Journal of Mechanical Design Volume 135 Issue 7 research-article

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1 Journal of Mechanical Deign Volume 135 Iue 7 reearch-article A Sampling Approach to Extreme Value Ditribution for Time- Dependent Reliability Analyi Zhen Hu Department of Mechanical and Aeropace Engineering Miouri Univerity of Science and Technology zh4hd@mt.edu Xiaoping Du, Ph.D. Correponding Author Department of Mechanical and Aeropace Engineering Miouri Univerity of Science and Technology 290D Toomey Hall 400 Wet 13th Street Rolla, MO (voice) (fax) dux@mt.edu 0

2 Abtract Maintaining high accuracy and efficiency i a challenging iue in time-dependent reliability analyi. In thi work, an accurate and efficient method i propoed for limittate function with the following feature: The limit-tate function i implicit with repect to time. There i only one tochatic proce in the input to the limit-ate function. The tochatic proce could be either a general trength or a general tre variable o that the limit-tate function i monotonic to the tochatic proce. The new method employ a ampling approach to etimate the ditribution of the extreme value of the tochatic proce. The extreme value i then ued to replace the correponding tochatic proce. Conequently the time-dependent reliability analyi i converted into it timeinvariant counterpart. The commonly ued time-invariant reliability method, the Firt Order Reliability Method, i then applied to calculate the probability of failure over a given period of time. The reult how that the propoed method ignificantly improve the accuracy and efficiency of time-dependent reliability analyi. Keyword: Time-dependent reliability, tochatic proce, extreme value, firt Order reliability method 1

3 1. Introduction Time-dependent reliability analyi quantifie the probability that a component or ytem urvive after it ha worked for a certainty time t or over the period [0, t ] [1]. Time-dependent reliability in general decreae with time for two reaon. Firt, there exit time-dependent uncertaintie. For example, the wave loading on offhore tructure alway change it amplitude and direction over time [2]. For thi kind of time-dependent uncertainty, it global extreme over time varie with the length of the time period. More pecifically, a longer time period will more likely correpond to a larger global extreme. On the other hand, the trength of a component or ytem generally deteriorate over time. The other reaon i that a limit-tate function, which predict the tate of afety or failure, may alo be a function of time. For example, the motion error of a mechanim varie at different intant of motion input [3]. Becaue of time-dependent uncertaintie, a product alway exhibit time-dependent failure, and the aociated reliability i almot alway decline over time. Limit-tate function for the time-dependent reliability analyi can be claified into the following three categorie: Limit-tate function have time-invariant random variable X and time t in it input. A function in thi category i therefore in the form of G = g( X, t), where g () i the limit-tate function, and G i a repone variable. Limit-tate function are in the form of G = g( X, Y( t)) with time-invariant random variable X and time-dependent tochatic procee Y () t, but are implicit with repect to time t. 2

4 Limit-tate function are explicit with repect to time t and are function of X and Y () t. Thi i the mot general cae where G = g( X, Y ( t), t). In the pat decade, many progree have been made in time-dependent reliability analyi methodologie. They include the Monte Carlo imulation (MCS) [4], the Gamma proce method [5], the Markov chain method [6], the extreme value ditribution method [7, 8], the upcroing rate method [9, 10], and the compoite limit tate method [1]. Amongt them, the mot popular method i the upcroing method. Thi method i baed on the Rice formula [11], whoe key tep i the computation of the upcroing rate. Thi method ha advantage over other method for it efficiency. But it accuracy i poor for problem whoe reliability i low. The accuracy iue come from the aumption that all the upcroing are independent. Many improvement on the Rice formula have been made to improve it accuracy. The recent development include the work of Sudret [12], where an analytical upcroing rate i derived. Thi upcroing rate i further derived by Zhang and Du [13] later for kinematic reliability analyi of function generator mechanim. Mejri and Cazuguel [14] alo combine the upcroing rate method with the finite element analyi calculation to ae the time-variant reliability of marine tructure. In pite of many progree, maintaining high accuracy and efficiency i till a challenge. The MCS method can evaluate the time-dependent reliability accurately, but the required computational cot may not be affordable. A mentioned previouly, the widely ued upcroing rate method i relatively efficient, but it accuracy may be poor [15]. It aume that all the upcroing are tatitically independent. The aumption i conervative and may produce large error. Several empirical modification have been 3

5 made by Vanmarcke [16] and Preumont [17] to remedy the drawback of the independent upcroing aumption. Thee modification, however, are limited to pecial problem. For pecial problem where there i only one loading tochatic proce in the input, the extreme value of the tochatic proce can be ued to replace the proce. Thi work i concerned with a ampling approach to the extreme value of a tochatic proce Yt, ( ) which can be a load or a trength. The baic idea i to replace Yt ( ) in the limit-ate function with it extreme value and then convert the time-variant problem into it timeinvariant counterpart. The ample of the extreme value of Yt ( ) are obtained from Monte Carlo imulation. It i then ued to etimate the ditribution of the extreme value. Saddplepoint approximation are employed [18] for the ditribution etimation. The new method i developed for pecial limit-tate function in category two. An eligible limit-tate function for the new method i in the form of G= g( X, Yt ( )); and the tochatic proce Yt ( ) i either a general trength variable, meaning that g () decreae when the proce increae, or a general tre variable, meaning that g () increae when the proce increae. Thi pecial type of limit-tate function appear in many important application. For example, the limit-tate function aociated with the following repone variable fall into the pecial type: the tre of hydrokinetic turbine blade under random wave loading [19], the damage of bridge under random traffic loading [20], and the deflection of beam under loading with tochatic diturbance [21]. Once the extreme value of Yt ( ) over [0, t ] i available, it i ued to replace Yt. ( ) Since the extreme value i a random variable, the limit-tate function become time invariant. Any time-invariant reliability method can then be applied for the reliability 4

6 evaluation. In thi paper, we ue FORM a an example for the time-invariant reliability analyi ince it i the mot widely ued reliability analyi method. For thoe limit-tate function, which are very non-linear and have multiple mot probable point (MPP), other time-invariant reliability method can be employed to ubtitute FORM. The major contribution of thi work are multifold. (1) With the ue of the extreme value of a general trength or general tre variable, the independent upcroing aumption of the Rice formula i completely removed, and then the accuracy of timedependent reliability analyi i alo improved ignificantly. (2) We employ the Saddlepoint Approximation in etimating the ditribution of the extreme value of the general trength or general tre procee. The employment make the ampling procedure not only accurate but alo robut. (3) Given the good accuracy and efficiency, the ue of the propoed method will alo make time-dependent reliability-baed deign more accurate and efficient. In Section 2, we review time-dependent reliability analyi. We then introduce the new method in Section 3. In Section 4, we provide two example followed by concluion in Section Methodology Review In thi ection we review the definition of time-dependent reliability. We alo dicu the upcroing rate method. Thi method i mot commonly ued and will be compared with the propoed method in the two example. 5

7 2.1. Time-Dependent Reliability Reliability i the probability that a component or ytem perform it intended function under intended condition over a certainty period of time. A limit-tate function G = g( XY, ( t) ) define the intended condition; it alo determine if the intended function i performed properly a follow: if g () < 0, the intended function i performed properly; if g () > 0, the intended function i not performed properly, and then a failure occur. Let the time period be [0, t ]. The time-dependent reliability i given by { )) < 0 [ } R (0, t ) = Pr g ( XY, ( t, " t Î 0, t ] (1) The aociated time-dependent probability of failure i { ) > ] } p (0, t ) = Pr g( XY, ( t 0, $ t Î[0, t ( 2) f It hould be note that p (0, t ) in Eq. (2) i defined for a given time period [0, t ]. It will change a the time period change. f It i well-known in reliability engineering that the time-dependent reliability can be derived from the failure rate by [22] = - -ò 0 t { l } p (0, t ) 1 exp ( t ) dt (3) f where l () t i the failure rate at time intant t. l () t i defined by the following limit of a conditional probability: l( t) = lil Pr{ t < T < t +D t) T > t} / Dt ( 4) D t 0 where T i the time to failure, which i apparently a random variable. Given the limittate function, we alo have 6

8 l( t) = lil Pr{ g( t +D t) > 0 g( t) < 0+ t Î[0+ t ]} / Dt (5) D t 0 where the notation g( t ) i ued for g( XY, ( t) ). The conditional probability in Eq. (4) or (5) i the probability of failure at t given that no failure have occurred prior to t. 2.2.Upcroing Rate Method Eq. (3) indicate that p (0, t ) i obtainable by integrating the failure rate l () t over f [0, t ]. The upcroing method ue the upcroing rate v + () t to approximate the failure rate l () t ; namely + l() t» v () t (6) an upcroing i defined a an event when the limit-tate function croe the limit tate from the afe region g () < 0 to the failure region g () > 0. We now briefly review how to compute the upcroing rate uing FORM, and more detail can be found in [13, 23]. With FORM, the limit-tate function i tranformed a follow: G = g( XY, ( t) ) = g( T( U ), T( U() t )) = h( U, U ( t)) (7) where T () tand for the tranformation. X Y X Y Then FORM earche for the Mot Probable Point MPP). The limit-tate function i then linearized at the MPP U () t = ( U, U ()) t, which i the hortet ditant point * * * X Y + between h () = 0 and the origin. The upcroing rate v () t i then computed by [11]: 7

9 ( ) ( ) + v (( t = w t f b(({ t fb [ ((/ t w((] t -[ b ((/ t w((] t F- [ b ((/ t w((]} t (8) in which f () and F () are the probability denity function (PDF) and cumulative ditribution function (CDF) of a tandard normal variable, repectively. w () t, b( t ), and b () t are variable computed uing function r ( t, t ) of Y (t ) [23]. 1 2 U * () t and the correlation + Once v () t i obtained, we can ue Eq. (3) to calculate the probability of failure p (0, t ). However, the equation i baed on the aumption that the initial probability of f failure at t = 0 i zero. After accounting for the initial probability of failure, p (0), we have the following equation [13] = - - -ò 0 t + { } p (0+ t ) 1 [1 p ( 0)] exp v () t dt (9) f f The upcroing method ha two drawback. It accuracy may not be atifactory becaue of the independent upcroing aumption, which may not hold for many application, epecially when the time interval i long and when there are many upcroing. The other drawback i the demanding computation. FORM mut be performed at all the time intant required by the integral in Eq. (9). In the next ection, we develop a new method to overcome the two drawback. f 3. The Sampling Methodology The central idea i to convert the time-dependent problem into a time-invariant 8

10 problem for limit-tate function with only one tochatic proce. Then the timeinvariant reliability method, uch a the FORM, can be ued. We firt give the general principle of the new ampling method and then provide it detailed tep Overview To ue the time-invariant reliability method, we make the following aumption: The limit-tate function i implicit of time. In other word, time t doe not explicitly appear in g (). The tochatic proce Yt () i either a general trength variable or a general tre variable. A general trength variable i a variable related to reitance to loading. For example, the yield trength of material i a trength variable. When the general trength variable increae, g () decreae. A general tre variable i a variable related to load. For example, the external force or moment i a tre variable. When the general tre variable increae, o doe g (). The above aumption i reaonable for a wide range of engineering application. For example, in many dynamic ytem, time t doe not appear explicitly in the dynamic equation, but the equation are till time-dependent becaue they involve time-dependent coefficient Yt. () For many engineering application, a large number of tochatic procee may be rarely encountered. Obtaining the complete information about a tochatic proce i cotly and time conuming. In many cae, only the mot important tochatic proce, uch a a load, i conidered. Then reource are allocated for collecting neceary data for the elected proce. With only one tochatic proce, it i eay to identify whether it i a general trength or a general tre variable before the 9

11 reliability analyi i performed. With the aumption, a limit-tate function i given by G= g( X, Yt ( )) (10) The probability of failure can then be evaluated by { X ) > } { ax g( X, Y( t)) > 0, t [0, t ] } { g X W > } p (0, t ) = Pr g(, Y( t 0, $ t Î[0, t ] f = Pr m " Î = Pr (, ) 0 where W i the global extreme value of Yt ( ) over [0, t ], It i given by ( 11) for a general trength variable, and W= min{ Yt ( ), tî [0 t]} (12), W= max{ Yt ( ), tî [0 t]} (13) for a general tre variable. We have now converted the time-variant problem into a time-invariant one a indicated in Eq. (11). Then a time-invariant reliability method can be applied., 3.2.Procedure Two tage are involved in the new method. The firt i to obtain the ditribution of the extreme value of the tochatic proce. The econd i the time-invariant reliability analyi. The procedure i ummarized in Fig. 1, and the detail are given below. Place Fig. 1 here Stage 1: Etimate the ditribution of extreme value of Yt () by MCS 10

12 Since we do not need to call the limit-ate function in thi tage, we can imply ue MCS. Denote the CDF of W by F ( w ), which can be obtained with the following three tep: W Step 1: Divide the time interval [0, t ] into dicrete point ( t, t,, t = t ) or N 0 1 N equal mall interval. Yt ( i ) ( i = 0,1, ¼, N ) i a random variable at t i. Generate n random ample for Yt ( i ). The ample or trajectorie of Yt () are denoted by Y = ( y ( t ), y ( t ) y ( t )) ( j = 1, 2,, n ), or Y = ( y ( t )), where n i the j j i i= 0,1, ¼, N j j 0 j i j N number of imulation. Step 2: Identify the extreme value of the trajectorie of Yt. () If Yt () i a general trength variable, the minimum value i found by w j = m i n( y j ( t i )) i = 0,1, ¼, N. If Yt () i a general tre variable, the maximum value i found by w j = m a x( y j ( t i )) i = 0,1, ¼, N. Step 3: Etimate the CDF F ( w ) from the ample w ( j = 1, 2, ¼, n ). W j Once the ditribution of W are available, we replace Yt () with W, and the limittate function become i i G = g( X, W) (14) The detail of thi tage will be dicued in Subection 3.3 and 3.4. Stage 2: Time-invariant reliability analyi In thi work, we ue FORM. Other reliability method can alo be ued. The ue of FORM will be preented in Subection

13 3.3. Sampling Approach to the Extreme Value of Stochatic Procee Our tak now i to obtain ample of Yt () and ue them to etimate the CDF of the extreme value F ( w ). We take a Gauian proce a an example. For thi proce, we W ue the Expanion Optimal Linear Etimation method (EOLE) [24] to generate ample. The approach approximate Yt () with a finite et of random variable. Let the mean function and tandard deviation function of Yt () be m () t and t () t, repectively, and Y Y aume that the autocorrelation coefficient function i r ( t, t ). At firt, the time interval Y 1 2 [0, t ] i divided into time point ( t ) = ( t = t, t, ¼, t ). After the dicretization, i i= 1, 2,, the correlation matrix of the time point ( t ) = 1, 2,, i i i computed by ( t ) t ) ( t ) t ) ( t ) t 1 ) ( t ) t 2 1) ( t ) t 2 2) ( t ) t 2 ) ρ ρ ρ ρ ρ ρ = ρ ρ ρ Y Y Y S Y Y Y S ( t ) t1) ( t ) t2) ( t ) t ) Y S Y S Y S S (15) Let i and i T φ i be the eigenvalue and eigenvector of the matrix, repectively. Then the expanion of the tochatic proce i given by [24] in which ( t ) p U Y t = t ) tå t (16) i T ( ) m () () φρ ( ) Y Y i Y i= 1 hi ρ i a vector of correlation coefficient with j-th element be Y Y( tt, j) r, j = 1, 2,, p, U ( i = 1, 2,, p ) are independent tandard normal random i variable. We can then generate random ample of U and ue them to reproduce ample i 12

14 curve (trajectorie) of Yt () uing Eq. (16). More detail about the EOLE can be found in [24]. After n imulation, we obtain n trajectorie of the tochatic proce ample. Fig. 2 how uch a trajectory, from which we can eaily find the extreme value. Place Fig. 2 here With the obtained extreme value ample w ( j = 1, 2, ¼, n ), we now dicu how to j etimate the CDF F ( w ), which i needed in FORM for the random variable W tranformation. The tail area F ( w ) are uually involved, and thi require the etimation of mall probabilitie and a large of number of imulation. W The mot traightforward way i to ue the empirical CDF by orting w uch that j w w w (1) (2) ( n ) < <¼<. Then we can ue the following equation: F W ( w ) () i i = (17) n For a variable w not ampled, w ( i 1) < w < w - ( i), we can ue a linear interpolation given by F ( w ( ( ) ( 1) ( ( ) - F w ) ) ) W i W i- F w = F w + ( w -w ) w - w W W ( i-1) ( i-1) ( i) ( i-1) ( 18) There are three problem with the above empirical CDF when it i ued in FORM. During the variable tranformation, a given value of W may be beyond the range of the ample w ( j = 1, 2, ¼, n ), reulting in a breakdown in the FORM algorithm. The j 13

15 econd problem i the non-uniformity of the ample may make the tail CDF etimation enitive to the ample ize and alo caue the convergence difficulty. The lat problem i that the derivative of F w ) at ome point may not exit, and thi will alo caue a W ( convergence difficulty. To deal with thee problem, we ue the Saddlepoint Approximation method that follow Saddlepoint approximation (SPA) The addlepoint approximation method [25] i widely ued to approximate a CDF, epecially the tail of the CDF, with excellent accuracy [26, 27]. The baic requirement of uing SPA i to know the cumulant generating function (CGF) of the correponding CDF. Detail about SPA can be found in [28]. No cloed-form CGF i available for the extreme value W, and then we generate ample for W to approximate the CGF [29]. We firt etimate the moment of W, which are then converted into the correponding cumulant. The cumulant are ued to approximate the CGF. Given a et of ample w ( j = 1, 2, ¼, n ), the cumulant are j computed with the following equation [30]: ìï 1 k = 1 n ï n k = 2 nn ( - 1) 3 2 ï 2-3n + n ík = 3 nn ( -1)( n-2) ï n -3 n( n -1) k = 4 nn ( -1)( n-2)( n-3) 2 ï - 4 n( n + 1) + n ( n + 1) ï ïî nn ( -1)( n-2)( n-3) (19) 14

16 where r ( r = 1, 2, 3, 4 ) are the um of the r-th power of the ample and are given by n r = r åw (20) i i= 1 Here we only ue the firt four cumulant. Baed on our numerical experiment, the order i good enough. Then the CGF i approximated by [25] K W r i x () x = å k (21) i i! i= 1 To obtain the CDF F ( w ), we mut olve for the addlepoint x, which i the olution to the following equation: W K x = w (22) ' () W where K ' () W i the derivative of the CGF. The above equation can be rewritten a where K ' W 2 3 x x x () x = k + k + k + k = w ( 23) ! 2! 3! It i a polynomial function, and it root can be eaily found. The CDF F ( w ) i then approximated with the following equation [31]: W æ1 1ö F ( w) = Pr{ W < w} =F () z + f() z - W ç z v çè ø ( 24) { é ù} 1/2 z = ign( x ) 2 ( ) êë xw -K x W úû ( 25) v = x ék " ( x ) ù ê ë W ú û 1/2 ( 26) 15

17 in which ign( x ) = +1, 1, or 0, depending on whether x i poitive, negative, or zero. K '' () W i the econd derivative of the CGF and i given by K 4 " ( x ) = k + k W 2 j j= 3 j-2 x å ( 27) ( j - 2)! SPA can give an accurate CDF etimation, epecially in a tail area of the CDF. Additionally, the CDF from SPA i continuou, and it can therefore avoid the abovementioned three problem when an empirical CDF i ued. 3.5.Time-Invariant reliability analyi After the CDF of the extreme value of Yt () i obtained, the time-dependent limittate function can be converted into a time-invariant function a indicated in Eq. (11). Then a time-invariant reliability method may be applied becaue the probability of failure i computed by { } p (0, t ) = Pr g( X, W) > 0 (28) f If FORM i ued, we tranform X and W into tandard normal variable U and X U W, repectively. The tranformation of W i given by W = F -1 ( F( U )) (29) W W where F - 1 () i the invere function of the CDF of W. Thi involve the invere W addlepoint approximation [32]. to U W Eq. (29) indicate that we need to olve for the aociated addlepoint correponding. The addlepoint x i obtained by olving the following equation: u 16

18 æ 1 1 ö F ( U ) = F ( z ) + f( z ) W u u ç - çèz v ø u u ( 30) where é ' ù { } 1/2 z = ign( x ) 2 ( ) ( ) u u êx K x -K x ë u W u W u úû ( 31) v = x ék " ( x ) ù ê ë ú û u u W u 1/2 ( 32) After obtaining the addlepoint x, we have u W = K ( x ) ( 33) ' W u After the tranformation, the following optimization model i olved ìï min ( u, u ) ï( u, u ) X W W W í ï ubject to g( T( u ), T( u )) ³ 0 ïî X W (34) The olution i the MPP and i denoted by * U. The reliability index b i calculated by b = U * () t (35) Then the probability of failure i etimated by p (0, t ) =F- ( b ) (36) The propoed new method ha the following characteritic: f There i no need to call the limit-tate function in the imulation for F ( w ). W The limit-tate function g () i only called by FORM. Only one MPP earch i needed. 17

19 Therefore, the efficiency of the propoed new method i equal or imilar to that of FORM. If we ue the number of function call to meaure the efficiency, the timedependent reliability analyi i a efficient a that of time-independent reliability analyi if FORM i employed. Previou tudie have aeed the efficiency of FORM [33-35]. Their concluion are applicable to the propoed method. 4. Example We now provide two example to demontrate the new time-dependent reliability method A Beam under Time-Variant Random Loading The firt example i a beam problem adapted from [10]. The beam hown in Fig. 3 i ubjected to an external force F acting at it midpoint. The limit-tate function i defined by 2 2 FtL () abl abt t u g( X, Yt ( )) =, - (37) in which t i the denity of teel, L i the length of the beam, a 0 and b 0 are the width and height of the cro ection, repectively, and i the ultimate tre of the material. u F i a Gauian proce (GP) with an autocorrelation function given by æ 2 ( t - t ) 2 1 ö r( t, t ) = exp ç è z ø (38) where z = 0.5 year and i the correlation length. Place Fig. 3 here 18

20 Table 1 preent the variable in the limit-tate function. In the table, Std and AC tand for tandard deviation and autocorrelation, repectively. Place Table 1 here In thi problem, the tochatic proce i the load F. It i eay to claify it a a general tre variable. It maximum value hould therefore be conidered. The time-dependent probabilitie of failure over different time interval up to [0, 30] year were computed uing the following three method: (1) The MCS: The purpoe of uing MCS i to ae the accuracy of the other method. The MCS olution i regarded a the accurate olution if the number of imulation i large enough. (2) The traditional upcroing method (UC) with FORM [13]: It i repreented by UC in the table and figure of the reult. (3) The propoed method: The method i repreented by Propoed in the table and figure of the reult. For the propoed method, 10 4 ample were generated for etimating the CDF of the maximum load over the time interval. Fig. 4 how the etimated CDF of the maximum load obtained from the propoed method. The figure indicate that the CDF curve hift to the right when t increae. Thi mean that the right tail of the ditribution become longer and that higher extreme load appear with a longer time interval. Place Fig. 4 here 19

21 The reult are given in Table 2 and Fig. 5. The lat two column of Table 2 how the point olution and their 95% confidence interval from MCS. The olution were reulted from 10 6 imulation. The reult indicate that the traditional upcroing method produced large error, epecially when the probability of failure i high. Thi mean that the upcroing method i too conervative. The propoed method generated very accurate reult. Place Fig. 5 here Place Table 2 here We ue the number of limit-tate function call to meaure the efficiency. Table 3 give uch number. The table indicate that the new method i alo much more efficient than the traditional upcroing method. The reaon i that the former method called the MPP earch once and the latter method called the MPP earch everal time. Place Table 3 here Moreover, it hould be noted that even if the limit-tate function of thi problem in Eq. (37) i given in an explicit form, the propoed method i alo applicable for more complex problem with implicit limit-tate function (black-box model). 20

22 4.2.Hydrokinetic Turbine Blade under Time-Variant River Flow Loading The tochatic proce in the lat example i pecial becaue it mean and tandard deviation function are contant. In thi example, the tochatic proce i the river velocity, which ha time-variant mean, tandard deviation, and autocorrelation function. With the eaonal characteritic in the river velocity, the external load in thi example i more complicated than the one in the lat example. Since the river velocity govern the blade tree, it i a general tre variable. Fig. 6 depict the implified cro ection of the hydrokinetic turbine blade. The blade under river flow loading i hown in Fig. 7. Place Fig. 6-7 here With a fixed tip peed ratio, the flapwie bending moment of the turbine blade i calculated a [36] in which velocity, M 1 2 = r () flap m 2 v tc (39) M tand for the flapwie bending moment, vt () i the random river flow flap 3 3 r = 1 10 kg/m i the river flow denity, and C = i the coefficient m of moment, which i obtained from the blade element momentum theory. Baed on the hitorical river dicharge data of the Miouri River from 1897 to 1988 at the Hermann tation in Miouri [23, 37] and the relationhip between river dicharge and river velocity, we fitted the mean and tandard deviation of the monthly river velocity a function of t a follow: 21

23 4 t = a m b m t + c m v i i i i= 1 n ( ) å in( ) (40) t 4 2 ( t) = a exp{ -[( t -b ) / c ] } v j j j j= 1 å (41) in which a, b, and c are contant. Similar to the wave loading [20], the monthly river velocity i aumed to be a v t v t narrowband Gauian proce (GP) with mean m (), tandard deviation t (), and autocorrelation coefficient function r (, t ). The auto-correlation coefficient function 1 2 v t r (, ) i given by t t v 1 2 ( t, t ) = co(2 p ( t - t )) (42) v The limit-tate function i defined by g 2 M t flap 1 rvt () C t m 1 = - f = - f (43) allow allow EI 2EI where e i the allowable train of the material, E = 14 GPa i the Young modulu, allow and I i the moment of inertia at the root of the blade, which i computed by 2 I = l ( t t 1 2 ) (44) 3 in which l 1, t 1 and t are the dimenion variable a hown in Fig. 6. Table 4 preent 2 the variable in thi example. We computed the probabilitie of failure of the hydrokinetic turbine blade over different time interval up to [0, 12] month uing MCS, UC, and the propoed method. The number of performed imulation and generated ample are alo the ame a thoe in Example 1. 22

24 Place Table 4 here Fig. 8 preent the etimated CDF of the maximal river velocity over different time interval obtained from the propoed method. The river velocity i a general tre variable. Place Fig. 8 here Table 5 and Fig. 9 how the probabilitie of failure of the turbine blade over different time interval obtained from the three method. The 95% confidence interval of the MCS olution are alo given in the lat column of Table 5. Place Table 5 here Place Fig. 9 here The reult how that the propoed method i much more accurate than the traditional upcroing rate method. The reult alo demontrate that the propoed method i applicable for time-dependent reliability analyi with a non-tationary tochatic proce. The new method i alo more efficient than the traditional method a indicated by the number of function call in Table 6. 23

25 Place Table 6 here 5. Concluion A new ampling method to time-dependent reliability method ha been propoed for limit-tate function that are implicit with repect to time and are function of a general trength or tre tochatic proce. The method employ a ampling approach to etimating ditribution of the extreme value of the tochatic proce over the time interval under conideration. The ditribution i etimated by Saddlepoint Approximation. The extreme value i then ued to replace it correponding tochatic proce. Then the time-dependent problem become a time-invariant problem, and a timeinvariant reliability analyi method can be ued. In thi work, we ued FORM. The new method ha advantage over the traditional upcroing method in the following two apect: The number of evaluation of the limit-tate function i ignificantly reduced. The method i therefore much more efficient. The accuracy of reliability analyi ha alo been improved ignificantly. In addition to FORM, other reliability method can alo be ued after the timedependent problem ha been tranformed into a time-invariant one. 24

26 The propoed method i limited to the limit-tate function that are implicit with repect to time. The input tochatic proce hould be either a general trength variable or a general tre variable. Our future work include incorporating the propoed method into reliability-baed deign optimization, uing the idea to more general limit-tate function, and invetigating the poible extenion of the method to problem with multiple tochatic procee. Acknowledgment Thi material i baed upon work upported in part by the Office of Naval Reearch through contract ONR N (Program Manager Dr. Michele Anderon), the National Science Foundation through grant CMMI , and the Intelligent Sytem Center at the Miouri Univerity of Science and Technology. Reference [1] Singh, A., Mourelato, Z. P., and Li, J., 2010, "Deign for Lifecycle Cot Uing Time- Dependent Reliability," Journal of Mechanical Deign, Tranaction of the ASME, 132(9), pp [2] Nielen, U. D., 2010, "Calculation of Mean Outcroing Rate of Non-Gauian Procee with Stochatic Input Parameter - Reliability of Container Stowed on Ship in Severe Sea," Probabilitic Engineering Mechanic, 25(2), pp [3] Sergeyev, V. I., 1974, "Method for Mechanim Reliability Calculation," Mechanim and Machine Theory, 9(1), pp

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29 [21] Yang, H. Z., and Zheng, W., 2011, "Metamodel Approach for Reliability-Baed Deign Optimization of a Steel Catenary Rier," Journal of Marine Science and Technology, 16(2), pp [22] Sheu, S. H., Yeh, R. H., Lin, Y. B., and Juang, M. G., 2001, "Bayeian Approach to an Adaptive Preventive Maintenance Model," Reliability Engineering and Sytem Safety, 71(1), pp [23] Hu, Z., and Du, X., 2012, "Reliability Analyi for Hydrokinetic Turbine Blade," Renewable Energy, 48, pp [24] Li, C. C., Kiureghian, A. D., 1993, "Optimal Dicretization of Random Field," Journal of Engineering Mechanic, 119(6), pp [25] Daniel, H., 1954, "Saddlepoint Approximation in Statitic," The Annal of Mathematical Statitic, 25(4), pp [26] Du, X., and Sudjianto, A., 2004, "Firt-Order Saddlepoint Approximation for Reliability Analyi," AIAA Journal, 42(6), pp [27] Marh, P., 1998, "Saddlepoint Approximation for Noncentral Quadratic Form," Econometric Theory, 14(05), pp [28] Du, X., 2010, "Sytem Reliability Analyi with Saddlepoint Approximation," Structural and Multidiciplinary Optimization, 42(2), pp [29] Huang, B., and Du, X., 2006, "A Saddlepoint Approximation Baed Simulation Method for Uncertainty Analyi," International Journal of Reliability and Safety, 1(1), pp. 9. [30] Fiher, R. A., 1928, "Moment and Product Moment of Sampling Ditribution," Proceeding of London Mathematical Society, 30(2), pp

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31 Lit of Table Caption Table 1 Variable and Parameter in Example 1 Table 2 Table 3 Time-Dependent Probabilitie of Failure Number of Function Evaluation by the Three Method Table 4 Variable and Parameter in Example 2 Table 5 Table 6 Time-Dependent Probabilitie of Failure Number of Function Evaluation Lit of Figure Caption Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Flowchart of the New Time-Dependent Reliability Method A Trajectory of a Stochatic Proce A Beam Under Random Loading CDF of Maximum Load over Different Time Interval Probability of Failure of the Beam over Different Time Interval Cro Section at Root of the Turbine Blade River Flow Loading on the Turbine Blade CDF of Maximum River Velocity over Different Time Interval Probability of Failure of the Hydrokinetic Turbine Blade 30

32 Table 1 Variable and Parameter in Example 1 Mean Std Ditribution AC a m 0.01m Lognormal N/A b m m Lognormal N/A u Pa Pa Lognormal N/A Ft () 4500 N 1050 N GP Eq. (38) L 5 m 0 Determinitic N/A t 78.5 KN/m 3 0 Determinitic N/A 31

33 Table 2 Time-Dependent Probabilitie of Failure [0, t ] (year) UC (10-3 ) Propoed (10-3 ) MCS (10-3 ) [0, 4] [1.215, 1.219] [0, 8] [1.987, 1.991] [0, 12] [2.574, 2.578] [0, 16] [3.096, 3.100] [0, 20] [3.567, 3.571] [0, 24] [3.954, 3.958] [0, 28] [4.312, 4.316] 32

34 Table 3 Number of Function Evaluation by the Three Method [0, t ] (year) UC Propoed MCS [0, 4] [0, 8] [0, 12] [0, 16] [0, 20] [0, 24] [0, 28]

35 Table 4 Variable and Parameter in Example 2 Variable Mean Std Ditribution AC v (m/) m () t t () t v v GP In Eq. (42) l (m) Gauian N/A t (m) Gauian N/A t (m) Gauian N/A e allow Gauian N/A 34

36 Table 5 Time-Dependent Probabilitie of Failure [0, t ] (month) UC (10-3 ) Propoed (10-3 ) MCS (10-3 ) [0, 4] [0.079, 0.080] [0, 5] [1.113, 1.117] [0, 6] [1.155, 1.159] [0, 7] [2.815, 2.819] [0, 8] [2.815, 2.819] [0, 9] [2.815, 2.819] 35

37 Table 6 Number of Function Evaluation [0, t ] (month) UC Propoed MCS [0, 4] [0, 5] [0, 6] [0, 7] [0, 8] [0, 9]

38 Define time-dependent limit-tate function G= g( X, Yt ( )) Obtain time-invariant limit-tate function G = g( X, W) Perform FORM MPP Compute p ( t, t ) f 0 Sampling of the tochatic proce Sample Yt () over [0, t ] Obtain ample of min max Y or Y Saddlepoint Approximation CDF of W Fig. 1 Flowchart of the New Time-Dependent Reliability Method 37

39 Y Global extreme value O t Fig. 2. A Trajectory of a Stochatic Proce 38

40 L L/2 F A-A A b 0 a 0 A Fig. 3. A Beam under Random Loading 39

41 CDF 0.4 [0, 7] t [0, 13] 0.2 [0, 1] [0, 4] [0, 10] F max (N) over different time interval [0, t] year Fig. 4. CDF of Maximum Load over Different Time Interval 40

42 0.015 MCS UC Propoed Probability of failure Time interval [0, t] (year) Fig. 5. Probability of Failure of the Beam over Different Time Interval 41

43 t 1 t 2 l1 Fig. 6. Cro Section at Root of the Turbine Blade 42

44 vt () R M flap Fig. 7. River Flow Loading on the Turbine Blade 43

45 1 0.8 t 0.6 CDF 0.4 [0, 3] [0, 7] [0, 5] 0.2 [0, 1] v max (m/) over Different Time Interval [0, t] Month Fig. 8 CDF of Maximum River Velocity over Different Time Interval 44

46 6 x MCS UC Propoed Probability of failure Time interval [0, t] (month) Fig. 9. Probability of Failure of the Hydrokinetic Turbine Blade 45