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Journal of the Brazilian Society of Mechanical
Sciences
Print versionISSN0100-7386
J. Braz. Soc. Mech. Sci. vol.23 no.4 Rio de Janeiro 2001
http://dx.doi.org/10.1590/S0100-73862001000400014
A N e w M ode l t o D e t e r m in e t h e
D ispe r sion of Fa t igu e D a m a ge
Ev a lu a t ion s
J. L. A. Fe r r e ir a
Department of Mechanical Engineering – UnB 70910-900 Bras•lia. Brazil
[email protected] J. L. F. Fr e ir e
Department of Mechanical Engineering – PUC-Rio 22453-900 Rio de Janeiro, RJ. Brazil
Reliable pr edict ions of r em aining lives of civil or m echanical st r uct ur es subj ect ed t o fat igue dam age are ver y difficult t o be m ade. I n gener al, fat igue dam age is
ext r em ely sensit ive t o t he r andom var iat ions of m at er ial m echanical pr oper t ies, envir onm ent and loading. These var iat ions m ay induce lar ge disper sions w hen t he st r uct ur al fat igue life has t o be pr edict ed. Wir sching ( 1970) m ent ions disper sions of t he or der of 30 t o 70 % of t he m ean calculat ed life. The pr esent ed paper int r oduces a m odel t o est im at e t he fat igue dam age disper sion based on know n st at ist ical dist r ibut ions of t he fat igue par am et er s ( m at er ial pr oper t ies and loading) . The m odel is developed by expanding int o Taylor ser ies t he set of equat ions t hat descr ibe fat igue dam age for cr ack init iat ion.
Ke y w or ds: Fat igue, r eliabilit y, life pr edict ion, r andom loading
Introduction
w her e di is t he calculat ed dam age aft er a t ot al of ni cycles of sam e st r ess (Ds i = st r ess r ange, s mi = m ean st r ess) and Ni is t he t ot al life cor r esponding t o each st r ess pair as it w as t he only
one solicit ing t he point in consider at ion of t he st r uct ur e. I n expr ession ( 1) nt is t he t ot al
num ber of cycles in t he loading hist or y and fi is t he r elat ive fr equency of a cycle of som e
st r ess ( = ni/ nt ) .
When ni is a r andom var iable and t he m at er ial has st r engt h var iat ions such t hat Ni is also a
r andom var iable, it is im por t ant t o det er m ine t he st at ist ical behavior of t he dam age ( m ean, var iance, m ax im um dam age, cum ulat ive dist r ibut ion, et c) . Defining t he uncer t aint ies of di, ni , and Ni r espect ively as d di , d ni , and d Ni , and assum ing t hat t hese ar e r easonably sm all, d di can be calculat ed as funct ion of d ni and d Ni using t he linear t er m s of a Taylor 's expansion;
and t he dam age uncer t aint y w ill be w r it t en as
The next sect ions w ill deal w it h t he calculat ion of d di , d ni , and d Ni in a m or e explicit and
r igor ous w ay .
Nom en clat u r e
= aver age num ber of cycles count ed for t he K loading hist or ies
= m ean r elat ive fr equency of t he st r ess r ange based on field dat a
1 = colum n vect or
2N = num ber of r ever sion b = fat igue st r engt h exponent c = fat igue duct ilit y exponent d = dam age incr em ent D = t ot al dam age E = Young m odulus
f = r elat ive fr equency of a cycle of som e st r ess r ange
F = covar iance m at r ix
f = dat a m at r ix of t he fr equencies associat ed t o st r ess hist ogr am s
I = ident it y m at r ix
K = num ber of hist ogr am s K' = cyclic st r engt h coefficient
Kf = fat igue st r ess concent r at ion fact or
n = num ber of cycles applied associat ed t o t he sam e st r ess r ange
N= t ot al life associat ed t o a const ant st r ess r ange
n' = cyclic st r engt h exponent
X0= vect or of t he aver age values of t he fat igue and m echanical pr oper t ies, of t he st r ess r ange, and t he fat igue st r ess concent r at ion fact or
Gr e e k Sy m bols
V = vect or of t he m ean values of t he m echanical pr oper t ies v = gener ic var iable
D S = nom inal st r ess r ange
De = st r ain r ange
Ds = st r ess r ange
G 0
= vect or of t he aver age value of t he fat igue pr oper t ies and t he st r ain r ange
c 0
= vect or of t he m ean value of t he m echanical pr oper t ies and t he st r ess r ange
d d = uncer t aint y of t he dam age incr em ent
d D = uncer t aint y of t he t ot al dam age
d n = uncer t aint y of t he num ber of cycles
d N = uncer t aint y of t he t ot al life associat ed t o a st r ess r ange
e 'f = fat igue duct ilit y coefficient s m = m ean st r ess
s ' = fat igue st r engt h coefficient
Su bscr ipt s
f = r elat ive t o fat igue pr oper t ies
t = r elat ive t o num ber t ot al of cycles
ut = r elat ive t o t ensile st r engt h
y = r elat ive t o yield st r engt h P = r elat ive t o num ber of class of t he hist ogr am
The fat igue analysis of a st r uct ur al point r equir es t he char act er izat ion of t w o gener al aspect s. The first aspect involves load descr ipt ion in t er m s of a t ypical hist or y and it s possible
var iat ion. The second aspect involves t he est ablishm ent of t he m at er ial fat igue const it ut ive equat ions and it s var iabilit y in t er m s of t he m at er ial fat igue par am et er s.
Dam age Calculat ion - Loadin g Var iabilit y
Each st r ess hist or y is unique and is dependent on: (i) m inut e var iat ions of t he geom et r y of t he st r uct ur al com ponent ; (ii) t he specific var iat ions of t he loading t r aj ect or y or ot her loading par am et er s. For exam ple, it can be obser ved t hat a st r uct ur al par t of a aut om ot ive vehicle suffer s st r ess var iat ion due t o: t he inst ant aneous dead w eight of a vehicle, dr iving speed, envir onm ent t em per at ur e, skills of dr iver , et c. Ther efor e, r ecor ded t apes of differ ent r uns of t he sam e t ype of vehicle in t he sam e r oad pr esent a r andom behavior . This behavior w ill be br oadened if ot her r oads w it h differ ent per cent ages of usage are incor por at ed in a big set of hist or ies. I n t he appr oach suggest ed in t his paper , each loading hist or y is acquir ed and com pact ed in t er m s of a hist ogr am t hr ough a st r ess or st r ain- cycle count ing- t echnique such as t he " r ain- flow " or " pagoda" m et hod ( Mat suishi, 1968) . Aft er t he applicat ion of t he st r ess-cycle count ing- t echnique t o K loading hist or ies, t he r elat ive fr equency of t he st r ess r anges, f (D Si) , can be est im at ed fr om t hese hist or ies t hr ough equat ions ( 4.1) and ( 4.2) below .
w her e fij is t he m ean r elat ive fr equency of occur r ences of D Si in t he K hist or ies and nj(D Si) is
equal t o t he num ber of occur r ences of t he st ress- r ange D Si in t he jt h loading hist or y; nt is t he
t ot al num ber of cycles w hich w er e count ed in t he jt h loading hist or y.
Equat ions 5.1 t o 5.3 below quant ify t he var iabilit y of t he t ypical hist or y t hr ough t he var iance of t he fr equency of occur r ence of t he st r ess- r anges. The est im at es of var iance can be
calculat ed t hr ough t he follow ing expr essions:
Classical Est im at or :
w her e i is t he m ean r elat ive fr equency of t he st r ess- r ange in t he it h class of t he hist ogr am
and K is t he num ber of t he loading hist or ies acquir ed.
Analyt ical Est im at or , Bendat ( 1983) :
w her e t is t he aver age of t he t ot al num ber of cycles count ed for t he K hist or ies, .
w her e f is t he (K x P) dat a m at r ix of t he fr equencies for each class (i = 1, ..., P) in t he K hist ogr am s, 1 is a colum n vect or of K ones, and I is t he (K x K) ident it y m at r ix.
A t ypical hist ogr am of t he m ean r elat ive fr equency of st r ess- r ange and of t he est im at es of st andar d deviat ions calculat ed t hr ough t he equat ions 5.1 and 5.2 are show n in figur e 1.
Dam age Calculat ion - Mat er ial Var iabilit y
Once t he disper sion of t he loading par am et er s, d ni, w ill be evaluat ed by VAR[ni] or VAR[fi] ,
t he calculat ion of t he disper sion of t he accum ulat ed dam age, d D, also r equir es t he
det er m inat ion of t he disper sion associat ed w it h fat igue st r engt h, d N. This sect ion show s how t he disper sions associat ed w it h t he m at er ial pr oper t ies influence t he fat igue life N. Calculat ion of VAR[Ni] w ill allow t he det er m inat ion of d D. The det er m inat ion of VAR[Ni] w ill be based on
Coffin- Manson's exNfat igue expr essions descr ibed by equat ions ( 6) , ( Fat igue Design Handbook, 1988) .
w her e Kf is t he fat igue st r ess concent r at ion fact or , D S and Ds ar e r espect ively t he nom inal
and m ax im um st r ess- r ange, De is t he m ax im um st r ain- r ange; E is t he Young Modulus, and K', n', s 'f, b, e 'f, and c ar e m at er ial fat igue pr oper t ies.
few st eps of t he pr oposed developm ent ( Fr eir e and Fer r eir a, 1995, Fer r eir a, 1997) . Consider ing equat ion ( 6.1) it is possible t o det er m ine,
w her e
and X0 is t he vect or of t he aver age values of t he m echanical pr oper t ies of t he m at er ial, of t he nom inal st r ess- r anges, and t he fat igue st r ess concent r at ion fact or .
Fr om equat ion ( 6.2) ,
w her e c = c [xq] = [ E, K', n', Ds ]T, c 0 is t he vect or of t he aver age values of t he m echanical
pr oper t ies of t he m at er ial and t he st ress- r ange, V = [ E, K', n']T, and COV[Ds , V l] is t he covar iance bet w een Ds and t he m at er ial pr oper t ies ut ilized in t he Equat ion ( 6.2) . This st at ist ical m easur e can be est im at e t hr ough of t he expr ession:
Applying t he sam e t echnique t o equat ion ( 6.3) , but using a second or der expansion, it is possible t o obt ain : ( Fr eir e and Fer r eir a, 1995; Fer r eir a, 1997) ;
w her e
w her e , and
Dam age Un cer t ain t y
The t ypical dam age value is calculat ed fr om t he m ean r elat ive fr equencies of t he K loading hist or ies ( i, eq. ( 4.1) ) and t he t ypical life values, Ni, det er m ined by applying t he aver age
values of loading and m at er ial pr oper t ies t o equat ion 6.3.
w her e nt is t he t ot al num ber of cycles per block.
Using first and second or der expansion, t he m ean dam age can be calculat ed, r espect ively, t hr ough expr essions ( 8.2) and ( 8.3) .
w her e i and i and VAR[Ni] w er e defined r espect ively by expr essions 4.1, 7.5 and 7.6.
Ex panding t he equat ion ( 1) , linear ly using a Taylor 's ser ies, t he uncer t aint y of dam age m ay be r epr esent ed by t he var iance of dam age, VAR[D] , given by
w her e VN= [s[N1] ,s[N2] ,...,s[NP] ]T, s[· ] is t he st andar d deviat ion, and Df, DNand F ar e given
Nu m er ical Result s
This sect ion show t he r esult s det er m ined using t he pr oposed m odel for dam age uncer t aint y est im at ion and t heir com par ison w it h t he dir ect applicat ion of t he Mont e Car lo Met hod ( Har r , 1987) t o t he sam e dat a condit ions. I n t his analysis r andom var iat ions w er e consider ed t o be pr esent in t he loading hist or ies and in t he m echanical pr oper t ies of t he m at er ial. Dam age uncer t aint y w as gener at ed by t he com binat ion of t he var iat ions of t he loading hist or y and m at er ial pr oper t ies. The m echanical pr oper t ies of t he m at er ial MANTEN st eel ar e given in t he t able 1.
I t w as assum ed t hat t he m echanical pr oper t ies of t he MANTEN st eel pr esent ed coefficient s of var iat ion of t he or der of 7,5% . I n ot her w or ds, it w as assum ed t hat all m at er ial const ant s had st andar d deviat ions equal t o 7,5% of t heir m ean values. I t w as also assum ed t he pr esence of a not ch w it h a st r ess concent r at ion fact or const ant and equal t o 3.
I n or der t o evaluat e t he pr oposed m odel, 18 differ ent loading hist or ies r epr esent ed by t heir one- sided pow er spect r al densit y ( PSD) w er e used. Each one of t hese 18 PSDs w er e used t o gener at e 400 loading blocks t hr ough Gaussian sim ulat ion ( Fer r eir a and Fr eir e, 1995) , each block cont aining 3,000 ext r em es ( picks and valley) . I n t his w ay, hist ogr am s w it h t he est im at es of t he m ean r elat ive fr equency and t heir r espect ive uncer t aint ies, calculat ed
t hr ough equat ions ( 4.1) and ( 4.2) , w er e based on about 1,200,000 ext r em es for each of t he 18 differ ent hist or ies.
The m at er ial and loading gener at ed above ( st r ess- r ange hist ogr am s and m at er ial's pr oper t ies) w er e used t o calculat e dam age r esult s t hr ough expr essions 8.2 - 8.4. The
est im at es calculat ed fr om t hese equat ions w er e com par ed w it h est im at es gener at ed fr om t he applicat ion of a Mont e Car lo t echnique.
The com par ison bet w een t he est im at es of t he m ean of t he dam age obt ained analyt ically and t hr ough t he Mont e Car lo sim ulat ion ar e pr esent ed in figur e 2. I t is ver ified t hat t he r esult s obt ained t hr ough t he second or der m et hod, eq. ( 8.3) , allow a quit e pr ecise evaluat ion of t his st at ist ic.
The com par ison bet w een t he est im at es of t he m edian values of t he dam age, obt ained by Mont e Car lo sim ulat ion, w it h t he t ypical dam age, calculat ed t hr ough t he equat ions 6.1 - 6.3, show a ver y good cor r elat ion as it can be obser ved in t he figur e 3. This is an int er est ing r esult because it allow s t he evaluat ion of anot her st at ist ic t hat descr ibes t he dam age behavior , in a ver y easy w ay .
The pr edict ions of t he st andar d deviat ion of t he dam age show t hat t he first and second or der m et hods pr esent ed biased est im at es w hen com par ed t o t he r espect ive r esult s obt ained by sim ulat ion of Mont e Car lo, as it can be obser ved in figur e 4.
How ever , as it can be obser ved in figure 5, t he biased behavior of t he dam age disper sion is at t enuat ed w hen t he coefficient of var iat ion of t he dam age is calculat ed using t he analyt ical est im at es of first or der of t he st andar d deviat ion and of t he m ean.
Using t he r esult s of t he figur es 4 and 5 a quit e efficient for m of evaluat ing in an unbiased w ay t he st andar d deviat ion is t o consider t he pr oduct of t he coefficient of var iat ion and t he m ean value of t he dam age, calculat ed r espect ively t hr ough t he equat ions of first and of second or der . A com par ison bet w een t he r esult s obt ained t hr ough t his w ay and t hat calculat ed t hr ough t he Mont e Car lo m et hod is pr esent ed in t he figur e 6.
Con clu sion s
This paper descr ibes a set equat ion t o det er m ine t he basic st at ist ical par am et er s of fat igue dam age evaluat ions. The pr edict ions equat ions ar e based on Palm gr en- Miner 's r ule and t he e -N m et hod. I t allow s for t he com bined use of r andom loading and r andom m at er ial pr oper t ies. The developed m odel w as applied t o 18 dam age exam ples and t he r esult s obt ained have been com par ed sat isfact or ily w it h ot her s det er m ined by st andar d Mont e Car lo pr edict ion t echniques.
Fat igue Design Handbook, 1988, Second Edit ion, Design Handbook Division of t he SAE Fat igue Design and Evaluat ion Technical Com m it t ee, 1988. [ Lin k s]
Fer r eir a, J. L. A. and Fr eir e, J. L. F., 1995 " Sim ulação de Car r egam ent os Com plex os par a Cálculo do Dano Por Fadiga" , Pr oc. Br azilian Congr ess of Mechanical Engineer ing COBEM -CI DI M, Belo Hor izont e, 1995. ( I n Por t uguese) . [ Lin k s]
Fer r eir a, J. L. A., 1997, " Um Modelo Par a a Pr evisão da I ncer t eza do Dano por Fadiga" , Thesis Dept . Eng. Mecânica PUC- Rio, 1997. ( I n Por t uguese) . [ Lin k s]
Fr eir e, J. L. F. and Fer r eir a, J. L. A., 1995, " An Analyt ical Model t o Det er m ine t he Disper sion of Fat igue Dam age Evaluat ions" , Pr oc. St r uct . Mech. in React or Technology, August , 1995, Rio de Janeir o, pp. 11 - 20. [ Lin k s]
Har r , M. E., 1987, " Reliabilit y - based Design in Civil Engineer ing" , McGr aw - Hill I nc., New Yor k, 1987. [ Lin k s]
Kam , J. C. P. and Bir kinshaw , M., 1994, " Reliabilit y - Based Fat igue and Fr act ur e Mechanics Assessm ent Met hodology for Offshor e St r uct ur al Com ponent s" , I nt . J. of Fat igue, Vol. 16, Apr il, 1994, pp. 183 - 192. [ Lin k s]
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Mat shuishi, M. and Endo, T., 1968, " Fat igue of Met als Subj ect ed t o Var ying St r ess" , Japan Societ y of Mechanical Engineer s, Fukuoka, Japan, 1968. [ Lin k s]
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Wir sching, P. H. and Yao, T. P., 1970, " St at ist ical Met hods in St r uct ur al Fat igue" , Jour nal of t he St r uct . Division, ASCE, Vol. 96, No. ST6, June, 1970, pp. 1201 - 1219. [ Lin k s]
Paper or iginally pr esent ed at t he 15t h Br azilian Congr ess of Mechanical Engineer ing ( XV COBEM) , São Paulo, Novem ber 22- 26, 1999.
COBEM Edit or s: R. G. dos Sant os, M. H. Rober t , A. C. Dannw ar t , J. R. B. Cr uz. Associat e Edit or : J. R. F. Ar r uda.
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