A Proposed Method for Reliability Analysis in Higher Dimension

Transcription

1 A Proposed Method or Reliabilit Analsis in Higher Dimension S Kadr Abstract In this paper a new method is proposed to ealate the reliabilit o stochastic mechanical sstems This techniqe is based on the combination o the probabilistic transormation methods or mltiple random ariables (MPTM) and the inite element method (EM) The transormation techniqe ealates the Probabilit Densit nction (PD) o the sstem response b the se o the Jacobian o the inerse mechanical nction This approach has the adantage o giing directl the whole densit nction o the response - in a closed orm - which is er helpl in reliabilit analsis Inde Terms Probabilistic method inite Element Method Reliabilit Analsis ORM Monte-Carlo simlation I INTRODUCTION Man mechanical applications reqire the consideration o stochastic properties o materials geometr and loads The basic representation o ncertain parameters in the nderling models is obtained b introdcing random ariables or ields Dierent kinds o analses acconting or ncertainties can be carried ot The second moment analsis aims to characterize the means and ariances o response qantities (displacements strain and stress components etc) in terms o the inpt ariable moments The pertrbation method [] and the weighted integral method [] belong to this categor On the other hand the reliabilit methods ocs on the calclation o the probabilit o ailre associated with a gien limit-state nction irst and second order reliabilit methods (ORM / SORM) and arios simlation methods are commonl sed in reliabilit analsis [] To accont or the spatial ariabilit o ncertain qantities (eg material properties) a characterization in terms o a random ield is sall emploed Throgh a process o discretization it is possible to represent the random ield b a ector o random ariables One o the methods mentioned aboe ma then be sed to carr ot second-moment or reliabilit analses The spectral stochastic inite element method (SSEM) proposed b Ghanem and Spanos [] is an approach well sited to analses inoling random ields It is based on two tpes o discretization o the sstem o stochastic partial dierential eqations goerning the problem nder consideration: one in the spatial domain and one Manscript receied ebrar 7 SKadr ebanese Uniersit Post Bo No:- Phone: 96-7; skadr@gmailcom in the probabilistic domain The response (eg the random ector o nodal displacements) is cast as a series epansion in terms o standard normal ariables This can be interpreted as an intrinsic representation o the random response rom which qantities sch as statistical moments can be compted b post-processing either analticall or b sampling The aim o this paper is to generalize the probabilistic transormation methods (PTM) introdced recentl b the athors [] or one-dimensional This new method is presented in order to ealate the stochastic mechanical response The method is based on the combination o the probabilistic transormation methods (PTM) or mltiple random ariables (eg Yong s modls and load) and the deterministic inite element method (EM) The transormation techniqe ealates the Probabilit Densit nction (PD) o the sstem response b mltipling the inpt PD with the Jacobean o the inerse mechanical nction II REIABIITY ANAYSIS The reliabilit methods aim at ealates the probabilit o ailre o strctral sstems sbjected to randomness The design o strctres and the prediction o their good nctioning lead to the eriication o a certain nmber o rles reslting rom the knowledge o phsical and mechanical eperience o designers and constrctors These rles tradce the necessit to limit the loading eects sch as stresses and displacements Each rle represents an elementar eent and the occrrence o seeral eents leads to a ailre scenario The objectie is then to ealate the ailre probabilit corresponding to the occrrence o critical ailre modes In addition to the ector o deterministic ariables sed in the sstem control and optimization the ncertainties are modeled b a ector o stochastic phsical ariables Y aecting the ailre scenario The knowledge o these ariables is not at best more than statistical inormation and we admit a representation in the orm o random ariables or a gien design rle the basic random ariables are deined b their probabilit distribtion associated with some epected parameters; the ector o random ariables is noted herein Y whose realizations are written The saet is the state where the strctre is able to lill all the nctioning reqirements: mechanical and serice abilit or which it is designed To ealate the ailre probabilit with respect to a chosen ailre scenario a limit state nction G( is deined b the

2 condition o good nctioning o the strctre The limit between the state o ailre G( and the state o saet G( > is known as the limit state srace G ( (igre ) The ailre probabilit is then calclated b: [ G ( ) ] P Pr ( ) d dn () X G( ) where P is the ailre probabilit () is the joint densit nction o the random ariables X and Pr[] is the probabilit operator The ealation o integral () is not eas becase it represents a er small qantit and all the necessar inormation or the joint densit nction are not aailable or these reasons the irst and the Second Order Reliabilit Methods ORM/SORM has been deeloped The are based on the reliabilit inde concept ollowed b an estimation o the ailre probabilit The inariant reliabilit inde β was introdced b working in the space o standard independent Gassian ariables instead o the space o phsical ariables The transormation rom the phsical ariables to the normalized ariables is gien b: T ( and T ( ) () This transormation T() is called the probabilistic transormation In this standard space the limit state nction takes the orm: H ( ) G( ) () or practical engineering eqation () gies sicientl accrate estimation o the ailre probabilit ig : Reliabilit analsis methodolog A Approimate reliabilit methods (orm & Sorm) The irst Order Reliabilit Method (ORM) ses the closest point on the limit state nction H() to the origin in the standard normal space as a measre o the reliabilit This point is called design point * and its beta ale β * speciies the reliabilit leel; the ailre probabilities P Φ( β ) where Φ () denotes the standard cmlatie distribtion nction Second Order Reliabilit Methods (SORM) approimate the limit state nctions b an incomplete second order polnomial which assmes in a rotated space the simple orm n H H ~ + k ( ) ( ) β i i () i The orm is incomplete since the terms {( β ) } n i i leading to a complete second order approimation arond the design point are missing An eact reslt in orm o a one dimensional integral has been deried in [6] or the incomplete representation [6] and an asmptotic reslt sicientl accrate or large β ale has been deeloped [7] B Importance Sampling Importance Sampling has been one o the most prealent approaches in the contet o Simlation based methods or the estimation o strctral reliabilit [] The nderling concept is to draw samples o the ector o random parameters X rom a distribtion X () which is concentrated in the important region o the random parameter space C ine Sampling An alternatie qite sitable or large dimension X and eicient in contet o is the ollowing approach denoted as line sampling It reqires an important direction as starting point deining the direction along the limit state G( will be determined Each point ( in the standard point is decomposed into the one dimensional space X [9] D Some actors o comparison The procedres or estimating the ailre probabilit deeloped oer the last twent ears like ORM/SORM Importance Sampling and all its ariants lack robstness or comptational eicienc as the nmber o random ariables tends to ininit Howeer the robst and simple straight orward ine Sampling procedres is able to oercome most diiclties encontered in traditional procedres Smmarizing the argments the ollowing can be conclded: ORM proides a point estimate sbject to linearization errors withot conidence Moreoer it reqires the ealation o the design point which becomes diiclt in high dimensions or nonlinear limit state nctions in the normalized space The eorts to compte the design point grows proportional with the nmber o ariables SORM reqires in addition to the design point the main cratres which cannot be obtained in a easible manner or high dimensions The procedre implies that the domain close to the design point is

3 the important domain which is not the case or high dimensions Importance Sampling is more robst and accrate than ORM / SORM bt not competitie to ine Sampling becase it is generall impossible to sample according to the optimal sampling densit The approach has diiclties to deal with mltiple ailre domains i the are not well separated ine Sampling is capable to take adantage o simple lat limit states in standard normal space and samples in the most important domain withot assming a linear or qadratic limit state srace or reqiring a design point comptation III TRANSORMATION METHOD The theor o the Probabilistic Transormation Method or PTM is based on the ollowing theorem []: Theorem (Single inpt-single otpt sstem): Sppose that X is a random ariable with PD (probabilit densit nction) () and A R is the one dimensional space where ( ) > Consider the random ariable (nction o ) Y ( X ) where ( ) deines a one-to-one transormation that maps the set A onto a set B R so that the eqation ( ) can be niqel soled or in terms Then the PD o Y is: o sa ( ) g [ ] J B ( ( where transormation d d ( J is the Jacobean o the d d The mathematical condition or this theorem is that the transormation mst be one-to-one Problems reqentl arise when we wish to ind the probabilit distribtion o the random nction YU(X) when X is a continos random ariable and the transormation is not one-to-one So the two major problems or limitations o PTM are: the transormation nction (U) shold be bijectie the determinant o Jacobean shold be not nll To aoid these limitations o PTM a n n sstem can be deined sch as ) : n n U IR IR Y Y U ( X ) ( n with ( n) U ( X ) i i i n () : J n IV PROPOSED TECHNIQUE: MPTM-EM The proposed techniqe is a combination o the deterministic inite element method and the random ariable transormation techniqe or mltiple ariables (MPTM) sing the preios theorem In this techniqe the dierential eqation is soled irstl b sing the deterministic theor o inite element (a) which ields to accrate nodal soltions These soltions are then sed to obtain the "eact" PD sing the random ariable transormation (b) between the inpt random ariables and the otpt ariable The accrac o the soltion is increased when increasing the nmber o elements in the EM as sal The organization chart o this techniqe is shown in igre and the algorithm is shown in igre a) the EM leads to: [ K ] U i e U K S (where S is the leibilit matri) S b) Using PTM techniqe U S U Stochastic Eqation o Moement EM PTM Probabilit Densit nction o response ig : Organizaton chart The aboe nction U is reersible i and onl i the determinant o the Jacobean is not nll Indeed

4 START Inpt: PD() () where () is calclated sing EM Introdcing some ictios ariables and calclate ind - Where - ( ind J determinant o the Jacobean ind PD( sing : PD( J PD() END ig : algorithm o MPTM-EM V APPICATION In this eample a 6-bar trss strctre with random Yong's modls E and random load It is reqired to ind the pd o the response o node (igre ) which is a nction o two inpt random ariables (load and Yong's modls) ig : 6-bar trss strctre To se the MPTM-EM techniqe we mst hae the nmber o otpt random ariables eqal to the nmber o inpt random ariables So we shold introdce a ictitios random otpt which is an arbitrar nction o the random inpts The choice o this nction doesn't aect the pd o the soltion process (response) The MPTM-EM is applied according to the ollowing steps: Models the trss sing EM to obtain the displacement o the irst and the second nodes or the trss bars the stiness matrices are gien b: Bar - Bar - Bar - Bar - Bar - Bar - The Assembl Theorem leads to the global eqilibrim sstem: Where the soltion gies: 6

5 6 the displacement o the node : where is the bar length m A is the cross-section Am E is the Yong's modls and is the load and E are phsicall two independent random ariables that hae some probabilit distribtions Using the MPTM-EM we can ind the pd o the response i E and are random ariables In this problem we hae two inpts (E) and one otpt ( ) So we introdce a ictitios random ariable sa ZE as the nd otpt The transormation eqations between inpts and otpts becomes: 6 Z E where the domain o E will be completel deined b the distribtions o E and The Jacobean o the transormation is E E Z ZA J Z 6 Z ZA i ( Z) The pd o the irst node is: 6 ZA ZA Z ( ) ; Otherwise Otherwise 66 ; 7 66 ; igre shows the pd o inpt and igre 6 shows the otpt ariables The reslt is eriied b Monte-Carlo simlation 66 7 Using the PTM techniqe we can get: Z Z) E [ E ] J ( In or problem rom the independenc between the material propert and the load we can simpl sa: E ( E ) E ( E) ( ) inall the pd o the response is compted rom the ollowing integral: ( ) Z ( z dz D z ) which is the eact pd o Now we let s consider the eample where and E are niorml distribted in the ranges [ ] and [ ] respectiel In this case the joint probabilit densit nction o and E is: E ( E ) i E Otherwise ig : PD o inpt E and The joint distribtion or ( Z) is:

6 VI CONCUSION In this paper the reliabilit analsis o mechanical sstems with parameter ncertainties has been considered The method is based on the combination o the probabilistic transormation method and the deterministic inite element method The application is gien or a simple 6-bar trss where the PTM-EM gies the closed orm epression o the response densit nction To proo the perormance o the proposed method the reslt is compared with Monte Carlo simlations ig 6: PD o otpt Reliabilit Analsis et s sppose the limit displacement is l 6 reqired to ind the ailre probabilit P P ) ( P ( ) 6 ( l ) l mm it is d 66 Table : comparison o P with Monte Carlo simlation Proposed Monte Carlo Method simlation() P 7 REERENCES [] i W-K Beltschko T Mani A: International J o Nmer Meth Engng (): 96 [] Deodatis G : J Engng Mech 7(): 6 99 [] Ditlesen O Madsen H Strctral reliabilit methods Chichester: Wile 996 [] Ghanem R-G Spanos P-D Stochastic inite elements a spectral approach Berlin: Springer 99 [] Kadr S Chateane A El-Tawil K in Proceedings o The Eighth International Conerence on Comptational Strctres Technolog BHV Topping G Montero and R Montenegro (Editors) Ciil-Comp Press Stirlingshire United Kingdom paper 6 [6] Tedt : J o Engineering Mechanics 6(6) [7] Breitng K Asmtotic approimations or probabilit inegrals ectre Notes in Mathematics 9 Berlin: Springer 99 [] Scheller G 99 Strctral Reliabilit - Recent Adances - redenthal ectre Proceedings o the 7th International Conerence on Strctral Saet and Reliabilit (ICOSSAR 97) N Shiraishi M Shinozka and Y Wen eds AA Balkema Pblications Rotterdam The Netherlands [9] Kotsorelakis P Pradlwarter H and Sch eller G: J Probab eng Mech ol 9 n o pp 9-7 [] Papolis A Pillai S U Probabilit Random Variables and Stochastic Processes th Edition McGraw-Hill Boston USA S Kadr receied his Masters degree in Modeling and Intensi Calcls () He receied his PhD (-7) rom Department o Mechanics Blaise Pascal Uniersit and IMA rance rom he worked as Head o Sotware Spport Unit and Analsis He pblished man papers in national and international jornals His research interests are in the areas o Compter Science Nmerical Analsis and Stochastic Mechanics ig 7: probabilit o ailre (red region)