Decomposition methods for structural reliability analysis

Transcription

1 Probablstc Engneerng Mechancs 0 (005) Decomposton methods for structural relablty analyss H. Xu, S. Rahman* Department of Mechancal Engneerng, The Unversty of Iowa, 140 Seamans Center, Iowa Cty, IA 54, USA Avalable onlne 19 July 005 Abstract A new class of computatonal methods, referred to as decomposton methods, has been developed for predctng falure probablty of structural and mechancal systems subject to random loads, materal propertes, and geometry. The methods nvolve a novel functon decomposton that facltates unvarate and bvarate approxmatons of a general multvarate functon, response surface generaton of unvarate and bvarate functons, and Monte Carlo smulaton. Due to a small number of orgnal functon evaluatons, the proposed methods are very effectve, partcularly when a response evaluaton entals costly fnte-element, mesh-free, or other numercal analyss. Seven numercal examples nvolvng elementary mathematcal functons and sold-mechancs problems llustrate the methods developed. Results ndcate that the proposed methods provde accurate and computatonally effcent estmates of probablty of falure. q 005 Elsever Ltd. All rghts reserved. Keywords: Relablty; Decomposton; Unvarate method; Bvarate method; Response surface; Stochastc fnte-element and mesh-free methods 1. Introducton A fundamental problem n the tme-nvarant relablty analyss entals calculaton of a mult-fold ntegral [1 3] ð P F hpðx U F Þ Z f X ðxþdx; (1) U F where XZfX 1 ;.; X N g T R N s a real-valued, N-dmensonal random vector defned on a probablty space (U, F, P) comprsng the sample space U, the s-feld F, and the probablty measure P; U F s the falure doman; and f X (x) s the jont probablty densty functon of X. In structural relablty analyss, X typcally represents loads, materal propertes, and geometry and P F s the probablty of falure. For component relablty analyss, U F Z{x:y(x)!0}, where y(x) represents a sngle performance functon. For system relablty analyses nvolvng m performance functons, U F Zfx :g m kz1y ðkþ ðxþ!0g and U F Zfx :h m kz1y ðkþ ðxþ!0g for seres and parallel systems, respectvely, where y (k) (x) represents the kth performance functon. Nevertheless, for most practcal problems, * Correspondng author. Tel.: C ; fax: C E-mal address: rahman@engneerng.uowa.edu (S. Rahman). URL: /$ - see front matter q 005 Elsever Ltd. All rghts reserved. do: /j.probengmech the exact evaluaton of ths ntegral, ether analytcally or numercally, s not possble because N s large, f x (x) s generally non-gaussan, and y(x) or y (k) (x) are hghly nonlnear functons of x. Whle research s ongong, approxmate methods, such as the frst- and second-order relablty methods (FORM/SORM) [1 8] and smulaton methods [9 18] are commonly employed to estmate the falure probablty. FORM/SORM are based on lnear (FORM) or quadratc approxmaton (SORM) of the lmt-state surface at a most probable pont (MPP). Experence has shown that FORM/SORM are suffcently accurate for engneerng purposes, provded that the MPP s accurately found, the lmt-state surface at MPP s close to beng lnear or quadratc, and no multple MPPs exst. The MPP can be located by varous gradent-based optmzaton algorthms, whch n turn requre frst- and/or second-order (also needed n SORM) response senstvtes or gradents, for whch effcent means of calculaton are also requred. If these senstvtes can be calculated analytcally, FOR- M/SORM are qute effcent. Otherwse, FORM/SORM can be neffectve, for nstance, when response senstvtes are not avalable or when senstvty analyss s computatonally ntensve. A prme example s a multdscplnary desgn envronment, where multple analyss codes from thrd-party sources are frequently employed wthout any knowledge of gradents. In that case, FORM/SORM may yeld naccurate relablty solutons

2 40 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) or create computatonally neffcent results when usng gradents from fnte-dfference approxmatons. Furthermore, for hghly nonlnear performance functons, whch exst n many structural problems, results based on FORM/SORM must be nterpreted wth cauton. If the Rosenblatt transformaton, frequently used to map non- Gaussan random nput nto ts standard Gaussan mage, yelds a hghly nonlnear lmt-state, nadequate relablty estmates by FORM/SORM may result [1,13]. Furthermore, the exstence of multple MPPs could gve rse to large errors n standard FORM/SORM approxmatons [3, 8]. In that case, mult-pont FORM/SORM along wth the system relablty concept s requred for mprovng component relablty analyss [8]. Smulaton methods nvolvng samplng and estmaton are well known n the statstcs and relablty lterature. Drect Monte Carlo smulaton [9] s the most wdely used smulaton method and nvolves the generaton of ndependent samples of all nput random varables, repeated determnstc trals (analyses) to obtan correspondng smulated samples of response varables, and standard statstcal analyss to estmate probablstc characterstcs of response. Ths method generally requres a large number of smulatons to calculate low falure probablty, and s mpractcal when each smulaton nvolves expensve fnteelement, boundary-element, or mesh-free calculatons. As a result, researchers have developed or examned faster smulaton methods, such as quas-monte Carlo smulaton [10], mportance samplng [11], drectonal smulaton [1 14], and others [15 18]. Whle smulaton methods do not exhbt the lmtatons of approxmate relablty methods, such as FORM/SORM, they generally requre consderably more extensve calculatons than the latter methods. Consequently, smulaton methods are useful when alternatve methods are napplcable or naccurate, and have been tradtonally employed as a yardstck for evaluatng approxmate methods. Ths paper presents a new class of computatonal methods, referred to as decomposton methods, for predctng relablty of structural and mechancal systems subject to random loads, materal propertes, and geometry. The dea of decomposton n multvarate functons, orgnally developed by the authors for statstcal moment analyss [19,0], has been extended for relablty analyss, whch s the focus of the current paper. The proposed relablty methods nvolve a very small number of exact or numercal evaluatons of the performance functon at selected nput, generaton of approxmate values of the performance functon at arbtrarly large number of nput usng the decomposton technque, and subsequent response surface approxmatons. Fnally, the relablty s evaluated usng the Monte Carlo smulaton. Seven numercal examples nvolvng elementary mathematcal functons and sold-mechancs problems llustrate the proposed method. Whenever possble, comparsons have been made wth alternatve approxmate and smulaton methods to evaluate the accuracy and computatonal effcency of the proposed methods.. Multvarate functon decomposton Consder a contnuous, dfferentable, real-valued functon y(x) that depends on xzfx 1 ;.; x N g T R N. Suppose that y(x) has a convergent Taylor seres expanson at an arbtrary reference pont xzcz{c 1,.,c N } T, expressed by yðxþ Z yðcþ C XN or jz1 yðxþzyðcþc XN 1 j! C XN j 1 ;j O0 jz1 1 X N v j y ðcþðx j! vx Kc Þ j CR ; () Z1 X N Z1 v j y ðcþðx vx Kc Þ j 1 X y vj1cj j 1!j! 1! vx j 1 1 vx j ðcþðx 1 Kc 1 Þ j1 ðx Kc Þ j CR 3 ; ð3þ where the remander R denotes all terms wth dmenson two and hgher and the remander R 3 denotes all terms wth dmenson three and hgher..1. Unvarate approxmaton Consder a unvarate approxmaton of y(x), denoted by ^y 1 ðxþh ^y 1 ðx 1 ;.;x N Þ Z XN Z1 yðc 1 ;.;c K1 ;x ;c C1 ;.;c N ÞKðN K1ÞyðcÞ; where each term n the summaton s a functon of only one varable and can be subsequently expanded n a Taylor seres at xzc, yeldng ^y 1 ðxþ Z yðcþ C XN jz1 1 X N j! Z1 ð4þ v j y ðcþðx Kc Þ j : (5) vx j Comparng Eqs. () and (5) ndcates that the unvarate approxmaton leads to the resdual error yðxþk ^y 1 ðxþzr, whch ncludes contrbutons from terms of dmenson two and hgher. For suffcently smooth y(x) wth convergent Taylor seres, the coeffcents assocated wth hgherdmensonal terms are usually much smaller than that wth one-dmensonal terms. In that case, hgher-dmensonal terms contrbute less to the functon, and therefore, can be neglected. Furthermore, Eq. (4) represents exactly the same functon as y(x) when yðxþz P y ðx Þ,.e. when y(x) can be addtvely decomposed nto functons y (x ) of sngle varables.

3 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) Bvarate approxmaton In the smlar way, consder a bvarate approxmaton ^y ðxþ h X yðc 1 ;.; c 1 K1; x 1 ; c 1 C1;.; c K1; x ; c C1;.; c N Þ 1! KðN KÞ XN C Z1 ðn K1ÞðN KÞ yðcþ; yðc 1 ;.; c K1 ; x ; c C1 ;.; c N Þ 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 xzc, yeldng ^y ðxþ ZyðcÞ C XN 1 X N v j y ðcþðx j! vx Kc Þ j jz1 Z1 C XN 1 X j v 1 Cj y ðcþðx j j 1 ;j O0 1!j! vx 1! 1 vx 1 Kc 1 Þ j 1 ðx Kc Þ j : ð7þ Agan, the comparson of Eqs. (3) and (7) ndcates that the bvarate approxmaton leads to the resdual error yðxþk ^y ðxþzr 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 leads to hgher accuracy than the unvarate approxmaton. Furthermore, Eq. (6) represents exactly the same functon as y(x) when yðxþz P!j y j ðx ; x j Þ,.e. when y(x) can be addtvely decomposed nto functons y j (x, x j ) of at most two varables..3. 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 1%S%N. ð6þ The generalzed S-varate approxmaton of y(x) s! ^y S ðxþh XS N KS C K1 ðk1þ Z0! X yðc 1 ;.; c k1 K1; x k1 ; c k1 C1;.; c ksk K1; x ksk ; k 1!/!k SK c ksk C1;.; c N Þ: ð8þ If y R hyðc 1 ;.; c k1 K1; x k1 ; c k1 C1;.; c kr K1; x kr ; c kr C1;.; c N Þ; 0%R%S, a multvarate functon decomposton theorem developed by the authors leads to [0]! y R Z XR N Kk t k ; 0%R%S; (9) kz0 R Kk where t 0 ZyðcÞ t 1 Z P 1 X N v j 1 y j 1 j 1! 1 Z1 vx j ðcþðx 1 Kc 1 Þ j1 1 1 t Z P 1 X y vj1cj j 1 ;j j 1!j! 1! vx j 1 1 vx j ðcþðx 1 Kc 1 Þ j1 ðx Kc Þ j : «t S Z P 1 X j 1 ;.;j S j 1!.j S! vx j 1 1.vx j ðcþðx 1 Kc 1 Þ j1.ðx S Kc S Þ js S S 1!/! S vj1c/cjs y (10) Usng Eqs. (9) and (10), t can be shown that ^y S ðxþ n Eq. (8) conssts of all terms of the Taylor seres of y(x) that have less than or equal to S varables [0]. The expanded form of Eq. (8), when compared wth the Taylor expanson of y(x), ndcates that the resdual error n the S-varate approxmaton s yðxþk ^y S ðxþzr SC1, where the remander R SC1 ncludes terms of dmenson SC1 and hgher. When SZ1, Eq. (8) degenerates to the unvarate approxmaton (Eq. (4)). When SZ, t becomes the bvarate approxmaton (Eq. (6)). Smlarly, trvarate, quadrvarate, and other hgher-varate approxmatons can be derved by approprately selectng the value of S. In the lmt, when SZN, Eq. (8) converges to the exact functon y(x). In other words, the proposed decomposton generates convergent sequence of approxmatons of y(x)..4. Remarks The decomposton of a general multvarate functon y(x) can be vewed as a fnte sum yðxþzy 0 C XN y ðx ÞC XN y 1 ðx 1 ;x ÞC/C XN y 1. S ðx 1 ;.;x s ÞC/Cy 1.N ðx 1 ;.;x N Þ; (11) Z1 fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 1 ; Z1 1 ;.; S Z1 Z^y 1 ðxþ 1! 1!/! S fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z^y ðxþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z^y S ðxþ where y 0 s a constant, y (x ) s a unvarate component functon representng ndependent contrbuton to y(x) by

4 4 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) nput varable x actng alone, y 1 ðx 1 ;x Þ s a bvarate component functon descrbng cooperatve nfluence of two nput varables x 1 and x, y 1. S ðx 1 ;.;x S Þ s an S-varate component functon quantfyng cooperatve effects of S nput varables x 1 ;.;x S, and so on. By comparng Eqs. (4) and (6) wth Eq. (11), the proposed unvarate and bvarate approxmatons provde two- and three-term approxmants, respectvely, of the fnte decomposton. In general, the S- varate approxmaton n Eq. (8) yelds the SC1-term approxmant of the decomposton. The fundamental conjecture underlyng ths work s that component functons arsng n proposed decomposton wll exhbt nsgnfcant S-varate effects cooperatvely when S/N, leadng to useful lower-varate approxmatons of a hgh-dmensonal functon. Indeed, ths s the major motvaton of the relablty methods developed. It s worth notng that the unvarate and bvarate approxmatons should not be vewed as frst- or secondorder Taylor seres expansons nor do they lmt the nonlnearty of y(x). Accordng to Eqs. (5) and (7), all hgher-order unvarate and bvarate terms of y(x), respectvely, are ncluded n the proposed approxmatons. Furthermore, the approxmatons contan contrbutons from all nput varables. If for x 1 Zx ðj 1Þ 1 y 1 ðx ðj 1Þ 1 ; x ðj Þ Þ and x Zx ðj Þ, n functon values hyðc 1 ;.; c 1 K1; x ðj 1Þ 1 ; c 1 C1;.; c K1; x ðj Þ ; c C1;.; c N Þ; j 1 Z 1; ;.; n; j Z 1; ;.; n ð15þ are gven, the functon value y 1 ðx 1 ; x Þ for arbtrary pont ðx 1 ; x Þ can be obtaned usng the Lagrange nterpolaton as y 1 ðx 1 ; x Þ Z Xn X n j Z1 j 1 Z1 f j1 ðx 1 Þf j ðx Þy 1 ðx ðj 1Þ 1 ; x ðj Þ Þ; (16) where shape functons f j1 ðx 1 Þ and f j ðx Þ are defned n Eq. (14). Note that there are n and n performance functon evaluatons nvolved n Eqs. (13) and (16), respectvely. Therefore, the total cost for unvarate approxmaton entals a maxmum of nnc1 functon evaluatons, and for bvarate approxmaton, N(NK1)n /CnNC1 maxmum functon evaluatons are requred. More accurate multvarate approxmatons can be developed n the smlar way. However, because of much hgher cost, only unvarate and bvarate approxmatons wll be examned n ths paper. 4. Monte Carlo smulaton 3. Response surface generaton Consder the unvarate terms y ðx Þhyðc 1 ;.; c K1 ; x ; c C1 ;.; c N Þ n Eqs. (4) and (6). If for x Zx ðjþ, n functon values 4.1. Component relablty analyss For component relablty analyss, the Monte Carlo estmates P F,1 and P F, of the falure probablty employng unvarate and bvarate approxmatons, respectvely, are y ðx ðjþ Þ Z yðc 1 ;.; c K1 ; x ðjþ ; c C1 ;.; c N Þ; j Z 1; ;.; n (1) P F;1 Z 1 N S X N S Z1 I½^y 1 ðx ðþ Þ!0Š (17) are gven, the functon value for arbtrary x can be obtaned usng the Lagrange nterpolaton as y ðx Þ Z Xn jz1 f j ðx Þy ðx ðjþ Þ; (13) where the shape functon f j (x ) s defned as f j ðx ÞZ ðx ðjþ ðx Kx ð1þ Kx ð1þ Þ.ðx Kx ðjk1þ Þ.ðx ðjþ Kx ðjk1þ Þðx Kx ðjc1þ Þðx ðjþ Kx ðjc1þ Þ.ðx Kx ðnþ Þ Þ.ðx ðjþ Kx ðnþ Þ : (14) By usng Eq. (13), arbtrarly many functon values of y (x ) can be generated f n functon values are gven. The same dea can be appled to the bvarate terms y 1 ðx 1 ; x Þh yðc 1 ;.; c 1 K1; x 1 ; c 1 C1;.; c K1; x ; c C1;.; c N Þ n Eq. (6). P F; Z 1 N S X N S Z1 I½^y ðx ðþ Þ!0Š; (18) where x () s the th realzaton of X, N S s the sample sze, and I½$Š s an ndcator functon such that IZ1 fx () s n the falure set (.e. when ^y 1 ðx ðþ Þ!0 for unvarate approxmaton and when ^y ðx ðþ Þ!0 for bvarate approxmaton of the performance functon) and zero otherwse. 4.. System relablty analyss For system relablty analyss nvolvng unon and ntersecton of m falure sets, smlar decomposton and response surface approxmatons can be performed for each performance functon. Let U F Zfx :g m kz1y ðkþ ðxþ!0g and U F Zfx :h m kz1y ðkþ ðxþ!0g denote component falure sets n seres and parallel systems, respectvely, where y (k) (x) s the kth performance functon. Hence, the Monte Carlo estmates P F,1 and P F, usng unvarate and bvarate approxmatons,

5 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) respectvely, are 8 1 X NS >< I g m N kz1^yðkþ 1 ðxðþ Þ!0 ; seres system S Z1 P F;1 Z 1 X NS >: I h m N kz1^yðkþ 1 ðxðþ Þ!0 ; parallel system S Z1 (19) 8 1 X NS >< I g m N kz1^yðkþ ðxðþ Þ!0 ; seres system S Z1 P F; Z 1 X ; NS >: I h m N kz1^yðkþ ðxðþ Þ!0 ; parallel system S Z1 (0) where I½$Š s another ndcator functon such that IZ1 f x () s n the system falure doman and zero otherwse. The decomposton methods nvolvng unvarate and bvarate approxmatons, are respectvely, defned as unvarate and bvarate methods n ths paper. Snce the proposed methods facltate explct lower-dmensonal approxmatons of a general multvarate functon, the embedded Monte Carlo smulaton can be conducted for any sample sze. However, the accuracy and effcency of the falure probablty calculatons usng Eqs. (17) (0) depend on both the decomposton and response surface approxmatons. They wll be evaluated usng several numercal examples n Secton Numercal examples Two sets of numercal examples, one nvolvng explct mathematcal functons (Examples 1 and ) and the other nvolvng sold-mechancs/structural problems (Examples 3 7), are presented to llustrate the proposed decomposton methods. Whenever possble, comparsons have been made wth alternatve approxmate (FORM/SORM) and several smulaton methods to evaluate the accuracy and computatonal effcency of the proposed decomposton methods. All numercal results of decomposton methods are based on the expanson at the mean pont. For response surface generaton, n (Z3, 5, 7 or 9) unformly dstrbuted ponts m K(nK1)s /, m K(nK3)s /,., m,., m C(nK3)s /, m C(nK1)s / were deployed at x -coordnate wth mean m and standard devaton s, leadng to (nk1)nc1 and (nk 1) N(NK1)/C(nK1)NC1 functon evaluatons by unvarate and bvarate methods, respectvely. In all examples, response surface approxmatons were constructed n the standard Gaussan space. When comparng computatonal efforts by varous methods, the number of orgnal performance functons 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 the proposed decomposton methods, they are dfferent, because the Monte Carlo smulaton (although wth the same sample sze as n drect Monte Carlo smulaton) embedded n decomposton methods are conducted usng 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 or mesh-free analyss, as n Examples 4 7. However, the dfference becomes trval when solvng problems nvolvng explct performance functons, as n Examples 1 3. Hence, the computatonal effort expressed n terms of functon evaluatons alone should be carefully nterpreted for problems nvolvng explct 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 1. Consder a component relablty problem wth a performance functon [1] yðxþ ZK 1 X 5 X KX b 6 Cb; (1) Z1 where X 1N(0,1), Z1,.,6 are ndependent, standard Gaussan random varables and bz4. Theexact value of the falure probablty s P F ZPyðXÞ!0 Z1:30!10 K3 [1]. Recently, Yonezawa et al. [], who developed a new smulaton method wth lmted samplng regon, predcted the falure probablty to be 1.8!10 K3, nvolvng 10,000 samples, When ths problem was solved by the proposed unvarate method, a falure probablty of P F,1 Z1.3! 10 K3 was calculated usng only 13 functon evaluatons (nz3, NZ6). Hence, the unvarate method s not only accurate, but also sgnfcantly more effcent than some exstng smulaton methods (e.g. smulaton wthn lmted samplng regon []). Snce the unvarate method exactly represents the unvarate performance functon n Eq. (1), there s no need to pursue the bvarate approxmaton. However, n many cases, bvarate approxmaton s needed to acheve mproved accuracy and wll be llustrated n forthcomng examples. Example. Consder a system relablty problem n whch the falure regon s bounded by the followng two performance functons [13,14] p y ð1þ ðxþ ZKX 1 KX KX 3 C3 ffff 3 ; y ðþ ðxþ ZKX 3 C3; () where X 1N(0,1), Z1 3 are ndependent, standard Gaussan random varables. Both seres and parallel systems are consdered.

6 44 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) Fg. 1. Falure doman for x Zconstant; (a) seres system; (b) parallel system. Seres system For the seres system, the system falure regon s defned as U F Zfx : y ð1þ ðxþ!0gy ðþ ðxþ!0g, whch s sketched n Fg. 1(a) n the x 1 Kx 3 space when x Zconstant. Katsuk and Frangopol [14] computed Dtlevsen s [3] secondorder bounds of the probablty of falure to be:.53734! 10 K3 %P F %.61864!10 K3. Table 1 compares the falure probablty obtaned by the proposed unvarate method, second-order bounds, drectonal smulaton usng Fekete ponts by Ne and Ellngwood [13], and the drect Monte Carlo smulaton wth 10 7 samples. Accordng to Ne and Ellngwood, the drectonal smulaton ncludng Fekete ponts generates the most accurate and effcent estmaton of the falure probablty. As can be seen n Table 1, the proposed unvarate method provdes dentcal result (P F,1 Z.585!10 K3 ) of drect Monte Carlo, but usng only 7 (nz3, NZ3) functon evaluatons. Ths s because both performance functons n Eq. () are unvarate functons, whch are exactly represented by ther response Table 1 Falure probablty for seres system Method Falure probablty (!10 K3 ) Number of functon evaluatons a Unvarate method b Second-order bounds c Drectonal smulaton d Fekete pontsz36.57 Fekete pontsz Fekete pontsz7.570 Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Drect Monte Carlo smulaton ,000,000 a Total number of tmes the orgnal performance functons s calculated. b (3K1)!3C1Z7. c See Refs. [3,14]. d See Ref. [13]. For each Fekete pont, several functon evaluatons are needed to determne the radus of hypersphere. The total number of functon evaluatons s not reported. surfaces. In contrast, to obtan the same accuracy, the drectonal smulaton nvolved drectons; for each drecton the radus of the hypersphere segment has to be obtaned through numercal methods, whch usually takes several functon evaluatons [13]. Hence, sgnfcantly fewer functon evaluatons are needed n the unvarate method. Parallel system For the parallel system, the system falure doman s defned as U F Zfx : y ð1þ ðxþ!0hy ðþ ðxþ!0g and s also depcted n Fg. 1(b). The second-order bounds [3] of the probablty of falure are: !10 K5 %P F %1.6595! 10 K4 [3,13]. The falure probablty reported by Katsuk and Frangopol [14] usng Hohenbchler s approxmaton [3] of multnormal ntegrals s 1.411!10 K4. Ne and Ellngwood [13] also produced varous estmatons usng the drectonal smulaton wth Fekete ponts. Table compares Table Falure probablty for parallel system Method Falure probablty (!10 K4 ) Number of functon evaluatons a Unvarate method b Second-order bounds c Multnormal ntegrals d 1.4 Drectonal smulaton e Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Drect Monte Carlo smulaton ,000,000 a Total number of tmes the orgnal performance functons s calculated. b (3K1)!3C1Z7. c See Refs. [3,14]. d See Refs. [14,3]. e See Ref. [13]. For each Fekete pont, several functon evaluatons are needed to determne the radus of hypersphere. The total number of functon evaluatons s not reported.

7 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) falure probablty calculated by the proposed unvarate method, second-order bounds, multnormal ntegrals, Ne and Ellngwood s drectonal smulaton usng Fekete ponts, and drect Monte Carlo smulaton wth 10 7 samples. For the same reason as n the seres system, the probablty estmates (P F,1 Z1.303!10 K4 ) by the unvarate method and drect Monte Carlo smulaton are dentcal. Accordng to Table, only 7 functon evaluatons are needed by the unvarate method. To acheve the same level of accuracy, the drectonal smulaton requred 96 smulatng drectons. Hence, the proposed unvarate method s more effcent than the drectonal smulaton for both seres and parallel systems n ths example. 5.. Example Set II structural and sold-mechancs problems Example 3. Rgd-Plastc Portal Frame Structure. Dtlevsen [4] and Ne and Ellngwood [13] studed the rgdplastc frame structure of Fg. by the drectonal smulaton method. Ths structure can be analyzed as a seres system of three lmt-state functons (collapse mechansms), whch, accordng to the prncple of vrtual work, are y ð1þ ðxþ Z X CX 3 CX 4 KGb y ðþ ðxþ Z X 1 CX CX 4 CX 5 KFa y ð3þ ðxþ Z X 1 CX 3 CX 4 CX 5 KFa KGb; (3) where y (1) (X), y () (X) and y (3) (X) are component lmt-state functons for beam, sway and combned mechansms, respectvely. The yeld moments X j, jz1,.,5, at hnge ponts are ndependent and dentcally dstrbuted lognormal random varables, wth unt mean and 5% coeffcent of varaton. The lateral force F, vertcal force G and dstances a and b are constants, wth GbZ1.15 Table 3 Falure probablty for portal frame Method Falure probablty (!10 K5 ) Number of functon evaluatons a Unvarate method b Frst-order bounds c Second-order bounds c Drectonal smulaton d 5.45 Drectonal smulaton e Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Fekete pontsz Drect Monte Carlo smulaton ,000,000 a Total number of tmes the orgnal performance functons s calculated. b (9K1)!5C1Z41. c See Refs. [3,13]. d A soluton by large-scale drectonal smulaton; see Ref. [13]. e See Ref. [13]. For each Fekete pont, several functon evaluatons are needed to determne the radus of hypersphere. The total number of functon evaluatons s not reported. and FaZ.4. The falure doman s defned as: U F Zfx : y ð1þ ðxþ!0gy ðþ ðxþ!0gy ð3þ ðxþ!0g. Table 3 lsts the system falure probabltes of the frame, whch are calculated by frst- and second-order bounds [3, 13], a large-scale drectonal smulaton reported n Ref. [13], Ne and Ellngwood s drectonal smulaton nvolvng Fekete ponts [13], proposed unvarate method nvolvng 41 (nz9, NZ5) functon evaluatons, and drect Monte Carlo smulaton nvolvng 10 8 samples. The results n Table 3 ndcate that the unvarate method provdes dentcal result (P F,1 Z5.544!10 K5 ) of the drect Monte Carlo smulaton. The drectonal smulatons also yeld falure probabltes close to the drect Monte Carlo result. But, the proposed method requres very few functon evaluatons to generate an accurate result. Example 4. Ten-Bar Truss Structure (Lnear-Elastc). A ten-bar, lnear-elastc, truss structure, shown n Fg. 3, was studed n ths example to examne the accuracy and effcency of the proposed relablty method. The Young s modulus of the materal s 10 7 ps. Two concentrated forces of 10 5 lb are appled at nodes and 3, as shown n Fg. 3. The cross secton area X, Z1,.,10 for each bar s normally dstrbuted random varable wth mean mz.5 n. and standard devaton sz0.5 n. Accordng to the loadng condton, the maxmum dsplacement [(v 3 (X 1,.,X 10 ))] occurs at node 3, where a permssble dsplacement s lmted to 18 n. Hence, the lmt-state functon s yðxþ Z 18 Kv 3 ðx 1 ;.; X 10 Þ: (4) Fg.. A portal frame as rgd-plastc system. Table 4 shows the falure probablty of the truss structure, calculated usng the proposed unvarate method, FORM, three varants of SORM due to Bretung [4], Hohenbechler [5] and Ca and Elshakoff [6], and drect

8 46 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) n 360 n 5 6 (a) σ D=(0,L) E=(L,L) 360 n L Crcular Hole x C=(0,a) A=(a,0) x 1 B=(L,0) ,000 lb 100,000 lb a Fg. 3. A ten-bar truss structure wth random cross-sectonal areas. Monte Carlo smulaton (10 6 samples). For the unvarate method, seven unformly dstrbuted ponts between mk3s and mc3s were deployed for functon evaluatons at each dmenson. As can be seen from Table 4, the unvarate method predcts the falure probablty more accurately than FORM and all three varants of SORM, yet the number of functon evaluatons s less than FORM and much less than SORM. Furthermore, the proposed method does not requre any response dervatves as requred by FORM/SORM n fndng the MPP. Example 5. Stochastc Mesh-Free Analyss of Plate wth a Hole (Lnear-Elastc). Consder a square plate wth a centrally located crcular hole, as shown n Fg. 4(a). The plate has a dmenson of LZ40 unts, a hole wth dameter az unts, and s subjected to a unformly dstrbuted load of magntude s N Z1 unt. The Posson s rato n was selected to be 0.3. The elastc modulus was assumed to be a homogeneous random feld and symmetrcally dstrbuted wth respect to x 1 - and x -axes [see Fg. 4(a)]. The modulus of elastcty E(x) was represented by a homogeneous, lognormal translaton feld E(x)Zc a exp[a(x)], wth mean m E Z1 unt and standard devaton s E Z0. or 0.5 for whch a(x) sazero-mean, homogeneous, Gaussan random feld Table 4 Falure probablty for ten-bar truss Method Falure probablty Number of functon evaluatons a Unvarate method b FORM SORM (Bretung) c SORM (Hohenbchler) d SORM (Ca and Elshakoff) e Drect Monte Carlo smulaton ,000,000 a Total number of tmes the orgnal performance functons s calculated. b (7K1)!10C1Z61. c See Ref. [4]. d See Ref. [5]. e See Ref. [6]. (b) σ L Fg. 4. A square plate wth a hole subjected to unformly dstrbuted tenson; (a) geometry and loads; (b) meshless dscretzaton. p wth standard devaton s a Z ffffffffffffffffffffffffffffffffffffffffffffffffffffff lnð1cs E =m E Þ, an exponental covarance functon represented by G a ðxþ Z E½aðxÞaðx CxÞŠ Z s aexp K jx 1j Cjx j ; bl cx; x CxD; (5) where D3R s the doman of the random feld represented by the shaded area n Fg. 4(a), and qffffffffffffffffffffffffffffffff c a Z m E expðks a=þ Z m E= m E Cs E: (6) Due to symmetry, only a quarter of the plate, represented by the regon ABEDC and shaded n Fg. 4(a), was

9 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) analyzed. Fg. 4(b) presents a meshless dscretzaton of the quarter plate wth 90 nodes. The random feld a(x) was parameterzed usng the Karhunen Loève expanson [5] aðxþy XN X j jz1 qffffffff f j ðxþ: (7) l j where X j 1N(0,1), jz1,.,n are standard and ndependent Gaussan random varables and {l j,f j (x)}, jz1,.,n are the egenvalues and egenfunctons, respectvely, of the covarance kernel. Mesh-free shape functons were employed to solve the assocated ntegral equaton needed to calculate the egenvalues and egenfunctons [6]. Based on the correlaton parameter bz0.5, a value of NZ8 was selected to adequately represent a(x). Hence, the nput uncertanty was represented by an 8-dmensonal standard Gaussan vector X1N(0,I), where 0R 8 and I LðR 8! R 8 Þ are the null vector and dentty matrx, respectvely. A stress-based falure crteron at a crtcal pont was employed to calculate the relablty of the plate. If (a) Falure Probablty of Plate, P(σ A > Sy) Falure Probablty of Plate, P(σ A > Sy) (b) Unvarate Method (33) Bvarate Method (481) Smulaton (10 5 Samples) µ E= 1.0 σe = Yeld Strength, S y Unvarate Method (33) Bvarate Method (481) Smulaton (10 5 Samples) µ E= 1.0 σ E = Yeld Strength, S y Fg. 5. Falure probablty of square plate wth a hole; (a) s E Z0.; (b) 0.5. s A (X 1,.,X 8 ) denotes the von Mses equvalent stress at pont A [see Fg. 4(a)], the lmt-state functon assocated wth the von Mses yeld crteron s yðxþ Z S y Ks A ðx 1 ;.; X 10 Þ; (8) where S y s a determnstc yeld strength of the materal. Fg. 5(a) and (b) present falure probabltes for varous yeld strengths, predcted by the proposed decomposton methods, as well as by the drect Monte Carlo smulaton (10 5 samples). As can be seen n Fg. 5(a), when the uncertanty of elastc modulus s lower (s E Z0.), both the unvarate and bvarate methods provde satsfactory results n comparson wth the smulaton results. However, when a hgher uncertanty s consdered (s E Z0.5), Fg. 5(b) shows that the accuracy of the falure probablty from the bvarate method s slghtly hgher than that from the unvarate method. The number of functon evaluatons by proposed methods wth unvarate and bvarate approxmatons are only 33 and 481, respectvely, when nz5 and NZ8. A comparson of total CPU tmes, shown n Fg. 6, ndcates that both decomposton methods are far more effcent than the Monte Carlo smulaton. In calculatng the CPU tmes, the overhead cost due to random feld dscretzaton, random number generaton, and response surface approxmatons are all ncluded. The overhead cost s comparable to the cost of evaluatng the structural response n ths partcular problem. For ths reason, the ratos of CPU tmes by bvarate and unvarate methods and by Monte Carlo and unvarate methods, are respectvely, only 8 and 1080, as compared wth 15 (z481/33) and 3030 (z100,000/33), when functon evaluatons alone are compared. For more complex problems requrng expensve response evaluatons, the overhead cost s neglgble. In that case, the CPU rato should approach the rato of functon evaluatons. Hence, the proposed methods are effectve when a response evaluaton entals costly mesh-free or fnte-element analyss. Normalzed CPU tme Normalzed = CPU 1 Unvarate 1 Method CPU CPU by Unvarate Method 8 Bvarate Method 1080 Monte Carlo Smulaton (10 5 Samples) Fg. 6. Comparson of CPU tme by varous methods.

10 48 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) (a) Falure Probablty of Arch, P(u > u 0 ) Unvarate Method (17) Bvarate Method (113) Smulaton (10 5 Samples) P = 400 N µ = 80 GPa E σ = 16 GPa E Fg. 7. A shallow arch subject to a concentrated load at mdspan; (a) geometry and loads; (b) fnte-element dscretzaton. Example 6. Stochastc Fnte-Element Analyss of Shallow Arch (Nonlnear, Large-Deformaton). In ths example, the proposed decomposton methods were employed to solve a nonlnear problem n sold-mechancs. Fg. 7(a) llustrates a shallow crcular arch, wth mean radus RZ 100 mm, rectangular cross-secton wth depth hz mm, thckness tz1 mm, and arc angle bz8.18. The arch, fxed at both ends, was subjected to a concentrated load PZ400 N at the center. The Posson s rato was zero n ths example. A fnte-element mesh employng 30 8-noded quadrlateral elements was used to model the arch, as shown n Fg. 7(b). The stress analyss nvolved largedeformaton behavor for modelng the geometrc nonlnearty of the arch. A plane stress condton was assumed. The modulus of elastcty E(x) was represented by a homogeneous, lognormal translaton feld E(x)Zc a- exp[a(x)], wth mean m E Z80 kn/mm and standard devaton s E for whch a(x) s a zero-mean, homogeneous, Gaussan p random feld wth standard devaton s a Z ffffffffffffffffffffffffffffffffffffffffffffffffffffff lnð1cs E =m E Þ, an exponental covarance functon G a ðxþze½aðxþaðxcxþšzs a exp½kjjxjj=ðblþš, pcx, ffffffffffffffffffffffffffffffff xc xd, bz0.1; and c a Zm E expðks a=þzm E= m E Cs E. The Karhunen Loève expanson was employed to dscretze the random feld a(x) nto four standard Gaussan random varables. Due to uncertanty n the elastc modulus, the deflecton u at center pont of ths arch s stochastc. A dsplacementbased falure crteron at mdspan was employed to calculate the relablty of the plate. If u(x 1,.,X 4 ) denotes the mdspan dsplacement of the arch, the lmt-state functon s yðxþ Z u 0 KuðX 1 ;.; X 4 Þ; (9) where u 0 s a determnstc threshold of dsplacement. The proposed unvarate and bvarate methods were employed to predct the falure probablty of the arch. To evaluate these methods, drect Monte Carlo smulaton was performed to generate benchmark solutons. The results, plotted as a functon of dsplacement threshold u 0, are (b) Falure Probablty of Arch, P(u > u 0 ) Permssble Dsplacement, u 0 Unvarate Method (17) Bvarate Method (113) Smulaton (10 5 Samples) P = 400 N µ = 80 GPa E σ = 40 GPa Permssble Dsplacement, u 0 Fg. 8. Falure probablty of shallow arch; (a) s E Z16 GPa; (b) 40 GPa. presented n Fg. 8(a) and (b) for two cases of statstcal nput: (a) s E Z16 GPa and (b) s E Z40 GPa, representng small and large uncertantes of elastc modulus. The results ndcate that the unvarate and bvarate methods provde excellent estmates of falure probabltes for both cases of nput. For each problem case, the unvarate and bvarate methods, respectvely, requre only 17 and 113 analyses (nz5, NZ4), as opposed to 10 5 analyses n Monte Carlo smulaton. Although the accuracy of the unvarate method slghtly decreases as the uncertanty ncreases, the bvarate approxmaton generates very accurate predcton even when the coeffcent of varaton s 0.5, as ndcated by Fg. 8(b). Example 7. Stochastc Fracture Mechancs (Lnear- Elastc). The fnal example nvolves a nonhomogeneous, functonally graded, edge-cracked plate, presented to llustrate mxed-mode probablstc fracture-mechancs analyss usng the decomposton method. As shown n Fg. 9(a), a functonally graded plate of length LZ16 unts was fxed at the bottom and subjected to a far-feld normal E

11 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) Table 5 Statstcal propertes of random nput for functonally graded edge-cracked plate Random varable Mean Standard devaton Probablty dstrbuton a Unform a W Unform b s N Gaussan t N Gaussan q Gaussan E Lognormal E Lognormal b Lognormal a Unformly dstrbuted over (.8,4.). b Unformly dstrbuted over (7,8). Fg. 9. A functonally graded edge-cracked plate subject to mxed-mode deformaton; (a) geometry and loads; (b) fnte-element dscretzaton. stress s N and a shear stress t N appled at the top. The elastc modulus was assumed to vary smoothly accordng to a hyperbolc tangent functon, gven by Eðx 1 ; x Þ Z E 1 CE C E 1 KE tanh bðx 1 cos q Cx sn qþ; (30) where (x 1,x ) are spatal coordnates [see Fg. 9(a)], E 1, E, b and q are modulus parameters. The followng eght ndependent random varables were defned: (1) crack length a; () plate wdth W; (3) far-feld normal stress s N ; (4) far-feld shear stress t N ; (5) modulus angle parameter q; (6) modulus at the left end E 1 ZE(0,x ); (7) modulus at the rght end E ZE(W,x ); and (8) modulus parameter b. The statstcal property of the random nput XZ {a,w,s N,t N,q,E 1,E,b} T s defned n Table 5. Due to far-feld stresses the plate s subjected to mxedmode deformaton nvolvng fracture modes I and II [7]. The mxed-mode stress-ntensty factors K I (X) and K II (X) were calculated usng an nteracton ntegral method [8]. The plate was analyzed usng the fnte-element method nvolvng a total of 83 8-noded, regular, quadrlateral elements and 48 6-noded, quarter-pont (sngular), trangular elements at the crack-tp, as shown n Fg. 9(b). The falure crteron s based on a mxed-mode fracture ntaton usng the maxmum tangental stress theory [7], whch leads to the performance functon yðxþ ZK Ic K K I ðxþcos QðXÞ K 3 K IIðXÞsn QðXÞ!cos QðXÞ ; ð31þ where K Ic s a determnstc fracture toughness, typcally measured from small-scale fracture experments under mode-i and plane stran condtons, and Q c (X) s the drecton of crack propagaton, gven by Q c ðxþ 8 p tan K1 1K ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff! 1C8½K II ðxþ=k I ðxþš ; f K 4K II ðxþ=k I ðxþ II ðxþo0 >< Z p tan K1 1C ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff! : 1C8½K II ðxþ=k I ðxþš >: ; f K 4K II ðxþ=k I ðxþ II ðxþ!0 ð3þ The falure probablty P F ZP[y(X)!0] was predcted usng the proposed decomposton methods and compared

12 50 H. Xu, S. Rahman / Probablstc Engneerng Mechancs 0 (005) Probablty of Fracture Intaton Fracture Toughness (K Ic ) wth the drect Monte Carlo smulaton, as shown n Fg. 10. In Fg. 10, the falure probablty s plotted as a functon of fracture toughness K Ic. Usng nz3 and NZ8, the unvarate and bvarate methods requred only 17 and 19 functons evaluatons (fnte-element analyses), respectvely, whereas 10 5 fnte-element analyses were performed by the Monte Carlo smulaton. The results clearly show that both the unvarate and bvarate methods can calculate probablty of fracture ntaton accurately and effcently. 6. Conclusons New computatonal methods, referred to as decomposton methods, were developed for predctng relablty of structural and mechancal systems subject to random loads, materal propertes, and geometry. The methods nvolve a novel functon decomposton that facltates unvarate and bvarate approxmatons of a general multvarate functon, response surface generaton of unvarate and bvarate functons, and Monte Carlo smulaton. Due to a small number of orgnal functon evaluatons, the proposed methods are very effectve, partcularly when a response evaluaton entals costly fnte-element, mesh-free, or other numercal analyss. The methods can solve both component and system relablty problems. Seven numercal examples nvolvng elementary mathematcal functons and soldmechancs problems llustrate the proposed method. Results ndcate that the methods developed provde accurate and computatonally effcent estmates of probablty of falure. Acknowledgements Smulaton (10 5 Samples) Unvarate Method (17) Bvarate Method (19) Fg. 10. Probablty of fracture ntaton n functonally graded edgecracked plate. The authors would lke to acknowledge the fnancal support by the US Natonal Scence Foundaton under Grant No. DMI References [1] Madsen HO, Krenk S, Lnd NC. Methods of structural safety. Englewood Clffs, NJ: Prentce-Hall; [] Rackwtz R. Relablty analyss a revew and some perspectves. Struct Saf 001;3(4): [3] Dtlevsen O, Madsen HO. Structural relablty methods. Chchester: Wley; [4] Bretung K. Asymptotc approxmatons for multnormal ntegrals. ASCE J Eng Mech 1984;110(3): [5] Hohenbchler M, Gollwtzer S, Kruse W, Rackwtz R. New lght on frst- and second-order relablty methods. Struct Saf 1987;4: [6] Ca GQ, Elshakoff I. Refned second-order relablty analyss. Struct Saf 1994;14: [7] Tvedt L. Dstrbuton of quadratc forms n normal space applcaton to structural relablty. ASCE J Eng Mech 1990;116(6): [8] Der Kureghan A, Dakessan T. Multple desgn ponts n frst and second-order relablty. Struct Saf 1998;0(1): [9] Rubnsten RY. Smulaton and the Monte Carlo method. New York: Wley; [10] Nederreter H, Spaner J. Monte Carlo and quas-monte Carlo methods. Berln: Sprnger-Verlag; 000. [11] Melchers RE. Importance samplng n structural systems. Struct Saf 1989;6:3 10. [1] Bjerager P. Probablty ntegraton by drectonal smulaton. ASCE J Eng Mech 1988;114(8): [13] Ne J, Ellngwood BR. Drectonal methods for structural relablty analyss. Struct Saf 000;: [14] Katsuk S, Frangopol DM. Hyperspace dvson method for structural relablty. ASCE J Eng Mech 1994;10(11): [15] McKay MD, Conover WJ, Beckman RJ. A comparson of three methods for selectng values of nput varables n the analyss of output from a computer code. Technometrcs 1979;1(): [16] Glks WR, Rchardson S, Spegelhalter DJ. Markov chan Monte Carlo n practce. London: Chapman-Hall; [17] Au SK, Beck JL. Estmaton of small falure probabltes n hgh dmensons by subset smulaton. Probab Eng Mech 001;16(4): [18] Schuëller GI, Pradlwarter HW, Koutsourelaks PS. A crtcal apprasal of relablty estmaton procedures for hgh dmensons. Probab Eng Mech 004;19(4): [19] Rahman S, Xu H. A unvarate dmenson-reducton method for multdmensonal ntegraton n stochastc mechancs. Probab Eng Mech 004;19(4): [0] Xu H, Rahman S. A generalzed dmenson-reducton method for mult-dmensonal ntegraton n stochastc mechancs. Int J Numer Methods Eng 004;61: [1] Harbtz A. An effcent samplng method for probablty of falure calculaton. Struct Saf 1986;3: [] Yonezawa M, Okuda S, Park YT. Structural relablty estmaton based on smulaton wthn lmted samplng regon. Int J Prod Econ 1999;60 61: [3] Hohenbchler M, Rackwtz R. Relablty of parallel systems under mposed stran. ASCE J Eng Mech 1983;109(3): [4] Dtlevsen O, Melchers RE, Gluver H. General mult-dmensonal probablty ntegraton by drectonal smulaton. Comput Struct 1990; 36(): [5] Davenport WB, Root WL. An ntroducton to the theory of random sgnals and nose. New York: McGraw-Hll; [6] Rahman S, Xu H. A meshless method for computatonal stochastc mechancs. Int J Comput Methods Eng Sc Mech 005;6: [7] Anderson TL. Fracture mechancs: fundamentals and applcatons. nd ed. Boca Raton, Florda: CRC Press Inc.; [8] Rao BN, Rahman S. Meshfree analyss of cracks n sotropc functonally graded materals. Eng Fract Mech 003;70(1):1 7.