A univariate decomposition method for higherorder reliability analysis and design optimization

Transcription

1 Unversty of Iowa Iowa Research Onlne Theses and Dssertatons 2006 A unvarate decomposton method for hgherorder relablty analyss and desgn optmzaton Dong We Unversty of Iowa Copyrght 2006 Dong We Ths dssertaton s avalable at Iowa Research Onlne: Recommended Ctaton We, Dong. "A unvarate decomposton method for hgher-order relablty analyss and desgn optmzaton." PhD (Doctor of Phlosophy) thess, Unversty of Iowa, Follow ths and addtonal works at: Part of the Mechancal Engneerng Commons

2

3

4

5 A UNIVARIATE DECOMPOSITION METHOD FOR HIGHER-ORDER RELIABILITY ANALYSIS AND DESIGN OPTIMIZATION by Dong We A thess submtted n partal fulfllment of the requrements for the Doctor of Phlosophy degree n Mechancal Engneerng n the Graduate College of The Unversty of Iowa July 2006 Thess Supervsor: Professor Sharf Rahman

6

7 To L and Malcom

8 ACKNOWLEDGEMENTS I would lke to thank my advsor, Professor Sharf Rahman, for hs professonal gudance and fnancal support durng my Ph.D studes. I apprecate hs valuable advce and encouragement n academc and professonal aspects n my lfe. I would also lke to thank Professor Kyung K. Cho, Professor M. Asghar Bhatt, Professor Ja Lu and Professor Pavlo Krokhmal, for ther wllngness to revew my research, helpful comments, and servng on my thess commttee. Specal thanks are due to Mr. R. Watkns for hs edtoral help to refne ths thess. I want to acknowledge many former and current colleagues of structural relablty group n Center for Computer-Aded Desgn: Dr. N.H. Km, Dr. B.N. Rao, Dr. H. Xu, Dr. T. Zhang, Dr. B.D. Youn, Mr. L. Du and many others, for ther unselfsh help and frutful dscussons. Last but not least mportant, I would lke to thank all my famly members for ther love and support.

9 ABSTRACT The objectve of ths research s to develop new stochastc methods based on most probable ponts (MPPs) for general relablty analyss and relablty-based desgn optmzaton of complex engneerng systems. The current efforts nvolves: () unvarate method wth smulaton for relablty analyss; (2) unvarate method wth numercal ntegraton for relablty analyss; (3) mult-pont unvarate for relablty analyss nvolvng multple MPPs; and (4) unvarate method for desgn senstvty analyss and relablty-based desgn optmzaton. Two MPP-based unvarate decomposton methods were developed for component relablty analyss wth hghly nonlnear performance functons. Both methods nvolve novel functon decomposton at MPP that facltates hgher-order unvarate approxmatons of a performance functon n the rotated Gaussan space. The frst method entals Lagrange nterpolaton of unvarate component functons that leads to an explct performance functon and subsequent Monte Carlo smulaton. Based on lnear or quadratc approxmatons of the unvarate component functon n the drecton of the MPP, the second method formulates the performance functon n a form amenable to an effcent relablty analyss by multple one-dmensonal ntegratons. Although both methods have comparable computatonal effcency, the second method can be extended to derve analytcal senstvty of falure probablty for desgn optmzaton. For relablty problems entalng multple MPPs, a mult-pont unvarate decomposton method was also developed. In addton to the effort of dentfyng the MPP, the unvarate methods requre a small number of exact or numercal functon evaluatons at v

10 selected nput. Numercal results ndcate that the MPP-based unvarate methods provde accurate and/or computatonally effcent estmates of falure probablty than exstng methods. Fnally, a new unvarate decomposton method was developed for desgn senstvty analyss and relablty-based desgn optmzaton subject to uncertan performance functons n constrants. The method nvolves a novel unvarate approxmaton of a general multvarate functon n the rotated Gaussan space; analytcal senstvty of falure probablty wth respect to desgn varables; and standard gradentbased optmzaton algorthms. In both relablty and senstvty analyses, the proposed effort has been reduced to performng multple one-dmensonal ntegratons. Numercal results ndcate that the proposed method provdes accurate and computatonally effcent estmates of the senstvty of falure probablty and leads to accurate desgn optmzaton of uncertan mechancal systems. v

11 TABLE OF CONTENTS LIST OF TABLES... LIST OF FIGURES... x x CHAPTER. INTRODUCTION.... Background and Motvaton....2 Objectve of the Study Organzaton of the Thess STATE-OF-THE-ART REVIEW Generaltes Probablty Space Random Varables Random Vector Relablty Analyss Basc Random Varables and Lmt State Functon Components and System Relablty Analyss Early Relablty Analyses Methods of Relablty Analyss Mean-Value Methods Mean-Value Method Advanced-Mean Value Method Advanced-Mean Value Method Frst-, Second-, and Hgher-Order Relablty Methods Transformaton Independent Random Varables Dependent Random Varables-Rosenblatt Trnsformaton Dependent Random Varables-Nataf Tranformaton Transformaton between Dependent and Independent Normal Varables Most Probable Pont (MPP) Early Approaches Hasofer-Lnd Method Improved Hasofer-Lnd Method Others Frst-Order Relablty Method Second-Order Relablty Method General Quadratc Approxmaton Parabolc Approxmaton v

12 Falure Probablty Evaluaton-Asymtopotc Soluton Least-Squares' Non-Asymptotc Soluton Other Non-Asymptotc Solutons Hgher-Order Relablty Method Multple MPP Problems Multpont FORM/SORM System Relablty Method Samplng-based MPP Search Multple Lnearzaton "Barrer" Optmal Method Global Response Surface Method Smulaton Methods Drect Monte Carlo Smulaton Importance Samplng Stratfed Samplng Drectonal Samplng Latn Hypercube Samplng Markov Chan Monte Carlo Smulaton Response Surface Methods Basc Prncple of Response Surface Method Buldng the Response Surface Varous Types of Response Surface Approaches Mean-Pont-based Decomposton Methods Mult-Varate Functon Decomposton Unvarate Approxmaton Bvarate Approxmaton Generalzed S-Varate Approxmaton Remarks Response Surface Generaton Monte Carlo Smulaton Senstvty Analyss Dervatve of Relablty Index-One Parameter Dervatve of FORM Approxmaton to Falure Probablty Relablty-Based Desgn Optmzaton Introducton FORM-Based Optmzaton Methods Double-Loop Approach Sngle-Loop Approach Sequental Methods Smulaton-based Optmzaton Methods Sample Average Approxmaton Response Surface Approxmaton Others Summary of Revew and Future Research Needs Conclusons form Exstng Research Need for Fundamental Research MPP-BASED UNIVARIATE METHOD WITH SIMULATION v

13 3. Multvarate Functon Decomposton at MPP Unvarate Approxmaton Bvarate Approxmaton Generlzed S-varate Approxmaton Remarks Response Surface Generaton Monte Carlo Smulaton Numercal Examples Example Set I -Elementary Methematcal Functons Example Set II - Sold Mechancs Problems Fatgue Relablty Applcatons Problem Defnton and Input Fatuge Relablty Analyss Results MPP-BASED UNIVARIATE METHOD WITH NUMERICAL INTEGRATION Unvarate Decomposton at MPP Unvarate Integraton for Falure Probablty Analyss Lnear Approxmaton of yn ( v N) Quadratc Approxmaton of yn ( v N) Unvarate Integraton Computatonal Effort and Flow Numercal Examples Example Set I - Explct Performance Functons Example Set II - Implct Performance Functons MULTIPLE MPP PROBLEMS Performance Functon Decomposton at the mth MPP Mult-Pont Unvarate Decomposton Method Unvarate Decomposton of Performance Functon Lagrange Interpolaton and Return Mappng Monte Carlo Smulaton Computatonal Effort Numercal Examples Example - Mathematcal Functons wth Gaussan Random Varables Example 2 - Mathematcal Functons wth Non- Gaussan Random Varables Example 3 - Sesmc Dynamcs of a 0-story Buldng-TMD System RELIABILITY-BASED DESIGN OPTIMIZATION BY UNIVARIATE DECOMPOSITION Relablty-Based Desgn Optmzaton Generalzed RBDO Problem v

14 6..2 Specal RBDO Problem Unvarate Decomposton Method Relablty Analyss Desgn Senstvty Analyss Unvarate Numercal Integraton for Relablty and Senstvty Analyss Computatonal Flow and Effort Numercal Examples Example Set I - Desgn Senstvty Analyss Example Set II - Relablty-Based Desgn Optmzaton CONCLUSIONS AND RECOMMENDATIONS Summary and Conclusons MPP-Based Unvarate Method wth Smulaton MPP-Based Unvarate Method wth Numercal Integraton Mult-Pont Unvarate Decomposton Senstvty Analyss and RBDO wth Unvarate Method Recommendatons for Future Work APPENDIX A. APPROXIMATE EVALUATIONS OF σ m AND ε APPENDIX B. POROSITY FIELD AND DEFECT SIZE APPENDIX C. BARRIER METHOD APPENDIX D. SENSITIVITY ANALYSIS BASED ON FORM/SORM REFERENCES x

15 LIST OF TABLES Table 3. Falure probablty for cubc performance functon Falure probablty for quartc performance functon Statstcal propertes of random nput for rotatng dsk Falure probablty of rotatng dsk Falure probablty of ten-bar truss structure Statstcal propertes of random nput for an edge-cracked plate Statstcal propertes of random nput for lever arm Probablty of fatgue falure of lever arm at locatons,2 and Falure probablty of cubc perfomance functon Falure probablty of quartc performance functon Statstcal propertes of random nput for rotatng dsk Falure probablty of rotatng dsk Falure probablty of ten-bar truss structure Statstcal propertes of random nput for an edge-cracked plate Frame element propertes Statstcal propertes of random nput for frame structure Correlaton coeffcents of random nput for frame structure Falure probablty of frame structure Falure probablty of parabolc performance functon Falure probablty of cubc perfomance functon Falure probablty of quartc performance functon Falure probablty of Example 2 wth transformaton T Falure probablty of Example 2 wth transformaton T x

16 5.6 Statstcal propertes of random varable nput for Example Falure probablty of Example Gradents of two mathematcal constrant functons Gradents of the constrant n 0-bar truss Computatonal efforts for 0-bar truss Optmzaton results by varous methods for mathematcal functons Statstcal propertes of random nput for cantlever beam Optmzaton results by varous methods for the cantlever beam Falure probabltes for cantlever beam Optmzaton results by varous methods for the 0-bar truss...94 x

17 LIST OF FIGURES Fgure. Uncertanty propagaton and probablstc analyss MPP at the 2D standard normal space Schematc flowchart for HORM Multple MPPs problems Performance functon approxmatons by varous methods Approxmate performance functons by varous methods Rotatng annular dsk subject to angular velocty A ten-bar truss structure An edge-cracked plate subject to mxed-mode deformaton Probablty of fracture ntaton n an edge-cracked plate A wheel loader under cyclc loads Fnte element analyss of a lever arm Falure lfe-based relablty ndex contour of lever arm Porosty feld of lever arm from castng smulaton Flowchart of the MPP-based unvarate method wth numercal ntegraton Probablty of fracture ntaton n an edge-cracked plate A three-span, fve-story frame structure subject to lateral loads A performance functon wth multple most probable ponts Flowchart of the mult-pont unvarate decomposton method Quadratc lmt-satte surface n Case I (Example ) Cubc lmt-state surface n Case II (Example) Quartc lmt-state surface n Case III (Example ) Lmt-state surface of Example x

18 5.7 A 0-story buldng-tmd system (Example 3) Normalzed pseudo-acceleraton response spectrum Varous approxmatons of the performance functon of kth constrant Flowchart of the proposed RBDO process A 0-bar truss structure (Repeatng Fgure 3.4) Hstory of mathematcal objectve functon A cantlever beam subjected to end loads Hstory of objectve functon for cantlever beam Hstory of objectve functon for 0-bar truss Intal desgn of torque arm geometry at mean values of shape parameters Locatons of ponts for prescrbng constrants Contour of von Mses stress at mean values of shape parameters for ntal desgn Contour of von Mses stress at mean values of shape parameters for RBDO desgn Optmzaton Hstory of objectve functon for torque arm Contour of von Mses stress at mean values of shape parameters for rskgnorng optmum desgn C. Sucessve uses of bulges to fnd multple MPPs...28 C.2 Profle of the bulge...29 C.3 Defnton of cone contanng the bulge...29 x

19 CHAPTER INTRODUCTION. Background and Motvaton Balancng the requrements of relablty and cost s a dlemma for engneers n the desgn of complex engneerng structures. Smply ncreasng the cost or weght of a structure does not always yeld an mprovement n relablty. Although tradtonal determnstc analyss and optmzaton technques are well defned, they provde lttle or no assstance to a desgner n consstently characterzng relablty. In other words, there s no systematc way for a determnstc desgn approach to predct structural safety probablstcally. Furthermore, optmzaton technques based on a determnstc approach usually lead to a cost and/or weght savngs, but an unrelable and/or unsafe desgn. There are many uncertantes, such as loads, materal propertes, geometry, and manufacturng tolerances that exst n engneerng structures. The ablty to accurately characterze and propagate these uncertantes s ncreasngly mportant n order to evaluate ther effects on the probablstc response and relablty of complex engneerng structures. Fgure. shows uncertanty propagaton and probablstc analyss n a physcs-based smulaton of such complex systems. Once desgners are able to model uncertantes and predct ther effects on response, relablty-based desgn optmzaton or robust desgn optmzaton can be conducted to solve the dlemma of cost and relablty. Unfortunately, stochastc methods embedded n current relablty-based/robust desgn processes are naccurate and/or computatonally neffcent when: () the nput-

20 2 output relatonshp s hghly nonlnear; (2) the number of nput random varables or felds s large; and (3) there are large statstcal varatons n nput. For example, the most common approach to predct the falure probablty nvolves frst- and second- order relablty methods (FORM/SORM), whch are not adequate for hghly nonlnear problems. Smulaton methods, whch are usually employed for obtanng benchmark results, are not computatonally effcent, and are not sutable to beng embedded n optmzaton desgn processes. A new stochastc method s thus needed wth greater accuracy and/or better effcency than tradtonal methods. Such a method should be able to be ntegrated nto the relablty-based optmzaton process to solve realstc desgn problems..2 Objectves of the Study The prmary objectve of ths study s to develop a new stochastc method to solve hghly nonlnear relablty problems, referred to as the most probable pont (MPP)-based decomposton method, for relablty analyss and subsequent desgn optmzaton of complex engneerng systems. The followng four research drectons have been pursued: () development of an MPP-based unvarate method wth smulaton; (2) development of an MPP-based unvarate method wth numercal ntegraton; (3) development of an MPP-based unvarate method for solvng multple MPPs problems; (4) senstvty analyss and relablty-based desgn optmzaton nvolvng the new unvarate method. The proposed MPP-based decomposton method s new and wll address hghly nonlnear nput-output transformaton, unlmted number of dependent or correlated

21 3 random varables, and a large uncertanty of random nput. The method nvolves a novel decomposton at the MPP that facltates a unvarate approxmaton of the general multvarate functon, and probablty estmaton by smulaton or numercal ntegraton. Ths unvarate approxmaton can be hghly nonlnear, whch ncludes all hgher-order unvarate terms, so t should provde better approxmaton around MPP than the lnear (FORM) or quadratc (SORM) approxmaton. In addton to the effort of dentfyng the MPP, the method developed requres a small number of exact or numercal evaluatons of performance functon at a selected nput. Hence, the proposed method wll not only provde accurate solutons, but also create computatonally effcent results compared wth exstng methods..3 Organzaton of the Thess Chapter 2 presents mathematcal generaltes and notatons requred by relablty analyss and a state-of-the-art revew of methods for relablty analyss and relabltybased-optmzaton-desgn. The need for fundamental research s emphaszed. Chapter 3 presents an MPP-based unvarate method wth smulaton. The followng topcs are dscussed: multvarate functon decomposton at MPP, response surface generaton by Lagrange nterpolaton, and Monte Carlo smulaton. Chapter 4 proposes an MPP-based unvarate method wth numercal ntegraton. Ths method nvolves unvarate decomposton of the performance functon and unvarate ntegraton for falure probablty estmaton.

22 4 Chapter 5 dscusses the extenson of the MPP-based unvarate method n multple MPPs problems. Ths applcaton nvolves a global optmzaton method entalng the barrer method, unvarate decomposton at multple MPPs, and system relablty analyss. Chapter 6 presents relablty-based desgn optmzaton wth unvarate decomposton. The proposed RBDO process nvolves relablty analyss by unvarate decomposton, desgn senstvty analyss by unvarate decomposton, and standard gradent-based desgn optmzaton. Chapter 7 provdes conclusons from the present work and recommendatons for the future studes.

23 5 PDF of Input Random Varable/ Feld Data Analyss Statstcs Dstrbuton Calculate Relablty CDF Second Moments Confdence Interval Importance Factors Math. Model Fatgue/Fracture Crashworthness MEMS Mcroelectroncs NVH PDF of Output Random Varable/ Feld Probablstc Methods FORM SORM AMV Monte Carlo Imp. Samplng Dr. Smulaton Valdaton Statstcs Dstrbuton Fgure. Uncertanty propagaton and probablstc analyss

24 6 CHAPTER 2 STATE-OF-THE-ART REVIEW 2. Generaltes Ths secton ntroduces mathematcal generaltes and notatons that are requred by the probablstc methods presented n subsequent sectons. 2.. Probablty Space The observaton of a random phenomenon s classcally referred to as a tral. All possble outcomes of a tral from the sample space of the phenomenon are denoted by Ω. An event s defned as a subset of Ω contanng all outcomes outcome n one event, then the event s null set, and denoted by ω Ω. If there s no. Events A and B are dsjont events f A B =. Events A and B are equal f and only f A B and B A. Probablty theory ams at assocatng numbers wth events,.e., ther probablty of occurrence. Let P denote the probablty measure. An σ-algebra F s a nonempty collecton of subsets of Ω such that the followng holds: () The empty set s n F. (2) If A s n F, then so s the complement of A. (3) If A, =, 2, s a sequence of elements of F, then the unon of A s n F. The probablty space constructed by these notons s denoted by a trple ( Ω, F,P) Random Varable Consder a probablty space ( Ω, F,P) and a real-valued random varable X defned on ths space. The cumulatve dstrbuton functon (CDF), denoted by FX ( x ),

25 7 of the random varable X s defned by the mappng X : Ω and the probablty measure P,.e., F ( x) P( X x). If F ( x) s contnuous n, then the probablty X X densty functon (PDF), denoted by f ( x ), s ( ) ( ) X f x = df x dx. X X ( ) l For functon g X = X, where g s contnuous n, the lth statstcal moment or moment of order l of X s defned as + ( ) l l ml E X x fx x dx, (2.) where s the expectaton operator. If functon g X = X m s consdered, then the results n Equaton (2.) defne the central moments of order l of X. The moments µ X m, E ( ) ( ) σ [ X µ ] 2 X X E, ν σ µ, [ ] 3 3 γ X µ σ X X 3 l E, and X X 4 [ ] 4 4 X γ E µ σ X X are called the mean, varance, coeffcent of varaton, coeffcent of skewness, and coeffcent of kurtoss, respectvely. These moments can be calculated by drect ntegraton, as expressed n Equaton (2.). The postve square root of the varance s called standard devaton and s denoted by σ X Random Vector Let X = { },, T N X X N be a real-valued random vector on the probablty space ( Ω, F,P). The jont cumulatve dstrbuton functon, denoted by ( x), of X s F X defned by the mappng X : Ω N and the probablty measure P,.e.,

26 ( = ) N X ( x) { }. If F ( x) s such that ( ) = N ( ) F P X x ( ) then x s called the jont probablty densty functon of X. f X Consder contnuous functon g( ) = = X 8 f X x FX x x x N exsts, N l X X, where l 0, =,, N are ntegers. The lth statstcal moment or moment of order N l = l of X s = m ( X ) E E. (2.2) N l,, l l g X N = For example, the frst and second moment propertes, such as mean µ X of correlaton ρ of (, X X, covarance γ j of (, j ) j j ) easly obtaned from Equaton (2.2) as [ X ] m,, µ X l N X X, and varance σ 2 of X X, X, can be E = l for l =, l = 0, j, (2.3) j and ρ E XX = m for l = lj =, lk = 0, k, j, (2.4) j j l,, l N ( X )( X ) j γ E µ µ = ρ µ µ j X j X j X X j, (2.5) 2 2 σ X X µ γ X = E. (2.6) 2.2 Relablty Analyss 2.2. Basc Random Varables and Lmt State Functon Consder a system wth uncertan mechancal characterstcs that s subject to random loads. Denote by X an N-dmensonal vector of basc random varables wth

27 9 components {X,, X N } descrbng the randomness n geometry, materal propertes and loadng. To assess the relablty of the structural system, a lmt state functon g that depends on basc random vector X defned as follows: g ( X ) > 0, whch defnes the safe state of the structure, and g ( ) 0 X defnes the falure state. The values of X satsfyng g ( X ) = 0 defne the lmt state surface of the structure n the orgnal space Component and System Relablty Analyses A fundamental problem n tme-nvarant component relablty analyss entals calculaton of a mult-fold ntegral (Madsen, et. al, 986) ( ) g ( x) < 0 P 0 ( F P g X < = fx x) dx, (2.7) where X { X,, X } T N = N s a real-valued, N-dmensonal random vector defned on a probablty space ( Ω, F,P ) comprsng the sample space Ω, the σ-feld F, and the probablty measure P; g(x) s the performance functon, such that g ( x) < 0 represents the falure doman; PF s the probablty of falure; and f X ( x) s the jont probablty densty functon of X, whch typcally represents loads, materal propertes, and geometry, respectvely. In general, any engneerng system has to satsfy more than one performance crteron. System relablty evaluatons are used to consder multple falure modes and/or multple component falures. A complete relablty analyss ncludes both

28 0 component- and system-level estmates. If there are M falure modes or multple component falures, seres, parallel and mxed system are descrbed by seres system M F = F, (2.8) = parallel system parallel systems n seres and seres systems n parallel F = F, (2.9) = F = F, m = M M l m l = j = j =, (2.0) l m l, = j= j =, (2.) F = F m = M where F g ( X ) < 0 s the falure event of the th system component, g ( X ) s the th performance functon, and F s the system falure event. If Equaton (2.0) s a mnmal set, t can be denoted by a mnmal cut set. Cut sets are mnmal f they contan no other cut sets as a genune subset. Analogously, Equaton (2.) s called a te set. Such sets are mnmal f no te set contans another te set as a genune subset. For a seres system made of M ndependent events, the falure probablty s gven by M M ( )) PF = P F = P F = =. (2.2) (

29 Smlarly, for a parallel system consstng of M ndependent events, the falure probablty s In the case of fully dependent events, and M M F = = = = ( ) P P F P F { ( )}. (2.3) P M F = P F max P F,,, M = = =, (2.4) { ( )} P M F = P F mn P F,,, M = = =. (2.5) For arbtrary cases of seres system (falure doman s gven by the ntersecton of componental falure domans), the falure probablty can be estmated by P M F ( β ; R), (2.6) = P Φ F = M Φ () { } where s the jont CDF of an M-dmensonal Gaussan vector, β = β,,β s a n vector of relablty ndces obtaned by FORM/SORM (wll be dscussed n secton M T 2.4.2) for each falure event, and R s the correlaton matrx. Furthermore, f the falure events can be reduced to the mnmal cut set and the cut sets all have small falure probabltes, then the narrow probablty bounds can be derved as PFL, PF P FU,, (2.7a) where the lower bound P F,L and the upper bound P F,U are M M P = P( F ) + max 0, P( F) P( F F ) (2.7b) FL, j = 2 = 2

30 2 and { { )} j< } M ( ) ( ) max ( P = P F + P F P F F. (2.7c) FU, j = Early Relablty Analyses Early structural relablty analyss amed at determnng the falure probablty n terms of second moment statstcs of resstance and load varables. Suppose that performances of a structural system can be lumped nto two random varables denoted by resstance R and load S respectvely. The safety margn s defned by Cornell s relablty ndex (Cornell, 969) s then defned by Z R S. (2.8) µ Z β C =, (2.9) σ Z where µ Z and σz are mean and standard devaton respectvely of Z. It can be gven the followng nterpretaton: f R and S are jontly normal, so s Z. The falure probablty s gven by Z Z Z PF = P( Z 0) = P µ µ Φ( β σz σz C ), (2.20) where Φ () s the standard normal cumulatve dstrbuton functon. In ths case, β C can be descrbed as a functon of the second moment statstcs of R and S, gven by µ R µ S β C =, (2.2) σ +σ 2 ρ σ σ 2 2 R S RS R S

31 3 where µ, R σ R are mean and standard devaton respectvely of R, µ S, σ S are mean and standard devaton respectvely of S, and ρ RS s the correlaton coeffcent of R and S. The general case would be that Z s a lmt state functon of random vector X, X, where {,, } T N X = X X N, and the mean vector Z g( ) µ X and covarance matrx R are known. If g-functon s nonlnear, usng Taylor expanson around the mean and only keepng the lnear term wll lead to the so-called mean value frst order relablty ndex Where { } β = MVFOSM,, T = X X N. g ( g ) ( µ ) T X R g X= µ X X= µ X. (2.22) Methods of Relablty Analyss For most practcal problems, the exact evaluaton of the ntegral n Equaton (2.7), ether analytcal or numercal, s not possble because N s large, ( x) s generally non-gaussan, and g(x) s hghly nonlnear functon of x. Therefore, some approxmaton and smulaton methods have been developed, whch wll be dscussed n detal n subsequent sectons. These methods nclude mean-value methods, frst-, second-, and hgher-order relablty methods, smulaton methods, response surface methods, and recently developed decomposton methods. f X

32 4 2.3 Mean-Value Methods Usng mean values as the approxmaton pont s a conventonal method for estmatng the mean and standard devaton of the response, and s the basc dea behnd mean-value methods. These methods usually provde an approxmate CDF analyss Mean-Value Method Assumng that a Z-functon s contnuous and smooth around the mean-values pont, the frst-order Taylor s seres expanson s N Z Z ( X ) = Z( µ X ) + ( X ). (2.23) MV µ X = X X = µ X X = { X,, X } T N N T where, and µ X = {µ X,,µ X N } s the mean vector of X. Snce the ZMV functon s lnear and explct, ts CDF, as well as the relablty analyss, can be computed effectvely. For nonlnear g-functons, the soluton based on (2.23) s, n general, not adequately accurate. Hgher-order expansons need to be consdered, for example, the second-order approxmatons and Z Z Z X N 2 2 MV 2( X) = MV( X) + 2 ( µ X ), (2.24) 2 = X X = µ X N 2 Z Z ( X) = Z ( X) + ( X µ )( X µ MV 3 MV 2 X X j, j= X X j j X = µ X ), (2.25) where Z MV2 and Z MV3 are partal and full second order Taylor expansons. Thrd and hgher-order approxmatons are not recommended because of a lack of effcency and

33 5 numercal ssues. Based on Equaton (2.23), or (2.24) and (2.25), the mean value probablstc soluton s defned as mean-value (MV) method Advanced-Mean Method The Advanced Mean-Value (AMV) method was proposed by Wu (990) prmarly to mprove the MV soluton wth slghtly more computatonal effort. By usng a smple correcton term, AMV compensates for the expanson truncaton error. The AMV model can be smply expressed by ( ) = ( ) + ( ) Z X Z X H Z, (2.26) AMV MV MV where H( Z MV ) s defned as the correcton term for hgher order expanson terms. H( Z MV ) denotes the dfference between the exact value of Z computed at the MPP, and the approxmaton of Z computed at the MPP determned by the MV method. The accuracy of AMV depends on the accuracy of the approxmate MPP Advanced-Mean Value Method+ The AMV procedure can be consdered an MV method n the frst teraton when the lnearzaton s performed at the mean pont. If subsequent teratons are carred out to mprove results, the AMV procedure becomes the so-called AMV+. The AMV+ procedure uses the MPP, but not the mean pont n the orgnal x-space as the expanson pont n subsequent teratons (Wu et al., 994). Iteratons wll contnue to perform untl the approxmate MPP converges to the exact value.

34 6 2.4 Frst-, Second-, and Hgher-Order Relablty Methods Independent Random Varables 2.4. Transformaton Consder a random component X wth CDF F ( x ), =,, N. Let U be a standard normal random varable wth ts CDF Φ ( u ). From the defnton X If ( ) [ ] F x P X x = p, (2.27) X ( ) P[ ] Φ u U u = p, (2.28) then the mappng between x and u can be obtaned from whch yelds or the nverse mappng ( u ) F ( x ) Φ, (2.29) X ( ) =Φ (2.30) u FX x ( ) x = FX Φ u. (2.3) As long as F X ( x ) can be nverted, ether analytcally or numercally, a performance functon descrbed n the x-space can easly be mapped onto u-space Dependent Random Varables Rosenblatt Transformaton Consder an N-dmensonal random vector X wth a generc jont dstrbuton functon F X ( x ). Let T : X U denote a transformaton from x-space to u-space,

35 7 where U s an N-dmensonal standard Gaussan random vector. Accordng to Rosenblatt (952), the transformaton s gven by u = Φ FX ( x ) u2 = Φ FX ( x 2 2 x) T :, (2.32) un = Φ FX ( x, 2,, ) N N x x x N where FX ( x x, x2,, x ), = 2,, N s the CDF of X condtonal on X = x, X2 = x2,, X = x and ( ) Φ s the CDF of a standard Gaussan random varable. The condtonal dstrbuton functon FX ( x x, x2,, x ) can be obtaned from x f XX 2 X ( x, x2,, x, ) ξ dξ FX ( x, 2,, ) x x x =, (2.33) f ( x, x,, x ) XX 2 X 2 where f ( x, x,, x ) s the jont probablty densty functon of XX 2 X 2 { X, X2,, X } T. The nverse transformaton can be obtaned n a stepwse manner as T [ ( )] x = FX Φ u x2 = FX Φ( 2 2 ) : u x. (2.34) xn = FX Φ( u, 2,, ) N n x x xn Dependent Random Varables Nataf Transformaton Consder a dependent random vector X, for whch the margnal cumulatve dstrbuton functons F ( x ), =,, N and the correlaton coeffcent matrx X

36 { ρ j } P = are known. It may have been descrbed by an approxmate but completely X specfed jont probablty dstrbuton functon F X ( x ). X may also be transformed to the standard normal random vector Y n y-space, gven by 8 y ( x ) F. (2.35) =Φ X where Y = { Y,, Y N } T s an N-dmensonal standard normal random vector wth jont probablty densty functon φn (, ) correlaton coeffcent matrx P { ρ Y = j } y P havng zero means, unt standard devatons, and Y. Then, gven the usual rules for transformaton of random varables, the approxmate jont densty functon 962) f X ( x) n x-space s (Nataf, ( ) f ( x X ) = φ N y, P Y J (2.36) wth J ( y, y2,, yn ) (,,, ) ( ) ( 2) ( ( ) ( ) φ( ) f X x f X x f 2 X N x N = = x x x φ y φ y y 2 N 2 N ). (2.37) To solve Y P n Equaton (2.36), consder any two random varables (, j) correlaton coeffcent between them as where Z = ( X ) X X X and the γj ρ j = = ZZ j = zzjφ2 ( y, yj; ρ j ) d j σ σ E ydy, (2.38) µ σ X X j obtaned from the known X = { ρ j }. Here the correlaton coeffcent matrx X P { ρ Y = j } P teratvely from (2.38). can be

37 9 Once P Y s determned for any par of (, j) X X, Equaton (2.35) can be used to obtan the correlated standard normal dstrbuton n y-space. Furthermore, an orthogonal transformaton can be used to obtan ndependent standard normal dstrbuton n u-space, whch wll be dscussed n the followng Transformaton between Dependent and Independent Normal Varables Let = ( X,, X N ) X be an N-dmensonal normal random vector wth jont probablty densty functon φ (, ) n xc havng mean vector µ X, and covarance matrx X C X. Let U = ( U,, UN ) be an N-dmensonal ndependent standard random vector. Then, the transformaton between X and U can be expressed by X = AU + µ X. (2.39) Snce C X s postve defnte and symmetrc, the orthogonal transformaton matrx ( T A = A ) can be defned by T CX = AA. (2.40) Because C X s known, Equaton (2.40) can be obtaned by Cholesky decomposton Most Probable Pont (MPP) A most probable pont (MPP) s defned as the pont u on the lmt state surface closest to the orgn n standard normal space (Fgure 2.). Ths pont leads to the defnton of the relablty ndex β as

38 20 The determnaton of MPP defned by β = u. (2.4) u can be formulated as a constraned optmzaton problem, mn u N u R, (2.42) s. t. g = 0 U ( u) where N = u u s the Eucldean L 2 -norm of the N-dmensonal vector u and ( ) = 2 g U u s the transformed performance functon n u-space Early Approaches The constraned optmzaton problem defned n (2.42) s equvalent to N u R, λ R ( u λ) mn L,, (2.43) where λ s the Lagrange multpler and 2 L( u, λ ) = u +λg U ( u ). (2.44) 2 Assumng the optmzaton soluton s (, λ ) functon nvolved, the partal dervatves of L(, λ) Hence and u and suffcent smoothness s found for the u have to be zero at the soluton pont. ( ) 0 u + λ g u = (2.45) U ( ) = 0 g u. (2.46) U

39 2 The postve Lagrange multpler λ can be obtaned from (2.45), and then substtuted n the same equaton. Ths yelds the frst order optmalty condton: ( ) gu ( ) 0 u g u + u u =. (2.47) U Ths condton means that the normal to the lmt state surface at the MPP should pont towards to the orgn of u-space. Many standard algorthms are avalable for solvng the Lagrange optmal problem defned n (2.43). However, the frst-order method may converge to an nfeasble pont; Newton s method requres second-order nformaton. These dffcultes suggest that the Lagrange method may not be a good optmzaton technque for the relablty problem Hasofer-Lnd Method Hasofer and Lnd (974) proposed an teratve algorthm to solve (2.43), whch was later used by Rackwtz and Fessler (978) n conjuncton wth probablty transformaton technques. Ths algorthm generates a sequence of ponts u from the recursve rule u + ( T ) ( ) g ( u ) ( ) ( u ) g u u g u g u = g U U U U U. (2.48) At the current teratve pont u, the lmt state surface s lnearzed,.e. replaced by the trace n the u-space of the hyperplane tangent to ( ) g U u at u= u. Equaton (2.48) s the soluton to ths lnearzed optmzaton problem, whch corresponds to the orthogonal projecton of u onto the trace of the tangent hyperplane. The Hasofer-Lnd method s wdely used due to ts smplcty. However, t may not converge n some cases.

40 Improved Hasofer-Lnd method Zhang and Der Kureghan (995,997) proposed an mproved verson of the Hasofer-Lnd method for whch uncondtonal convergence could be proven. It s based on the followng reformulaton of the recursve defnton of Equaton (2.48): u = u +λd +, λ = (2.49) and d ( T ) ( ) g ( u ) ( ) ( u ) g u u g u g u U U U = U g U u, (2.50) where d and λ are the searchng drecton and the step sze respectvely. The orgnal Hasofer-Lnd method can be mproved by computng an optmal step sze λ. For ths purpose, a mert functon m( u) s ntroduced. Durng each teraton, after computng (2.50), a lne search s carred out to fnd λ such that mert functon s mnmzed. That s, fnd λ to satsfy mnmzng m( u + λd ). The unconstraned optmzaton problem (Equaton (2.43)) s replaced by the problem of fndng a value λ such that the mert functon s suffcently reduced (f not mnmal). The so-called Armjo rule (Luenberger, 986) s an effcent technque, whch s wrtten by k k k { b m( u b d ) m( u ) ab m( u ) } 2 λ = max + k R, (2.5) where ( ab, 0, ) are pre-selected parameters and k s an nteger. Zhang and Ker Kureghan (995,997) proposed the followng mert functon: 2 m( u) = u + c gu ( u ). (2.52) 2

41 23 Ths expresson has two propertes: () The search drecton d defned n (2.50) satsfes: T u u, m( u) d 0 provdes c> g U u. (2.53) (2) The mert functon attans ts mnmum at the MPP provded that the same condton s fulflled on c. Both propertes are suffcent to ensure that the global algorthm defned by (2.49), (2.50) and (2.5) s uncondtonally convergent (Luenberger, 986). ( ) Others Wth the excepton of the Hasofer-Lnd method and ts varants, Lu and Kureghan (986,99) dscussed other algorthms n structural relablty analyss, ncludng the gradent projecton method, the augmented Lagrange method, and sequental quadratc programmng (SQP) method. Based on results of numercal examples, they recommended the SQP and mproved/modfed Hasofer-Lnd method because of ther convergence and computatonal effcency. Other recently proposed ntellgent algorthms for relablty analyss nclude neural networks (Shao and Morutso, 997), and evolutonary algorthms (Elegbede, 2005). These methods stll need further nvestgaton.

42 Frst-Order Relablty Method The frst order relablty method (FORM) s based on the frst-order lnear approxmaton of the lmt state surface g(x) = 0, tangent to the closest pont of the surface to the orgn. The determnaton of ths pont nvolves nonlnear constraned optmzaton, and s usually performed n the standard Gaussan mage of the orgnal space, whch can be obtaned by the Rosenblatt transformaton (Rosenblatt, 952). The FORM algorthm nvolves several steps; they wll be descrbed brefly assumng a generc N-dmensonal random vector X. Frst, the space x of uncertan parameters X s transformed nto a new N- dmensonal u space of ndependent standard Gaussan varables U. The orgnal lmt state g(x) = 0 then becomes mapped nto the new lmt state g U (u) = 0 n u space. Second, the pont u * on the lmt state g U (u) = 0 that has the shortest dstance to the orgn of the u space s determned by usng an approprate nonlnear optmzaton algorthm. Ths pont s referred to as the desgn or beta pont, and has a dstance b HL (known as the relablty ndex) to the orgn of u space. Thrd, the lmt state, g U (u) = 0, s approxmated by a hyperplane ( lnear or frstorder), g L (u) = 0, tangent to t at the desgn pont. Fnally, the probablty of falure P F s thus approxmated by P ( ) F, P gl 0 = U < n FORM and gven as: ( ) ( ) PF, = P gl U < 0 =Φ β HL, (2.54) where

43 25 Φ = 2π 2 u 2 ( u) exp ξ d ξ (2.55) s the cumulatve dstrbuton functon (CDF) of a standard Gaussan random varable Second-Order Relablty Method Second order relablty methods (SORM) are proposed as a natural extenson of FORM. The dea s to approxmate the lmt state surface by a quadratc surface whose probablstc content s known analytcally General Quadratc Approxmaton For a standard Gaussan random vector (, ) N ( ) =φ ( ) = ( 2π) 2 exp N U N 0 I, ts jont PDF s T f u u U 2 u u. (2.56) Assumng that g U (u)s contnuous, smooth, and at least twce dfferentable, ts secondorder Taylor seres expanson about MPP (u * ) s g g g g 2 where * * T * * T * * ( u) ( u ) + ( u ) ( u u ) + ( u u ) H( u )( u u ) ( u) U U U Q ( * * gu u ) s the gradent vector and ( ) ( ) * at the MPP. Snce g u = 0, U, (2.57) H u s the Hessan matrx, both evaluated T * * T * T * T T gq( u) = gu u + u Hu ( gu + u H) u+ u Hu, (2.58) 2 2 n whch the argument * ( u ) has been dropped for notatonal convenence. At the MPP

44 26 u β * HL g g U = = U α *, (2.59) where * * β HL = u and α s unt vector to MPP. Dvdng equaton (2.59) by g U yelds 2 ( ) β gq u HL * T H * * T * T H T H = β HL + α α α +β HL α u + u g U 2 g U g U 2 gu u. (2.60) Parabolc Approxmaton Construct an orthogonal matrx ( N N R ) * L, whose Nth column s α,.e., R R α, (2.6) where N N ( ) R L satsfes * T α R = 0 ( N ). (2.62) The matrx R can be obtaned by Gram-Schmdt orthogonalzaton. Consder the orthogonal transformaton and parttonng u= Rv, (2.63) v v =, (2.64) v N where Hence, equaton (2.60) becomes N (, ) V N 0 I and V N 0,. (2.65) N N N ( )

45 27 2 ( ) β gq u HL * T H * * T H = β HL + α α vn +β HL α Rv + v T Av g 2 g U U g, (2.66) U T R HR 2 g N N where A = ( ) U L. Consder the partton of A A N A N A =, (2.67) A N A NN N L ( ) ( ) where A N N N, A T N = A N L partton, the general quadratc equaton reads 2 ( ) β, and A. In vew of ths g u H H = v +β + v A v+ α α β α Rv+ 2v A 2 v + A v N. (2.68) Q T HL * T * * T N HL N HL N N NN gu 2 gu gu Madsen et al. (986) proposed a parabolc approxmaton of (2.68) by neglectng NN any cross terms and second order terms of vn and β HL. Only the frst three terms are left, leadng to g Q g U v +β + v A v. (2.69) T N HL N Ths parabolc approxmaton has been used by many researchers, such as, Bretung (984), Hohenbechler and Rackwtz(988), Tvedt (990), Ca and Elshakoff(994), Koyluoglu and Nelsen (994), Adhkar (2004), and others.

46 Falure Probablty Evaluaton - Asymptotc Solutons Usng (2.69) as the parabolc approxmaton of the falure surface, the second order estmate of the falure probablty s g Q T PFII, = P < 0 P V N>β HL+ N g V A V U. (2.70) Defne a random varable Z : N by Z T V A V, (2.7) N. N as the quadratc mappng of a standard Gaussan vector V N ( 0, I ) Therefore [ ] ( ) N N PFII, P VN >β HL+ Z = E Φ βhl Z. (2.72) Unfortunately, the exact probablty densty functon of the quadratc form Z s n general not avalable n closed form. For ths reason, t s dffcult to calculate the expectaton E Φ ( β Z ) HL analytcally. ) The functon Φ( βhl Z s contnuous and dfferentable (of any order) for z. Expandng ln Φ( βhl Z ) at Z = 0 and keepng only the lnear term where ( ) 2 ( HL ) ( ) φβ ln Φ( βhl Z ) ln Φ( βhl ) Φ β HL Z. (2.73) φ u = exp u s the probablty densty functon (PDF) of a standard 2π 2 Gaussan random varable. Hence,

47 29 The moment generaton functon M ( ) ( HL ) ( ) φβ Φ( βhl Z ) Φ( βhl ) exp Φ β Z s of a random varable Z s defned as ( ) exp( ) HL Z. (2.74) M Z s E sz. (2.75) For a quadratc form Z V T V AN of Gaussan varables, t s elementary to show 2 E exp N 2 ( ) ( ) MZ s sz = I sa N. (2.76) From Equaton (2.72) P FII, ( Z) E Φ βhl ( HL ) ( ) φβ E Φ( βhl ) exp Z Φ β. (2.77) HL ( HL ) ( ) φβ =Φ( β ) I + 2 Φ β HL N N HL A 2 Let a, =,,N be the egenvalues of A N, t can be proved that egenvalues of I N + 2( φ( βhl) Φ( βhl) ) A N are ( ( ) ( )) Equaton (2.77) can be rewrtten as P + 2 φβ Φ β a, =,, N. Thus, Φ( β ) + 2 HL HL n 2 ( HL ) ( ) φβ a FII, HL = Φ βhl. (2.78) Hohenbchler and Rackwtz (988) gave the mproved asymptotc soluton as P FII, HL = n 2 ( HL ) ( ) φβ Φ( β ) +κ Φ β HL, (2.79)

48 30 whch s the same formula as (2.78), f the prncpal curvatures at the MPP are denoted by κ = 2a. The sgn conventon s such that curvature s postve when the surface curves are away from the orgn. Further, consder when β HL, so that ( HL ) ( ) φβ Φ β HL β HL. Equaton (2.79) s smplfed as the same asymptotc soluton gven by Bretung (984), whch s P n ( ) ( +κβ ) Φ β FII, HL HL = 2. (2.80) Least-Squares Non-Asymptotc Soluton Consder an approxmaton of ( Z ) ) Φ β by HL ( ) exp( ) Φ βhl Z c c2z, (2.8) such that the error n ths approxmaton s n some sense mnmzed. The error n representng Φ( βhl Z by (2.8) s gven by Defne the objectve functon ( Z; c, c ) ( Z) c exp( c Z) ε =Φ β. (2.82) 2 HL 2 (, ) ( ;, ) ( ) exp( ) 2 2 Ψ c c = ε z c c dz = Φ β z c c z dz. (2.83) 2 2 HL 2 To mnmze (2.83) and smultaneously satsfy Ψ ( c, c ) Ψ ( c, c ) c = 0 and = 0. (2.84) c 2 2 2

49 3 Further smplfcaton of Equaton (2.85) reveals that a system of nonlnear equatons must be solved to obtan optmal parameters c and c2. An exact soluton of (2.84) does not exst. However, a numercal soluton can be obtaned easly usng wdely avalable nonlnear equaton solvers (MATLAB, IMSL, et al.), Hence, the SORM falure probablty estmate s (Adhkar, 2004) N 2 FII, N N = +κ 2 = ( ) P c I c A c c, (2.85) 2 where κ, =,, N are the prncpal curvatures. The least-squares method degenerates to Bretung s and Hohenbchler s asymptotc solutons when HL (Hohenbchler s) or c2 β HL (Bretung s). β, c ( ), c ( ) ( ) Φ βhl 2 φ βhl Φ β HL Other Non-Asymptotc Solutons For parabolc falure surface, the only sources of error n Equaton (2.72) are from the approxmaton of Φ( βhl Z ) by frst order Taylor expanson, whch s asymptotcally correct when β HL. If ths condton s not well satsfed, other methods exst to mprove ths approxmaton. Tvedt (990) extended Bretung s asymptotc soluton to obtan a three-term soluton gven by

50 32 N ( ) ( ) 2 FII, Φ β HL +κβ HL = P + β Φ β φ β +κβ + β + κ N N 2 ( ) ( ) ( ) 2 HL HL HL HL ( ( HL ) ) = = N N +β 2 ( ) ( ) ( ) ( ) 2 HL + βhlφ βhl φβ HL +κβ HL Re ( +β ( HL + ) κ ) = =, (2.86) where =. It can be shown that second and thrd terms of (2.86) vansh when βhl, whch results n only the frst term remanng: Bretung s asymptotc soluton (2.80). Tvedt also derved an alternatve formulaton n a complex doman for a general quadratc falure surface; however, because ths formulaton s not n closed form, numercal ntegraton s needed. Koyluoglu and Nelsen (994) derved a seres soluton for SORM usng hgherorder approxmatons of ( Z ) P F,II where N ( HL) Φ β +κ = 0, Φ β. c HL 2 2 N N N κ k κ k κ k + c, + c 2, k= +κ k c0, 4 k= +κ k c0, k= +κk c0, N N N κ k κ k κ N k + c κ 3, + 2 k k= +κ k c0, k= +κ k c0, k= +κk c0, k = +κk c0, c,j are coeffcents that can be expressed solely n terms of β HL., (2.87) Ca and Elshakoff (994) also derved a seres soluton for a parabolc falure surface based on the Taylor expanson of ( Z ) Φ β as HL P FII, 2 β ( ) exp HL Φ β HL + ( D+ D2 + D + ), (2.88) 3 2π 2

51 33 where D D D 2 3 N = κ = N N 2 = βhl 3 κ + κκ j 2 =, j=, j 2 N N N 3 2 = ( βhl ) 5 κ + 9 κ κ j + κκ jκk 6 =, j=, j, j, k=, j k. (2.89) The seres soluton also converges to an exact soluton of falure probablty for a parabolc falure surface Hgher-Order Relablty Method For those cases, n whch the lmt-state surface has a large curvature (a hgh nonlnearty) around the MPP, both FORM and SORM can have large errors n ther estmates of falure probablty. For example, f the MPP s an nflecton pont (cubc form), or the lmt-state surface s a flat hyperplane around the MPP, then curvature-ftted parabola of SORM s reduced to the tangent hyperplane, thus provdng no mprovement over FORM. Wang and Grandh (998) proposed a hgher-order relablty method (HORM), whch can be used for problems wth hghly nonlnear lmt state functons, or wth an nflecton pont of MPP. The flow chart n Fg. 2.2 llustrates the procedure of ths method. In the rotated standard Gaussan space, two-pont adaptve nonlnear approxmaton was used to approxmate hghly nonlnear lmt state functons, and then based on the approxmaton gven by Koyluoglu and Nelsen (994) for ( Z ) ( ) Φβ + Φβ HL, the falure probablty by numercal ntegratons was calculated. HL

52 34 In Fgure 2.2, when m = 2, ths method s smplfed to the same SORM soluton proposed by Koyluoglu and Nelsen (994) Multple MPP Problems The prevously dscussed methods only consder the case of a sngle MPP. If multple MPPs exst, or f there are contrbutons from other regons around local mnmums besdes the regon around a sngle MPP (Fgure 2.3), these methods can fal to provde a correct estmaton of falure probablty. The exstence of multple MPPs may cause the followng problems: () the optmzaton algorthm may converge to a local MPP (local mnmum), n whch case FORM/SORM and the MPP-based method wll mss the regon of domnant contrbuton to the falure probablty; (2) even f the global MPP s found, other sgnfcant contrbutons to the falure probablty may exst n the neghborhoods of other local MPPs; and (3) suppose that all MPPs are dentfed successfully, system FORM/SORM may not be suffcent accurate f there are hghly nonlneartes around these MPPs. To handle multple MPPs problems, two steps need to be nvestgated: () global optmzaton technques to fnd all MPPs, and (2) system relablty that consder the correlaton of the pecewse approxmaton lmt-state surface based on these MPPs. The followng sectons dscuss possble strateges to solve multple MPPs problems Mult-Pont FORM/SORM The arbtrary falure set defned n Equaton (2.7) may exhbt large calculaton dffcultes. As suggested by Dtlevsen and Madsen (996), ths set can be approxmated

53 35 by multple frst-order or second order approxmatons. A relablty calculaton can then be undertaken wth less dffculty for a smpler falure set. To use ths method, all local mnmums need to be determned n advance and the qualty of the soluton depends on the accuracy of these approxmatons System Relablty Method Once multple MPPs are dentfed successfully, system relablty methods can be used to estmate the falure probablty. If the falure regon s gven by an ntersecton of falure domans, system relablty can be estmated usng Equaton (2.6). If the falure doman s a unon of falure domans, a bound technque can be used, as descrbed n Equaton (2.43). More complcated falure regons can also be consdered, whch are dscussed n detal n secton Samplng Based MPP Search Thacker (200) proposed a smple samplng technque to locate all MPPs for a gven level, and then the system approach can be used. The procedure runs as follows: () estmate the probablty of falure usng coarse samplng; (2) convert the realzatons n the falure regon to u-space (standard Gaussan space); (3) evaluate the dstance to the orgn for each realzaton; and (4) sort the results and report those realzatons wth the shortest dstance. Although ths method s not very effcent, snce the samplng technque s much easer and more stable, t was mplemented wth structural relablty software such as NESSUS.

54 Multple Lnearzaton Mahadevan and Sh (200) proposed a smple multpont lnearzaton method for nonlnear relablty analyss n order to mprove the falure probablty approxmaton of FORM. The method s based on four man concepts: () approxmate the lmt state usng multple hyperplanes; (2) search for the multple lnearzaton ponts on the lmt state; (3) seek computatonal effcency through the nvestgaton of correlated hyperplane; (4) estmate the falure probablty through unon and/or ntersecton operatons, dependng on the lmt state defnton. The dffculty wth ths method s how to select and search the multple lnearzaton ponts on the lmt state Barrer Optmal Method In optmzaton theory, a common trck to fnd multple solutons for a problem s to construct barrers around prevously found solutons, thereby forcng the algorthm to seek a new soluton. In relablty problems, the objectve functon s the dstance from the lmt state surface to the orgn. A barrer around the frst soluton can be constructed by movng the lmt state n the neghborhood away from the orgn. Consequently, followng barrers can be constructed to fnd the new solutons. Der Kureghan and Dakessan (998) proposed ths method to solve multple MPPs problems and suggested a barrer functon B ( u ) lke the followng: B s r u u, u u r = 0, elsewhere 2 2 ( ) ( ) 2 u (2.90)

55 37 where r s the radus of the bulge and s s a postve scale factor, th soluton. u s the vector form of Global Response Surface Method The basc dea behnd ths method, proposed by Gupta and Manohar (2004), s to construct a response surface for the lmt state by usng global nformaton, rather than the local nformaton around a sngle MPP. The algorthm s descrbed as follows: () defne a new set of coordnates to be dentfed for each pont, whch ncludes translatng the orgn a prescrbed dstance along one of the axes and the number of shftng orgns depends on the number of ponts one wshes to dentfy; (2) the Bucher (990) approach s subsequently used to dentfy the desgn pont of the new performance functon n the new coordnate system; (3) a polynomal response surface s obtaned, whose coeffcents are determned by a least square regresson analyss; and (4) Monte Carlo smulatons are carred out on the response surface to obtan estmates of falure probablty. 2.5 Smulaton Methods Smulaton methods that nvolve samplng and estmaton are well known n the statstcs and relablty lterature. Drect Monte Carlo smulaton (MCS) s the most wdely used smulaton method, whch nvolves the generaton of ndependent samples of all nput random varables, repeated determnstc trals to obtan correspondng smulated samples of response varables, and standard statstcal analyss to estmate probablstc

56 38 characterstcs of response. In order to mprove the computatonal effcency of drect MCS, many strateges are appled to reduce the number of samplngs Drect Monte Carlo Smulaton For structural relablty analyss nvolvng a random vector X = { X X },, N wth jont densty f X ( x), the probablty of falure defned by lmt state functon ( ) g X can be estmated from M ndependent samples x, x generated from the, M densty functon f X ( x). If there are an M * number of samples among them satsfyng g ( x) 0, then the probablty of falure s approxmated by P F M = P g( X ) 0. (2.9) M and the relablty s P S M M = PF. (2.92) M By drect MCS, a large number of samples are requred to estmate accurately for the small probablty of falure Importance Samplng The dea of mportance samplng s to generate samples, not from the probablty densty functon f X ( x) of random vector X, but from another samplng dstrbuton. It s expected that better approxmaton can be obtaned for probablty of falure from the

57 39 new densty functon. The probablty of falure defned n Equaton (2.7) can be rewrtten as where w( ) = f ( ) h( ) X F g X ( ) ( ) ( ) ( x) 0 g( x) 0 P = f x d x = w x h x d x, (2.93) x x x. By usng the mportance samplng method wth an M number of samples, the probablty of falure s gven by PF = w x g x. (2.94) M ( ( ) ( ) 0 M = ) A smple and wdely employed approach of mportance samplng s to move the samplng center from the orgn n standard Gaussan space to the desgn pont (MPP) on the falure surface (Scheller and Stx, 987). Other samplng dstrbuton and samplng centers were dscussed by Engelund and Rackwtz (993) Stratfed Samplng In stratfed samplng, the doman of ntegraton s dvded nto several regons (Melcher, 999). Emphass can be attrbuted by mplementng more smulaton n regons that contrbute to the falure event. Consequently, the total doman of ntegraton can be dvded nto m regons,.e., R, R2,, Rm. Usng the total probablty theorem, the probablty of falure can be estmated as m N j = ( j) ( x), (2.95) PF P R j= N j =

58 40 where ( x) ( x) ( x) 0, f g > 0 =, f g < 0 s the ndcaton functon wth respect to the performance g ( x ) ( j ) functon, P R s the falure probablty of regon Rj, and N j s the number of smulatons performed n regon R j. Snce the falure regon may not be known n advance, a tral and error method s necessary to mplement ths strategy Drectonal Smulaton Consder a relablty problem wth a lmt state functon g ( X ) nvolvng N normally dstrbuted random varables. If length R and drecton A of a vector X = RA are defned, then R 2 2 s Ch-square dstrbuted wth the CDF ( r ) χ havng N degrees of freedom. Probablty of falure defned n Equaton (2.7) can be gven by an ntegraton of the condtonal falure probablty n the drecton A = a as (Bjerager, 988) ( X) ( A) PF = P g 0 = P g R 0 All drecton = χ All drecton ( a) 0 A a ( a) = P g R = fa da, (2.96) ( ) ( a) r f da 2 2 N a A where a s a realzaton of a random unt vector A unformly dstrbuted on the N dmensonal unt hypersphere Ω N and centered around the orgn, and f ( ) unformly densty functon, gven by f A ( ) ( N /2) N A a s the Γ a = =, (2.97) N /2 S 2π

59 4 where S s the surface area of Ω N, Γ( ) s the Gamma functon, and r a s the dstance from the orgn to the lmt state surface n the drecton a. Suppose the unt hypersphere s dvded evenly nto M subsurfaces wth the same area S / M and a representatve drecton a. From Equaton (2.97), the falure probablty can be approxmated by P F M ( r ) M 2 2 χ N a. (2.98) = The key ssue for ths method s to generate M evenly dstrbuted a drectons. There are two approaches avalable: () generate M vectors of N ndependent Gaussan random varables and normalze these vectors to unt length; or (2) generate M vectors of N ndependent unform random varables by usng the rejecton method, such that a vector s retaned only f ts length s no greater than one, n whch case all vectors are normalzed. The drectonal smulaton method s very effcent, because the dstance from the orgn to the lmt state surface n any drecton can be obtaned effcently, as ponted out by Dtlevsen (990) Latn Hypercube Samplng Latn hypercube samplng, frst proposed by McKay (979) and further developed by Sten (987) and Olsson and Sandberg (2002), uses a stratfed samplng procedure to sample the values of the random varables from ther probablty densty functons. For a problem nvolvng N random varables, f M s the requred number of samples, then an M N matrx P can be created, n whch each of the N columns s a random permutaton

60 42 of, 2,, M, and an M N matrx R of ndependent random numbers from the unform (0,) dstrbuton are establshed by standard MCS. A matrx S s obtaned as S = ( P R ). (2.99) M Then, realzaton of the random vectors become n whch x F ( S =, where ( ) j x j j ) x j ( ) x,,,,, = x xn = M, (2.00) F s the nverse of the CDF for random varable X j. There s a rsk that some spurous correlaton wll appear, and Olsson (2003) proposed some methods to reduce t Markov Chan Monte Carlo Smulaton The Markov chan MCS method, also called subset smulaton, was appled by Au and Beck (200) to estmate small falure probablty usng a modfed Metropols algorthm (Metropols, 953, Fsherman, 996). Ths method transfers the evaluaton of falure probablty to the evaluaton of a sequence of smulatons of more frequent events n condtonal probablty spaces. For a gven falure event F, construct F F F = F as a decreasng sequence of falure events, so that 2 m k F = F, k =,2,, m. Accordng to the defnton of condtonal probablty, falure k = probablty becomes

61 43 [ ] m PF = P Fm = P F = m m m = = = P F F P F [ ]. (2.0) = P F P F F P F F P F F m m The major task s to smulate the condtonal samples effcently, whch was acheved usng the modfed Metropols algorthm (Au and Beck, 200). Ths method s found robust up to the number of random varables and effcent n computng small probabltes. However, the proposed PDFs nvolved n the Metropols algorthm have to be carefully chosen because the spread affects the sze of the regon covered by the Markov chan samples and, consequently, the effcency. A small spread tends to ncrease the dependence between two successve samples due to ther proxmty, and an excessvely large spread may reduce the acceptance rate and ncrease the number of repeated Markov chan samples. In both cases, convergence could be slow. 2.6 Response Surface Methods The practcalty of relablty methods for a specfc lmt state depends on the complexty of the formulaton of the lmt state. Often the lmt state functon s not avalable n explct form, but rather defned mplctly through a complcated numercal procedure, gven for example by the fnte element analyss. For such lmt state formulatons, the needed calculatons may requre prohbtve large computatonal efforts. One way to solve such complex problem s to approxmate the lmt state surface n a numercal-expermental way by usng a surface n explctly mathematcal form, and then mplementng a relablty analyss. Ths procedure s called the response surface method.

62 44 Let = { X X } 2.6. Basc Prncple of Response Surface Method X,, N be the vector of basc random varables. The central dea of the response surface method s to approxmate the exact lmt state functon g ( X ), whch s usually known through an algorthmc procedure, by a polynomal functon ĝ ( X ). In practce, quadratc functons are used n the form 2 ( ) ˆ ( ) 0 N N N N g x g x = a + a x + a x + a x x, (2.02) where the set of coeffcents a = { a,,, 0 a a aj} j j = = = j=, j square, and cross terms, respectvely, are to be determned., whch correspond to the constant, lnear, A lmted number of evaluatons of the lmt state functon are requred to buld the surface. A relablty analyss can then be performed by means of the analytcal expresson n Equaton (2.02), nstead of the true lmt state functon. Ths approach s partcularly attractve when smulaton methods such as mportance samplng (Bucher and Bourgund, 990) are used to obtan the relablty results Buldng the Response Surface The determnaton of the unknown coeffcents a s performed by usng the least squares method. After choosng a set of fttng ponts xk, k =,, n, for whch the exact value yk ( ) = g x k s computed, the error ε( a ), defned by n 2 a ( k k ) = ( ) y gˆ ( x ) ε =, (2.03)

63 45 s mnmzed wth respect to a. Reformulatng Equaton (2.03) n the form 2 T T ( ) { } { } ( ) gˆ X =, x, x, x x a, a, a, a V x a, (2.04) j 0 j where, j =,, N and j. The least squares problem becomes: mn n ( ) T 2 yk V ( xk) a. (2.05) = After some basc algebra (Faravell, 989), the soluton to the above problem yelds T ( ) T a = νν ν y, (2.06) where ν s the matrx whose rows are the vectors V ( x ) and y s the vector whose k ( ) components are y = g x. k k The varous response surface methods proposed n the lterature dffer only n the terms retaned n the polynomal expresson (2.02), and the selecton of the coordnates of the fttng ponts,.e., the expermental desgn used n the regresson analyss. It s emphaszed that n N s requred to solve (2.05). Furthermore, the fttng ponts have to be chosen n a consstent way n order to get ndependent equatons Varous Types of Response Surface Approaches Early applcatons of the response surface method nvolved the so-called factoral expermental desgn. For each random varable X, lower and upper values of realzatons ( x, x + ) are selected. Overall, 2 N fttng ponts are defned by all possble combnatons

64 { x,, xn } ± ±. The number of fttng ponts ncreases exponentally wth the number of random varables N nvolved n the relablty problem under consderaton. In order to reduce the number of fttng ponts for cases n whch N s large, Bucher and Bourgund (990) proposed a smplfed quadratc expresson wthout cross terms, whch s defned by only ( 2N + ) coeffcents a = { a a a },, In the frst step, the mean vector µ s chosen as the center pont of the response surface. Exact ( 2N + ) X fttng ponts are selected along the axes, descrbed by x = µ X x2 = µ X f σ e, =,, N, (2.07) x2+ = µ X + f σ e, =,, N where σ s the standard devaton of the th random varable, e s the th bass vector of the space of parameters, whose coordnates are { 0,,0,,0, 0}, and f s an arbtrary number (set to 3 by Bucher and Bourgund (990)). From ths frst response surface, an estmate of the desgn pont x s computed. Then, a new center pont x M s obtaned as a lnear nterpolaton between µ X and x by x M ( x X) = µ + µ X g g ( µ ) X ( µ X ) g( x ). (2.08) A second response surface s then generated around x M. As a whole, the approach requres only ( 4N + 3) evaluatons of the lmt state functon, and can thus be carred out for structural systems nvolvng a great number of random varables. Fnally, mportance samplng s used to obtan the relablty results.

65 47 Later, Rajashekhar and Ellngwood (993) consdered the same approach by Bucher and Bourgund (990) as the frst two steps of an teratve procedure untl full convergence. They also added cross terms to the response surface defnton, obtanng better results n numercal examples. Km and Na (997) observed that n prevous research, the fttng ponts are selected around a preselected pont (.e., the mean value of the basc random vector) and arranged along the axes or dagonals of the space of parameters, wthout consderng the orentaton of the orgnal lmt state surface. The authors argued that n some cases these procedures mght not converge to the true desgn pont. Alternatvely, they proposed to determne a seres of lnear response surfaces as follows: n each teraton, the fttng ponts used n the prevous step are projected onto the prevous response surface, and the projecton ponts that are obtaned (whch are closer to the actual lmt state surface) are used for generatng the next response surface. In each teraton, an approxmate relablty ndex s readly avalable, snce the response surface s lnear. In some sense, ths method fnds the desgn pont wthout solvng the mnmzaton problem usually assocated wth FORM. Ths method s called the vector projecton method. Startng from the dea of Km and Na (997), Das and Zhang (2000) proposed enhancng the lnear response surface by addng square terms. The fttng ponts defnng the fnal lnear response surface are reused to produce the quadratc surface. SORM analyss s then performed. Lemare (997) presents a synthetc summary of the response surface method and draws the followng conclusons: () t s better to cast the response surface n standard

66 48 normal space, rather than n the orgnal space because regresson can be controlled better; (2) provded enough fttng ponts, the choce of the type of expermental desgn s not fundamental; and (3) the qualty of the response surface has to be checked. Dfferent ndcators are proposed to estmate the accuracy: () the back-transformaton of the fttng ponts from standard normal space to the orgnal space, n order to exclude nonphyscal ponts; (2) the condtonng of the expermental matrx ννn Equaton (2.06); T (3) the qualty of the regresson measured by a correlaton coeffcent; and (4) the extent to whch the obtaned desgn pont belongs to the orgnal lmt state surface. 2.7 Mean-Pont-based Decomposton Methods Recently, Rahman and Xu (2004) developed new decomposton methods that can solve hghly nonlnear relablty problems more accurately or more effcently than FORM/SORM and smulaton methods. A major advantage of these decomposton methods over FORM/SORM, so far they are based on the mean pont of random nput as a reference pont, s that hgher-order approxmatons of performance functons can be obtaned wthout calculatng the MPP or the gradents Multvarate Functon Decomposton Consder a contnuous, dfferentable, real-valued functon y(x) that depends on x N = {x,,x N } R. Suppose that y(x) has convergent Taylor seres expanson at an T arbtrary reference pont x = c = { c,, c N }, expressed by N j y j y( x) = y( c) + ( c )( x c) + R2 (2.09) j! x j j= =

67 49 or N j y j y( x) = y( c) + ( c) ( x c ) j j= j! = x, (2.0) j+ j2 y j j2 + ( c) ( x c ) ( ) x c 2 + R 2 3 j! j! x x j j2 j, j2> 0 2 < 2 2 where the remander R 2 denotes all terms wth dmenson two and hgher and the remander R 3 denotes all terms wth dmenson three and hgher Unvarate Approxmaton Consder a unvarate approxmaton of y(x), denoted by N yˆ ( x) yˆ ( x,, x ) = y( c,, c, x, c,, c ) ( N ) y( c), (2.) N + N = where each term n the summaton s a functon of only one varable and can be subsequently expanded n a Taylor seres at x = c, yeldng N j y j yˆ ( x) = y( c) + ( c )( x c). (2.2) j! x j j= = A comparson between Equatons (2.09) and (2.2) ndcates that unvarate approxmaton leads to the resdual error y( x) yˆ ( x ) = R2, whch ncludes contrbutons from terms of dmenson two and hgher. For suffcently smooth y(x) wth a convergent Taylor seres, the coeffcents assocated wth hgher-dmensonal terms are usually much smaller those wth one-dmensonal terms. In that case, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected.

68 Bvarate Approxmaton In a smlar way, consder the bvarate approxmaton yˆ ( x) y( c,, c, x, c,, c, x, c,, c ) N < 2 N ( N )( N 2) ( N 2) y( c,, c, x, c+,, cn) + y( c) 2 =, (2.3) of y(x), where each term on the rght hand sde s a functon of at most two varables, and can be subsequently expanded n a Taylor seres at x = c, yeldng N j y yˆ 2( x) = y( c) + ( c) ( x c) j! x j j= = j+ j2 y + ( c )( x c ) ( x c ) j! j! x x j j2 j, j2> 0 2 < 2 2 j j j (2.4) Agan, a comparson of Equatons (2.0) and (2.4) ndcates that the bvarate approxmaton leads to the resdual error y( ) ˆ2( ) x y x = R 3, n whch remander R 3 ncludes terms of dmenson three and hgher. The bvarate approxmaton ncludes all terms wth no more than two varables, thus leadng to a hgher rate of accuracy than the unvarate approxmaton Generalzed S-Varate Approxmaton The procedure for unvarate and bvarate representatons descrbed above can be generalzed to an S-varate representaton for any nteger S N. The generalzed S- varate approxmaton of y(x) s S N S+ yˆ S( x) ( ) ys ( x), (2.5) = 0

69 5 where y R R N k = tk; 0 R S k = 0 R k, (2.6) wth t 0 = y() c N j y t = ( c)( x c ) j! x j j = j+ j2 y t2 = ()( c x c )( x c ) j! j! x x j j2 j, j2 2 < 2 2 j j j (2.7) y t x c j+ j2+ + js j j2 js S = ()( c )( ) ( ) 2 S x c 2 x j j j 2 c S S j j! j2! js! x x x, j2,, js < 2< < S 2 S Usng a multvarate functon theorem, t can be shown that yˆ ( x) n Equaton (2.5) S conssts of all terms of the Taylor seres of y( x) that have less than or equal to S varables. The expanded form of Equaton (2.5), when compared wth the Taylor expanson of y( x), ndcates that the resdual error n the S-varate approxmaton s x ˆ x, where the remander R S + ncludes terms of dmenson S + and y( ) ys( ) = R S + hgher. When S =, 2, Equaton (2.5) degenerates to unvarate and bvarate approxmaton. Smlarly, trvarate, quadrvarate, and other hgher-varate approxmatons can be derved by approprately selectng the value of S. At the lmt, when S = N, Equaton (2.5) converges to the exact functon y( x). In other words, the proposed approxmaton generates convergent representaton of y( x).

70 Remarks The decomposton of a general multvarate functon y( x) can be vewed as a fnte sum y( ) y y ( x ) y ( x, x ) y ( x,, x ) y ( x,, x ), (2.8) N N N x = S + + s 2 N N =, 2=,, S = < 2 < < S = yˆ ( x) = yˆ 2 ( x) = yˆ S ( x) where y 0 s a constant, y ( x ) s a unvarate component functon representng an ndependent contrbuton by nput varable v, y ( x, x ) s a bvarate component 2 2 functon descrbng cooperatve nfluence of two nput varables x and x, 2 y ( x,, x ) S S nput varables x s an S-varate component functon quantfyng cooperatve effects of S x,, S, and so on. By comparng Equatons (2.) and (2.3) wth Equaton (2.8), the unvarate and bvarate approxmatons provde two- and threeterm approxmants, respectvely, of the fnte decomposton. In general, the S-varate approxmaton n Equaton (2.5) yelds the S+-term approxmant of the decomposton. The fundamental conjecture underlyng ths work s that component functons arsng n the proposed decomposton wll exhbt nsgnfcant hgherdmensonal effects. It s worth notng that the unvarate approxmaton n Equaton (2.) should not be vewed as frst- or second-order Taylor seres expansons and does not lmt the nonlnearty of y(x). Accordng to Equaton (2.2), all hgher-order unvarate terms of y(x) are ncluded n the proposed approxmaton. In fact, the unvarate component

71 53 functon y ( x ) can be hghly nonlnear and, therefore, n general should provde hgherorder representaton of a performance functon than those by FORM or SORM. Furthermore, the approxmatons contan contrbutons from all nput varables Response Surface Generaton Consder the unvarate terms y( x) y( c,, c, x, c+,, cn) n Equatons (2.2) and (2.3). If for x = x, n functon values ( j ) y ( x ) = y( c,, c, x, c,, c ); j =,2,, n ( j) ( j) + N (2.9) are gven, the functon value for arbtrary x can be obtaned usng the Lagrange nterpolaton as where the shape functon φ ( x ) s defned as j n ( j) y ( x ) = φ ( x ) y ( x ), (2.20) j j = () ( j ) ( j+ ) ( n) ( x x ) ( x x )( x x ) ( x x ) j( x) ( j) () ( j) ( j ) ( j) ( j+ ) ( j) ( n) ( x x ) ( x x )( x x ) ( x x φ =. (2.2) ) By usng Equaton (2.20), many functon values of y ( x ) can be arbtrarly generated f n functon values are gven. The same dea can be appled to the bvarate terms y ( x, x ) y( c,, c, x, c,, c, x, c,, c ) N n Equaton (2.6). If for x ( j2 ) = x and x = x, n 2 functon values ( j ) 2 2 y ( x, x ) y( c,, c, x, c,, c, x, c,, c ); j j2 j j N j =,2,, n; j =,2,, n 2 (2.22)

72 54 are gven, the functon value y ( x, x ) 2 2 for arbtrary pont ( x, x ) can be obtaned 2 usng the Lagrange nterpolaton as n n j j2 y ( x, x ) = φ ( x ) φ ( x ) y ( x, x ), (2.23) 2 2 j j j2= j= where shape functons φ ( x ) and j φ ( x ) are defned n Equaton (2.2). Note that j2 2 there are n and n 2 performance functon evaluatons nvolved n Equaton (2.9) and Equaton (2.22), respectvely. Therefore, the total cost for unvarate approxmaton entals nn + functon evaluatons, and for bvarate approxmaton, 2 N( N ) n /2 nn + + functon evaluatons are requred. More accurate multvarate approxmatons can be developed n a smlar way, but wth a much hgher cost Monte Carlo Smulaton For component relablty analyss, the Monte Carlo estmates P F, and P F,2 of falure probablty, employng unvarate and bvarate representatons, respectvely, are P N S () = yˆ ( ) < 0 N = x (2.24) F, S and where P N S () = yˆ ( ) < 0 N = x, (2.25) F,2 2 S () x s the th realzaton of X, NS s the sample sze, and [ ] s an ndcator () functon such that = f () x s n the falure set (.e., when y ˆ ( x ) < 0 for unvarate

73 55 representaton and when () ˆ 2( ) y x < 0 for bvarate representaton of the performance functon) and zero otherwse. Snce unvarate or bvarate representatons facltate lower-dmensonal response surface approxmatons, the subsequent Monte Carlo smulaton can be conducted for any sample sze. However, the accuracy and effcency of falure probablty calculatons usng Equatons (2.24) and (2.25) depend on both the decomposton and response surface approxmaton. 2.8 Senstvty Analyss Senstvty analyss provdes a measure of a specfc nput varable s mportance to the relablty results. The determnstc nput varables and the parameters n the dstrbutons of random nput varables are denoted as nput parameters. The senstvty of the relablty measure wth respect to changes n these parameters s mportant for relablty-based desgn optmzaton. Ths can easly be evaluated as a change of relablty for a gven change n the desgn. In combnaton wth an optmzaton procedure that ams at mnmzng total cost, the senstvtes can be also used wth teratve soluton methods Dervatve of Relablty Index One Parameter Consder a performance functon g ( u ; θ), where θ s a sngle parameter, for an MPP u on the lmt-state surface n the standard Gaussan space, the followng equatons exst u = α β, (2.26)

74 56 α = g( u ; θ) g( u ; θ), (2.27) g T αα =, (2.28) ( ; ) u θ = 0, (2.29) where ={,, T u u N }. It follows that β s also a functon of θ. The senstvty of β wth respect to changes n θ s measured by the dervatve dβ dθ. From Equatons (2.26) and (2.28) T β= u α, (2.30) whch, on takng dervatve wth respect to θ, gven T dβ dα T du = u + α. (2.3) dθ dθ dθ u u = The frst term on the rght-hand sde of Equaton (2.3) s zero because dα dθ and αare mutually orthogonal and u = α β. The orthogonalty s verfed drectly by dfferentaton of Equaton (2.28). Dfferentaton of Equaton (2.29) gves dg( u ; θ) g T du = + g = 0. (2.32) dθ θ dθ u u = Ths, after dvded by g on both sdes, yelds T g g du + = 0, (2.33) g θ g d θ u = u By comparng Equaton (2.33) and (2.3), gves dβ g =. (2.34) dθ g θ

75 57 That s the senstvty of relablty ndex wth respect to parameter θ. There are two dfferent cases of applcaton for Equaton (2.34). () Case I: θ s a determnstc nput varable Hence, θ s a parameter that concerns the defnton of the lmt state. The lmt-state functon n standard Gaussan space and orgnal space satsfy ( u ; θ ) = ( x ); θ = ( x θ ) g g T G ;, (2.35) where u= T ( x) s the gven transformaton. It then follows that g θ= G θ. (2) Case II: θ s a dstrbuton parameter Such a parameter has no nfluence on the lmt state n orgnal space, but has nfluence on the lmt state n the standard Gaussan space through the transformaton u=t ( x ; θ). That means: g ( ; θ ) = G( ) u x, (2.36) where the rght sde s ndependent of. Therefore, the partal dervatve of the left hand sde of Equaton (2.36) wth respect to θ s zero, wrtten as T g g u + 0. (2.37) θ θ By applcaton of (2.27), Equaton (2.34) gves dβ = dθ u T α. (2.38) θ u = u

76 Dervatve of FORM Approxmaton to Falure Probablty The FORM based approxmaton to the probablty of falure s The dervatve of the falure probablty s F, ( ) P = Φ β. (2.39) dpf, dβ = φ( β) dθ dθ. (2.40) Correspondngly, the dervatve of the natural logarthm to P F, s ( ) ( ) For large values of β the asymptotc formula φ( β) / ( ) results of (2.4) can be smplfed. dlogp F, φ β dβ = dθ Φ β dθ. (2.4) Φ β β s vald, whereby the If P F, s vewed as a functon of an nput parameter, the mage s n most cases strongly curved. However, f β or log P F, s mapped as a functon of the nput parameter, most often the mage s only slghtly curved. Assume that the probablty of falure s known for a value θ of the nput parameter. We want to determne the probablty of falure correspondng to the parameter value θ + θ. A calculaton based on P dp P + dθ F, θ+ θ F, θ F, θ, (2.42) often wll be qute naccurate except for very small values of θ. However, a calculaton based on

77 59 P F, θ+ θ dβ = Φ( ( β+ β) ) Φ β+ θ dθ (2.43) s often a reasonable approxmaton even for large values of θ. Smlarly another reasonable approxmaton s P dlog P F, θ exp log P + θ. (2.44) dθ F, θ+ θ F, θ 2.9 Relablty-based Desgn Optmzaton 2.9. Introducton Stochastc optmzaton s a mathematcal framework for solvng general optmzaton problems n the presence of uncertanty, typcally manfested by the probablstc descrpton of objectve and constrant functons. A specal case of stochastc optmzaton, frequently encountered n structural desgn, s referred to as relablty based desgn optmzaton (RBDO). Accordng to Royset et al. (200), there are three types of formulaton for solvng an RBDO problem: Type I: Mnmze the cost of the desgn, subject to relablty and structural constrants; Type II: Maxmze the relablty of the desgn, subject to cost and structural constrants; and Type III: Mnmze the ntal cost of the desgn plus the expected cost of falure, subject to relablty and structural constrants.

78 60 Of these three types, the frst formulaton s wdely studed, typcally nvolvng a tme-nvarant relablty analyss. The basc mathematcal model of Type I RBDO can be wrtten as mn c ( d) K d D R 0 ( X d) subject to c( d) P g ; 0 R; =,, d d d l u n c, (2.45) where d = { },, T d d k s a K-dmensonal vector of desgn varables wth a nonempty closed set { } K R ; X = R s an N-dmensonal random,, T N X X N vector wth mean and jont probablty densty functon defned on a probablty space (,,P); f s the objectve functon that depends on d; Ω F ( Xd) g ;, =,, n c s the th performance functon that depends on d; and 0 R, =,, n are target probabltes. c The desgn vector d can be determnstc parameters of objectve and constrant functons and/or dstrbuton parameters of X (e.g., the mean). The lower and upper bounds of d are denoted by dl and du, respectvely. Accordng to Equaton (2.45), the objectve and constrant functons are both determnstc; however, the evaluaton of constrants requres a relablty analyss FORM-based Optmzaton Methods Tradtonally, RBDO formulaton s based on the FORM, due to ts smplcty and computatonal effcency. The constrant of Equaton (2.45) can then be expressed by ( ) ( ) ( ) P g Xd ; 0 = Fg 0 Φ β,, t = n c, (2.46)

79 6 where β t s the target relablty ndex and the cumulatve dstrbuton functon ( 0) descrbed as ( ) g = X ( 0 ) ( x) g ( Xd) ; 0 F s F f dx =,, n, (2.47) and x s the jont probablty densty functon (JPDF) of X. f X Double-Loop Approach The classcal double-loop RBDO method employs two nested optmzaton loops: the desgn optmzaton loop (outer) and the relablty assessment loop (nner). The nner loop s needed to evaluate each probablstc constrant n Equaton (2.45), whch together wth the outer loop make the double-loop RBDO computatonally very expensve. The probablstc constrant n Equaton (2.45) can be further expressed through nverse transformatons n two alternatve ways: ( ( F ( 0) )) β = Φ β (2.48) g g ( ( )) P F g t t = Φ β 0 (2.49) c g where β and g P are called the safety ndex and the probablstc performance measure for the th probablstc constrant, respectvely. If Equaton (2.48) s employed to descrbe the probablstc constrant n equaton (2.45), t s called relablty ndex approach (RIA). The relablty ndex β s the mnmum dstance of a pont u on the lmt state ( u ) 0 g = U from the orgn of the standard normal space. Therefore, t can be calculated from the followng relablty mnmzaton problem

80 62 β= mn u u st.. g ( u) = 0 U. (2.50) Smlarly, Equaton (2.49) can replace the probablstc constran wth the performance measure, whch s referred to as the performance measure approach (PMA). In PMA, the performance measure s calculated from the followng relablty mnmzaton problem st.. g P u = mn g =β u t U ( u). (2.5) Methods to solve (2.50) are dscussed n secton Methods to solve (2.5) were summarzed by Youn, et al.(2003), whch ncluded the varatons of mean-value methods Sngle-Loop Approach An equvalent formulaton of the general RBDO problem (2.45) can be stated as mn K d R c 0 ( d ) ( Xd) R subject to g ; 0, =,, n, (2.52) d d d where g R s the R-percentle of the constrant g ( X; d ). It s defned as l ( ; ) R P g g R u c Xd =, (2.53) R where R s the target relablty for the constrant. If 0, P g Xd ; 0 R. g ( ) Therefore, R g 0 provdes an equvalent determnstc expresson of the probablstc constrants n Equaton (2.45). The R-percentle g R s evaluated usng the PMA method

81 63 (Lang, et al., 2004) of Equaton (2.5). After the MPP s calculated, the R-percentle s gven by g R ( ; ) = g X d. (2.54) MPP Thus, the optmzaton problem of (2.52) becomes mn K d R c 0 ( d ) ( X d) subject to g ; 0, =,, l MPP d d d u n c, (2.55) where µ = X U σ and X, U are the MPP for the th constrant n x and u X MPP MPP MPP MPP spaces, respectvely and σ s the vector of standard devatons. Usng the PMA approach, an nner loop of the double-loop method solves the optmzaton problem descrbed by Equaton (2.5). At the optmal pont, the followng Karush-Kuhn-Tacker (KKT) optmalty condton s satsfed U ( u) + λ H( u) 0 g =, (2.56) where H ( u) = u β s an equalty constrant and λ s the correspondng Lagrange multpler. The dervaton gves where = g ( X; d) / g ( X; ) U U u = β t α, (2.57) α d s the constran normalzed gradent n u space. The transformaton of x to u space yelds the followng relatonshps X = µ X σβ t α, (2.58)

82 64 where = g ( X; d) / g ( X; ) α σ σ d. Usng (2.58), the RBDO formulaton of X X Equaton (2.55) can be transformed to the followng sngle-loop, equvalent determnstc optmzaton problem mn K d R c 0 ( d ) ( X d) subject to g ; 0, =,, ( ) g ( ) n c, (2.59) where X = µ X σβ tα, α = σ g Xd ; / σ Xd ;, dl d du, and βt s the X X target relablty ndex for the th constrant, α s the normalzed gradent of the th constrant. The sngle loop method does not search for the MPP of each constrant at each teraton. Instead, the MPP of the actve constrants are correctly dentfed at the optmum. Ths dramatcally mproves the effcency of the proposed sngle loop method wthout compromsng accuracy Sequental Methods for RBDO To avod a nested optmzaton problem, sequental RBDO methods have been developed, whch decouple the upper level desgn optmzaton from the relablty analyss (Agarwal, 2004). The desgn optmzaton and the search of the MPPs are performed separately and the procedure s repeated untl a desred convergence s acheved. The dea s to fnd a consstent relable desgn at lower computatonal cost as compared wth the nested approach (double-loop). A consstent relable desgn s a feasble desgn that satsfes all the relablty constrants. The relablty analyss s employed to check f a gven desgn meets the desred relablty level. In most

83 65 sequental technques of RBDO, a desgn obtaned by performng a determnstc optmzaton s updated based on the nformaton obtaned from the relablty analyss, and the updated desgn s used as a startng pont for the next cycle. Chen and Du (2002) proposed a sequental optmzaton and relablty assessment methodology (SORA). In SORA, boundares of the volated constrants are shfted nto the feasble drecton based on the relablty nformaton obtaned by prevous teraton. Both RIA and PMA can be used for relablty assessment f FORM s deserved adequately. The PMA approach was reported to be computatonally more effcent than RIA approach. In SORA, a frst order relablty analyss s performed to obtan the MPP for each falure drven constrant. Therefore, a consstent relable desgn provdes an approxmate soluton. However, a true local optmum cannot be guaranteed, because the MPP for actve constrants are obtaned from the prevous desgn pont. Consequently, an MPP update has been suggested, but t may lead to spurous optmal desgn Smulaton-based Optmzaton Methods In RBDO, a partcular source of dffculty s constructng approxmatng expressons for the falure probablty that can be used n conjuncton wth optmzaton algorthms. Two approaches for such approxmatons are descrbed as follows Sample Average Approxmaton A sample average approxmaton method s constructed by replacng the falure probabltes n the orgnal RBDO problem wth Monte Carlo samplng estmates. The results assocated wth such approxmatons gve asymptotc propertes of mnmzes of

84 66 sample average approxmaton problems as the number of samples goes to nfnty, and gve error estmates for fnte sample szes. Royset and Polak (2004) descrbed the RBDO problem wth component falure probabltes as mn N x R 0 K ( x) + ( x) ( x) c c p k = ( x) ( x) k subject to p pˆ, k =,, K, (2.60) k k f 0, j =,, J j k where f j ( x) s the jth determnstc contnuously dfferentable, constrant functon; c ( x), k =, K k s the contnuously dfferentable cost functon assocated wth the falure of the kth component; pˆ k s the pre-defned bound for the kth component falure probablty and c0 ( x) s the ntal cost functon. The dffculty assocated wth solvng (2.60) s that the falure probabltes cannot be computed exactly and hence has to be approxmated. In addton, expressons, f they exst, are dffcult to obtan for the gradents of the falure probabltes and ther approxmatons. Due to ths reason, a drect applcaton of the standard optmzaton algorthm s mpossble. Royset and Polak (2004) gave approxmatons of the ntegrals for probablty of falure and ts gradent by usng samplng technques. To mprove effcency, they prefer mportance samplng rather than orgnal drect Monte Carlo smulaton. Through these estmates, combned wth the Polak-He algorthm (2004), a new algorthm was proposed and proved to converge wth a sample sze that tends to nfnty.

85 Response Surface Approxmaton Smulaton-based methods are easy to mplement nto a relablty analyss. However, these methods also produce nosy responses that make them dffcult to use n gradent-based optmzaton algorthms. Response surface approxmatons can help solve two problems of smulaton-based methods: smulaton cost and the nose from random samplng. To solve RBDO of composte lamnates n cryogenc envronments, Qu and Haftka (200, 2003) proposed a method based on response surface approxmatons. Two types of response surfaces need to be created. The frst type s analyss response surface (ARS), whch s ftted to the performance response n terms of both desgn varables and random varables. When ARS s used, the probablty of falure at each desgn pont can be calculated effcently. The second type s desgn response surface (DRS), whch s ftted to probablty of falure as a functon of desgn varables. The DRS s created to flter out nose generated by smulaton-based methods and s used to calculate the relablty constrant n the desgn optmzaton procedure Others Xu and Rahman (2004) recently proposed new decomposton methods to solve RBDO problems. The applcaton nvolves lower-dmensonal approxmatons of general mult-varate functons, response surface approxmatons, and Monte Carlo smulaton. Snce the probablty of falure s estmated from the mean pont based decomposton approxmaton of falure surface, there s no need for an nner optmzaton loop of the relablty assessment. The new method does not depend on FORM/SORM to conduct

86 68 relablty analyss, and t requres only a small number of functon evaluatons. Numercal examples show the proposed method s both accurate and effcent. The cost for functon evaluatons n the proposed method can be predetermned, whle other optmzaton methods cannot predct the cost for optmzaton procedure. Recently, Zou and Mahadevan (2006) proposed a new decouplng approach, for decouplng the optmzaton and relablty analyss teratons n tradtonal nested formulatons. The relablty constrants are approxmated by frst-order Taylor seres expanson based on relablty analyss results, so that the outer loop only performs determnstc optmzaton. The advantage of ths method s that any relablty methods can be employed n nner loop for relablty analyss. The computatonal effcency depends on the relablty method used and accuracy of the approxmaton for relablty constrants. 2.0 Summary of Revew and Future Research Needs 2.0. Conclusons from Exstng Research Accuracy and effcency are the two major concerns n exstng structural relablty and probablstc desgn research. In practcal applcatons, the number of desgn/random varables s large, the lmt state/constrant functon could be hghly nonlnear, and local mnmums n optmal desgn could exst, n addton to other possble complcatons. The followng provdes a summary of revew of exstng methods for both relablty analyss and relablty-based desgn optmzaton.

87 69 Relablty Analyss () Mean-Value methods are desrable for computatonal effcency and usually used to predct the CDF of the response. Snce orgnal MV methods are frst/second order Taylor expansons of the performance functon at the mean pont, they can only solve lnear or slghtly nonlnear problems. Although AMV compensates for the expanson truncaton error by usng a correcton term, the dfference between the approxmate MPP and exact MPP can be enormous. Because AMV+ uses Taylor expanson based on an exact MPP, relablty estmaton can be much mproved. However, snce AMV+ s a second order estmaton, t cannot produce acceptable solutons n hghly nonlnear problems. (2) Classcal FORM/SORM methods are wdely used n relablty calculatons. The advantage of FORM s computatonal effcency and the nformaton provded by MPP. For most relablty problems, SORM can mprove FORM by provdng curvature nformaton around the MPP, whch ncreases the computatonal effort needed to calculate the second dervatves of performance functon wth respect to random varables. For hghly nonlnear problems, the lnear/quadratc approxmaton n FORM/SORM can cause errors n the estmaton of probablty. For example, f the MPP s an nflecton pont, or f the falure surface around the MPP s very flat, then FORM/SORM can lead to a very large error n probablty calculaton. HORM was proposed to mprove pont-ftted SORM and approxmates the hghly nonlnear surface by usng so-called two pont adaptve nonlnear approxmaton. Ths method can solve smple, hghly nonlnear problems. However, for practcal applcatons,

88 70 when the number of random varables s large and response calculaton s tme consumng, the computatonal effcency of HORM s low, and may not be useful for solvng ndustral-scale structural relablty problems. (3) Smulaton methods are appled when no feasble analytcal soluton s possble, or when some approxmate methods need to be verfed. Drect Monte Carlo smulaton generally requres a large number of smulatons to calculate small probablty, and s mpractcal when each smulaton nvolves expensve fnte-element, boundaryelement, or mesh-free calculatons. As a result, researchers have developed or examned faster smulaton methods (see secton 2.5). The most dffculty of these samplng methods s, they need to determne the most probable falure regon, or chose an approprate PDF n advance, dependng on the falure regon. Hence, smulaton methods are useful when alternatve methods are napplcable or naccurate, and have been tradtonally employed as a benchmark for evaluatng approxmate methods. (4) For most practcal applcatons, the performance/response s usually mplct. Such a determnstc predcton can be very tme consumng (e.g., large-scale fnte element analyss), and prevents the mplementaton of smulaton methods. The response surface method provdes an approxmaton of performance/response by usng ftted polynomals. Then, based on explct functons, a broad range of analytcal methods/smulaton methods can be used effcently. Snce most response surface methods are second order, they may not be adequate for hghly nonlnear problems; the model s accuracy cannot be adequately assessed and controlled outsde selected

89 7 data regons. The requred number of orgnal model evaluatons ncreasng dramatcally for those response surface methods wth full cross terms n the case of a large number of random varables. (5) The major advantage of mean-pont-based decomposton methods over FORM/SORM s that hgher order approxmatons of a performance functon can be acheved wthout calculatng the MPP or gradents. Thus, these methods can solve hghly nonlnear relablty problems more accurately and/or more effcently. However, for a certan class of relablty problems these methods may requre computatonally demandng hgher-varate (bvarate, trvarate, etc.) decomposton to adequately represent performance functon, whch wll add computatonal effort sgnfcantly. Relablty-based Desgn Optmzaton (6) Solvng RBDO wth the double-loop approach s expensve because of the nherent computatonal expense requred for a relablty analyss n the nner loop. For a large-scale multdscplnary system wth a large number of random/desgn varables and falure modes, ths method s not practcal due to hgh computatonal costs. Although the decoupled method (e.g. sequental optmzaton relablty assessment) and the sngle-loop approach wth the KKT condton mprove the computatonal effcency, these methods may not yeld requred accuracy and convergence. If the relablty problem s hghly nonlnear or multple MPPs exst, then FORM-based RBDO methods may obtan an undesrable probablstc optmal desgn.

90 72 (7) Smulaton-based RBDO approaches yeld better accuracy than FORM-based RBDO approaches. However, the huge computatonal cost of samplng only makes them applcable to smple problems. Addtonally, the optmzaton algorthm assocated wth the outer loop may face dffculty n searchng the optmum, due to the samplng errors of dervatves of probablstc constrants. Even though response surface approxmaton of the dervatves may overcome some of these dffcultes generated from samplng, another source of approxmaton s nvolved. (8) The exstng mean-pont-based decomposton methods ntroduced recently n RBDO can mprove both effcency and accuracy for hghly nonlnear problems. The cost of functon evaluatons n the proposed method can be predetermned, whle most exstng optmzaton methods cannot predct the cost of the optmzaton procedure apror. Based on the same comment assocated wth the relablty analyss for the mean-pont-based decomposton method, a hgher-varate approxmaton, f requred, may ncrease the computatonal costs of RBDO sgnfcantly Need for Fundamental Research Based on the revew descrbed n the precedng sectons, the followng fundamental research should be pursued: () The decomposton methods for relablty analyss depend on the selected reference or expanson pont. It s elementary to show that an mproper or careless selecton of the reference pont can spol the approxmaton. Past work ndcates that the mean pont of random nput s a good canddate for defnng the reference pont. However,

91 73 for certan class of relablty problems, exstng mean-pont-based decomposton methods may requre computatonally demandng bvarate or trvarate decompostons to adequately represent performance functons. Hence, developng unvarate methods, capable of producng computatonally effcent, yet suffcently adequate performance functons, s a major motvaton of the current work. The present work s motvated by the argument that usng MPP as the reference pont may provde an mproved functon approxmaton, however, wth the addtonal expense of dentfyng the MPP. (2) The mean pont- or MPP-based decomposton methods nvolve further layers of approxmatons, due to both response-surface generaton of unvarate or bvarate component functons and Monte Carlo smulaton. However, the MPP-based unvarate decomposton, f approprately cast n the rotated Gaussan space, permts an effcent evaluaton of the falure probablty by closed-form solutons. In other words, t s possble to perform a general falure probablty analyss, whch represents a mult-dmensonal ntegraton over an arbtrary regon, by multple onedmensonal ntegratons. Therefore, developng closed-form solutons of relablty wthout relyng on response surface generaton or Monte Carlo smulaton s proposed. (3) The exstence of multple MPPs n relablty analyss can result n large errors by currently avalable methods. Even f all MPPs can be dentfed, the hgh nonlnearty around some or all MPPs may lead to nadequate accuracy or unacceptable effcency by usng exstng mult-pont FORM/SORM. Therefore, developng a unvarate

92 74 decomposton method that can handle multple MPPs and yeld superor accuracy or computatonal effcency than exstng methods s hghly desrable. The proposed research wll extend the MPP-based unvarate method for solvng multple MPP problems. (4) A major by-product of formulatng closed-form solutons for determnng falure probablty by MPP-based unvarate decomposton method s the lkelhood of developng analytcal senstvtes of relablty wth respect to desgn varables. Such senstvtes are useful for subsequent relablty-based desgn optmzaton and should be developed. (5) The ultmate goal of a relablty analyss s desgn optmzaton of mechancal and structural systems n the presence of uncertantes. If the results of both relablty and senstvty are accurate and/or computatonally effcent, the assocated relablty-based desgn optmzaton wll also be effectve. Therefore, the fnal goal of the proposed research s to develop a new RBDO methodology employng the MPP-based unvarate decomposton method.

93 75 u 2 v 2 Falure set g U (u) < 0 FORM [g L (u) = 0] g U (u) = 0 β HL MPP (u * or v * ) SORM [g Q (u) = 0] u v Fgure 2. MPP at the 2D standard normal space

94 76 Search for MPP Coordnates Transformaton Y=RU Select two addtonal ponts, and construct an adaptve approxmaton y n Construct =β+ n = a y m Estmate of falure probablty by Gauss Hermte ntegral m- even a - postve m- even a -negatve m- even a -postve & negatve m- odd a -all postve or all negatve m- odd a -postve & negatve Fgure 2.2 Schematc flowchart for HORM

95 77 u 2 u 2 β O β β β u O β β u 2 MPPs 4 MPPs u 2 u 2 β β O β 2 u O u MPP, but 2 mportant regons Infnte MPPs Fgure 2.3 Multple MPPs

96 78 CHAPTER 3 MPP-BASED UNIVARIATE METHOD WITH SIMULATION 3. Multvarate Functon Decomposton at MPP Consder a contnuous, dfferentable, real-valued performance functon g(x) that depends on x = { x,, x } T N N. If u = { u,, u } T N N s the standard Gaussan space, let { u u N } u * = * * denote the MPP or beta pont, whch s the closest pont on,, T the lmt-state surface to the orgn. The MPP has a dstance β HL, whch s commonly referred to as the Hasofer-Lnd relablty ndex (Madsen et al., 986), s determned by a standard nonlnear constraned optmzaton. Construct an orthogonal matrx R N N whose Nth column s α * u * βhl,.e., N N R R α, where R satsfes * = α R = 0 * T N. The matrx R can be obtaned, for example, by Gram-Schmdt orthogonalzaton. For a orthogonal transformaton u= Rv, let v = { v,, v } T N N represent the rotated Gaussan space wth the assocated MPP T { v v v } { T v =,,, = 0,,0, β }. The transformed lmt states h ( u ) = 0 and * * * * N N HL y ( v ) = 0 are therefore the maps of the orgnal lmt state g ( x) = 0 n the standard Gaussan space (u space) and the rotated Gaussan space (v space), respectvely. Fgure 3. depcts FORM and SORM approxmatons of a lmt-state surface at MPP for N = 2. Suppose that y(v) has a convergent Taylor seres expanson at MPP { v v N } v * = * * and can be expressed by,, T

97 79 j y j y( v) = y v + v v v + R N * * * ( ) j ( )( ) j! v j= = 2 (3.) or j y y( v) = y v + v v v j! v N * * * ( ) j ( )( ) j= = j+ j2 y + j! j! v v j j2 j, j2> 0 2 < 2 2 j * * * ( v )( v v ) ( v v ) j2 2 2 j + R 3, (3.2) where the remander R 2 denotes all terms wth dmenson two and hgher and the remander R 3 denotes all terms wth dmenson three and hgher. 3.. Unvarate Approxmaton Consder a unvarate approxmaton of y( v), denoted by N * * * * * ( v) ˆ(,, N) = (,,,, +,, N ) ( ) ( v ) = yˆ y v v y v v v v v N y, (3.3) where each term n the summaton s a functon of only one varable and can be subsequently expanded n a Taylor seres at v = * v, yeldng N j * * * j yˆ ( v) = y( v ) + ( v )( v v ) j j= j! = x y. (3.4) Comparson of Equatons (3.) and (3.4) ndcates that the unvarate approxmaton leads to the resdual error y( v) y ( v) = R, whch ncludes contrbutons from terms of ˆ 2 dmenson two and hgher. For suffcently smooth y(v) wth convergent Taylor seres, the coeffcents assocated wth hgher-dmensonal terms are usually much smaller than that wth one-dmensonal terms. As such, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected. Nevertheless, Equaton (3.4) ncludes all

98 80 hgher-order unvarate terms, as compared wth FORM and SORM, whch only retan lnear and quadratc terms, respectvely. Hence, yˆ ( v) yelds more accurate representaton of y( v) than FORM/SORM. Furthermore, Equaton (3.4) represents exactly the same functon as y( v ) when y( v ) = y ( v ),.e., when y(v) can be addtvely decomposed nto functons y (v ) of sngle varables Bvarate Approxmaton In a smlar manner, consder a bvarate approxmaton * * * * * * ( + + N ) yˆ ( v) = y v,, v, v, v,, v, v, v,, v < 2 N ( N )( N 2) ( N 2) y v,, v, v, v,, v + y 2 = * * * * * ( + N ) ( v ) (3.5) of y(v), where each term on the rght hand sde s a functon of at most two varables and can be expanded n a Taylor seres at v = * v, yeldng y y yˆ ( v) = y v + v ( v v ) + v v v v v. 2 N j j+ j2 j * * * j * * ( ) j ( ) j j ( )( ) ( j * j= j! = v j, j2 0! 2! ) > j j < v 2 v 2 (3.6) Agan, the comparson of Equatons (3.2) and (3.6) ndcates that the bvarate approxmaton leads to the resdual error y( v) yˆ 2( v) = R 3, n whch the remander R 3 ncludes terms of dmenson three and hgher. The bvarate approxmaton ncludes all terms wth no more than two varables, thus yeldng hgher accuracy than the unvarate approxmaton. Furthermore, Equaton (3.6) exactly represents y( v) y ( v, v ), = j j

99 8.e., when y(v) can be addtvely decomposed nto functons y j (v, v j ) of at most two varables Generalzed S-varate Approxmaton The procedure for unvarate and bvarate approxmatons descrbed n the precedng can be generalzed to an S-varate approxmaton for any nteger S N. The generalzed S-varate approxmaton of y(v) s S N S + * * * * * * yˆ S( v) ( ) y( v,, vk, vk, v k+,, vk, v,,, ) S k v S k v S +. N = 0 k < < ks (3.7) * * * * * * If y y( v,, v, v, v,, v, v, v,, v ); 0 R S, a R k k k+ kr kr kr+ N multvarate functon decomposton theorem, developed by the frst author s group, leads to (Xu and Rahman, 2004) y R R N k = tk; 0 R S k = 0 R k, (3.8) where t 0 * ( v ) N y t = v v j! v j j = j j * * ( v )( ) j j2 j, j2 2 < 2 2 y t2 = v v v v j! j! v v t S = y j+ j2 j2 j js j,, js S < < S S j * * * ( v )( ) ( ) 2 2 y = j! j! v v j + + js j * * * ( v )( v v ) ( v v ) S S js. (3.9)

100 82 Usng Equatons (3.8) and (3.9), t can be shown that yˆ ( v) S n Equaton (3.7) conssts of all terms of the Taylor seres of y( v) that have less than or equal to S varables (Xu and Rahman, 2004). The expanded form of Equaton (3.7), when compared wth the Taylor expanson of y( v), ndcates that the resdual error n the S-varate approxmaton s y( v) y ˆS( v) = R S +, where the remander R S + ncludes terms of dmenson S + and hgher. When S =, Equaton (3.7) degenerates to the unvarate approxmaton (Equaton (3.3)). When S = 2, Equaton (3.7) becomes the bvarate approxmaton (Equaton (3.5)). Smlarly, trvarate, quadrvarate, and other hgher-varate approxmatons can be derved by approprately selectng the value of S. In the lmt, when S = N, Equaton (3.7) converges to the exact functon y( v). In other words, the decomposton technque generates a convergent sequence of approxmatons of y( v) Remarks The decomposton of a general multvarate functon y( v) can be vewed as a fnte sum N N N v = S + + s 2 N N =, 2=,, S = < 2 < < S = yˆ ( v) y( ) y y ( v ) y ( v, v ) y ( v,, v ) y ( v,, v ) = yˆ 2 ( v) = yˆ S ( v), (3.0)

101 83 where y 0 s a constant, y ( v ) s a unvarate component functon representng ndependent contrbuton to y(v) by nput varable v actng alone, y ( v, v ) 2 2 s a bvarate component functon descrbng cooperatve nfluence of two nput varables v and v, 2 y ( v,, v ) S S nput varables s an S-varate component functon quantfyng cooperatve effects of S v,, v, and so on. By comparng Equatons (3.3) and (3.5) wth S Equaton (3.0), the unvarate and bvarate approxmatons provde two- and three-term approxmants, respectvely, of the fnte decomposton. In general, the S-varate approxmaton n Equaton (3.7) yelds the S+-term approxmant of the decomposton. The fundamental conjecture underlyng ths work s that component functons arsng n the proposed decomposton wll exhbt nsgnfcant hgher-dmensonal effects cooperatvely. It s worth notng that the unvarate approxmaton n Equaton (3.3) should not vewed as frst- or second-order Taylor seres expansons nor does t lmt the nonlnearty of y(v). Accordng to Equaton (3.4), all hgher-order unvarate terms of y(v) are ncluded n the proposed approxmaton. In fact, the unvarate component functon y ( v ) can be hghly nonlnear and therefore should provde n general hgherorder representaton of a performance functon than those by FORM or SORM. Furthermore, the approxmatons contan contrbutons from all nput varables. Fnally, the decomposton presented here depends on the selected reference pont. It s elementary to show that an mproper or careless selecton of the reference pont can spol the approxmaton. The authors past work ndcates that the mean pont of random

102 84 nput s a good canddate for defnng the reference pont (Xu and Rahman, 2005). Ths present work s motvated by the argument that usng MPP as the reference pont may provde an mproved functon approxmaton, however, wth the addtonal expense of dentfyng the MPP. 3.2 Response Surface Generaton Consder the unvarate component functon y ( v ) y( v,, v, v, v,, v * ) * * * + N n Equaton (3.4). If for v = v ( j), n functon values ( j) * * ( j) * * y ( v ) = y( v,, v, v, v,, v ); j =,2,, n + N (3.) are gven, the functon value for arbtrary nterpolaton as v can be obtaned usng the Lagrange where the shape functon φ ( v ) j n ( j) y ( v ) = φ ( v ) y ( v ), (3.2) j j= s defned as φ ( v ) = k=, k j j n n ( k ) ( v v ) ( j) ( k) ( v v ) k=, k j. (3.3) By usng Equatons (3.3) and (3.4), arbtrarly many values of y ( v ) can be generated f n values of that component functon are gven. The same procedure s repeated for all unvarate component functons,.e., for all y ( v ), =,, N. Therefore, the total cost

103 85 for the unvarate approxmaton n Equaton (3.4), n addton to that requred for locatng MPP, entals a maxmum of nn + functon evaluatons. More accurate bvarate or multvarate approxmatons (e.g., Equatons (3.6) or (3.8)) can be developed n a smlar way. However, because of much hgher cost of multvarate approxmatons, only the unvarate approxmaton wll be examned n ths paper. 3.3 Monte Carlo Smulaton For component relablty analyss, the Monte Carlo estmate P F, of the falure probablty employng the proposed unvarate approxmaton s P N S () = I yˆ ( ) < 0 N = v, (3.4) F, S where () v s the th realzaton of V, NS s the sample sze, and I[] s an ndcator () functon such that I= f () v s n the falure set (.e., when y ˆ ( v ) < 0) and zero otherwse. Smlar falure probablty estmates can be developed usng hgher-varate models f requred. In addton, smlar approxmatons can be employed for system relablty analyss (Xu and Rahman, 2005). The decomposton method nvolvng unvarate approxmaton (Equaton (3.4)), n-pont Lagrange nterpolaton (Equatons (3.3) and (3.4)), and Monte Carlo smulaton (Equaton (3.5)) s defned as the MPP-based unvarate method n ths chapter. Snce the unvarate method leads to explct response-surface approxmaton of a performance functon, the embedded Monte Carlo smulaton can be conducted for any sample sze.

104 86 However, the accuracy and effcency of the resultant falure-probablty calculaton depend on both the unvarate and response surface approxmatons. They wll be evaluated usng several numercal examples, as follows. 3.4 Numercal Examples Four numercal examples nvolvng explct functons from mathematcal or soldmechancs problems (Examples and 2) and mplct functons from structural or soldmechancs problems (Examples 3 and 4), are presented to llustrate the MPP-based unvarate response-surface method developed. Whenever possble, comparsons have been made wth exstng mean-pont-based unvarate response-surface method, FORM/SORM, and smulaton methods to evaluate the accuracy and computatonal effcency of the proposed method. For the MPP-based unvarate response-surface method, n (= 3, 5, or 7) unformly dstrbuted ponts v ( n ) 2, v ( n 3) 2,, v,, v + ( n 3) 2, v + ( n ) 2 were deployed at v - * * * * * coordnate, leadng to locatng the MPP. ( n ) N functon evaluatons n addton to those requred for When comparng computatonal efforts by varous methods, the number of orgnal performance functon evaluatons s chosen as the prmary metrc n ths paper. For the drect Monte Carlo smulaton, the number of orgnal functon evaluatons s the same as the sample sze. However, n unvarate response-surface methods, they are dfferent, because the Monte Carlo smulaton (although wth same sample sze as n drect Monte Carlo smulaton) embedded n the proposed method s conducted usng

105 87 ther response surface approxmatons. The dfference n CPU tmes n evaluatng an orgnal functon and ts response surface approxmaton s sgnfcant when a calculaton of the orgnal functon nvolves expensve fnte-element analyss, as n Examples 3 and 4. However, the dfference becomes trval when analyzng explct performance functons, as n Examples and 2. Hence, the computatonal effort expressed n terms of functon evaluatons alone should be carefully nterpreted for explct performance functons. Nevertheless, the number of functon evaluatons provdes an objectve measure of the computatonal effort for relablty analyss of realstc problems Example Set I Mathematcal Functons (Example ) Consder a cubc and a quartc performance functons (Grandh and Wang, 999), expressed respectvely by g( X, X2) = ( X+ X2 20) + ( X X2) (3.5) and 4 g( X, X2) = 5 + ( X+ X2 20) 33 ( X X 2), (3.6) where X N(0,3), =,2 are ndependent, Gaussan random varables, each wth mean µ = 0 and standard devaton σ = 3. From an MPP search, v * = {0,2.2257} T and * β HL = = * v for the cubc functon and v = {0, 2.5} T and * β HL = = v 2.5 for the quartc functon, as shown n Fgures 3.2(a) and 3.2(b), respectvely. In addton, Fgures 3.2(a) and 3.2(b) plot exact lmt-state surfaces and ther varous approxmatons by FORM/SORM (Bretung, 984; Hohenbchler et al., 987; Ca and Elshakoff, 994),

106 88 mean-pont-based unvarate response-surface method (Xu and Rahman, 2005), and proposed MPP-based unvarate response-surface method. For unvarate responsesurface methods, a value of n = 5 was selected, resultng 9 functon evaluatons. Accordng to Fgures 3.2(a) and 3.2(b), the MPP-based unvarate response-surface method yelds exact lmt-state equatons, snce both performance functons consdered are unvarate functons and at most consst of fourth-order polynomal n the rotated Gaussan space. For the cubc functon, the lmt-state equaton by mean-pont based unvarate method matches the exact equaton only at MPP. However, for the quartc functon, the mean-pont-based lmt-state equaton s non-negatve, leadng to a null falure set. FORM and SORM yeld grossly naccurate representaton of both lmt-state equatons, due to zero (nflecton pont of the cubc functon) or very small (hghly nonlnearty of the quartc functon) curvatures at MPP. Tables 3. and 3.2 show the results of the falure probablty calculated by FORM, SORM due to Bretung (984), Hohenbchler (987), and Ca and Elshakoff (994), mean-pont-based unvarate response-surface method (Xu and Rahman, 2005), proposed MPP-based unvarate response-surface method, and drect Monte Carlo smulaton usng 0 6 samples. The MPP-based unvarate response-surface method predcts exact probablty of falure. The unvarate response-surface method usng mean pont, whch yelds poor approxmatons of performance functons [see Fgures 3.2(a) and 3.2(b)], underpredcts (cubc functon) or fals (quartc functon) to provde a soluton. Other commonly used relablty methods, such as FORM and SORM, underpredct falure probablty by 3 percent and overpredct falure probablty by 7 percent when

107 89 compared wth drect Monte Carlo results. The SORM results are the same as the FORM results, ndcatng that there s no mprovement over FORM for problems nvolvng nflecton ponts or hgh nonlnearty Example Set II Sold Mechancs Problems Example 2 Burst Margn of a Rotatng Dsk Consder an annular dsk of nner radus R, outer radus R o, and constant thckness t R o (plane stress), as shown n Fgure 3.3. The dsk s subject to an angular velocty ω about an axs perpendcular to ts plane at the center. The maxmum allowable angular velocty ωa when tangental stresses through the thckness reach the materal ultmate strength S u factored by a materal utlzaton factor α m s (Bores and Schmdt, 2003) ( ) 3α m S u R o R ω a = 3 3 ρ( Ro R ) 2, (3.7) where ρ s the mass densty of the materal. Accordng to an SAE G- standard, the satsfactory performance of the dsk s defned when the burst margn M b, defned as M b 2 ω a 3α m S u ( R o R ) = ( Ro R ) ω ρω, (3.8) exceeds a crtcal threshold of (Penmetsa and Grandh, 2003). If random varables X =αm, X 2 = Su, X 3 = ω, X 4 = ρ, X 5 = Ro, and X 6 = R, and have ther statstcal propertes defned n Table 3.3, the performance functon becomes g( X ) = M ( X, X, X, X, X, X ) , (3.9) b

108 90 Table 3.4 presents predcted falure probablty of the dsk and assocated computatonal effort usng MPP- and mean-pont-based unvarate response-surface methods, mean-pont-based bvarate response-surface method, FORM, Hohenbchler s SORM (Hohenbchler et al., 987), and drect Monte Carlo smulaton (0 6 samples). For unvarate and bvarate response-surface methods, a value of n = 7 was selected. The results ndcate that the proposed MPP-based unvarate method and mean-pontbased bvarate method produce the most accurate soluton. The mean-pont based unvarate method sgnfcantly overpredcts the falure probablty, whereas FORM and SORM slghtly underpredct the falure probablty. The MPP-based unvarate responsesurface method surpasses both the accuracy (although margnally) and effcency of SORM and mean-pont-based bvarate response-surface method n solvng ths relablty problem Example 3 0-Bar Truss Structure A ten-bar, lnear-elastc, truss structure, shown n Fgure 3.4, was studed n ths example to examne the accuracy and effcency of the proposed relablty method. The Young s modulus of the materal s 0 7 ps. Two concentrated forces of 0 5 lb are appled at nodes 2 and 3, as shown n Fgure 3.4. The cross-sectonal area X, =,,0 for each bar follows truncated normal dstrbuton clpped at x = 0 and has mean µ = 2.5 n 2 and standard devaton σ = 0.5 n 2. Accordng to the loadng condton, the maxmum dsplacement [( v ( X,, X )] occurs at node 3, where a permssble dsplacement s 3 0 lmted to 8 n. Hence, the performance functon s

109 9 ( ) g( X ) = 8 v X,, X. (3.20) 3 0 From the MPP search nvolvng fnte-dfference gradents, the relablty ndex s * β HL = v = Table 3.5 shows the falure probablty of the truss, calculated usng the proposed MPP-based unvarate response-surface method, mean-pont based unvarate response-surface method (Xu and Rahman, 2005), FORM, three varants of SORM due to Bretung (984), Hohenbechler et al. (987) and Ca and Elshakoff (994), and drect Monte Carlo smulaton (0 6 samples). For unvarate responsesurface methods, a value of n = 7 was selected. As can be seen from Table 5, both versons of the unvarate response-surface method predct the falure probablty more accurately than FORM and all three varants of SORM. Ths s because unvarate methods are able to approxmate the performance functon more accurately than FORM and SORM. A comparson of the number of functon evaluatons, also lsted n Table 3.5, ndcates that the mean-pont-based unvarate response-surface method s the most effcent method. The number of functon evaluatons by the MPP-based unvarate response-surface method s slghtly larger than FORM, but much less than SORM Example 4 Fracture Mechancs of Functonally Graded Materal The fnal example nvolves an edge-cracked plate, presented to llustrate mxedmode probablstc fracture-mechancs analyss usng the unvarate response-surface method. As shown n Fgure 3.5(a), a plate of length L = 6 unts, wdth W = 7 unts was fxed at the bottom and subjected to a far-feld and a shear stress τ appled at the top. A 2b µ 2b 2 doman wth 2b = 2b 2 = 3.5 unts, requred to calculate the M-ntegral. The

110 92 elastc modulus and Posson s rato were unt and 0.25, respectvely. A plane stran condton was assumed. The statstcal property of the random nput = { a/ W, τ, K Ic } T X s defned n Table 3.6. Due to the far-feld shear stress τ, the plate s subjected to mxed-mode deformaton nvolvng fracture modes I and II (Anderson, 995). The mxed-mode stress-ntensty factors KI ( X ) and KII ( X ) were calculated usng an nteracton ntegral method (Yau et al. 980). The plate was analyzed usng the fnte element method (FEM) nvolvng a total of noded, regular, quadrlateral elements and 48 6-noded, quarter-pont (sngular), trangular elements at the crack-tp, as shown n Fgure 3.5(b). The falure crteron s based on a mxed-mode fracture ntaton usng the maxmum tangental stress theory (Anderson, 995), whch leads to the lmt-state equaton 2 Θ( X ) 3 Θ( X ) g( X) = KIc KI ( X)cos KII ( X)sn Θ( X ) cos 2 2, (3.2) 2 where K Ic s a determnstc fracture toughness, typcally measured from small-scale fracture experments under mode-i and plane stran condtons, and Θc ( X ) s the drecton of crack propagaton, gven by [ KII ( X) KI ( X) ] 2tan, f KII ( X ) > 0 4 KII ( X) KI ( X) Θ c ( X ) =. (3.22) 2 8 [ KII ( ) KI ( ) ] 2tan + + X X, f KII ( X ) < 0 4 KII ( X) KI ( X)

111 93 Falure probablty estmates of P = P[ g( X ) < 0], obtaned usng the proposed F MPP-based unvarate method, mean-pont-based unvarate and bvarate methods, FORM, Hohenbechler s SORM, and drect Monte Carlo smulaton, are compared n Fgure 3.6 and are plotted as a functon of E [ τ ], where E s the expectaton operator. For each relablty analyss (.e., each pont n the plot), FORM and SORM requre 29 and 42 functon evaluatons (fnte-element analyss). Usng n = 9, the mean-pont-based and MPP-based unvarate methods requre only 25 and 53 ( = ) functon evaluatons, respectvely, whereas 2 and 50,000 fnte-element analyses are needed by the mean-pont-based bvarate method and Monte Carlo smulaton, respectvely. The results clearly show that MPP-based unvarate method s more accurate than other methods, partcularly when the falure probablty s low. The computatonal effort by MPP-based unvarate method s much lower than that by mean-pont-based bvarate or smulaton methods. 3.5 Fatgue Relablty Applcatons The objectve of ths secton s to llustrate the effectveness of the proposed unvarate response-surface method n solvng a large-scale practcal engneerng problem. The problem nvolves mechancal fatgue durablty and relablty analyses of a lever arm n a wheel loader Problem Defnton and Input Fgure 3.7(a) shows a wheel loader commonly used n the heavy constructon ndustry. A major structural problem entals fatgue lfe evaluaton of lever arms, also

112 94 depcted n Fgure 3.7(a). The loadng and boundary condtons of a sngle lever arm are shown n Fgure 3.7(b). The load F E at pn E can be vewed as an nput load due to other mechancal components of the wheel loader. The determnstc constant-ampltude load cycles at pn E vary from -800 to 3200 kn and are shown n Fgure 3.7(c). The lever arm s made of cast steel wth determnstc elastc propertes, as follows: () Young s modulus E = 203 GPa, (2) Posson s rato ν = 0.3. In general, the random nput vector X, whch comprses castng defect characterstcs and materal propertes, nclude defect radus r, ultmate strength S u, fatgue strength coeffcent σ, fatgue strength exponent b, f fatgue ductlty coeffcent ε f, and fatgue ductlty exponent c. Table 3.7 defnes statstcal propertes of X. The objectve s to predct fatgue durablty and relablty of the lever arm. A value of n = 3 was selected for the proposed unvarate method Fatgue Relablty Analyss The von Mses stran-lfe method was employed for fatgue durablty analyss (Stephens et al., 200). Accordng to ths method, the Coffn-Manson-Morrow equaton for determnng fatgue crack-ntaton lfe N f at a pont s (Stephens et al., 200) ε σ f σm b = ( 2N f ) +ε f ( 2N f ) c, (3.23) 2 E where ε s the equvalent stran range and σ m s the equvalent mean stress, both of whch depend on stran and stress felds. Appendx A provdes a bref exposton of calculatng ε and σ, whch requres results of lnear-elastc fnte-element stress m

113 95 analyss. Appendx B descrbes how defect sze can be estmated from castng smulaton. Once ε and σ are calculated, the fatgue lfe Nf (X), whch depends on random m nput X, can be calculated by solvng Equaton (3.23). The fatgue falure s defned when N f (X) exceeds a desgn threshold n 0. Hence, the performance functon becomes g( X) = N ( X ) n. (3.24) f 0 A value of 7 n 0 = 0 cycles was employed n ths study Results Wthout Defects Fgure 3.8(a) shows a three-dmensonal fnte-element mesh of the lever arm nvolvng 77,54 tetrahedral elements and 7,089 nodes, whch was generated usng the ABAQUS commercal software (ABAQUS, 2002). Usng the FEM-based stress analyss and followng the procedure descrbed n Appendx A, Fgures 3.8(b) and 3.8(c) present contours of equvalent alternatng stran ( = half of equvalent stran range = ε 2 ) and equvalent mean stress ( σ m ), respectvely, of the lever arm. The MPP-based unvarate response-surface method was appled to calculate the probablty of fatgue falure PF P N f ( X ) < n 0. Snce no defects are consdered ntally, only four random varables comprsng fatgue strength coeffcent, fatgue strength exponent, fatgue ductlty coeffcent, and fatgue ductlty exponent are requred. Fgure 3.9 shows the contour plot of the relablty ndex β Φ ( P ) of the F

114 96 entre lever arm. Results ndcate that the relablty ndces are relatvely small (.e., falure probabltes are relatvely large) n Regon A where there are large strans (see Fgure 3.8(a)) or n Regon B where there are large mean stress (see Fgure 3.8(b)) and are expected. A further comparatve analyss ndcates that largest falure probabltes n Regons A and B are and , respectvely. Therefore, f the lever arm s redesgned, a natural tendency s to modfy the shape or sze of Regon A untl the falure probablty s lowered to a target value Wth Defects The probablstc analyss descrbed n the precedng can also be employed when castng-nduced shrnkage defects are consdered. However, any detrmental effect of defect sze on ε and σ and two addtonal random varables, such as defect radus and m ultmate strength Su, (see Appendx A) must be accounted for n subsequent relablty analyss. Fgure 3.0 shows the contour plot of porosty dstrbuton n the lever arm and was generated usng the MAGMASOFT commercal software (MAGMASOFT, 2002). The MAGMASOFT smulaton predcts larger porosty n Regon B n ths partcular lever arm. By followng the procedure of Appendx B, the mean radus (µ r ) of equvalent sphercal defects at three nternal (near surface) locatons, 2, and 3, sketched n Fgure 0, are estmated to be 4.4 mm, 5.5 mm, and.5 mm, respectvely. The 0 percent coeffcent of varaton and lognormal dstrbuton of r were defned arbtrarly. Table 3.8 presents predcted falure probabltes at locatons, 2, and 3, calculated wth and wthout consderng castng-nduced shrnkage porosty. Results

115 97 suggest that the presence of defect can alter falure probablty by 3 to 4 orders of magntude. It s nterestng to note that the largest falure probablty of , whch occurs n Regon B due to the presence of defect, has now become larger than the largest falure probablty of n Regon A. In other words, larger falure probablty may occur at other seemngly non-crtcal regons when castng-nduced defects are consdered. Therefore, mechancal fatgue desgn processes that do not account for castng-nduced defects may nether mprove desgn nor provde a truly relable soluton.

116 98 Table 3. Falure probablty for cubc performance functon Method Falure Probablty Number of functon evaluatons (a) MPP-based unvarate method (b) Mean-pont-based unvarate (c) method (Xu and Rahman, 2005) FORM SORM (Bretung, 984) SORM (Hohenbchler, 987) SORM (Ca and Elshakoff, 994) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 2 + ( n ) N = 2 + (5 ) 2 = 29 (c) ( n ) N + = (5 ) 2+ = 9 Table 3.2 Falure probablty for quartc performance functon Method Falure Probablty Number of functon evaluatons (a) MPP-based unvarate method (b) Mean-pont-based unvarate _ (c) _ (c) method (Xu and Rahman, 2005) FORM SORM (Bretung, 984) SORM (Hohenbchler, 987) SORM (Ca and Elshakoff, 994) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 2 + ( n ) N = 2 + (5 ) 2 = 29 (c) Fals to provde a soluton.

117 99 Table 3.3 Statstcal propertes of random nput for rotatng dsk Random Standard Probablty Varable Mean Devaton Dstrbuton α m Webull (a) S u, ks Gaussan ω, rpm Gaussan ρ, lb-sec 2 /n /g (b) /g (b) Unform (c) R o, n Gaussan R, n Gaussan (a) Scale parameter = ; shape parameter = (b) g = n/sec 2 (c) Unformly dstrbuted over (0.28,0.3). Table 3.4 Falure probablty of rotatng dsk Method Falure Probablty Number of functon evaluatons (a) MPP-based unvarate method (b) Mean-pont-based unvarate (c) method (Xu and Rahman, 2005) Mean-pont-based bvarate (d) method (Xu and Rahman, 2005) FORM SORM (Hohenbchler, 987) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 3 + ( n ) N = 3 + (7 ) 6 = 67 (c) ( n ) N + = (7 ) 6+ = 37 (d) N N n + n N + = + + = 2 2 ( ) ( ) /2 ( ) 6 (6 ) (7 ) /2 (7 ) 6 577

118 00 Table 3.5 Falure probablty of ten-bar truss structure Method Falure Probablty Number of functon evaluatons (a) MPP-based unvarate method (b) Mean-pont-based unvarate (c) method (Xu and Rahman, 2005) FORM SORM (Bretung, 984) SORM (Hohenbchler, 987) SORM (Ca and Elshakoff, 994) Drect Monte Carlo smulaton 0.394,000,000 (a) Total number of tmes the orgnal performance functons s calculated. (b) 27 + ( n ) N = 27 + (7 ) 0 = 87 (c) ( n ) N + = (7 ) 0+ = 6 Table 3.6 Statstcal propertes of random nput for an edge-cracked plate Random Standard Varable Mean Devaton Probablty Dstrbuton a/w Unform (a) τ Varable (b) 0. Gaussan K Ic Lognormal (a) Unformly dstrbuted over (0.3, 0.7). (b) Vares from 2.6 to 3..

119 0 Table 3.7 Statstcal propertes of random nput for lever arm Random Varable (a) Mean Coeffcent of Varaton Probablty Dstrbuton σ f, MPa, Lognormal b Lognormal ε f Lognormal c Lognormal S u, MPa Lognormal r, mm Varable (b) 0. Lognormal (a) Random varables S u and r are actve only when castng defects are consdered. (b) Vares as follows: 4.4, 5.5, and.5 mm at locatons, 2, and 3, respectvely (see Fgure 0). Table 3.8 Probablty of fatgue falure of lever arm at locatons, 2, and 3 Locaton Mean Defect Radus, mm Probablty of Fatgue Falure Wthout Wth Defect (a) Defect (b) (a) Random nput vector: (a) Random nput vector: X = { σ, b, ε, c} T f f 6 X = { σ, b, ε, c, S, r} T f f u 4

120 02 MPP-based unvarate method u 2 [ŷ (v) = 0] v 2 Falure set y(v) < 0 FORM y(v) = 0 β HL MPP (u * or v * ) SORM u v Fgure 3. Performance functon approxmatons by varous methods

121 03 4 v 2 u Mean-pontbased unvarate method Falure set y(v) < 0 MPP (u* or v*) β HL = FORM & SORM y(v) = 0 and MPP-based unvarate method v u (a) v 0 - β HL = 2.5 MPP (u* or v*) FORM & SORM -2 u y(v) = 0 and MPP-based unvarate method Falure set y(v) < u Fgure 3.2 Approxmate performance functons by varous methods; (a) cubc functon; (b) quartc functon (b) v 2

122 04 R R o ω Fgure 3.3 Rotatng annular dsk subject to angular velocty

123 n 360 n n ,000 lb 00,000 lb Fgure 3.4 A ten-bar truss structure

124 06 τ L/2 2b Crack a 2b 2 L/2 Integral Doman (a) W (b) Fgure 3.5 An edge-cracked plate subject to mxed-mode deformaton; (a) geometry and loads; (b) fnte-element dscretzaton

125 Probablty of fracture ntaton Smulaton (50,000 Samples) FORM (29) SORM (42) Unvarate Method at Mean (25) Bvarate Method at Mean (2) Unvarate Method at MPP (53) E[ τ ] Fgure 3.6 Probablty of fracture ntaton n an edge-cracked plate

126 08 Pn E Pn F Two lever arms Pn G (a) u y =u z =0 F E y 7.5 Pn F x Pn G Pn E u x =u z =0 F E (b) 3200 kn -800 kn cycles Fgure 3.7 A wheel loader under cyclc loads; (a) two lever arms; (b) loadng and boundary condtons of a lever arm; (c) constant-ampltude cyclc loads at pn E (c)

127 09 (a) (b) (MPa) (c) Fgure 3.8 Fnte element analyss of a lever arm; (a) mesh (77,54 elements; 7,089 nodes);(b) equvalent alternatng stran ( ε/2) contour; (c) equvalent mean stress (σ m ) contour

128 0 Regon A (Maxmum P F = 0.027) Regon B (Maxmum P F = ) Fgure 3.9 Fatgue lfe-based relablty ndex contour of lever arm

129 Locaton (µ r = 4.4 mm) Locaton 2 (µ r = 5.5 mm) Locaton 3 (µ r =.5 mm) Fgure 3.0 Porosty feld of lever arm from castng smulaton

130 2 CHAPTER 4 MPP-BASED UNIVARIATE METHOD WITH NUMERICAL INTEGRATION 4. Unvarate Decomposton at MPP The general case of multvarate functon decomposton at MPP s descrbed n Chapter 3. In ths Chapter, a unvarate decomposton at MPP s dscussed further. Consder a unvarate approxmaton of N y( v), denoted by * * * * * yˆ ( v) yˆ ( v,, v ) = y( v,, v, v, v,, v ) ( N ) y( v ), (4.) N + N = where each term n the summaton s a functon of only one varable and can be subsequently expanded n a Taylor seres at MPP T { v v v } { * * * * T N N HL v =,,, = 0,,0, β }, yeldng N j * * * j yˆ ( v) = y( v ) + ( v )( v v ) j j= j! = x In contrast, the Taylor seres expanson of y(v) at * = { v * v * N } y. (4.2),, T v can be expressed by N j * y * * j y( v) = y( v ) + j ( v )( v v ) + R j! v j= = 2 (4.3) where the remander R 2 denotes all terms wth dmenson two and hgher. A comparson of Equatons (4.2) and (4.3) ndcates that the unvarate approxmaton of yˆ ( v) leads to a resdual error y( v) y ( v) = R, whch ncludes contrbutons from terms of ˆ 2 dmenson two and hgher. For suffcently smooth y(v) wth convergent Taylor seres, the coeffcents assocated wth hgher-dmensonal terms are usually much smaller than

131 3 that wth one-dmensonal terms. As such, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected. Nevertheless, Equaton (4.) ncludes all hgher-order unvarate terms, as compared wth FORM and SORM, whch only retan lnear and quadratc unvarate terms, respectvely. It s worth notng that the unvarate approxmaton n Equaton (4.) should not vewed as frst- or second-order Taylor seres expansons nor does t lmt the nonlnearty of y(v). Accordng to Equaton (4.2), all hgher-order unvarate terms of y(v) are ncluded n the proposed approxmaton. 4.2 Unvarate Integraton for Falure Probablty Analyss The proposed unvarate approxmaton of the performance functon can be rewrtten as N * yˆ ( v) = y ( v ) + y ( v ) ( N ) y( v ), (4.4) N N = where * * * * y ( v ) y( v,, v, v, v,, v ); =, N. Due to rotatonal transformaton + N of the coordnates (see Fgure (3.)), the unvarate component functon y N ( v ) N n Equaton (4.4) s expected to be a lnear or a weakly nonlnear functon of vn. In fact, y ( v ) s lnear wth respect to vn n classcal FORM/SORM approxmatons of a N N performance functon n the v space. Nevertheless, f y N ( v ) N s nvertble, the unvarate approxmaton yˆ ( v) can be further expressed n a form amenable to an effcent relablty analyss by one-dmensonal numercal ntegraton. In ths work, both lnear

132 4 and quadratc approxmatons of are derved, as follows. y N ( v ) N and resultant equatons for falure probablty 4.2. Lnear Approxmaton of y ( v ) Consder a lnear approxmaton: yn( vn) = b0 + bv N, where coeffcents b 0 N N and b (non-zero) are obtaned by least-squares approxmatons from exact or numercally smulated responses () ( n) { yn( vn ),, yn( vn )} at n sample ponts along the v N coordnate. Applyng the lnear approxmaton, the component falure probablty can be expressed by N * P [ ( ) 0 ] [ ˆ F P y V < P y( V) < 0 ] P b0 + bv N + y( V) ( N ) y( v ) < 0, (4.5) = whch on nverson yelds P F * N ( N ) y( v ) b0 PV N < yv ( ), f b > 0 b b = * N ( N ) y( v ) b0 PV N yv ( ), f b < 0 b b =. (4.6) Snce V N follows standard Gaussan dstrbuton, the falure probablty can also be expressed by P F * N ( N ) y( v ) b0 E Φ y( V), (4.7) b b =

133 5 where u 2 E s the expectaton operator and Φ ( u) = ( 2π) exp( ξ 2) dξ s the cumulatve dstrbuton functon of a standard Gaussan random varable. Note that Equaton (4.7) provdes hgher-order estmates of falure probablty f unvarate component functons y ( v ), =, N are approxmated by hgher than second-order terms. If y ( v ), =, N retan only lnear and quadratc terms ftted wth approprately selected sample ponts, Equaton (4.7) can be further smplfed to degenerate to the well-known FORM and SORM approxmatons Quadratc Approxmaton of y ( v ) The lnear approxmaton descrbed n the precedng can be mproved by a 2 quadratc approxmaton: yn( vn) = b0 + bv N + b2vn, where coeffcents b0, b, N N and b2 (non-zero) are also obtaned by least-squares approxmatons from exact or numercally smulated responses at n sample ponts along the vn coordnate. Smlarly, the quadratc approxmaton of y N ( v ) N employed n Equaton (4.4) leads to N 2 * P [ ] [ ˆ F P y( V) < 0 P y( V) < 0 ] P b0 + bv N + bv 2 N + y( V) ( N ) y( v ) < 0. (4.8) = N * By defnng B( V ) b + y ( V) ( N ) y( v ), where V = { V,, } T N s an N-- 0 = dmensonal standard Gaussan vector, the followng solutons are derved based on two cases: V

134 6 (a) Case I Trval Soluton (b 2 4b 2 B < 0; no real roots): P F 0, f b2 > 0, f b2 < 0. (4.9) (b) Case II Non-Trval Soluton (b 2 4b 2 B 0; two real roots): P F 2 2 b b 4 bb 2 ( V ) b+ b 4 bb 2 ( V ) P < VN <, f b2 > 0 2b2 2b2 2 2 b+ b 4 bb 2 ( V ) b b 4 bb 2 ( V ) P < VN <, f b2 < 0 2b2 2b2, (4.0) yeldng P F 2 2 b2 b 2 b b 4 b2b( ) b b 4 b2b( ) + V V + E Φ E Φ. (4.) 2 2b 2 2b2 Both Equatons (4.7) and (4.) can be employed for non-trval solutons of falure probablty. Improvement of the accuracy of results, f any, depends on how strongly y ( v ) depends on vn. Furthermore, t s possble to develop a generalzed verson of N N Equaton (4.) when y N ( v ) N s hghly nonlnear (e.g., polynomal of an arbtrary order), but nvertble. However, due to the rotatonal transformaton from the x space to the v space, t s expected that the lnear approxmaton of y N ( v ) N (Equaton (4.7)) should result n a very accurate soluton. Hence, the present study s lmted to only lnear and quadratc approxmatons of y ( v ). It s worth notng that unlke Equaton (4.7), N N Equaton (4.) cannot be reduced to FORM/SORM equatons as second-order term. y N ( v ) N ncludes a

135 Unvarate Integraton The falure probablty expressons n Equatons (4.7) and (4.) nvolve calculaton of expected values of several multvarate functons of an N- dmensonal standard Gaussan vector V = { V,, } T N. A generc expresson of such calculaton V requres determnng ( f ( V )) N E Φ, where f : s a general mappng of V and depends on how unvarate component functons y ( v ), =, N are approxmated. Unfortunately, the exact probablty densty functon of f ( V ) s n general not avalable n ( ( ) closed form. For ths reason, t s dffcult to calculate E Φ f ) V analytcally. Numercal ntegraton s not effcent as ( f ( )) Φ v s a multvarate functon and becomes mpractcal when the dmenson exceeds three or four. In reference to Equaton (4.), consder agan a unvarate approxmaton of ( ) ln Φ f ( v ), expressed by N ( ) ( ) ( ) ( * ln Φ f( v ) ln Φ f( v) N 2 ln Φ f( ) v, (4.2) = ) where f ( v ) f( v,, v, v, v,, v * ) * * * + N are unvarate component functons and * * * f( v ) f( v,, vn ). Hence, { } ( f( v ) ) exp ln ( f( v ) ) Φ = Φ N * exp ln Φ( f( v) ) ( N 2) ln Φ( f( )) v = = N = Φ Φ ( f ( v )) * ( f ( v )) N 2. (4.3)

136 8 yeldng N N + E ( f V ) ( f v ) ( f v ) Φ ( ) Φ ( ) φ( v ) dv E ( f ( )) Φ V = Φ ( ) Φ ( ) = = * N 2 * N 2 ( f v ), (4.4) whch nvolves a product of N- unvarate ntegrals wth φ( ) denotng standard Gaussan probablty densty functon. Usng Equaton (4.4), the nontrval expressons of falure probablty n Equatons (4.7) and (4.) are P F N + Φ b * ( N ) y( v ) b0 y( v) = N * * ( N ) y( v ) b0 y( v ) = Φ b φ( v) dv N 2 (4.5) and P F + N 2 b b 4 b2b( v ) + Φ φ b 2b2 2 b = 2 N * b+ b 4 b2b( v ) + N 2 = 2 Φ 2b + 2 b b 4 b2b( v ) Φ φ( v) dv 2b N 2 2 * b b 4 b2b( v ) Φ 2b 2 ( v ) dv (4.6) respectvely, where * * * B( v ) B( v,, v, v, v,, v * ). The unvarate ntegraton + N nvolved n Equatons (4.5) or (4.6) can be easly evaluated by standard one-

137 9 dmensonal Gauss-Hermte numercal quadrature (Abramowtz, 972). The decomposton method nvolvng unvarate approxmaton (Equaton (4.)) and unvarate ntegraton (Equatons (4.5) or (4.6)) s defned as the MPP-based unvarate method wth numercal ntegraton n ths chapter. 4.3 Computatonal Effort and Flow Consder y ( v ) y( v *,, v *, v, v *,, v * ); =, N, for whch n functon + N values y ( v ) y( v,, v, v, v,, v ); j =,, n are requred to be evaluated ( j) * * ( j) * * + N at ntegraton ponts v = v ( j) to perform an n-order Gauss-Hermte quadrature for th ntegraton n Equatons (4.5) or (4.6). The same procedure s repeated for N- unvarate component functons,.e., for all y ( v ), =,, N. Therefore, the total cost of the proposed unvarate method ncludng n functon values of for ts lnear or quadratc approxmaton entals a maxmum of y N ( v ) N nn + requred functon evaluatons. Note that the above cost s n addton to any functon evaluatons requred for locatng the MPP. Fgure 4. shows the computatonal flowchart of the MPP-based unvarate method wth numercal ntegraton. The proposed effort n evaluatng the falure probablty has been transformed nto numercally calculatng unvarate component functons at selected nput determned by sample ponts n the v N -coordnate and Gauss- The orders of numercal ntegraton and the number of functon values of yn( vn) need not be the same. In addton, dfferent orders of ntegraton can be employed f desred.

138 20 Hermte ntegraton ponts n the v -coordnate ( =,N-). Compared wth the prevously developed unvarate method (Rahman and We, 2006), no Monte Carlo smulaton s requred n the present method. The accuracy and effcency of the new method depend on both the unvarate approxmaton and numercal ntegraton. They wll be evaluated usng several numercal examples n a forthcomng secton. In performng n-order Gauss-Hermte quadratures n Equatons (4.5) or (4.6), ( j) two optons for evaluatng y ( v ) are proposed. Opton nvolves calculatng y ( v ) at ntegraton ponts * * ( ) * * ( v,, v, v j, v,, v ); j, n + N = from drect numercal analyss (e.g., fnte element analyss). When computng y ( v ) s expensve, the frst opton s neffcent f n s requred to be large for accurate numercal ntegraton. The second opton nvolves developng frst a unvarate response-surface approxmaton of y ( v ) from selected samples ponts n the v-coordnate, followed by numercal ntegraton of the response-surface approxmaton. Opton 2 s computatonally effcent, because no addtonal numercal analyss (e.g., fnte element analyss) s requred f the order of ntegraton s larger than the number of sample ponts. However, an addtonal layer of response surface approxmaton s nvolved n the second opton. Both optons were explored n numercal examples, as follows. 4.4 Numercal Examples Fve numercal examples nvolvng explct performance functons from mathematcal or sold-mechancs problems (Examples and 2) and mplct performance

139 2 functons from structural or sold-mechancs problems (Examples 3, 4, and 5), are presented to llustrate the MPP-based unvarate method wth numercal ntegraton. Whenever possble, comparsons have been made wth the prevously developed MPPbased unvarate method wth smulaton (Rahman and We, 2006), FORM/SORM, and drect Monte Carlo smulaton to evaluate the accuracy and computatonal effcency of the new method. To obtan lnear or quadratc approxmaton of y N ( v ), n (= 5, 7 or 9) sample N ponts v ( n ) 2, v ( n 3) 2,, v,, v + ( n 3) 2, v + ( n ) 2 were * * * * * N N N N N deployed along the v N -coordnate. The same value of n was employed as the order of Gauss-Hermte quadratures n Equatons (4.5) or (4.6) of the proposed unvarate method wth numercal ntegraton. Furthermore, opton was used n Examples 3 and 4 and opton 2 was nvoked n Examples,2 and 5. When usng opton 2, an nth-order polynomal equaton was employed for generatng response-surface approxmaton of varous component functons y ( v ), =, N. For a consstent comparson, the same value of n was also employed as the number of sample ponts n the prevously developed unvarate method wth smulaton (Rahman and We, 2006). Hence, the total number of functon evaluatons requred by both versons of the unvarate method, n addton to those requred for locatng the MPP, s ( n ) N. When comparng computatonal efforts by varous methods, the number of orgnal performance functon evaluatons s chosen as the prmary metrc n ths paper.

140 Example Set I: Explct Performance Functons Example Elementary Mathematcal Functons Consder a cubc and a quartc performance functons, expressed respectvely by (Rahman and We, 2005) g( X, X2) = ( X+ X2 20) + ( X X2) (4.7) and 4 g( X, X2) = 5 + ( X+ X2 20) 33 ( X X 2), (4.8) where X N = 2 (0, 3 ),, 2 are ndependent, Gaussan random varables, each wth mean µ = 0 and standard devaton σ = 3. From an MPP search, v * = {0,2.2257} T and * β HL = = * v for the cubc functon and v = {0, 2.5} T and * β HL = = v 2.5 for the quartc functon. For both varants of the unvarate method, a value of n = 5 was selected, resultng 9 functon evaluatons. Snce both performance functons n the rotated Gaussan space are lnear n v 2, the proposed method nvolvng Equaton (4.6) was employed to calculate the falure probablty. Tables 4. and 4.2 show the results of the falure probablty calculated by FORM, SORM (Bretung, 984; Hohenbchler, et al., 987; Ca and Elshakoff, 994), MPPbased unvarate method wth smulaton (Rahman and We, 2006), proposed MPP-based unvarate method wth numercal ntegraton, and drect Monte Carlo smulaton usng 0 6 samples. The unvarate method wth smulaton, whch yelds exact lmt-state

141 23 equatons n ths partcular example, predcts the same probablty of falure by the drect Monte Carlo smulaton. The unvarate method wth numercal ntegraton also yelds exact lmt-state equatons and predcts very accurate estmates of falure probablty when compared wth smulaton results. A slght dfference n the falure probablty estmates by two versons of the unvarate method s due to approxmatons nvolved n Equatons (4.5) and (4.3) of the proposed method. Nevertheless, other commonly used relablty methods, such as FORM and SORM, underpredct falure probablty by 3 percent and overpredct falure probablty by 7 percent when compared wth the drect Monte Carlo results. The SORM results are the same as the FORM results, ndcatng that there s no mprovement over FORM for problems nvolvng nflecton pont (cubc functon) or hgh nonlnearty (quartc functon) Example 2 Burst Margn of a Rotatng Dsk Consder an annular dsk of nner radus R, outer radus R o, and constant thckness t R o (plane stress), as shown n Fgure 3.3. The dsk s subject to an angular velocty ω about an axs perpendcular to ts plane at the center. The maxmum allowable angular velocty ωa when tangental stresses through the thckness reach the materal ultmate strength S u factored by a materal utlzaton factor α m s (Bores and Schmdt, 2003) ( ) 3α m S u R o R ω a = 3 3 ρ( Ro R ) 2, (4.9) where ρ s the mass densty of the materal. Accordng to an SAE G- standard, the satsfactory performance of the dsk s defned when the burst margn M b, defned as

142 24 M b 2 ω a 3α m S u ( R o R ) = ( Ro R ) ω ρω, (4.20) exceeds a crtcal threshold of (Penmetsa and Grandh, 2003). If random varables X =αm, X 2 = Su, X 3 = ω, X 4 = ρ, X 5 = Ro, and X 6 = R, and have ther statstcal propertes defned n Table 3, the performance functon becomes g( X ) = M ( X, X, X, X, X, X ) (4.2) b Table 4.4 presents predcted falure probablty of the dsk and assocated computatonal effort usng new and exstng MPP-based unvarate methods, FORM, Hohenbchler s SORM (Hohenbchler, et al., 987), and drect Monte Carlo smulaton (0 6 samples). For unvarate methods, a value of n = 7 was selected. For the unvarate method wth numercal ntegraton, falure probabltes based on lnear (Equaton (4.5)) and quadratc (Equaton (4.6)) approxmatons are almost dentcal, whch verfes the adequacy of the lnear approxmaton of y N ( v ) N n ths example. The results also ndcate that the unvarate methods usng ether smulaton or numercal ntegraton produce the most accurate soluton. FORM and SORM slghtly underpredct the falure probablty. Both unvarate methods surpass the effcency of SORM n solvng ths partcular relablty problem Example 3 Ten-Bar Truss Structure A ten-bar, lnear-elastc, truss structure, shown n Fgure 3.4, was studed n ths example to examne the accuracy and effcency of the proposed relablty method. The Young s modulus of the materal s 0 7 ps. Two concentrated forces of 0 5 lb are

143 25 appled at nodes 2 and 3, as shown n Fgure 3.4. The cross-sectonal area X, =,,0 for each bar follows truncated normal dstrbuton clpped at x = 0 and has mean µ = 2.5 n 2 and standard devaton σ = 0.5 n 2. Accordng to the loadng condton, the maxmum dsplacement [( v ( X,, X )] occurs at node 3, where a permssble dsplacement s 3 0 ( ) lmted to 8 n. Hence, the performance functon s g( X ) = 8 v X,, X. 3 0 From the MPP search nvolvng fnte-dfference gradents, the relablty ndex s v * β HL = = Table 4.5 shows the falure probablty of the truss, calculated usng the proposed MPP-based unvarate method wth numercal ntegraton, MPP-based unvarate method wth smulaton (Rahman and We, 2006), FORM, three varants of SORM due to Bretung (984), Hohenbchler (987) and Ca and Elshakoff (994), and drect Monte Carlo smulaton (0 6 samples). For unvarate methods, a value of n = 7 was selected. As can be seen from Table 4.5, both versons of the unvarate method predct the falure probablty more accurately than FORM and all three varants of SORM. Ths s because unvarate methods are able to approxmate the performance functon more accurately than FORM/SORM. The unvarate method wth numercal ntegraton nvolvng the quadratc approxmaton of y N ( v ) N yelds slghtly more accurate result than that based on ts lnear approxmaton. A comparson of the number of functon evaluatons, also lsted n Table 4.5, ndcates that the computatonal effort by the MPP-based unvarate methods s slghtly larger than FORM, but much less than SORM.

144 Example Set II: Implct Performance Functons Example 4 Mxed-Mode Fracture-Mechancs Analyss The fourth example nvolves an sotropc, homogeneous, edge-cracked plate, presented to llustrate mxed-mode probablstc fracture-mechancs analyss usng the proposed unvarate method. As shown n Fgure 3.5(a), a plate of length L = 6 unts, wdth W = 7 unts s fxed at the bottom and subjected to a far-feld and a shear stress τ appled at the top. The elastc modulus and Posson s rato are unt and 0.25, respectvely. A plane stran condton was assumed. The statstcal property of the random nput = { aw, τ, K Ic } T X s defned n Table 4.6. Due to the far-feld shear stress τ, the plate s subjected to mxed-mode deformaton nvolvng fracture modes I and II (Anderson, 995). The mxed-mode stress-ntensty factors KI ( X ) and KII ( X ) were calculated usng an nteracton ntegral (Yau, et al., 980). The plate was analyzed usng the fnte-element method nvolvng a total of noded, regular, quadrlateral elements and 48 6-noded, quarter-pont (sngular), trangular elements at the crack-tp, as shown n Fgure 3.5(b). The falure crteron s based on a mxed-mode fracture ntaton usng the maxmum tangental stress theory (Anderson, 995), whch leads to the performance functon 2 Θ( X ) 3 Θ( X ) g( X) = KIc KI ( X)cos KII ( X)sn Θ( X ) cos 2 2, (4.22) 2

145 27 where K s statstcally dstrbuted fracture toughness and Θ ( X ) s the drecton of Ic c crack propagaton. Falure probablty estmates of P = P[ g( X ) < 0], obtaned usng the proposed F unvarate method wth numercal ntegraton, unvarate method usng smulaton, FORM, Hohenbchler s SORM, and drect Monte Carlo smulaton, are compared n Fgure 4.2 and are plotted as a functon of E [ τ ], where E s the expectaton operator. For each relablty analyss (.e., each pont n the plot), FORM and SORM requre 29 and 42 functon evaluatons (fnte-element analyss). Usng n = 9, the MPP-based unvarate methods requre only 53 (= ) functon evaluatons, whereas 50,000 fnte-element analyses were employed n the drect Monte Carlo smulaton. The results show that both versons of the unvarate method are more accurate than other methods, partcularly when the falure probablty s low. The computatonal effort by unvarate methods s slghtly hgher than that by FORM/SORM, but much lower than that by the drect Monte Carlo smulaton Example 5 Three-Span, Fve Story Frame Structure The fnal example examnes the accuracy and effcency of the proposed unvarate method for solvng relablty problems nvolvng correlated random varables. A three-span, fve-story frame structure, orgnally studed by Lu and Kureghan (986), s subjected to horzontal loads, as shown n Fgure 4.3. There are totally 2 random varables: () three appled loads, (2) two Young s modul, (3) eght moments of nerta, and (4) eght cross-sectonal areas. Tables lst the statstcal propertes of these

146 28 varables. The lognormally dstrbuted load varables are ndependent and all other random varables are assumed to be jontly normal. Falure s defned when the horzontal component of the top floor dsplacement u ( ) lmt-state functon g( ) = 0.2 u ( ) X X. X exceeds 0.2 ft, leadng to the The MPP-based unvarate methods wth numercal ntegraton and smulaton, FORM, Hohenbchler s SORM, and drect Monte Carlo smulaton were employed to estmate the falure probablty and are lsted n Table 4.0. For the relablty analyss, FORM and SORM requre 474 and 43 functon evaluatons (frame analyss), respectvely. Usng n = 7, the unvarate methods requre 595 functon evaluatons, whereas,000,000 frame analyses are needed by the drect Monte Carlo smulaton. For the unvarate method wth numercal ntegraton, the lnear approxmaton of y N ( v ) N was employed. The results clearly show that both versons of the unvarate method provde more accurate results than FORM and SORM. In terms of computatonal effort, the method developed s slghtly more expensve than FORM, but sgnfcantly more effcent than SORM. In all numercal examples presented, the number of functon evaluatons requred by both versons of the unvarate method s the same. However, the present unvarate method developed does not requre any Monte Carlo smulaton embedded n ts prevous verson. Instead, explct forms of falure probablty requrng only one-dmensonal ntegratons have been formulated. Hence, the new method should be useful n dervng senstvty (gradents) of falure probablty for relablty-based desgn optmzaton, whch s a subject of current research by the authors.

147 29 Table 4. Falure probablty for cubc performance functon Method Falure probablty Number of functon evaluatons (a) MPP-based unvarate method wth (b) numercal ntegraton MPP-based unvarate method wth (b) smulaton (Rahman and We, 2006) FORM SORM (Hohenbchler et al., 987) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 2 + ( n ) N = 2 + (5 ) 2 = 29 Table 4.2 Falure probablty for quartc performance functon Method Falure probablty Number of functon evaluatons (a) MPP-based unvarate method wth (b) numercal ntegraton MPP-based unvarate method wth (b) smulaton (Rahman and We, 2006) FORM SORM (Hohenbchler et al., 987) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 2 + ( n ) N = 2 + (5 ) 2 = 29

148 30 Table 4.3 Statstcal propertes of random nput for rotatng dsk Random Standard Probablty varable Mean devaton dstrbuton α m Webull (a) S u, ks Gaussan ω, rpm Gaussan ρ, lb-sec 2 /n /g (b) /g (b) Unform (c) R o, n Gaussan R, n Gaussan (a) Scale parameter = ; shape parameter = (b) g = n/sec 2 (c) Unformly dstrbuted over (0.28,0.3). Table 4.4 Falure probablty of rotatng dsk Falure probablty Number of functon evaluatons (a) Method MPP-based unvarate method wth numercal ntegraton Lnear (Equaton (4.5)) (b) Quadratc (Equaton (4.6)) (b) MPP-based unvarate method wth (b) smulaton (Rahman and We, 2006) FORM SORM (Hohenbchler, et al., 987) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. (b) 3 + ( n ) N = 3 + (7 ) 6 = 67

149 3 Table 4.5 Falure probablty of ten-bar truss structure Falure probablty Number of functon evaluatons (a) Method MPP-based unvarate method wth numercal ntegraton Lnear (Equaton (4.5)) (b) Quadratc (Equaton (4.6)) (b) MPP-based unvarate method wth (b) smulaton (Rahman and We, 2006) FORM SORM (Bretung, 984) SORM (Hohenbchler, et al., 987) SORM (Ca and Elshakoff, 994) Drect Monte Carlo smulaton 0.394,000,000 (a) Total number of tmes the orgnal performance functons s calculated. (b) 27 + ( n ) N = 27 + (7 ) 0 = 87 Table 4.6 Statstcal propertes of random nput for an edge-cracked plate Random Standard Probablty varable Mean devaton dstrbuton a/w Unform (a) τ Varable (b) 0. Gaussan K Ic Lognormal (a) Unformly dstrbuted over (0.3, 0.7). (b) Vares from 2.6 to 3..

150 32 Table 4.7 Frame element propertes Element Young s modulus Moment of nerta Cross-sectonal area B E 4 I 0 A 8 B 2 E 4 I A 9 B 3 E 4 I 2 A 20 B 3 E 4 I 3 A 2 C E 5 I 6 A 4 C 2 E 5 I 7 A 5 C 3 E 5 I 8 A 6 C 4 E 5 I 9 A 7

151 33 Table 4.8 Statstcal propertes of random nput for frame structure (a) Random Standard Probablty varable Mean devaton dstrbuton P 30 9 Lognormal P Lognormal P Lognormal E 4 454,000 40,000 Normal E 5 497,000 40,000 Normal I Normal I Normal I Normal I Normal I Normal I Normal I Normal I Normal A Normal A Normal A Normal A Normal A Normal A Normal A Normal A Normal (a) The unts of P, E, I, and A are kp, kp/ft 2, ft 4, and ft 2, respectvely.

152 34

153 35 Table 4.0 Falure probablty of frame structure Method Falure Probablty Number of functon evaluatons (a) MPP-based unvarate method wth (b) numercal ntegraton MPP-based unvarate method wth (b) smulaton (Rahman and We, 2006) FORM SORM (Hohenbchler, et al., 987) ,43 Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functons s calculated. (b) ( n ) N = (7 ) 2 = 600

154 36 Start Locate MPP (v * ) Develop Unvarate Approxmaton at MPP N * * * * + N = = y( v) yˆ ( v) y( v,, v, v, v,, v ) ( N ) y( v * ) Develop Lnear Approxmaton of y ( v ) N y ( v ) = b + bv N N 0 N N Develop Quadratc Approxmaton of y ( v ) y ( v ) = b + bv + b v 2 N N 0 N 2 N N N Perform Unvarate Numercal Integraton (Optons or 2) Perform Unvarate Numercal Integraton (Optons or 2) P F N + Φ * ( N ) y( v ) b0 y( v) = N * * ( N ) y( v ) b0 y( v ) = Φ b b φ( v) dv N 2 F + N 2 b b 4 b2b( v ) + Φ φ b 2b2 2 b = 2 N * b+ b 4 b2b( v ) P + N 2 = 2 Φ 2b ( v ) dv + 2 b b 4 b2b( v ) Φ φ( v) dv 2b N 2 2 * b b 4 b2b( v ) Φ 2b 2 Fgure 4. Flowchart of the MPP-based unvarate method wth numercal ntegraton

155 Probablty of fracture ntaton Smulaton (50,000 Samples) aaaaaaaaaaaaaa FORM (29) SORM (42) MPP-based Unvarate Method wth Smulaton (53) MPP-based Unvarate Method wth Numercal Integraton (53) E[ τ ] Fgure 4.2 Probablty of fracture ntaton n an edge-cracked plate

156 38 P C B B 2 B C 2 C 2 C 2 ft P C B B 2 B C 2 C 2 C 2 ft P C 2 B 2 B 3 C 3 C 3 B 2 C 2 2 ft P 2 C 2 B 2 B 3 C 3 C 3 B 2 C 2 2 ft P 3 B 3 B 4 B 3 C 3 C 4 C 4 C 3 6 ft 25 ft 30 ft 25 ft Fgure 4.3 A three-span, fve-story frame structure subjected to lateral loads

157 39 CHAPTER 5 MULTIPLE MPP PROBLEMS Ths chapter presents extenson of the unvarate method for multple most probable pont (MPP) problems. If all MPPs can be dentfed, and a hgh nonlnearty exsts n a lmt state around some or all of these MPPs, then exstng analytcal methods, such as multpont FORM/SORM, and smulaton-based methods may not be accurate or provde computatonally effcent solutons. It has been demonstrated that the MPPbased unvarate decomposton method (Chapters 3 and 4) s more accurate than analytcal methods, and more effcent than smulaton methods. In ths chapter, an ntegrated decomposton method wth the barrer method to locate all MPPs, s proposed as a new strategy to solve multple MPP problems. The barrer method s descrbed n Appendx C. 5. Performance Functon Decomposton at the mth MPP Consder a contnuous, dfferentable, real-valued performance functon g(x) that depends on {,, } T N x = x x N. The transformed lmt state h ( u) = 0 s the map of g ( x) = 0 n the standard Gaussan space (u space), as shown n Fgure 5. for N = 2. Let the performance functon contan M number of MPPs u,, M * * u wth correspondng, M dstances β, β, as shown n Fgure 5.. For the mth MPP, defne an assocated local coordnate system vm = { vm,,, vm, N}, where v mn, s the coordnate n the drecton of the MPP, as depcted n Fgure 5.. In the vm space, denote the mth MPP by v = {0,, 0, β } and the * m m

158 40 performance functon by y ( v ) = 0, whch s also a map of the orgnal lmt state m m g ( x ) = 0. A decomposton of a general multvarate functon y ( v ), descrbed by m m N N y ( v ) = y + y ( v ) + y ( v, v ) + + y ( v,, v ) m m m,0 m, m, m, 2 m, m, 2 m,2 N m, mn, =, 2= < 2 = yˆ m, ( vm) = yˆ m,2 ( vm) (5.) can be vewed as a fnte herarchcal expanson of an output functon n terms of ts nput varables wth ncreasng dmenson, where y m,0 s a constant, y ( v ): s a m, m, unvarate component functon representng ndvdual contrbuton to y m ( v ) m by nput 2 varable actng alone, y ( v, v ): s a bvarate component functon v m, m, 2 m, m, 2 descrbng cooperatve nfluence of two nput varables vm, and v, and so on. If m, 2 N N N yˆ ( v ) = y + y ( v ) + y ( v, v ) + + y ( v,, v ) (5.2) ms, m m,0 m, m, m, 2 m, m, 2 m, S m, m, s =, 2=,, S = < 2 < < S represents a general S-varate approxmaton of y ( v ), the unvarate (S = ) m m approxmaton yˆ ( v ) m, m provdes a two-term approxmant of the fnte decomposton n Equaton (5.). Smlarly, bvarate, trvarate, and other hgher-varate approxmatons can be derved by approprately selectng the value of S. The fundamental conjecture underlyng ths work s that component functons arsng n the functon decomposton wll exhbt nsgnfcant S-varate effects cooperatvely when S N, leadng to useful lower-varate approxmatons of ym( v m). In the lmt, when S = N, yˆ m,( vm) converges to the exact functon y m ( v ). In other words, Equaton (5.2) generates a herarchcal and m convergent sequence of approxmatons of y ( v ). Readers nterested n the m m

159 4 fundamental development of the decomposton method are referred to authors past work. 5.2 Mult-Pont Unvarate Decomposton Method 5.2. Unvarate Decomposton of Performance Functon At the mth MPP, consder a unvarate (S = ) approxmaton of y m ( v ), denoted m by N * ˆ m, ( vm) m, ( m,,, m, N) = m(0,,0, m,,0,, βm) ( ) m( vm ) = = ym, ( vm, ) = ym,0 yˆ y v v y v N y, (5.3) where * ym( vm ) ym(0,,0, βm) and ym, ( vm, ) ym(0,,0, vm,,0,, βm). Usng a multvarate functon theorem, t can be shown that the unvarate approxmaton yˆ ( v ) m, m leads to the resdual error y ( v ) yˆ ( v ), whch ncludes contrbutons from terms of m m m, m dmenson two and hgher. For a suffcently smooth y m ( v ) m wth a convergent Taylor seres, the coeffcents assocated wth hgher-dmensonal terms are usually much smaller than that wth one-dmensonal terms. As such, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected. Nevertheless, Equaton (5.3) ncludes all hgher-order unvarate terms. In contrast, FORM and SORM also ental unvarate approxmatons, but retan only lnear and quadratc unvarate terms, respectvely. Hence, Equaton (5.3) should provde n general a hgher-order approxmaton of the performance functon than those by commonly employed FORM/SORM.

160 Lagrange Interpolaton and Return Mappng Consder the unvarate component functon ym, ( vm, ) ym(0,,0, vm,,0,, βm) n Equaton (5.3). If for sample ponts ( j) vm, = vm, ; j =,, n, n functon values ( j) A y (0,,0, v,0,, β ); j =,, n m, j m m, m are gven, the functon value for an arbtrary where v m, can be obtaned by the Lagrange nterpolaton n n ( j) m, ( m, ) = φj( m, ) m(0,,0, m,,0,, β m) = m, jφj( m, j= j= y v v y v A v ), (5.4) φ ( v ) = k=, k j j m, n n ( k ) ( vm, vm, ) ( j) ( k) ( vm, vm, ) k=, k j (5.5) s the shape functon. By usng Equatons (5.4) and (5.5), arbtrarly many values of y ( v ) can be generated f n values of that component functon are gven. The same m, procedure s repeated for all unvarate component functons,.e., for all y ( v ), =,, N, leadng to an explct unvarate approxmaton m, m, N n yˆ ( V ) A φ ( V ) ( N ) m, m m, j j m, m,0 = j= A, (5.6) where A * m,0 ym( vm ). By developng smlar decompostons at all MPPs (.e., for all m=,, M ), unvarate approxmatons yˆ,( V ),, yˆ M,( V M ) assocated wth M number of MPPs can be generated. The functons yˆ,( V ˆ ),, y M,( V M ) represent M local approxmatons n vcntes of MPPs * * v,, vm of v,, vm spaces, respectvely. To descrbe these approxmatons

161 43 n a common space, such as the u space, consder a return mappng u= R v m m, where R = [ R ]; k, =,, N s an N N orthogonal rotaton matrx assocated wth the m m, k mth MPP. Consequently, M local approxmatons of the performance functon n the u space become N n N ˆ hm, ( U) = Am, jφj Rm, kuk ( N ) Am,0 ; m=,, M = j= k=, (5.7) as schematcally depcted n Fgure 5.. Therefore, the actual falure doman, defned by { : ( ) 0 } { : ( ) 0} ΩF x g x < = u h u < (5.8) and represented by the shaded area n Fgure 5. can be approxmated by a unon of M falure sub-domans hˆ ˆ, ( u) < 0,, h M,( u) < 0, thereby yeldng the unvarate approxmaton M ˆ ˆ Ω F = u: hm,( u) < 0. (5.9) m= Note that the boundary of the falure doman Ω ˆ F can be hghly nonlnear, whch depends on how yˆ m,( v m) or h ˆ ( ) m, u are constructed. In contrast, FORM/SORM produce only mult-lnear or mult-quadratc boundares, also plotted n Fgure 5.. Therefore, the falure doman defned by Equaton (5.9) wth a Lagrange nterpolaton order n > 2 should provde a hgher-order approxmaton than that by the mult-pont FORM/SORM Monte Carlo Smulaton Once the Lagrange shape functons φ j( v m, ) and determnstc coeffcents Am, j; j =,, n are generated for all =,, N and m=,, M, Equaton (5.7)

162 44 provdes explct local approxmatons of the performance functon n terms of the random nput U. Therefore, any probablstc characterstcs of a response, ncludng ts moments and probablty densty functon, can be easly evaluated by performng Monte Carlo smulaton on Equaton (5.7). For a component relablty analyss, the Monte Carlo estmate of the falure probablty employng the proposed unvarate approxmaton s () ( ) N S M ˆ l U I, u ˆ PF P ΩF hm ( ) 0 N S l = <, (5.0) m = where () l u s the lth realzaton of U, NS s the sample sze, and I[] s an ndcator () functon such that I = f l l u s n the falure set (.e., when u Ω F ) and zero () ˆ otherwse. Snce Equatons (5.7) and (5.9) are explct and do not requre addtonal numercal evaluatons of response (e.g., solvng governng equatons by expensve fnte element analyss), the embedded Monte Carlo smulaton can be effcently conducted for any sample sze. The proposed method nvolvng mult-pont unvarate approxmaton, n-pont Lagrange nterpolaton, and Monte Carlo smulaton s defned as the mult-pont unvarate decomposton method n ths paper. Fgure 5.2 shows the computatonal flowchart of the method developed. 5.3 Computatonal Effort The mult-pont unvarate decomposton method requres evaluaton of coeffcents A * m,0 = ym( vm ) and A = y (0,,0, v,0,, β ); for j =,, n; ( j) m, j m m, m =,, N and m=,, M. Hence, the computatonal effort requred by the proposed

163 45 method can be vewed as numercally evaluatng the orgnal performance functon at several determnstc nput defned by user-selected sample ponts. For each MPP, there are n numercal evaluatons of y m ( v ) m nvolved n Equaton (5.4). Therefore, the total cost for the mult-pont unvarate method entals a maxmum of MnN+ [ ] functon evaluatons n addton to those requred for locatng all MPPs. If the sample ponts nclude a common pont n each coordnate (see the forthcomng secton), the number of functon evaluatons reduces to M[( n ) N + ]. 5.4 Numercal Examples Three numercal examples nvolvng explct performance functons from mathematcal problems (Examples and 2) and an mplct performance functon from a structural dynamcs problem (Example 3) are presented to llustrate the mult-pont unvarate decomposton method. Comparsons have been made wth exstng multpont FORM/SORM and drect Monte Carlo smulaton to evaluate the accuracy and effcency of the new method. For the mult-pont unvarate decomposton method, n (= 3 or 5) unformly dstrbuted ponts v ( n ) 2, v ( n 3) 2,, v,, v + ( n 3) 2, v + ( n ) 2 were * * * * * m, m, m, m, m, deployed at the v m, -coordnate of the mth MPP, leadng to M[( n ) N + ] functon evaluatons n addton to those requred for locatng all MPPs. A barrer method developed by Der Kureghan and Dakessan (998) and the Hasofer-Lnd-Rackwtz- Fessler algorthm (Rackwtz, 200) were employed to fnd multple MPPs. The mult- * The numerc nsde the parenthess s due to a functon evaluaton Am,0 = ym( vm ). It should be removed f an MPP search algorthm already has the nformaton.

164 46 pont FORM/SORM also nvolved Monte Carlo estmates of the falure probablty usng ther respectve approxmate falure domans. When comparng computatonal efforts by varous methods, the number of orgnal performance functon evaluatons s selected as the prmary metrc n ths paper Example Mathematcal Functons wth Gaussan Random Varables Consder the performance functon p ( ) ( ) ( ) 2 g = A+ B X+ D C X+ D X2 X, (5.) where X = { X, X } T 2 2 s a bvarate standard Gaussan random vector wth the mean 2 vector µ X E[ X ] = 0 and the covarance matrx Σ E [( X µ )( X µ ) ] = I T 2 2 X X X ; A, B, C, D are real-valued determnstc parameters; and p s an nteger-valued determnstc parameter. By approprately selectng these determnstc parameters, component relablty problems nvolvng a sngle MPP or multple MPPs can be constructed. Three cases nvolvng quadratc, cubc, and quartc functons, each contanng two MPPs, were studed, as follows. Case I: A = 5, B = 0.5, C =, D = -0., p = 2 (Quadratc): For Case I, the quadratc lmtstate surface has two MPPs: u = (2.96,.036) wth the Hasofer-Lnd relablty ndex β = 3.094, and u 2 = ( 2.74,0.966) wth the ndex β 2 = 2.906, as shown n Fgure 5.3. The falure probablty was estmated by the proposed unvarate method (n = 3), FORM, curvature- and pont-ftted SORM, and drect Monte Carlo smulaton (0 6 samples). The results consderng ether one MPP (sngle-pont) or two MPPs (mult-pont) and

165 47 assocated computatonal efforts are lsted n Table 5.. Compared wth the benchmark result of the Monte Carlo smulaton, all sngle-pont methods generate a large amount of error regardless of whether unvarate, FORM, and SORM are employed. Ths s because both MPPs have sgnfcant contrbutons to the falure probablty. Therefore, when two MPPs are accounted for, the proposed mult-pont unvarate method (P F ) and both varants of the mult-pont SORM (P F and ) yeld hghly accurate results. The mult-pont FORM (P F ), whch slghtly underpredcts the falure probablty, s also farly accurate. Ths s because, errors n approxmatng falure domans by varous methods occur far away from the orgn. Snce the performance functon s parabolc, no meanngful dfference was observed between the results of the mult-pont unvarate method and the mult-pont SORM. However, a comparson of computatonal efforts shows slghtly or sgnfcantly better effcency of the proposed unvarate method when compared wth the pont-ftted or the curvature-ftted SORM, respectvely. Case II: A = 5, B = 0.5, C =.5, D = 2, p = 3 (Cubc): As shown n Fgure 5.4, the lmtstate surface for Case II also has two MPPs: u = (0,3) wth β = 3, and u 2 = ( 3.43,0.466) wth β 2 = Table 5.2 presents smlar comparsons of results and computatonal efforts by varous methods stated earler. For the unvarate method, a value of n = 5 was selected to capture hgher-order terms of the performance functon. The results obtaned from sngle-pont and mult-pont methods show a smlar trend as n Case I. However, snce the performance functon n ths case takes on a cubc form, the mult-pont SORM no longer predcts hghly accurate results as n Case I. Compared

166 48 wth the Monte Carlo smulaton (0 6 samples), the mult-pont FORM overestmates the falure probablty by 32 percent and the mult-pont SORM underestmates the falure probablty by 9 percent. Snce the performance functon s a unvarate functon (u space) and the frst MPP les on the u 2 axs (.e, no rotaton), the sngle-pont unvarate approxmaton at that MPP and the mult-pont unvarate approxmaton yeld the exact falure doman. Hence, both sngle-pont (frst MPP) and mult-pont unvarate methods predct the same falure probablty estmated by the drect Monte Carlo smulaton. The mult-pont unvarate method s more accurate than ether varant of the mult-pont SORM and requres only a lttle more computatonal effort than the mult-pont FORM. Case III: A = 3, B = 2, C =, D = -0., p = 4 (Quartc): The fnal case nvolves a quartc lmt-state functon that also has two MPPs, as shown n Fgure 5.5. The MPPs are: u = (0.544,2.88) wth β = 2.932, and u 2 = ( 0.364,2.877) wth β 2 = 2.9. The falure probablty estmates by varous methods and ther computatonal efforts are lsted n Table 5.3. For the unvarate method, a value of n = 5 was selected. Due to hgher nonlnearty of the performance functon n Case III than that n Cases I and II, the multpont FORM/SORM fal to provde an accurate soluton. Compared wth the benchmark result of the Monte Carlo smulaton (0 6 samples), the errors n calculatng the falure probablty by the mult-pont FORM and the mult-pont SORM are 39 and percent, respectvely. The mult-pont unvarate method s more accurate (error 2 percent) than the mult-pont FORM/SORM wth a computatonal effort slghtly hgher than that requred by the mult-pont FORM. The hgher accuracy of the unvarate method s attrbuted to a hgher-order approxmaton of the falure boundary that permts

167 49 an accurate representaton of the flat regon between two MPPs (see Fgure 5.5). The underpredcton of the mult-pont SORM s due to ts second-order approxmaton, whch cannot capture the flatness of the falure boundary n that regon. Although the mult-pont FORM approxmates that flat regon well, t fals to capture the nonlnearty of the performance functon on other sdes of the MPPs, leadng to a sgnfcant overpredcton of the falure probablty. Addtonal cases entalng hgher-order nonlnearty of the performance functon can be created n a smlar manner to show a progressve loss of accuracy by the mult-pont FORM/SORM. The results of Cases I-III demonstrate that the mult-pont unvarate method can consstently handle hgher-order relablty problems wth multple MPPs. For all three cases, the boundares of falure domans plotted n Fgures ndcate that the unvarate method yelds a better approxmaton than FORM/SORM, especally when the performance functon s hghly nonlnear. The pont-ftted mult-pont SORM exhbts a smlar computatonal effcency of the mult-pont unvarate method, because the analyss performed s only two-dmensonal. For hgher-dmensonal relablty problems, the computatonal effort by the pont-ftted SORM should grow larger than that by the unvarate method. Nevertheless, the results of the mult-pont SORM (curvatureor pont-ftted), whch captures at most a second-order approxmaton, should be carefully nterpreted when a relablty problem s hghly nonlnear Example 2 Mathematcal Functon wth Non-Gaussan Random Varables A well-known performance functon, orgnally ntroduced by Hohenbchler and Rackwtz (98) and subsequently dscussed by others (Madsen, 986; Der Kureghan, et al., 998), s

168 50 g ( ) X = 8 3 2X X 2, (5.2) where X = { X, X } T 2 2 s a bvarate random vector wth the jont cumulatve probablty dstrbuton functon F XX 2 ( x, x ) 2 ( ) ( ) ( ) exp x exp x2 + exp x+ x2 + xx2, x, x2 0 = 0, otherwse. (5.3) Due to the symmetry n Equaton (5.3) between x and x 2, there are two dstnct Rosenblatt transformatons (952) dependng on the orderng of varables { x, x 2} and { x2, x }, whch lead to mappngs { exp( x )} ( ) exp ( x ) u =Φ T ( x, x2) ( u, u2) : u2 =Φ + x2 x2 + { } (5.4) and { exp( x )} ( ) exp ( x ) u =Φ 2 T2 ( x2, x) ( u, u2) : u2 =Φ + x x + 2 { }, (5.5) respectvely, where u 2 ( u) ( 2 )exp( 2) Φ = π ξ dξ s the cumulatve dstrbuton functon of a standard Gaussan random varable. Due to the nonlnearty of transformatons, the lnear lmt-state surface n the x space becomes nonlnear functons n the u space, as depcted n Fgures 5.6(a) and 5.6(b) for transformatons T and T 2, respectvely. Regardless of the transformaton, each lmt-state surface possesses two dstnct MPPs whch are: u = (2.782,0.0865) wth β = 2.784, and u 2 = (.296,3.253) wth β 2 = 3.50 for transformaton T; and u = (.24,2.399) wth β = 2.649, and

169 5 u 2 = (3.630,0.42) wth β 2 = for transformaton T 2. The unvarate decomposton method and FORM/SORM entalng sngle and multple MPPs were appled to obtan estmates of the falure probablty, whch are presented n Tables 5.4 and 5.5 for transformatons T and T 2, respectvely. Also lsted s the reference soluton obtaned by the drect Monte Carlo smulaton nvolvng 0 6 samples. For the unvarate method, a value of n = 3 was selected. The tabulated results ndcate that the falure probablty estmates based on a sngle MPP strongly depend on the selected transformaton and the partcular MPP that s found. If an optmzaton algorthm can fnd only one (e.g., the second MPP) of these two MPPs, results based on that MPP may contan sgnfcant errors regardless of the relablty method employed. The mult-pont FORM usng the transformaton T yelds an excellent result, but also produces an erroneous result when the transformaton T 2 s chosen. In contrast, the mult-pont unvarate method and the mult-pont curvature-ftted SORM yeld excellent estmates of the falure probablty regardless of the transformaton nvoked. The maxmum errors by the mult-pont unvarate method, mult-pont FORM, and mult-pont SORM are.7, 40.9, and 2.7 percent, respectvely. Although the unvarate method and SORM have comparable accuraces, the mult-pont unvarate method s more computatonally effcent than the mult-pont SORM Example 3 Sesmc Dynamcs of a Ten-Story Buldng-TMD System In ths example, consder a 0-story shear buldng subjected to sesmc ground moton wth a tuned mass damper (TMD) placed on the roof, as shown n Fgure 5.7. A smlar problem has been dscussed by Der Kureghan and Dakessan (998) and Gupta

170 52 and Manohar (2004). The buldng has random floor masses M, =,,0 and random story stffness K, =,,0. The TMD has a random mass M0 and random stffness K 0. The combned system has random modal dampng ratos ζ, = 0,,0. The nput moton s defned by a pseudo-acceleraton response spectrum A( T, ) SH( ) a( T) ζ = ζ, where T s the perod, S = 0.6 s a scale factor, H ( ζ ) s a dampng dependent correcton factor defned by the Appled Technology Councl (US Army Corps ( ) Engneerng, 995), and at s the pseudo-acceleraton response spectrum shape for a 5 percent dampng, as shown n Fgure 5.8. The TMD s effectve n reducng the dynamc response of the buldng over a narrow band of frequences, provdng best results when ts natural frequency ω = k 0 0 m 0 s perfectly tuned to the fundamental frequency of the buldng. In realty, due to uncertantes n mass, stffness, and dampng propertes, perfect tunng between the TMD and the buldng may not occur. As a result, the TMD can be over-tuned or under-tuned, leadng to two dstnct MPPs when conductng relablty analyss of a combned buldng-tmd system. For the present relablty analyss, consder the lmt-state functon ( ) = ( ) g X V V X, (5.6) 0 base where T 33 { M, M,, M, K, K,, K,,,, } X = ζ ζ ζ s a random vector consstng of 33 ndependent random varables, Vbase ( ) X s the base shear response of the buldng whch s an mplct functon of X, and V 0 = 000 kp s an allowable threshold. Each of these random varable s lognormally dstrbuted wth respectve means and coeffcents of varatons lsted n Table 5.6. The base shear s computed by

171 53 combnng modal responses of the -DOF buldng-tmd system usng the CQC rule (Clough, 993). Each realzaton of X nvolves an egenvalue analyss of the system, the computaton of the modal contrbutons to the base shear, and ther combnaton accordng to the CQC rule. Startng from the mean nput, the frst MPP was found wth a value of the Hasofer-Lnd relablty ndex β =.37 (over-tuned). The second MPP was located wth the correspondng ndex β 2 =.846 (under-tuned). Table 5.7 summarzes varous estmates of the falure probablty, based on sngle- and mult-pont unvarate decomposton method and FORM/SORM. These results are compared wth the soluton usng the drect Monte Carlo smulaton employng 5000 samples. For the unvarate method, a value of n = 3 was selected. Falure probablty estmates by all methods that are based on a sngle MPP mprove when both MPPs are consdered. Both the multpont SORM (curvature-ftted) and mult-pont unvarate method provde very accurate results. However, by comparng the number of functon evaluatons, also lsted n Table 5.7, the mult-pont unvarate decomposton method s more computatonally effcent than the mult-pont SORM.

172 54 Table 5. Falure probablty for quadratc functon n Example (Case I) MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Mult-pont unvarate method Mult-pont FORM Mult-pont SORM (curvature-ftted) Mult-pont SORM (pont-ftted) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. Table 5.2 Falure probablty for cubc functon n Example (Case II) MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Mult-pont unvarate method Mult-pont FORM Mult-pont SORM (curvature-ftted) Mult-pont SORM (pont-ftted) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated.

173 55 Table 5.3 Falure probablty for quartc functon n Example (Case III) MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM (curvature-ftted) Sngle-pont SORM (pont-ftted) Mult-pont unvarate method Mult-pont FORM Mult-pont SORM (curvature-ftted) Mult-pont SORM (pont-ftted) Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated.

174 56 Table 5.4 Falure probablty for Example 2 (Transformaton T ) MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Mult-pont unvarate method Mult-pont FORM Mult-pont SORM Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated. Table 5.5 Falure probablty for Example 2 (Transformaton T 2 ) MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Mult-pont unvarate method Mult-pont FORM Mult-pont SORM Drect Monte Carlo smulaton ,000,000 (a) Total number of tmes the orgnal performance functon s calculated.

175 57 Table 5.6 Statstcal propertes of random nput for Example 3 Random varable Mean Coeffcent of varaton Probablty dstrbuton M,,M 0 93 kp/g 0.2 Lognormal K,,K kp/n 0.2 Lognormal M 0 58 kp/g 0.2 Lognormal K 0 22 kp/n 0.2 Lognormal ζ 0,, ζ Lognormal Table 5.7 Falure probablty for Example 3 MPP st MPP (u * ) 2nd MPP (u 2 * ) Both MPPs (u * and u 2 * ) Number of Relablty Method Falure probablty functon evaluatons (a) Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Sngle-pont unvarate method Sngle-pont FORM Sngle-pont SORM Mult-pont unvarate method Mult-pont FORM Mult-pont SORM Drect Monte Carlo smulaton (a) Total number of tmes the orgnal performance functon s calculated.

176 58 u 2 Ω F = {u: h(u) < 0} h(u) = 0 ĥ m, (u) = 0 v m,2 Unvarate SORM ĥ, (u) = 0 st MPP (u * ) FORM β mth MPP (u m * ) ĥ M, (u) = 0 β m β M Mth MPP (u M * ) u v m, Fgure 5. A performance functon wth multple most probable ponts

177 59 Start Locate all MPPs (u m * or v m *, m =,,M) Calculate all response coeffcents ( j) Am,0 = ym(0,,0, β m); Am, j = ym(0,,0, vm,,0,, β m ) for =,, N; j =,, n; and m=,, M Develop local unvarate approxmatons at all MPPs n the v m space N n yˆ ( V ) A φ ( V ) ( N ) A ; m=,, M m, m m, j j m, m,0 = j= Develop local unvarate approxmatons at all MPPs n the u space N n N ˆ hm, ( U) = Am, jφj Rm, kuk ( N ) Am,0 ; m=,, M = j= k= Defne approxmate falure doman M ˆ : ˆ Ω F = u hm,( u) < 0 m= Calculate falure probablty () ( ) N S M ˆ l U I, u ˆ PF P ΩF hm ( ) 0 N S l = < m = Fgure 5.2 Flowchart of the mult-pont unvarate decomposton method

178 60 0 Ω F = u: { h(u) < 0} 0 Ω F = u: { h(u) < 0} 8 6 FORM SORM Unvarate 8 6 SORM Exact and Unvarate u Exact u2 4 2 st MPP FORM 0 2nd MPP st MPP 0 2nd MPP u u Fgure 5.3 Quadratc lmt-state surface Fgure 5.4 Cubc lmt-state surface n Case I (Example ) n Case II (Example )

179 Ω F = {u: h(u) < 0} Exact Unvarate u SORM 2nd MPP st MPP FORM u Fgure 5.5 Quartc lmt-state surface n Case III (Example )

180 62 u Ω F = {u: h(u) < 0} 2nd MPP h(u) = 0 (transformaton T ) 0 st MPP u (a) 6 5 Ω F = {u: h(u) < 0} 4 3 h(u) = 0 (transformaton T 2 ) u 2 2 st MPP 0 2nd MPP u Fgure 5.6 Lmt-state surface of Example 2; (a) transformaton T ; (b) transformaton T 2 (b)

181 63 TMD: M 0, K 0, ζ 0 Buldng: Floor mass: M, Story stffness: K Modal dampng rato: ζ Fgure 5.7 A ten-story buldng-tmd system (Example 3) 0 Normalzed acceleraton [a(t)/g] Perod T, s Fgure 5.8 Normalzed pseudo-acceleraton response spectrum

182 64 CHAPTER 6 RELIABILITY-BASED DESIGN OPTIMIZATION BY UNIVARIATE DECOMPOSITION Ths chapter presents a unvarate decomposton method for relablty-based desgn optmzaton of mechancal systems. The method nvolves: () hgher-order, unvarate approxmaton of performance functons for relablty analyss; (2) analytcal senstvtes of falure probablty wth respect to desgn varables; (3) standard gradentbased optmzaton algorthms. The chapter begns wth a bref exposton of the RBDO formulaton n Secton 6.. Secton 6.2 brefly summarzes the MPP-based unvarate decomposton method for relablty analyss, presents new senstvty equatons for desgn varables, and desgn optmzaton. The computatonal flow and effort are descrbed n Secton 6.3. Two sets of examples, each nvolvng mathematcal functons and structural/sold-mechancs problems, llustrate the senstvty analyss and RBDO method developed n secton 6.4. Comparsons have been made wth alternatve FORM/SORM and smulaton-based methods to evaluate the accuracy and computatonal effcency of the new RBDO method. Fnally, Secton 6.5 provdes conclusons and future outlook. 6. Relablty-Based Desgn Optmzaton 6.. Generalzed RBDO Problem The mathematcal formulaton of a general stochastc optmzaton problem P nvolvng a sngle objectve functon and K < constrant functons entals the statement

183 65 [ f X d ] mn c0( d) B 0( ; ) M d D P : subject to ck( d) P X ΩF, k( d) pk; k =,, K dl d du, (6.) n whch d = { d,, d } T M D s an M-dmensonal desgn vector wth non-empty closed set M D ; {,, } T N X = X X N s an N-dmensonal random vector wth jont probablty densty functon f X ( x) defned on a probablty space (Ω, F, P), where Ω s the sample space, F s the σ-algebra, and P s the probablty measure; ΩFk, ( d) Ω, k =,, K s the kth falure doman that may depend on d; and 0 p, k =,, K are target falure probabltes, and d and d are lower and k upper bounds of d. The desgn vector d can be determnstc parameters of objectve and constrant functons and/or dstrbuton parameters of X (e.g., mean of X). The objectve functon c0 s obtaned by applyng an approprate rsk functonal B : f on a l u X random state functon f ( X; d ). For example, a common characterzaton of c 0 0, obtaned by applyng the expectaton operator E : f, s c ( d) = E[ f ( X; d )], whch nvolves X 0 0 statstcal moment analyss. In contrast, the constrant functons c k, depcted n Equaton 6., requres relablty analyss. For component relablty analyss, the falure doman Ω Fk, = { x: gk( x; d ) 0}, where gk ( xd ; ) s a sngle performance functon for each constrant. Smlar performance functons can be defned for system relablty analyss. Equaton (6.) defnes a generc sngle-objectve RBDO problem.

184 Specal RBDO Problem In engneerng applcatons, RBDO s commonly formulated assumng a determnstc state functon n the objectve functon and component falure probabltes n constrant functons, leadng to the problem P 2 wth the mathematcal statement mn c0 ( d) M d D R P2 : subject to ck( d) P gk ( X; d) < 0 pk; k =,, K dl d du, (6.2) whch s a specal case of Problem P. Solvng Problem P 2 requres only component relablty analyss n evaluatng constrants, and s the focus of the current research. The scope of Problem P 2 can be expanded by ncludng constrants nvolvng systemrelablty analyss, but they were not consdered n ths study. The optmal soluton s denoted by * M d. 6.2 Unvarate Decomposton Method Consder a contnuous, dfferentable, real-valued performance functon g ( xd ; ) = 0 that depends on {,, } T N M x = x x N and d = { d,, d } T M. If k u = {,, } T u u N N s the standard Gaussan space, let * u k denote the MPP or beta pont, whch s the closest pont on the lmt-state surface to the orgn. The MPP has a dstance βk, whch s commonly referred to as the Hasofer-Lnd relablty ndex (Madsen, et al., 986), s determned by a standard nonlnear constraned optmzaton. Construct an orthogonal matrx N N Rk whose Nth column s α * * k uk βh L,.e., * Rk = Rk, α k, where N N Rk, satsfes * T N αk R k, = 0. The matrx k, R can

185 67 be obtaned, for example, by Gram-Schmdt orthogonalzaton. For an orthogonal transformaton u= R v, let {,,,, } T N vk = vk vk N represent the rotated Gaussan k k space wth the assocated MPP T { v, v, v, } { * * * * T k = k,, kn, kn = 0,,0, βk} v. The transformed lmt states h ( ud ; ) = 0 or y ( v ; d ) = 0 are therefore the maps of the k orgnal performance functon g ( xd ; ) = 0 n the standard Gaussan space (u space) and k the rotated Gaussan space (v k space), respectvely, as shown n Fgure 6. for N = 2. k k 6.2. Relablty Analyss MPP-based Unvarate Decomposton of Performance Functon At the MPP, consder a unvarate approxmaton of y ( v ; d), denoted by k k N * ˆ k, ( vk; d) k, ( k,,, k, N; d) = k(0,,0, k,,0,, βk; d) ( ) k( vk ; d) = yˆ y v v y v N y, (6.3) where y * k( vk ; d) yk(0,,0, βk; d) and k, k, d k k, k y ( v ; ) y (0,,0, v,0,, β ; d); =, N. From author s past work (Rahman and We, 2006), t can be shown that the unvarate approxmaton yˆ k,( vk; d ) leads to the resdual error y ˆ k( vk; d) yk,( vk; d), whch ncludes contrbutons from terms of dmenson two and hgher. For a suffcently smooth y ( v ) k k wth a convergent Taylor seres, the coeffcents assocated wth hgherdmensonal terms are usually much smaller than that wth one-dmensonal terms. As such, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected. Nevertheless, Equaton (6.3) ncludes all hgher-order unvarate terms. In contrast, FORM and SORM also ental unvarate approxmatons, but retan only lnear and quadratc unvarate terms, respectvely. Hence, Equaton (6.3) should provde n

186 68 general a hgher-order approxmaton of the performance functon than those by commonly employed FORM/SORM Falure Probablty Analyss for Constrant Evaluatons The unvarate approxmaton n Equaton (6.3) can be rewrtten as N * yˆ ( v ; d) = y ( v ; d) + y ( v ; d) ( N ) y ( v ; d), (6.4) k, k kn, kn, k, k, k k = where, due to the rotatonal transformaton of the coordnates (see Fgure 6.), the unvarate component functon y ( v ; d) kn, kn, n Equatons (6.3) or (6.4) s expected to be a lnear or a weakly nonlnear functon of vn. In fact, y ( v ; d) s lnear wth respect to kn, kn, vk,n n classcal FORM/SORM approxmatons of a performance functon n the v k space. Hence, consder a lnear and quadratc approxmaton: y ( v ; d) = b ( d) + b ( d) v kn, kn, k,0 k, kn, 2 and ykn( vkn; d) = bk ( d) + bk ( d) vkn+ bk ( d ) vkn, where coeffcents bk,0, bk,,,,0,,,2, and bk,2 (non-zero) are obtaned by least-squares approxmatons from exact or numercally smulated responses { () ( n y ) kn, ( vkn, ),, ykn, ( vkn, } ) at n sample ponts along the v k,n coordnate. Applyng the lnear and quadratc approxmaton respectvely and notng that V k,n follows standard Gaussan dstrbuton, the component falure probablty embedded n the kth constrant can be expressed by N * ( N ) yk( vk ; d) bk,0 ( d) yk, ( Vk, ; d) = ck( d) = P yk ( Vk; d) < 0 E Φ, (6.5a) bk,( d) and

187 69 bk,2 bk,2 ck( d) = P yk ( Vk; d) < bk, ( ) + bk, ( ) 4 bk,2 ( ) B( k; ) + Φ d d d V d E, (6.5b) 2 bk,2( d) b ( d) b ( d) 4b E Φ 2 bk,2( d) 2 k, k, k,2 ( V k d) ( d) B ; z 2 where Φ ( z) = ( 2π) exp( ξ 2) dξ s the cumulatve dstrbuton functon of a standard Gaussan random varable. In Equaton (6.5b) ( ) ( ) N ( ) ( = B V * k, d bk d + yk Vk d N ) yk vk d, where {,0,, ; ( ; V k = Vk,,, Vk, N } ) T s an N--dmensonal standard Gaussan vector. Note that Equaton (6.5) provdes hgherorder estmates of falure probablty than that by FORM/SORM f unvarate component functons y ( v ), =, N k, k, are approxmated by hgher than second-order terms. By ntegratng wth respect to {,,,, } T N v k = vk vk N, c and k k k k,0 k, k, N = k ( d) Φ φ( vk, ) k, bk,( d) = N N * ( N ) y ( v ( β ); d) b ( d) y ( v ; d) fk ( v ) dv, (6.6a)

188 70 c ( ) b ( d) + b ( d) 4 b ( d) B V ; d 2 b N k,2 b k,2 k, k, k,2 k k ( d) + Φ φ( vk, ) k, 2 2 bk,2( ) dv d = N ( ) 2 b,( ), ( ) 4,2 ( ) ; N k d bk d bk d B Vk d Φ φ( v ) dv 2 bk,2( ) d = N k, k,, where ( v ( ) ( 2 k, ) d ( vk, ) dvk, 2 exp vk, 2) (6.6b) φ Φ = π s the probablty densty functon of a standard Gaussan random varable Desgn Senstvty Analyss In gradent-based optmzaton algorthms, dervatves of both objectve and constrant functons wth respect to each desgn varable are requred. For the RBDO problem P 2, calculatng such dervatves of the objectve functon s trval. However, the formulaton of gradents for a constrant functon s dependent on how the underlyng relablty analyss s performed. A new analytcally derved senstvty analyss of general constrant functons c ( d ); k =, K was conducted as follows. k The ntegrand of the mult-dmensonal ntegraton (Equaton (6.6)) depends on βk ( d ) and v k ( d), each of whch n turn depends on desgn d. By applyng the chan rule n Equaton (6.6), the partal dervatve of the constrant functon ck ( d) wth respect to desgn varable d s N c ( d k ) ck( ) βk ck( ) v = d + d d β d v d k, j k j= k, j, (6.7) whch nvolves four partal dervatves descrbed as follows.

189 Partal Dervatve of c k (d) wth Respect to β k Usng the unvarate approxmaton of y ( v ; d) k k n Equaton (6.4) and lnear approxmaton of the component functon y ( v ; d), kn, kn, * ( v ( ); d) y ( v ; d) y β = β k k k k k k β βk k * v k=vk b + b v + y v N y N * k,0 ( d) k, ( d) k, N k, ( k, ; d) ( ) k( vk ; d) = * v k=vk, (6.8) whch yelds * ( v ( ); d) y N k k β k * bk,0 ( d) + bk, ( d) β k + yk, (0; d) ( N ) yk ( vk ; d) β. (6.9) k βk = For a varaton n β k, coeffcents b ( k,0 d ), b ( ) k, d, and yk, (0; d) employed n representng the component functon y ( v ; d) kn, kn, s not expected to change sgnfcantly. In addton, the term * ( N ) y k ( v k ; d) n Equaton (6.4) s a constant, whch does not change wth βk n Equaton (6.9). Hence, t can be assumed that bk,0 β k = bk, β k = yk, (0; d) β k = yk( v ; d ) β k = 0 n Equaton (6.9), yeldng y * ( v ( β ); d) k k k β k b k, ( d ). For quadratc approxmaton of the component functon (6.0a) y ( v ; d), wth the kn, kn, assumpton bk,0 β k = bk, β k = bk,2 β k = yk, (0; d) β k = yk( v ; d ) β k = 0, Equaton (6.0a) s also satsfed. Further dervaton leads to ( ; ) * ( ( ); ) B Vd k yk vk βk d ( N ) ( N ) b k,( d). (6.0b) β β k k

190 72 Takng partal dervatve of Equaton (6.6) wth respect to β k, N * y ( N ) y ( ( β ); ) b ( ) y ( v ; ) ( N ) c ( ) v d d d d β φ βk bk, ( d) bk,( d) and N * ( v ( β ); d) k k k k,0 k, k, k = k N N = φ( v ) dv k, k, ( V ; d) ( V ; d) 2 k, k, k,2 k βk 2 k, k,2 k k k (6.a) B 2 ( ) ( ) b,( ), ( ) 4,2 ( ) ; N k bk bk B c d + d d V k k d d βk φ φ( v ) dv 2 2 k bk,2( ) β d b ( ), ( ) 4,2 ( ) ; k d bk d B V = k d b ( d) b ( d) 4 b ( d) B( V ; d ) φ 2 bk,2( d) N whch when combned wth Equaton (6.0), yelds N b B ( k ) ( d) 4 b ( d) B V ; d N = k, k, φ( v ) dv N * ( N ) y ( ( β ); ) b ( ) y ( v ; ) ck ( ) v d d d d φ φ β k and k, k,, (6.b) k k k k,0 k, k, N = ( N sgn ) ( bk, ( d) ) ( vk, ) dv k, bk,( d) = fk ( v ) (6.2a)

191 73 N = ( ) 2 bk, ( ) + bk, ( ) 4 bk,2 ( ) B ; k φ d d d V d 2 bk,2( d) 2 2 c ( ) b ( ), ( ) 4,2 ( ) ( ; ) k, ( ) bk, ( ) 4 bk,2 ( ) B k; d d d V d bk d bk d B V k k d d. ( N ) bk, ( d ) +φ β k 2 bk,2( d) fk ( v ) N φ( vk, ) dvk, (6.2b) Partal Dervatve of c k (d) wth Respect to v k,j The partal dervatve of falure probablty defned by constrant functon wth respect to the realzaton vk,j of the th rotated Gaussan random varable V k,j s and N * yk, j( vk, j; d) ( N ) yk ( k ( βk); ) bk,0 ( ) yk, ( vk, ; ) ck ( ) v d d d d v = k, j φ v k, j bk, ( ) bk,( ) d d N fk ( v ) N * ( N ) yk( vk ( βk); d) bk,0 ( d) yk, ( vk, ; d) N = Φ v, φ( v, ) d bk,( ) d v = fk ( v ) ck ( d). k j k k, (6.3a)

192 74 B ( V k ; d) 2 ( ) bk, ( ) bk, ( ) 4 bk,2 ( ) B( ; c k ) k d + d d V d d vk, j φ v 2 k, j 2 bk,2( ) d bk, ( d) 4 bk,2 ( d) B( V k; d ) N fk ( v ) 2 b ( ),( ), ( ) 4,2 ( ) ; N k d + bk d bk d B V k d Φ v φ( v ) dv 2 bk,2( ) d = fk ( v ) fk ( v ) k, j k, k, B ( V k ; d) 2 bk, ( d) bk, ( d) 4 bk,2 ( d) B( V k; d ) vk, j φ 2 b 2 k,2( d) bk, ( d) 4 bk,2 ( d) B ( V k ; d) N 2 bk, ( d) bk, ( d) 4 bk,2 ( d) B( V k; d ) N Φ vk, j φ( vk, ) dv 2 bk,2( ) d = fk ( v ) k,. (6.3b) Partal Dervatves of β k and v k,j wth Respect to d Of the two remanng gradents, the partal dervatve of relablty ndex β k wth respect to desgn varable d s T β k hk( ud ; ) u = d hk( ud ; ) d u = u * k, (6.4) where {,, T h = h u h u }, s the norm, and the vector dervatve k k k N T u d = { u d,, un d } s obtaned from the x-u transformaton. In Equaton L 2

193 75 (6.4), h ( ud ; ) T represents a vector of structural response senstvtes, s problem k dependent, and s calculated ether analytcally or numercally by a fnte-dfference approxmaton. Fnally, the partal dervatve v d ncluded n Equaton (6.7) s k, j obtaned from the x-v k transformaton. Both x-u and x-v k transformatons depend on the probablty dstrbuton of X and hence on a specfc RBDO problem to be solved Unvarate Numercal Integraton for Relablty and Senstvty Analyses The expressons of constrant functon n Equaton (6.6) and ther partal dervatves n Equatons (6.2) and (6.3) nvolve multvarate ntegratons over = N. A N generc evaluaton of these ntegrals requres calculatng f ( ) (, ) N k v qv k dv, where f k : N s the multvarate part of the ntegrand and depends on how unvarate component functons y ( v ), =, N are constructed, and q : s k, k, the remanng unvarate part of the ntegrand. The exact calculaton of ths ntegral s not possble n general. Numercal ntegraton s not effcent as fk ( v ) s a multvarate functon and becomes mpractcal when the dmenson exceeds three or four. In reference to Equaton (6.3), consder agan a unvarate approxmaton of [ ] ln f ( v ) at k * v {0,, 0} T = 0 N, expressed by N [ f ] f v ( N ) [ ln ( v ) ln ( ) 2 ln f k ( 0 ), (6.5) k k, k, = ] where f ( v ) f (0,,0, v,0,,0) k, k, k k, are unvarate component functons and f ( 0) f (0,,0). Hence k k

194 yeldng k N = { [ fk v ]} f ( v ) = exp ln ( ) N exp ln fk, ( vk, ) ( N 2) ln [ fk( 0) ] = = k f ( v ) k, k, f ( 0) N 2 76, (6.6) N + f ( v ) q( v ) dv N k, k, k, k, = f ( ) (, ) N k v q vk d v N 2 = fk ( 0), (6.7) whch nvolves a product of N- unvarate ntegrals. Usng Equaton (6.7) wth approprately defned f ( v ) n Equatons (6.6), (6.2), and (6.3), the falure probablty by lnear approxmaton of component functon y ( v ; d) and ther dervatves kn, kn, becomes c ( d) k N + * ( ) ( N ) yk vk ( βk); d bk,0 ( d) yk, (0; d) Φ φ( v ) dv b ( d) = k, N * ( N ) yk ( vk ( βk); d) bk,0 ( d) yk, (0; d) = Φ bk,( d) k, k, N 2, (6.8) ( ) ( d ) * ( ) + N ( N ) yk vk ( βk); d bk,0 ( d) yk, ( vk, ; d) k, φ k, vk, = bk,( d) N N 2 N sgn b ( ) φ ( v ) d ck ( d) βk * ( N ) yk ( vk ( βk); d) bk,0 ( d) yk, (0; d) = φ bk,( d), (6.9) and

195 77 * ( ) + N ( N ) yk vk ( βk); d bk,0 ( d) yk, ( vk, ; d) φ φ( vk, ) dvk, c ( ), j k,( ) k d = b d N 2 v N k, j * ( N ) yk ( vk ( βk); d) bk,0 ( d) yk, (0; d) = + N φ bk,( d) yk, j( vk, j; ) * ( N ) yk ( vk ( k); ) bk,0 ( ) yk, j( v, ; ) d β d d k j d v k, j φ φ( vk, j) dvk, j bk, ( d) bk,( d) + * ( ) ( N ) yk vk ( βk); d bk,0 ( d) yk, ( vk, ; d) Φ φ( v ) dv b ( d) =, j k, + N * ( N ) yk ( vk ( βk); d) bk,0 ( d) yk, (0; d) = Φ bk,( d) * ( ) N 2 k, k, ( N ) yk vk( βk); d bk,0 ( d) yk, j( vk, j; d) Φ vk, jφ( vk, j) dvk, j, bk,( d) (6.20) respectvely. The unvarate ntegraton nvolved n each of Equatons (6.8), (6.9), and (6.20) can be easly evaluated by standard one-dmensonal Gauss-Hermte numercal quadrature. The falure probablty by quadratc approxmaton of component functon y ( v ; d) and ther dervatves can be evaluated by the smlar forms. Equaton (6.7) kn, kn, wth partal dervatves formulated n Equatons (6.9) and (6.20) provde desgn senstvtes for a gradent-based desgn optmzaton.

196 Computatonal Flow and Effort In summary, the overall process for solvng the RBDO problem P 2 can be descrbed by the followng steps: () Defne an ntal desgn wth d = d 0. Use the fnal result of mean- or other relevant reference-pont-based optmzaton f avalable. (2) Evaluate both objectve and constrant functons for the current desgn vector. For constrant functons, use the proposed unvarate decomposton method (Equaton (6.8)) for relablty analyss. (3) Evaluate gradents of both objectve and constrant functons for the current desgn vector. For gradents constrant functons, use the proposed unvarate decomposton method (Equatons (6.7), (6.9), and (6.20)) for desgn senstvty analyss. (4) Perform determnstc optmzaton to solve Equaton (6.2) by a selected gradent-based algorthm. (5) Check for the convergence of the objectve functon and desgn vector. If the convergence s reached, stop. If not, update the desgn vector to fnd the next desgn vector and repeat steps 2 through 4. Fgure 6.2 depcts the flowchart of the proposed RBDO process. New methods were developed n the shaded areas. For determnng computatonal effort, consder y ( v ; d) y (0,,0, v,0,,0; d) ; =, N, for whch n functon values k, k, k k, ( j) ( j) y ( v ; d) y(0,,0, v,0,,0; d) ; j =,, n are requred to be evaluated at k, k, k, ntegraton ponts v = v ( j) k, k, to perform an n-order Gauss-Hermte quadrature for th

197 79 ntegraton n Equatons (6.8)-(6.20). The same procedure s repeated for N- unvarate component functons for each constrant,.e., for all y ( v ; d), =,, N k, k, and for K constrant functons,.e., for all y ( v ; d), k =,, K. Therefore, the total cost of the k k proposed unvarate method entals a maxmum of nnk functon evaluatons. Note that the above cost s n addton to any functon evaluatons requred for locatng the MPP n each constrant. Desgn senstvtes usng FORM/SORM approxmatons of falure probablty n evaluatng constrants are descrbed n Appendx D. For lnear approxmatons of at MPP, Equaton (6.7) can be further smplfed to degenerate to FORM senstvty equatons. ck ( d) 6.4 Numercal Examples Two example sets, one nvolvng two desgn senstvty problems, and the other nvolvng four RBDO problems, are presented to llustrate the proposed unvarate decomposton method. Constrants assocated wth both mathematcal functons (Examples and 3) and structural/sold-mechancs (Examples 2, 4, 5, and 6) problems were employed. Whenever possble, comparsons have been made wth the FORM/SORM, and drect Monte Carlo smulaton to evaluate the accuracy and effcency of the new method. In solvng RBDO problems (Examples 4-6), all approxmate methods employ the nested double loop for desgn and relablty teratons. No sngle-loop FORM-based methods, although avalable n the current lterature, were ncluded, as the objectve was to determne how the accuracy and effcency of a relablty analyss nfluence the optmzaton process. All structural senstvtes

198 80 ( h ( ud ; ) T ) were obtaned by the fnte-dfference method nvolvng percent k perturbatons. To obtan lnear approxmaton of ykn, ( vkn, ; d ); k =, K, n (= 5 or 7) sample ponts v ( n ) 2, v ( n 3) 2,, v,, v + ( n 3) 2, v + ( n ) 2 * * * * * kn, kn, kn, kn, kn, were deployed along the v k,n -coordnate. The same value of n was employed as the order of Gauss-Hermte quadratures n Equatons (6.6), (6.2), or (6.3) of the proposed unvarate method. Hence, the total number of functon evaluatons requred by the unvarate method, n addton to those requred for locatng the MPP, s ( n ) NK. When comparng computatonal efforts by varous RBDO methods, the number of orgnal performance functon evaluatons was chosen as the prmary metrc n ths work. The optmzaton algorthms employed were sequental quardratc programmng n Example 3, 4, and 6; and sequental lnear programmng n Example Example Set I Desgn Senstvty Analyss Example Elementary Mathematcal Functons Consder two constrant functons c ( ) = P[ g ( ; ) < 0] k d X d ; k =, 2, where the cubc and quartc performance functons are respectvely expressed by g( X; d) = ( X( d) + X2( d2) 20 ) + ( X( d) X2( d 2) ) (6.2) and g2( X; d) = + ( X( d) + X2( d) 20 ) ( X( d) X2( d 2) ), (6.22) k

199 8 n whch, Xd ( ) = { X ( d ), X ( d )} T 2 2 s a bvarate, ndependent, Gaussan random vector wth means µ = 0 and standard devatons σ = 3; =,2. From an MPP search, * v = {0,2.2257} T and * * β = v = for the cubc functon and v2 = {0, 2.5} T and β = v = for the quartc functon. For the unvarate method, a value of n = 5 was * selected, resultng 9 functon evaluatons. The desgn vector s T d = { µ, µ } = {0,0} 2 T for both functons. Table 6. presents partal dervatves c ( ) ;,2 d d = and c ( ) ;,2 2 d d =, calculated by FORM/SORM, proposed unvarate decomposton method, and Monte Carlo smulaton usng 0 6 samples. The unvarate method yelds very accurate estmates of gradents of both constrants wth a maxmum error of less than percent when compared wth smulaton results. In contrast, exstng FORM/SORM for ths partcular example contans maxmum errors of 64 and 3 percent for cubc and quartc performance functons, respectvely. The SORM results are the same as the FORM results, ndcatng that there s no mprovement over FORM for problems nvolvng nflecton pont (cubc functon) or hgh nonlnearty (quartc functon) Example 2 Ten-Bar Truss Structure A ten-bar, lnear-elastc, truss structure, shown n Fgure 6.3, was studed n ths example to examne the accuracy and effcency of the proposed unvarate method for calculatng gradents. The Young s modulus of the materal s 0 7 ps. Two concentrated forces of 0 5 lb are appled at nodes 2 and 3. The cross-sectonal area X ( d) for each bar s ndependent, follows normal dstrbuton, and has means µ = 2.5 n 2 and standard

200 82 devaton σ = 0.5 n 2 ; =,,0. Accordng to the loadng condton, the maxmum dsplacement [( v ( X ( d ),, X ( d ))] occurs at node 3, where a permssble dsplacement s lmted to 8 n. Hence, the constrant functon s ( ) c( d) = P 8 v3 X( d),, X0( d) < 0. From an MPP search, the relablty ndex s * β= v = Table 6.2 lsts ten gradents of the falure probablty of the truss,.e., c( d) d ; =,,0, whch were calculated usng the proposed unvarate method (lnear and quadratc approxmaton), FORM, SORM, and drect Monte Carlo smulaton (0 6 samples). For the unvarate method, a value of n = 7 was selected. As can be seen from Table 6.2, both SORM and the unvarate method predct dervatves of the falure probablty more accurately than FORM. Ths s because unvarate methods and SORM are able to approxmate the performance functon embedded n the constrant more accurately than FORM. The computatonal efforts to obtan these senstvtes are descrbed n Table 6.3. For all methods, no addtonal functon evaluatons other than that for relablty analyss were requred n obtanng these senstvtes. In other words, the same computatonal effort s needed to obtan both relablty and senstvty results Example Set II Relablty-based Desgn Optmzaton Example 3 Mathematcal Functons Consder a mathematcal example wth two ndependent Gaussan random varables and three nonlnear constrants. The RBDO problem s defned by

201 83 mn c ( d) = d + d 2 d X ( d) X2( d2) subject to c ( d) = P < 0 Φ( 3) 20 c ( X ( d ) + X ( d ) 5 ) ( X ( d ) X ( d ) 2) ( d) = P + < 0 Φ( 3) c3( d) = P < 0 Φ( 4) 52 ( X ( d ) + 8 X 2( d 2) + 5) 0 d 0; =,2, (6.23) where Xd ( ) = { X ( d ), X ( d )} T s an ndependent, bvarate, Gaussan random vector wth means µ and standard devatons σ = 0.3; =,2. The desgn vector s T T d = { d, d } = { µ, µ }. 2 2 Usng the ntal desgn pont { } d0 = 5,5 T, Fgure 6.4 depcts the optmzaton hstory when the constrants are evaluated by the proposed unvarate decomposton method, FORM, SORM, and Monte Carlo smulaton nvolvng 0 6 samples for each falure probablty calculaton. The detaled results presented n Table 6.4 suggest that all four methods are able to reach an optmum state n 4-6 teratons, whch yeld very close optmal solutons. Hence, each method can be used to solve ths optmzaton problem. It s nterestng to note that SORM requres fewer functon evaluatons than FORM, whch s somewhat counter-ntutve because relablty analyss by SORM s generally more expensve than that by FORM. However, an excepton may occur, when SORM leads to fewer desgn teratons than FORM n the outer loop, whch was observed n ths partcular RBDO problem. Nevertheless, the unvarate method s more effcent than

202 84 FORM or SORM, because of the fewest number of functon evaluatons requred to solve ths example Example 4 Cantlever Beam In ths example, the desgn of a fxed cantlever beam, whch has a determnstc length L = 00 nches, a random vertcal load X, a random lateral load X 2, shown n Fgure 6.5, was studed. The beam s made of a materal wth random unaxal yeld strength X 3 and random elastc modulus X 4. The wdth d and heght d 2 of the prsmatc cross-secton are two desgn varables. The objectve s to mnmze the area of the beam cross-secton so that ts total volume s mnmzed. Two nonlnear falure modes were examned. The frst falure mode s due to yeldng at the fxed end of the cantlever; and the second falure mode s assocated wth the tp dsplacement exceedng a permssble value of 2.5 nches. The RBDO problem s stated as mn c ( d) = d d 2 d X X = + < Φ, (6.24) X X 2 c2 ( d) = P < 0 ( 3.5) 4 4 Φ Xdd 4 2 d2 d 0 d 5 nches; =,2 2 subject to c( d) P X3 0 ( 2.5) dd 2 d2 d where X = { X, X, X, X } T s an ndependent, four-dmensonal, Gaussan random vector, n whch each random varable has mean and standard devaton lsted n Table 6.5. The desgn vector s d = { d, d } T. The proposed RBDO method starts wth 2 { } the ntal desgn vector d0 = 2, 4 T n.

203 85 Fgure 6.6 llustrates the optmzaton hstory of the proposed unvarate method, FORM, SORM, and Monte Carlo smulaton. Table 6.6 compares the accuracy and effcency of three approxmate methods by usng the Monte Carlo benchmark soluton. The results suggest that all three methods attan the same optmum value ( 9.2 nch 2 ) of the objectve functon. The unvarate method s slghtly more expensve than FORM, because of () addtonal functon evaluatons requred after locatng MPPs and (2) larger desgn teratons nvolved n ths partcular example. Even f the numbers of desgn teratons are the same, the unvarate method wll requre slghtly more functon evaluatons than FORM. In ths example, both FORM and unvarate methods are more effcent than SORM, a trend that s expected unless the number of desgn teratons requred by SORM s sgnfcantly fewer than others. Snce the unvarate method and FORM/SORM entals approxmate relablty analyss, the constrants assocated wth the optmal desgn generated by each method were evaluated usng the Monte Carlo smulaton (0 6 samples). Table 6.7 presents the values of falure probablty embedded n each constrant. It appears that both FORM and SORM slghtly volates the second constrant wth a maxmum error of 8 and 9 percent n calculatng the falure probablty. In contrast, no such volatons were observed n the unvarate method. Ths s because the proposed unvarate method s more accurate than FORM/SORM n performng relablty analyss n ths example.

204 Example 5 0-Bar Truss A ten-bar truss, llustrated n Fgure 6.3, was desgned by mnmzng ts total volume gven that the truss relablty s no less than a target value of Φ (2) = The RBDO formulaton s 0 d ( ) ( ) mn c ( d) = 360 d + d + d + d + d + d + 2 d + d + d + d subject to c ( d) = P 4 v3 X( d),, X0( d0) < 0 Φ( 2), (6.25) 0 d 5 nches; =,0 where X = { X ( d ),, X ( d )} T s an ndependent, Gaussan random vector wth each component representng a random cross-secton of the truss. The random varable X follows Gaussan dstrbuton, and has means µ and standard devaton σ = 0.2 n 2 ; T T. The desgn vector s d = { d,, d } = { µ,, µ }. The ntal desgn =,,0 0 0 T 2 { } pont s d0 = 3,,3 n. Fgure 6.7 and Table 6.8 present the optmzaton hstory and optmzaton results by varous methods. The optmal volumes acheved by the unvarate method, SORM, and Monte Carlo vary from 9327 to 9340 nch 3. In contrast, FORM leads to a lower optmal volume, whch s 9282 nch 3. A Monte Carlo relablty analyss at optmal desgns obtaned by FORM, SORM, and unvarate method reveals that the falure probablty estmates have assocated absolute errors of 55%, 5%, and %, respectvely. Hence, FORM volates the constrant leadng to the lower optmum volume of the truss. Both SORM and unvarate method satsfy the constrant and hence provde acceptable desgns. However, the unvarate method proposed s more effcent than SORM n solvng the truss problem.

205 Example 6 Torque Arm The fnal example nvolves desgnng a torque-arm, where eght random shape parameters X ( d ); =,8 descrbe ts outer and nner boundares, as shown n Fgure 6.8 for the mean nput at the ntal desgn. The left hole of the structure s fxed and two determnstc forces F = 2789 N and F 2 = 5066 N are appled at the center of the rght hole. The torque-arm materal has mass densty ρ = 7800 kg/m 3, elastc modulus E = 207 GPa, Posson s rato ν = 0.3, and unaxal yeld strength S y = 400 MPa. The objectve s to mnmze the mass of the structure m( d) by changng the shape of the geometry (.e., by 8 Xd ( ) ) such that the von Mses stresses at fve selected ponts do not exceed Sy. Locatons of these fve ponts, marked as fnte element nodes 90, 98, 06, 73, and 75, are llustrated n Fgure 6.9. Mathematcally, mn c ( d) = m( d) 0 d ( X d) subject to ck( d) = P Sy σ k, e ; < 0 Φ( 3); k =,5 mm d mm mm d mm mm d mm 2 mm d mm 5 mm d mm 0.5 mm d 2 mm mm d 6 mm 0.5 mm d mm, (6.26) where σ, ( X; d ) s the von Mses equvalent stress at the kth selected pont. The fnte ke element mesh ncludes 657 nodes and 77 eght-noded quadrlateral elements. A plane stress condton was assumed. The ndependent random vector X, whch represents manufacturng varablty, follows Gaussan dstrbuton. The components X has means

206 88 µ and standard devatons σ = 0.2 mm; =,,0. The desgn vector s d = { d,, d } = { µ,, µ } T T 8 8. The ntal desgn pont s { } d0 = 0,,0 T mm wth the correspondng fnte element mesh depcted n Fgure 6.8. Followng lnear-elastc stress analyss, Fgure 6.0 presents the contour plot of the von Mses stress at the ntal desgn when shape parameters assume ther mean values. Due to conservatve ntal desgn, the maxmum von Mses stress of 30 MPa, whch occurs at node 98, s much lower than the unaxal yeld strength (Sy = 400 MPa). Durng desgn teratons, the movement of nodes, whch control shape parameters X ( d ); =,8, was performed by desgn velocty feld nvolvng an soparametrc mappng (Cho and Chang, 994). For computatonal effcency, the optmal desgn was obtaned n two steps. In the frst step, a coarse RBDO was performed usng the ntal desgn d = { } 0 0,,0 T and an approxmate relablty method, known as the mean-value frst-order second moment method. The resultant desgn after 0 teratons n the frst step (coarse RBDO) s { } T d 0 = 0.427,, 0.063, 2, 0.327, 2, 0.234, mm. In the second step, a refned RBDO nvolvng the proposed unvarate method and the result of step as the ntal desgn (.e., d 0 = d 0 ) was employed. After 9 teratons, the fnal desgn was attaned, { } * whch s d = 0.709, 0.72, 0.077, 2, 0.247, 2, 0.258, mm wth the correspondng mean shape presented n Fgure 6.. The optmal mass of the torque arm s kg a 30 percent reducton from the ntal mass of 2.95 kg. Fgure 6. also dsplays the contour plot of the von Mses stress at the optmal desgn when the shape parameters assume ther mean values. Compared wth the conservatve ntal desgn of T

207 89 Fgure 6.0, larger stresses, for example 256, 247, and 226 MPa at nodes 98, 06, and 73, respectvely, can be safely tolerated n the fnal desgn of Fgure 6.. The larger area of the slotted hole and movement of outer boundares have led to sgnfcant alteraton of the shape of the ntal desgn. Fgure 6.2 shows the optmzaton hstory of the objectve functon. If the uncertanty of X s gnored and the constrants n Equaton 6.26 s replaced by ( ) Sy σ k, e dd ; < 0; k =, 5, as commonly adopted n tradtonal desgn optmzaton, { } * 3 teratons led to d =,, 0.062, 2,.785, 2,.676, mm and a correspondng optmal mass of.86 kg a 36 percent reducton from the ntal mass. Therefore, a tradtonal rsk-gnorng optmzaton process may lead to a smaller mass than that obtaned from RBDO, however wth the hgher stresses, as depcted n the T contour plot of Fgure 6.3. If uncertantes are ncluded, the optmal desgn n Fgure 6.3 s hghly lkely to volate the relablty constrants. By comparng optmal desgns from RBDO (Fgure 6.) and rsk-gnorng optmzaton (Fgure 6.3), t appears that the outer boundares generated by both desgns are smlar. However, the nner slot from the RBDO s smaller than that from the rsk-gnorng optmzaton. The prmary reason s that the latter optmzaton does not account for varablty of shape parameters and of the performance functon. In addton, the senstvty of the von Mses stress wth respect to shape parameters n the nner boundary s much larger than that n the outer boundary. In summary, the unvarate method consstently provdes very accurate RBDO solutons. Of the three methods studed, the FORM-based RBDO s the most effcent method; however, t may lead to nfeasble or naccurate desgns. Both SORM and unvarate method have comparable accuraces, but the unvarate method s less

208 90 expensve than SORM. Nevertheless, for ndustral-scale desgn applcatons, further research s requred n makng the proposed unvarate method computatonally more effcent by potentally decouplng the desgn and relablty teratons or explorng the possblty of sngle-loop formulatons.

209 9 Table 6. Gradents of two mathematcal constrant functons Methods Gradents FORM SORM Unvarate Monte Carlo (a) c ( c ( ) d d 2 c ( 2 d c ( ) 2 d d 2 (a) Sample sze = 0 6 for each smulaton; fnte dfference wth % perturbaton. Table 6.2 Gradents of the constrant n 0-bar truss Methods Gradents FORM SORM Unvarate Unvarate Monte Carlo (a) (lnear) (Quadratc) c( d ) d c( d ) d c( d ) d c( d) d c( d ) d c( d ) d c( d ) d c( d) d c( d ) d c( d ) d (a) Sample sze = 0 6 for each smulaton; fnte dfference wth % perturbaton.

210 92 Table 6.3 Computatonal efforts for 0-bar truss Methods Number of functon evaluatons FORM 27 SORM 365 Unvarate (a) 87 Monte Carlo 0 6 (a) ( n ) N ( ) 27 + = = 87. Table 6.4 Optmzaton results by varous methods for mathematcal functons Methods (a) FORM SORM Unvarate Monte Carlo No. of teratons No. of functon evaluatons Fnal desgn: d * = {d *,d 2 * } T * d * d Constrant functons: c (d * ) - Φ(-3) c 2 (d * ) - Φ(-3) c 3 (d * ) - Φ(-4) Objectve functon: c 0 (d * ) (a) Intal desgn d 0 = {5,5} T.

211 93 Table 6.5 Statstcal propertes of random nput for cantlever beam Random Standard Probablty varable Mean devaton dstrbuton X, lb Gaussan X 2, lb Gaussan X 3, ps 40, Gaussan X 4, ps Gaussan Table 6.6 Optmzaton results by varous methods for the cantlever beam Methods (a) FORM SORM Unvarate Monte Carlo No. of teratons No. of functon evaluatons Fnal desgn: d * = {d *,d * 2 } T d *, n d * 2, n Constrant functon: c (d * ) - Φ(-3) c 2 (d * ) - Φ(-3.5) Objectve functon: c 0 (d * ), n (a) Intal desgn d 0 = {2,4} T n.

212 94 Table 6.7 Falure probabltes for cantlever beam Methods c (d * ) c 2 (d * ) FORM SORM Unvarate Monte Carlo Table 6.8 Optmzaton results by varous methods for the 0-bar truss Methods (a) FORM SORM Unvarate Monte Carlo No. of teratons No. of functon evaluatons Fnal desgn: d * = {d *,,d * 0 } T d *, n d * 2, n d * 3, n 2 d * 4, n 2 d * 5, n d * 6, n d * 7, n d * 8, n d * 9, n d * 0, n Constrant functon: c (d * ) - Φ(-2) Objectve functon: c 0 (d * ), n (a) Intal desgn d 0 = {3,,3} T n 2.

213 95 MPP-based unvarate method u 2 [ŷ k, (v k ;d) = 0] v k,2 Falure set y k (v k ;d) < 0 FORM y k (v k ;d) = 0 β HL MPP (u k * or v k * ) SORM u v k, Fgure 6. Varous approxmatons of the performance functon of kth constrant

214 96 Defne PDF of random nput (X) Set j = 0; Intalze desgn varables (d j ) (Step ) Relablty analyss by unvarate decomposton method Evaluate objectve [c 0 (d j )] & constrants [c k (d j ), k =, K] (Step 2) Desgn senstvty analyss by unvarate decomposton method Evaluate gradents of objectve & constrants wth respect to d (Step 3) Perform standard gradentbased desgn optmzaton (Step 4) Update system: j = j + Stop Yes Converge? (Step 5) No Fgure 6.2 Flowchart of the proposed RBDO process

215 n 360 n n ,000 lb 00,000 lb Fgure 6.3 A ten-bar truss structure (Repeatng Fgure 3.4)

216 Objectve Functon Monte Carlo FORM SORM Unvarate Method Iteraton Number Fgure 6.4 Hstory of mathematcal objectve functon

217 99 X X d 2 X 2 00 n d Fgure 6.5 A cantlever beam subjected to end loads

218 200 Objectve Functon Iteraton Number Monte Carlo FORM SORM Unvarate Method Fgure 6.6 Hstory of objectve functon for cantlever beam

219 Objectve Functon Monte Carlo FORM SORM Unvarate Method Iteraton Number Fgure 6.7 Hstory of objectve functon for 0-bar truss

220 202 X 2 X X 6 X 8 X 4 X 3 X 5 X N 5066 N Fgure 6.8 Intal desgn of torque arm geometry at mean values of shape parameters Node 90 Node 98 Node 73 Node 06 Node 75 Fgure 6.9 Locatons of ponts for prescrbng constrants

221 203 Maxmum von Mses stress at node 98 Fgure 6.0 Contour of von Mses stress at mean values of shape parameters for ntal desgn node 06: 247 MPa node 98: 256 MPa node 73: 226 MPa Fgure 6. Contour of von Mses stress at mean values of shape parameters for RBDO desgn

222 204 Objectve Functon Iteraton Number Fgure 6.2 Optmzaton hstory of objectve functon for torque arm

223 205 Maxmum stress at node 98 and 73 Fgure 6.3 Contour of von Mses stress at mean values of shape parameters for rskgnorng optmum desgn