Fatigue damage due to non-Gaussian responses

INTRoDUcTIoN

FATIGUE DAMAGE DUE TO NoN-GAussI

RESPONSES By J. Juncher Jensen' TECHNISCHE. UNIVERSITEIT Laboratorium voor ScheepshydromeChaflica Archief Mekelweg 2, 2628 CD DeIft Tel.: O15-786873 - Fax: Ó15 - 781836

This technical note presents a formula for fatigue-damage estimation ap-plicable to weakly non-Gaussian responses The method makes use of a Charher series expansion in cumulants of the joint probability density function of the response signal and its time derivative. Thereby, crossing levels are deter-mined and used for estimating the peak-value distnbution of the response The desired formula is then derived from Miner's law. Two examples il-lustrate the applicability of the formula.

WEAKLY NON-GAUSSIAN RESPONSES

Following Longuet-Higgins (1963, 1964), the joint probability density function p (, ) of a stationary, stochástic variable with zero mean, and its time-derivative , can be wntten as a Charlier senes

eh/2)(ff

_{I i}

p(,) =

i ± - [X30H3(f)H0(f) + 3X21H2(f)H1(f1) 2irocr t. 3! i + 3X12H1(f)H2(f1) + X03H0(f)H3(f)} + - [XH4(f)H0(f) + 4X31H3(f)H,(f1) ± 6X22H2(f)H2(f) + 4X,3H1(f)H3(f) + X04H0(f)H4(fJJ + where

f=--;

ft=1

and where H(f) = the Hermite polynomial of order n

n n(n

Ï)f2

n(n l)Çn - 2)(n

34

Furthermore

Knm

crtcr

'Assoc. Prof., Dept. of Ocean Engrg., Tech. Univ. of Denmark, Bldg. 101E,

DK-2800 Lyngby, Denmark. r

Note. Discussion open until June 1, 1990. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 4, 1988

This paper is part of the Journal of Engineering Mechanics, Vol 116, No 1, January, 1990. ©ASCE, ISSN 0733-9399/90/00010240/$1 .00 + $. 15 per page.

Paper No. 4269, .

240

where K denote 'the curnulánts ófp(,, and where o and o = the

stan-dard deviation of and , respectively Eq 1 also contains quadratic and

higher-order terms in X,,m, which here will be assumed small and therefore omitted.

The cumulants can be expressed in terms of the central moments

= E{[ - E()]m[

E()]"} ' ' '

'(5)

Thus, for instance

K 3m0

K = m22 - 2m - m02,n20

implying that m

X40 =j--3

m20 m22 - 2m1 X22 = (7a) i (7b)

If the response spectrum is' narrowbanded, the probability distribution

F () for the peak values of

can be approximated by

N()

F() =

N(0)

(8)

where N()

the average number of crossings of the level = per unit time

N()

=

J

p(,)IId

' ' '

(9)

In Eq. 8 it is assumed that the mostoften crossed level is

= O. This

holds for a symmetric response, but is only an approximation for an

asym-metric response even with a zero mean (S R Winterstein, private

corn-mumcation, l989a) An example concerning an asymmetric response is given at the end of this techñical note

Substitution of Eq. 1 into Eq. 8 yield (Longuet-Higgins 1964) with f

r ' i

N()

e ° i + - [XH3(f) + 3X12H1(f)] i i.

6.

[X40H4(J) + 6X22H(f) -' 0410(1)] + . . .}

...(10)

using 1

2'

Jç1/2)flIfvIHn(f:)df,

.0 n odd ' '

(il)

(2H_2(0) ' n even

sto-chastic variáble is determined. Similarly, the distribution for the negative peaks is found by replacing f with

-/

in. Eq. 10.

FATIGUE DAMAGE

For use in fatigue-damage calculations where the stress ranges, rather than the stress peaks, are important, a stress amplitude distribution Fa() is de-fined as

n i * n

Fa()[F()+F()]

(12)

which then becomes, neglecting tenns in X where m + Ñ 6

[X(f4

-

612 + 3) + 622(J2 - 1)

Fa()

i -

24

1 1 - (13)

i

8 4 24

The expected value of the cumulative damage AD per cycle is estimated using Miner's law

n i .n

D=J P())m4

(14)

assuming S-N curves of the type, with rn <O

N= a(2)m

(15)

and with the probability density function p() given by dFa()

p()=

(16) d Upon integration

AD=

---

- (17) Ò(2\/o)m 242

where Fx) is the Gamma function.

For asymmetnc responses, another definition of the cumulative damage is (Winterstein and Manuel 1989)

AD =

f:

₍₁₈₎

J0 N(0) d

a(

where the value ₌

) <C) of the trough is the one with the sam

up-crossing rate as trie pak

[ie

,

N() = N() This defimtion takes into

acçount the strong dependence found in narrow-banded processes between a peak and the following trough.

However, use of Eq 18 requires that the amplitudes of the troughs are determined numerically as function of the peaks , due to the çomplexity of

Eq. 10 for the uperossing rate. Therefore, in the present technical note, the cumulative damage AD will be based on Eq. 17 rather than on Eq. 18. For symmetric responses, the two formulations yield identiáal results.

The influence of the non-Gaussian behavior on the fatigue damage appears through the coefficients of kurtosis X, X22, and X of the stress signal and its timederivative. If these coefficients are small compared to unity, the expression cän be reduced to

fi

fl--rn

\

2

_i

₁ AD

= a(2f2cr)m

_t!

₊

-

[m2X + m(4X - 6X22)]

which resembles the expression (Winterstein 1985)

r'(i

m

\

2/[

i

i

= - _.

I

1 + - (m2Xo + mX) I

...

(20)

a(2\/o)m

_L 24 _J

derived using a very different approach based on first-order Hermite models and Eq. 18.

A full discussion of the similarities and differences between the Charlier series and the Hermite model approaches is given by Winterstein (1989b). In the present note, it is only noted that the tail distribution in the Charlier

series can be very inaccurate with even negative probability densities (Jensen and Pedersen 1978; Ochi and Wang 1984). The Charlier series approach is therefore not especially useful for extreme value estimates, but is better suited for fatigue analysis. The use of Hermite models seems to yield much better tail distribution and are therefore applicable for higher nonlinearities or smaller probability of exceedence than the Charliet series approach..

As stated previously, quadratic and higher terms are omitted in the joint probability density function (Eq. 1) when carrying out Eq. 19 for the fatigue damage. However, for some stochastic processes, the square of the skewness

(X) is of the same order as the kurtosis. X. In such cases, which, for

instance, can appear for wave loads (Ochi and Wang 1984; Jensen 1989), the quadratic terms also need to be included in the derivation of the formula for the fatigue damage. This is discussed in the second example tò follow. EXAMPLESYMMETRIC RESPONSE

Consider the problem

(21)

where x are statistically independent, Gaussian-distributed variables with zero means and variances equal to unity, and where n is an odd, positive integer.

Then, using Eqs. 5-7

=

= i + 2p..

- (22a)

= 2(n - 1)+e + [(2n3 +

6n2 - 4n)=2

-

(8n2 - 4n)

ii+1ie2 + O()

. (22b)

(4D

= 4(n - 1)p+1e +

[(24n2 - l2n)_2

. (16n2 4n)p_1p.+1e2 +

O()

.. (22c) where . . P2i =

13 .5

...

(2i - 1)

...

. (23)

Thus, for«1

.1

K40 . (24)

Thereby, the new formula? Eq. 19, coincides asymptotically with Wi ter-stein's (1985) Eq. 20. It should be noted that all coefficients of skewness are zero for the present example. Generally, it can be shown (S. R. Win-terstein, pnvate communication, 1989a) that the two formulations, Eqs 19 and 20, yield the same asymptotic result for sufficient small nonlinearities.

Finally, it is seen that

r

= 1 + L_m +

(m + m)(n - 1)

)E=o 6

EXAMPLEASYMMETRIC RESPONSE

- Consider the problem

=x+2ExX

244

p'

+ Q(2)(25)

(26)

and thus, like in the previous example, X, (i-/2)X40, prov.ided is

suf-ficiently small However, here the quadratic terms in

the coefficients of skewness X30, X21, X12, and X03 should also be included as

they are of the

same, order

(2)

as the kurtosis.. A consistent derivation is given by Jensen (1989), and the resulting formula fOr D including all terms of order 2 is

/

1

r

1 :

- _.

+ -. (rn3 - rn2

m) ± 0(4)j

... . (30)

a(2V2X)m 2

where the standard deviation X of 'the linear term

is unity for the present

example

Taking m

3, both, Eqs: 19 and 20 yield.

where x are statistically independent and Gaussian-distributed variables with zero means and variances equal to unity.

Then, using Eqs. -5-7

m20 o i + 2e2 (27)

,242 + 32

Xfl-.

. (28) (i + 22)(1 .+

42)

482 + 48

X40= -' . (29) (1 + 2e2)2

TABLE 1. Numerical integration of Eq. 14 Using Exact Probabifity Density

Func-ifons

For this example, exact distribution functions of the peaks and the troughs exist (Lin 1963) Assuming no peaks below and no troughs above mean level , a numerical integration of Eq. 14 using the exact probability density func-fions yields results in very close agreement with Eq. 30, even for high values of E, as shown in Table 1.

For m = 3, it is seen that omission of the quadratic terms in the skew-ness results in a pronounced overestimation of the influence of the

nonlin-arities, whereas for m = 4, the skewness terms are of minor importance and become zero for m = 4.302. However, only Eq. 30 yields an accurate estimate of the fatigue damage for all values of m.

This indicates that the Charlier series can be used for fatigue-damage cal-culations in spite of its limited applicability for extreme value estimations.

APPENDIX I. REFERENCES

Jensen, J. J. (1989). "Fatigue analysis of ship hulls under non-Gaussian wave loads." DCAMM Report No. 389, Tech. Univ. of Denmark, Lyngby, Denmark.

Jensen, J J , and Pedersen, P T (1978) "On the calculation of the joint probability

density of slightly non-linear stochastic processes." ZAMM, 58, T481-482.

Longuet-Higgins, M S (1963) "The effect of non-lineanties on statistical distri-butions in the theory of sea waves " J Fluid Mech, 17, Part 3, Nov , 459-480

Longuet-Higgins, M. S. (1964). "Modified Gaussian distribution for slightly non-linear variables." Radio Sci., 68D(9), 1049-1062.

Ochi, M. K., and Wang, W. C. (1984). "Non-Gaussian characteristics of coastal waves." 19th Conf. Coastal Engrg., ASCE, Aug.

Winterstein, S. R. (1985). "Non-normal responses and fatigue damage,". J. Engrg.

Mech., ASCE, 111(10), 1291-1295.

Winterstein, S. R. (1989b). "Hermite moment analysis of nonlinear random vibra-tion" Computational mechanics of reliability analysis, Liu and Belytshko, eds,

Elmepress mt., Lausanne, Switzerland.

Wintérstein, S. R., and Manuel, L. (1989). "Non-Gaussian response of offshore

m E Eqs. 19 and 20 Eq. 30 Eq. 12 'exact'

(1) (2) (3) (4) (5)

3

0.1 1.150 1.105 1.104 0.2 1.600 1.420 i .402 0.4 3.400 2.680 2.704

4

0.1 1.280 1.260 1.263 0.2 2.120 2.04.0 2.086 0.4 5.480 5.160 5760 I'(2.5) LD

(2'ñ)3[1 ±

152

+ 0()]

...

(31) a

whereas Eq. 30 gives

F(2.5)

platforms: latigue," by S. R Winterstein and L Manuel, J. Str. Engrg., ASCE, 115(3), 749-752.

APPENDIX II. NOTATION

The following symbols are used in this paper.

a

coefficient in the S-N curve;

R()

= expected value of ;

F() =

probability distribution function of the peak values of ,

F0()

= stress amplitude distribution function,

f =

dimensionless stochastic stress variable (/o), ft

=

dimensionless time derivative of

e (/o);

J =

dimensionless peak value

f

H(f) =

Hermite polynomium of order n;

Kmn = cumulants of the j oint probability density function p(, )

m exponent in the S-N curve;

mm,, central moments of , ;

N()

= average number of crossings of the level per unit time;

p(, )

= joint probability density function of , ;

= expected value of the cumulative damage per cycle;

e nonlinearity parameter; ['(u)

= Gamma function;

dimensionless cumulants; = stQchastic stress variable; = _{time derivative of}

;

= peak values of ;

=

standard deviation of ,

= standard deviation of ; and