Int J Fatigue 15 No 4 (1993) pp 265-272
Rainflow analysis: Markov method M. Frendahl and I. Rychlik
In this paper we discuss rainflow cycle counting methods and linear fatigue damage accumulation for stationary loads. The expected damage is computed by approximating the sequence of local extremes by a Markov chain. The algorithm is implemented as part of a 'fatigue toolbox'. Several examples illustrate the results. Key words: amplitude; crossings; fatigue; Markov chain of turning points; peaktrough count; rainflow count
D(T)
Notation L(t)
(x,y),, S, M, mi
{tk; '}
#{.) Nr(u, v)
E[x] load ith cycle counted at time ti with peak x and trough y amplitude of the ith cycle; Si x - y height of the ith peak of a load height of the ith trough of a load height of the trough of the ith ralnflow cycle set of points tk fulfilling the condition '" the number of elements in the set (.} the cycle counting distribution of a cycle count; Nr(u, v) = #{(x,Y)t3 ti E [0,T] and X > u ~> v > y} =
The general approach in fatigue life prediction is to relate the fatigue life of a construction, subjected to a random load, to laboratory fatigue experiments of simple specimens subjected to constant amplitude load: so called S - N data. Therefore it is necessary to define amplitudes of equivalent 'load cycles' Sk, which are functions of the sequence of peaks and troughs in the load, and assume a damage rule, that is, a method to measure the damage caused by each simple cycle. A commonly used damage rule, due to Palmgren and Miner, postulates that the total damage caused by a stress history {Si} is
DCT)= ,,.~rNsi E !
fx(x) f(k)
/z(u, v) v,~C(u, v) Izvr(u, v) /~rr(u, v) cu
the total damage at time T mathematical expectation of the random variable X the density function of the random variable X the spectral density of a load L(t) at the angular frequency k the intensity of a cycle count; E[Nr(u, v)]/T the intensity of the rainflow count the intensity of the peak and the following trough count the intensity of the peak and the preceding trough count the intensity of peaks; the expected number of peaks in unit time
line' identify the closed hysteresis loops. The following definition, given in Ref. 4, is more convenient for statistical analysis. Definition 1. Let L(t), 0 <~ t <~ T, be a load and denote by Mi the peaks of L(t) at times ti, tl < ti+~. Define half cycles (Mi, mF), (Mi, m+), where
m;- = min L(t),
tF = sup {t; 0 < t <
ti,
L(t) > M i } ,
t~-
(1)
where tl is the time when the ith cycle is counted and N, is the cycle life obtained from $ - N data. Fatigue failure occurs when D(T) exceeds 1. In the literature one can find several definitions of cycle counting procedures) Dowling 2 studied the accuracy of predictors of the fatigue life based on the eight most commonly used counting methods, and found that only the rainflow count leads to prediction agreeing with actual lives. In the present paper we shall study the expected damage obtained using rainflow count and the Palmgren and Miner rule (Equation 1). The original definition of rainflow count given by Endo et al) had a complicated sequential structure in order to 'on
Rrc I
t7
I
I
t,
q
i.
Fig. 1 Definition of rainflow cycle
0142-1123/93/040265-08 © 1993 Butterworth-Heinemann Ltd Int J Fatigue July 1993
265
m/~ = min L(t),
t;- = inf {t; t~ < t < T, L(t) >t M~} ,
ti
(if t7 and/or t + does not exist, then t7 = 0 and/or t,+. = T). Then the rainflow cycle counted at time ti is (Mi, rr~/c)ti where
m/RFC [max(m/-,mT) if t/- > 0 /tm+
otherwise
See Fig. 1. The algorithm for the rainflow count given in Definition 1 is equivalent to the original method proposed by Endo et a/. The peaks and troughs of L(t) which are the endpoints of closed hysteresis loops are identified by both methods and paired into cycles. The remaining peaks and troughs form the so-called residual (or a set of half cycles) (see Fig. 2). The difference between the original rainflow method and the new one by Rychlik is the way that half cycles are treated. In Endo's definition the set of half cycles is found, while in the new definition the peaks in the residual are paired with the following trough (in the residual) to form cycles. The other frequently used counting method is the so called peak-trough count. 1 Definition 2 Let L(t), 0 ~ t ~ T, be a load and denote by Me mi the peaks and the following troughs of L(t), respectively. Let ti be the time for the occurrence of the peak Mi. Then the peak-trough cycle is defined as (Mi, mi),~. Obviously in both counting methods the number of cycles is equal to the number of peaks in the load. Therefore the variability of a cycle count can be described using the cumulative distribution (or histogram) of cycle amplitudes:
Total damage
number of cycles with amplitude less than s Fr(s) = total number of peaks Next, assuming that L(t) is a path of a stationary and ergodic random process, the probability distribution Fs of a cycle amplitude, i.e. the limit of Fr as the interval in which cycles are counted tends to infinity exists, and is given by Fs(s) --- E[number of cycles with amplitude less than s]
E[total number of peaks] (2) The expected damage is then given by E[D(T)] = T ' c M " E [ N f 1]
and Gluver. 8'9) However, since the expected total number of peaks in Equation 2 is equal to cMT, then in order to compute the expected damage we first divide by cMT in Equation 2 and then multiply by the same factor in Equation 3. This is obviously redundant and sometimes even prohibiting. For example, some loads used in applications have infinite peak intensity cM: for example, the Gaussian process with JONSWAP spectrum described later. Usually one counts cycles in sampled and smoothed loads, and hence in practice cm is always finite. However, if sampling frequency increases then cM "-->0o while E[N~ 1] ---> 0 which can lead to errors in computed E[D(T)] as defined by Equation 3. This indicates that the use of the empirical distribution Fs(S) to describe the variability of cycles is not always the best choice. In Ref. 10 a different way of describing the distribution of cycles was introduced (see the next section). This will be used, together with the Markov chain approximation for the sequence of peaks and troughs, to construct a new algorithm to compute the expected damage defined for a rainflow count. One of the advantages of the approach is that the method does not require knowledge of the intensity of the peaks but the intensity of downcrossings of the mean level. The algorithm becomes more stable and results are more accurate. We shall show in some examples that the Markov chain approximation to the sequence of turning points gives excellent agreement between computed expected damage and simulated damage for Gaussian as well as for some non-Ganssian loads. The results indicate that the description of the variability of the turning points of the load as a Markov chain has ~ wide range of applications. The algorithm for calculations of the expected damage have been implemented as part of a 'fatigue toolbox', presented in Ref. 11.
(3)
where cM is the intensity of peaks, that is, the average number of peaks in unit time. This is a standard approach, and in previous papers 5-7 we have presented a method to approximate the rainflow distribution Fs based on a Markov chain approximation for the sequence of peaks and troughs. (This approach was also used by Bishop and Sherratt, and Krenk
D(T)
In this section we shall briefly review some of the results given in Refs 10 and 12 necessary for the presentation. We assume that a cycle is defined by specifying its highest and lowest points, x, y, respectively, and the time at which the cycle is counted, t say; ie a cycle is a time-indexed pair (x,y)r, x > y. A cycle count is a procedure that transforms a load function L(t), 0 ~< t ~< T, into a set of cycles {(x,y)~); for example, see Definitions 1 and 2. The subscript ti indicates that we consider only countable sets of cycles. Denote by f(x,y) damage caused by a cycle (x,y)~. Assume f(x,y)>~ 0 and 02f(x,y)/OxOy ~ 0 for all x/> y. For linear cumulative damage theory f(x,y)= N~-2y, where N, is obtained from constant-amplitude fatigue data. In applications one often finds an explicit expression for N,_y, eg Nx-y = c(x-y) -~, 13I> 1. By Equation 1 the total damage D(T) is given by D(T) = ~ f((x,y),,)
(4)
ti~ T
Obviously, given a cycle count {(x,y),), Equation 4 is the computationally most convenient. However, for theoretical considerations and computation of expected damage, if the function f is known explicitly, it is more convenient to rewrite Equation 4. We begin with a definition of a counting distribution of a cycle count.
Fig. 2 Example of a trough with peaks and troughs which do not form closed hysteresis loops marked by • 266
Definition 3 Let L ( t ) , 0 ~ t ~ < T , be a load and {(x,y),) a cycle count. Let N~(u, v ), u >! v, denote a counting distribution function of a cycle count {(x,y)t) defined as follows: Int J Fatigue July 1993
Nr(u,v)=#((x,y),,;ti~Tandx>u>--v>y}
(5)
ie the number of cycles with top higher than u and trough lower than v. In the following we shall denote the counting distribution of the rainflow and peak-trough counts by N ~ c and Nrvr respectively.
Theorem 1 Consider a countable cycle count {(x,y)t) obtained from a load L(t), 0 ~ t <~ T, with a counting distribution Nr(u, v) and let the damage caused by a cycle (x,y)~, fix,y), be a twice continuously differentiable function such that #2f(x,y)/#xOy ~ 0 for all x t> y and fix,x) = O. Then the total damage D(T) is finite and given by
for all u ~ v), the expected damage is given by
E[D(T)]=-fl -f?
E[N:r(U'v)]d2f(n'V)OuOvdudv
E[Nz(u,u)]~],.
du
(8)
Consequently, while the counting distribution Nr(u, v) defines the total damage, the expectation
t.*r(u, v) = E[Nr(u, 'v)] will define the expected total damage. Obviously, for a stationary load u,~u, v) = T. u,,(., v)
D(T) = -
Nr(u, v) ~
dudv
~i~qO
The proof is given in the Appendix. It is an important fact that the diagonals Nr~C(u,u) and N~r(u,u) are identical and equal to the number of downcrossings of the level u by the load L(t). Further, for fixed values of u, v the counting distribution N~r(u, v) of a peak-trough count is equal to the number of peaks higher than u such that the following trough is lower than v. It is less obvious that for the rainflow count N~r C(u, v) is equal to the number of downcrossings of the level u by L(t) followed by a downcrossing of the level v without passing u in between. (The proof of the last statement can be found in Ref. 12.) Consequently N~rr(u, v) ~ N~a:rC(u, v)
(7)
for all u ~ v. By combining Equations 6 and 7 we obtain the result that the damage due to rainflow count always bounds the damage due to the peak-trough count. We turn now to the expected damage for random loads. Assume L(t), 0 ~< t ~< T, is a random load, and {(x,y),} is a cycle count with counting distribution Nr(u, v). The total damge D(t) is an increasing random process. Since the fatigue failure time Tf is defined by D(T f) = 1, the failure probability is given by 1"[7~ ~ 73 = I'[ D( T) ~ 1] and hence the distribution of D(T) is of great importance. However, in our setting of the problem, we define a probabilistic structure of the load L(t) and since D(T) is a complicated nonlinear function defined on L by Equation 4 or 6 the problem of computing P[D(T) >i 1] is extremely hard. The simplest estimator if, say, of 7~ is obtained by using the moment method; that is, ff is the solution of the moment equation E[D(ff)] = 1 and for stationary loads if=
1 E[D(1)]"
In the following we shall call E[D(1)] the damage intensity. Now, the expected damage is obtained by computing the expectation of the integrals in Equation 6. By changing the order of expectation and integration (this is allowed since 02
Ouovf(u, v) ~ 0
Int J Fatigue July 1993
In the following we shall omit the subscript 1. Since t~T(u, v)/T is the expected number of cycles with peaks higher than u and troughs lower than v in unit time, g.(u, v) is called the count intensity. In what follows, we assume that the loads are stationary processes and denote by/z~C(u, v), /zrr(u, v) the counting intensities of the rainflow and peak-trough counts, respectively: lzl"~C(u, v) = E[N~r C(u, v)]/T
~ ( u , v) = E[N~rr (u, v)]/T
Algorithm for method
IxaFC(u,
(9)
V): Markov chain
Consider a stationary, ergodic, load L(t) and let {Ms mi}, i = 1, 2 . . . . . be a sequence of pairs of peaks and the following trough of the load L and ti the times of the occurrence of peaks Mi. Obviously {(M~ mi),) forms a peaktrough count with counting distribution N~(u, v) and intensity ~rr(u, v). Similarly we can consider the cycles formed by the peak Mi and the preceding trough mi-1, that is, {(M~ mi-t)} wkh an intensity ~r'r(u, v), say. For timereversible loads /zrr(u, v ) = ~l'r(u, v). Assume that the intensities /zrC(u, v), ).rr(u, v) are known. For example, they can be estimated from the load or computed from the probabilistic properties of L, as is possible for Ganssian loads, for example, where /fir(u, v) can be accurately computed from the power spectrum of the load using the regression method. 13 The problem is to approximate the damage intensity E[D(1)] due to the rainflow count. Since in practical applications one measures the load in a discretized form we assume that the sequence of turning points can take only a finite number of discrete values. Let u~, ui > ui+t, i = 1, 2, . .., n, be the discrete levels and let P = (p#), 1~ = ~#) be (n, n)-matrices, where p#, ## are the transition probabilities from peak to the following trough and from trough to the following peak, respectively: P0 = P[mk = uilMk = u~]
fi# = P[Mk = uilmk-, = ui] The matrices P and P can be obtained from the intensities /z~(u, v), /~,~(u, v), respectively; see algorithm given in Ref. 11. Now assuming that the sequence of turning points is a Markov chain (see Ref. 14 for the discussion of various properties of Markov chain models), the rainflow intensity
267
tz~c(u, v) is obtained by using simple matrix manipulations. We turn now to the presentation of the Markov method. For fixed indices (i, j), i < j , define the following submatrices of P and P: A=(pkt)
i
<~k<~j - 1
B = ~kl)
i+l<~k<~j
i + l<-l<~j i
(10) E M " k ° v [ D ( 1 ) ] = - - f l . ~,~ t2(u'v)O2f(u'V)dudvouOv
<~l<~j-1
The matrix A contains the probabilities that peaks at levels u~ ui+l . . . . . ui_l are followed by troughs at levels ui+l, ui+2. . . . . ui and B the probabilities that troughs at ui+l, ui+2. . . . . ui are followed by peaks at levels ui, ui+l, •
.
.,
/4~j~ 1 •
Further, for a fixed level ui, let P = [Pk] =
p,, ,
k = 1,2 ..... n
(11)
The vector p/contains the conditional probabilities that a maximum with height uk is followed by a minimum smaller than ui. Theorem 2 Under the general assumptions of this section, if the peak-trough intensities/~rr(u, v), /2rr(u, v) are known then for fixed values (u, v ) = (u, u/), the rainflow intensity/zm~C(u, v) is given by /~grC(u, v) =/zVr(u, v) + qB(I - AB)-le (12) where A and B are defined by Equation 10, eT = [pi = [tzVr(u~ /¢/-1)- ]-LPT(Igi, UI)]' i+l<~l<~j.
pi+l" ".Pj-l] and q
The proof is given in the Appendix.
Examples In this section we shall use the Markov approximation, Equation 12, to compute t~Rrc and E[D(1)] for different load processes, such as a Gaussian load and Morison forces• The examples are computed as follows. First, we simulate a long sample path of a random load with specified probabilistic structure. The rainflow and peak-trough counts are carried out for that sample path. The rainflow count is used to compute an estimate of the rainflow intensity, such as /zr'rC(u, v) ~- N~r c(u, v ) / T =/Q(u, v ) ,
(13)
The damage intensity E[D(1)] is estimated by
e[o(1)]=-ff. _
du
(14)
D(T) T
The accuracy of the numerical integration in Equation 14 is checked by comparing the results with an alterative way of compute D(T)/T; that is, using Equation 4 to calculate D(T). The intensities tzrr and t2rr are estimated from the sample path by replacing N ~ c in Equation 13 by N~rv and N~v, respectively. The Markov transition matrices P, P are computed from estimates of tzrr and /2rr, respectively. Finally, the Markov approximation/2(u, v), say, of #V.FC is computed using Theorem 2. In order to check the correctness of the Markov approach
268
The JONSWAP spectrum 15 is designed to fit empirical wave data from the North Sea and has the explicit form f(a)
=
As
I
[a[ ~< am
v.
for
crb
otherwise
We choose g = 9.81, a = 0.0029, fl = 1.25, Am = 0.075, 3' = 1, or, = 0.07, crb= 0.09 and we truncate the spectrum at 0.2, ie a E [0,0.2], in order to guarantee the existence of the second derivative of the process (it will he needed in the definition of Morison force). Then we renormalize the spectrum so that variances oar = Oar' = 1. The results presented in Fig. 3 show the applicability of the Markov method for the load. In Ref. 11 we present numerous examples showing that the Markov chain approximation to the sequence of turning points gives excellent agreement between computed expected damage and simulated damage for Gaussian loads with various spectral densities, as well as for some non-Gaussian loads like ~-loads and a-stable symmetric loads, which are loads with significant tails.
Morison force A stochastic process F(t) is called a Morison force if it is generated from a stationary Ganssian process Y(t) in the following way: F(t) = K, Y'(t) + Ka Y(t)
IY(t)l,
where /(1 and K2 are positive constants. See Ref. 15 for discussion of the formula for the Morison force. We consider two examples of the Morison force as follows.
Y(t) a Gaussian process with truncated JONSWAP spectrum
~o fit(u,v)oZf(u'V)dudvouov
af(u,v) - f~olQ(u,u)~,=
We shall assume throughout this section that the damage function is flu, v) = (u - v) ~, u ~ v, 2 ~ ~ ~ 5.
Gaussian load with JONSWAP spectrum
J
LI=j+1
the counting intensities/V(u, v) and tg(u, v) are compared. We also present plots of the estimated damage intensity, D[D(1)], given by Equation 14, and computed expected damage intensity, EM,rkov [D(1)], using the Markov method:
Let Y(t) be a stationary, twice continuously differentiable Ganssian process with truncated JONSWAP spectrum described earlier and let E[Y(t)] = 0, tr//= oar, = 1. For K, = K2 = 1, define the Morison force by F(t) = Y'(t) + Y(t)[Y(t)[. The results presented in Fig. 4 show the applicability of the Markov method for the force.
Y(t) a Gaussian process with rectangular spectrum Let Y(t) be a stationary, twice continously differentiable Gaussian process with rectangular spectrum f(a) = [
0,
otherwise
Int J Fatigue July 1993
Cycles 1
4 3
• . ".: : • . :...-~
2
;':..:"
~.. .
. . "... ; , ; . ~ : . '
.-
..
•.
.x~.;~.... " .'.':~Z'~'¢ :
1
" •"-:~'~ g . . • ... • ~."',:.,,:"::.-'..... "~'* q ?,~.'t "."
8.
0 -1 -2 -3 -4 J
4 a
trough
Damage intensity
Counting intensity 4
100
i
90
3
80
2 70 1
60
..~ 0 40
-1
30
/
-2
20
-3
10
-4 -4
b
6
I
I
,
v-~veb
2
2.5
C
315
;
415
5
b
Fig. 3 Gaussien load process with JONSWAP spectrum: (a) rainflow count; (b) contour plots of N(u, v) (solid line) and /z(u, v) (dashed line); (c) estimated expected damage (solid line) and computed damage using Markov approximation (dashed line) For KI : K2 = 1, define the Morison force as before by F(t) = Y'(t) + Y(t)[Y(t)[. The results presented in Fig. 5 show the applicability of the Markov method for the force.
Acknowledgement We should like to express our appreciation to Professor Georg Lindgren for giving his time to the discussion of this paper.
Conclusions A new algorithm is proposed for computation of the expected damage for stationary loads due to rainflow count, based on a Markov chain approximation of the sequence of turning points of the load. The results indicate that the description of the variability of-the turning points of the load as a Markov chain has a wide range of applications.
Int J Fatigue July 1 9 9 3
Appendix:
p r o o f s of t h e o r e m s
Theorem 1 Consider a smooth load L(t), 0 ~< t <~ T, and let {(x, y),} be a countable cycle count with distribution Nr(u, v). A simple sufficient condition for a cycle count to be countable
268
Cycles 15 10
:"'.i '..;i%-,. ' "'.V:,~.'" 0 -5 -10 -15
' -15
' -10
-~
a
6
5'
1'0
' 15
trough
Counting intensity
x 104
Damage intensity
6 15 5 10 5 ¢1 >
4
0
~
3
-5
2
-10 1
-15
' -15
b
-i0
-;
0'
;
' 10
1;
v-levels
o2
, 2.5
3, - - - - ~3.5
C
4,
, 4.5
5
b
Fig. 4 Morison force when Y(t) is a Gaussian process with truncated JONSWAP spectrum: (a) 2054 rainflow cycles; (b) contour plots of /0(u, v) (solid line) and /~(u, v) (dashed line); (c) estimated expected damage (solid line) and computed damage using Markov approximation (dashed line)
is that Nr(u, u) is a b o u n d e d function of u. ~2 Denote by A(x, y), x > y, the following set:
a(x,y)=
((u,v);x>u>~v>y}
Let fix, y) be a damage caused by a cycle x) = 0, we have that for any x I> y
(x, y)t.
Since f(x,
f(x,y)=-ff.~o la(x,y)(u,v)°Zf(u'V)dudvouOv - f~ l~,(~,,y,(u,u)~ 270
,~=,,du
where 1A(x.y)(U, v) is an indicator function of the set A(x, y). N o w , using Equations 4 and 15:
D(T) = ~ f((x, y)t) ti~T
= (15)
,~r la(x,y)~ (u, v)
- f~=( ~rla(x,Y',,(u'u)) ~
==.du
Int J Fatigue July 1993
Cycles
10 •
.
. .': .
. .
• . '..'...~...:".. • - ....-.?,f':-. • " • : =:. : ~;~:::b4t.
• "
/
:. 0
-5
-10
-io
I
I
-5
0
,'o
a
trough Counting
intensity
xl0 4
Damage intensity
5
I0
4.5
4 5
3.5
3
-,°>
~
:
2.5
::
2 1.5
-5
I
-I0 0.5
-10
-5
b
0
5
ll0
,-le,e~
02
c
2:5
3
3.5
4
4.5
5
b
Fig. 5 Morison force when YIt) is a Gaussian process with rectangular spectrum: (a) 1958 rainflow cycles; (b) contour plots of
N(u, v) (solid line) and /.L(u, v) (dashed line); (c) estimated expected damage (solid line) and computed damage using Markov approximation (dashed line)
since o2f(u,v)/OuOv <~ O, Of(u,v)/Ovl.=. ~ 0 for all u t> v, and we can change the integration and summation operations. This finishes the proof, since
NT(u, v) = ~
l~(.,y),, (u, v), u ~ v
ti~ T
Theorem
2
As mentioned above after the statement of Theorem 1, the rainflow counting distribution Nr~C(u, v), u t> v is equal to the number of u-downcrossings followed by the downcrossing of v without crossing u in between. Consider fixed values (u, v) = (u~ uj), i < j. Using the concept of turning points it is easy to see that
Int J F a t i g u e J u l y 1 9 9 3
h~rFC(u, v) = Nr~Tr(u, v) + ~/
r~_rl~ "<,v/ K~(u, qr(1)v;l)
l=i+l
where NPT(u, v) is the peak-trough counting distribution; Q~(I) is the number of peaks greater than u such that the following trough is equal to ua K~(u, v; l) is the number of peaks greater than u such that the following trough is equal to u~ and the subsequent turning points cross v before u. Now, by ergodicity of L(t) we have N~r (,,, v) J Qr(/) Kr0,, v;/) ~aFC(u'v)= T-~®lim T + l-i+, ~ rlim T Qr(l)
271
/ = ~."~(~,, v) + Y~ (a~r(., "t-,) - ~ ( ~ ' , "l))" l=i+1
[{mi}i~,2 crosses v before I 1 [{Mi}iu.2 crosses u M1 > u, m t = ut
(16)
4.
For general loads the conditional probability in Equation 16 is difficult to compute. However, if (Mu rni} is a Markov chain then
5.
"P
J
P [ {ml}~'2 cr°sses v bef°re I
[ (~/i}i~'2 crosses u
]
6.
M, > u, ml = ut
7.
]
8.
= p [{mi}i~z crosses v before . {Mi}i~.2 crosses u rnl - uzj o~
= ~ pk (,,l), k=o
9.
where, for fixed (u, v) p,(ut) = l~m2 < v, M2 < u lm~ = m] = [Be]~
10.
pk(ut) = P[rnk+, < v, Mk+, < u, v ~ m i < M i < u for allj = 2,3 . . . . .
klm, = ~,,]
= [B(AB)k-te]z
11.
12.
Consequently, since 2~=0pk(ul) = [B(I - AB)-le]I / /z~C(u, v) =/zVr(u, v) + ~
qCl)[B(I - A a ) - l e ] t
13.
l=i+ 1
-- ¢Lr'T(u, V) + qB(I - AB)-~e
14.
which finishes the proof. 15.
References 1. 2. 3.
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Collins, J.A. Failure of Materials in Design (WileyInterscience, New York, 1981) Dowling, N.E. 'Fatigue prediction for complicated stess-strain histories' J Mater 7 (1972) pp 71-87 Endo, T., Mitmunage, K. and Nekagewa, H. 'Fatigue of
metals subject to varying stress - prediction of fatigue lives' Preliminary Proc Chugoku-Shikoku District Meeting, 1967 (Japan Society of Mechanical Engineers, November 1967) pp 41-44 Rychlik, I. 'A new definitioon of the rainflow cycle counting method' Int J Fatigue 9 (1987) pp 119-121 Lindgren, G. and Ry©hlik, I. 'Rainflow cycle distribution for fatigue life prediction under Gaussian load processes' Fatigue Fract Eng Mater Struct 10 (1987) pp 251-260 Rychlik, I. 'Rainflow cycle distribution for ergodic load processes' SIAM J Appl Math 48 (1988) pp 662-679 Rychlik, I. 'Simple approximations of the rain-flow-cycle distribution for discretized random loads' Probabilistic Eng Mech 4 (1989) pp 40-48 Bishop, N.W.M. and Sherratt, F. 'A theoretical solution for the estimation of "rainflow" ranges from power spectral density data' Fatigue Fract Eng Mater Struct 13 (1990) pp 311-326 Krenk, S. and Gluver, H. 'A Markov matrix for fatigue load simulation and rainflow range evaluation' Res Report 1989:388 (The Danish Centre for Applied Mathematics and Mechanics Lyngby, Denmark, 1989) Rychlik, I. 'Rainflow cycles in Gaussian loads' Fatigue Fract Eng Mater Struct 15 (1992) pp 57-72 Frendahl, M. and Rychllk, I. 'Rainflow analysis - Markov method' Stat Res Report 1992:6 (Dept of Mathematical Statistics, University of Lund 1992) pp 1-60 Rychlik, I. 'Note on cycle counts in irregular loads' Stat Res Report 1992:1 (Dept of Mathematical Statistics, University of Lund 1992) pp 1-15; to be published in Fatigue Fract Eng Mater Struct 18 Undgren, G. end Rychlik, I. 'Slepian models and regression approximations in crossing and extreme value theory' Int Star Rev 59 (199t) pp 195-225 ~inlar, E. Introduction to Stochastic Processes (PrenticeHall International, Englewood Cliffs, NJ, 1975) Madsen, H.O., Krenk, S. end Und, N.C. Methods of Structural Safety (Prentice-Hall International, Englewood Cliffs, NJ, 1974)
Authors The authors are with the Department of Mathematical Statistics, University of Lund, PO Box 118, S-22100 Lund, Sweden. Received 15 June 1992; revised 15 January 1993.
Int J Fatigue J u l y 1993