On the prediction of gfatigue life under random loading

by E. D. Poppleton

FEBRUARY 1962

ff{ H. CH WJG

OEtFf

VtrEGTUIGBCU t.INL'é Mid,itl d.c iUltl'rW \f

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oan

UTIA REPORT NO. 82 AFOSR 2258

.,

ON THE PREDICTION OF FA:rIGUE LIFE UNDER RANDOM LOADING by

E . D. Poppleton

('

FEBRUARY 1962 UTIA REPORT NO. 82

This work was supported by the United States Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638) -548.

•

SUMMARY _,

A review is given of some current methods of estimating fatigue damage and a new damage equation is derived based on the work of Corten and Dolan, and Torbe. This equation is applied to the case of a stationary Gaussian stress history and a discussion is given of the para-meters appearing in the resulting equation for the fatigue life.

1. Il. IIl. IV.

V.

TABLE OF CONTENTS

PRINCIPAL NOT ATION INTRODUCTION

DERIVATION OF A DAMAGE LAW 2.1 Discussion of Some Previous Work 2.2 Derivation of a New Damage Law APPLICATION TO RANDOM LOADING

3. 1 General Case 3.2 Gaussian Loading

DISCUSSION OF THE PARAMETERS IN THE DAMAGE EQUATION

CONCLUDING REMARKS REFERENCES

APPENDIX A: EVALUATION OF DAMAGE DUE TO

Page ii 1 3 3 6 9 9 11 14 18 20

GAUSSIAN RANDOM LOADING 22

APPENDIX B: SOME STATISTICAL PROPERTIES OF

(ii)

PRINCIPAL NOT ATION

A/"5/c)]).

E)

t,

k'rtfunctions defined on pages 30 and 31

B~ coefficient in representation of effect on

N

of

SM

(p.7)

D

fatigue damage function

d

exponent in Eq. 11

coefficient defined on p. 28 expected value of ( )

{ ) +'2...

functions of

SA

'<SM defined on p. 7

Cr

_ss

L~) power spectral density of

5

J

joint probability density

k

exponent of

5:

in assumed damage law Eq. (14)

N

(SA)

fatigue life in constant amplitude test

VL

(I-)

num ber of fatigue cycles applied prior to instant considered

.

..

probability that

S ,

$)

CS

!ie within certain limits (p.ll)

k?!>s er)

autocorrelation function of

S

S

stress excursion from

SM

*

'SA

stress amplitude in constant amplitude test

*

"SM

mean stress in constant amplitude test

*

stress excursion from apparent mean

_*

Sc:t..

apparent stress amplitude

*

SIM apparent mean stress

*

(Also in Appendix B, time average of

S

over one cycle)

•

reference stress

*

function defined on p. 12 ith time interval

time

r

complete gamma function exponent in dam age Eq. 1

cl~N/d ~

<SA

fatigue damage function

(=-])

y'( )

"...A<-- damage nucleus parameter

Y

number of times stress exceeds a certain value

P

damage rate parameter

0-

root mean square stress fluctuation about

'51V\

*

tJ

_o

i(average circular frequency of zero crossings

J

ave rage circular frequency of stress maxima average circular frequency of the mean stress Subscripts and Superscripts

•

(

conditions appropriate to reference stream

.-L

_t0p

d~t

maximum stress which has occurred prior to instant considered and conditions appropriate to this stress

d/ds

mean square value

*

All stresses are made non-dimensional by dividing by «SR

(1 ) 1. INTRODUCTION

A large research effort is being directed at the problem of metal fatigue, in many research establishments, the character of the investi-gations ranging from fundamental studies of the basic fatigue mechanism to ad hoc fatigue tests on specific structures. Considerable progress has been made in the understanding of the fatigue mechanism during the past decade,

and plausible theories have been proposed to explain qualitatively a number of the phenomena accompanying cyclic straining. Nevertheless, the funda-mental research has not, as yet, made any significant impact on quantitative methods for the design of structures subject to fluctuating loads of a compli-cated nature, and, for all but the most trivial loading conditions, the engineer must rely on "laws" of very limited applicability.

Such laws are based to some extent on the results of the latest fundamental research, but inevitably at some stage, some mathematical formulation must be introduced, containing parameters to be obtained from large scale tests. The predictions of the resulting law can thus be used with complete confidence only in the circumstances wherein the parameters were evaluated and the only way to avoid the proliferatiön or" laws to cover numerous special circumstances is a greater emphasis on basic research in the physics of the process. However, the large number of circumstances which a fatigue law is expected to cover is, to some extent, artificially generated by the fact that the majority of fatigue testing equipment, at least until recently, has been capable only of applying to a specimen a programme of sinusoidal loading (see Ref. 1). Fatigue laws have been formulated, there-fore, in an effort to describe the results of tests made with a type of loading which is probably the exception rather than the rule in a large number of practical applications. It is probable that there is considerably more hope of deriving a "law" of fairly general applicability for loadings which, like many service loadings, are truly random in character. By definition, there can be no effect due to the order in which loads are applied, and it is felt that a number of concepts common to previous analyses may be u~ed with perhaps more success in th is case.

One such concept is that of "fatigue damage". This term is used to describe the gradual "deterioration" of a metal during cyclic strain-ing, and a damage function

D

is postulated, the value of which increases from zero to unity during the fatigue test. The function J) may be looked upon as describing, in a r'ather vague way, the many complicated physical changes which take place in a metal during the initial formation of striations on the surface, the initiation of microcracks and their propagation to final critical size.

The simplest design rule based on ideafL of fatigue damage is that of Palmgren and Miner (Refs. 2 and 3) and this can be shown to be de-rivable from any assumption regarding the rate of damage accumulation, provided that it is assumed to be continuous and to depend only on the instan-taneous value of the stress (Refs. 4 and 9, for example, and see Sec. 2.1).

In spite of the known general unreliability of the Palmgren-Miner rule, it is still used extensively, firstly on account of its attractive simplicity and, secondly, because more sophisticated rules have yet to be shown to be gen-erally applicable to all types of service loading. Moreover , a body of ex-perience has been built up (see Ref. 6, for example) which enables an esti-mate to be made of the error to be expected in any set of circumstances.

Two other more recent techniques for the ca1culation of fatigue life has been shown to give good results under certain conditions . These are due to Freudenthal and Heller (Ref. 7) and Corten and Dolan (Ref. 8), both analyses reducing to the use of the Palmgren-Miner rule together with a fictitious

'SA -

N

curve. The former was derived from tests in which blocks of sinusoidal loading were applied in random order in rotp.ting beam testing machine. The probability distribution function of the stress amplitude (or

so-called "load spectrum") was the commonly assumed exponential gust "spectrum" and the resulting damage law contains a parameter which is ex-pressed not only in terms of the material properties but also those of this particular load distribution. Care would be required in its application to fatigue loads having other probability distributions . The damage rule of

Corten and Dolan is derived from tests in which a regular programme of loads was applied to the specimen, in a special rotating beam machine, and here the only parameter is a material constant and is independent of the load pro-gramme.

The concern of the present report is with the prediction of fatigue life under random loading and it is felt that Torbe has made a signi-ficant contribution to the subject with his analysis of the fatigue process in terms of the actual stress history, rather than by implicity replacing it by a series of sinusoidal stress cycles (Refs. 4 and 5). As far as is known, all previous work (and certainly that referred to above) has analysed the damage in terms of the damage accumulated per sinusoidal cycle, and has not been concerned with the distribution of this damage within the cycle. This

automaticallY precludes the analysis of random loading in any terms other than as a sequence of sine waves, with a constant mean value. In practice, the stress history may differ considerably from this picture (see Fig. 1).

In the general method of analysis proposed by Torbe, the pro-bability that failure will take place at a given instant is expressed as a power

•

series in

5 ,

the stress, and

S ,

the stress gradient. This requires the experimental evaluation of a fairly large number of parameters from tests which may not be very well conditioned, particularly if "historicai" or "stress interaction" effects are to be included. Moreover , only small historical

effects are covered. It is proposed here, therefore, that the general ideas of Torbe should be retained, but in a modified form in which some of the features of the Corten-Dolan analysis are incorporated. The most important of these features is the simple method of accounting for the interaction of the previous stress history on the current rate of damage accumulation. It is

(3 )

hoped that the number of parameters required in the resulting damage law will be reasonably small and that a fairly· simple test ·program will suffice to determine them . . It must be recognised, of course, th at the law will have its restricted rang!= of application, which cannot be expected to be much wider than the range wherein the necessary parameters are determined. We shall

not expect, for example, to be able to predict the life under artificial programmes where blocks of sinusoidal loading are repeated only a few times. We shall ex-peet, however, to be able to predict the fatigue behaviour over a fairly wide range of random loadings.

Il. DERIV A TION OF A DAMAGE LA W

2. 1 Discussion of Some Previous Work

. As a preliminary to the derivation o{ the proposed damage law, let us consider the formulation usedin a number of analyses of sinusoidal loading, viz.

D=

(1)

where VI... is the number of stress cycles applied to the specimen,

N

is the number of cycles required to break it, and

Y

may be a function of

'SA

(see Refs. 10 and 11, for example).

If

.D

is looked upon as a m eas ure of the length of a dom inant fatigue crack then we see that

Y

'> 1 agrees qualitatively with observations, giving a slow initial rate of growth with a precipitate approach to failure.

In order to apply this equation tb a fatigue loading which con-sists of blocks of sinusoidal loads, as shown in the sketch below, the reason-able assumption is generally made that the value of..::o at the beginning of one block is the same as that at the end of the block immediately p;receding it.

Hence, we may interpret Eq. (1) as requiring that the damage accumulated during the

i

th block of loading be given by

l

vt\.+t1~

/

)Yi..

N,:

VJ.~I

;

Nt.

(Z

p

'

\

Y

y (..

where

d="I } )

=

number of cycles at stress level ~~ required to do the same dam age as all the previous

(i -

I)

blocks.

Hence af ter the jth block, we have

(2)

Failure occurs at the end of the jth block if the 'quantity in the square brackets has the value unity and we see that if

i

is independent of '

SA , then we recover the Palmg~en-Miner law viz.

=1

±

V\i

INZ

(3)

It should be noted that th is result differs from that of Ref. 11, where Eq. (1) is interpreted rather differently.

The condition that a damage law should reduce to the Palmgren-Miner criterion for failure under programmed loading is given by T<;>rbe in Ref. 4. Since, in general, service loadings are not composed of sinusoidal blocks, Torbe proposes to analyse damage as a function of the instantaneous stress and stress gradient, and in terms of a damage function rather than the probability of failure used in the original paper, this leads to

where

(4) It is now assumed that the change in damage during one cycle ) is small, so that we can write, for sinusoidal loading,

I :z.

)\1'2

5

=

-S

ALL -

~~t

(5)

Torbe now distinguishes between his.torical fatigue in which the rate of damage accumulation depends on previous history and non-historical fatigue where the function ~ is separable into a product of two functions

'tJ

(SA)

and ~(J)). We then have

~ ~(1)) ~(SA)

db _

dj) (5)

where

L0.

=

J~(l:»

=

another damage function, which depends only on the

stress amplitude of the cyc1e considered, ,and .is thus independent of the

pre-vious history. Putting Eq. (1) in the form of Eq. (5) we-get

"(-I

b?

'6

-

(7)

N

,

~-I

which is an historical equation since

1)7

In fact, failure occurs when Dj..

=

1 in Eq.

however, Eq. (7) can be written

is a function of both

D

and 'SA .

(2).

lfY

is independent of ~À

(8)

-

_~

where

This is the non-historical theory of Palmgren and Miner, leading to Eq. (3). The method developed by Torbe for dealing with historical effects (Ref. 5) assumes that these effects are smal! and requires the

evalua-tion of numerous data. Since the fictitious

SA-

N

curve has proved

success-ful in the case of programmed loading (ReL 12) and of randomised loading (Ref. 7) it is proposed that this idea should be fitted to Torbe's general frame-work in the hope that this wil! lead to a reduction in the number of experi-mental parameters, and in the test programme required to determine them.

It wil! also allow a large amount of stress interaction to be incorporated.

The work of Corten and Dolan seems to be most suitable for this purpose and

we now examine their damage equation (Ref. 8): - ] )

=

\.Ao'\.~

1AY'a..

independent of in the form

Their test results indicated that it is reasonable to take

a..

as

CSA-

and, as suggested by Eq.' (8), we shall use the equation

~

_

b'/c:L _

~Y-.

'1'-!Io.

~

=-

~f

"l

Now if~ and

f

were independent of

-SA ,

this would yield

the Palmgren-Miner rule but the essence of the Corten-Dolan technique is

that the value of VVl (and hence of~ ) is a measure of the number of active

"damage nuclei" and th is is determined by the maximum stress amplitude that has occurred prior to the instant considered. The "rate" parameter

depends only on the current value of 'SA so that we have for programmed

loading the equation

(9) ' /\

where~

is the value of

~appropriate

to the maximum stress

"SA

which has occurred prior to the instant considered.

Since at failure we have

~

-=

1=

/'1

fN

.

we may reduce Eq. (9) to the form of the Palmgren-Miner· equation, thus

A. I

I

=

~-;::j

-

- /

.,..AA

N

(10)

where

NI

=

~

=

life at the stress considered on a fictitious

-:>A -

N

I diagram

(Ref. 12).

;:::,A

2.2 Derivation of a New Damage Law

It has been shown(Refs. 8, 11, 12) that the mean value of the

rate parameter

f:

in Eq. (9), at two stress levels, is given by

(f,/f,)

=-

l

~2/S, J~

where

ol

is a material constant which is independent of the values of .::;, and

-:5

_{2 '} and which may be determined from a series of programmed load tests.

lf we now work in terms of a reference stress amplitude ~~

and make all stresses non-dimensiong.l with respect to this stress, the

equa-tion for

p

may be written

(11)

and since, at failure .L:j

=- (

we have

~2

fll.

Nfl.

:= land / ' "

f

N :::

where

Nr<.

and

N

.

are the fatigue

live~

at

S~

and

$A

respectively.

Assumin~ that the log

SA

-log

N

curve is linear, the relationship

be-tween

N

and

N

R is ~A.'óN

-

Nf

and we may therefore

derive

(12)

Hence, if there is no variation in the mean stress, we may write Eq. (9) in the

form

dA

/~

)6-d

~~

- ==- ~ f(

L

~A ,:), ...

~

From Eq. (12) we see that the fictitious life defined by Eq. (10)

may now be written I

/.c

)~.~

...

I

N

=

l~~/SA I~ or

N

'5A~ --

No

2

A

oI.-b

(7 )

anel 'hence, the fictltious

SA~N

I curve' pÓasses

th~ough

the point

SA~ ~A

and

h~s ~

slope

'

d

~tr-à

'

t-.l/d~~

<:>l-ot

(Hef

~

f2). '

In order to analyse results at different values of mean stress

we now write , ,

~A~

N

~

te

SA/SM)

'

=

NR

'

and we put

-FésA

J

D)

=

1 so that

NI(

is the life at the reference level with zero

mean stress. The funCtion

..f-(SA)

SM)

may be determined from

experiment-al curves of N//'J

e

plotted against

"5

M for constant values of

SA

.

For

convenience in subsequent work, however, we shall assume th at the result of this operation m ay be represented by the equations

where and hence

=

flff(

=

+,

l

SA,

SM)

-:)A

S-d.

cL

+2.

L

SA

SM)

'S

A

*,

("SA)

0)

=

-t-

L ($A) ()) = (

+'-

LS

4 ₎

$1-\)

=

z.2

~f>CV

p 0;

ti

(

S

~)

SH

)

~

.(

~

l

SA

j

S

H .) We may now write Eq. (9) in the form

~ .1\

d

2-ClfJ ~

~

~f-

)

,/fA11.

fll

~A

zZ

l)IpOV

S,...

'SM

~

.

~:

p 2.'4;

-

- ) --

L3 Z

()p~

-SM SA

.,AA~

Nfl

t'=' ~ (13 )

and in order to follow the analysis of Ref. 4, we require to express this in terms of the rate of damage accumulation with increase in stress.

We now assume a series of damage functions which give a plausible variation of damage throughout the cycle anel yield Eq. (13) as the damage for a complete cycle. With sinusoidal loading, specification of the stress amplitude and the instantaneous value of the difference between the stress and the mean stress fixes the value of the stress gradient and hence,

a function of

S

and

S,

which gives the correct damage per cycle is

~

=

«;.-2

la

Y

d

\~.

I

~

,"-20

-

1

I

iS

I

i

cs

s...)

(14)

d

OS

LA

12

)--4

NR

S

1.

A.~

In this equation the factor

Is/loS

is incorporated to

en-sure that

d

..ó/

o

t

(=

Np ;:,

êL~I/d

S)

positive and the exponent of

SA

is

made even for later convenience. It should be noted that it is reasonable to

limit

k

to values such that

d-

21z _-I ~ 0 so that the damage gradient is

In addition to the functions of the form of Eq. (14,) we can also

choose sets of functions which add nothing to the net damage:.· per cycle, but

only redistribute the damage within the cycle.

It is shown in Appendix A, tha t the following functions have

this property for a Gaussian random stress history and it is probabie that this will also be true for any Ioading with a symmetrical probability density

I

_d.,D

(.-á )

_SA

2k?

_ot-2~-1

_I

_s(

_{1'5 I}

.fz

(~A)

SM)

ds

-

.M~

4N12

I

sI

-.-

s

5

{~()

sIte

Is

l

d-2P-( -4NR odd IAA. even vvt even (15)

We may now express the rate of damage, accumulation by means

of a series of functions such as those given in Eqs. (14,)and(15) and the resulting expres sion will be constrained to yield a fatigue life in agreement with experi-ment for sinusoidal and programmed Ioading of the type used in Ref. 8. A num-ber of such damage functions is illustrated in the following sketch.

(9)

Some of the above results depend on the assumption that the

)A-N curve for the material,with zero mean stress, is a str.aight line on a logarithmic plot, and that

?:::

FR.

st .

If the SA,-N curve, or the curve defiriing the rate function ~ turns out to be better represented by a straight line in a semi-logarithmic plot, then this fact is fairly easily incorporated in the analysis, and in fact, some of the subsequent calculations may be somewhat simpler, in that event.

IIl. APPLICA TION TO RANDOM LOADING 3. 1 General Case

In the previous section we have obtained an expression which is capable of describing a wide variety of plausible variations of damage through a fatigue cycle. This was based on a harmonie variation of stress with time and, for random loading, we now require to derive a function appli-cable to a more general stress variation. However, we may reduce the

generality of this variation appreciably, and still cover a useful range of

applicability, if we recognise that, in practice, the stress variation will arise from the response of an elastic system to random excitation. This will mean that the power spectrum of the stress will be concentrated around a number of natural frequencies, and, in some cases, this number may be quite small. An example of such a limited response is that of an aircraft wing to atmos-pheric turbulence, where the power spectrum of the input is such that the response is frequently concentrated in the rigid body and fundamental wing bending mode.

Stress histories corresponding to power spectra of the type just mentioned have the appearance of modulated sine waves (see Fig. 1) and it is reasonable to assume that, at any instant, the rate of damage accumula-tion is the same as that on a sine wave which corresponds closely to condi-tions at that instant. We define such correspondence as shown in the sketch, and obtain the following expression in terms of the instantaneous apparent "mean" stress (

CS"", )

and stress amplitude (

"SQ,.

):-I

s

{ ( \ L - - - J - - - -________________ ~ ______________ ~---~---~~--rIME

(16)

where

?tJj::l

.

is the average circular frequency of the peaks in the stress

his-tory. From this we obtain

•• <S V\o\. - 'S

+

~M

+

.

-s

• "2.. •• "2-"S -+- S (17) •• S

+

'S M - <S~ = - . "S

.

.

.

where CS and ~ are time derivatives, made non-dimensional by dividing by

AJ P and hJ~ respectively.

We may now use the damage expressions derived for sinusoidal loading by making the substitutions

A

but, in addition, we must now make some estimate of the value of

(~~R)

at any instant.

In the work of Corten and Dolan with repeated blocks of

sinu-soidal loading, the value of~ was that appropriate to the highest load in

the programme. Under random loading, however, continually higher peak loads appear in the stress history as the test progresses and consequently

we expect the value

of~

to vary with time. It is not possible to say at the

moment whether this variation of~ is of importance or whether it is

sufficiently accurate to assum e, as a mean value, the value appropriate to the stress corresponding to some chosen low frequency of occurrence, as

recommended in Ref. 13, If the variation of~with time has to be taken into

account, an experimental programme wi11 be necessary to determine what

combination of "So.. and ~IN\ is required to fix the value of..A-A-, and whether

one or more applications of this combination are required to "aetivate" new "damage nuclei" and henee ehangeA-.

For the present, we shall proe~ed on the assumption that

onee a certain value of..-A-t-has been fixed ( =.,.AAi... say). an interval of time

-;z

elapses before a new eombination of ~~ and~Wl has oeeurred a

sufficient number of times to change the value of~ to~:'4'. We may now

evaluate the damage aeeumulated during the time interval

ï1

using an

expression for

otA/d6

whieh contains a series of functions of the type given

in Eqs. (14) and (15), in whieh are made the substitutions for

'SM

,SA

and

'S

given in Eqs. (17.) If we write this series of functions as

b't...

('5)

S,

s)

and substitute

d~

_

d."E.

dl-

_

AJ~-&dt

e>lt-we have

_{o\A -}

~L(S)S)g),wt> ~

de

,

(11)

.

.

..

.

the amount of time during wh.ich ~ ... ~ )

S •

lie sim,;~taneou~ly b~tween the

lirn its (S and

s+d

S ), ( 5 and ~+,d S ) and ( S and S +

d!. )

respectively, in a time interval

IZ '

is given by

oU-:=

Tc:.

:r

L

s)

i:,)

oS

J

J.

~

cl

s

J

s

• • •

If we further restriet the values o~~.> CS and ~ to lie between the limits

-+

S+

and-

S- ,

+

~

and -

~-

, -I-

S·

and -

~-

respectively, these values being chosen to give the combination of

Sc:t

and S,,-\ which when exceeded

a given number of times will change the value of ~ from~ .. to ~t·.1 ' we obtain :=-

TL

J

{~)

5)

~)

ot

~ ~ ~

cl

G

+g~ +!... ~~+

f

J

fJ(~)S)~)d5d:5dS

-

~-

-1·

~_ ., - $ where

?

t

=

We may therefo~e write the damage increment during the interval as +

1+

~

t...

+ $ ....

b i

~

.N,,-::;:

f

I

I

A~

h,s)s)r(s)!.)s)

$

c:l.sdf,cls·

(18)

11.

-3-

-;-

-~-and ~he failure time is found by adding these and equating the result to unity i. e. ,

3. 2 Gaussian Loading

We can go no further without assuming a probability density andthe most obvious choice is a three-dimensional normal dis'tribution. Not only is there a large body of literature concerned with this distribution but a number of important structural inputs appear to be good approximations to a Gaussian process (e. g. atmospheric turbulence, Ref. 14). Moreover, a Gaussian random nois-e- generator is a convenient device for exciting labora-tory fatigue testing machine.

Evaluating the terms in the joint normal distribution (see p. 118, Ref. 15 for example) we find that

5

is statistically. independent of

S

and

S

and we obtain

-:JI..

SJ {,

,S)

=t"r53/·I ....

i~-"'4Eit"

f

H"

S'42.H~SS

+

H31 S

2

+

H·.,

S'3J

where

~

1 \ .

~. ~~)

H

\~.:

("?J-)

7. ) . tv\.!3:

~.

.g

M,

z ;

S~. S~

-(1:l.J

~

1"4

I

=

sz.

[~1..-g;1

-

C~

)

J

This may be cast in a m~re convenient form, if we introduce the ratio of the number of zero crossings to the number of peaks, which gives an approximate measure of the irregularity of the loading (Ref. 16).

If

c;.:s~(t.:l)

is the power spectral density of

S ,

we have the following results "2

f~0""

Gns[wj cLv

tUI'

=

_{l~ .07-(ï$S (w) d N} o

s

- 0-2. (say) ."'2... S

_-

_I oè

f

tAJL

bss

{w)d

t0

N'2.. ~ 0 I

f

0 t.)-4

L::;s",,(w)d

L-U

-h,)"" ~ D which give

_S

., 2

_-

...-;.

_-s

_-

₁₂

2.

_r:::r

2 and

There is no great difficulty in evaluating the integral, in Eq. (18), provided th at the range of integration is made infinite and we ob-tain the very simple result (see Appendix

A):-(r even)

r

= complete gamma function

Due to the symmetry of

J ,

all the correction distributions given by Eq. (15) contribute nothingto the total damage and, since the result is independent of

k ,

it is therefore independent of the nature of the series of which Eq. (14)is a representative term. It appears that the details of the damage distribution within a cyCle are irrelevant in th is case and only the overall damage per sinusoidal cycle is important. Bearing in mind that

Np

TI.

/2

ïr is the number of random "cycles " during the time

-r:. ,

Eq. (20) is seen to be remarkably similar to its constant amplitude counterpart Eq. (13). The stress amplitude in Eq. (l3,}is replaced by (2.

\Te

(.J2x.RMS of the stress excurs~on from the instantaneous mean, $""" ) and

"SM

is re-placed by ~\110'\ • The latter term is a function of

"5

_M and

f2 (,-

cz2)

Yz

V

(13)

( J2)<..

RMS of ~ \M ) and has the following lim iting behaviour

-=

The use of infinite limits in the integration of Eq. (18) does not restriet the validity of the result unduly, since the integrals converge quite rapidly and the difference between the value with limits at =40-(say) and that with infinite limits will be quite small. Hence, provided the com-bination of <S;;'et and 'S~ dictating the value of~ requires values of ~

,cS ,

S

: 7 4 V , the result given in Eq. (20) will be satisfactory. In a Gaussian process, the stress exceeds

L4

CT after a relatively short time, since we have (p. 127, Ref. 15) for the time to exceed

S

once, on the ave rage

"-z.

:2Tf

5/7.,,"2--e

A.Jo

The number of random "cycles",

.A, 0- is thus

Al

p

-c.

=== .-<.J 0

-;;tr

;;L-

-rr

~cr ~

rr

7< on the average, g"

-é'/e

-

.

.

before the stress exceeds

~O()O

Consequently, provided

V

is small and the fatigue life is relatively large (>:;::. 3000), the error involved in assuming that Eq. (20) is va lid throughout the lifetime will be small. In these circumstances, the average failure time

(Ti )

is given by

L.LI"

=

1

= {

0;

erl?

)-'1

f

~

BP<j.-

r (

'1'+

~

+I).J:

(Jz

0-11.

r

j~~

'Z

~

1;

(21)

4=~T;,

If K is put equal to unity in Eq. (20), we recover the same

result as would be obtained if the Rayleigh distribution SA were substituted in E q. (13) i. e.

bi-

=

I~;.

.

)

AJpïi

(~;-'J)d

Z

Z

_BpCV

f1 (qt+

~

+

IjC::;~

(J2

q-'t"

( ..,A-AI!... 2;;

Ne

P 9;

a re sult which is applicable to a system with a very narrow band response.

' .

.

A

The relationship between~ ànd

-r

requires an experim ental /I

investigation 'and a plausible hypothesis might be that' the value of /-"'i is de-termined by a stress rriaximum

Si

which has occurred

Vi.

times previously. The time interval

-r:

is then the time between this event and the occurrence of the. event

whic~

ch.arige·s the value

o~~ f

.

rom~~

to

A~+,.

The time

~.~

1

elapsmg before Sc: lS exceeded

V,;

times lS, on the ave rage :z.1T"VL./t.Jo

.e..

Z<!

and this may be used in Eq. (21). Alternatively, it might be assumed that a given combination of the stress excursion (.c1 ) and '5...,. must be exceeded a certain number of times in order to fix the value of ~. Some notes on the statistical properties of SIM are given in Appendix B, and these may be used in the interpretation of an experimental investigation of the behaviour of~ (see Section 4).

...

As another alternative, we might consider ~ to be a con-tinuous variable and replace the summation by an integration, thus

~(;~;;

)7,

"ti

(~,)

cd

(J-e

;Zh<T»

where

S+

is the limit required to

make~ ~~

=

1.

,/\ If it turns out that it is sufficiently accurate to assume that

~~f!. ü:; a constan~ however, and appropriate to some known combination of

~

and

~"-\

( "'::>"

and~...

say) then we have

which, in view of Eqs. (17)and(19,) we may write =

for a given normal loading. Consequently, Eq. (20) becomes

A discussion of the parameters appearing in the above equa-tions is presented in the next section.

IV. DISCUSSION OF THE PARAMETERS IN THE DAMAGE EQUATION One of the most important parameters appearing in Eq. (21)

A

is ~, since

it

is this that describes any 'historical' or 'stress interaction' effects that may be present. lf the ideas of Corten and Dolan are applicable to the random case, it may turn out that, in some circumstances, it is rea-sonable to consider...--M- to be a constant for long periods in the later life of a specimen. The reason for this may be seen from the following table, which gives the average number of cycles before a given stress is exceeded once in a narrow band-width Gaussian stress history.

Stress l-s/cr 4.0 4.5 5.0 5. 5 6. 0 6.5 (15)

"

No. of cycles af ter which (5 ) is exceeded once, on averáge

3.0 x 10 3 2, 5 x 104 2.7 x 10 5 3.7 x 106 6.5 x 107 15.0 x 108

Let us consider, for example, a specimen subject to a narrow band random load with zero mean (

K'-,

I ) 5_M .=-0 ) and having an average life of 3. 7 x 10 6 cycles. We see that, during the last 90% of its life (i. e. from 2.7 x 10 5 to 3.7 x 10 6 cyc1es) a stress amplitude of 4.5

cr

will be exceeded about 136 times, 5

cr-

will be exceeded 13 times and 5.5 Cl once, on the average . It seems very likely that a repetition of 136 cyc1es will be sufficient to fix the value of~, but it is also possible that the stress of 5 () wil! have been exceeded sufficiently often for this purpose. If we were to say that ~ was a constant during th is period and appropriate to

S

=

5

cr

then we would expect our estimate of the damage during this, the major part of the fatigue life to lie between the following fractions of the true damage

(

6-d

0' \

and

~,5" )

since

The latter fraction is a rather pessimistic estimate since it assumes that the true life is dictated by a stress amplitude which occurs only once on the average. Nevertheless, we see that, provided the exponent

(

~

-cl )

is not too large, the damage is not.undlily sensitive to the exact value

of~ , and since in Ref. 12 it was found that (~-d )

=

0.4 for aluminum and

1. 2 for steel, in these cases, at least, it appears that a precise estimate of

A .

/V'- may be unnecessary. If this were found to be generaUy true, then

Eq. (20) might be used in place of Eq. (21),to estimate the fatigue life of a specimen, provided the life were of sufficiently long duration. The error involved in the calculation of the damage incurred during the first 10% or so of the life would probably then be within the experimental accuracy of any fatigue test, in this case.

The above remarks were based on the assumption that the band width of the stress history was very narrow and the mean stress was zero, and, in addition to an experimental investigation of that, rather un-usual, case, it wiU be necessary to establish the general behaviour of the parameter ~. A possible experimental technique for determining its varia-tion with time is to interrupt a random load test af ter a known time has

elapsed and subsequently to break the specimen in a constant amplitude

fatigue test. A program in which systematic variatlons are made in the RMS stress and the value of

7?

of the random load, and in the am plitude of the subsequent sinusoidal load, would enable the value ofP at any instant to be

identified with that occurring in constant amplitude fatigue at a known level. Further, a correlation could be sought with the known average number of high loads which had occurred during the initial random loading, using the results obtained in Appendix B.

The function of V and

K

appearing in Eq. (21) and (22) may

also have a profound effect on the rate of damage accumulation. Consideration

of Eq. (22) shows that, if the function were to vary only very slowly with cr- ,

the

q---ç:

curve would be very nearly parallel to the SÁ -

N

curve, on a logarithmic plot. This is certainly not generally true, as may be seen from Refs. 17 and 18 which give the results of bending tests on small notched speci-mens, with very narrow band loading. Moreover, the results of constant amplitude fatigue tests with a steady mean value also suggest that these func-tions may well have a significant effect on the damage.

It seerns verllikely, however, that in general the values of

the parameters in.t, and

+2.

will differ _ from those measured in a fatigue

test with a constant mean stress. This is substantiated" to some extent by

the results of bending tests on notched steel specimens reported in Ref. 19, where it is shown that the life of a specimen was less with a stress history

s

==

than with a history

'S;- ~l ~ t;)\> I-- +

'::;2-The factor involved was about three but, in all cases, the maximum stress

( ~, + Sz. ) was high, being greater than the yield stress of the material.

Nevertheless, the fluctuation of the mean was quite slow

(t:AJ

p . . 215.e0"",)

so that the change in the mean during a single stress cyc1e was smal!, and a further investigation of this remarkable result would be of great interest in the present connection.

In view of the above, we expect that "B/:>'1f rnay be a function of

the ratio tV..,..". /

t.J

'p and it seems that the most satisfactory method of

deter-rn ining these param"eters is from a program of tests in which the mean

fluctuates. This might be a program of random load tests, using a machine

such as that described in Ref. 20, which enables a range of 1( to be covered.

A series of tests would be made in which

K

is held constant and the

varia-tion of the fatigue life with 0- is measured. These results would be analysed

on the basis of Eq. (21), in which a limited num ber of terms would be retained.

'"

The value of./"'" would have been previously determined in the manner indicated earlier in this section. Knowing the shape of the power spectrum, the mean

value of the ratio NM/Wp may be determined as indicated in Appendix Band,

hence, the relationship between 5\>q,t and

w ... /

_Np obtained from the solution

of a number of sirnultaneous algebraic equations.

An alternative programme for determining the values of the

E:>

Ioq.\

is one similar to that used in Ref. "19, in which the stress history

i ' . (17)

(23)

The analysis of the results is somewhat more complex in this

case but it has the advantage that the parameters may be found from tests

using commercially available equipment. This may be a matter of sorne con-sequence if it can be shown that Eq. (14) gives an adequate description of the

fatigue process for a large range specimens. From Eq. (23) we derive the instantaneous 'mean' stress, using Eq. (17) and obtain

The test results may then be analysed, using a damage equation including terms like Eq. (14), viz

Terms like Eq. (15) are not included, since the loading is symmetrical and it is not expected that such terms will affect the total damage. Moreover, from the results of the analysis with the sym metrical random load, we expect that

the final damage equation will also be independent of

R ,

although this seems

to be difficult to establish in general.

In order to facilitate the analysis, suppose that the tests are arranged so that the ratio AJj:>/.:0I.M is an integer and the phase angle is zero. We may then express ~tJ~r as a terminating power series in ~

NI.Nll-and

d

S in terms of d. (~N...,I-) thus

ot

~

=

(-SI

Z

C\.."

~?0...,

I-

-+

<'::.2.

J

c;Á

~

0"""

t-

((;>C!L'Ö)

Eq. (24) may now be integrated over one complete cycle of the mean stress on the assumption that this is a small fraction of the total life expressed in terms of cycles of the mean ( N"'I)' We then obtain

Damage per cycle of the mean

=

~ I

(25)

/_:;;t

ol-2.l>-1

(~+S-+SHj\-=>

0-1

oAs

~

The value of'::" would be constant af ter the first few cycles, as in the pro-grammed tests of Corten and Dolan, and this could be obtained experirnentally by tests of the type suggested earlier in this section.

The integrand consists of polynomials in

~?UM

t- .

'

The integration of this with respect to ~ tJ ... 1- may be carried out numerically and the computation of tables of this integral for various values of parameters is not an unduly difficult task. Failure occurs when

I

=

d:..

~: ~

LJ

2i

.2

~rcr

c

I

~M

.

cs~'p-P

-

1;

~ ~ ~

rJ

fl- \'Z- y:. q, ~

r--v

in which C~ is the proportion of damage due to the rat~. of damage accumula-tion given by Eq. (24.) This parameter does not appear in the analysis of ran-dom load fatigue and it is felt that T.\l~ will, at most, be weakly dependent on

R ,

so that it may be possible to omit the sumrnation on

R .

The dependence of the f\\IIS on A) ...

~\?

may now be deter-mined from a series of tests in which St-I/S, ) <;I/S2. ~.J N~/AJp are varied, and the results are analysed using a limited number of terms in fq. :(25.) It

would then be reasonable to assume that these values of the E>p~5 would be applicable to fatigue under a random loading having the same mean value of

tJl.<-1./~", calculated in the manner of Appendix B.

v.

CONCLUDING REMARKS

Using the ideas of Torbe (Refs. 4 and 5) and Corten and Dolan (Ref. 8) as a basis, a damage equation has been derived (Eq. 21) which may be used in the analysis of fatigue tests in which the stress is a stationary Gaussian random process. Experimental determination of the parameters in the equation will then allow the application of the analysis to a more general type of random loading.

As an example of a more general application, Eq. (21) may be used to determine the damage with a non-stationary Gaussian loading of the type proposed in Ref. 14 as a description of atmospheric turbulence. Here, the stress history would consist of blocks of stationary Gaussian loading, each block having a distinct mean square stress and the order of application of the blocks being random with a known distribution function. The value of

-no

appearing in Eq. 21 would then be the length of time during which

cr

remain-ed at a given value, provided that.--C--remained constant during that time. If

a loading block were such that the value of~ changed during the block, then two or more consecutive values of

Ti.

would be associated with the appro-priate value of

v .

It is thus possible to ca1culate the fatigue life due to any desired "load spectrum" and one is not restricted to that associated with the response of the structure to stationary random excitation. For example, with very narrow band response, a varietyof."load spectra", differing markedly from the Rayleigh distribution, are available.

The analysis mayalso be applied to a stationary random load-ing other than Gaussian, provided that the joint probability density,

"Tt~.)

5)0

is available for substitution in Eq. (18), and the power spectrum is known for the evaluation of the average frequencies.

- - - -- - - -- - - -- - - "

(19)

A programme has been initiated at the Institute of Aerophysics, for the investigation of the parameters in the damage equation. Once these are established, for a given material, it is felt that it wiU be possible to pre-dict the fatigue life for a wide range of random loadings, provided that condi-tions do not depart markedly from those in which the parameters were

evaluat-ed. For instance, it would not be possible to take into account temperature

fluctuations nor is it expected that the analyses would apply to a load history containing a large number of stresses weU into the plastic range. Nevertheless, this stiU leaves a large number of possible histories of an extremely practical

kind, namely, random oscillations, of fairly low RMS stress, about a mean

1. Freudenthal, A. M. 2. Palmgren, A. 3. Miner, -M. A. 4. Torbel 1. 5. Torbel 1. 6. Naumann, E. C. Hardrath, H. F. Guthrie , D. E. 7. Freudenthal, A. M. Heller , R. A. 8. Corten, H. T. Dolani T. J. 9. Shanley, F. R. 10. Marcol S. M. Starkey, W. L. 11. Liu, H. W. Corten, H. T. 12. Liu, H. W. Corten, H. T. REFERENCES

Fatigue of Materials and Structures under Random Loading "WADC - U of Minnesota Conference on Acoustical Fatigue", March

19611 WADC Tech. Report 59-676.

Die Lebensdauer von Kugellagern:

Z. V. D. 1. I Bd. 681 No. 141 1924.

Cumulative Damage Fatigue , Jour. Appl.

Mech. I Vol. 121 No. 31 Sept. 1945.

A New Framework for the Calculation of Cumulative Damage in Fatigue. Part I: Non-Historical Theory, University of

Southampton Report No. 61 April 1959.

A New Framework for the Calculation of Cumulative Fatigue Damage in Fatigue.

Part II: Historical Theory, University of

Southampton U. S. A. A. Report No. 1111

July 1959.

Axial Load Fatigue Tests of 2024-T3 and 7075-T6 Aluminum AHoy Sheet Specimens under Constant and Variable Amplitude

Loads , NASA TN D-2121 Dec. 1959.

On Stress Interaction in Fatigue and a

Cumu-lative Damage Rule , J. Aero. Sci. I Vol. 26₁

No. 71 July 1959.

Cumulative Fatigue Damage. Proceedings of the International Conference on Fatigue

of Meials 1956. Inst. Mech. Eng. London.

Strength of Materials , McGraw Hill Book Co. 1957.

A Concept of Fatigue Damage , Trans.

ASME, Vol. 76, No. 41 May 1954.

Fatigue Damage During Complex Stress

Histories. NASA TN D-2561 Nov. 1959

Fatigue Damage Under Varying Stress

13. Corten, H. T. 14. Press, H. Meadows, M. T. Hadlock, 1. 15. Bendat, J. S. 16. Kowalewski, J. 17. Fralich, R. W. 18. Fralich, R. W. 19. Nishihara, T. Yamada Toshiro 20. (21)

Design Considerations for Limited Fatigue Life Under Complex Stress Histories. SESA Design Clinic. SESA Annual Meeting Detroit 1959.

A Re-evaluation of Data on Atmospheric Turbulence and Airplane Gust Loads for Application in Spectral Calculations.

NACA Report 1272, 1956.

Principles and Applications of Random Noise Theory. John Wiley and Sons, 1958.

Fatigue Lives Under Random and Pro-grammed Loading. FuH Scale Fatigue Testing of Aircraft Structures. Pergamon Pre ss, 1961.

Experim ental Investigation of Effects of Random Loading on the Fatigue Life of Notched Cantilever-Beam Specimens of 7075-T6 Aluminum AHoy. NASA Memo

4-12-59L, June 1959.

Experimental Investigation of Effects of Random Loading on the Fatigue Life of Notched Cantilever-Beam Specimens of SAE 4130 Normalised Steel. NASA TN D-663, Feb. 1961.

Fatigue Life of Metals under Varying Re-peated Stresses. Proceedings 6th Japan National Congress for Appl. Mech. 1956. Bulletin and Annual Progress Report 1960. Institute of Aerophysics, University of Toronto, Oct. 1960.

APPENDIX A

EVALUATION OF DAMAGE DUE TOGAUSSIAN RANDOM LQADING

We require to evaluate the 'integral in Eq. (18) for the special case where the loading is Gaussian. The damage rate

61-

contains terms like those given in Eqs. (14) and (15), and we express all these in the form

b,d

~~ (~)J-(r;lj~(~)b ~ ~ B~~ S~ S~

where, for Eq. (14) and the first of Eqs. (15),

j.

may not be an integer but is counted as being an even number, and generally, the exponents

a...

and

b

m ay be zero or unity.

We substitute from Eqs. (17) and consider only one term in the summation. We thus require the value of

SSS

(~'+;';'

)1>+'\1

(-;.;)j.

('~'r(~

t ("

+

'i,.+

SM

')'p

~

JlS

,

5.,

S)

cls.<s.J

s

where

and investigate only one term of this summation, e. g., the term

-s~ ...

_SSS

és~+

s'

1''1'

(s+./(-

5jt'

ttf(~f

$

J(S, S, S)dsd

s,J.5

Considering first the integration with respect to

S

and assum-ing that the, as yet unspecified, limits of the other integrals are independent

of

"S ,

we have ' .

J~:+:+:,;5'" 4 [- :>-~'<T'

{Cè>S';_?:,":;'

+!,' +-

;,~)] d~

s~-1.-"ó;4-g ...

~ r(~~sJ-t-~L--

(S-/-5)'-

_

J .

.

',..

".2-( 1-·p}fl"l. .,=-~- ,

'

. '

'

wpere

-x

=

(23)

(S+-

:SJhe

l-e'-.)CJL.

We now wish to investigate the limits of X wheri

"S

has the (large) values+

g-l-

and -

S- .

and to do this we consider the following sketch showing possible extreme values of

"5

_M

'

5~~-,----Bearing in mind that S -t-

5

-=

"S ... -

SIV\. we see that for points like A and B) S +S=-o )since the instantaneous mean stress is approximately

SM • if

S+

does not differ greatly fr om

,5,-

.

For C • however. we have S ... , SM

+

g+

so that ~ +--

5.

'.

~+- • Similarly. we have the following approxim áte'välues:

'S -+- ;S

.

A At E .l-_-:z. 'S

.. .

_-~

"'-At F ~+~

-

.

S At G OS +

..

s- .!-

.

...L ~ "'-+ OS d A~ A_

an • consequently. we see that if S is limited to the range from +$ to -5 then the (~+ ~) is also lim ited to the sam e range. We now suppose that the value of the two limits is several times the RMS stress fluctuation

cr- ,

and hence deduce that the limits of

:c

will be fairly large numbers. being the square of this quanti~ydivided by

(1-

e

2

.,) • Inspection of tables of the

incomplete gamma function shows that the value of the integrals in Eq. (Al) approach that of the complete gamma function quite rapidly. and we assume that the values of ~ are such that we may make this approximation.

Hence, the integration with respect to S results in

[ .;---+ I -

S--+ ;;: 2. ï.î I .-{--:t...-I- I )

"2. / _c 1-

e

2 _..J \~2.J _~ --::;c _-.;... D 2 ~~c::r"2. " l _• reven _,

the value being zero when r is odd. We now have to evaluate

S~-"- [2(1-i'JrJ~ I'(k~,

)

SSé."

+

'$>j~"v

{, {-

;:fCi

~~(~J~ ~:~~ol sJs

and expanding the binominal we have

".l. "2

~

(~-I-CV)/

'S

r-""'t-:;/

,-e2fl~1

~

r

('!:!.I

)JJ

~

2.c:y42\z-2L-+1

~:2

L (-

5)d-(Ii.I)ct~~0b~~~~~d

G

ZJl'/(I..' ~~\I M " - l J -,... , , ..

, . ""'Cj. 1:;, :s ~

(..=-o

Assuming again that the limits may be taken as infinite.

respect to ~ is

the integral with

...tI ,

106

~

2

~-p

, - 2

~

(I

!

I)

C{

~

if a

=

1

o

if a

=

0

The last integral is .'2 .

ob '

!i

"1)6 -

$~

r/o-"1.

J

;S2~

(-

~)C-

l!

t).e

~

d

S

- o ó

and we see that this is zero for the following cases

,

cf odd. b

=

0

•

cr

even, b

=

1

For the other cases (

J-

odd. b = 1. and

ct

even. b = 0) we

obtain the value

We have shown. therefore. that~ for functions like the first

five of Eqs. (15) the net damage is zero with Gaussian loading. while for functions like Eq. (14) and each of the terms in the last of Eq. (15) we obtain

(25)

Now k; q, i are integers so

r(e-fq.t

-l..

+1)

=

(k-f.cy-iJ.I

and, by expanding the summation and inspecting the terms, we see that

2:

(1).;'t:,1

r

(i

+

./:~f)

-

(b

q, ...

J{:-I)

r (

J--î'

+

"-+'1')

t

=0 ~.

Cc +0/2.

Substitution of these relationships into the last of Eqs. (15) confirms that no net damage arises from this term, while the damage result-ing from the use of Eq. (14) is found by puttresult-ing

ct

.=.

d

-2

e-I

in the above

formulae. This yields ~

~

p/

S

P--t-"

[2 I

_r

-e2jçr2

7"'~'

rti:±l)

2.(2-'J7e2J7:+~+(1

11-

i

.ta'j

~

oe

~

--t.'

(p--t)

( M l

.J

7-

l~

-2. 2

J

2-

-v

and, on incorporating all the omitted coefficients, from Eqs. (14) and (18), the

factor (

d,-z

~

) cancels to give

bi..

=

A~pTL C[ï<rg)~ 66 5p~ r(9J+~+I)J~

(JîrgfCV

,./'l( 2

-rr

N~

p CV

In the above,

-p

has been put equal to unity since infinite integration limits ~

have been used, and

APPENDIX B

SOME STATISTICAL PROPERTIES OF THE MEAN STRESS

B.1 Expected number of Zeros of the Mean Per Unit Time

It is suggested in Section IV that some of the parameters in the damage equation may be functions of the average frequency of the "rnean" stress. To evaluate this frequency, we require a quantity that gives an over-all description of the fluctuating m ean stress. It has been found that, instead of the instantaneous mean given in Eqs. (17), the use of the foÜowing expression gives good agreement with experirnental results:

-N - t-+i0

vp

rt-"""~JP

J

"S

~

(

l:-)

=

î!ir

L

loO

-S

cl l- - --.l.l "5

eH-The syrnbol

'5""'1.

has been retained for this rnean stress, since confusion with the instantaneous mean of Eqs. (17) is unlikely.

in which

The non-dirnensional time derivative of "S1.4.<\

is:-Now, the expected frequency of

'S"",

is given by (Ref. 15)

E

[{d--k'

t]

E"

C

cs,.!: ]

T

11-

d,

r

ts

(l-

+-"%.) -

~,(t--1f-1.J.)Y-

oH-/47Cü - T .. . "'f'"

iJ-

;T

r

[s·

(c+

TJfv,

j-:.-

s(

f-FJl1J, ')

5

(f--1Y

_Np

j+~2 (f--~.

)Jolf-\-170()

)T

for a stationary random process,

(27)

where

~S(I)

autocorrelation function of the stress. In terms of the power spectrum therefore we have

i{l-

~

~';;' )~,;,;{4l)

C!;.)

Similarly

000

~~

E

(S~)

-

J.

l

1-

~2~~) ~1(WJJ~

where G,X'x(

tJ)

is the power spectral density of:C . However, we have two alternative expressions for the autocorrelation function {(ss viz.

"E!:><i. (""'IJ:-

s:..o

~~

(w)

~t>-")

N'

~

?V

72~~

l'l)

=

~~m

(1') :: -

Ç)2-~~~)

=

t~~ ~~(~)

~

~JY

cJt:,.)

Hence, taking the Fourier cosine transform, we get

Gr""s(~)

= 2.

F...::le~(\') ~t-'--?N'YcÁ,

-;r

Jo

so that we may write

Finally we have

This formula has been applied to the results of which the two right hand traces of Fig. 1 are samples. The power spectrum of this stress history is shown dotted in Fig. 2, and, using this, the following ave rage fre-quencies were obtained

Frequency of peaks

=

55 cps

=

t-

_p Frequency of zeros = 43 cps =

+0

Frequency of me.an

=

22 cps

=

+-'-If a rnean stress curve is sketched on this trace, by eye, it will be seen that the above results are in fair agreement with the average values which may be derived from these limited samples viz.

+p

=

65 cps

to

=

41 cps

f~

=

20 cps

A

B. 2 Expected Number of Times

S

Exceeds

S

While,5-...., is Greater :x

Than 6~ in a Given Time Interval ' ...

First we form the joint normal probability density function of

5 , S

and

CS>\.!

thus

where in the notation of Ref. 15,

IMI

=

J

1l

d'2

-cÁ'l

J.,~

d,!>

d1..\

d2.2...

d'Z.3

d?>,

Oh2 d3.5

E

L

~z)

_-

-E(~

S')

-R~cr"J..

-r

iJ- /,

f:j.

$'

d

t-

-=-0

-

J.

at T-?t>C -T

,..

~

4

EeSS~)=-

'U2'TI

[X(I--I"r)S(t-)

-T((~-~)~((-)7dt-T~cXJ

'

'j

-y -== '1f4,Jf:>

=

1J--:>.~

t'[SfN'r)

s(l.) -

s(? -

'Y)

i,f

t-

j]

dt-I~~

-r A)p

Np

'Y",- ;r~.)

r

_ 'b

21, ( [

sfl-)

S

LI--'Y) -

sf

f-')

M

1c--t'YJ]

clt-T-z7C>Ó _,

Np

~b

'Y=

~Jp

(sinee process is stationary)

(29) 06

d

_'3

_J.-[Y

_~~(~)~

_~~

_otw

-r - . ~ I

=-

, .. 11 #.1.

V

()

A.>~

_

... -.~. , .

0\31

..

o(Z'l.

;=

E

(

~l)

-

iu!sJGr ..

(.:J)dN

and, in a manner sirnilar to the derivation of

otlJ

we obtain

oI>~ =E(sS~):

--t

[(f;fJ&.J~)~~ ~olA) ~ oI~2.

J

33

-=

E

(5:')

=

~2 i(~)2.

(I -

~t>-j2ïïAJ

\

~~(t.j) d~

o Np )

The M's are the co-factors of the determinant

l ""

I

so we

have

tv\.

/1 ~ -

d

22

d

33

dL::

M'7-

::::

ol

'l-'".;.

d

_{3 ,} :

Hz./

M'3

:=-

- d-z.

_z

01

3 / ::;.

1'1

31 M.1-2- ==

2-ol"

d

3S

- 01'0

H2-3 ~ - d i l

d

32. =

M

32 ~

oC,dz7-We now form the conditional joint probability density function

f(s,~)

=

r

TU,s,

5~) dS~

~

l"'d'$~

7iQ

7..0 Td"

d ".

s.... J~ J--o ~ ~

which gives the probability that

-;,W

and

s·

simultaneously lie in the range

S

to

6

4

d/S

and

'-5

to i,' -+ 0

s;

respectively while

'5 ...

is greater than

,.,.

::5... .

Fro.m this we rnay then determine the expected number of

, f . A . . -r- L

times that

.s

exceeds

5

in a time: interval , while --.::J ... is greater

""

than 6,...., from the formula.

<=><>

'-J

=

T

f

I

~î ~

(g

~

)

d

~

_ 0 6

s ...