utmis.org.loopiadns.comutmis.org.loopiadns.com/...Fatigue_Stockholm_08.pdf · • the stress concentration in weldments, • the nominal stress, the hot spot stress and the peak stress
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6) The Fracture Mechanics approach• basics of fatigue crack growth analysis• stress intensity factors for cracks in weldments,• the weight function approach – 2-D and 3-D solutions,• the residual stress effect,• modeling of the fatigue crack growth of irregular planar cracks, • quantification of the scatter and the Monte Carlo simulation),
7) Simple methods for improvement of fatigue performance of welded structures• decrease of the stress concentration, • elimination of the local bending effect, • modification of the stress path, • optimization of the global geometry of a welded structure, • introduction of favorable residual stresses• live examples
8) Recent developments in the fatigue crack growth analysis • The UniGrow fatigue crack growth model for spectrum loading
11. K. Iida and T. Uemura, “Stress Concentration Factor Formulas Widely Used in Japan”, Document IIW XIII-1530-94, The International Welding Institute, 1994.
12. Bahram
Farhmand, G. Bockrath
and J. Glassco, Fatigue and Fracture Mechanics of High Risk Parts, Chapman @Hall, 1997
13. Sandor, R.J., Principles of Fracture Mechanics,
Prentice Hall, Upper Saddle River, 2002.
14. Socie, D.F., and Marquis, G.B., Multiaxial
Fatigue, Society of Automotive Engineers, Inc., Warrendale, PA, 2000.
22. Collins, J.C., Mechanical Design of Machine Elements and Machines, John Wiley & Sons, New York, 2003.
23. Shigley, J.E., and Mischke, C.R., Mechanical Engineering Design, McGraw-Hill, New York, 2001
24. Juvinal, R.C., and Marshek, K.M., Fundamentals of Machine Components Design, John Wiley & Sons, New York, 2000.
25. Norton, R.L., Machine Design –
An Integrated Approach, Prentice Hall, New Jersey, 2000
26. Hamrock, B.J., Jacobson, Bo and Schmid, S.R., Fundamentals of Machine Elements, McGraw-
Hill, Boston, 1999.
27. Orthwein, W., Machine Component Design, West Publishing Company, New York, 1990.
28. J.-L. Fayard, A. Bignonnet
and K. Dang Van, “Fatigue Design Criterion for Welded Structures”, Fatigue and Fracture of Engineering Materials and Structures, vol. 19, No.6, 19996, pp723-729.
29. V.A. Ryakhin
and G.N. Moshkarev, Durability and Stability of Welded Structures in Earth Moving Machinery”, Mashinostroenie, Moscow, 1984 (in Russian)
30. 7. V. I. Trufyakov
(editor), The Strength of Welded Joints under Cyclic Loading, Naukova
Dumka, Kiev, 1990 (in Russian).
31. J.Y. Young and F.V. Lawrence, ”Analytical and Graphical Aids for the Fatigue Design of Weldments”, Fracture and Fatigue of Engineering Materials and Structures, vol. 8, No.3, 1985, pp. 223-241.
37. E. Niemi, Structural Stress Approach To Fatigue Analysis Of Welded Components,
The International Institute of Welding, Doc. XIII-WG3-06-99, 1999
38. D. Pingsha, A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints, International Journal of Fatigue, vol. 23, 2001, p. 865-876
39. R.E. Little andJ.C. Ekvall, Statistical Analysis of Fatigue Data, ASTM STP 744, 1981
40. J.D. Burke and F.V. Lawrence, The Effect of Residual Stresses on Fatigue Life, FCP Report no. 29, University of Illinois, College of Engineering, 1978
1) Brief review of American and European 1) Brief review of American and European rules concerning static strength analysis rules concerning static strength analysis of of weldmentsweldments,,
When permanent joints are an appropriate design solution, welding
is often an economically attractive alternative to threaded joints. Most industrial welding processes involve local fusion
of the parts to be joined, at their common interfaces, to produce a weldment.
In the design of welded joints in a structure, two most common types of welds are used, i.e. butt welds
and fillet welds.
A butt weld
is usually used to join two plates of the same thickness and is
considered to be an integral part of the loaded component. Calculations are carried out as if stresses and trains in the weld and in the base metal were the same. The influence of the weld reinforcement (overfill) is ignored.
Fillet welds
are non-integral in character, and have such a shape and orientation relative to the loading that it almost disallows application of simple stress analysis. Conventional practice in welding engineering design has always been to quantify the size of the weld depending on the stress acting in the weld throat cross sectional area. Thus, fillet weld sizes
are determined by reference to allowable shear stress in the throat cross section.
Fillet welds
being parallel
to the direction of the loading force are called longitudinal fillet welds.
Fillet welds normal
to the direction of the loading force are called transverse fillet welds.
The stress state
in the weld throat plane consists in general of three stress components, i.e. two shear components τ1
Stress concentration & stress distributions in weldments
Various stress distributions in a butt weldment;
r
A
B
C
DE
P
M
F
σpeak
σhs
σn
σpeak
t
C
σn
• Normal stress distribution in the weld throat plane (A), • Through the thickness normal stress distribution in the weld toe plane (B), • Through the thickness normal stress distribution away from the weld (C),• Normal stress distribution along the surface of the plate (D),• Normal stress distribution along the surface of the weld (E), • Linearized normal stress distribution in the weld toe plane (F).
Various stress distributions in a T-butt weldment with transverse fillet welds;
r
t
t1
ED
BC
A
σpeak
σn
σhs
FP
M
C
Θ
• Normal stress distribution in the weld throat plane (A), • Through the thickness normal stress distribution in the weld toe plane (B), • Through the thickness normal stress distribution away from the weld (C),• Normal stress distribution along the surface of the plate (D),• Normal stress distribution along the surface of the weld (E), • Linearized normal stress distribution in the weld toe plane (F).
Stress concentration & stress distributions in weldments
Static strength analysis of weldments•The static strength analysis of weldments requires the determination of stresses in the load carrying welds. •The throat weld cross section is considered to be the critical section and average normal and shear stresses are used for the assessment of the strength under axial, bending and torsion modes of loading. The normal and shear stresses induced by axial forces and bending moments are averaged over the entire throat cross section carrying the load. •The maximum shear stress generated in the weld throat cross section by a torque is averaged at specific locations only over the throat thickness
but not over the entire weld throat cross section area.
Stress components in the weld throat cross section plane in a Stress components in the weld throat cross section plane in a TT--
butt weldment with loadbutt weldment with load--carrying transverse fillet weldscarrying transverse fillet welds(nominal stress components in the weld throat cross section!!)(nominal stress components in the weld throat cross section!!)
(simplified combination of stresses in the weld throat cross sec(simplified combination of stresses in the weld throat cross section according tion according
to the customary American method !!)to the customary American method !!)
σ
= 0 !!;
τ1
= σX
=P/A;
τ2
=R/A
Calculation of stresses in a fillet weld
τ1
σxτ2
RP
L
t
RP
σn
α
(It is assumed that only shear stresses act in the weld throat !)
Fillet welds in primary shear and bending:Fillet welds in primary shear and bending:the American customary method of combining the primary shear andthe American customary method of combining the primary shear and
bending stressesbending stresses
(according to (according to R.C.JuvinalR.C.Juvinal
& K.M. & K.M. MarshekMarshek
in Fundamentals of Machine Component Design, Wiley, 2000)in Fundamentals of Machine Component Design, Wiley, 2000)
Fillet welds in primary shear and bending:Fillet welds in primary shear and bending:the ISO/IIW method of combining the primary shear and bending shthe ISO/IIW method of combining the primary shear and bending shear stressesear stresses
The AWS method: It is assumed that the weld throat is in shear for all types of load and the shear stress in the weld throat is equal to the normal stress induced by bending moment and/or the normal force and to the shear stress induced by the shear force and/or the torque. There can be only two shear stress components
acting in the throat plane -
namely τ1
and τ2
. Therefore the resultant shear stress can be determined as:
2 21 2τ τ τ= +
The weld is acceptable if :
3ys
ys
στ τ< =
Where:
τys
is the shear yield strength of the: weld metal for fillet welds
and parent metal for butt welds
Static Strength Assessment of Fillet WeldsStatic Strength Assessment of Fillet Welds
Stress components in the weld throat cross section plane in a Stress components in the weld throat cross section plane in a TT--
butt weldment with loadbutt weldment with load--carrying transverse fillet weldscarrying transverse fillet welds(correct solid mechanics combination of stresses in the weld thr(correct solid mechanics combination of stresses in the weld throat oat ––
International Welding Institute employs a method in which the stresses are resolved into three components across the weld throat. These components of the stress tensor are the normal stress component σ
perpendicular to the throat plane, the shear stress τ2
acting in the throat parallel to the axis of the weld and a shear stress component τ1
acting in the throat plane and being perpendicular to the longitudinal axis of the weld. The proposed formula for calculating the equivalent
stress is:
( )2 2 21 1 23eqσ β σ τ τ= + +
The weld is acceptable if : eq ysσ σ<Where: σys
is the yield strength of the: weld metal for fillet welds and parent metal for butt welds The β
coefficient is accounting for the fact that fillet welds are slightly stronger that it is suggested by the equivalent stress σeq
.
β
= 0.7
for steel material with the yield strength σys
< 240 MPa
β
= 0.8
for steel material with the yield strength 240 < σys
< 280 MPa
β
= 0.85
for steel material with the yield strength 280 < σys
< 340 MPa
β
= 1
for steel material with the yield strength 340 < σys
Idealization of welds in a TIdealization of welds in a T--
butt welded joint; a) geometry and loadings, b) butt welded joint; a) geometry and loadings, b) and c) position of weld lines in the model for calculating stresand c) position of weld lines in the model for calculating stresses under ses under
axial, torsion and bending loadsaxial, torsion and bending loads
The StrainThe Strain--life and the Cyclic Stresslife and the Cyclic Stress--Strain Curve Obtained from Strain Curve Obtained from Smooth Cylindrical Specimens Tested Under Strain Control Smooth Cylindrical Specimens Tested Under Strain Control
states that if the nominal stress histories in the structure and in the test specimen are the same, then the fatigue response in each case will also be the same and can be described by the generic S-N curve. It is assumed that such an approach accounts also for the stress concentration, loading sequence effects, manufacturing etc.
K0K1K2K3K4K5St
ress
am
plitu
de, Δ
σ n/2
or Δ
σ hs/2
Number of cycles, N0
The Similitude Concept in the The Similitude Concept in the SS--NN
states that if the local notch-tip strain history in the notch tip and the strain history in the test specimen are the same, then the fatigue response in the notch tip region and in the specimen will also be the same and can be described by the material strain-life (ε-N) curve.
The Similitude Concept in the The Similitude Concept in the εε
states that if the stress intensity K for a crack in the actual component and in the test specimen are the same, then the fatigue crack growth response in the component and in the specimen will also be the same and can be described by the material fatigue crack growth curve da/dN
-
ΔK.
a) Structure
Q
H
F
ab) Weld detail
c) Specimena
P
P
ΔK 10-12
10-11
10-10
10-9
10-8
10-7
10-6
1 10 100
Cra
ck G
row
th R
ate,
m/c
ycle
mMPa,KΔ
The Similitude Concept in the The Similitude Concept in the da/dNda/dN
What stress parameter is needed for the Fracture What stress parameter is needed for the Fracture Mechanics based (Mechanics based (da/dNda/dN--ΔΔKK) fatigue analysis?) fatigue analysis?
x
a0
T
ρ
Stre
ss σ
(x) ( )
2x
xKσπ
=
S
SThe Stress Intensity Factor K
characterizing the stress field in the crack tip region is needed!
The
K
factor can be obtained from:-
ready made Handbook solutions (easy to use but often inadequate for the analyzed problem)
-
from the
σ(x)
near crack tip stress or
displacement data obtained from FE analysis of a cracked body (difficult)
-
from the weight function by using the FE stress analysis data of un-cracked body (versatile and suitable for FCG analysis)
Loads and StressesLoads and StressesThe load, the nominal stress, the local peak stress and the streThe load, the nominal stress, the local peak stress and the stress concentration factorss concentration factor
Loads and StressesLoads and StressesThe load, the nominal stress, the local peak stress and the streThe load, the nominal stress, the local peak stress and the stress concentration factorss concentration factor
stress concentration factor (net or gross, net Kt ≠
gross Kt
!!
)
σpeak
–
stress at the notch tip
σn
-
net nominal stress
S -
gross nominal stress
Stress Concentration Factors in Fatigue AnalysisStress Concentration Factors in Fatigue AnalysisThe nominal stress and the stress concentration factor in simpleThe nominal stress and the stress concentration factor in simple
Loads and StressesLoads and StressesThe load, the nominal stress, the local peak stress and the streThe load, the nominal stress, the local peak stress and the stress concentration factorss concentration factor
How to establish the nominal stress history?How to establish the nominal stress history?a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, Mb
=1, T=1 in order to establish the proportionality factors, kP
, kM
, and kT
such that:
;;= ⋅ = ⋅ = ⋅P M Tn n nP M Tbk P k M k Tσ σ τ
b) The peak and valleys of the nominal stress history σn,,i
are determined by scaling the peak and valleys load history Pi
, Mb,I
and Ti
by appropriate proportionality factors kP
, kM
, and kT
such that:
, , ,;,= ⋅ = ⋅ = ⋅P M Tn i n i n i iiP M Tb ik P k M k Tσ σ τ
c) In the case of proportional loading the normal peak and valley stresses can be added and the resultant nominal normal stress history can be established. Because all load modes in proportional loading have the same number of simultaneous reversals the resultant history has also the same number of resultant reversals as any of the single mode stress history.
;,, i Mi Pn b ik P k Mσ += ⋅ ⋅
d) In the case of non-proportional loading the normal stress histories (and separately
the shear stresses) have to be added as time dependent processes. Because each individual stress history has different number of reversals the number of reversals in the
resultant stress history can be established after the final superposition of all histories.
Superposition of nominal stress histories induced by two Superposition of nominal stress histories induced by two independent loading modesindependent loading modes
How to establish the linear elastic peak stress, How to establish the linear elastic peak stress, σσpeakpeak
,,
history?history?
a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, Mb
=1, T=1 in order to establish the proportionality factors, kP
, kM
, and kT
such that:
;;peak peak peaP M Tkbk P k M k Tσ σ τ= ⋅ = ⋅ = ⋅
b) The peak and valleys of the notch tip peak stress history σpeak,,i
are determined by scaling the peak and valleys load history Pi
, Mb,I
and Ti
by appropriate proportionality factors kP
, kM
, and kT
such that:;, , ,, iipeak i peak i peak iP M Tb ik P k M k Tσ σ τ= ⋅ = ⋅ = ⋅
c) In the case of proportional loading the normal peak and valley stresses can be added and the resultant notch tip normal peak stress history can be established. Because all load modes in proportional loading have the same number of simultaneous reversals the resultant history has also the same number of resultant reversals as any of the component single mode stress history.
;, ,iP Mp i bk iea k P k Mσ += ⋅ ⋅
d) In the case of non-proportional loading the normal stress histories (and separately
the shear stresses) have to be added as time dependent processes. Because each individual stress history has different number of reversals the number of reversals in the
resultant stress history can be established after the final superposition of all histories.
Superposition of linear elastic notch tip stress histories inducSuperposition of linear elastic notch tip stress histories induced ed by two independent loading modesby two independent loading modes
a) Ground loads on the wings, b) Distribution of the wing bending moment induced by the ground load, c) Stress in the lower wing skin induced by the ground and
flight loads
Characteristic load/stress history in the aircraft wing skinCharacteristic load/stress history in the aircraft wing skin
How to establish the link between the fluctuating load history How to establish the link between the fluctuating load history and the stress distribution, and the stress distribution, σσ((x,yx,y)),,
in the potential crack plane?in the potential crack plane?a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, Mb
=1, T=1 in order to establish the link between the load and the
stress distribution, kP
, kM
, and kT
such that:
;( , ) ( , ) ( ,( , ) ; ( , ) () , )P M Tbk x yx y x y x yP k x y M k x y Tσ σ τ= ⋅ = ⋅ = ⋅
b) The fluctuating stress distributions corresponding to instantaneous peaks and valleys of the load history are determined by scaling the reference stress distributions kP
(x,y), kM
(x,y), kT
(x,y) by appropriate magnitudes of the load history Pi
, Mb,I
and Ti
such that:
Where: σ(x,y) –
stress distribution in the x-y
plane (crack plane)
kP
(x,y), kM
(x,y), kT
(x,y)
–
reference stress distributions induced by unit loads P=1, Mb
=1, T=1
;,( , ) ; (( , ) ( , ) ( ,) )), ( ,i i ii iP M Tb ik x y P k x y M kx y x y xx y y Tσ σ τ= ⋅ = ⋅ = ⋅
c) In the case of proportional loading
the stress distributions corresponding to peaks and valleys of the load history can be added and the resultant stress distributions can be established. The nominal stress history can be also used as the stress distribution calibration parameter.
;,( , ) ( , )( , ) ii P M b ik x y P kx y x y Mσ += ⋅ ⋅
, ,, , ;( , ) ( , )( , )n P n Mi ii n P n Mx yy k x y k xσ σσ σσ += ⋅ ⋅
How to get the resultant stress distributionHow to get the resultant stress distribution
from the from the Finite Element stress data? Finite Element stress data? (Notched shaft under axial, bending load)(Notched shaft under axial, bending load)
Determination of approximate stress distributions in notched Determination of approximate stress distributions in notched bodies bodies (for simultaneous axial and bending load)(for simultaneous axial and bending load)
Details of the stress history a3799_c01; visible repeatable workDetails of the stress history a3799_c01; visible repeatable working cycles; ing cycles; ((Right Hand Axle Torque history Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the histowith removed ranges less than 5% of the largest one in the history)ry)
NonNon--dimensional stress history a3799_01; dimensional stress history a3799_01;
((Right Hand Axle Torque history Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the histowith removed ranges less than 5% of the largest one in the history)ry)
Statistical data for the scaling peak stress Smax
Log-Normal Probability distribution, LN (4.1975, 0.1098)
Mean Value Standard Deviation Coefficient of variation
Cycle Counting ProcedureCycle Counting Procedureand Presentation of Resultsand Presentation of Results
The measured stress, strain, or load history is given usually in
the form of a time series, i.e. a sequence of discrete values of the quantity measured in equal time intervals. When plotted in the stress-time space the discrete point values can be connected resulting in a continuously changing signal. However, the time effect on the fatigue performance of metals (except aggressive environments) is negligible in most cases. Therefore the excursions of the signal, represented by amplitudes or ranges, are the most important quantities in fatigue analyses. Subsequently, the knowledge of the reversal point values, denoted with large diamond symbols in the next Figure, is sufficient for fatigue life calculations. For that reason the intermediate values between subsequent reversal points can be deleted before any further analysis of the loading/stress signal is carried out. An example of a signal represented by the
reversal points only is shown in slide no. 141.The fatigue damage analysis requires decomposing the signal into
elementary events called ‘cycles’. Definition of a loading/stress cycle is easy and unique in the case of a constant amplitude signal as that one shown in the figures. A stress/loading cycle, as marked with the
thick line, is defined as an excursion starting at one point and ending at the next subsequent point having exactly the same magnitude and the same sign of the second derivative. The maximum, minimum, amplitude or range and mean stress values characterise the cycle.
Unfortunately, the cycle definition is not simple in the case of
a variable amplitude signal. The only non-
dubious quantity, which can easily be defined, is a reversal, example of which is marked with the thick line in the Figures below.
Constant and Variable Amplitude Stress Histories;Constant and Variable Amplitude Stress Histories; Definition of the Stress Cycle & Stress ReversalDefinition of the Stress Cycle & Stress Reversal
Stress Reversals and Stress Cycles in a Variable Stress Reversals and Stress Cycles in a Variable Amplitude Stress HistoryAmplitude Stress History
The reversalreversal
is simply an excursion between two-consecutive reversal points, i.e. an excursion between subsequent peak and valley
or valley and peak.
In recent years the rainflowrainflow
cycle counting method has been accepted world-wide as the most appropriate for extracting stress/load cycles for fatigue analyses. The rainflowrainflow
cycle is defined as a stress excursion, which when applied to a deformable material, will generate a closed stressstress--strain hysteresis loopstrain hysteresis loop. It is believed that the surface area of the stress-strain hysteresis loop represents the amount of damage induced by given cycle. An example of a short stress history and its rainflowrainflow
The The Direct Direct MMethod:ethod:1. Find the absolute maximum revesal point; 1. Find the absolute maximum revesal point;
2. Add the absolute maximum at the end of the history, 2. Add the absolute maximum at the end of the history, i.e. make it to be the last reversing point in the history;
3. Start counting form the reversal no. 3 (always); 3. Start counting form the reversal no. 3 (always);
The stress ranges are often grouped into classes and the final result of the rainflow
counting is presented in the form of an exceedance
diagram or a cumulative frequency distribution. The character of the exceedance
diagram depends on the loading conditions and the dynamics property of the system within which the given component is working. It is generally known that the character of the frequency distribution diagram does not depend strongly on the loading conditions and it remains almost the same for a given system or machine. Therefore, some kind of standard frequency distribution (exceedance) diagrams can be found for similar types of machines and structures such as cranes, aeroplanes, offshore platforms, etc. The frequency distribution diagram can also be interpreted as a probability density distribution when presented in nj
/NT
vs. Δσj
/Δσmax
co-ordinates. The experimental probability density, f(xj
), for stress range, Δσj
, is determined as:
The discrete experimental probability density function, represented by the rectangular bars can be approximated by a continuous function. The attempt is usually made to fit one of the well-known theoretical probability density distributions such as the Normal, Beta, Log-Normal, Weibull, or others. The probability density data can also be converted into the sample cumulative probability diagram according to the relation:
a) Probability Diagram Obtained from the Experimental a) Probability Diagram Obtained from the Experimental Cumulative Spectrum; b) Cumulative Probability Cumulative Spectrum; b) Cumulative Probability F(xF(x<<xxjj
))
Relative stress range, xj
= Δσj /Δσmax
Prob
abili
ty f(
x j)=
n j/N
T
0j=7 j=6 j=4
0.3
0.5 1.0
0.1
0.2
j=5 j=3 j=2 j=1
a)
0j=7 j=6 j=4
0.5 1.0
0.25
j=5 j=3 j=2 j=1
0.50
0.75
1.00
nn
nn
nn
NT
76
54
32
++
++
+
nn
nn
nNT
76
54
3+
++
+
nn
nn
NT
76
54
++
+
Relative stress range, xj
= Δσj /Δσmax
Cum
ulat
ive
prob
., F(
x<x j
)=Σn
j/NT b)
Continuous probability distribution function: p(x), p(∆σ/ ∆σmax
Diagram obtained from the Diagram obtained from the ExperExper. . Cumulative Spectrum; b) Cumulative Probability Cumulative Spectrum; b) Cumulative Probability F(xF(x<<xxjj
))
Continuous probability density distribution: p(x), p(∆σ/ ∆σmax
A Frequency Domain approach to fatigue A Frequency Domain approach to fatigue analysis of random cyclic stress processes analysis of random cyclic stress processes In the case of vibration induced stresses and strains the determination of the stress-
or strain-time history as discussed earlier is rather difficult and very often the stress signal/process can not be determined in a deterministic form.
Therefore concepts taken from random process theory, such as the
power spectral density (PSD),
root mean square (RMS)
and the Fourier transform
analysis are used to characterize random load and stress histories.
The task of the analyst is determine the most probable stress spectrum or the probability density distribution of stress amplitudes of the analysed object knowing the statistical properties (PSD, RMS,
etc.) of the stress process at the critical location.
The stress spectrum determined in the form of discrete classes of stress amplitudes or in the form of a probability density distribution is subsequently used for the determination of cumulative fatigue damage and fatigue durability.
Therefore some elements of the random process theory need to be reviewed including the physical interpretation of basic parameters used for the characterisation of a random process such as the stress fluctuations induced by a random excitation signal.
The technique was pioneered by electronics engineers in the 1940s. They used it to characterise noise in electronic circuits. They were interested in the average amplitude of noise at different frequencies but couldn’t calculate the Fourier Transform at that time..