General probability weighted moments for the three-parameter Weibull Distribution and their application in S-N curves modelling

Transcription

1 General probability weighted moments for the three-parameter Weibull Distribution and their appliation in S-N urves modelling Paul Dario Toasa Caiza a,, Thomas Ummenhofer a a KIT Stahl- und Leihtbau, Versuhsanstalt für Stahl, Holz und Steine, Karlsruher Institut für Tehnologie (KIT) Karlsruhe, Germany Abstrat Modelling the statistial fatigue behavior, based on fatigue test results is still a hallenge for researhers and engineers. The model proposed by Castillo et al. is one alternative to desribe this phenomenon, whih is based on a threeparameter Weibull Distribution whose parameters should be estimated. There are several methods to determine these parameters, however there is no onsensus about whih is the most appropriate. In this artile, a general formulation of the Probability Weighted Moments of the Weibull Distribution is presented in order to estimate the parameters mentioned above. Finally, an appliation with experimental data from onrete speimens and simulated data is presented. Keywords: Cyle loading, S-N urves, Weibull, Probability weighted moments, Estimation. Introdution The presene of fatigue failures in metalli strutures is a ommon tehnial problem, and the modelling of the fatigue life of these strutures is still a researh hallenge in the materials siene sine the 9th entury []. At that time, August Wöhler reognized, that applying a single load, whih is muh lower than the stati strength of a struture, does not damage it, but if this load is applied several times, it ould indue a omplete failure of the struture. The failure begins Corresponding author. Tel.: ; Fax: address: paul.toasa@it.edu (Paul Dario Toasa Caiza ) German engineer who made a systemati researh of fatigue Preprint submitted to International Journal of Fatigue July 25, 2

2 Model Wöhler urves Basquin (9) logn = A Blog σ; σ σ Stomeyer (94) logn = A Blog( σ σ ) Bastenaire (972) N = σ E A exp[ C( σ E)] B Pasual & Meeer (999) logn = A Blog( σ σ ) Kohout & Věhet (2) log( σ σ ) = log ( ) b N+N N+N 2 Table : Common models to represent the S-N urves. with the ourrene and growth of miro-ras whih are basis for maro ras leading to ollapse. With the stress based approah, several models have been proposed to represent the S-N urves [2], [3], [4], [5], [6], [7], [8], whih play a ruial role for the strutural design; some of them are listened in table. Most of them only onsider physial arguments and empiri data, and unfortunately, they only represent an elementary geometri approah whih offers a limited judgement of the experimental results. Moreover, from the statistial point of view, some models do not propose a ummulative distribution funtion, so that, they are not suitable to extrapolate the results into the high yle fatigue region [9], [], []. In other words, it is not possible to predit with a probability p the fatigue life of a struture under a signifiant lower stress value as the tested experiments. Based on a Weibull Distribution, Castillo et al. [2] proposed a probabilisti methodology for prediting the number of load yles leading to failure of strutural details. In ontrary the traditional methods, this proposal emphasizes the stohasti nature of fatigue by onsidering both the stress range and the lifetime (number of load yles) as random variables [3], [4], ensures a dimensional onsisteny [5] and onsiders the influene the run-outs obtained during the experiments. The methodology mentioned above demands the estimation of the three parameters of the Weibull Distribution. In this artile, a general formulation of the Probability Weighted Moments (PWM) of the Weibull Distribution is presented and applied to estimate these parameters. The appliation is based on (a) Experimental data from onrete speimens. (b) Data from omputer simulation. 2

3 2. Weibull model The three-parameter Weibull Distribution denoted by W(a, b, ) is a member of the family of extreme value distributions. The umulative distribution funtion (CDF) (also alled life distribution or failure distribution) of a random variable x whih follows a W(a,b,) [6] is given by where F(x a,b,) = exp [ ( x a a R : Loation parameter (minimum life) b > : Sale parameter (harateristi life) > : Shape parameter (slope of F(x a, b, )) b ) ], x a () In the ase of fatigue modelling, Castillo et al. onsider that the onditional CDF of the random variables lifetime and stress range are not independent and must satisfy a ompatibility ondition whih leads to a funtional equation [3]. The solution of this funtional equation leads to the following probabilisti fatigue model for a onstant stress range, whih is based on a Weibull Distribution [4], where the random variable is x = (logn B)(log σ C). { [ ] (logn B)(log σ C) a } Q(N, σ) = exp (2) b where σ : stress range during the test N : number of load yles up to failure during the test B : threshold value of lifetime N C : endurane limit for σ The model given by Eq.(2) has two geometrial parameters B and C, and three Weibull parameters a, b and, whih should be estimated. The estimation of these parameters is done in two steps. First, the geometrial parameters are determined and seond the Weibull parameters. 3

4 3. Parameter estimation Consider that n experimental data points from stress ranges and load yles are given as follows N i = N,N 2,...N n σ i = σ, σ 2,... σ n The geometrial parameters B, C and the mean 2 µ of W(a,b,) are estimated by solving the following non-linear optimization problem [2]. min B,C,µ R n i= ( logn i B ) µ 2 (3) log σ i C Within this paper, the PWMs introdued by Greenwood at all [7] and applied to the Extreme Value Distribution [8] are onsidered. The PWMs of a random variable X with CDF F are the quantities M p,r,s = [ x(f) ] pf r ( F) s df, (4) where p,r,s N. In the subsequent subsetions, the dedution and properties of M,,s, M,r,, M,r,s and M p,r,s from a Weibull Distribution are presented. 3.. Moments M,,s The inverse funtion of Eq.() is given by x(f) = a + b [ log( F) ], (5) replaing p = and r = in Eq.(4) gives the following PWMs. M,,s = a ( F) s df + b [ log( F) ] ( F) s df. (6) 2 Theoretially the mean of W(a,b,) is given by µ = a + bγ( + /) 4

5 The first integral of Eq.(6) is equal to a s+. Substituting u = log( F) in the seond integral of Eq.(6) leads to substituting x = (s + )u in Eq.(7) leads to b (s + ) + b u e (s+)u du, (7) x e x b dx = (s + ) + where, Γ(z) is de Gamma funtion defined by Γ(z) = ( Γ + ), (8) x z e x dx, z >. (9) Therefore, the PWM M,,s for the three-parameters Weibull Distribution are given by M,,s = a ( s + + b Γ + ), >. () (s + ) + Now, the first three PWMs are onsidered in order to estimate the Weibull parameters a, b and. Denoting M s = M,,s, s =,, 2 and Γ = Γ( + ) in Eq.() the following system of equations is obtained M = a + bγ () M = a 2 + b Γ (2) 2 + M 2 = a 3 + b 3 + Γ. (3) Combining Eq.() and (2) gives 5

6 ( ) 2M M = bγ 2. (4) Combining Eq.() and (3) gives ( ) 3M 2 M = bγ 3. (5) Then dividing Eq.(5) by Eq.(4), leads to an equation for the parameter. 3M 2 M = 3 2M M 2 whih should be solved by numerial methods. From Eq.(4), the value of the parameter b is (6) From Eq.(), the value of the parameter a is b = 2M M (2 )Γ. (7) a = M bγ. (8) It is neessary to now the value of the PWMs M, M, M 2 to solve the equations (6)-(8). For this reason, their estimators whih depend on the ordered experimental data are used. Let x < x 2 <... < x n be the order sample of the experimental data points. Then the estimators of the first three PWMs M s [6] are given by M 2 = M = M = n n i= n n(n ) i= n 2 n(n )(n 2) i= x i (9) (n i)x i (2) (n i)(n i )x i. (2) By substituting the estimators given by equations (9)-(2) in the equations (6)-(8) the value of the Weibull parameters an be determined. 6

7 3.2. Moments M,r, Replaing the Eq.(5) into Eq.(4) for p = and s = gives the following PWMs. M,r, = a F r df + b The first integral of Eq.(22) is equal to r+ a. [ log( F) ] F r df. (22) Substituting u = log( F) in the seond integral of Eq.(22) gives b u ( e u ) r e u du (23) now, applying the binomial theorem given by ( ) n ( a) n = a ( ), (24) it an be proved that ( e u ) r = n = r = Then, the integral of Eq.(23) beomes r b =( ) ( ) r u e (+)u du = b =( ) ( r ( ) r e u ( ). (25) r ) Substituting x = ( + )u in the integral of Eq.(26) gives ( + ) + x e x dx = ( + ) + u e (+)u du. (26) ( Γ + ). (27) Thus, the integral of Eq.(23) beomes ( I 4 = bγ + ) r ( ) ( r). (28) = ( + ) + 7

8 Finally, the PWM M,r, for the three-parameters Weibull Distribution are given by M,r, = a ( r + + bγ + ) r ( ) ( r), >. (29) = ( + ) + Similar to the previous ase, onsider the following first three PWMs. Denoting M r = M,r,, r =,, 2 and Γ = Γ( + ) in Eq.(29) the following system of equations is obtained M = a + bγ (3) M = a ( 2 + ) bγ 2 + (3) M 2 = a ( ) bγ (32) Combining Eq.(3) and (3) leads to ( 2M M = Combining Eq.(3) and (32) gives 3M 2 M = 2 ( ) bγ. (33) ) + 3 bγ. (34) By dividing Eq.(34) by Eq.(33) an equation for the parameter is obtained 3M 2 M = , (35) 2M M 2 whih should be solved by numerial methods. From Eq.(33) the value of the parameter b is obtained b = 2M M ( 2 )Γ. (36) From Eq.(3) the value of the parameter a is given by 8

9 a = M bγ. (37) Finally, it is neessary again to now the values of the PWMs M, M, M 2 to solve the equations (35)-(37). Thus, their estimators whih depend on the ordered experimental data points are used. Let x < x 2 <... < x n be the order sample of the experimental data points. Then, the estimators of the first three PWMs M r [9], [2] are given by M 2 = M = M = n n(n ) n(n )(n 2) n i= n i= n i= x i (38) (i )x i (39) (i )(i 2)x i. (4) By substituting the estimators given by equations (38)-(4) in the equations (35)-(37) the value of the Weibull parameters an be determined Moments M,r,s Replaing Eq.(5) in Eq.(4) for p = gives the following quasi general equation of the PWMs. M,r,s = a F r ( F) s df + b [ log( F) ] F r ( F) s df. (4) Applying the binomial theorem given by Eq.(24), the first integral of Eq.(4) beomes a F r s = ( ) s F ( ) df = a s = ( ) s ( ) + r +. (42) On the seond integral of Eq.(4) maing the substitution u = log( F) and applying also the binomial theorem results in 9

10 b u ( e u ) r e (s+)u du = b = b r = u e (s+)u ( r r = )( ) ( ) r e u ( ) u e (+s+)u du. Substituting x = ( + s + )u in the integral of Eq.(43) gives ( + s + ) + then, Eq.(43) beomes x e x dx = ( + s + ) + ( bγ + ) r ( ) r ( ) = ( + s + ) + (43) ( Γ + ), (44) (45) Therefore, the PWM M,r,s for the three-parameters Weibull Distribution are obtained s ( ) s ( ) M,r,s = a + r + = ( + bγ + ) r = ( r ) ( ) ( + s + ) + > Moments M p,r,s Replaing Eq.(5) into Eq.(4) gives a general equation of the PWMs. M p,r,s = (46) { a + b [ log( F) ] } p F r ( F) s df. (47) Maing the substitution u = log( F) into Eq.(47) results in M p,r,s = [ a + bu ] p( e u ) r e (s+)u du. (48)

11 Applying the binomial theorem given by Eq.(24) in the first term of the integral of Eq.(48) leads to M p,r,s = p i= ( p )a p i b i u i ( e u ) r e (s+)u du. (49) i Then, applying again the binomial theorem, the integral of Eq.(49) beomes = = u i e (s+)u r = r = r = ( ) r e u ( ) ( r )( ) u i e (+s+)u du ( ) r ( ) Γ ( + i ). (s + + ) + i (5) Finally, the general PWM M p,r,s for the three-parameters Weibull Distribution an be alulated from p ( p M p,r,s = )a p i b i r ( ) r ( ) Γ ( + i ), >. (5) i (s + + ) + i i= 4. Appliation and results = In this setion the method of the PWMs is applied in order to estimate the three parameters of a Weibull Distribution. The data arise from two different soures. a) The experimental data reported by Holmen (979). b) Simulated data whih were generated with Matlab. In every ase, a omparison between the results is presented. 4.. Experimental data from Holmen These 75 data are desribed by Holmen [2]. A sample of 5 onrete speimens was tested at eah of the five levels σ i = {.675,.75,.825,.9,.95}. Here σ i is the ratio S max /S f where S max is the maximum applied stress and S f is the stress leading to stati failure. The lifetimes are measured as the number of yles to failure divided by. The data are shown in Table 2.

12 The values of the geometrial parameters are already given as B = and C =.67, whih are taen from [2]. The S-N urves obtained from Castillo-Hadi method [9] in Figure and from the PWM method in Figure 2 have a similar geometry, however the S-N perentiles in Figure 2 are more narrow than the perentiles in Figure. σ Holmen Data N Table 2: Holmen data. Fatigue results with onrete speimen Simulated data In this subsetion, 75 data points of a Weibull Distribution W(8,.5,3) obtained through a Matlab numerial simulation are onsidered. In order to ompare easily the results of the PWM method, the simulation of the data too into onsideration the same values of the geometrial parameters given by B = and C =.67 from Holmen s data [2]. The simulated data are shown in Table 3. In this ase the S-N urve based on the PWM method has a quite similar geometry to the simulated urve. Partiularly, the estimation of the loation parameter a and sale parameter b do not differ too muh from the theoretial values. 2

13 ..5 Weibull Parameters Castillo Hadi Method a = b = = Log( σ) C Log(N) B Figure : S-N urves from Holmen data from Table 2 based on the Castillo-Hadi estimators...5 Weibull Parameters PWM Method a = b =.699 = Log( σ) C Log(N) B Figure 2: S-N urves from Holmen from Table 2 based on the PWMs estimators. 3

14 σ Simulated Data N Table 3: Simulated data from a Weibull distribution W(8,.5,3)..5 Weibull Parameters Theoretial Values a = 8 b =.5 = Log( σ) C Log(N) B Figure 3: S-N urves from the simulated data from Table 3. 4

15 ..5 Weibull Parameters PWM Method a = 8.88 b =.4898 = Log( σ) C Log(N) B Figure 4: S-N urves from the simulated data from Table 3 based on the PWMs estimators. 5. Conlusions and future wor The PWM method offers a good alternative to estimate the three parameters of the Weibull Distribution. The obtained estimations do not differ signifiantly from the values of the simulated distribution, see Figures 3, 4. The orresponding S-N urves of both appliations are quite similar in the high yle region. These urves desribe properly the fatigue s properties suh as the existene of endurane limit σ. The endurane limit and the extrapolation of the S-N urves in the high yle region are estimated in a better way than in the traditional linear regression model [6] whih is urrently used to represent the S-N urves and the fatigue data respetively. This linear regression do not allow to onsider the existene of the endurane limit. In other words, the asymptoti behaviour obtained from the low stress values is negleted. The Weibull model produes good results beause of its statistial approah, whih an be extrapolated if the experimental data points are abundant and ome from the high yles region as it is seen in Figures -4. One of the problems regarding fatigue experiments on nothed steel details is that the speimens and their tests are very expensive and tae a very long time. This situation beomes more ompliated if we want to apply the Weibull model, 5

16 beause it demands more data partiularly from the high yle region. In the future in order to overome this problem, the following situation is assumed. Several run-outs are obtained from speimens whih resisted the first fatigue test after a predetermined maximum number of applied load yles. Then, these resistant speimens are re-tested again with a higher stress value. If the speimen breas, an additional value of failure is obtained. Otherwise, the speimen an be re-tested one again with a higher stress value. Therefore, it would be interesting to establish if it is possible, to use the information obtained from the subsequent experiments in order to improve the estimation of the Weibull fatigue model. For this purpose it is mandatory to define two aspets: (a) A suitable definition of a damage aumulation funtion whih onsiders (where pertinent): the influene of the oaxing effet, material properties, frequeny of the test mahine, et. This funtion will allow taing into aount the aumulated damage on a run-out for the subsequent tests. (b) A ensoring sample model and the method to estimate the Weibull parameters based on it. 6. Anowledgement The authors are grateful to Prof. Enrique Castillo of the Department of Applied Mathematis and Computational Sienes at the University of Cantabria for his assistane in reviewing this artile. Referenes [] W. Shütz, Zur Geshihte der Shwingfestigeit, Materialwissenshaft und Werstofftehni 24 (6) (993) [2] F. Bastenaire, New method for the statistial evaluation of onstant stress amplitude fatigue-test results, Probabilisti Aspets of Fatigue STP (5) (972) [3] G. Marquis, M. Huther, A. Galtier, Guidane for the appliation of the best pratie guide on statistial analysis of fatigue data, IIW-WG [4] F. Pasual, W. Meeer, Estimating fatigue urves with the random fatiguelimit model, Tehnometris 4 (4) (999) [5] J. Shijve, Fatigue of Strutures and Materials, Springer Verlag, 29. 6

17 [6] Sedlae, Hobbaher, Nussbaumer, Stötzel, st draft of the baground doument pren , Euroode 3: Design of steel strutures. [7] C. Stromeyer, The determination of fatigue limits under alternating stress onditions, Proeedings of the Royal Soiety of London 9 (62) (94) [8] J. Kohout, S. Věhet, A new funtion for fatigue urves haraterization and its multiple merits, International Journal of Fatigue 23 (2) (2) [9] T. Lassen, P. Daris, N. Reho, Fatigue behavior of welded joints part - statistial methods for fatigue life predition, Welding Journal (25) [] T. Lassen, N. Reho, Proposal for a more aurate physially based S-N urve for welded steel joints, International Journal of Fatigue 3 () (29) [] T. Lassen, N. Reho, Life predition of welded steel joints - auray and shape of the S-N urves under onstant and variable amplitude loading, Fatigue Design 29. [2] E. Castillo, A. Fernández-Canteli, A Unified Statistial Methodology for Modeling Fatigue Damage, Springer Verlag, 29. [3] E. Castillo, V. Esslinger, A. Fernández-Canteli, Statistial model for fatigue analysis of wires, strands and ables, in: IABSE Proeedings, Vol. 9, IABSE, IABSE Periodia, 985. [4] E. Castillo, J. Galambos, Lifetime regression models based on a funtional equation of physial nature, Journal of Applied Probability 24 () (987) [5] E. Buingham, On physially similar systems; illustrations of the use of dimensional equations, Phys. Rev 4 (94) [6] H. Rinne, The Weibull Distribution, Cr Pr In, 28. [7] J. Greenwood, J. Landwehr, N. Matalas, J. Wallis, Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resoures Researh 5 (5) (979)

18 [8] J. Hosing, J. Wallis, E. Wood, Estimation of the generalized extreme-value distribution by the method of probability-weighted moments, Tehnometris 27 (3) (985) [9] E. Castillo, A. S. Hadi, Parameter und quantile estimation for the generalized extreme-value distribution, Environmetris 5 (994) [2] D. J. Dupuis, Parameter and quantile estimation for the generalized extremevalue distribution: a seond loo, Environmetris (999) [2] J. Holmen, Fatigue of onrete by onstant and variable amplitude loading, Ph.D. thesis, University of Trondheim (979). 8