P. H. WRSCHING Associate Professor. M em_ASM E
A. MOHSEN SHEHATA Research Associate. Aerospace and Mechanical Engineering, The University of Arizona. Tucson. Ans.
Introduction
Fatigue is a primary mode of failure for metallic structural
and mechanical components subjected to oscillatory stress
proc-esses. However an engineering description of fatigue behavior is extraordinarily complicated for several reasons including the
following: (a) the fatigue process itself is not well understood and consists of two distinct phases (crack initiation and propaga-tion), (b) there exist many factors which significantly influence fatigue behavior, i.e., temperature, surface finish, residual stresses, etc. (e) there is substantial variability in failure data even in well controlled experiments (d) stress processes are often very complicated (i.e., nonstationary random),(e) there is a paucity of fatigue data for materials under random loading. While many uncertainties in the fatigue problem exist, recently developed probabilistic design methods (e.g., reference [2, 7])'
hold promise of providing a mechanism of treating such
un-certainty in a rational manner.
The goal of this study is to develop a fatigue design procedure for a structural component subjected to a stationary and ergodic random stress process X(t) which in general can be wide band. Stich a procedure could be useful in design of components for land-based, sea-based, aerospace vehicles, offshore platforms, marine equipment, etc. The proposed algorithm should have the following conditions (a) provides an accurate description of
the physical phenomena (b) has a probability-basis, and (c) is relatively easy to use.
A random fatigue design algorithm for stationary gaussian
Numbers in Irrackets designate References at end of paper.
Contributed iv the ritcnials Division for pnihlicatir,n n th ,JomJt,N.u_ o?
Ea,ac,;stae M.TCRt.L.O .ao Trct,aor.oey. Manuscript rrceivet by the
Materials Division June10, 1070; revised manuscript receiver! .\pril 1J, 1077.
Journa' of EngineeringMaterials and Technology
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Fatigue Under
Wide Band
Random Stresses
Using the Rain-Flow Method
A fatigue design procedure is proposedforthe structural elements subjected to a stress process, modeled as stationary wide-band gaussian. This procedure, restricted to high
cycle fatigue design, uses available constant amplitude material fatigue data aral a modified, probability based, Palm gren-.liiner (Pif) rule. Statistical uncertainty in
fatigue behavior and nonstati.stical uncertainty in the Pif rule are implicitly accounted
for by treating ¿he Pi! index at failure as a random variable. The rain flow met/md
of counting stress cycles is used. The fatigue design algorithm requires only that the RuS and irregularity factor of the stress process be specified in addition to the constant amplitude S-N curve for the material.
processes has been proposed by Wirsching and Haugen [19, 20]. This procedure is based upon the assumption that each tensile peak of the same magnitude causes an equal amount of fatigue damage. Both wide band and non-zero mean processes could be considered. Recent work of investigators in random fatigue
de-sign include that of Ang and Morose [1] and Moses [12] on
bridges, Nolte and I{ansford [13] on offshore platforms, Land-graf and LaPoint [91 and Wetzel [17] on automotive structures,
and Dowling [4], Gallagher and Stalnaker [6], and Yang and
Trapp [21] on aircraft structures.
Description of the Problem
Stress Process. Because X(t) is assumed tobestationary, the spectral density function ¡V(f) exists. A realization of X(t) from a typical IV(f) is shown in Fig. t. Define Og as the standard
deviation or RMS value of X(t). Define m0 and n0 as the ex-pected rate of peaks and zero crossings, respectively (see refer-ence [10] for evaluation). An index of spectral shape is the ir-regularity factor a defined as,
ne
-m0
with O < a 1, and a = i for X(t) narrow band.
It is assumed that X(t) is a gaussian process and that fatigue damage will be independent of the frequency. Furthermore it will be assumed that the stresses ovili be primarily in the elastic range so that only high cycle fatigue is considered. Residual
stresses are not present. Combined loading can be possible if X(t) is interpreted as the equivalent design stress.
Fatigue Behavior. There exists of plethora of constant afn
plittide S-N data (eg. Fig. 2) on a wide variety of metallic ma-tenaIs. These data may be (a) actual, (b) hypothesized from JULY 1977 / 205 (1)
0.6 0.4 0.2 0.0 - 0.2 - 0.4 -06 XC?) kN 0 20 30 40 50 60 70 80
Freq u ecc yf, Hz
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T,ne Sec. Fig. i RealizatIon of a force process X(f) from a typical W(f)
static properties, or (c) code requirements (e.g., ASME Boiler
Code, AWS and AISC Codes). On the other hand, relatively
little random fatigue data are available (see reference [3, 8, 15,
16, 23[). Moreover development of an engineering model which relates fatigue crack initiation and propogat.ion to random stress processes ha been found to be difficult. The general approach
herein will be to utilize constant amplitude data with a linear
damage accumulation rule. Therefore it is necessary to be able tu count equivalent stress cycles (magnitude and number). The rain-flow method will be employed for this counting procedure. Given that X() is a random process, where e(O, T), the number
N ornen c a tu re 206
/ JULY 1977
Pf Q(a, m) Q(t) S (O, T)Y = number of cycles to failure in
fatigue test
no = expected rate of zero
cross-ings, crossings/s = probability of failure = correction factor for D = applied random load
= random variable denoting
stress amplitude estimate of L.g
SS = estimate of os
SDF = Spectral Density Function T = design life
IV(f) = spectral density function of
random process X(i) .V(i) = random Stress process
Y = raro mm va ri able (le no ti u z t he
- - load producing
environ-NSm k
m
N
Cycles to Failure,N
Fig. 2 (a), (b). The S-N curve (mean of S-Nor "minimum"), (a) a typical curve. (b) a special case often used for high Cycle fatigue
of stress cycles in life T will be a random variable n arid the
stress cycle magnitudes S will be a random variable. S is a
cori-tinuous random variable with probability density function
fs(s).
Assuming a linear damage rule, it has been shown that 'dam-age" after rs stress cycles can be written s.s (IO),
ment range
a
irregularity factorT(.)
gamma functionô = \Veibull distribution
param-eter
= deterministic value of fatigue
damage
T) = included in the range rom
zero to T
o- = standard deviation
= Weibull distribution param-et er transformation coefficient in equation (5) = trarisformatiots coefficient in equation (5) -'l'K = randomplia.se angle -w = frequenc in radians/s -= mean value
Transactions of the ASME
D=n
["n Is(s) (2)where Y(s) is the number of cycles to failure at stress range s in a constant amplitude test. Equation (2) can be written s.s,
D = nE []
(3)where E(.) is the expected value.
A form for D has been developed by Nolte and 1-fansford [131 based on the following assumptions:
C, coefficient of variation of the random variable X CDF
Cumulative Distribution
Function
D = fatigue damage E(.) = expected value
fa(s) = probability density function
of S
f
= frequency, (Hz)Fy(y) distribution function of Y
C(w) = spectral density function of
frequency w
K = defined by equation (G)
L = rantilever beam length 'n negative reciprocal slope of
t he SN curve --
-n, = expected rateof peaks, peaks/s
ro random variable denoting
number- uf.sr &'cIc'
ut-Stoess Raroe,
SR
1.0
I The load producing environment is a random variable with
rouges Y that have a \Ve,bull distribution with parameters (, 6). The distribution function of Y has the form,
Fy(y) =
i - exp {
-
() }
(4)2 The transformation from Y to stress range is assumed to be of the form,
(5) 3 The "characteristic' fatigue curve of Fig. 2 is linear (on a log-log basis) and is of the form
N(S) =
KS--
(6)and is valid for all S > O.
Then it can be shown that equation (2) reduces to,
D =
o-w-r
(1
±
m (7)where T (.) = gamina function
Tri the special case where X(t) is a narrow band gaussian
process, S has a Rayleigh distribution and equation (7) reduces
to,
D=
(2/)-r
(
+
nl (S) The critical issue in this study becomes one of deriving Is(s)fur the case where X(t) is gaussian but not narrow band, in
which case the distribution of S is not known. A secondary issue investigated herein is the possibility of simplifying the fatigue
descript inn h' characterizing the frequency content of X(t) with a, a real number, rather than by the function TV(f).
Palmgren-Mìner Hypothesis
According to the Palmgren-Miner (PM) hypothesis, failure occurs when D - 1. However it is known that the Pt rule fails
to provide consistently accurate predictions of fatigue behavior. Nevertheless the PM rule seems to be much easier to apply than other methods which have been proposed, and it is widely used in design practice (e.g., AS1E and A\VS Codes).
Wirsching and Yao [IS] have proposed a probabilistic approach to variable amplitude fatigue which maintains the simplicity of application of the PMrule. At the same time this approach
pro-Fig. 3 Mean damage as a function of risk for various values of the
coefficient of variation of damage
-Journal of Engineering Materials and Technology
vides s description of the inaccuracies of the PM rule as ss-gil as variability of fatigue behavior inherent in the material. Dedne
as the value of D at failure. To describe the variability in ob-served values of , \Virsching and Vao have modeled -as a
lognormallv distributed random variable with mean and standard deviation o equal to 1. Their assumption was based upon an amalgamation of test results from marty investigators. The probability of failure, pj is then defined as,
pj=P(D)
(9)Assuming that D is also a lognormally distributed random vari-able, p.r s.s a function of btD and C0 (coefficient of variation of D) was derived and plotted in Fig. 3. For example, given a basic design requirement of pj 1O and an assumed C0 0.50,
-the design should be such that 15D < 0.040. Irs turn j.i is
ap-proximately equal to the right-hand side of equatioui (7) with the mean values of all the parameters (providing that the variances of each are small).
The Rain-Flow Method for Cycle Counting
To apply the PM rule, or a modified version thereof, to a
process A(i), it is necessary to specify the statistical distribution of S and n- To obtain a sample of S and n from a sample function x(t:l,several cycle counting methods for fatigue damage analysis have been proposed [4]. But the methods which seem to provide the most accurate prediction of material behavior are the range pair, rain flow methods, and a cycle counting algorithm proposed
\Vetzel [17]. Tests on 2024T351 have demonstrated that the latter method performs better than the rain flow method [9, 17]. Nevertheless, Dowling [4] presents stroDg supporting
evidence, based on tests of S3 specimens of 2024T4 that the
rain-flow method, originally suggested by Matsuishi and Endo [11], is superior to the others which have been proposed prior to Wetzel's work. Some of Dowling's specimens had large plastic
strains, others had predominantly elastic strains. Predicted lives were within a factor of three of the actual lives for ali of the tests, arid within a factor of two for 90 percent of the tests.
The value of the rain-flow method is that it identifies events in a complex strain sequence which are compatible with constant-amplitude fatigue data, i.e., it identifies strain ranges associated with closed hysteresis loops [9]. Furthermore it is capable of
identifying stress range cycles associated with low frequency
components. A basic criticism of this method is that it ignores the sequence or continuity of application of stress. Clearly a need exists for more experimental data, but the available evidence suggests that the rain-flow method holds considerable promise.
The rain-flow method is used to count stress cycle magnitudes
(i 1, n) and n. from a realization of X(t). A sample of X(t) is converted to a point process of peaks and troughs as shown in Fig. 4 with the peaks identified as the even numbers. The time axis is then oriented vertically with the positive direction
down-ward. Consider the process as a sequence of roofs with rain
falling on them. The rain-flow paths are defined according to
the following rules.
i A rain-flow is started at each peak and trough.
2 When a rain-flow path started at a trough comes to a tip
of the roof, the flow stops if the opposite trough is more negative
than that at the start of the path under consideration, (e.g., Fig. 4, path [1S], path [9-10], .. .).
For a path started at a
peak, it is stopped by a peak which is more positive than that at the start of the rain path under consideration, (e.g., Fig. 4,
path [2-31, path [4-5], path [6-7[).
3 If the ritiri flowing down a roof intercepts flow from
pre-vious path, the present path is stopped, (e.g, .Fig. 4, [3-3a[,
).
-.1 A new path is not started until the path urider
-Half cycles of trough originated stress range magnitudes, S,
would be, for example, projected distances on the stress axis, (e.g., Fig.4, [1S], [3-3a[, [5Sa], etc.)
It should be noted that for X(t) sufficiently long, any trough originated half cycle will be followed by another peak originated half erde for the same range» For simplicity, stress ranges and
cycles are estimated herein by starting paths at the troughs only.
The Simulation
Because of the complexity of the rain-flow algorithm, it would be extraordinarily difficult to derive fa(s) from a given ¡VU).
However Monte-Carlo methods can be used to simulate X)
and estimatefs(s).
-A digital computer program with the capabilities shown in the flow chart Fig. 5 has been developed. Given an arbitrary W(f), a sample of X(t) is simulated using the following form (22)
X(t) = [2G(K)o,}U2 cos (w
+
) (10)G(w) is the one sided spectral density function in ternis of f
re-quency o (rad/s). [G(w) = ¡V(f)/2r]. Frere-quency is defined over the interval (0, ui,) with partitions of length ¿WK such that
X(t)
10Stress
/
3 2 3a 4-is a random phase angle, uniformly d-istributed in the interval (0, 2w). The number of harmonic components J is arbitrary, but
in this study values of 20 < J < 120 were used.. However
J
= 20 was found to provide adequate results.
If all are equal, X() will be periodic with a period of the reciprocal of the minimum frequency of the input spectral
den-sity. This problem is avoided by using random intervals for
.½. typical simulation of X(e) fromIV(f) is shown in Fig. l-a, and b.
After X() is simulated for life T, it is converted to point
proc-esses of peaks and troughs. A sample of S and n is obtait.ed
using the rain flow cycle counting method.
Results
Sjmulations of X(t) for eleven different forms of ¡V(f) at values of a ranging from 0.218 to 0.99S were performed. The
¡vif)
used were mostly of regular and smooth form (rectangular and triangular shape), typical forms of which are shown in Fig. 6. Also the process of Fig. i was used. For each simulation S(i
= 1, n) and n were recorded. The n was found to be within 5
percent of moT in all cases. Therefore n is assumed to be deter-ministic and equal to moT.
.ss andOgare estimated by S andSg, respectively, where
Input Spectral Density
Function 5(w)
valuate ,a0,a and
fron 5(w)
Senerate Randan Phase An1e,
Cirnulato X(t)
Cstjnate Statistics of
Transform x(t) to Faint
Processes of Teaks & Trouhs
tise The Rain Plow Cycle Counting Method to Count
The Stress Cycles (Magnitude
and Wumber n )
Sort S, and
valuatel_i5 and
(12)
T
±
in F 11 13 14I
CDFig. 4 Applying rain flow to a sample ofX(t)
a: p' ti-p: ti-. si. dì tF si s li:
T
-t-t Obtain The EmpericalDirtribution Function F3(s)
Evaluation of
Eva)-
For Various Values
of mFig. S Flow chart for stress siriTulation progràm
arid,
S= [7t..1
(S - S)2]
1iii (13)Fig. 7 shows the nondimensional values of
/or and ss/a
plotted as functions of a. The curves are drawn by eye through
the points. In the case where a = 1, the distribution of S
ap-proaches the Raleigh distribution, a special case of the \Veibull, the distribution function of which was defined in equation (4).
The fact that there exists such limited scatter in the data
suggests that for fatigue purposes the function 1V(f) can be characterized by a single real number a.
The Rayleigh distribution has = 2.0 and 5 = v'S
ar. For
the Rayleigh distribution (a
i), S/c.r
2.506 and saiS = 0.52 which are plotted (triangles) on Fig. 7. Simulationre-sults are in good agreement with theory. In the case of a 0,
S will approach zero. The S/o-.v and a relationship appears to be
linear.
To evaluateD (equation (3)) it is necessary to know the dis-tribution of S. The possibility of fitting the Weibull disdis-tribution to S (for a different from 1) was investigated. The Weibu.1l was considered because the S is Rayleigh where a = 1, and because the \Veibull, a compliant distribution, is widely used in engineer-ing application. Fig. S shows the empirical distribution functions of S for various simulations plotted on Weibull probability paper. It is suggested from Fig. 8 that the distribution of. S in general
is not \Veibu.11. However the possibility of modeling S with a
Weibuil is explored.
The Weibull parameters , and S are estimated by the
follow-ing forms
valid for as/S < 2.0, and
=
Fig. 9 shows the estimated nondimensional Weibull parameters
and 5/o plotted as functions of a. In case where a 1,
= 2 and S/or =
These two points (triangles) are plottedon Fig. 9 and they are also in a good agreement with the data
obtained.
For the case where the S-N curve could be represented by
= K the damageD becomes (from equation (3)),
D =
E(S") (16)E(S") is estimated from the simulation program for various a
and for m = 3, 4, 5, 7, 9, and 11 by the form,
E(S»') = Si"' (17)
The corresponding E(S") can be computed assuming that S is Weibull.
E(S-)1= sr
(i
±
where and S are obtained from Fig. 9 for a given a. The ratio,
E(S"')
Q(a, tu) =
Journal of Enginee;ingMaterialsand Technology
fl2 t
WI f
Will
Fig. S Typical forms of the spectral density functions used in simula.
tion WI I
Will
f f Wj II f f 0.5 1.0 Irregfflart Facfor,QFig. 7 Relationship of S/.t and (s2/S with
was computed for values of a from 0.5 to 1, and for the range of m given above. Q(a, m) is plotted in Fig. 10.
A generalized form of the Nolte-Harisford equation can now be written as
/
m\
(10) D = Q(a, m)1 +
- )
(20) JULY 1977 / 209(
5gk s)
(14) (15) 3.-2 s oxr ( i±
i9g 90 50 20 lo s -5 FJ % 10 u;
Fig. S F.g(s) for different a plotted on Welbuil probability paper
3
lo
0 0.5 1.0
(regulan!? Factör.Q Fig. 9 Weibull parameters are function of
Q is a correction factor for using the \Veibull for S when S in
fact is not Weibull. ExampLe
Consider a cantilever beam as shown in Fig. 11 with the di-mensions given, and a concentrated load Q(1) applied at the end.
1l'f) of Q(t) is given earlier iii this paper in Fig. 1 for which = 0.840, hg 3S.6 crossing/s, mg-= 460 peak/a, and.OQ
0.200 kV. The mean value of the load is0 0.15 \7 It is
required to calculate the mi'nimunc required beam .tliicknesst, according to the following requirements,
-(a The cies jn life N is to he at least Ti'XiD6 cyclea. o lo lo 1.0 '5 .3 .2 Oto 'n r,,= 3 IRREGULARITY 'ACTOR, O
Fig. 10 Correction factor to be applied to Nolte-Hanstord model
820 Stress Range, t M Pal 200 9 11 0.8 0.9 .0 bu4cm H 3 4 5 6 10 10 10 10 N Cycles
Fig. 11 The cantilever beam dimensions and the mean S-N curve of AST M-36
(b) Maximum allowable p, 10 (assuming that C0 0.50,
this requirement is equivalent to /.xo = 0.04; see Fig. 3).
(e) The material used in ASTM A-36; mean yield strength
= 290 MPs (42 kai) mean ultimate strength = 470 MPa (68 kai). ThemeanSN curve is shown in Fig. 11. All
factors which in6uence fatigue, e.g., surface 6nish,
en-vironment, etc., are included in the SN curve. The
straight line portion of the curve is modeled as .VS"'= K, for which ra = 4.895, and K = 1.S4 X 1017 MPa".
The beam thickness t is obtained by applying equation (20). First using the stated values of a and mn with Fig. 10, Q(a, ra) is equal to 0.68. The maximum stress S due to bending is given by
S(t) = 'I'Q(t) (21)
is equal tu unity because stress is- assumed to be directly
proportional to load.... . . -.
Therefore '1' is, (using the basic beam stressJorin'ula
g V e ç a F o S a: -V' d n: Pt w C st pr PC fe st: se' ari f o-pr Cr: 1V sp, ch: Os WC-he:
210 / JULY 1977
Transactions of the ASME
Jca -.37 .22 .50 .63 .72 .83 .95 o .998 s ox 0.5 0.6 0.7
S,,, =
= 6jL/bt2
(23)To account for a mean stress different from zero, the S-N curve
is adjusted downward using a smaller value of K, namely K',
given by (5)
K' = K
(
-\. S!
where S,,, is the mean stress given by equation (23), and K is for the zero mean case. S! is the true fracture stress, estimated
empirically as (5),
S1
S, + 345 = 815 MPa
(25)Substituting equation (23) and (25) into (24), and equation (22) and (24) into (20), the fatigue equation (equation (20)) becomes,
6L/bt "'
p(
m
1+
"I
-
) (26) 815From Fig. 9, for a = 0.840;
= 1.53, (â/o-Q) = 2.30; thus ô = 0.481 kNSubstituting the values of m, k,
, D, j, and â in
equation (26) and letting D = 0.04 it follows thatt > 2.64 cm
For simplicity, S can be assumed conservatively to have a
Weibull distribution in which case Q(a, m) 1, and the resulting
design requirement would be t > 2.64 cm. Use of the simple
narrow band formula of equation (8) to approximate the damage provides conservative results. In this example the requirement
would be t > 2.76 cm.
Conclusions
A procedure for fatigue design under stationary random
stresses X(t) has been proposed. The procedure employs a probability-based modified form of the Palmgren-Miner hy-pothesis and the rain-flow method to count cycles. The main features of the method are:
i X(t) can in general be wide band.
2 Because the linear damage accumulation rule is used,
non-stationary X(t) can be easily treated considering X(t) as a
sequence of stationary processes.
3 Uncertainties in the accuracy of the Palmgren-Miner rule and variability in the fatigue behavior can be explicitly accounted
for.
4 The procedure has a probability basis. Maximum allowable probability of failure or risk can be specified as the basic design criterion.
5 The procedure requires only that a be computed from IF(f).
The major result of this study is that, for fatigue design, the
spectral density function TV(f) of the stress process X(t) can be characterized by the irregularity factor a, a single real number. Otherwise the analyst, instead of only referring to Fig. 9 and 10, would have to perform a simulation of X(t) (as the one described herein) to evaluate the distribution of stress cycles. Moreover,
Qn.
D = - o"'
K
(
- bt/l2 - ht2
The mean stress is
(24)
Journal of Engineering Materials and Technology
the assumption that the magnitude of the stress range cycles have a \Veibuli distribution will lead to conservative results.
This approximation would not lead to unreasonable results for typical values of m encountered in practice (i.e., 3 to 5).
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