Structural reliability analysis by univariate decomposition and numerical integration

Transcription

1 Probablstc Engneerng Mechancs 22 (2007) Structural relablty analyss by unvarate decomposton and numercal ntegraton D. We, S. Rahman Department of Mechancal and Industral Engneerng, The Unversty of Iowa, Iowa Cty, IA 52242, Unted States Receved 22 July 2005; receved n revsed form 17 Aprl 2006; accepted 1 May 2006 Avalable onlne 16 June 2006 Abstract Ths paper presents a new and alternatve unvarate method for predctng component relablty of mechancal systems subject to random loads, materal propertes, and geometry. The method nvolves novel functon decomposton at a most probable pont that facltates the unvarate approxmaton of a general multvarate functon n the rotated Gaussan space and one-dmensonal ntegratons for calculatng the falure probablty. Based on lnear and quadratc approxmatons of the unvarate component functon n the drecton of the most probable pont, two mathematcal expressons of the falure probablty have been derved. In both expressons, the proposed effort n evaluatng the falure probablty nvolves calculatng condtonal responses at a selected nput determned by sample ponts and Gauss Hermte ntegraton ponts. Numercal results ndcate that the proposed method provdes accurate and computatonally effcent estmates of the probablty of falure. c 2006 Elsever Ltd. All rghts reserved. Keywords: Relablty; Probablty of falure; Decomposton methods; Most probable pont; Unvarate approxmaton; Numercal ntegraton 1. Introducton A fundamental problem n tme-nvarant component relablty analyss entals calculaton of a mult-fold ntegral [1 3] P [g(x) < 0] = f X (x) dx, (1) g(x)<0 where X = {X 1,..., X N } T R N s a real-valued, N-dmensonal (N 2) 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; 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. The most common approach to compute the falure probablty n Eq. (1) nvolves the frst- and second-order relablty methods (FORM/SORM) [1 8], whch are respectvely based on lnear Correspondng author. Tel.: ; fax: E-mal address: rahman@engneerng.uowa.edu (S. Rahman). URL: rahman (S. Rahman). (FORM) and quadratc (SORM) approxmatons of the lmtstate surface at a most probable pont (MPP) n the standard Gaussan space. When the dstance β HL between the orgn and the MPP, a pont on the lmt-state surface that s closest to the orgn, approaches nfnty, FORM/SORM strctly provdes asymptotc solutons. For non-asymptotc (fnte β HL ) applcatons nvolvng a hghly nonlnear performance functon, ts lnear or quadratc approxmaton may not be adequate and therefore resultant FORM/SORM predctons must be nterpreted wth cauton [9]. In the latter cases, an mportance samplng method developed by Hohenbchler and Rackwtz [10] can make the FORM/SORM result arbtrarly exact, but t may become expensve f a large number of costly numercal analyses, such as large-scale fnte element analyss embedded n the performance functon, are nvolved. Furthermore, the exstence of multple MPPs may lead 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 to mprove component relablty analyss [8]. Recently, the authors have developed new decomposton methods, whch can solve hghly nonlnear relablty problems more accurately or more effcently than FORM/SORM and /$ - see front matter c 2006 Elsever Ltd. All rghts reserved. do: /j.probengmech

2 28 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) y(v) = y 0 + y (v ) + }{{} =ŷ 1 (v) 1, 2 =1 1 < 2 y 1 2 (v 1, v 2 ) }{{} =ŷ 2 (v) + + } {{ } =ŷ S (v) y 1 S (v 1,..., v s ) + + y 12 N (v 1,..., v N ), 1,..., S =1 1 < < S Box I. smulaton methods [11,12]. A major advantage of these decomposton methods, so far based on the mean pont [11] or MPP [12] of a random nput as reference ponts, over FORM/SORM s that hgher-order approxmatons of performance functons can be acheved usng functon values alone. In partcular, an MPP-based unvarate method developed n the authors prevous work nvolves unvarate approxmaton of the performance functon at the MPP, n-pont Lagrange nterpolaton n the rotated Gaussan space, and subsequent Monte Carlo smulaton [12]. The present work s motvated by an argument that the MPP-based unvarate approxmaton, f approprately cast n the rotated Gaussan space, permts an effcent evaluaton of the component falure probablty by multple one-dmensonal ntegratons. Ths paper presents a new and alternatve MPP-based unvarate method for predctng the component relablty of mechancal systems subject to random loads, materal propertes, and geometry. Secton 2 gves a bref exposton of a novel functon decomposton at the MPP that facltates a lower-dmensonal approxmaton of a general multvarate functon. Secton 3 descrbes the proposed unvarate method, whch nvolves unvarate approxmaton of the performance functon at the MPP and unvarate numercal ntegratons. Secton 4 explans the computatonal effort and flowchart of the proposed method. Fve numercal examples nvolvng elementary mathematcal functons and structural/sold-mechancs problems llustrate the method developed n Secton 5. Comparsons have been made wth alternatve approxmate and smulaton methods to evaluate the accuracy and computatonal effcency of the new method. 2. Multvarate functon decomposton at MPP Consder a contnuous, dfferentable, real-valued performance functon g(x) that depends on x = {x 1,..., x N } T R N. If u = {u 1,..., u N } T R N s the standard Gaussan space, let u = { u 1,..., } T u N denote the MPP or beta pont, whch s the closest pont on 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 [1 3], determned by a standard nonlnear constraned optmzaton. Construct an orthogonal matrx R R N N whose Nth column s α u /β HL,.e., R = [ R 1 α ], where R 1 R N N 1 satsfes α T R 1 = 0 R 1 N 1. The matrx R can be obtaned, for example, by Gram Schmdt orthogonalzaton. For an orthogonal transformaton u = Rv, let v = {v 1,..., v N } T R N represent the rotated Gaussan space wth the assocated MPP Fg. 1. Performance functon approxmatons by varous methods. v = { v1,..., v N 1, } T v N = {0,..., 0, βhl } T. The transformed lmt states h(u) = 0 and 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. Fg. 1 depcts FORM and SORM approxmatons of a lmt-state surface at the MPP for N = 2. Consder a decomposton of a general multvarate functon y(v), whch can be vewed as a fnte sum [11 13] (see Box I), where y 0 s a constant, y (v ) s a unvarate component functon representng an ndvdual contrbuton to y(v) by nput varable v actng alone, y 1 2 (v 1, v 2 ) s a bvarate component functon descrbng the cooperatve nfluence of two nput varables v 1 and v 2, y 1 S (v 1,..., v S ) s an S-varate component functon quantfyng the cooperatve effects of S nput varables v 1,..., v S, and so on. If ŷ S (v) = y 0 + y (v ) , 2 =1 1 < 2 y 1 2 (v 1, v 2 ) y 1 S (v 1,..., v s ) (2) 1,..., S =1 < < represents a general S-varate approxmaton of y(v), the unvarate (S = 1) and bvarate (S = 2) approxmatons

3 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) ŷ 1 (v) and ŷ 2 (v) respectvely provde two- and three-term approxmants of the fnte decomposton n the equaton gven n Box I. Smlarly, trvarate, quadrvarate, and other hgher-varate approxmatons can be derved by approprately selectng the value of S. In the lmt when S = N, ŷ S (v) converges to the exact functon y(v). In other words, the decomposton n Eq. (2) generates a convergent sequence of approxmatons of y(v). Readers nterested n the fundamental development of the decomposton are referred to Xu and Rahman [13]. 3. New unvarate method 3.1. Unvarate decomposton of performance functon Consder a unvarate approxmaton of y(v), denoted by ŷ 1 (v) ŷ 1 (v 1,..., v N ) = y(v1,..., v 1, v, v+1,..., v N ) (N 1)y(v ), (3) where each term n the summaton s a functon of only one varable and can subsequently be expanded n a Taylor seres at the MPP, v = { v 1,..., v N 1, v N } T = {0,..., 0, βhl } T, yeldng ŷ 1 (v) = y(v ) + j=1 1 j! j y ( v ) v j (v v ) j. (4) In contrast, the Taylor seres expanson of y(v) at v = { v 1,..., v } T N can be expressed by y(v) = y ( v ) + j=1 1 j! j y ( v ) ( v j v v ) j + R2 (5) where the remander R 2 denotes all terms wth dmensons two and hgher. A comparson of Eqs. (4) and (5) ndcates that the unvarate approxmaton of ŷ 1 (v) leads to a resdual error y(v) ŷ 1 (v) = R 2, whch ncludes contrbutons from terms of dmenson two and hgher. For suffcently smooth y(v) wth a convergent Taylor seres, the coeffcents assocated wth hgher-dmensonal terms are usually much smaller than those wth one-dmensonal terms. As such, hgher-dmensonal terms contrbute less to the functon, and therefore can be neglected. Nevertheless, Eq. (3) ncludes all hgher-order unvarate terms, compared wth FORM and SORM, whch only retan lnear and quadratc unvarate terms, respectvely. If the unvarate decomposton s not suffcent, bvarate or hgher-varate decompostons may be requred [11]. Because of the hgher cost, these were not ncluded n ths study. In addton to the MPP as the chosen reference pont, the accuracy of the unvarate approxmaton n Eq. (3) may depend on the orentaton of the frst N 1 axes. In ths work, the orentaton s defned by the matrx R. However, an mproved approxmaton may be possble by selectng an orentaton that s optmal n some sense. Ths ssue was not consdered n the present work Unvarate ntegraton for falure probablty analyss The proposed unvarate approxmaton of the performance functon can be rewrtten as ŷ 1 (v) = y N (v N ) + N 1 y (v ) (N 1)y(v ), (6) where y (v ) y(v1,..., v 1, v, v+1,..., v N ); = 1, N. Due to rotatonal transformaton of the coordnates (see Fg. 1), the unvarate component functon y N (v N ) n Eq. (6) s expected to be a lnear or a weakly nonlnear functon of v N. In fact, y N (v N ) s lnear wth respect to v N n the classcal FORM/SORM approxmaton of a performance functon n the v space. Nevertheless, f y N (v N ) s nvertble, the unvarate approxmaton ŷ 1 (v) can be further expressed n a form amenable to an effcent relablty analyss by one-dmensonal numercal ntegraton. In ths work, both lnear and quadratc approxmatons of y N (v N ) and the resultant equatons for falure probablty are derved, as follows Lnear approxmaton of y N (v N ) Consder a lnear approxmaton: y N (v N ) = b 0 + b 1 v N, where coeffcents b 0 R and b 1 R (non-zero) are obtaned by least-squares approxmatons { from exact } or numercally smulated responses y N (v (1) N ),..., y N (v (n) N ) at n sample ponts along the v N coordnate. The least-squares approxmaton was chosen over nterpolaton, because the former mnmzes the error when n > 2. Applyng the lnear approxmaton, the component falure probablty can be expressed by P [y(v) < 0] = P [ ŷ 1 (V) < 0 ] = P [ b 0 + b 1 V N + N 1 y (V ) (N 1)y(v ) < 0 whch, on nverson, yelds [ P V N < (N 1)y(v ) b 0 1 b 1 b 1 = [ f b 1 > 0 P V N (N 1)y(v ) b 0 1 b 1 b 1 f b 1 < 0. N 1 N 1 y (V ) y (V ) ] ] ],,,(7) Snce V N follows standard Gaussan dstrbuton, the falure probablty can also be expressed by ( )] (N 1)y(v = E [ ) b 0 1 N 1 y (V ), (9) b 1 b 1 ( where E s the expectaton operator and (u) = 1/ ) 2π u exp ( ξ 2 /2 ) dξ s the cumulatve dstrbuton functon of a standard Gaussan random varable. Note that Eq. (9) provdes hgher-order estmates of falure probablty f unvarate component functons y (v ), = 1, N 1, are approxmated (8)

4 30 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) by terms hgher than second order. If y (v ), = 1, N 1, retan only lnear and quadratc terms ftted wth approprately selected sample ponts, Eq. (9) can be further smplfed to degenerate to the well-known FORM and SORM (pont-ftted) approxmatons Quadratc approxmaton of y N (v N ) The lnear approxmaton descrbed n the precedng can be mproved by a quadratc approxmaton: y N (v N ) = b 0 + b 1 v N + b 2 v 2 N, where coeffcents b 0 R, b 1 R, and b 2 R (non-zero) are also obtaned by least-squares approxmatons from exact or numercally smulated responses at n sample ponts along the v N coordnate. Agan, the least-squares approxmaton was selected due to error mnmzaton when n > 3. Smlarly, the quadratc approxmaton of y N (v N ) employed n Eq. (6) leads to P [y(v) < 0] = P [ ŷ 1 (V) < 0 ] = P [b 0 + b 1 V N + b 2 VN 2 N 1 + y (V ) (N 1)y(v ) < 0 ]. (10) By defnng B(Ṽ) b 0 + N 1 y (V ) (N 1)y(v ), where Ṽ = {V 1,..., V N 1 } T s an N 1-dmensonal standard Gaussan vector, the followng solutons are derved on the bass of two cases: (a) Case I Trval Soluton (b 1 2 4b 2 B < 0; no real roots): = { 0, f b2 > 0 1, f b 2 < 0. (11) (b) Case II Non-Trval Soluton (b 2 1 4b 2 B 0; two real roots): P b 1 b 2 1 4b 2 B(Ṽ) < V N < b 1 + b 2 1 4b 2 B(Ṽ), f b 2 > 0 = (12) 1 P b 1 + b 2 1 4b 2 B(Ṽ) < V N < b 1 b 2 1 4b 2 B(Ṽ), f b 2 < 0, yeldng 1 b 2 / b 2 = + E b 1 + b 2 1 4b 2 B(Ṽ) 2 E b 1 b 2 1 4b 2 B(Ṽ). (13) Both Eqs. (9) and (13) can be employed for non-trval solutons of falure probablty. Improvement n the accuracy of results, f any, depends on how strongly y N (v N ) depends on v N. Furthermore, t s possble to develop a generalzed verson of Eq. (13) 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 ) (Eq. (9)) should result n a very accurate soluton. Hence, the present study s lmted to only lnear and quadratc approxmatons of y N (v N ). It s worth notng that, unlke Eq. (9), Eq. (13) cannot be reduced to FORM/SORM equatons, as y N (v N ) ncludes a second-order term Unvarate ntegraton The falure probablty expressons n Eqs. (9) and (13) nvolve the calculaton of the expected values of several multvarate functons of an N 1-dmensonal standard Gaussan vector Ṽ = {V 1,..., V N 1 } T. A generc expresson for such a calculaton requres the determnaton of E[ ( f (Ṽ))], where f : R N 1 R s a general mappng of Ṽ and depends on how unvarate component functons y (v ), = 1, N 1, are approxmated. Unfortunately, the exact probablty densty functon of f (Ṽ) s, n general, not avalable n closed form. For ths reason, t s dffcult to calculate E[ ( f (Ṽ))] analytcally. Numercal ntegraton s not effcent, as ( f (ṽ)) s a multvarate functon and becomes mpractcal when the dmenson exceeds three or four. In reference to Eq. (3), agan consder a unvarate approxmaton of ln [ ( f (ṽ)) ] N 1 = ln [ ( f (v ))] (N 2) ln [ ( f (ṽ ) )], (14) where f (v ) f (v1,..., v 1, v, v+1,..., v N 1 ); = 1, N 1 are unvarate component functons; and f (ṽ ) f (v1,..., v N 1 ). Hence ( f (ṽ)) = exp { ln [ ( f (ṽ)) ]} { N 1 = exp yeldng = (N 2) ln [ ( f (ṽ ) )]} N 1 E[ ( f (Ṽ))] = ( f (v )) ln [ ( f (v ))], (15) ( f (ṽ N 2 )) = N 1 N 1 E [ ( f (V ))] ( f (ṽ )) N 2 + ( f (v ))φ(v ) dv ( f (ṽ )) N 2, (16)

5 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) whch nvolves a product of N 1 unvarate ntegrals wth φ( ) denotng the standard Gaussan probablty densty functon. Usng Eq. (16), the nontrval expressons of falure probablty n Eqs. (9) and (13) are = and N 1 ( ) + (N 1)y(v ) b 0 y (v ) b 1 φ(v ) dv 1 b 2 / b 2 = 2 N 1 + N 1 (N 1)y(v ) b 0 N 1 b 1 y (v ) N 2 ( ) + b 1 + b 2 1 4b 2 B (v ) φ(v ) dv [ ( b 1 + b 1 2 4b 2 B(ṽ ) ( + b 1 b 2 1 4b 2 B (v ) [ ( b 1 + b 1 2 4b 2 B(ṽ ) )] N 2 (17) ) φ(v ) dv )] N 2 (18) respectvely, where B (v ) B(v1,...,v 1,v,v+1,..., v N 1 ). The unvarate ntegraton nvolved n Eqs. (17) or (18) can be evaluated easly by standard one-dmensonal Gauss Hermte numercal quadrature [14]. The decomposton method nvolvng unvarate approxmaton (Eq. (3)) and unvaratentegraton(eqs.(17)or(18))sdefnedasthemppbased unvarate method wth numercal ntegraton n ths paper. 4. Computatonal effort and flow Consder y (v ) y(v1,..., v 1, v, v+1,..., v N ), = 1, N 1, for whch n functon values y (v ( j) ) y(v1,..., v j) 1, v(, v+1,..., v N ), j = 1,..., n, are requred to be evaluated at ntegraton ponts v = v ( j) to perform an n-order Gauss Hermte quadrature for the th ntegraton n Eqs. (17) or (18). The same procedure s repeated for N 1 unvarate component functons,.e. for all y (v ), = 1,..., N 1. Therefore, the total cost of the proposed unvarate method, ncludng the n functon values of y N (v N ) requred for ts lnear or quadratc approxmaton, entals a maxmum of nn functon evaluatons. 1 Note that the above cost s n addton to any functon evaluatons requred for locatng the MPP. Fg. 2 shows the computatonal flowchart of the MPPbased unvarate method wth numercal ntegraton. The mplementaton of the method requres the calculaton of y N (v N ) at a user-selected nput to obtan coeffcents b 0, b 1, 1 The orders of numercal ntegraton and the number of functon values of y N (v N ) need not be the same. In addton, dfferent orders of ntegraton can be employed f desred. and/or b 2 and the calculaton of y (v ), = 1,..., N 1, at Gauss Hermte ntegraton ponts to perform the numercal ntegraton. Both calculatons ental the evaluaton of unvarate component functons, whch are condtonal responses. Hence, the proposed effort n determnng the falure probablty can be vewed as numercally calculatng condtonal responses at a selected nput. Compared wth the prevously developed unvarate method [12], 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 Eqs. (17) or (18), two optons for evaluatng y (v ) are proposed. Opton 1 nvolves calculatng y (v ( j) ) at ntegraton ponts (v1,..., v j) 1, v(, v+1,..., v N ), j = 1, n, from drect numercal analyss (e.g. fnte-element analyss). When the computaton of 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 sample 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. 5. Numercal examples Fve numercal examples nvolvng explct performance functons from mathematcal or sold-mechancs problems (Examples 1 and 2) and mplct 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 MPP-based unvarate method wth smulaton [12], FORM/SORM, and drect Monte Carlo smulaton to evaluate the accuracy and effcency of the new method. To obtan the lnear or quadratc approxmaton of y N (v N ), n (= 5, 7 or 9) sample ponts v N (n 1)/2, v N (n 3)/2,..., v N,..., v N + (n 3)/2, v N + (n 1)/2 were deployed along the v N -coordnate. The same value of n was employed as the order of Gauss Hermte quadratures n Eqs. (17) or (18) of the proposed unvarate method wth numercal ntegraton. Furthermore, opton 1 was used n Examples 3 and 4 and opton 2 was nvoked n Examples 1, 2 and 5. When usng opton 2, an nth-order polynomal equaton was employed for generatng the response-surface approxmaton of varous component functons y (v ), = 1, N 1. A 10-order Gauss Hermte ntegraton was nvoked n opton 2. 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 [12]. Hence, the

6 32 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Fg. 2. Flowchart of the MPP-based unvarate method wth numercal ntegraton. total number of functon evaluatons requred by both versons of the unvarate method, n addton to those requred for locatng the MPP, s (n 1)N. When comparng computatonal efforts by varous methods, the number of orgnal performance functon evaluatons s chosen as the prmary metrc n ths paper Example 1 Elementary mathematcal functons Consder cubc and quartc performance functons, expressed respectvely by [12] g(x 1, X 2 ) = (X 1 + X 2 20) (X 1 X 2 ) (19) g(x 1, X 2 ) = (X 1 + X 2 20) (X 1 X 2 ), (20) where X, = 1, 2, are ndependent, Gaussan random varables, each wth mean µ = 10 and standard devaton σ = 3. From an MPP search, v = {0, } 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 n nne functon evaluatons. Snce both performance functons n the rotated Gaussan space are lnear n v 2, the proposed method nvolvng Eq. (17) was employed to calculate the falure probablty. Tables 1 and 2 show the results of the falure probablty calculated by FORM, SORM [4 6], the MPP-based unvarate method wth smulaton [12], the proposed MPP-based unvarate method wth numercal ntegraton, and drect Monte Carlo smulaton usng 10 6 samples. The unvarate method wth smulaton, whch yelds exact lmt-state 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 the approxmatons nvolved n Eqs. (7) and (14) of the proposed method. Nevertheless, other commonly used relablty methods, such as FORM and SORM, underpredct the falure probablty by 31% and overpredct the falure probablty by 117% 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

7 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Table 1 Falure probablty for cubc performance functon Method Falure probablty Number of functon evaluatons a MPP-based unvarate method wth numercal ntegraton b MPP-based unvarate method wth smulaton [12] b FORM SORM [4 6] Drect Monte Carlo smulaton ,000 a Total number of tmes that the orgnal performance functon s calculated. b 21 + (n 1) N = 21 + (5 1) 2 = 29. Table 2 Falure probablty for quartc performance functon Method Falure probablty Number of functon evaluatons a MPP-based unvarate method wth numercal ntegraton b MPP-based unvarate method wth smulaton [12] b FORM SORM [4 6] Drect Monte Carlo smulaton ,000 a Total number of tmes that the orgnal performance functon s calculated. b 21 + (n 1) N = 21 + (5 1) 2 = 29. problems nvolvng an 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 Fg. 3. The dsk s subject to an angular velocty, ω, about an axs perpendcular to ts plane at the centre. The maxmum allowable angular velocty, ω a, when tangental stresses through the thckness reach the materal s ultmate strength, S u, factored by a materal utlzaton factor α m, s [15] [ ] 1/2 3α m S u (R o R ) ω a = ρ ( Ro 3 ), (21) R3 where ρ s the mass densty of the materal. Accordng to an SAE G-11 standard, the satsfactory performance of the dsk s defned when the burst margn M b, defned as [ ] 1/2 M b ω a ω = 3α m S u (R o R ) ρω 2 ( Ro 3 ), (22) R3 exceeds [16]. If random varables X 1 = α m, X 2 = S u, X 3 = ω, X 4 = ρ, X 5 = R o, and X 6 = R have ther statstcal propertes defned n Table 3, the performance functon becomes g(x) = M b (X 1, X 2, X 3, X 4, X 5, X 6 ) (23) Table 4 presents the predcted falure probablty of the dsk and the assocated computatonal effort usng new and exstng MPP-based unvarate methods, FORM, Hohenbchler s SORM [5], and drect Monte Carlo smulaton (10 6 samples). For unvarate methods, a value of n = 7 was selected. For the Fg. 3. Rotatng annular dsk subject to angular velocty. unvarate method wth numercal ntegraton, the falure probabltes based on lnear (Eq. (17)) and quadratc (Eq. (18)) 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 Fg. 4, was studed n ths example to examne the accuracy and effcency of the proposed relablty method. The Young s

8 34 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Table 3 Statstcal propertes of random nput for rotatng dsk Random varable Mean Standard devaton Probablty dstrbuton α m Webull a S u, ks Gaussan ω, rpm Gaussan ρ, lb s 2 /n /g b /g b Unform c R o, n Gaussan R, n Gaussan a Scale parameter = ; shape parameter = b g = n./s 2. c Unformly dstrbuted over (0.28, 0.3). Table 4 Falure probablty of rotatng dsk Method Falure probablty Number of functon evaluatons a MPP-based unvarate method wth numercal ntegraton Lnear (Eq. (17)) b Quadratc (Eq. (18)) b MPP-based unvarate method wth smulaton [12] b FORM SORM (Hohenbchler) [5] Drect Monte Carlo smulaton ,000 a Total number of tmes that the orgnal performance functon s calculated. b (n 1) N = (7 1) 6 = 167. Ca and Elshakoff [6], and drect Monte Carlo smulaton (10 6 samples). For unvarate methods, a value of n = 7 was selected. From Table 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 a slghtly more accurate result than that based on ts lnear approxmaton. A comparson of the number of functon evaluatons, also lsted n Table 5, ndcates that the computatonal effort by the MPP-based unvarate methods s slghtly larger than FORM, but much less than SORM. Fg. 4. A ten-bar truss structure. modulus of the materal s 10 7 ps. Two concentrated forces of 10 5 lb are appled at nodes 2 and 3, as shown n Fg. 4. The cross-sectonal area X, = 1,..., 10 for each bar follows a 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 3 (X 1,..., X 10 )] occurs at node 3, where the permssble dsplacement s lmted to 18 n., leadng to g(x) = 18 v 3 (X 1,..., X 10 ). From an MPP search nvolvng fnte-dfference gradents, β HL = v = Table 5 shows the falure probablty of the truss, calculated usng the proposed MPPbased unvarate method wth numercal ntegraton, MPPbased unvarate method wth smulaton [12], FORM, three varants of SORM due to Bretung [4], Hohenbchler [5] and 5.4. 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 Fg. 5(a), a plate of length L = 16 unts and 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 1 unt and 0.25, respectvely. A plane stran condton was assumed. The statstcal property of the random nput X = {a/w, τ, K I c } T s defned n Table 6. Due to the far-feld shear stress τ, the plate s subjected to mxed-mode deformaton nvolvng fracture modes I and II [17]. The mxed-mode stress-ntensty factors K I (X) and K I I (X) were calculated usng an nteracton ntegral [18]. The plate was analysed usng the fnte-element method, nvolvng

9 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Table 5 Falure probablty of ten-bar truss structure Method Falure probablty Number of functon evaluatons a MPP-based unvarate method wth numercal ntegraton Lnear (Eq. (17)) b Quadratc (Eq. (18)) b MPP-based unvarate method wth smulaton [12] b FORM SORM (Bretung) [4] SORM (Hohenbchler) [5] SORM (Ca and Elshakoff) [6] Drect Monte Carlo smulaton ,000 a Total number of tmes that the orgnal performance functon s calculated. b (n 1) N = (7 1) 10 = 187. Table 6 Statstcal propertes of random nput for an edge-cracked plate Random varable Mean Standard devaton Probablty dstrbuton a/w Unform a τ Varable b 0.1 Gaussan K I c Lognormal a Unformly dstrbuted over (0.3, 0.7). b Vares from 2.6 to 3.1. Fg. 6. Probablty of fracture ntaton n an edge-cracked plate. Fg. 5. An edge-cracked plate subject to mxed-mode deformaton: (a) geometry and loads; (b) fnte-element dscretzaton. a total of noded, regular, quadrlateral elements and 48 6-noded, quarter-pont (sngular), trangular elements at the crack-tp, as shown n Fg. 5(b). The falure crteron s based on a mxed-mode fracture ntaton usng the maxmum tangental stress theory [17], whch leads to the performance functon [ g(x) = K I c K I (X) cos 2 Θ(X) 2 3 ] 2 K I I (X) sn Θ(X) cos Θ(X), (24) 2 where K I c s the random fracture toughness and Θ c (X) s the drecton of crack propagaton. Falure probablty estmates of = P[g(X) < 0], obtaned usng the proposed unvarate method wth numercal ntegraton, unvarate method usng smulaton, FORM, Hohenbchler s SORM, and drect Monte Carlo smulaton, are compared n Fg. 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 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 drect Monte Carlo smulaton Example 5 Three-span, fve-story frame structure The fnal example examnes the proposed unvarate method for solvng relablty problems nvolvng correlated random varables. A three-span, fve-story frame structure, studed by Lu and Der Kureghan [19], s subjected to horzontal loads, as shown n Fg. 7. There are 21 random varables: (1) three appled loads, (2) two Young s modul, (3) eght moments of nerta, and (4) eght cross-sectonal areas. The random

10 36 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Fg. 7. A three-span, fve-story frame structure subjected to lateral loads. Table 7 Frame element propertes Element Young s modulus Moment of nerta Cross-sectonal area B 1 E 4 I 10 A 18 B 2 E 4 I 11 A 19 B 3 E 4 I 12 A 20 B 4 E 4 I 13 A 21 C 1 E 5 I 6 A 14 C 2 E 5 I 7 A 15 C 3 E 5 I 8 A 16 C 4 E 5 I 9 A 17 varables assocated wth frame elements are defned n Table 7. Tables 8 and 9 lst statstcal propertes of all random varables. The lognormally dstrbuted load varables are ndependent and all other random varables are jontly normal. Falure s defned when the horzontal component of the top-floor dsplacement u 1 (X) exceeds 0.2 ft, leadng to g(x) = 0.2 u 1 (X). 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 10. FORM and SORM requre 474 and 1143 functon evaluatons (frame analyss), respectvely. Usng n = 7, the unvarate methods requre 600 functon evaluatons, whereas 1,000,000 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 effort, the method developed s slghtly more expensve than FORM, but sgnfcantly more effcent than SORM. Table 8 Statstcal propertes of random nput for frame structure a Random varable Mean Standard devaton Probablty dstrbuton P 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. 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 does not requre any Monte Carlo smulatons embedded n ts prevous verson. Instead, explct forms of falure probablty requrng only one-dmensonal ntegratons are nvolved. Hence, the new method should be useful n dervng the senstvty (gradents) of falure probablty for relablty-based desgn optmzaton, whch s a subject of current research by the authors.

11 D. We, S. Rahman / Probablstc Engneerng Mechancs 22 (2007) Table 9 Correlaton coeffcents of random nput for frame structure P 1 P 2 P 3 E 4 E 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 I 13 A 14 A 15 A 16 A 17 A 18 A 19 A 20 A 21 P P P E E I I I I I I I I A A A A A A A A Table 10 Falure probablty of frame structure Method Falure probablty Number of functon evaluatons a MPP-based unvarate method wth numercal ntegraton b MPP-based unvarate method wth smulaton [12] b FORM SORM (Hohenbchler) [5] Drect Monte Carlo smulaton ,000 a Total number of tmes that the orgnal performance functon s calculated. b (n 1) N = (7 1) 21 = Conclusons A new unvarate method was developed for predctng the component relablty of mechancal systems subject to random loads, materal propertes, and geometry. The method nvolves novel functon decomposton at the most probable pont, whch facltates unvarate approxmaton of a general multvarate functon n the rotated Gaussan space and one-dmensonal ntegratons for calculatng the falure probablty. Based on lnear and quadratc approxmatons of the unvarate component functon n the drecton of the most probable pont, two mathematcal expressons of the falure probablty were derved. In both expressons, the proposed effort n evaluatng the falure probablty nvolves calculatng condtonal responses at a selected nput determned by sample ponts and Gauss Hermte ntegraton ponts. The results of fve numercal examples nvolvng elementary mathematcal functons and structural/sold-mechancs problems ndcate that the proposed method provdes accurate and computatonally effcent estmates of the probablty of falure. Compared wth the authors prevous work, no Monte Carlo smulaton s requred n the present verson of the unvarate method that has been developed. Although both versons of the unvarate method have comparable computatonal effcences, the new method should be useful n dervng the senstvty of the falure probablty for relablty-based desgn optmzaton, whch s the ultmate goal of probablstc mechancs. Acknowledgment The authors would lke to acknowledge the fnancal support of 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, Inc.; [2] Rackwtz R. Relablty analyss A revew and some perspectves. Structural Safety 2001;23(4): [3] Dtlevsen O, Madsen HO. Structural relablty methods. Chchester: John Wley & Sons Ltd.; [4] Bretung K. Asymptotc approxmatons for multnormal ntegrals. ASCE Journal of Engneerng Mechancs 1984;110(3): [5] Hohenbchler M, Gollwtzer S, Kruse W, Rackwtz R. New lght on frstand second-order relablty methods. Structural Safety 1987;4:

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