Reliability-Based Design Optimization with Equality. Constraints

Transcription

1 Internatonal Journal for Numercal Methods n Engneerng, 7(), pp Relablty-Based Desgn Optmzaton wth Equalty Constrants Dr. Xaopng Du Department of Mechancal and Aerospace Engneerng Unversty of Mssour Rolla Correspondng author Phone: Fax: E-mal: dux@umr.edu Beqng Huang Graduate Research Assstant, PhD Canddate Department of Mechancal and Aerospace Engneerng Unversty of Mssour Rolla E-mal: beqng.huang@gmal.com

2 SUMMARY Equalty constrants have been well studed and wdely used n determnstc desgn optmzaton, but they have rarely been addressed n relablty-based desgn optmzaton (RBDO). The ncluson of an equalty constrant n RBDO results n the dependency (correlaton) among random varables. Theoretcally, one random varable can be expressed n terms of the remanng random varables gven an equalty constrant; and the equalty constrant can then be elmnated. However, n practce, elmnatng an equalty constrant may be dffcult or mpossble because of complextes such as couplng, recurson, hgh dmensonalty, nonlnearty, mplct formats, and hgh computatonal costs. The objectve of ths work s to develop a methodology to model equalty constrants and a numercal procedure to solve an RBDO problem. A sequental optmzaton and relablty analyss strategy s proposed to solve RBDO wth equalty constrants. The Frst Order Relablty Method (FORM) s employed for relablty analyss. The proposed method s llustrated by a mathematcal example and a two-member frame desgn problem. KEY WORDS: Optmzaton; relablty; equalty constrants; probablstc constrants

3 . INTRODUCTION Wth the advancements of computatonal technologes, desgn optmzaton has been ncreasngly used n engneerng desgn. Combned wth mathematcal models and smulaton tools such as fnte element analyss, desgn optmzaton enables engneers to reach an nexpensve and optmal desgn soluton n an automatc manner. In real-world problems, uncertantes such as varatons n desgn varables and model parameters always exst. Determnstc optmzaton wthout consderng uncertantes usually pushes the desgn to the lmts of constrants, leavng lttle or no room for accommodatng uncertantes n modelng and smulaton and manufacturng mperfectons. Consequently, determnstc optmzaton could lead to unrelable decsons. To meet the need of hgher product qualty and safety, optmzaton under uncertanty has been ncreasngly appled as an alternatve to determnstc optmzaton. Relablty-based desgn optmzaton (RBDO) [-4] s one of the representatve methods of optmzaton under uncertanty. RBDO mantans desgn constrant satsfacton at expected probablty (relablty) levels. There are two common RBDO formulatons: relablty ndex approach (RIA) [4, 5] and performance measure approach (PMA) [, ]. In RIA, desgn feasblty s formulated as the probablty of constrant satsfacton equal to or greater than the desred relablty. In PMA, desgn feasblty s formulated by a percentle constrant functon value that corresponds to the desred relablty. PMA has a couple of numercal advantages over RIA [6]: () PMA s more robust n terms of convergence; and () PMA s more effcent n relablty analyss because t performs relablty assessment only up to a necessary level.

4 An RBDO problem s generally formulated from a determnstc optmzaton problem by convertng determnstc constrants nto probablstc ones whle the objectve functon s evaluated at the mean values of random varables. A general determnstc optmzaton problem nvolves both nequalty and equalty constrants. However, wth the presence of uncertanty, formulatng an equalty constrant s much more complcated [7, 8]. Only few researchers have addressed the equalty constrants wth random varables n robust desgn optmzaton, whch s another optmzaton methodology under uncertanty. In Das s work [9], equalty constrants are elmnated by solvng out the dependent random varables from equalty constrants. Yu and Ish [0] formulate equalty constrants at the mean values of random varables. In Mattson and Messac s work [8], a comprehensve dscusson on equalty constrants wth random varables s provded. They classfy the treatments of equalty constrants under uncertanty nto three categores: (a) to relax equalty constrants, (b) to satsfy equalty constrants n a probablstc sense, and (c) to remove the equalty constrants through substtuton. All of the aforementoned approaches are conducted under the framework of robust desgn optmzaton. Handlng equalty constrants n RBDO s stll a rarely touched area. In ths work, a general method of modelng and handlng equalty constrants for RBDO s developed. A sequental optmzaton and relablty analyss (SORA) strategy and the Frst Order Relablty Method (FORM) are appled to solve an RBDO problem wth equalty constrants. The rest of the paper s organzed as follows. Secton revews the general models of determnstc optmzaton and RBDO wth equalty constrants. Secton 3 classfes equalty constrants nto two types: physcs-based equalty constrants and demand-based 3

5 equalty constrants. Secton 4 formulates an RBDO problem for the two types of equalty constrants. In Secton 5, a computatonal algorthm s developed to solve RBDO problems nvolvng physcs-based equalty constrants. A mathematcal example and a two-member frame desgn problem are used to llustrate the effectveness of the proposed method n Secton 6. Secton 7 concludes ths research work.. DETERMINISTIC OPTIMIZATION AND RELIABILITY-BASED DESIGN OPTIMIZATION WITH EQUALITY CONSTRAINTS In ths secton, we frst brefly revew the model of determnstc desgn optmzaton wth an emphass on equalty constrants. Then, we dscuss the RBDO model wth equalty constrants.. Determnstc desgn optmzaton model A determnstc optmzaton model s gven by mn f ( d) d st.. g( d) 0, =,,, n hj( d) = 0, j =,,, n g h () In the above model, = ( d, d,, d nd ) d s the vector of desgn varables; f ( d ) s the objectve functon; g ( d ) are nequalty constrant functons; hj ( d ) are equalty constrant functons; n d s the number of desgn varables; n g s the number of nequalty constrants; and n h s the number of equalty constrants. 4

6 An equalty constrant mposes a functonal relatonshp on desgn varables. Theoretcally, gven an equalty constrant, one desgn varable can be solved out and be expressed n terms of the remanng desgn varables. In other words, elmnatng one equalty constrant means elmnatng one desgn varable. If we can elmnate all the equalty constrants by solvng the smultaneous equalty equatons, h ( d) = 0 ( j =,,, n ), the number of ndependent desgn varables wll be j h n d n. However, n practce, elmnatng an equalty constrant may be very dffcult or h mpossble because of complextes such as couplng, hgh dmensonalty, nonlnearty, mplct functonal relatonshps (block-boxes), and hgh computatonal costs. Furthermore, n collaboratve multdscplnary desgn optmzaton [], equalty constrants are artfcally added to mantan the consstency among dscplnes.. Relablty-based desgn optmzaton model In relablty-based desgn optmzaton, random varables are used to account for uncertantes from varous sources, such as varatons n materal s propertes, manufacturng processes, and operatng envronments. By convertng determnstc model n Equaton () nto a RBDO model, we obtan mn f ( d, μp ) d s.t. Pr { g ( dp, ) 0 } R, =,,, n hj( dp, ) = 0, j =,,, n h g () In the above model, = ( P, P,, P np ) P s the vector of random varables and n p s the number of random varables. P are treated as desgn parameters that are out of desgners control. The objectve functon f ( d, μ P ) s evaluated at the means, μ P, of the 5

7 random varables P. Pr{} denotes a probablty, and R stands for the desred relablty for constrant. Equalty constrants hj ( dp, ) nvolve random parameters P. At each desgn pont = ( d, d,, d nd ) calculate the relablty Pr { g ( dp, ) 0} k d, relablty analyss s needed n order to. Typcal relablty analyss methods [, 3] or Monte Carlo Smulaton [5] can be used for the relablty analyss purpose. It s worthwhle to explan and dfferentate the concepts of the desgn space (determnstc space) and relablty analyss space (random space) used n RBDO. The desgn space conssts of determnstc desgn varables = ( d, d,, d nd ) d. It s an n d - dmensonal space. The relablty analyss space conssts of random varables ( P, P,, P np ) P =. It s an np - dmensonal space. 3. CLASSIFICATION OF PROBABILISTIC EQUALITY CONSTRAINTS To properly formulate equalty constrants n an RBDO problem, t s necessary to study ther features and then classfy them nto dfferent categores. Adoptng Mattson and Messac s dea [8], we classfy equalty constrants nto two categores: physcs-based equalty constrant and demand-based equalty constrant. Statstcally, these two types are fundamentally dfferent. 6

8 3. Type : Physcs-based equalty constrant A physcal-based equalty constrant functon s determned by a physcal prncple; the equalty condton always holds regardless of the varatons of ts consttutng varables. For nstance, Newton s second law states the equalty relatonshp among the mass, m( P ), of a partcle, external resultant force, F( P ), actng on the partcle, and the acceleraton, a( P 3 ), of the partcle. The equaton s gven by h( P ) = h( P, P, P ) = F ma = 0 (3) 3 If F and m are random, a s also a random varable dependent on F and m. The above equaton should be always satsfed durng a numercal mplementaton. In a cantlever beam desgn problem, four random varables P, P, P 3, and P 4 are nvolved as shown n Fgure. P P P 3 P 4 Fgure. A cantlever beam. The total length P 4 s the sum of the other varables. P4 = P+ P + P3 (4) Then, an equalty constrant consstng of P, P, P 3, and P 4 s gven by h( P ) = P + P + P P = 0 (5) 3 4 Ths equalty constrant s determned by the materal contnuty of the beam. 7

9 In the example of a cubc contaner desgn (see Fgure ), there are three random parameters P, P, and P 3. P s the length and wdth, P s the heght, and P 3 s the volume, whch s gven by P = PP (6) 3 P P P Fgure. A cubc contaner. Therefore, the equalty constrant s expressed by h( P ) = PP P= 0 (7) 3 The above equalty constrant s determned by a geometrc relatonshp. Features of a physcal based equalty constrant nclude: () The equalty condton should always hold. Snce an equalty constrant represents a physcal prncple, the equalty condton should always hold. In other words, the probablty of the equalty constrant satsfacton should always be.0, no mater how large uncertantes are. The volaton of a physcsbased constrant wll result n an nfeasblty desgn. For example, the volaton of the equalty constrant n Equaton (5) mples the breakage of the beam. 8

10 () The equalty condton holds n the random analyss space. From Equatons (3) and (7), we see that the equalty condton holds n the random space, no matter how random varables vary. suffcent. (3) Satsfyng an equalty condton n determnstc desgn space may not be As dscussed n Secton, n some lterature an equalty constrant s formulated at the means of random varables n the determnstc desgn space as h ( μ ) = 0 (8) p Ths equaton may or may not guarantee the equalty condton n the random space. For the cantlever beam problem n Fgure, let P, P, and P 3 be ndependently normally dstrbuted wth ther means, µ, µ, and µ 3, and standard devatons, σ, σ, and σ 3, respectvely. Snce P 4 s a lnear combnaton of the other normally dstrbuted varables, P 4 s also normally dstrbuted wth ts mean and ts standard devaton µ 4 = µ + µ + µ 3 (9) 4 3 σ = σ (0) = Wth Equaton (8), the equalty constrant n the determnstc desgn space s gven by h ( μ ) = µ + µ + µ µ = 0 () 3 4 Ths equaton guarantees the same equalty condton n the random analyss space when the standard devaton of P 4 s determned by Equaton (9). 9

11 However, for the cubc contaner desgn problem, let P and P be ndependently normally dstrbuted wth ther means, µ and µ, and ther standard devatons, σ and σ, respectvely. The mean of the volume P 3 s gven by µ = ( µ + σ ) µ () 3 The equalty constrant n the determnstc desgn space s µ = µµ (3) 3 Snce the standard devaton σ s not zero, Equaton () conflcts wth Equaton (). Ths ndcates that the equalty constrant specfed n Equaton (3) n the determnstc desgn space cannot guarantee that the equalty constrant s satsfed n the random analyss space. Generally, whether an equalty constrant n determnstc desgn space ensures the equalty condton n the random analyss space depends on the constrant functon form and the dstrbuton types of random varables. varables. (4) A physcs-based equalty constrant mposes correlatons among random A physcal-equalty constrant nexplctly reduces one degree of freedom of random varables. For example, for the cantlever problem, there are correlatons between P 4 and P, P 4 and P, and P 4 and P 3. The correlatons ncrease the complexty of computng the probablstc characterstcs of equalty and nequalty constrants. 0

12 3. Type : Demand-based equalty constrant A demand-based equalty constrant s determned by desgners preferences or desres, not by any physcal prncple. For example, n the above cubc contaner desgn problem, the desgn varables are chosen as the length and wdth ( P ) and heght ( P ), and they are mutually ndependent. A desgner may wsh the volume to be as close as possble to a target value of m 3. Ths s a demand-based equalty constrant. Features of a demand based equalty constrant nclude: () The equalty condton may not be requred to be strctly satsfed. Snce a demand-based equalty constrant s not determned by physcal prncples, a slght constrant volaton does not necessarly mean an nfeasble desgn. Instead, ths type of constrant s somewhat flexble snce t only reflects desgners preferences or desres. analyss space. () A demand-based equalty constrant cannot always be satsfed n random In the above cubc contaner desgn problem, we wsh the volume to be m 3. Then, we could express our preferences n the random analyss space as or PP= (4) h( P ) = PP = 0 (5) Snce the volume PP s a contnuous random varable and the probablty of the volume beng equal to a specfc value s zero. Equatons (4) or (5) wll never be satsfed n the random analyss space. We must fnd other means to formulate ths type of constrant (see the next secton).

13 varables. (3) A demand-based equalty constrant may not mpose a correlaton on the random A demand-based equalty constrant may not necessarly ndcate the reducton of a degree of freedom of random varables that are nvolved n an equalty constrant. For nstance, n the above example, P and P can be changed ndependently even though we wsh the volume to be a certan value. Therefore, there s no correlaton between P and P. 4. MODELING EQUALITY CONSTRAINTS IN RBDO We have dscussed the features of two fundamentally dfferent types of equalty constrants. Next, we dscuss how to formulate both types of constrants. 4. Formulatng a demand-based equalty constrant A demand-based equalty constrant wll never be satsfed. Therefore, we have to relax the deal equalty condton. The smplest treatment s to formulate t just at the mean values of random varables. The optmzaton model s therefore gven by mn f ( d, μp ) d st.. Pr { g ( dp, ) 0 } R, =,,, n hj( d, μp) = 0, j =,,, n h g (6) Another treatment s to relax and replace the equalty constrants h ( dp, ) = 0, j =,,, n, as shown n Equaton () by the followng two equalty j constrants: h { h δ } Pr ( δp, ) R, (7) j j j

14 and where δ j, j ' { hj δ j} Rj Pr ( δp, ), (8) R and R are a small tolerance and two desred relabltes, respectvely. ' j By use ether of the above two approaches, there s no equalty constrant. Therefore, the problem can be solved by any exstng RBDO algorthm. Next, we thus focus on formulatng and solvng RBDO problems wth physcs-based equalty constrants. 4. Formulatng a physcs-based equalty constrant As we dscussed n Secton 3, a physcs-based equalty constrant must be satsfed wth a probablty of.0 n the random analyss space. However, n the optmzaton model, we can only formulate an equalty constrant n the determnstc desgn space, and an equalty constrant formulated n the determnstc desgn space may not guarantee the equalty condton n the random analyss space. To ths end, t s nearly mpossble to satsfy a physcs-based probablstc equalty constrant n the determnstc desgn space. The only way to ensure the probablty of.0 s to elmnate the equalty constrants by elmnatng dependent random varables. Snce elmnatng equalty constrants may not be practcal, we develop the followng numercal procedure based on the prncple of varable elmnaton. Let equalty constrant functons n the random analyss space be h( dp, ) = 0 =,,, n (9) d are determnstc desgn varables and are treated as constants n the random analyss space. The ncluson of equalty constrants n RBDO results n the dependency h 3

15 (correlatons) among random varables. In ths work we assume that there are n P n ndependent random varables. We partton random varables P nto ndependent varables X and dependent varables Y such that (, ) P= X Y (0) Solvng the smultaneous equatons n Equaton (9) yelds (, ) where G represents the relatonshp between X and Y. Then, an RBDO problem s formulated as Y= G dx () h mn f ( d, μ X ) d s.t. Pr { g dx,, G( X) 0 } R =,,, n () where equalty constrants are elmnated. In practce, elmnatng equalty constrants s dffcult. We propose to use the Frst Order Relablty Method [] to formulate and solve the above RDBO model. As dscussed n the ntroducton, performance measure approach (PMA) [, ] has advantages over the tradtonal relablty ndex approach (RIA). We therefore use PMA to formulate the relablty constrants n Equaton (0). Suppose the Most Probable Pont (MPP) [] of the constrant functon g,, ( ) dx G X s found at accordng to RIA, the relablty constrant g ( ) ( ) * x gven the desred relablty R, then { } * * g dx,, G x 0. Hence the RBDO model can be rewrtten as Pr dx,, G X 0 R s equvalent to mn f ( d, μ X ) d ( ) = * * s.t. g dx,, G x 0,,, n (3) 4

16 Usng the concept of varable elmnaton but not actually elmnatng dependent varables, we nclude the mean values * y at the MPP * x as desgn varables as well. The RBDO model then becomes μ Y, of the dependent varables Y and ther values * Y can be calculated by (, ) Y = G dx (4) * * mn f ( d, μ * X, μy) d, μy, y * * s.t. g dx,, G( x ) 0 =,,, n hj( d, μx, μy) = 0, j =,,, nh h = j = n = n * * j( dx,, y) 0,,,, h,,,, g (5) In the above model, snce each nequalty constrant has ts own MPP, the totally number of equalty constrants at MPPs s n h n g. Solvng ths model needs to search the MPP x for the relablty constrant g 0. The MPP search for relablty analyss tself s * also an optmzaton problem [, ], and therefore, drectly solvng the above RBDO problem wll nvolve an expensve double-loop procedure. Next, we develop a numercal procedure that decouples the optmzaton outer loop from the relablty analyss nner loop wth the sequental sngle-loop strategy. 5. SOLVE RBDO WITH PHYSICS-BASED EQUALITY CONSTRAINTS In ths secton, we frst brefly revew the recently developed sequental sngle-loop strategy for RBDO wthout equalty constrants. Then, we develop the formulaton and numercal procedure for RBDO wth physcs-based equalty constrants. 5

17 5. Sequental sngle-loop strategy for RBDO RBDO wthout equalty constrants has been extensvely studed. Tradtonal approaches to RBDO wthout equalty constrants requre a nested double-loop process. Under the optmzaton outer loop, the nner loop of relablty analyss calculates the relablty for each of probablstc constrants. The optmzaton outer loop searches for the optmal soluton by updatng desgn varables and callng the relablty analyss repeatedly. Because of the nested framework, ths process s computatonally ntensve. To mprove computatonal effcency, sngle loop methods have been developed. The orgnal approaches can be found n Chen and Hasselman [6], Wu and Wang [7] and Wu, et al. [8]. The sequental optmzaton and relablty assessment (SORA) [] method s one of later developed methods. SORA conssts of a few cycles as shown n Fgure 3. In each cycle, there are two decoupled parts: determnstc optmzaton (DO) and relablty analyss (RA); determnstc optmzaton s performed frst, followed by relablty analyss. Cycle Cycle Cycle DO RA DO RA DO RA Fgure 3. Sequental sngle-loop strategy for RBDO. In each cycle, optmzaton and relablty analyss are decoupled from each other and run sequentally. The determnstc optmzaton s performed to acheve an optmal desgn soluton; the relablty analyss s conducted after optmzaton to verfy the 6

18 satsfacton of relablty constrants and also provdes mprovement drecton for updatng desgn soluton wthn optmzaton n the next cycle. If the process does converge, a new determnstc optmzaton model s formulated for the next cycle based on the relablty analyss results just obtaned. The new optmzaton formulaton shfts the constrant boundares of unsatsfed relablty constrants toward the feasble regon and therefore guarantees the relablty mprovement. 5. Determnstc Optmzaton The determnstc optmzaton n each cycle s gven by Equaton (5), where n addton to the orgnal desgn varables d, two new groups of varables, μ Y and * y, are also added as desgn varables. The nequalty constrants are evaluated at the MPPs * * (, ) x y of ndependent random varables X and dependent random varables Y. The MPPs * x of ndependent varables are the results from the relablty analyss n the prevous cycle. Equalty constrants hold at both the means and the MPPs. 5.3 Relablty analyss The Frst Order Relablty Method (FORM) s employed to calculate the MPPs correspondng to the desred relablty. Two man steps are nvolved. () Transformaton: Rosenblatt transformaton [9] s used to convert the ndependent random varables X n each of the constrant functons g nto standard normal varables U. The transformaton s gven by u = F F ( x ), j =,,, n j X j j X (6) 7

19 where Φ [ ] s the nverse cumulatve dstrbuton functon (CDF) of the standard normal dstrbuton, F () s the CDF of X j. After the transformaton, the constrant functon X j g dxy n the orgnal space becomes another functon * (,, ) g duy n the transformed * (,, ) normal space wth regardng to X. d and determnstc optmzaton n the prevous cycle. * y are known because they are obtaned n the () MPP search: For constrant, a maxmzaton (optmzaton) procedure s used to search the MPP u * n the transformed normal space. The model s gven by for all g, =,,, n g st * max g (,, ) uy, u duy = Φ R. ( ) h duy = j = n * j(,, ) 0,,,, h (7) Dfferent from the conventonal MPP search wthout any equalty constrant, the above model ncludes all the equalty constrants to ensure ther satsfacton at the MPP. The soluton The MPP * u to the above problem s the MPP for constrant n the transformed space. * x n the orgnal space can be obtaned by usng Eq. (6). The overall flowchart s depcted n Fgure 4. 8

20 Start Intalzaton: * * d, μ, y, x Y Determnstc Optmzaton d Relablty Analyss c = c+ f converges & all g 0? Stop Yes No New u * for all constrants Fgure 4. Flowchart of RBDO wth Equalty Constrants. 6. NUMERICAL EXAMPLES In ths secton, we use two examples to demonstrate the proposed method, ncludng a mathematcal problem and a desgn of a two-member frame. For the two-member frame desgn, the result from the proposed method s also verfed wth that from the elmnaton method, n whch equalty constrants are elmnated. 6. A mathematcal problem A determnstc optmzaton problem s gven by 9

21 d and two equalty The problem nvolves desgn varables = ( d, d, d, d, d, y, y ) constrants. The last two desgn varables (, ) equalty constrants y y can be elmnated because of the two To demonstrate the effectveness of the proposed method, ths determnstc optmzaton problem s reconfgured as an RBDO problem, n whch (,,...,5) d = are consder as random desgn varables, whch follow normal dstrbutons ( ) N µ, σ, σ = 0.. To use the RBDO model as shown n Equalty (5), a random desgn varable s splt nto two parts, namely, X ndependent random varables, and (, ) µ +, where ~ ( 0, ) X N σ. X can be treated as y y become the dependent random varables ( Y, Y ). Therefore, d = ( µ, µ, µ, µ, µ ), P= ( X, Y ), = ( X, X, X, X, X ) ( Y, Y ) Y =, and the RBDO model s gven by X, mn f( d, d, d3) = d + d + d3-dmy d= ( d, d, d3, d4, d 5) st : -( d+ X + ( d4 + X 4) + ( d5 + X 5) 0.4 Y) 0 -( ( d+ X) + ( d + X) + ( d3+ X3) + ( d + X)( -Y) ) 0 Y = ( d+ X) + ( d + X) -( d3+ X3) + Y Y = ( d+ X)( d4 + X4) + ( d4 + X4) + ( d5 + X5) + Y 0 d 0, =,,...,5 The desred relablty s 0.9 for the two nequalty constrants. The results from the proposed method are gven n Table I. The entre RBDO procedure converges wth three cycles. The fnal optmal desgn soluton s 0

22 ( d, d, d, d, d ) (.494, , , , 8.480) d = = s Table I. The results for the mathematcal RBDO problem. Cycle Desgn soluton = ( d, d, d, d, d ) 3 4 5, and the objectve value d Objectve (.335, 0.380,7.8, , 6.990).0000 (.489, , 7.505, , 8.467) (.494, , , , 8.480) RBDO for a two-member frame [0] A two-member frame s subject to out-of-plane load as shown n Fgure 5. Such frames are commonly encountered n automotve, aerospace, mechancal and structural engneerng applcatons. The desgn s to mnmze the volume of the frame wth the stress constrants such that the maxmzaton stresses should be less than or equal to the allowable materal strengths. Three desgn varables are the wdth (d), the heght (h), and the wall thckness (t) of the member, namely, d = (d, h, t). l z P U l 3 t h x U U 3 y b Fgure 5. A two-member frame. The volume of the structure s gven by

23 f l dt ht t ( d ) = ( + 4 ) The members are subjected to both bendng and shear stresses, and the combned stress constrant needs to be mposed at ponts and. Accordng to von Mses yeld condton, the two constrants are g = σ + 3τ.0 0 σ a σ a ( ) g = σ + 3τ.0 0 ( ) where σ, and σare the maxmum bendng stresses at ponts and, respectvely; and τ s the shear stress n the members. To calculate the stresses, the vertcal dsplacement U at Pont, the rotaton U about lne 3- and the rotaton U 3 about lne - need to be frst computed usng fnte element analyss procedure, whch nvolves solvng the followng three equalty constrants, EI 3 L 4 6L 6L U P GL 6L 4L + L 0 U 0 EI =, U3 0 GL 6L 0 4L + L EI where ( )( ) 3, 3 I (moment of nerta) = dh d t h t G (polar moment of nerta) = ( ) ( ) ( d + h) t t d t h t, and A (area for calculaton of torsonal shear stress) = - ( d t)( h t) -.

24 Once the U, U, U 3 have been solved out, the torque T, and bendng moments M and M at ponts and for the member - can be calculated as T = - GJ U 3, L EI M = ( 3U+ UL ), L and EI M = ( 3U+ UL ). L Usng the torque and moments, the torsonal shear and bendng stresses can be calculated as T t =, At σ = ( Mh ), I and σ = ( Mh ). I For more detals about ths example, refer to [0]. Fve ndependent random varables are nvolved. They consst of the length L, the modulus of elastcty E, the shear modulus G, the allowable stress (materal strength) σ, and the external force P. Ther dstrbuton parameters are gven n Table II. a Table II. Dstrbutons of random varables. 3

25 Varable Dstrbuton type Mean (μ) Standard devaton(σ) L Normal 00 n n E Normal 30,000,000 ps 3,000,000 ps P Normal -0, 000 lb,000 lb G Normal,540,000 ps,000,000 ps σ Normal 40,000 ps 4,000 ps a The formulaton wth both nequalty and equalty constrants s brefly shown as s a 4 ( ) mn f ( d) = L dt + ht 4t d st : g( dp, ) = ( s + 3t ).0 0 s a g ( dp, ) = ( s + 3t ).0 0, 4 6L 6L U P EI GL 6L 4L + L 0 U 3 = 0 L EI U3 0 GL 6L 0 4L + L EI.5 h 0;.5 d 0; 0. t.0 (9) P XY. The ndependent n whch d = ( dht,, ) and = (, ) = ( ELGP,,,, σ, U, U, U ) random varables are = ( ELGPσ ) ( U, U, U ) Y =. 3 a 3 X,,,, a and the dependent random varables are To show the effectveness of the proposed method, we frst solve the RBDO problem wth three equalty constrants and two probablstc nequalty constrants, whch s converted from the determnstc optmzaton model shown n Eq. (9). The desred relablty s 0.9 for both nequalty constrants. The results are shown n Table II. The entre RBDO process converges wth three seral cycles. The fnal desgn soluton s d = (d, 4

26 h, t) = (9.758, , 0.000) (n). The total volume of the two-member frame structure s n 3. Table III. Results from the proposed RBDO wth equalty constrants. Cycle Desgn soluton d = (d, h, t ) (n) Objectve (7.7987, , 0.000) (9.757, , 0.000) (9.758, , 0.000) Ths problem can also be formulated as an RBDO problem wth only two nequalty constrants. To verfy the proposed method, we also use the RBDO wth only two nequalty constrants. The result from the RBDO model wth only two nequalty constrants s the same as the result from the proposed RBDO formulaton wth both nequalty and equalty constrants. 7. CONCLUSIONS Equalty constrants exst n many engneerng problems and have been welladdressed n determnstc optmzatons. Wth random varables, t s straghtforward to deal wth equalty constrants n RBDO. Equalty constrants are therefore overlooked n lterature or are smply treated as determnstc ones at the mean values of random varables. To approprately handle equalty constrants n RBDO, we classfy them nto two types. The frst type conssts of demand-based equalty constrants, whch are resulted from a desgner s preferences and desres. A demand-based equalty constrant may not be 5

27 satsfed when random varables exst. They can be treated as determnstc equalty constrants or as nequalty relablty constrants that bound an equalty constrant wth a small tolerance. The second type conssts of physcs-based equalty constrants, whch must be satsfed wth a probablty of.0. Ths type of constrants can be elmnated by solvng and expressng the dependent random varables n terms of ndependent random varables. However, n practce, varable elmnaton may not be feasble due to couplng, hgh dmensonalty, nonlnearty, mplct format, and hgh computatonal costs. A numercal procedure s proposed to handle and solve the RBDO problems wth physcs-based equalty constrants. A sequental determnstc optmzaton and relablty analyss strategy s employed. Formulatons for the determnstc optmzaton and relablty analyss are developed. To mantan the equalty relatonshps wthout elmnatng equalty constrants, addtonal desgn varables are added, ncludng the mean values and the MPPs of the dependent random varable. The two examples ndcate the feasblty and effectveness of the proposed method. ACKNOWLEDGEMENT The support from US Natonal Scence Foundaton Grant CMMI and the Intellgent Systems Center at the Unversty of Mssour-Rolla s gratefully acknowledged. 6

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