4102- Chap 5 VA.pdf

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Strength & Fracture

Chapter 5

Variable Amplitude Loading

Professor R. BellDepartment of Mechanical & Aerospace Engineering

Carleton University

© 2013

Chapter 5 - Variable AmplitudeLoading

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Variable Amplitude loading

Department of Mechanical &

Aerospace Engineering

In reality most structures

are subjected to variable

amplitude loading,

but before discussing Variable Amplitude

loading it is necessary to address the

pro em o oa n erac on an cracgrowth retardation.


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Load Interaction



an overload is much slower thanbefore the overload.

After a eriod of slower rowth the

original growth rate is gradually

resumed.

Overloads have a major effect

on the life of the structure.


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Overload and Underload




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Retardation



To account for retardation, consider

a crack subjected to constant amplitude

loading. After unloading there is a

compressive stress that is at least equal .


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Retardation



After overload occurs a more extensive residualstress field occurs.

This residual stress field acting against the

applied stress causes subsequent crack

growth to be slower (retarded).

After the crack has grown through the

enlarged residual stress field (plastic zone)

the original residual stress field is restored

and the crack growth rate returns to normal.


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1. Crack Tip Blunting

2. Com ressive Residual Stress at Ti

3. Closure Effects

1. Crack Ti Bluntin

Plastic Zone Size

2

r y y

This does not explain delayed retardation.It is observed that crack growth retardation

does not happen immediately after the overload but there is a delay


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2. Compressive Residual Stresses Models

In this case the retardation models attempt

to account for the residual stress field bysuperimposing it on the applied stress field.

be added to the SIF caused by the applied

load.

This is not an easy matter since it is difficult to

express the residual stresses in a quantitative

manner. Also all of the models are

2-dimensional sim lifications of a com lex

3-dimensional problem.


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Retardation Models



3. Crack Closure Models

remotely applied load is still tensile and do not reopen until a sufficientlyhigh tensile load is applied in the next cycle.

It was proposed that crack closure occurs because of crack tip plasticity.

As the crack grows a wake of plastically deformed material is developed

along the crack faces while the surrounding material remains elastic.

As the component is unloaded the plastically deformed materials causes.

Also, on the loading cycle a stress has to be applied before the crack

is just open.

This is called the opening stress op.openeff K K K max

or a gue crac grow o occur ecrack must be fully open.

Therefore there is an effective K for crack growth. open

K K

K K

thus

since min

minmax


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openeff

K K K

K K K

max

eff

open

K K

K K

thus

since min


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Crack Closure Model



Crack closure arguments successfully predict

delayed retardation after a single overload.1. Determine

eff i

eff K and hence

While the plastic zone is in front of the crack the

residual stresses do not affect the crack openingstress until the crack has propagated some distance

into the plastic zone.

eff

i i op

i

max

a

At this point the compressive stresses reduce the

stress applied to the crack causing retardation.

The crack propagates at a decreasing rate until

.

ada

dN f K i

i

eff i

Thus for the Paris Law

reac es a m n mum.

As the crack begins to grow out of the plastic zone

the crack growth increases until it resumes its normal rate.

ada

dN C K i

i

o eff i

m

n us ng a crac c osure mo e o pre c crac growthe difficulty is in obtaining the opening stress, op for

variable amplitude loading.

Once op is determined the following steps are followed.

where Co and ΔK eff correspond

to the same crack closure stress


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Crack Closure Model



3. Determine a i1

i i i1

This is repeated using cycle by cycle calculation- therefore long computer runs

● Co must be determined at the same

closure level as the effective SIF ΔKeff

● To Correct C for a new closure level

C C

U o m If C is determined at R = 0 ie Kmin = 0

Kop is assumed to be 0.3 Kmax

where U K

eff

K K K eff op max

ere ore

C

C o m

0 7.

The crack growth exponent, m, does not

K K K max min

.


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Wheeler Model



WHEELER MODELfor crack tip plasticity

The Wheeler model redicts that retardation

in the crack growth rate after an overload by

modifying the constant amplitude growth rate.

Pro ression of Wheeler Model

Wheeler introduces a retardation parameter R .

This parameter is based on the ratio of currentpi

zone size formed by the overload, r OL.

Overload at a crack of ao

strainPlane 6

stressPlane 2 2

22

2

2

YS

oo

YS

OOL

aK r


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Wheeler Model



When the crack has propagated further to a length ai the current plastic zone size is:

2

max

22

max iii

i

aK r

YS YS

Elastic - Constant Amplitude (CA)

dam

iOLO

i yi y

i R ar a

r r

thereforedN

Wheeler Model

da da

which can be adjusted to calibrate the

model.

(See later)

tarded Re Constant Amplitude

Where φR is the retardation parameter

The retardation arameter is a ower

dN

C K i

R i

m

Where:

relationship of the ratio of the currentplastic zone size and the remaining plastic

region formed by the overload, ρ.

i.e.

r yi

= cyc c p as c zone s ze ue o e oa ng cyc e

ap = sum of the crack length at which the overload

occurred and the plastic zone size

ai = crack length at the ith load cycle

γ = empirical shaping parameter


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Wheeler Model



In a constant amplitude test with a single

overload the retardation factor gradually

through the overload plastic zone.

Initial overload:

After the ith cycle of crack growth


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Wheeler Model



As soon as the boundary of the current

plastic zone touches the boundary of the

pi 1

, . .

i yi p

r

a a

yi

i 1

Crack Growth is summed as follows: a ada

dnr

i

r

i

0

1

0

ar = crack length after r cycles

da

= the crack growth during ith cycle.


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Wheeler Model



In the case where there is a second overload which

produces a plastic zone beyond the original plastic

zone e oun ary o s new p as c zone s use

and the instantaneous crack length becomes the new ao.

econ over oa

Problem: value of empirical parameter γ.

However this model predicts immediate max retardation


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Calibration of the Wheeler Model



It is important to calibrate the above model as follows.

An experimental test is carried out under variable amplitude loading.

The test result is predicted several times using proper da/dN - K data and proper

values using several different values of the adjustable parameter γ.

The value of the parameter which gives best coverage of the test data is the valueused in the predictions.

An example of this procedure is shown:

is not general. It depends on the load

history and spectrum.

A different s ectrum will re uire a different

calibration factor; calibration factors aresuitable for one type of spectrum only.

Failure to re-calibrate the model can lead to very erroneous results.


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1. Root Mean Square (Statistical Model)

This approach is based upon attempting to make predictions using the

constant am litude a roach.

The average crack growth rate is predicted from constant amplitude data

using the equations below.

da A K RMS

m

K K

k RMS

ii

k

2

1

Where a and m are constants


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= -

obtained using the Root Mean Square approach to crack growthprediction.

In addition, these predictions are given to demonstrate

the effects of stress ratio on welded structures (R=0.05 vs R = -1).

It is possible, using a life prediction program to investigate the

contribution that the compressive portion of a R = -1 loading cycle

makes to crack growth.


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Root Mean Square Example


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Prediction Models

SFC - straight fronted crack

SC1 - natural growth with single crack (ai/2c = 0.5)

SC2 - single crack with fixed aspect ratio (a/2c = 0.1)

MC1 - single crack with forcing function for a/2c to account for coalescence

MC2 - single crack with forcing function for a/2c to account for coalescence and edge cracking


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Predictions using Full Range Loading




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SFC Predictions – Various % of Loads



% of compressive

Stress included


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MC2 Predictions – Various % of Loads



% of compressive

Stress included

Thickness 78 mm


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SCF Predictions – Various % of Loads



% of compressive

Stress included

=


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f h l

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3. Crack Growth Analysis for VA Loading – Cycle by Cycle Calculation

In order to account for crack retardation in VA loadin it is necessar to carr out

cycle by cycle calculation of da/dN. It is also necessary to account for the actual

load history (see later). As an example we will consider crack growth calculation using the Wheeler Model

for retardation which was outlined above.

The information that must be included in a computer code is:

SIF calculation routines - factors

Crack growth rate equations for un-retarded growth e.g.max

29107.2 K K dN

a

Obtain and R from stress history

Calculate

ii

iiii

RK K

aK

1/max

Plastic zone size22 /

max YS yi K r

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Plastic zone size2

2

max

YS

yi

K r

Calculate new p pold pnew p

yiinew p

aa

aathenaaif

r aa

Calculate the retardation parameter (Wheeler Model)

i y

i R

r

Calculate the retarded growth rate dN dN

tarded

R i

Re

Constant Amplitude

1dN

daai

1

N N

aaa ii j

Calculate increase in a cycle by cycle.

Round-off with small numbers is often a roblem in these calculations.


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Parameters Affecting Fatigue Crack Growth Rates



THICKNESS

TYPE OF PRODUCT

TREATMENT

COLD DEFORMATION

TEMPERATURE

MANUFACTURER

BATCH TO BATCH VARIATION

ENVIRONMENT AND FREQUENCY



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Load Spectra and Stress Histories



For the application of damage tolerance procedures a reliable prediction

must be made of the number of load cycles which will propagate a crack

from its startin size to the ermissible size. To do this the followin is re uired:

crack propagation data geometry factors (SIF’s)

stress histor

Most load spectra can be represented by exceedance diagrams and these

can be used to develop the stress history for structure subjected to random

or semi-random loading.Most naturally occurring loading is termed semi-random because loads applied

to a ship or offshore structure are random in nature but there are high loads

(storms) which are clustered in certain periods (winter).

Load spectra are developed from measurements on structures in service

(strains, wave heights, wind forces etc.).

These spectra must be interpreted into an exceedance diagram and hence

to stress cycles using counting methods.



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These diagrams show how many times a stress or load is exceeded.

The exceedance are exceedances of peaks or ranges.

1 4 4

2 20 16

3 100 80

4 800 700

5 3000 2200

6 20000 17000



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C cle Countin Methods



1. Level Crossing counting

2. Peak Counting

3. Simple Range Counting

4. Rainflow Counting Original rainflow method

-

Hysteresis counting

Racetrack method

Ordered overall range counting Range-pair-range counting

In methods 1-3, once counts have been obtained they must be combined to

.

counting method take no consideration of the order in which cycles are applied.

Ignoring sequence effects can5have a significant effect on fatigue life.


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Rainflow Countin

p f


This gives the best representation of loading for fatigue calculations.

Rainflow counting has become a generic term for any counting methodwhich describes any counting method which attempts to identify closed

hysteresis loops.

A number of computer algorithms have been developed for rainflow

counting and are given as an ASTM Standard.



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Rainflow Countin

p f


Example

Depict the loading sequence as a function of time.

• use straight lines between min and max

Let “drops” start from every max and min and stop if:• it starts from max and passes a larger or equal max

• it reaches the run of another drop

Identify closed loops by joining drops

1 passes an equally large maximum

3 passes a larger maximum

4 reaches the run of drop 2

5 reaches the run of drop 1

“ ”

7 “falls out”8 reaches the run of drop 6

1 and 6, 2 and 5, 3 and 4; 7 and 8 form

2


. .


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Standardized Stress-Time Histories


GAUSS for general fatigue investigations

g er rcra oa ng an ar or a gue

HELIX-FELIX for articulated and semi-rigid rotors of helicoptersCOLD TURBISTAN for cold (low pressure) compressor disks of gas turbines

WHISPER for disks of as turbines

ENSTAFF for fighter aircraft components of carbon fibre

reinforced composites

WALZ for steel mill drive systems

WASH Wave Action Stress History


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A i i

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Fati ue Desi n Criteria


3. Fail - Safe Design.

Fail-safe desi n criteria was desi ned b aircraft en ineers who could

not tolerate extra weight required by large safety factors nor the danger

resulting from small safety factors.Fail - safe recognizes fatigue cracks will occur and arranges the structure

so that the cracks will not lead to failure of the structure before the are

detected and repaired.

i.e. multiple load paths and crack arresters. This was originally applied

to airframes now used in other applications.

Aero engines are fail - safe designs only in multi-engine aircraftLanding gear not fail safe but designed for safe life.

4. Damage Tolerant Design

Refinement of fail - safe philosophy Assumes that cracks exist and uses a Fracture Mechanics analysis to check

whether such cracks will grow large enough (critical size) to produce failures

before they are sure to be detected and repaired. (eg “Leak before Break”)

Chapter 5 - Variable Amplitude

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FM Approach to Crack Growth Aerospace Engineering


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. ,

Extraction of stress cycles (rainflow counting) or reversals from the stress history (Fig. e),


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FM A roach to Crack Growth Aerospace Engineering

Determination of stress

intensity factor

Indirect method (Fig. f):

analyze un-cracked weldment anddetermine the stress field, (x,y),

normalize the calculated stress

distribution with respect to the

nominal or any other reference

stress, i.e. (x,y)/n, c oose appropr a e we g unc on,

calculate stress intensity factor.

Direct method (Fig g):

determine the stress or

displacement field near the crack,

or the strain energy release rate,

calculate stress intensity factor.


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FM A r h Cr k r w h


Determination of crack increments for each stress cycle (Fig. h),

Determination of the number of cycles, N, necessary to grow

the crack from its initial size, a0, up to final size, af .


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p g g

J.M. Barsom and S.T. Rolfe, “Fatigue & Fracture Control in Structures”,ren ce a , .

Bannantine, J.A., Comer, J.J. and Handrock, J.L. "Fundamental of Metal FatigueAnalysis“ (Prentice Hall, 1990)

Almar-Naess, A. "Fatigue Handbook - Offshore Steel structures“. (TapirPublishers, Trondheim, Norway, 1985)

H. Neuber, “ Theory of Stress Concentration of Shear Strained Prismatic

Bodies with Arbitrar Non Linear Stress-Strain Law”, ASME Journal of Applied Mechanics, vol. 28, 1961, pp. 544-550.

K. Molski and G. Glinka, “ A Method of ElasticPlastic Stress and StrainCalculation at a Notch Root”, Material Science and Engineering, vol. 50, 1981,pp. 93-100.

“. , . . , -

and Stresses in Notches under Multiaxial Loading”, International Journal ofFracture, vol. 70, 1995, pp. 357-373.


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