Non-Normal Responses and Fatigue Damage

N O N - N O R M A L RESPONSES AND F A T I G U E D A M A G E

iNTnoOUCTION

Conventional results for fatigue damage rates and life statistics require II e stress response X{1), lo be a Gaussian random process. These relul s may be cons derabiy unconservative, however, if the Gaussian assump!

.on does not apply (3). In practice, deviations from normality may result rom nonl.near.fy n either structural behavior (e.g., stiffness or^damp-..ig) o.- ,ts interaction w i t h the loading enviror^ment (e.g., wave loTds 0.1 offshore structures). Even if X(0 Is Gaussian during an environ.^en

n n"^: ( ^ - ^ ^ " " l - ^ ' ^-^-.-O'^. -«c.), imcertahit; l X d inlensily eads to no.vnormalily when X(/) is sampled f.om event to event to Z Z f r e ^ " " ^ ' ' ' ^ " ' ' ' y " " ' approximations

Tl ^ •''"rf 'he fatigue damage they

m n m P . J c results require only limited, easily estimated response

moments, e.g. he skewness and kurtosis coefficients, a , and a, (In X ( n - ' i X r n ^ T ^ ' V l ! ' '/"^ "^^ s.anda;dized response fn. H",i I, ~ " ' ^ 1 / ; ^ - ) Analytical esfimates of fatigue damage rates are found to be supported by the simulation results of Ref. 3.

A P P R O X I M A T I N Q NoN-NoftMAL R E S P O I « J 3 E S

A flexible model of non-normal responses is obtained (1) by anplyinR a mo.iotonic function, g, to a standard normal process, U{1) ^ ^ m =^ mi)]

i'lrJlrn^^l'V"^

I"

moments (e.g., a , and a,) are available. ^ To address these problems, it is useful to construct a polynomial

ap-proximation to ^ based on the k n o w n response moments. WWle tWs polynomial may be expressed as a Taylor series, it is more conveniei lo arrange terms into a Hermile series:

(2)

(" = 0,1,...) (3) ^ ( 0 = 2 ''Mc„[U{l)]

in which Hc„{u) = (-1)" exp ( - ] ~ L p ( - - ]

M ' , " " 8 Assl. I'rof., Dept. of Civ. Engrg , Stanford Univ.,' Slanford, Cilif 94305 , n n n l r " " " " " P " " ^^S**' extend Ihe clos ng dale one m ~ l Y f n K''^"^ ASCE Manager of Joumals The Feb Ü r^^ submitted /or review and possible publication on n I M 7 i n n ^ ' I f o T c ^ i i e " / ' ° ' " ' ^ I"""'"' of Engineering Mechanics. Vol l^o 2M47 ' '^^'^ 0733-9399/e5/0010-l!91/$OI.OO. Pape

I'lir oxam|ilc, nc„{ii) = I, HC|(i() = ii, Hpj(i() = - 1, llcj{li) = H ' - 3/i, /•ƒ(.',,((() = ((' - 6//' + 3, clc, This ciioice is nclvantngeous because Ihe terms of lü]. 2 are uiicorrelaleii. As shown in Appeiulix I, this properly sim= plifle.s calibrnlion i.f I'll). 2 (i.e., selection of the coefficients n„) from knowledge of central moments.

Tor cases of mild non-norninlily, il Is convcnienl lo rearrange lïq. 2 ink) standardized form:

X(l) - III. .-,

X>'il) ^ " m + i//.'„[(./(/)] (4)

in which il nssumcd that e,,,, s i i j i r ^ « I (/i > 2), .so that 0(E„e,„) terms are negligible. Shifling subindicus is natural because e„ is directly related lo H „ (Appendix I):

III (5)

For example, applying Eq. 5 w i l l i He, ami lliu as given previously, one finds thai = u.i/6 and €4 = {in - 3)/24. X(/) may then be approximated from U(f) using Eq. 4;

X ( 0 - iiix (»i , Ul - 3 ,

= (.!(() I y\U'(l) - 11 -I- (L/\() - 3U(()1 (6)

f» 24

As expecled, the normal model is recovered if a , - 0 and a , - 3. More refined approximations may follow from liq. 4 by matching central mo-ments of higher orders; however, liq. 6 appears lo provide relatively accurate models of mildly nonlinear (and thus non-normal) belinvior.

The random process model in Eq. 6 is somewhat analogous to a Cor-nish-Fi.sher expansion for fracliles of a random variable (2). The same coefficients, e „ , also arise in Edgeworlh series for related deterministic functions, e.g., the density anil cumulative distribution functions of X(0. Note, however, that the use of lhe.se coefficients is fundamenlally dif-ferent here. In fact, exact analysis of Liq. 6 often provides more accurate estimates of probability dcnsilies and ci'ossing rales than the conven-tional Edgeworth results. Applications of Ihis model lo extremes w i l l be presenled in a follow-up paper.

F A T I G U E A N A L Y S I S

Assume that a cycle of stress w i l h range /{. causes Ihc fatigue damage, 0(1), lo increase by AD, = r«l' (e.g.. Miner's law). If X(/) is narrow-band and satisfies Iiij. I, the mean damage accumulation rate is approximately (-1)

t"(0('))

^in ^ —j— = O'MxiS) - ,V(-S)|") (7)

in which u„ = (rx/27r(rx, and S is distributed as peaks of U((), i.e., Ray-leigh distributed with £[S'l = 2. Note that t\r, governs fatigue damage and life statistics, and retains relevance under many nonlinear damage accumulation laws, e.g., the Paris law (3-5).

P , t r r . ' n / 7 ' L'^Suerre senes based on an exponential process. iMjs. 6 and 7 can be combined to yield

- 3)

24 (8)

e x n p r t 7 ' ' " ™ " ^ - " ; . r f ^ ' - ^i ^ave been neglected. As mighi be expected, skewness (i.e., nonzero a,) has no effect on stress ranges The siónses Ü 4T T . " " " ' ^ " ^ " " ^ ^ ••'^ "^^ ^ • ^ - 8 ^ ^'-"^ u n d e ï Z u s s n responses (3,4) The remaining term in square brackets is hence n m r rection factor that reflects the effects of non-normality

I lie behavior of this factor, denoted y is s b n w n i n 1 1. • dieted that ,he Gaussian resuU will be u L n s S v - X f ' „ ? . 3 nd I h ^ ; Hie effec of non-normality will increase w i t h both a, and the st ess- aw exponent, f. 1 he figure also shows the average behavior o m> ™ responses w i t h various bandwidths, obtained from s i Z i i \^^r '

lowcounle,l damage (3). I3y comparison, Eq. 8 appears . é fairly -curAte and mildly conservative. If the response bandwidth is k n o w n

K u r t o s i s , 0 O s c i l l a t o r « B a n d p a s s " M a r k o v " G a u s s i a n y = l - C , ( t . „ f l , ) - ^ . "•6 r , l.B Z : ^ ^ ' - 01 BendwIdlH on M e .

ir.l

m

I

illis coiisei viVtisMi cnn be iccliiccd by combining Eq. 8 w i l l i n second cor-rection fnclor (cr. Fig. 2 ) , wliich accounts for the reduction in damage late due to bantlwidth effects (4,5).

C O N C L U S I O N S

1. A response with arhitrar)' statistics (e.g., specified sicewness and kurtosis coefficients) can be formed through functional transformation of a normal process. This transformation is convenienlly defined through a I lerniite series expansion.

2. The non-normal model proposed herein leads to a correction factor for Ihe mean damage rale that is roughly linear in the kurtosis cocffl= cienl. This result is found lo be supported by simulations of rainflow-counled damage rales of various non-normal responses. Note that this concction (Fig. ]) generally dominates the Inore widely>sludled effect of response bandwidth (Fig, 2) on the damage rale.

3. Conventional rcsulis (or fatigue damage and life statistics become increasingly unconservative ns either the kurtosis, a , , or stress-law ex-ponent, increases. The effect of non-normality is thus le.ss w h e n the fatigue analysis is based on a fracture mechanics approach (in w h i c h typicnily 2 s h s 4), as opposed to a simple S-N curve (which typically involves somewhat larger values o f l ; ) .

A C K N O W L E D G M E N T S

This materinfis based' upon work supported by the National Science Foundntion under Grant No. CfïG-8410845.

A P P E N D I X I . — C A U o n A T i N G T H E H E R M I T E S E R I E S FHOM M O M E N T S

To Calibrate the standardized response model in Éq. 4, Iho coefficients^ e„ should be expressed in terms of central moments, a„ = EjXnl (Ihe argument / is omitted for brevity). Such nn expression follows by eval-uating //Pi(Xn) from Eq. 4;

Hc,(X„) = Hc, il t- |Wc„(U)

(9) The lalter result follows from a Taylor expansion, noting that Hi;l{u) -mci_,(") for^r > I. Because E\llc„(U)lk,„{U)\ = »i!8,„„, the expectation of Eq. 9 involves only the term in which n ^ k - \; solving for from this result lends lo Eq. 5.

A P P E N D I X M r — R E F E R E N C E S

1, C r i g o r i u , M . , "Cro.ssliigs of N o n - G i i u s s i n n T r a i i . i l a l i ó i i f r O c e s s e s , " loll mnt of Eiit^iiiceriiifi Mcclmincs, A S C E , Vof. 110, N o . <), A p r . , 1984, p p . 6 1 0 - 6 2 0 . 2. Joliii.son, N . L . , and K o l z , S . , Con/iiiiwii.'; Uiiimrinle Dhlrilmlions-^l, J o l u i W i l e y

& S o n s , I n c . , N e w Y o r k , N . Y . , 1970.

LINEAR STRESS ANALYSIS OF TOROSPHÈRICAL H E A D Dy Phillip L. Gould,' F. A S C n , Jhun-Sow L i n , ' S, M, A S C E

and J. Michael Rottér' INTRODUCTION

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