-
cracking in the 90° plies of a quasi-isotropic (45/0/-45/90) in Aoki et al. (2000).33
2s
4-2 Transverse strains of an angle-ply laminate (±25) under the same loading
4S
condition as Table A1..............................................................................................34
4-3 Strain allowables for IM600/133 at –423°F.............................................................34
4-4 Optimal laminates for different operational temperatures: ε u of 0.0110.................37
2
4-5 Optimal laminates for temperature dependent material properties with ε u of 0.0110
2
(optimized for 21 temperatures)...............................................................................38
4-6 Optimal laminate for temperature dependent material properties allowing partial
matrix cracking: ε u of 0.011 for uncracked plies and 0.0154 for cracked plies......39
2
4-7 Optimal laminates for reduced axial load of1, 200 lb./inch by using load shunting
cables (equivalent laminate thickness of 0.005 inch)...............................................40
5-1 Strain allowablesa for IM600/133 at –423°F............................................................42
5-2 Coefficients of variation (CV) of the random variables...........................................42
5-3 Range of design variables for analysis response surfaces........................................44
5-4 Quadratic analysis response surfaces of strains (millistrain)....................................44
5-5 Design response surfaces for probability of failure (Probability calculated by Monte
Carlo simulation with a sample size of 1,000,000)..................................................45
5-6 Comparison of reliability-based optimum with deterministic optima......................46
5-7 Refined reliability-based design [±θ] (Monte Carlo simulation with a sample size
S
of 10,000,000)..........................................................................................................47
5-8 Comparison of probability of failure from MCS based ARS and CLT....................47
5-9 Accuracy of MCS.....................................................................................................48
ix
5-10 Effects of quality control of ε u on probability of failure for 0.12 inch-thick (±θ)
2 S
laminates...................................................................................................................49
5-11 Effects of quality control of ε u, ε l, ε l, and γ on probability of failure of 0.12
1 1 2 12
inch-thick (±θ) laminates........................................................................................50
S
5-12 Effects of quality control of E , E , G , µ , T , α , and α on probability of
1 2 12 12 zero 1 2
failure of 0.12 inch-thick (±θ) laminates................................................................50
S
5-13 Effects of quality control of ε u on probability of failure for 0.1 inch-thick (±θ)
2 S
laminates...................................................................................................................51
5-14 Effects of quality control of ε u on probability of failure for 0.08 inch-thick (±θ)
2 S
laminates...................................................................................................................51
5-15 Sensitivity of failure probability to mean value of ε u (CV=0.09) for 0.12 inch-thick
2
l(±θ) aminates.........................................................................................................52
S
5-16 Sensitivity of failure probability to CV of ε u ( E(ε u)=0.0154 ) for 0.12 inch-thick
2 2
(±θ) laminates.........................................................................................................52
S
5-17 Maximum ε (millistrain) induced by the change of material properties E , E , G ,
2 1 2 12
µ , T , α , and α for 0.12 inch-thick [±25°] laminate.......................................54
12 zero 1 2 S
5-18 Probability of failure for 0.12 inch-thick [±25°] laminate with improved average
S
material properties (Monte Carlo simulation with a sample size of 10,000,000)....54
6-1 Random variables in the beam design problem........................................................62
6-2 Range of design variables for design response surface............................................72
6-3 Comparison of cubic design response surface approximations of probability of
failure, safety index and probabilistic sufficiency factor for single strength failure
mode (based on Monte Carlo simulation of 100,000 samples)................................73
6-4 Averaged errors in cubic design response surface approximations of probabilistic
sufficiency factor, safety index and probability of failure at 11 points on the curves
of target reliability....................................................................................................74
6-5 Comparisons of optimum designs based on cubic design response surface
approximations of probabilistic sufficiency factor, safety index and probability of
failure.......................................................................................................................75
6-6 Comparison of cubic design response surface approximations of the first design
iteration for probability of failure, safety index and probabilistic sufficiency factor
for system reliability (strength and displacement)...................................................76
x