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APPLICATIc

OF

IN FATIGUE ANALYSIS

January

1985

Turan

DIRLIK

Ph. D. Thesis Department University Coventry,

submitted

to

of Engineering of Warwick England

-1-

I providing guidance,

am most the but

sincerely

grateful subject

to

Dr. his

F.

Sherratt, generous

not

only

for and with

research also for

and for

most

advice

his

never-ending

encouragement

and patience

me throughout

my years

as a student.

I his

also

wish

to

thank

Mr. A.

Redhead,

the

Chief

Technician, and for his

for kind

expert

advice

and help

over many practical

issues,

and friendly

interest

in my work.

I wish her unwavering

to

express

my sincere and faith

gratitude in me.

to Miss

L.

H. Heah, for

moral

support

Lastly, at such short

I am much indebted notice

to Mrs.

C. Gow for

typing

this

thesis

and to Miss N. Gvero for

additional

assistance.

-ii-

SUMMARY

The availability reasonable appraisal reviews cost, of

of

minicomputers a significant

and microprocessors, stimulus methods. in This

at thesis

a

has provided fatigue testing

a critical

and analysis

and extends some of the recent fatigue analysis methods. Two in detail major areas investigated are cycle counting methods and methods for prediction to crack initiation. of fatigue lifeThe rainflow, traditional with them. at three which cycle to is recent avoid find counting the distortion methods out if the methods, range-pair, are described history Wetzel's from which and the and inaccuracy suffer,

counting

and compared between and ends identical

each other It

similarities loading three the

and differences starts give an

shown that peak,

a service all for

an extreme All

then

methods

count. histories

relevant are

methods history

the in

description connection for

of measured service with better methods information between fatigue life

reviewed in

critically

assessment,

service

regeneration rainflow use of

and simulation. fatigue like life Finite about rainflow and ergodic simulation a closed-form of rainflow A closed-form presented. life under ingredients are a

Confidence predictions Element component counting and analysis have and the the which ranges for Methods variable of the

the

method analytical

increased offering initiated

frequency a search density

domain for of in

a link

power spectral Using thesis defines

a stationary and digital shape of function density. ranges From the prediction how the the

random process. techniques, expression counted expression

a Monte Carlo presents a link the probability power fatigue are

approach density

for

any given

spectral

the distribution of predicting loading

of ordinary crack life

is also basic

initiation

amplitude local-strain

reviewed. various

approach, with a given to the for

procedures stress local

assembled are strain

methodically linked for.

regard load life,

to how the local level,

and strain stress effect and is

determined are

and how the

mean stress

accounted

Predictions published themselves is test data;

by made however

these

methods are

are

compared

with

the within It very

predictions

compared

mostly

in order under

to highlight certain A sensitivity

the differences circumstances, analysis are to determining is

between methods. some methods carried in out the give

shown that

erroneous how

results.

to examine material properties

sensitive

various

methods of

changes the

properties. from

A new procedure data

material

the experimental

is proposed.

-111-

Acknowledgements Summary ii

Chapter

1.

IN1l0OCTIO References

1 10

Chapter

2.

AN

OF CYCLE COUNT Miner's

AND RAINW W Rule

11

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Damage Accumulation: Cycle Counting Count Method Method of Cycle

12 13 19 22 26

Range-Pair Wetzel's Rainflow Overview

Counting

Methods

35 43 45

Conclusions References

Chapter

3.

LINK

BETWEEN PG ER SPECTRAL DENSITY AND

FATIGIE LIFE BASEa ON PJU2FWW 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 Introduction Some Aspects Description Narrow-Band Expected of Random Loading of the Problem Case Damage

47 47 48 51 52 56 58

Fatigue

The Wide-Band

Case

iv -

-

3.4 3.4.1 3.4.23.5.

Description

of the Simulation

62 62 6374

General Procedure SimulationResults

3.5.1 3.5.2 3.5.33.5.4 3.6 3.6.1 3.6.2 3.7 3.8

Peak-Trough Density Ordinary-Range Rainflow-RangeMoments Modelling Model for Model for Discussions

Functions Functions Function

74 76 8288 94

Density Density

Ordinary-Range Rainflow-Range

Densities Densities

94 107 122 127 132

Summary and Conclusion References

Chapter

4.

AN OVERVIEW OF FATIGUE DAMAGE CALCULATIONS

133

4.1 4.1.1 4.1.2 4.1.3 4.2 4.2.1 4.2.2 4.2.3 4.2.4

Nominal Nominal Nominal Nominal Local Cyclic

Stress Stress Stress Stress

Methods Method I Method II Method III Methods Curve Hysteresis Loops

134 135 136 138 139 140 142 144

Stress-Strain Stress-Strain

Simulation

of Stress-Strain

Notch Analysis Cyclic Strain-Life Properties

147

-v-

4.3

Various

Cumulative

Damage Calculations Approach

by150 157 162

using Local Stress-Strain 4.4 Discussion References

Chapter

5.

A SENSITIVITY

ANALYSIS OF FATIGUE DAWE

CALCUIJffIONS 5.1 Introduction

164 164

5.2 5.35.4 5.4.1 5.4.2 5.4.3 5.5

Load-Life

Estimations

166 170 175 176 180 'Old Properties' Methods to 191 Data 207 212 218 225 228 186

Nominal Stress MethodsLocal Stress-Strain Analysis Analysis Analysis, Methods

Load-Strain Neuber's Neuber's Sensitivity Material

of Prediction Properties

5.6 5.7 5.8 5.

Determination The Effect Discussion Conclusion

of the Fatigue of Mean Stress

References

Chapter

6.

([IJSIoN

AND SOME SUGGESTIONS FOR

229

FUTURE RESEARCH

CHAPTER1

Fatigue application a single that

in

materials less than

is

the

process

by

which

repeated

of a load, application,

the one needed to break leads level are to the

a component by failure of

eventually

mechanical

component.

At the microscopic which same

the process in to

generally

depends

on crystal straining Eventually the then crystal,

imperfections, and this at the

increased collected creates

number by repeated preferred discontinuity propagating tensile sites. in

time

slip-driven continues

process to

a major

which

grow and becomes a crack, mechanism linked to

by an opening

and closing

principal

stresses.

Many researchers mechanisms the atomic relevant scale of to

have aimed at clarifying fatigue failure, those ranging on the

the major from of

factors

and on on by

investigations a few metres scale

10-7mm to The

scale

engineering physicists can

structures. and metallurgists employed

research

on a near-atomic produced little

has as yet in practical and

knowledge designs. fatigue

which Recent has scale,

be usefully in both

engineering designing

progress

understanding

against

by investigations emerged mainly by using empirical

at more appropriate laws.

levels

of physical

Engineers fatigue will occur

have been faced in a component,

with

the

problem cyclic

of

predicting first

when caused

since

loading

-2-

unexpected 1800'slll*. believed that

metal In

failures the early existed the

in days

the of

railway fatigue

industry analysis, level of this fatigue so that

in it

the

middle

was widely below which limit, or

there

some critical specimen. the main

stress fatigue tests,

no damage was done to endurance design occur. safe-life the design it are is limit all

Finding target of this

was then applied there recent

and good

kept

stresses are years still

below

no damage would designed with the in

Although method,

some structures

have seen radical

changes of emphasis

of engineering

components. acceptable

In an increasing to manufacture

number of cases products requirements which of

no longer functional the and the

commercially but

overdesigned.

The conflicting

maintaining society pressure development components elements

demand on reliability drive for

imposed by the consumer-oriented design place The for the considerable in the

an economic techniques. approach of

on traditional of is a modern the accurate

design design

objective

critically-stressed damage caused by the

assessment

of the

service

history.

The simplest to put it in a test

way of machine

getting which

fatigue will

data load it

about

a component until

is it

repeatedly Nf, of

breaks. levels which of

Recording stress

of the number of cycles amplitude, referred is S, yields the

to failure, basic form

at different fatigue the test data load

is universally

as the S/N curve. sinusoidal. However,

Normally there

in obtaining

S/N curves

are very

*Numbers

in brackets

designate

References

at the end of each chapter.

-3-

few

cases

where

the

service

load

can

be

so easily In all is

described, other cases,

like the

eccentric description

masses rotating of the is fatigue

at a constant loading the

speed.

environment future

a complex

problem. which

The difficulty the unit

in foreseeing to survive.

range of service

loads

is expected

For many years the fatigue performance

the

experimental

procedures materials

used to

investigate have

of structural limited to

and configurations and

almost loadings. in

exclusively Fatigue

been testing

artificial greatly the

much simplified innovations the modern

has profited For example, machines has

from recent of to

equipment

technology. testing

introduction possible

electrohydraulic complex of the the

it made

meet more simulation

experimental actual service

requirements, conditions. has been

to produce Another the

a much better big step of in

understanding computers into

fatigue testing

phenomenon

introduction

fatigue

and analysis.

Applications Typical accuisition application

of ranges

computers cover

in

fatigue

are

quite

numerous. data

structural bulk of

and materials computer replace for

testing,

and reduction,

and the

usage,

cumulative single and of

damage calculations. purpose devices, but

Computers offer

do not only

conventional the

new possibilities In addition a realistic test the for

performance a variety actual

organization program conditions, temperature, While certainly

of fatigue

testing.

to providing of

waveforms,

including

simulation variables test

service strain,

a computer and also not a

can monitor ensures 'cure that all'

such as load,

conditions

are maintained. and complexities

many problems

-4-

associated implemented, improve procedures. of capability, advances cost of in

with

fatigue can

performance effectively of data

evaluation, extend testing

when

properly

computers and

facilities, and analysis in terms The recent size to and the

efficiency Computers

quality

acquisition approach

offer

an attractive reliability, technology modest

to fatigue

versatility, manufacturing to very

accuracy have levels brought giving

and cost. down the

processors

added impetus

application

of computers

in fatigue.

The microprocessors it makes analysis. the ways whole in

increasing and

availability associated peripherals

of at

minicomputers a relatively to fatigue low testing to

and cost and cover

necessary Obviously range which of

to it

re-evaluate is beyond

approaches the scope of

this

research the

computer

based applications, might assist is to only towards remain way of in

however fatigue

principal are fatigue and increasing a

computers Since the is

analysis realism, activity ever

investigated. performance computer

trend likely

improved a complex meeting the

analysis

approach

be the may

demands on reliability

and economy.

Cumulative fatigue lie in analysis; this area. fatigue design of life

damage calculations hence In life the order bulk to of

occupy computer

dominant a applications level of

sector in

of

fatigue

achieve

a high

structural stages of the

reliability, of the

predictions

have to be made at several An important is the ability aspect

development and fatigue and thus resistant anticipate

process. structures possible

development component

to predict

problems.

Good agreement

-5-

between tests

predicted

fatigue

lives

and actual in a design

fatigue

lives

obtained

from

gives

a greater

confidence

and may eliminate

the need

for

retesting

improved designs.

Due to of fatigue data extent published the

increased

research

activities for

of the past design

a large

amount To 7000 1975. vast the Readers

was produced of this on the

and offered Watson in

applications. record in the

show the papers

interest subject long

and Rebbeck[l] last fatigue 20 years, research, of

the of

Considering amount formidable may find of

180-year

history and the is

papers task various describes fatigue of

published literature

diversity not

interests, here.

survey

attempted in the

state-of-the-art randomized testing, looks sequence acoustic at the

surveys constant fatigue problems

literature. and actual crack

Swanson[2] random load propagation. and

amplitude

and random load in aircraft effects, discusses

Schijve[3] under fatigue flight prediction stress

structures crack some

materials

damage simulation

and

interaction Dowling[4] complicated effects. mechanics In more

propagation of the

loading. for

failure

methods

stress-strain The same effects point recent of the of view: papers, local in

histories, are

and man by

and load-sequence from a fracture crack the growth.

reviewed

Nelson[5] of

prediction Morrow and

methods Socie[61

fatigue trace for also

historical

evolution life types

stress-strain service. available need Sherrat of

approach Haibach[71 for use with

predicting reviews

the fatigue some basic

of components of fatigue

data

existing approaches

procedures in

and emphasizes fatigue

the practical assessment.

different in

making

t[81,

a series

of

4 articles,

provides

a survey

of

the

-6-

current traditional

position

on fatigue

life

estimation

methods, points

to parts

of of

design which are still

in use, and gives local

enough details

the new methods - fracture vibration to allow -

mechanics,

stress-strain

approach and

engineers

to use them properly.

The literature contains techniques. aware of the much

dealing

with

metal

fatigue on

and cumulative life

damage

new knowledge

bearing point

improved it is not

prediction to be affecting

From a practical fatigue

of view,

sufficient

phenomenon,

to know about

the parameters

the fatigueobserve knowledge in-service and

propertiesregister the

of materialsa manifestation terms

and structures.and needed to another put it

Itto

is one thingexpress work

to

this making

in life

pragmatic

to

predictions.

The situation life prediction stress

does

not

become easier the

due to literature, S/N data,

various

fatigue

procedures approach approach

recommended in requiring requiring cyclic

such as the data, the the local

nominal

conventional strain-life stress-strain

stress-strain mechanics

fracture

approach

requiring

and crack

propagation

data.

Furthermore,differing in the

eachin

of

theseterms of

proceduresaccuracy the

existsand/or

inin of

severalterms of

modifications, simplicity such as

application. design of aircraft, type

Considering ground of materials for

diversity

interests, off-shore surprising performance

vehicles, used, predicting it

bridges, is not

structures, that

and numerous accepted

no generally

procedure

fatigue

has been accepted

up to now.

Nor does it

seem possible

to

indicate

-7-

such

a

universal procedure level

procedure of of fatigue

in

the life

future. estimation available. both

Selection depends Life with to

of

the

appropriate extent

a large

on the

input

information

predictions and

made by new procedures with simpler, long

need to

be reconciled methods.

experience

established

In many cases, if at this all. was not so, it

the

present

commonly used methods are adequate; be possible is the to design against of fatigue component Just to as the

would not fatigue still

However, there methods ones, new ideas are are

since

commnest to of

cause

failure, present earlier those

evidently the result

problems of a series will

be solved. modifications

so improved which

techniques

come about Fatigue of these

by incorporating research problems, so that has to making the user range an life was

are relevant to

and useful. solve parts

be considered small additions

as attempting

and changes to established to him methods which Starting from of this fatigue on in the are valid

techniques, for

has available of problems.

an ever widening only one, but

philosophy, analysis, this crack

quite

important estimation, further life major life stated,

one,

aspect

namely This

fatigue subject of

was concentrated

research. propagation fracture

narrowed estimation, area of

down to exclude because it

part

fatigue is a

deemed that was own right. thesis

mechanics the term

research used

on its in

Therefore, refers to,

'fatigue otherwise

estimation' prediction

this life

unless

of fatigue

to crack

initiation.

In mechanics

any

cumulative there

damage calculation, is a stage where the

including load history

the

fracture has to be

approach,

-8-

reduced damaging described

into

discrete

cycles Three in

which widely 2.

can then used Their simpler

be used to calculate counting methods

their are and a are also

effect. in of not detail

cycle

Chapter the

use and limitations, cycle counting of point

comparison discussed

them with only from

other

methods but

a damage calculation

point

view,

from a service

history

regeneration

and simulation

of view.

Since counting fatigue fatigue

its

introduction has been shown In

10 years to be 3,

ago,

the

rainflow the

cycle best

method life life

superior a link range

and yields is proposed

estimates. estimation density the form

Chapter

between and the The

based on rainflow of a stationary of a distribution spectral

distributions

power spectral link ranges approach of is in for

and ergodic function density. probability

random process. of Using rainflow-counted a Monte

a given

power

Carlo

and digital ranges data

simulation are obtained

methods, for

density spectra,

functions from and

rainflow

various

power for the

observed probability study, between

a closed-form functions density peaks for is

expression proposed. of

rainflow-range of this

density

As a by-product ranges also is put

probability consecutive

functions and

ordinary are

the distance and a

troughs

determined forward.

closed-form

expression

the ordinary

ranges

Af ter estimating recent these

describing life, the local

the

traditional

nominal

stress ingredients are

methods

of

fatigue

in Chapter

4, the basic method,

of a more From

approach, ingredients

stress-strain life prediction

discussed. are local

various are

procedures relating the

assembled. stress and

The reasons

why there

many ways of

-9-

strain

to the

life

using

the

same set

of data

are investigated,

and the

choices

a would-be-user

has to make are outlined

methodically.

InChapter the other life

Chapter

5,

predictions

using

the

methodstest data

describedpublished with of

inin each the and the and in

4 are

compared with However the to gain

the well-documented predictions into highlight the

literature. in order

are compared mostly potential the variability

insight to

prediction

methods,

significance,

utility

implication prediction the detail. effect

of various methods of are

alternatives. to variations on life

The question in cyclic

of how sensitive properties are analysed

material

mean stresses

calculations

Although the end of each for

relevant chapter,

discussions overall

and conclusions implications of

are this

included research 6.

at and

directions

future

improvement

are outlined

in Chapter

-10-

1.

Watson, P., Rebbeck, R. G. "Modern Assessment", Railway Engineering-Journal, London, 1975

Methods of Fatigue Ins, of Mech. Engrs.,

2.

Swanson, S. R. "Random Load Fatigue Testing: State of the Art Survey", Materials Research and Standards, MTRSA, Vol. 8, No. 4 Apr. 1968 pp. 10-44,

3.

Schijve, J., "Cumulative Damage Problems in Aircraft The Aeronautical and Materials", Journal of Aeronautical Soc., Vol. 74, pp. 517-532, June 1970 Dowling, N. E., "Fatigue Failure Stress-Strain Histories", Journal No. 1, pp. 71-87, March 1972 Predictions of Materials,

Structures the Royal

4.

for Complicated JMLSA, Vol. 7,

5.

"Review of Fatigue Nelson, D. V., Crack Growth Prediction Methods", Experimental Mechanics, Vol. 17, No. 2, Feb. 1977"The Evolution Morrow, J., Socie, D. F., Crack of Fatigue Initiation Life Prediction Methods", '81 SEE Proc. of Fatigue Conference, held at Warwick University, England, March 1981

6.

7.

Haibach, Fatigue England,

E., "Fatigue Data for Design Appliations", Proc. '81 SEE Conference, held University, at Warwick March 1981

of

8.

F., Reprint Sherratt, Soc. of Environmental 1983

in the Jr. of 4 articles of published Engineers, Dec. 1982, March, Sept., Dec.,Methods" A Review of Traditional Mechanics" Using Simple Fracture by Local Stress-Strain Estimation

"Fatigue i) Estimation: Life "Fatigue ii) Estimation Life iii) Eaton, D., "Fatigue Life Methods"

iv)

"Vibration

and Fatigue:

Basic

Life

Estimation

Methods"

CHAPTER 2

AN OVERVIEW OF CYCLE QOONTIF3 AND RAINFU)W

An essential components strain process in the or

step

in is

the the

prediction reduction

of of

the

fatigue

life stress

of or This

structures to a series

a service

history is

of cycles

or half-cycles,

reversals. of large are which three avoid

known cycle

counting life used

and can be the source calculation. in recent the but There years

errors cycle the

subsequent methods

fatigue widely

counting distortion These cycle these

and inaccuracy methods for all exists

from which similar, of service

traditional not

methods suffer. identical each of of a The from a

three counts

produce types in

necessarily Moreover, The of cycle extract

histories. variations.

methods

several

choice counting. cycles

particular fundamental service however from obtain testing.

method also purpose in counting history of

depends on the purpose cycle to counting calculate are also is to their

history cycle

order

damage contribution, delete small cycles and to

methods for

used to test

a service relevant

an accelerated to generate

programme, signals for

information

loading

fatigue

Although as such, life a linear

it

has nothing

to do with rule

the cycle on which first.

counting most of After

methods fatigue a brief the order cycle their they are

damage accumulation methods are based the

prediction of

is presented cycle are and

description recent to

some of and their their are

traditional

counting in

methods, in the

methods

variations

examined

detail All

highlight

differences then

similarities. critically to for

counting limitations intended.

methods and

reviewed for

establish which

applicability

the

purposes

-12-

2.1

Damage Accumulation:

Miner's

Rule

The basis

of

nearly

all

working

methods

for

damage summation

is surprisinglyin 1924,

a simple hypothesisa linear The same it This to

put forwarddamage law idea the was for

firstthe

by Palmgren[l]estimation later of fatigue of by in

who proposed bearing in 1945, life.

roller [2J Miner structural

rediscovered type

who applied

ordinary accepted Rule,

components. usually load referred

widely

Palmgren-Miner assumes of that stress test, Or it if ni

hypothesis, cycles would the of

as Miner's to a component in

are applied at

at a level

which then can

cause fraction

failure of life

Ni cycles

a constant proportional

amplitude to ni.

used is

exactly when:

be stated

as that

failure

occurs

ni -=1 Ni

Eq. 2.1

Sometimes substituted conservatism The most for is obvious

an arbitrary unity on the

constant, right-hand There

generally side,

less

than

1.,

is of

hence

an element to this cycle of

introduced. one is

are many objections assume the first

rule. load cycle rule

that

to

applied applied survives

to a virgin near failure

specimen is

does the

same damage as a similar However the

an oversimplification. of use.

and has stood

up to a good deal

-13-

2.2

Cycle Counting

Whether stress-strainbound to cycle cycles

one

uses

the

nominal

stress

or

the

local there isthe

approach or the fracturecounting to reduce part in

mechanics analysis,each approach. load their history

be a cycle is

The aim of into

counting which

a complex

discrete

can then be used to calculate

damaging effect.

Earlier ten load below separate history.

reviews techniques Some of

of counting which the

methods[3'

41 have noted

at least

have been used to analyse more 2.1 notable ones are

irregular an briefly

covered

and illustrated

in Fig.

Peak count: All maximums above the mean and all the below minimums the

mean are counted. Mean crossing Similar successive Level crossing peak count: to peak crossings count, only the largest peak between

the mean is counted. of

count: passes are registered at levels above the Counts at

Only positive-going mean, and only each level Fatiguemeter count:

negative-going

below the mean. pass

are cumulative.

The- fatiguemetervariations

has been developed in aeronauticsSimilar to level

to recordexcept

in acceleration.

crossing

that

is accepted after a count

the load crosses a preset

-14-

Strain

Ctr-n

in

(a)

Peak

count

(b)

Mean crossing

peak

count

Strain

Strain

(c)

Level

crossing

count

(d)

Fatiguemeter

count

(d)

Range

count

(f)

Range-mean

count

Figure

2.1

Soir cycle

counting

methods

-15-

level

in

the

downward such

direction. noise,

In

this do not

way small influence

load the

variations,

as electrical

count.Range count: Each range, trough values i. e. the difference as a half between cycle. successive peak and

is counted

Range mean count: Not only range the value of the range, but the mean value of each

is also

recorded.

However, mainly apparent define "What three

the

more

recent

reviews[5,6,71 rainflow of these and Wetzel methods is

concentrate methods. that they

on The all

methods, for

range-pair, superiority stress-strain

reason cycles

as closed

hysteresis when the

loops. loading

The question is constant in the a

is

a cycle? " presents block loadings In programves shown in the

no problem or narrow Figure

amplitude case of question. reversals compared

band histories. it in is

However,

2.2,

no longer 2.2(a),

so trivial the small

sequence

shown

Figure

do some fatigue to the

damage that

may or may not large cycle

be significant they and are (b), peak is

damage done by the Peak counting gives

on which for (a)

superimposed. but (b) is likely

the

same result than all (a) the .

to the

be more damaging same result for

Mean crossing cases which

counting

gives

three

non-conservative closer counting in

in situations to the record

where interrupting large ones. of The

cycles range

become range-mean cycles with

amplitude methods

and

would

a series

low-amplitude

-16c+r-1 4rStrain

Cfn-n

in

(a)

(b)

(c)

Figure

2.2

Some

sequences

causing

problem

for

several

counting

methods[4].

.

Strain

D

rain

B

B'

Time

A

Figure

2.3

Interruption

of a major

cycle

by a minor

one.

rain

Strain 1 2" 3 5 5' :6 7Time

s2';

Figure

2.4

Stress loops.

and strain-time

history

and associated

hysteresis

-17-

gradually ignoring

changing the existence

mean,

for

Figure

2.2(a)

and

(b),

completely amplitude

of underlying

waves of much greater

and damage significance.

To show the the following steel[81, data to

extent is

of

the

effect from

of the

small

cycle

interruption curve subject for to

extracted the

strain-life the specimen

Man-Ten

predict

damage to

the history

shown in Figure

2.3.

Let 0.005 and

the 0.007 i. e.

amplitudes

of

AB,

BC, CD and AD be 0.003,0.001, then the corresponding as 4.60 damage x 104, the

respectively, reversals 104 x to failure

parameters 4.61

can be extracted 103 x respectively

x 106,1.12 curve.

and

5.62

from

strain-life would be:

The damage calculated

by an ordinary

range method

Dl

=DAB

+DBC+DCD

1+1+1 4.60 104 x 4.61 x 106 1 9000 In other words, the specimen On the half would other cycle fail hand, after if the 9000 repetitions same waveform by a small of is 1.12 x 104

such

a waveform.

considered BCB', then

as one large

AD interrupted

cycle

the damage would be:

-18-

D2=DAD

+2

xD

1 5.62 x 103 +2 4.61 x 106

1 5600

One way of is very to use a "gate early of of in the

handling level",

the

problem or

of

small

cycle

interruption It was found that of small a large level was

"threshold" of

"dead zone". techniques to a count

development on the record In

counting lead avoid preset

amounts number

noise small only

could to

ranges. ranges

order this

this level

a gate

introduced, For instance, then range level instead

exceeding 2.3,

would be counted. than BC been used, one large of the by gate many

as in Figure of three

had a gate ranges

greater

smaller

AB, BC and CD, only However, the gate the choice

AD would has

have been included. a further problem, the

posed

paradox

noted

observers are

[5,6,9].

Decreasing in less

level,

so that calculated

more events for a given the

counted,

may result

damage being above.

loading gate

waveform, level

as calculated the

Conversely, events counted to this

increasing but

may decrease damage. gate damage,

number sensible and use

of

increase is to the will

calculated try several

The only levels i. e.,

approach the the one

problem maximises

which

calculated

minimises

predicted

life.

This

givebut

thethis

most conservativeis not necessarily

life

estimate

possible

with

this

method

absolutely

conservative.

There are three

counting

techniques

mentioned

earlier

which

-19-

separate physically methods. counting

low

and

high way.

amplitude

cycles

and

record Wetzel

them

in

a

meaningful The is

These are range-pair, feature basis of of these the this

and rainflow is that

outstanding out

techniques

carried

on the

stress-strain point, a brief

behaviour complex response by

of the material strain-time are these Each shown in counting of these with are and

considered. stress-time 2.4. are is in

To exemplify history The cycles identified a closed constant based.

and the that

stress-strain would

Figure

be determined and 5-6-5'. hysteresis

methods cycles those invariably

by 1-4-7,2-3-2', stress-strain amplitude tests

loop, life

consistent predictions

on which

2.3

Range-Pair

Count

There Dowling's{51 a strain subsequent Each peak that

are at least interpretation

two variations of the as

of this

counting

technique.

range-pair if in it

counting

is that method with a

range

is

counted of in

a cycle magnitude

can be paired the opposite

straining is taken is

equal order if

direction. except

as the the

initial of

peak of the history

a range,

a peak it initial this the

skipped

part with

immediately counted is range. counted occurs

following If the

has already peak minimum of

been paired a range the is

a previously

a minimum, positive than is

a cycle

between before range. counted occurs

and

most

maximum which the initial

strain

becomes more negative if the initial

peak of the is

Conversely, between before this

peak the

a maximum, a cycle negative than the

maximum and

most

minimum which initial peak.

the strain

becomes more positive

-20-

Figure marked with lines. being positive and 6, between

2.5

illustrates

the

technique,

the

counted

ranges short

are

long

dashed lines a cycle

and the paired is counted

ranges with

dashed

For example, the most than and the

between peaks 2 and 3, peak 3 the is strain becomes more 5

negative

minimum Similarly

before a cycle

peak 2.

counted paired

between peaks with the

range between

peaks 1 and 4 is

range

peaks 4 and 7.

Livesey method instead equal is of levels,

and Webber's[31l different ranges. each level in

interpretation that counting strain is

of

the

range-pair

somewhat pairing and treated counters

done by counters is divided into Each

The full is

range with

associated range acting

a counter.

is maximum peak. All

as a positive within this

from the cocked;

most minimum those already is range range are

range are then

cocked are unaffected. measured triggered, from the

When a minimum detected maximum. Counters

the negative within this

previous completing all

thereby

the counting

action.

When the absolute The number of the count at

is detected minimum cycles the next at each strain higher

cocked counters range is obtained

are triggered. by subtracting

from the count range

at the

in question. range

The first through cocked. three which cocked, levels-2 The next counters. means that it is not

reversal

in

Figure in

2.6, level-4.

starts

in

level-l,

passes 1-4 are the at first

and 3 and stops reversal The reversal all counters travels

Hence counters triggering

three

levels

from peak 3 to peak 4 stops are cocked; The remaining since counter

level-5 already in

4 is

affected.

reversals

are examined

-21-

Strain

41_ ---

-" "'

Counted Paired

range range

i/

7

Time

Figure

2.5

Example of

range-pair

counting

method

rD

o= x= 5 4 3 2 1 o o oo

Cock counter Trigger x x2 5 31

4) bDc 4

+O) UU

Q) rHU

counter

o

511 410 332 230 13o by using counters.

x xx

o oo

x xx

o oo

x xx

Figure

2.6

Range-pair

counting

Peak`4

1234567H + + +

+22

1/2---

6%_ 5

+l

-

+ + +-

+ + ++

+ + ++H

i531

+-

++

-

1w4 215

-

Figure

2.7

Counting

of Fig.

2.5 by Wetzel's

method.

-22-

the

same way. Then

Total the

number of number of

triggering cycles at

action each level

at

each is

level

is

counted. yielding three

calculated, of

one cycle levels.

that

has a range of five

levels,

and two cycles

2.4

Wetzel's

Method

This similar method. "bands' l, reversal; for they in using occur order, to the The

counting Livesey full line

technique and Webber's range or

developed

by Wetzel191 of the into are equal

is

very

interpretation is divided

range-pair levels define or each

strain

Next, the the

segments

"elements" is equal from

used to

length

of each element follow:

to one band. each reversal with

The rules peak, the first, has for sign It an the is is as

elements

Starting

in sequence, to its sign" under to the full to

each element value indicate if if

is used starting Each is

available. the is element

element

"availability reversal opposite likely elements found. again

available its when reversal. history

question. sign of the

An element the slope for

available current

the of

that in At

during

evaluation over

a complex

several element is may

a row may be skipped a latter time for in use, the

until

an available the skipped

analysis

elements

become available

but

the decision

at the instant comes

of need.

Figure the five same, brief bands.

2.7 demonstrates history Since the

the application The full at the

of the

rules

above to into all

used before. history starts

is range absolute

divided minimum,

-23-

the to

elements peak 2,

are travels

initialized in first the

to

(-).

The first direction,

reversal, passing (+). sign

from peak 1 through four

positive are

bands. requires elements. elements.

Hence the three The (-)

elements

set

to the

The next of the first

reversal three (+)

elements

and changes from peak array

reversal the

3 to

peak

4 needs that for

four

Examining while

previous

reveals

elements having

1,2 or the same 4 is

3 are available sign. skipped, Therefore

element 1,2

4 is not available, 3 and 5 are set to

elements

(+),

element

and so on.

Elements 12345 +2 +1 Bands 0 1 -1 -2 I. 1 I .. -1 -2 1 +1 011 11 1 (b) -2 ; '1 11 1 12345 1 1 23 4 5

(a)

(c)

Figure

2.8

Barr-Element

matrix

for

Figure

2.7.

There the array cycle

are

two ways of One is the to

storing construct cycles

the

necessary

data

to

compile

count.

two-dimensional at is

band-element as the

to determine at the each

number of

each range as well to use a single The first

mean values sacrificing

range.

The other about

sum array of these

information

mean values.

-24-

ways

is

illustrated value reversal in

in

Figure

2.8. and also

If

one

starts that

at

the

largest for each to

absolute tensile

the history, there will

ends at

value,

occur

an equal

compressive

reversal

complete

the

cycle.matrix

For thisonly those

reason,elements

one needs to enterused during

into

thein

band-element

the excursions

the same direction;first reversal in

in thisFigure

case from a minimum to a maximum. Of the2.7, band (-2) uses element 1, band (-1) 4.

uses element These data 2.8(a).

2, band

(0) uses element

3, and band (1) uses element matrix

are now entered During the next

in the band-element positive going

in Figure shown from peak 3 to

excursion,

peak 4, band (-1) uses element uses element

uses element remembering

1, band that

(0) uses element

2, band (+1) band (+2)

3, and,

element

4 is skipped,

5, and so on.

The diagonals, shown in

data starting

are

removed

from

the

matrix

by

constructing diagonals, as

from the 2.8(c). which

largest In this in

one to the smaller example, band (-2) the

Figure a cycle with

largest

diagonal band of 0. (+2), The

describes

starts

in and ends a mean value

i. e. a cycle next mean, 0.5. diagonal the

of five a range a cycle

bands about with of

represents cycle

of three a range two bands

bands and zero of

last

has a range

and a mean value

In counting Figure highest

the

alternate number of

version, sign

"sum" a for from

array

is

constructed

by

the 2.9.

changes of data

each the

element, array

in as shown starts sign at the twice

The extraction which contains

element,

non-zero

count.

Changing

-25-

Sum 1 2 Elements 3 4 5 6 6 4i2 12*j . i4 4 2* 0 0 f j 0 2 2* 0 0

Figure

2.9

SUM array

for

Figure

2.7

is of two.

interpreted cycles at

as two reversals the highest at element that

or one full is is found

cycle.

Hence the number the count by in

by dividing from

Then the

count

level

subtracted

each value

the array.

The containing this case,

process non-zero the

is count

repeated until

for zeros of of

the occur

next for

highest every

element For five a

element. of

history with

consists a range

one cycle three,

with

a range

elements,

one cycle

and another

one with

range of two elements.

The Wetzel's counting, Elements element triggered, is

reader

has

probably

noted

that

these of

two the

methods, range-pair

method, are in very

and Livesey similar in

and Webber' s version that both to methods counters a counter is within

use the in is the the

same logic. other. cocked "An or

one method not available"

correspond means that if it

already counting

hence

skipped,

range.

-26-

However,method "the

there

is

one major

difference

which makes themethod, the

range-pairranges or difference reversals. a tensile peak and between counters to

give number

unreasonable of bands to

results. travel" for both

In Wetzel's are calculated

as the

between On the reversal the

successive other is hand, always

peaks in

tensile

and compressive range of that

the

range-pair as the If

method the distance

calculated value.

between

absolute

minimum peaks instead

a range find

was calculated the number of

successive be cocked exactly the

and used to then the

or

triggered, same result

range-pair method.

method would

have given

as Wetzel's

2.5

Rainflow

Method

One of been the the first

the

important most of the rainflow to attracted the

developments

in cycle last

counting decade. in 1972,

has Since the

emergence

in the method West by

introduction method has

Dowling[5] It

rainflow

much attention.

has been widely tests

by many investigators used to be superior to all it

and shown by extensive other counting

experimental

techniques[5,6,7,11,12, tool for fatigue life

13, ].

As a result,

has become a standard

predictionrainf loops, low

packages.method defines

The apparentcycles local as

reasonclosed

for

this

is

that

the

stress-strain analysis.

hysteresis

without

elaborate

stress-strain

There identical version, cycle

are

three

versions for all

of

the

rainflow

method which histories. uses the

give

counts

types

of service

The first imaginary

sometimes

referred

as Pagoda Roof method,

-27-

rain the

flow reason

on fictive for generic

multifarious names of

overlapped "rainflow" in

pagoda

roofs.

Hence, is work. this To

and "pagoda-roof" their original

analogy

used by Matsuishi

and EndoI101

illustrate2.10

the method,the time

the strain-timeaxis peaks is are vertically

history

is

plotted

in Figurelines pagoda

so that the

downward, to

and the of

connecting roofs .

strain

imagined

be a series

Rainflow and successively falling following until negative) stops rainflow it on the

starts at the

at

the beginning inside eaves of to Flow every the

of

the

strain-time

history, keep the down more also

peak. roofs at

The raindrops until one of

imaginary is

lower

conditions comes than

met.

initiating

drips a peak (minimum The flow length

opposite

a maximum more it

positive from. The the

the meets

(minimum) maximum rain from a roof cycle

started

it when is

above. range.

of each

recorded

as a half

Strain

Figure

2.10 Example of pagoda-roof

method

-28-

Figure history. initiatingopposite the range

2.10 demonstrates represent

the technique the raindrops.

on a short

strain-time

Dashed lines at peak-1,peak-5, from peak-2 stops rain

For example the rain and 4, and stopspeak-l. Hence

a minimum, fallspeak-5 to

on peaks-2

because peak-1

is more negative is extracted

than

peak-4

as a half-cycle. because the rain

Similarly from peak-2 However, from the

and peak-3 opposite starting

make one half-cycle, which

peak-4

is more positive at 2' where it

than peak-2. meets rain by the of

the roof

at peak-3

stops

above.

Note that already

when a half-cycle exists first

is extracted

second condition, equal magnitude

there extracted cycle.

a corresponding condition, 2-3

half-cycle

by the example,

and two together 3-2' make a

make one full full cycle.

For

range

and range

The second version, by the following original authors[11],

which

is called of

Maximum-Minimum Procedure repeated application of the

consists

procedure:

1. strain-time divides the

The history history

maximum under into

and

the

minimum

points

of

the This

whole action

consideration three front points. parts, and rear

are determined. the middle sections section

between the

extremum points, points the next and the stage 2. extremum point

and the extremum analysis. the

between the terminal is deferred to

The middle

section

For

front is

section, made towards

starting the front

from

the

bounding for the

a search

terminal

next

opposite

sign extremum point,

i. e.,

if

the extremum point

is a

-29-

minimum, terminal,

the

next

maximum versa.

point

is

searched

towards

the

front

or vice

Once again

the section

between the extremunl

points

is deferred3. Step

to the next stage analysis.(2) is repeated until the front terminal is reached.

4.

Steps (2) and (3) are repeated

for

the rear section.

The strain-time increase extremum standard cycle ordinal the

first history

application gives until is half the

of cycle

this

procedure Note bound

to that by the

the the

total ranges

ranges.

steadily points

maximum range then they to

absolute When the full even When

reached, is applied

decrease the

steadily.

procedure ranges are

again by

deferred

sections, having

determined The ranges are

picking are at

up the odd order step (1)

ranges are for

numbers.

which

ignored. the

extremum the for

points terminal

searched which reason.

deferred be

sections, disregarded

points

are maximum or minimum should

the obvious

Figure procedure the rules. to

2.11

shows

the

application

of

the

maximum-minimum to clarify as the history, the at

comparatively At step (1),

a longer peak-5

strain-time and peak-12 MAXO of is

history are the

located whole

absolute respectively. maximum of step (2).

minimum,

MIND and the front

maximum, section is

Since this

bound by a minimum, Peak-2 is (3) located is

section,

FMPXl, is

searched.

The front (4),

terminal

reached, rear

hence step

skipped. as of 1-2,

At step peak-18. the

the minimum of the is

section,

RMIN1, is determined application peaks

The end terminal procedure is

reached,

hence the first The ranges

standard

completed.

between

'. _)

2-5, shown history stage

5-12,12-13,13-15 by triangular between analysis. to the peaks

and marks 2-5,5-12

16-19 , in

are Figure

counted 2.12. are is

as half

cycles of

and the

The parts left to the

and 13-18 procedure

second for

When the section for this

standard

applied

again,

example

between section

peak 13-18 are

the

absolute Peak-17 no more

minimum and and peak-14 points left points, is

maximum points are located these (2), (3), for

searched. there points are

respectively. newly located

Since extrenum

between steps

and the

terminal

(4) are this

skipped. it is

Hence the divided

second stage three the stage

analysis

completed peaks between these are

section:

into

ranges, part of

between history Then

13-14,14-17 peaks ranges 14-17 are

and 17-18, is deferred

and furthermore to the ranges next

analysis.

ordered,

and the

having

odd or even orders range between analysis

shown by marks X and 0, is counted for

respectively. cycle. deferred of applied

Hence the

peaks 14-17 continues stage. deferred

as one full the rest is

The second stage sections

similarly The

from the first until all the

same procedure are exhausted.

repeatedly

sections

The Classification stress-strain smaller range one, 2-3,

justification Procedure, behaviour. in Figure as the interruption the plot is shape the of

for of the

the rainflow

third count range

version, follows is

Pattern from material by a

When a large 2.12(a) the

interrupted interrupted

1-4 is range a closed interrupted be if would

by the loop. on a

forms the

stress-strain excursion the interruption in

Furthermore, stress-strain not occurred.

same as it called

had

This

has been

"memory effect" a

material

-31-

Strain

MtYn

me

x

i, 10

Xi

Stage

'4

Is

OXIStage 3I IIiXi0Xx-0YX 0X

Stage

2

!, I

? '

L_]

Stage

l

--

Front

----- ob

Middle

_------

-- 4

__

Rear

Figure

2.11 Rainflow

Count using MaximmrMinimun Procedure

-32-

4.

- --

4 4

j 2 1 1 (a) D-I type 1(b) I-I

3type

3

-)

2

24

---4

1 3 (c) I-D type (d) D-D

3_1

type

Figure

2.12

Four patterns

of

strain-time

(left)

and stress

vs strain

(right)

.

behaviour, following interruption

the

material

"remembers" and

the

stress-strain this path

path

it

was the

when

interrupted

resumes

when

is over[9,141.

When four considered,

successive they

peaks form falls

of

a strain-time into

history types

are as

the pattern 2.12. the

one of the four

shown in Figure is smaller than than the

The type first one. type. one,

(a) shows that 1-2, and the

the second range, third range, is types to 3-4

2-3, is

larger

second D-I

Consequently The names of D-I hysteresis type

this the

pattern other

called follow a large patterns

decrease-increase, similarly. cycle

As discussed

earlier,

corresponds

interrupted

by one closed

loop.

The other

-33-

formversion

spiralof the for

type,the four

sharp

cornered,count

open hysteresisthree Figure

curves.

Thisas

rainflow

classifies shown in

consequent 2.12.,

ranges

one of taken

patterns[11J

and the

steps

each case are as follows:

In range peaks 2-3

the is

case

of

D-I

type,

Figure cycle

2.12(a),

the

interrupting the are

counted

as one full connected

and removed. next two peaks,

Accordingly 5 and 6,

1 and 4 are

and the

considered

to form a new pattern.

In 2-3 always

the

case of

I-I to

type,

Figure

2.12 (b) cycles

the ,

ranges

1-2 and of the

correspond ranges.

one-half

irrespective

following

Hence peaks 3,4,5

and 6 are checked next.

In the is always a half

case of cycle.

I-D

type,

Figure

2.12(c),

the

first

1-2 range

Hence peaks 2,3,4

and 5 are checked next.

In

the

case loop

of

D-D type,

Figure be said

2.12(d), without next,

whether further still

3-4

will

make a closed Therefore information the

or not cannot peak is first peak.

information. retaining the

following the

considered

about

When this Figure range 2.11, 1-2 is the

technique first

is

applied 1-2-3-4 cycle.

to is

the

strain-time type,

history therefore 2-3-4-5

in the is

pattern as one half

I-D

counted

The next

pattern

D- I

type,

hence theAfter

range

3-4

is

counted

one full as2-5-6-7

cyclewhich

andis

removed.

connecting

peaks 2 and 5, one gets

-34-

D-D type.

The next

two patterns

are also

D-D type,

hence retaining

peaks 2,5type, pattern 6-7-10-11 cycle. next cycle, increasing half 2.13. cycles the

and 6, the next patternrange 8-9 is is still type, extracted D-D type, the range which type, end

7-8-9-10as one full therefore 7-10 is yet is

is checked.cycle.

It

is D-I

The remaining considered. as one full D-D type. The

5-6-7-10

peak 11 is extracted another

makes D-I This leaves 5-6-11-12 so in on.

2-5-6-11, is D-I At the then

pattern and

hence the of the

range 6-11 is 6 ranges, are

one full first as

count, steadily, cycles,

magnitude,

decreasing as full

counted

and another

6 ranges

as shown in

Figure

rz

'

Figure

2.13 Deconposition half cycles

of the (left)

strain

history cycles

in Figure (right).

2.12 into

and full

-35-

2.6

Overview

of Cycle

Counting

Methods

Throughout and buildings with time.

their

service

life,

machines, the majority

equipment, of which

vehicles fluctuate there are

are subjected When considering two of aspects the load of

to loads, the effects the history growth. the

of random loading, The first life, is

generally significance crack aspect using safely

problem.

the

damage

on structural

as measured by important

initiation is the

and crack

The second and equally same fatigue

the means by which simulation of service

damage may be produced to design structures it is

loads.

In order of r,, iaterial of

without to (1)

unnecessary have

expenditure

and effort,

necessary for both

satisfactory

techniques of service

analysing and

random data (2) service

damage assessment and simulation.

history

spectrum

generation

There analysis, domain. knowledge subject service very to one

are in

two the

fundamentally time domain

different and the

approaches other in the

for

this

frequency and

The latter of the the

entails frequency

power spectrum response

analysis

of the signal of the

characteristics

system of a

loading. random has been receiving results obtained the

The power spectral more consideration by S. 0. Rice[151 of

presentation

history important to

because of the derived who a level is from an the

equation

calculate of

number of

crossings process,

power spectrum to or level

a stationary count in the

Gaussian time

which

equivalent stationary stress-time

crossing

domain.

By means of a series of

quasistationary

Gaussian

processes, with satisfactory

histories for

be described can

accuracy

as necessary fatigue

-36-

life

evaluation. signal

Stress-time generators which

histories and filters

can be generated or corresponding

by means of digital as the

random computer original

techniques ones.

have the

same statistical

properties

The main the to stresses

reason

why frequency from

domain analysis effects

works

is

that

resulting

environmental

usually

correspond for

random vibrations the power are

and can be treated presentation wind,

as continuous seems to

processes,

which

spectral of

be suited. or noise.

Typical The

examples opposite for

gusts

sea waves, stresses

vibration

to this

represents

due to the usage of a structure, truck, discontinuous processes one has all for to the

example the the

loading time

of a fork-lift becomes in

which analyse

averaging history

questionable. time domain to

Then

service relevant

the

extract

information

to the fatigue

behaviour

of the structure.

Ignoring are too small

fluctuations in magnitude are three

in

frequency,

which

in most situations effect on fatigue might be and this

to have any significant of the behaviour: Of numerous

damage, there expected the to

features fatigue

random signal amplitude, counting

which

influence of the

mean level methods in

sequence

signal.

category,

some notable

ones are described

in the earlier

section.

The oldest signal example, results is to count

and very

widely

used method event of level

to

analyse

a random for The

how often or

a defined a crossing of

has

occurred, level.

a peak, are plotted

a range, as

a given against

a histogram

cumulative Figure 2.14.

occurrences

known, cumulative or as commonly

frequency,

-37-

T.narl

_me e

Figure

2.14 Load-Cumulativemethod.

Frequency

distribution

for

level

crossing

Despite which

the either

known serious do not pass

flaws through

in

them, the

as a result or

of

small

cycles larger The

mean level

interrupt features.

amplitude first is

cycles,

these

have some advantageous methods which data can be collected

the ease with counting frequency

by means of simple Secondly, in many cases introduction of of the

automatic cumulative by

devices

and interpreted

graphically.

distributions

can be approximated functions[161. The the stresses

standardized

distribution distributions

standardized results

facilitates of allowable

extrapolation for design.

and the estimation

One logical

step

to

improve

one-parameter

counting

is methods

-38-

to

count

two eventsthe waveform

as one.is divided

In

the into

peak-trough a succession on the

or of

range-mean half-cycles, and mean data can axis or being in the

counting, the peaks

and their

subsequent simultaneously. dimensional trough,

troughs,

amplitude of overall three

values, be either range,

are counted a three

Presentation graphical display,

mean or

peak,

and number of Although this

occurrences,

form of a numerical details from the of of fluctuations that

matrix.

counting

technique still

retains suffers

in both

amplitude

and mean, it

fact

small waves

excursions of much Figure used

can completely greater 2.2

mask the existence and damage

underlying

amplitude However, for the

significance, counting standardized peak-trough a row value,

as shown in method load joint has been

and 2.3.

peak-trough of the along peak trough row in

extensively stationary is

generation If

sequences probability trough

with matrix value,

properties. a position representing of next

known, with

representing then

down a column the possible are given

once a peak is

chosen

values

and their the matrix,

relative Figure

probabilities 2.15.

by the appropriate process

Random selection

moves alternately sequence for

along fighter the

a row and down a column. aircrafts, FALSTAFF[171, probability However, be the

A standardized

loading

has been generated was obtained a recorded parts.

in this from

way where flight and over instance, al[18,19] and

peak-trough

matrix

actual over

recordings. again omit may not or

repeating best

history It

way to test

for may, et

exaggerate the

some sequence signals

effects. to

Sherratt drive

generated

random rigs

needed

servo-hydraulic

elect ro-magnetic techniques

by applying peak-trough

one or two-dimensional probability matrix.

random walk

to a joint

-39-

Trough65

Level

12345 1 2 X X 2 1 1

4

Peak3

3 4

X1 3 5 2

6X 2

11 X 1

2 -- -

51

6

Figure

2.15

Peak-trough

distribution

matrix

of a history

segment.

If

the

cycle to similar

counting estimate methods

procedure the crack

is

used

as a part life,

of then

the the

damage analysis rainflow to the and other and suffer

initiation which

have features

make them superior They or avoid the

techniques inaccuracy as a result

considered from of which small

earlier. one-parameter which amplitude

distortion methods through achieved may well

peak-trough do not pass is It

cycles larger

either

the mean level without be that resort

interrupt or to arbitrary

cycles.

This

dead zones or reset in magnitude counted for

levels. to contribute

some cycles damage, the they user

are too small are to

any significant and it is

nevertheless these speed.

and recorded, the purpose of

up to testing

eliminate

accelerated

or gains

in computing

-40-

For Wetzel'shistory

repeating alleither

histories, givethe

the

range-pair, results,peak or the

the

rainflow, that

and theThe

methodsstarts at

identicalhighest

providedlowest

trough.

reason forinterruptions highest effects cyclic repetitive constant typical extreme possible the first

starting

with

an extreme pointoutside the largest If

in the historyrange one

is that

no

can exist and the depend or

determined the

by the

peak which

lowest on the softening of looping

trough. initial

ignores

transient and then

stress-strain of the

co-ordinates material,

hardening

tendencies the load history over

application hysteresis fatigue points

produces most of the

a stable, life at which, of a

pattern 14].

specimen[9r assures starting will

Consequently on a stress-strain

starting path other,

one of with to

interruptions, after additional

terminate possible

at

the

and return 2.4.

interruptions,

as in Figure

If initiation does not any

a finite life, create

loading then the

history requirement

is analysed of starting

to estimate

the crack point of

of an extreme

too

difficulty. much history between troughs, are

When two consecutive considered,

blocks

repeatedly

applied defined lowest

a new block of the

be may highest

conveniently peaks or the

successive as in Figure

occurrences 2.16.

However, the for aforesaid the

for

non-repeating, give are of different

that

is,

open-ended, The rules at its any time at

histories described during the the first

methods

results. if

Wetzel's the

method

incomplete load

history peak.

absolute

value

exceeds

value

The second version

of the rainflow

method,

maximum-minimum

-41-

Time

Original

Block

Original

Block

i---

New Block

_

Figure

2.16 Repetition

of a block

history

to define

a new block.

procedure history range-pair with to

cannot

be used the

either, absolute a load of sort

because

it

requires

the

complete The

determine

maximum and minimum points. as a cycle or greater if it

method counts

range equal of

can be paired in the the

a subsequent direction.

loading This

magnitude

opposite

counting

completely

ignores

existencedoes not increasing rainflow unequal range-pair

of the sharp corneredform a closed or method ranges

spiral

type stress-strainis subjected strain or

curve whichto successively The

loop when a metal ranges these the less of

decreasing counts

stress[lll.

ranges

as half-cycles. of the the greater

By matching range, the

ignoring and estimates

remainder

method

damage than in situations are

rainflow

method. are only cycles

The difference a few cycles

may be significant to failure

where there

or where there

insignificant

minor

-42-

and most of the extreme

the

damage is done by a few major which the then is the same type is trough

cycles.

However, point,

when is peak, both

point, if

as the

starting

reached, if it is

i. e.

starting the

point lowest the

a peak then is achieved,

the highest thereafter

a trough

counting

methods give

exactly

same result.

Although affected concerned simplified points. patterns Then every forming search, by the the

none of the starting version, if

three point

versions as far

of the as

rainflow

method is cycles are is

extracted

third

Pattern history exceeds

Classification starts the at starting one of

Procedure the

considerably Since no other 2.12

the

extreme the four

point

point,

in Figure newly

are reduced

to two,

namely D-D and D-I a converging any length

types.

examined point converging the list

either

extends of

sequence, during loop. the

an extended or reduces

sequence

by forming

a closed

hysteresis

In within obviously number of

Wetzel's a full to

counting element

method,

if

any

history used.

point This

occurs

a band, leads

and band are which However storage counting

procedure

rounding

errors,

are minimized as the space is to

by increased bands is

elements computing If the

and bands. time purpose and of

number of for extract the

increased, increases. method this

band cycles,

array this of

becomes method is

inefficient that it

and uneconomic. facilitates nominal the

The major simulation history.

advantage of local

stress-strain

response

to a given

loading

-43-

2.7

Conclusions

Itthe basis

isof

ratherits ability In data

hazardous

to grade a cycle"better

countingfor

method onone type is of

to produce fatigue acquisition (b) regeneration of smaller initiation

results" a cycle

application for (a)

only. on-line

analysis, and

counting to

needed the

reduction

determine of the

loading history, estimation

environment, (c) of life. of what truncation crack

and simulation for and accelerated (e)

service (d) crack

loads life,

testing, of

estimation

propagation on the basis procedures

Hence a cycle is expected

counting

method ought

to be chosen Cycle counting

out of the analysis. rather than merely

need to be understood

followed.

in other

Frequency areas the of

domain analysis

techniques

are well

established analysis to

of engineering

be used in fatigue and may to measure and control and simulate are originate noise particularly

describe response history.

loading a test

environment, rig, and to

the dynamic the service

regenerate

Frequency that act

domain

techniques

effective

where loads for example,

upon a structure gusts of wind,

from the environment, and vibration.

sea waves,

easy results to

One-parameter implement in

counting hardware or

methods software,

are

simple

to

understand, to display

convenient them.

graphically

and interpret

and extrapolate

-

The peak-trough

counting

be used to method may

regenerate

standardized

load sequences with or random

stationary

properties.

-44-

differences and ends at rainflow

For

a given the

block use of point,

of

service

history levels, if

and ignoring the block

the starts

due to

discrete then the

an extreme

range-pair,

Wetzel's,

and the

methods all

give

identical

result.

the crack

These methods initiation is life,

are

best

suited they

for all

the

use of

estimating in of a the

because with the

extract

cycles

manner which material

consistent

stress-strain

behaviour

considered.

growth

The choice

of

a counting

method for model used.

predicting

the

crack

largely

depends on the growth

-45-

REFERENCES FOR CHAPTER 2

1.2.

Palmgren, Technical

"The A., Translation,

Service Life of NASA TT F-13460.in

Ball

Bearings",ASME, Journal

a

"Cumulative Miner, M. A., Damage Mechanics, 1945. Sept. of Applied Livesey, J., Webber, Strain Measurements Conference, 1972. D., in

Fatigue",

3.

"Recording Military

and Interpretation of Bridges", J. B. C. S. A.

4.

Haas, T., Mechanical 5,1962.

"Loading Design",

Statistics Engineers

as a Basis Digest, Vol.

of Structural 23, Nos. 3,4

and and

5.

"Fatigue Dowling, N. E., Stress-Strain Histories", 17, No. 1, March 1972.

Failure Journal

for Predictions of Materials,

Complicated JMLSA, Vol.

6.

"Cycle P., Dabell, Watson, B. J., SEE Symposium held Damage" Proc. of 1975. Power, Testing"

Counting and Fatigue Feb. at Warwick Uni., for Fatigue Feb. 1975.Damage

7.

"The Analysis E. M., of Random Data Proc. of SEE Symp. held at Warwick Uni."The SAE Cumulative No. 750038.

8.

S., Tucker, L., Bussa, SAE Paper Test Program",

Fatigue

9.

Wetzel, Thesis,

Damage Analysis", R. M., "A. Method of Fatigue Canada, 1971. Waterloo, University of

Ph. D.

10.

to Subjected "Fatigue Metals M., Endo, T., of Matsuishi, to Japan Soc. of Mech. Eng., Paper presented Stress", Varying Fukuoka, March 1968 (in Japanese). K. and K., Kobayashi, K., Takahashi, Mitsunaga, Endo, T., for Random or Metals "Damage Evaluation M., of Matsuishi, Behaviour of Proc. of Symp. on Mechanical Loading", Varying Kyoto, Japan, Aug. 1974. Science, Soc. of Material Materials,"The On-Site H., Anzai, T., Endo, Proc. Complex Load", Damage under 1978. April University, Warwick Indication of SEECO '78 of Fatigue held at

11.

12.

13.

0.,, "Predictions of Cumulative Fuchs, H. D. V., Nelson, SAE Paper No. 750045. Damage Using Condensed Load Histories", M., "Computer T. H., Sinclair, J. F., Topper, Martin, Behaviour with Cyclic Stress-Strain Simulation of Based Materials Research and Standards, to Fatigue", Applications MTRSA, Vol. 11 No. 2, Feb. 1977. G.

14.

-46-

15.

"Mathematical Rice, S. 0., Analysis of Random Noise", Selected Papers on Noise and Stochastic Processes, edited by Nelson Wax, Dover Pbl. Inc., New York, 1958. "Random Load Analysis 0., Buxbaum, Between as a Link Operational Stress Measurement and Fatigue Life Assessment", ASTM, STP, 671,1979. FALSTAFF, Description for Fatigue Evaluation, Sherratt, Realistic of Aircraft a Fighter ICAF Doc. No. 839 Loading Standard

16.

17.

18.

P. W., "Advances in Computer Controlled F., Davall, ASTM STP 613,1976. Fatigue Testing",

19.

"The Use of Small On-Line P. R., F., Edwards, Sherratt, Journal Testing", for Random-Loading Fatigue Computers of 13-4, issue 63, Dec. Vol. Engineers, Soc. of Environmental 1974.

CAMP ER 3

LINK

BETWEEN POWER SPECTRAL DENSITY AND FATIGUE LIFE BASED ON RAINFLOW

3.1

Introduction

Throughout and buildings time. structure different by a set are

their subjected task

service to in is

life loads,

machines, the majority of

equipment, of which service

vehicles, vary life in with of its a

An important or

the to

estimation analyse loads most the

the

a component

operational

loads

parts. of static

These operational forces. In

can seldom be characterized cases the loading is

practical

stochastic statistical

and may be a continuous methods are needed to

random process, describe describes the

hence conventional loads. is the A

operational loads

common statistical spectral density.

function

which

the

power

The power been Finite input density is to receiving Element forces, plots locate has

spectral

presentation as

of

a stress-time of Given predict

history used of

has of

more

consideration to analyse Element

a result

increased a set

methods a Finite

structures. program will

dynamic spectral the aim proposed has been is

power Often the

at any point the critical

on a component areas disastrous and to

or structure. see whether

structure done, useful,

potentially estimate

resonances. most severely spectral density

Once this stressed and life

a life

based on the between power

location estimation

hence links

are needed.

-48-

The procedure ergodic specific

purpose

of

this

study

is

to

develop to

a life

estimation and The is

for

a structural process

component which in

subjected general density

a stationary band. loads,

random stress aim, given

can be wide of operational cycle counting in the

the power spectral which

to produce the fatigue

an expression life. for which for

uses rainflow could act

to predict design of from or

Such a procedure the loads that

be useful

components the

upon them may originate of wind; sea waves, noise,

environment,

example

from

gusts

road roughness.

3.2

Sonne Aspects

of Random Loading

The theory physical processes, subejct definitions is systems are

of

random processes of this of depth type. probability in the

is

extensive,

because treatment necessary, 2]. of

so many random and the some

For a rigorous theory literature[" is

a knowledge covered in

However,

and results

are given

here to base the discussion

properly.

The ensemble random ensemble ergodic signal. does not

is

the entire y

history random

in time process is

and amplitude one in

of the the An is a

A stationar vary is the its

which time. sample provides

statistical one

properties in which

with one

process of

a stationary ensemble. to what degree

representative means of two

A correlation two functions correlation

function

determining x(t)

are related. function

Given is

functions as:

and y(t),

their

Rxy(T)

defined

Rxy(T)

= tim

T 1 Jx(t)Y(t

T-co 2T-T

+T)

dt

Eq. 3.1

An autocor relation function of T,

function

shows how x(t) of x (t) lim T-

is correlated

with

itself

as a

or time

average

x (t +T) . 1T fx(t)x(t+T)dt 2T-T

Rxx (T) =R)= Power spectral a sand of density describes

Eq. 3.2

the amount of average asT fx(t)e_iwtdt -T

power contained

in

frequencies,

and is defined G (w) = firn 1 T, 2T

Eq. 3.3

The autocorrelation lyr

function

and spectral co G(w) = R (T) =1G 21T

density

are a Fourier

transiorr:

i

P.('7)e'7 co (w)

dT

Eq. 3.4

wT-dw

Eq. 3.5

The autocorrelation

function

and' spectrald2nRft-)

densityco

are further

related[1] Eq. 3.6

1 fw2(w)ei1Fdw

{-1) n=-

nG

d-r2n

2 ,r

Defining

the ntn mordent of spectral1

density,CO

mn, as: Eq. 3.7

(w) dw wnG mit =-J 2, rT , then, with T=J, 3.6 1 ?ells e(,.f

JG(w)_co

00

R( )=1 no =

dw

Eq. 3.7. i

2 7T Iw2G(wl

r__;

")?

'` ;

Eq. 3.7. ii

00

G(r1) 2Tr _

Eq.

3.7. iii

--

Eq. 3.7. i may be re-written

as:co T

11 x2 =-G 2

-

(w) dw =1 im T+c2T

x2 (t) dt-T

Eq. 3.8

Thus the power' rrns) ,a, of

total the is

area process

of G(w)/27T is x(t), and its

nonnegative square the root

and equals (the

the

'average

root-mean-square, of a process.

a corirnon way of specifying

intensity

A narrow spectral frequencies centre density whose

band G(w) width of

process has is

is

a stationary values to

random process only the in

whose of the

significant small

a band of

compared A wide band

magnitude has

frequency over

the

band. range of time

process Fig.

significant two

power terms types

a wide

frequencies.

3.1 shows these

of processes

and their

domain representations.

G(w)

G(w)

W

W

(a)

(c)

-/

4- 1

t

t

Fig.

3.1

Examplestime-domain

of

two

spectral

densitynarrow-band

plots(a)

andand (b),

theirwide

representation,