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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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Variable Amplitude Loading
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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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Variable Amplitude Loading
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Root Mean Square Example
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Variable Amplitude Loading
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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
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% of compressive
Stress included
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MC2 Predictions – Various % of Loads
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% of compressive
Stress included
Thickness 78 mm
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SCF Predictions – Various % of Loads
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% of compressive
Stress included
=
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f h l
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Variable Amplitude Loading
Department of Mechanical &
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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
Chapter 5 - Variable AmplitudeLoading 30
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Variable Amplitude Loading
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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.
Chapter 5 - Variable AmplitudeLoading 31
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Parameters Affecting Fatigue Crack Growth Rates
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THICKNESS
TYPE OF PRODUCT
TREATMENT
COLD DEFORMATION
TEMPERATURE
MANUFACTURER
BATCH TO BATCH VARIATION
ENVIRONMENT AND FREQUENCY
Chapter 5 - Variable AmplitudeLoading 32
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Load Spectra and Stress Histories
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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.
Chapter 5 - Variable AmplitudeLoading 33
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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
Chapter 5 - Variable AmplitudeLoading 34
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C cle Countin Methods
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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
Aerospace Engineering
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.
Chapter 5 - Variable AmplitudeLoading 37
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Rainflow Countin
p f
Aerospace Engineering
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
Chapter 5 - Variable AmplitudeLoading 38
. .
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Standardized Stress-Time Histories
Aerospace Engineering
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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Fati ue Desi n Criteria
Aerospace Engineering
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”)
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FM Approach to Crack Growth Aerospace Engineering
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Aerospace Engineering
. ,
Extraction of stress cycles (rainflow counting) or reversals from the stress history (Fig. e),
Chapter 5 - Variable Amplitude
Loading
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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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Loading
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FM A r h Cr k r w h
Aerospace Engineering
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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Loading
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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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Loading
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