Chapter 6
• Failure resulting from fluctuating
load
Fluctuating load?
What is special about it?
1 M S Dasgupta BITS Pilani
• Variable loading results when the applied load or the induced stress on a component is not constant but changes with time
• In reality most mechanical components experience variable loading due to
-Change in the magnitude of applied load
-Change in direction of load application
-Change in point of load application
Fluctuating / Variable load
2
Stress variation: Sinusoidal
2stresseor variabl amplitude
2stressmean or midrange
stress of range
stress maximum
stress minimum
minmax
minmax
minmax
max
min
a
m
r
Completely Reversed : mean stress is zero; equal reversals on both sides; useful in conducting experiments
Repeated stress: minimum stress is zero; mean stress equal to half of the range stress
Fluctuating stress: maximum, minimum and mean stress are all non-zero and arbitrary
Idealized types of cyclic loading: Sinosoidal
Fatigue
• Fatigue is a phenomenon associated with
variable loading or more precisely to cyclic
stressing or straining of a material
• ASTM Definition of fatigue
– The process of progressive localized
permanent structural changes occurring in a
material subjected to conditions that produce
fluctuating stresses at some point or points
and that may result in cracks or complete
fracture after a sufficient number of
fluctuations. 5
Fatigue failure in Metals
Crack initiation at
the outer surface
Beach marks
showing the nature
of crack propagation
Final rupture occurs
over a limited area,
characterizing a very
small load required
to cause it
Crack initiation, propagation and rupture in a shaft subjected to repeated bending
6
Fatigue Life Prediction
predict the failure in number of cycles N to failure for a specific type of
loading 33 10 :(HCF) fatigue cycleHigh ;101:(LCF) fatigue cycle Low NN
• Stress life methods
– Based on stress levels only
– Least accurate of the three, particularly for LCF
– It is the most traditional because easiest to implement for a wide range of applications
– Has ample supporting data
– Represents high cycle fatigue adequately
• Strain life methods
– Involves more detailed analysis of plastic deformation at localized regions
– Good for LCF
– Some uncertainties may exist in results because several idealizations get compounded
– Hence normally not used in regular (special occasions)
• Linear elastic fracture mechanics methods (LEFM)
– Assumes that crack is already present and detected
– The crack location is then employed to predict crack growth and sudden rupture with respect to the stress nature and intensity
7
The S-N Diagram for steel (UNS G41300), normalized, Sut=812 MPa.
Endurance Limit,
It is the stress at which the
component can sustain
infinite number of cycles
S-N Diagram
R. R. Moore high-
speed rotating
beam machine.
Non-Ferrous materials tested up to 5*108 cycles
S’e
8
Sut – S’e relation
conditions loading actual in thelimit Endurance
bending reversein obtainedlimit Endurance
1460700
146050
'
'
e
e
ut
utut
e
S
S
MPa Sfor MPa
MPaSfor S.S
9
Se S’e relation
'
eedcbae SkkkkkS
factoronmodificatieffects ousmiscellane
factory reliabilit
factoron modificati etemperatur
factoron modificati load
factoron modificati size
factoron modificaticondition surface
f
e
d
c
b
a
k
k
k
k
k
k
10
b
uta aSk
Surface cond. Mod. factor (ka)
The surface modification factor depends on the quality of the
finish of the actual part surface and on the tensile strength of
the part material.
Size modification factor, kb
1. effect, size no loading axialFor
25451000837.0859.0
5179.224.162.7/
:only torsion and bendingin barscircular rotatingFor
107.0107.0
b
b
k
mmdifd
mmdifddk
Concept of Equivalent Diameter de
What happens when bars are not rotating but
say under bending.
Or non-circular bars like square, or I section?
Kb for non-rotating shapes
Effective dimension “de”
obtained by equating the
volume of material stressed
at and above 95 percent of
the maximum stress to the
same volume in the
rotating-beam specimen
Load modification factor, kc
torsion
axial
bending
kc
,59.0
,85.0
,1
Actually the kc is sensitive
to Sut of the material. Tables
6-11 to 6-14 (page no. 333)
in Text Book give the
details. The above values
are representative.
Temperature modifying factor, kd
Brittle fracture is a strong possibility when
operating temp is below RT
At temp. higher than RT, yielding should be
investigated first because the yield strength drops
off rapidly with temperature.
Creep at elevated temperature
Temperature modifying factor, kd
FT
where
TTTTk
o
F
FFFFd
100070
10595.010104.010115.010432.0975.0 41238253
For carbon and alloy steels experimental result
expressed as a fourth-order polynomial curve fit
to the data underlying
Or interpolate from a chart / table of
operating temp. vs tensile
ae zk 08.01
Reliability factor, ke
Based on standard
deviation of Endurance
strength data
Accounts for
– Residual stress
– Coating failure
– Frettage corrosion material of mating part.
– Synergic effect of corrosion and temperature
where is Se is function of frequency of loading.
Miscellaneous effects factor, kf
Kf is a reduced value of Kt and it is also called fatigue
strength reduction factor
Actual / Fatigue stress concentration factor, Kf
factor) (geometricfactor ionconcentrat stress lTheoretica
21)-6& 20-6 Fig.(from y valuesensitivit notch
tK
q
1111 tsshearfstf KqKorKqK
specimen free-notch in stress
specimen notched in stress maximumfK
Stress-concentration factors for a variety of geometries under
different loading conditions can be found in Table A–15
(page:1026-1032) 19
Notch Sensitivity
20
21
Estimation of Kf
Kf = 1+q(Kt -1).
•When q=0, the material has no sensitivity to notches, Kf=1.
•When q=1, or when notch radius is large for which q is
almost equal to 1, the material has full notch sensitivity, and
Kf = Kt.
•For all grades of cast iron, use q=0.20.
•Use the different graphs to obtain q for bending/axial and
torsional loading.
•Whenever the graphs do not give values of q for certain
combinations of data, use either Neuber equation or
Heywood equation.
22
Use the Neuber equation when the notch is circular/cylindrical.
Estimation of Kf
radiusnotch
strength. ultimate offunction i.e ),(
constant material a is andconstant Neuber is a where
11
1
1
r
Sfa
KqKand
r
aq
ut
tf
For steel, with Sut in kpsi, the Neuber constant can be
approximated by a third-order polynomial fit of data as
38253
38253
)10(67.2)10(35.1)10(51.219.0:
)10(67.2)10(51.1)10(08.3246.0:
ututut
ututut
SSSaTorsion
SSSaaxialorBending
100p
si =
0.6
89M
Pa
23
Use Heywood equation when the notch is NOT circular/cylindrical but is a
tranverse hole or shoulder or groove.
Estimation of Kf
335 page 15;-6 Table thein given are values
121
a
where
r
a
K
K
KK
t
t
tf
r= hole/ shoulder/groove size
24
Sn’
Goodman Method
m
a
-Sy
Sy
Sy Su
FATIGUE
FAILURE REGION
NO FATIGUE
FAILURE REGION
Goodman Line
0
Yield Line (Langer line)
1
u
m
n
a
SS
Predictor of failure in ductile materials
experiencing fluctuating stress
Sn’ = endurance strength
a = alternating stress
m = mean stress
25
Sn’/N
Sn’
Goodman Diagram
m
a
-Sy
Sy
Sy Su
FATIGUE
FAILURE REGION
Goodman Line
0
Yield Line
1
u
m
n
a
SS
Su/N
NSS u
m
n
a 1
Safe Stress Line
Safe Stress Line
SAFE ZONE
Sn’ =endurance strength
a = alternating stress
m = mean stress
26
Design under cyclic loading
fyt
m
e
a
nSS
1
1yt
m
e
a
SS
1
ut
m
e
a
SS
1
2
ut
m
e
a
SS
1
22
yt
m
e
a
SS
27
28
Different fatigue failure models
yielding)staticfor
checkingfor (only lineLanger 1
line EllipticASME1
lineGerber 1
line Goodman Modified1
line Soderberg1
222
2
yyt
m
yt
a
fyt
m
e
a
fut
mf
e
a
fut
m
e
a
fyt
m
e
a
nSS
nSS
nSn
S
nSS
nSS
M S Dasgupta BITS Pilani 29
Modified Goodman and
Langer Failure Criteria
Important Intersections in First Quadrant
30
Important Intersections in First Quadrant
Gerber and Langer
Failure Criteria
31
Important Intersections in First Quadrant
ASME-Elliptic and Langer
Failure Criteria