CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department of Cvl and Envronmental Engneerng Unversty of Maryland, College Park CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 1 Introducton The relablty of an engneerng system can be defned as ts ablty to fulfll ts desgn purpose defned as performance requrements for some tme perod and envronmental condtons. The theory of probablty provdes the fundamental bases to measure ths ablty.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. Introducton The relablty assessment methods can be based on 1. Analytcal strength-and-load performance functons, or. Emprcal lfe data. They can also be used to compute the relablty for a gven set of condtons that are tme nvarant or for a tme-dependent relablty. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 3 Introducton The relablty of a component or system can be assessed n the form of a probablty of meetng satsfactory performance requrements accordng to some performance functons under specfc servce and extreme condtons wthn a stated tme perod. Random varables wth mean values, varances, and probablty dstrbuton functons are used to compute probabltes.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 4 Frst-Order Second Moment (FOSM) Method. Advanced Second Moment Method Computer-Based Monte Carlo Smulaton CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 5 Advanced Second-Moment Method Z = Z( n 1,, K, ) = Supply - Demand (1a) Z = Z( 1,, K, n ) = Structural strength - Load effect (1b) Z = Z,,, n ) = ( 1 K R-L (1c) Z = performance functon of nterest R = the resstance or strength or supply L = the load or demand as llustrated n Fgure 1
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 6 Densty Functon Falure Probablty (Area for g < 0) Performance Functon (Z) Load Effect (L) Strength (R) Orgn 0 Random Value Fgure 1. Performance Functon for CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 7 Advanced Second-Moment Method The falure surface (or the lmt state) of nterest can be defned as Z = 0. When Z < 0, the element s n the falure state, and when Z > 0 t s n the survval state. If the jont probablty densty functon for the basc random varables s s f x x x 1,, K, ( n 1,, K, n), then the falure probablty P f of the element can be gven by the ntegral P = f L f,, K, ( x x K xn dx dx Kdx n 1,,, ) 1 () 1 n
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 8 Advanced Second-Moment Method Where the ntegraton s performed over the regon n whch Z < 0. In general, the jont probablty densty functon s unknown, and the ntegral s a formdable task. For practcal purposes, alternate methods of evaluatng Pf are necessary. Relablty s assessed as one mnus the falure probablty. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 9 Advanced Second-Moment Method Relablty Index Instead of usng drect ntegraton (Eq. ), performance functon Z n Eq. 1 can be expanded usng Taylor seres about the mean value of s and then truncated at the lnear terms. Therefore, the frst-order approxmaton for the mean and varance are as follows: µ Z Z µ, µ, K, µ ) (3) Z n n = 1 j= 1 ( 1 n Z ( Z )( j ) Cov(, j ) (4a)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 10 Advanced Second-Moment Method Relablty Index (cont d) Where µ = mean of random varable µ = mean of Z Z = varance of Z Z Cov(, Z = ) = covarance of j and partal dervatve evaluated at the 1 mean of random varable CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 11 Advanced Second-Moment Method Relablty Index (cont d) For uncorrelated random varables, the varance cab be expressed as n Z Z ( ) (4b) = 1 The relablty ndex β can be computed from: µ Z µ R µ L β = = (5) Z µ R + µ L = 1 Φ( β ) (6) P f If z s assumed normally dstrbuted.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 1 Load (L) Lmt State L = R Falure Regon L > R Survval Regon L < R Strength or Resstance (R) Fgure. Performance Functon for a Lnear, Two-Random Varable Case CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 13 Advanced Second-Moment Method Nonlnear Performance Functons For nonlnear performance functons, the Taylor seres expanson of Z n lnearzed at some pont on the falure surface referred to as the desgn pont or checkng pont or the most lkely falure pont rather than at the mean. Assumng varables are uncorrelated, the followng transformaton to reduced or normalzed coordnates can be used: µ Y = (8a)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 14 Advanced Second-Moment Method Nonlnear Performance Functons (cont d) It can be shown that the relablty ndex β s the shortest dstance to the falure surface from the orgn n the reduced Y-coordnate system. The shortest dstance s shown n Fgure 3, and the reduced coordnates are Y L Y µ µ R R L = R + (8b) L L CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 15 Falure Regon L > R L µ L YL = L R Lmt State: Y L = Y L R µ R µ L + L Desgn or Falure Pont β Survval Regon L < R µ R µ L Intercept = L R µ R YR = Fgure 3. Performance Functon for a Lnear, Two-Random Varable Case n Normalzed Coordnates R
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 16 Advanced Second-Moment Method Nonlnear Performance Functons (cont d) The concept of the shortest dstance apples for a nonlnear performance functon, as shown n Fgure 4. The relablty ndex β and the desgn pont, * * * ( 1,, K, n ) can be determned by solvng the followng system of nonlnear equatons teratvely for β: CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 17 Y L L µ L = L Desgn or Falure Pont Lmt State n Reduced Coordnates R = Resstance or Strength L = Load Effect β Y R R µ R = Fgure 4. Performance Functon for a Nonlnear, Two-Random Varable Case n Normalzed Coordnates R
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 18 Advanced Second-Moment Method Nonlnear Performance Functons (cont d) Z ( ) α = 1/ n Z (9) ( ) = 1 * = µ αβ * * * Z(,, K, n ) = 1 0 (10) (11) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 19 Advanced Second-Moment Method Nonlnear Performance Functons (cont d) Where α s the drectonal cosne, and the partal dervatves are evaluated at the desgn pont. Eq. 6 can be used to compute P f. However, the above formulaton s lmted to normally dstrbuted random varables. The drectonal cosnes are consdered as measure of the mportance of the correspondng random varables n determnng the relablty ndex β.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 0 Advanced Second-Moment Method Nonlnear Performance Functons (cont d) Also, partal safety factors γ that are used n load and resstance factor desgn (LRFD) can be calculated from * (1) γ = µ Generally, partal safety factors take on values larger than 1 loads, and less than 1 for strengths. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 1 Advanced Second-Moment Method Equvalent Normal Dstrbutons If a random varable s not normally dstrbuted, then t must be transformed to an equvalent normally dstrbuted random varable. The parameters of the equvalent normal dstrbuton are µ N and These parameters can be estmated by mposng two condtons. N
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. Advanced Second-Moment Method Equvalent Normal Dstrbutons (cont d) Frst condton can be expressed as * µ N * Φ( ) F ( ) N = (13a) Second condton can be expressed as * µ * φ( ) = f ( ) N N (13b) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 3 Advanced Second-Moment Method Equvalent Normal Dstrbutons (cont d) where = non-normal cumulatve dstrbuton functon F f Φ φ = non-normal probablty densty functon = cumulatve dstrbuton functon of the standard normal varate = probablty densty functon of the standard normal varate.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 4 Advanced Second-Moment Method Equvalent Normal Dstrbutons (cont d) The standard devaton and mean of equvalent normal dstrbutons are gve by * φ( Φ 1 [ F( )]) = * f ( ) N µ = * Φ 1 [ F ( * )] N N (14a) (14b) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 5 Advanced Second-Moment Method Equvalent Normal Dstrbutons (cont d) N N Once and µ are determned for each random varable, β can be solved followng the same procedure of Eqs. 9 through 11. The advanced second moment (ASM) method can deal wth Nonlnear performance functon, and Non-normal probablty dstrbutons
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 6 Advanced Second-Moment Method Correlated Random Varables A correlated (and normal) par of random varables 1 and wth a correlaton coeffcent ρ can be transformed nto noncorrelated par Y 1 and Y by solvng for two egenvalues and the correspondng egenvectors as follows: Y 1 = 1 µ t 1 1 1 + µ (15a) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 7 Y 1 1 µ µ 1 = t 1 where t = 0.5. The resultng Y varables are noncorrelated wth respectve varances that are equal to the egenvalues (λ) as follows: = λ = + ρ Y 1 1 1 (16a) (15b) = λ = ρ Y 1 (16b)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 8 For a correlated par of random varables, Eqs. 9 and 10, have to be revsed, respectvely, to α Y1 Z ρ t Z + t 1 1 1 + = Z Z ρ Z Z + 1 + 1 1 / 1 1 (17a) α Y Z ρ t Z t 1 1 1 = Z Z ρ Z Z + 1 + 1 1 / 1 1 (17b) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 9 and ( ) * = µ t β α λ + α λ 1 Y 1 Y 1 1 1 (18a) ( ) * = µ t β α λ α λ Y 1 Y 1 (18b) where the partal dervatves are evaluated at the desgn pont.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 30 Advanced Second-Moment Method Numercal Algorthms The advanced second moment (ASM) method can be used to assess the relablty of a structure accordng to nonlnear performance functon that may nclude non-normal random varables. Implementaton of the method requre effcent and accurate numercal algorthms. The ASM algorthms are provded n the followng two flowcharts for Noncorrelated random varables (Case a) Correlated random varables (Case b) CHAPTER 4a. RELIABILITY ASSESSMENT Start Slde No. 31 Assgn the mean value for each random varable as a startng pont value: * * *, L, = µ, µ,, µ ( 1 n ) ( L ), 1 n Case a: Non-correlated Random Varables Compute the standard devaton and mean of the equvalent normal dstrbuton for each non-normal random varable usng Eqs. 13 and 14 Z Compute the partal dervatve for each RV usng Eq. 9. Compute the drectonal cosne α for each random varable as gven n Eq. 9 at the desgn pont. Compute the relablty ndex β substtute Eq. 10 nto Eq. 11 satsfy the lmt state Z = 0 n Eq. 11 use a numercal root-fndng method. Compute a new estmate of the desgn pont by substtutng the resultng obtaned n prevous step nto Eq. 10 β No β converges? Yes Take β value End
CHAPTER 4a. RELIABILITY ASSESSMENT Start Slde No. 3 Assgn the mean value for each random varable as a startng pont value: * * *, L, = µ, µ,, µ ( 1 n ) ( L ), 1 n Case b: Correlated Random Varables Compute the standard devaton and mean of the equvalent normal dstrbuton for each non-normal random varable usng Eqs. 13 and 14 Z Compute the partal dervatve for each RV usng Eq. 9. Compute the drectonal cosne α for each random varable as gven n Eq. 9 at the desgn pont. For correlated pars of random varables Eq. 17 should be used Compute the relablty ndex β : substtute Eq. 10 (for noncorrelated) and Eq. 18 (for correlated) nto Eq. 11 satsfy the lmt state Z = 0 n Eq. 11 use a numercal root-fndng method. Compute a new estmate of the desgn pont by substtutng the resultng β obtaned n prevous step nto Eq. 10 (for noncorrelated) and Eq. 18 (for correlated) No β converges? Yes Take β value End CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 33 Example 1: Usng a Nonlnear Performance Functon The strength-load performance functon for a components s assumed to have the followng form: Random Varable Z = 1 3 where s are basc random varables wth the followng probablstc characterstcs: Mean Value (µ) Standard Devaton () Coeffcent of Varaton Case (a) Dstrbuton Type Case (b) Dstrbuton Type 1 1 0.5 0.5 Normal Lognormal 5 0.5 0.05 Normal Lognormal 3 4 0.80 0.0 Normal Lognormal
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 34 Example 1 (cont d): Usng a Nonlnear Performance Functon Usng frst-order relablty analyss based on frstorder Taylor seres, the followng can be obtaned from Eqs. 3 to 5: µ Z ( 1)(5) 4 = 5 = 3 Z = µ Z β (5) (0.5) + (1) (0.5) + ( 0.5/ 1.565+ 0.065+ 0.04 = 1.903 Z = 3 1.903 =.35 4) (0.8) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 35 Example 1 (cont d): Usng a Nonlnear Performance Functon These values are applcable to both cases (a) and (b). Usng advanced second-moment relablty analyss, the followng table can be constructed for cases (a) and (b): Random Varable Case (a): Iteraton 1 Falure Pont Z Drectonal Cosnes (α) 1 1.000E+00 1.50E+00 9.687E-01 5.000E+00.500E-01 1.937E-01 3 4.000E+00 -.000E-01-1.550E-01
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 36 Case (a): Iteraton 1 Example 1 (cont d): Usng a Nonlnear Performance Functon The dervatves n the above table are evaluated at the falure pont. The falure pont n the frst teraton s assumed to be the mean values of the random varables. The relablty ndex can be determned by solvng for the root accordng to Eq. 11 for the lmt state of ths example usng the followng equaton: Z ( µ α β )( µ α β ) µ α β 0 = 1 = 1 1 3 3 3 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 37 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β =.37735 for ths teraton. Random Varable 1 3 Case (a): Iteraton Falure Pont Z 4.4E-01 1.1E+00 4.885E+00 1.061E-01 4.95E+00-1.930E-01 Drectonal Cosnes (a) 9.841E-01 8.547E-0-1.555E-01
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 38 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β =.368 for ths teraton. Random Varable 1 3 Case (a): Iteraton 3 Falure Pont Z 4.187E-01 1.37E+00 4.950E+00 1.047E-01 4.94E+00-1.930E-01 Drectonal Cosnes (α) 9.846E-01 8.39E-0-1.536E-01 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 39 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β =.368 for ths teraton whch means that β has converged to.368. The falure probablty =1-Φ(β) = 0.009068. The partal safety factors can be computed as: Random Varable Falure Pont Partal Safety Factors 1 0.418378 0.418378 4.950849 0.99017 3 4.90389 1.07597
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 40 Example 1 (cont d): Usng a Nonlnear Performance Functon Case (b) The parameters of the lognormal dstrbuton can be computed for three random varables based on ther respectve means (µ) and devatons () as follows: Y = ln 1 + µ 1 µ Y = ln µ Y ( ) and CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 41 Example 1 (cont d): Usng a Nonlnear Performance Functon The results of these computatons are summarzed as follows: Random Varable Dstrbuton Type Frst Parameter (µ Y ) Second Parameter ( Y ) 1 Lognormal -0.0303131 0.46068 Lognormal 1.60818947 0.04996879 3 Lognormal 1.366684005 0.0
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 4 Example 1 (cont d): Usng a Nonlnear Performance Functon Case (b): Iteraton 1 Equvalent Normal Random Varable 1 3 Falure Pont 1.000E+00 5.000E+00 4.000E+00 Standard Devaton.46E-01.498E-01 7.9E-01 Mean Value 9.697E-01 4.994E+00 3.9E+00 Z N 1.31E+00.498E-01-1.980E-01 Drectonal Cosnes (α) 9.681E-01 1.965E-01-1.557E-01 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 43 Example 1 (cont d): Usng a Nonlnear Performance Functon The dervatves n the above table are evaluated at the falure pont. The falure pont n the frst teraton s assumed to be the mean values of the random varables. The relablty ndex can be determned by solvng for the root accordng to Eq. 11 for the lmt state of ths example usng the followng equaton: Z = N N N N N N ( µ α β )( µ α β ) µ α β 0 1 1 1 3 3 = 3
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 44 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β =.30530 for ths teraton. Case (b): Iteraton Equvalent Normal Random Varable 1 3 Falure Pont 4.0E-01 4.881E+00 4.06E+00 Standard Devaton 1.035E-01.439E-01 8.330E-01 Mean Value 7.718E-01 4.99E+00 3.91E+00 Z N 5.050E-01 1.05E-01 -.031E-01 Drectonal Cosnes (α) 9.118E-01 1.850E-01-3.667E-01 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 45 Example 1 (cont d): Usng a Nonlnear Performance Functon Random Varable 1 3 Therefore, β = 3.34 for ths teraton. Falure Pont 4.584E-01 4.843E+00 4.97E+00 Case (b): Iteraton 3 Equvalent Normal Standard Devaton 1.19E-01.40E-01 9.758E-01 Mean Value 8.00E-01 4.991E+00 3.803E+00 Z N 5.465E-01 1.109E-01 -.198E-01 Drectonal Cosnes (α) 9.118E-01 1.850E-01-3.667E-01
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 46 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β = 3.316 for ths teraton. Case (b): Iteraton 4 Equvalent Normal Random Varable 1 3 Falure Pont 4.61E-01 4.843E+00 4.989E+00 Standard Devaton 1.136E-01.40E-01 9.880E-01 Mean Value 8.041E-01 4.991E+00 3.789E+00 Z N 5.499E-01 1.116E-01 -.1E-01 Drectonal Cosnes (α) 9.118E-01 1.850E-01-3.667E-01 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 47 Example 1 (cont d): Usng a Nonlnear Performance Functon Random Varable 1 3 Therefore, β = 3.315 for ths teraton. Falure Pont 4.61E-01 4.843E+00 4.989E+00 Case (b): Iteraton 5 Equvalent Normal Standard Devaton 1.136E-01.40E-01 9.880E-01 Mean Value 8.041E-01 4.991E+00 3.789E+00 Z N 5.500E-01 1.116E-01 -.1E-01 Drectonal Cosnes (α) 9.118E-01 1.850E-01-3.667E-01
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 48 Example 1 (cont d): Usng a Nonlnear Performance Functon Therefore, β = 3.315 for ths teraton whch means that β has converged to 3.315. The falure probablty =1-Φ(β) = 0.0004619. The partal safety factors can be computed as: Random Varable Falure Pont Partal Safety Factors 1 0.461189 0.461189 4.843135 0.96867 3 4.988968 1.474 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 49 Monte Carlo Smulaton Methods Monte Carlo smulaton (MCS) technques are bascally samplng processes that are used to estmate the falure probablty of a component or system. The basc random varables n Eq. 1, that s Z = Z,,, n ) = ( 1 K R-L are randomly generated and substtuted nto above equaton.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 50 Monte Carlo Smulaton Methods Then the fracton of the cases that resulted n falure are determned to assess the falure probablty. Three methods are descrbed heren: 1. Drect Monte Carlo Smulaton. Condtonal Expectaton 3. The Importance Samplng Reducton Method CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 51 Monte Carlo Smulaton Methods Drect Monte Carlo Smulaton Method In ths method, samples of the basc noncorrelated varables are drawn accordng to ther correspondng probabltes characterstcs and fed nto performance functon Z as gven by Eq. 1. Assumng that N f s the number of smulaton cycles for whch Z < 0 n N smulaton cycles, then an estmate of the mean falure probablty can be expressed as N f Pf = (19) N
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 5 Monte Carlo Smulaton Methods Drect Monte Carlo Smulaton Method (cont d) The varance of the estmated falure probablty can be approxmately computed usng the varance expresson for a bnomal dstrbuton as: (1 Pf ) Pf Var( Pf ) = (0) N Therefore, the coeffcent of varaton (COV) of the estmated falure probablty s 1 (1 Pf ) Pf COV( Pf ) = (1) P N f CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 53 Monte Carlo Smulaton Methods Drect Monte Carlo Smulaton Method (cont d) Some of the advantages of ths method s that t s easy to mplement and understand. The dsadvantages nclude: Expensve n some cases, especally f the falure probabltes are small. Ineffcent The mportance samplng method (IS) s descrbed later for the purpose of ncreasng the effcency of the s method.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 54 Monte Carlo Smulaton Methods Condtonal Expectaton Ths method can also be used to estmate the falure probablty accordng to the performance functon of Eq. 1. The method requres generatng all the basc random varables n Eq. 1 except the random varables wth the hghest varablty (.e., COV), whch s used as a control varable, k. The condtonal expectaton s computed as the cumulatve dstrbuton functon. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 55 Monte Carlo Smulaton Methods Condtonal Expectaton (cont d) For the followng performance functon: Z = R L and for a randomly generated value of L or R, the falure probablty for each cycle s gven, respectvely, as (3) P = f F l R ( ) () P = F f 1 L ( r ) (4)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 56 Monte Carlo Smulaton Methods Condtonal Expectaton (cont d) In these equatons, L and R are the control varables. The total falure probablty P f can be estmated from P f Where N s the number of smulaton cycles. N N P f (5) = = 1 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 57 Monte Carlo Smulaton Methods Condtonal Expectaton (cont d) The accuracy of Eq. 5 can be estmated through the varance and coeffcent of varaton as gven by ( Pf P f ) = 1 Var( P f ) = N( N 1) COV N Var( P P (6) f ( P f ) = (7) f )
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 58 Monte Carlo Smulaton Methods Importance Samplng (cont d) To mprove the effcency of smulaton when estmatng the probablty of falure for a gven performance functon, Importance Samplng (IS) technques are used. In some performance functon, f the margn of safety Z s large and ts varance s too small, larger smulaton effort wll be requred to obtan suffcent smulaton runs wth satsfactory performances,.e., smaller falure probabltes requre larger number of smulaton cycles CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 59 Monte Carlo Smulaton Methods Importance Samplng (cont d) In ths method, the basc random varables are generated accordng to some carefully selected probablty dstrbutons,.e., Importance densty functon, h (x) Wth mean values that are closer to the desgn pont than ther orgnal (actual) probablty dstrbutons.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 60 Monte Carlo Smulaton Methods Importance Samplng (cont d) The fundamental equaton for ths method s gven by N 1 f ( x1, x,..., xn ) Pf = I (8) N h ( x, x,..., x ) = 1 1 N = number of smulaton cycles f (x 1,x,..., x n ) = orgnal jont densty functon of the basc random varables evaluated at the th generated values of the basc random varables h (x 1,x,..., x n ) = selected jont densty functon of the basc random varables evaluated at the th generated values of the basc random varables I = performance ndcator functon that takes values of ether 0 for falure and 1 for survval n CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 61 Monte Carlo Smulaton Methods Importance Samplng (cont d) The coeffcent of varaton of the estmate falure probablty can be based on the varance of a sample mean as follows: COV ( P f ) = 1 N( N 1) N =1 I f h ( x, x P f 1 1 ( x, x,..., x,..., x n n ) Pf ) (9)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 6 Monte Carlo Smulaton Methods Correlated Random Varables A correlated (and normal) par of random varables 1 and wth a correlaton coeffcent ρ can transformed usng lnear regresson transformaton as follows: = b0 + b11 + ε (30a) b 0 = ntercept of a regresson lne between 1 and b 1 = slope of the regresson lne ε = random (standard) error wth a mean of zero and a standard devaton as gven n Eq. 30d). CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 63 Monte Carlo Smulaton Methods Correlated Random Varables (cont d) These regresson model parameters can be determned n terms of the probablstc characterstcs of 1 and as follows:: b b 1 = ρ 1 = µ b µ 0 1 1 (30b) (30b) ρ ε = 1 (30d)
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 64 Monte Carlo Smulaton Methods Correlated Random Varables (cont d) Procedure for a correlated par of random varables: 1. Compute the ntercept of a regresson lne between 1 and (b 0 ), the slope of the regresson lne (b 1 ), and the standard devaton of the random (standard) error (ε) usng Eqs. 30b to 30d.. Generate a random (standard) error usng a zero mean and a standard devaton as gven by Eq. 30d. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 65 Procedure for a correlated par of random varables (cont d): 3. Generate a random value for 1 usng ts probablstc characterstcs (mean and varance). 4. Compute the correspondng value of as follows (based on Eq. 30a): x = b0 + b1x1 + ε where b 0 and b 1 are computed n step 1; ε s a generated random (standard) error from step ; and x 1 s generated value from step 3. 5. Use the resultng random (but correlated) values of x 1 and x n the smulaton-based relablty assessment methods.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 66 Monte Carlo Smulaton Methods Tme-Dependent Relablty Analyss Several methods for analytcal tme-dependent relablty assessment are avalable. In these methods, sgnfcant structural loads as a sequence of pulses that can be descrbed by a Posson process wth mean occurrence rate, λ, random ntensty, S, and duraton, τ. The lmt state of the structure at any tme can be defned as R(t) - S(t) < 0 (31) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 67 Monte Carlo Smulaton Methods Tme-Dependent Relablty Analyss (cont d) where R(t) s the strength of the structure at tme t and S(t) s the loads at tme t. The nstantaneous probablty of falure can then be defned at tme t as probablty of R(t) less than S(t). The relablty functon, L(t), was defned as the probablty that the structure survves durng nterval of tme (0,t) as t 1 L( t) = exp[ λ t[1 Fs( g( t) r) dt] fr ( r) dr t (3a) 0 0
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 68 Monte Carlo Smulaton Methods Tme-Dependent Relablty Analyss (cont d) where f R (r) s the probablty densty functon of an ntal strength, R, and g(t) s the tme-dependent degradaton n strength. The relablty can be expressed n terms of the condtonal falure rate or hazard functon, h(t) as or d ht () = ln Lt () dt L( t) = exp[ t h( ξ) dξ ] 0 (3b) (3c) CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 69 Monte Carlo Smulaton Methods Tme-Dependent Relablty Analyss (cont d) The relablty L(t) s based on the complete survval durng the servce lfe nterval (0,t). It means the probablty of successful performance durng a servce lfe nterval (0,t). Therefore, the probablty of falure, P f (t), can be computed as the probablty of the complementary event,.e., P f (t) = 1 - L(t) beng not equvalent to P[R(t) < S(t)].
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 70 Emprcal Relablty Analyss Usng Lfe Data Falure and Repar The basc noton of relablty analyss based on lfe data s tme to falure. The useful lfe of a product can be measured n terms of ts tme to falure. In addton to tme, other possble exposure measures nclude the number of cycles to falure of mechancal, electrcal, temperature or humdty. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 71 Emprcal Relablty Analyss Usng Lfe Data Falure and Repar (cont d) If the faled product s subject to repar or replacement, t s called reparable (n opposte to non-reparable objects). The respectve repar or replacement requres some tme to get done, whch s called tme to repar/replace The tme to falure s used for the nonreparable components or systems.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 7 Emprcal Relablty Analyss Usng Lfe Data Falure and Repar (cont d) For reparable products, there s another mportant characterstc, whch s called tme between falures. Ths s another random varable or a set of random varables. It can be assumed that the tme to the frst falure s the same random varable as the tme between the frst and the second falures, the tme between the second and the thrd falures, and so on. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 73 Emprcal Relablty Analyss Usng Lfe Data Types of Data Falure data often contan not only tmes to falure (the so-called dstnct falures), but also tmes n use (or exposure length of tme) that do not termnate wth falures. Such exposure tme ntervals termnatng wth non-falure are called tmes to censorng (TTC). Therefore, lfe data of equpment can be classfed nto two types, complete and censored data
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 74 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) Falure data often contan not only tmes to falure (the so-called dstnct falures), but also tmes n use (or exposure length of tme) that do not termnate wth falures. Such exposure tme ntervals termnatng wth non-falure are called tmes to censorng (TTC). Therefore, lfe data of equpment can be classfed nto two types, complete and censored data. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 75 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) The complete lfe data are commonly based on equpment tested to falure or tmes to falure based on equpment use,.e., feld data. Censored lfe data nclude some observaton results that represent only lower or upper lmts on observaton of tmes to falure.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 76 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) Censored data can be further classfed nto Type I or Type II Type I data are based on observatons of a lfe test, whch for economcal or other reasons, must be termnated at specfed tme t 0. As the result, only the lfetmes of those unts that have faled before t0 are known exactly. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 77 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) If, durng the tme nterval (0, t 0 ], s out of n sample unts faled, then the nformaton n the data set obtaned conssts of s observed, ordered tmes to falure as follows: t 1 < t <...< t s (33a) and the nformaton that (n s) unts have survved the tme t 0.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 78 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) In some lfe data testng, testng s contnued untl a specfed number of falures r s acheved,.e., the respectve test or observaton s termnated at the r th falure. In ths case, r s not random. Ths type of testng,.e., observaton or feld data collecton, results n Type II censorng. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 79 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) It ncludes r observed ordered tmes to falure t 1 < t <...< t r (33b) And the nformaton that (n r) unts have survved the tme t r. But, n opposte to Type I censorng, the test or observaton duraton t r s random, whch should be taken nto account n the respectve statstcal estmaton procedures.
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 80 Emprcal Relablty Analyss Usng Lfe Data Types of Data (cont d) In relablty engneerng, Type I rght-censored data are commonly encountered. Fgure 5 shows a summary of these data types. Other types of data are possble such as random censorng. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 81 Emprcal Relablty Analyss Usng Lfe Data Left Lfe Data Censored Complete Rght Other Type I Type II Other Types Fgure 5. Types of Lfe Data
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 8 Emprcal Relablty Analyss Usng Lfe Data Example : Data of Dstnct Falures In ths example, the followng complete sample of 19 tmes to falure for a structural component gven n years to falure s provded for llustraton purposes: 6, 7, 8, 9, 30, 31, 3, 33, 34, 35, 36, 37, 38, 39, 40, 4, 43, 50, 56 CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 83 Emprcal Relablty Analyss Usng Lfe Data Example 3: Rght Censored Data In ths example, tests of equpment are used for demonstraton purposes to produce observatons n the form of lfe data as gven n Table 1. The data n the table provde an example of Type I censored data (the sample sze s 1), wth tme to censorng equal to 51 years. If the data collecton was assumed to termnate just after the 8 th falure, the data would represent a sample of Type II rght
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 84 Emprcal Relablty Analyss Usng Lfe Data Example 3 (cont d): Rght Censored Data censored data wth the same sample sze of 1. The respectve data are gven n Table. Table 1. Example of Type I Rght Censored Data (n Years) for Equpment Tme Order Number Tme (Years) TTF or TTC 1 7 TTF 14 TTF 3 15 TTF 18 TTF 31 TTF 37 TTF 40 TTF TTF = tme to falure, and TTC = tme to censorng 4 5 6 7 8 46 TTF 9 51 TTC 10 51 TTC 11 51 TTC 1 51 TTC CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 85 Emprcal Relablty Analyss Usng Lfe Data Example 3 (cont d): Rght Censored Data Table. Example of Type II Rght Censored Data (n Years) for Equpment Tme Order Number 1 3 4 5 6 7 8 9 10 11 1 Tme (Years) 7 14 15 18 31 37 40 46 46 46 46 46 TTF or TTC TTF TTF TTF TTF TTF TTF TTF TTF TTC TTC TTC TTC TTF = tme to falure, and TTC = tme to censorng
CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 86 Emprcal Relablty Analyss Usng Lfe Data Example 4: Random Censorng Table 3 contans the tme to falure data, n whch two falure modes were observed. The data n ths example were generated usng Monte Carlo smulaton. The smulaton process s restarted once a falure occurs accordng to one of the modes at tme t, makng ths tme t for the other mode as a tme to censorng. CHAPTER 4a. RELIABILITY ASSESSMENT Slde No. 87 Emprcal Relablty Analyss Usng Lfe Data Example 4 (cont d): Random Censorng Year TTF (Years) Number of Occurrences of a Gven Falure Mode Strength (FM1) Fatgue (FM) 1984 1 0 0 1985 7 0 1986 3 6 0 1987 4 3 0 1988 5 0 0 1989 6 1 7 1990 7 1 1 1991 8 0 0 199 9 1 36 1993 10 1 47 1994 11 5 61 1995 1 3 33 1996 13 1 74 1997 14 65 1998 15 58 1999 16 44 Table 3. Partal Data Set From 0,000Smulaton Cycles for the Two Falure Modes of Strength and Fatgue for a Structural Component