MODULE 5_Deisign for Fatigue Loads (1).pdf

MHA2002 MECHANISMS AND DESIGN CONCEPTS
MODULE 5
Design for Fatigue Loads
By
Dr. T. CHRISTO MICHAEL
Winter 2021-22
School of Mechanical Engineering
B.Tech. – Mechatronics and Automation

Stress Concentration
 The elementary equations are based on a number of
assumptions.
 One of the assumptions is that there are no discontinuities
in the cross section of the component.
 In practice, discontinuities and abrupt changes in cross
section are unavoidable due to certain features of the
component such as oil holes and grooves, keyways and
splines, screw threads and shoulders.
4/9/2022 MEE3001: DESIGN OF MACHINE ELEMENTS 2

A plate with a small circular hole subjected to tensile stress is
shown in Figure.
The distribution of stresses
near the hole can be
observed by using photo-
elasticity technique.

 Stress concentration is defined as the localization of high
stresses due to the irregularities present in the component and
abrupt changes of the cross section.
 In order to consider the effect of stress concentration and find
out localized stresses, a factor called stress concentration
factor is used. It is denoted by Kt, and defined as

Causes of Stress Concentration
Variation in Properties of Materials
• Internal cracks and flaws like blow holes
• Cavities in welds
• Air holes in steel components
• Non-metallic or foreign inclusions
Load Application
• Machine components are subjected to forces
• These forces act either at point or over a small area
• Since the area is small, the pressure at these points is
excessive
• This results in stress concentration

Examples:
• Contact between the meshing teeth of the driving and the
driven gear
• Contact between the cam and the follower
• Contact between the balls and the races of ball bearing
• Contact between the rail and the wheel
• Contact between the crane hook and the chain

Causes of Stress Concentration
Abrupt Changes in Section
• In order to mount gears, sprockets, pulleys and ball bearings
on transmission shaft, steps are cut on the shaft and shoulders
are provided from assembly considerations
Discontinuities in the Component
• Certain features of machine components such as oil holes or
oil grooves, keyways and splines, and screw threads result in
discontinuities in the cross-section of the component
Machining Scratches
• Machining scratches, stamp mark or inspection mark are
surface irregularities, which cause stress concentration

Determinationof Kt
(Theoretical Stress Concentration Factor)
• Mathematical method based on theory of elasticity
• Experimental methods :
 Photo-Elasticity Technique

Mathematicalmethod based on theory of elasticity
 It is possible for simple geometry only
 Kt for a flat plate with elliptical hole subjected to tensile force is
given by

Charts for TheoreticalStress ConcentrationFactor
 The charts are based on the photo-elastic analysis or
theoretical or finite element analysis of the mathematical model
 The model is made of a different material than the actual
material of the component
 The ductility or brittleness and static or cyclic load affects the
stress concentration
 Therefore, there is a difference between theoretical and actual
stress concentration factor in a component

Chart 1
Nominal stress

Nominal stress
Chart 2

Nominal stress
Chart 3

Nominal stress
Where,
Chart 4

Nominal stress
Where,
Chart 5

Reductionof stress concentration
• Additional Notches and Holes in Tension Member
• Fillet Radius, Undercutting and Notch for Member in Bending
• Drilling Additional Holes for Shaft with Keyway
• Reduction of Stress Concentration in Threaded Members

Additional Notches and Holes in TensionMember
(a) Original Notch
(b) Multiple Notches
(c) Drilled Holes
(d) Removal of Undesired Materials

Fillet Radius, Undercuttingand Notch for Member in Bending
Original Component Fillet Radius
Undercutting Addition of Notch

Drilling Additional Holes for Shaft with Keyway

Reduction of Stress Concentration in Threaded Members
Original Component Undercutting
Reduction in Shank Diameter

Introduction to Fatigue
• In static loading, the load is applied gradually and the failures
give visible warning in advance
• In practice, other kinds of loading occurring in machine
members produce stresses that are called variable, repeated,
alternating, or fluctuating stresses
• Often, machine members are found to have failed under the
action of repeated or fluctuating stresses
• The actual maximum stresses were well below the ultimate
strength of the material, and quite frequently even below the
yield strength
• The failures due to the repeated stresses for a very large
number of times is called a fatigue failure
• Fatigue failure gives no warning! It is sudden and total, and
hence dangerous

a) Completely reversed cycle of stress
(sinusoidal)
b) Repeated stress
c) Fluctuating stress
Stress Cycles
𝜎𝑚 =
𝜎max + 𝜎min
2
𝜎𝑎 =
𝜎max − 𝜎min
2
𝜎m = Mean Stress
𝜎a = Amplitude stress
or Variable stress

Factors causing Fatigue Failure
Basic Factors
 A maximum tensile stress of sufficiently high value.
 A large amount of variation or fluctuation in the applied
stress.
 A sufficiently large number of cycles of the applied
stress.
Additional Factors
• Stress concentration • Corrosion • Temperature
• Overload • Metallurgical structure
• Residual stress • Combined stress

Fatigue-Life Methods
• Stress-life method
• Strain-life method
• Linear-elastic fracture mechanics method

Stress-Life Method
High-speed Rotating-
beam Specimen
𝝈𝒆
′
𝝈𝐮𝐭

Endurance Limit & Modifying Factors
• The strength corresponding to the knee of the S-N curve is
called endurance limit or fatigue limit (𝜎𝑒
′)
• The limiting stress below which the load is repeatedly applied
an indefinitely large number of cycles without causing failure
𝜎𝑒 = 𝐾 𝜎𝑒
′
Where, 𝐾 = 𝐾𝑎𝐾𝑏𝐾𝑐𝐾𝑑
Ka - Surface Finish Factor
Kb- Size Factor
Kc- Reliability Factor
Kd- Factor to account for stress concentration (𝐾𝑑 =
1
𝐾𝑓
)
𝜎𝑒
′ - Endurance limit stress of a
rotating beam specimen
𝜎𝑒 - Endurance limit stress of a
particular component

Derating Factors
Surface Finish Factor (Ka):
Mirror polished and non-ferrous metals, Ka = 1

Shigley and Mischke have suggested an exponential equation for
surface finish,
𝐾𝑎 = 𝑎 𝜎ut
𝑏 𝐼𝑓 𝐾𝑎 > 1, 𝑠𝑒𝑡 𝐾𝑎 𝑡𝑜 1
Surface finish factor for Steel
Surface finish factor for cast iron parts is 1

Size Factor (Kb):
𝑑 ≤ 7.5 𝑚𝑚 Kb= 1
7.5 𝑚𝑚 < 𝑑 ≤ 50 𝑚𝑚 Kb= 0.85
𝑑 > 50 𝑚𝑚 Kb= 0.75
Reliability Factor (Kc):
Reliability, % Kc
50 1
90 0.897
95 0.868
99 0.814
99.9 0.753
99.99 0.702
99.999 0.659

Fatigue Stress ConcentrationFactor and Notch Sensitivity
Fatigue stress concentration factor
𝑘𝑓 =
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑛𝑜𝑡𝑐ℎ𝑒𝑑 𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛
𝑠𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑛𝑜𝑡𝑐ℎ − 𝑓𝑟𝑒𝑒 𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛
Notch sensitivity
𝑞 =
𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑎𝑐𝑡𝑢𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑣𝑒𝑟 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑣𝑒𝑟 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
𝑞 =
𝑘𝑓𝜎o − 𝜎o
𝑘𝑡𝜎o − 𝜎o
=
𝑘𝑓 − 1
𝑘𝑡 − 1
𝑘𝑓 = 1 + 𝑞(𝑘𝑡 − 1) [q varies from 0 to 1]
When q = 0; 𝑘𝑓 = 1 and when q = 1; 𝑘𝑓 = 𝑘𝑡
𝝈𝒐 - Nominal stress

𝝈𝒆
′
𝒗𝒔 𝝈𝐮𝐭
𝜎𝑒
′
= 0.5 𝜎ut 𝑓𝑜𝑟 𝑠𝑡𝑒𝑒𝑙
𝜎𝑒
′ = 0.4 𝜎ut 𝑓𝑜𝑟 𝑐𝑎𝑠𝑡 𝑠𝑡𝑒𝑒𝑙 𝑎𝑛𝑑 𝑐𝑎𝑠𝑡 𝑖𝑟𝑜𝑛
𝜎𝑒
′ = 0.4 𝜎ut 𝑓𝑜𝑟 𝑤𝑟𝑜𝑢𝑔ℎ𝑡 𝑎𝑙𝑢𝑚𝑖𝑛𝑖𝑢𝑚 𝑎𝑙𝑙𝑜𝑦
𝜎𝑒
′ = 0.3 𝜎ut 𝑓𝑜𝑟 𝑐𝑎𝑠𝑡 𝑎𝑙𝑢𝑚𝑖𝑛𝑖𝑢𝑚 𝑎𝑙𝑙𝑜𝑦𝑠
Factor of Safety for Fatigue Loading
𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 =
𝐸𝑛𝑑𝑢𝑟𝑎𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡 𝑠𝑡𝑟𝑒𝑠𝑠
𝐷𝑒𝑠𝑖𝑔𝑛 𝑜𝑟 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑠𝑡𝑟𝑒𝑠𝑠
=
𝜎𝑒
′
𝜎𝑑
For steel, 𝜎𝑒
′ = 0.8 to 0.9 𝜎yt
𝜎𝑒
′ − 𝐸𝑛𝑑𝑢𝑟𝑎𝑛𝑐𝑒 𝑠𝑡𝑟𝑒𝑛𝑡ℎ 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚
𝑟𝑜𝑡𝑎𝑡𝑖𝑛𝑔 𝑏𝑒𝑎𝑚 𝑢𝑛𝑑𝑒𝑟 𝑏𝑒𝑛𝑑𝑖𝑛𝑔

Endurance Limit for other Loads
𝜎𝑒 𝑎 = 0.8 𝜎𝑒
𝜏𝑒 = 0.5 𝜎𝑒 According to maximum shear stress theory
𝜏𝑒 = 0.577 𝜎𝑒
According to distortion energy theory
Torsional Shear Stress
For Axial load

Fatigue Design
Completely reversed
Stresses
Fluctuating Stresses
Finite Life Infinite Life
For the given number of
cycles, the corresponding
fatigue strength is
obtained from S-N curve
(for steel).
Endurance limit
value is used for the
design.
Soderberg or Goodman
equation can be used
for design.

Problem1
A rod of linkage mechanism made of steel with σut = 550 MPa is
subjected to a completely reversed axial load of 100 kN. The rod
is machined on lathe and the expected reliability is 95%. There is
no stress concentration. Determine the diameter of the rod using
factor of safety of 2 for an infinite life condition.
Reversed Load - Infinite Life
d = 44.78 mm

Problem2
A component machined from a plate made of 45C8 (σut = 630
Mpa) is shown in Figure. It is subjected to a completely reversed
axial load of 50 kN. The expected reliability is 90% the factor of
safety is 2. The size factor is 0.85. Determine the plate thickness t
for infinite life, if the notch sensitivity factor is 0.8.
t = 27.6 mm

Problem3
A rotating shaft, subjected to a non-rotating force of 5 kN and
simply supported between two bearings A and E is as shown in
Fig. The shaft is machined from plain carbon steel 30C8 (σut = 500
MPa) and the expected reliability is 90%. The equivalent notch
radius at the fillet section can be taken as 3 mm. what is the life of
the shaft? The notch sensitivity factor is 0.8.
ReversedLoad – Finite Life

Problem 4
A cantilever beam made of cold drawn steel 20C8 (σut = 500 MPa)
is subjected to a completely reversed load 1000 N as shown in
Fig. The notch sensitivity factor can be taken as 0.85 and the
expected reliability is 90%. Determine the diameter d of the beam
for a life of 10000 cycles.

Soderberg andGoodman Equations
𝜎𝑚
𝜎𝑦𝑡
+
𝜎𝑎
𝜎𝑒
=
1
𝑛
Soderberg Equation is given by
𝒏- factor of safety
Fluctuating Stresses – Infinite Life

Modified GoodmanEquation
For Axial and Bending Stresses
𝜎𝑚
𝜎𝑢𝑡
+
𝜎𝑎
𝜎𝑒
=
1
𝑛
If fatigue is the governing
failure mode
If static failure governs, then
𝜎𝑎 + 𝜎𝑚 =
𝜎𝑦𝑡
𝑛

Modified GoodmanEquation
For TorsionalShear Stress
𝜏𝑎 =
𝜏𝑒
𝑛
If fatigue is the governing
failure mode
If static failure governs,
then
𝜏𝑎 + 𝜏𝑚 =
𝜏𝑦𝑡
𝑛
𝜏𝑒 = 0.5 𝜎𝑒 According to maximum shear stress theory
𝜏𝑒 = 0.577 𝜎𝑒
According to distortion energy theory
𝜏𝑦𝑡 = 𝜎𝑦𝑡/2

Problem 5
A cantilever beam made of cold drawn steel 40C8 (σut = 600 MPa
and σyt = 380 MPa) is shown in Fig. the force P acting at the free
end varies from -50 N to +150 N. The expected reliability is 90%
and the factor of safety is 2. The notch sensitivity factor at the fillet
is 0.9. Determine the diameter d of the beam at the fillet cross
section.

Problem6
A shaft of 760 mm length is simply supported at its ends. It is
subjected to a central concentrated cyclic load that varies from 12
to 36 kN. Determine the diameter of the shaft assuming a factor of
safety of 2, size correction factor of 0.8, and surface correction
factor of 0.85. The material properties are: ultimate strength = 500
MPa, yield strength = 280 MPa, and endurance limit = 250 MPa.
Fatigue stress concentration factor = 1.5.

Problem 7
A stepped shaft of diameters D and d is subjected to a variable
axial load P which cyclically varies between 0 and 10 kN. The
shaft is made of C20 steel, mirror polished with ultimate strength
= 500 MPa and yield strength = 260 MPa. Determine the
diameters D and d with D/d=1.5, factor of safety = 2, notch
sensitivity factor = 0.8 and r/d=0.2 where r is the shoulder fillet
radius.

Factors to be Consideredwhile Designing Machine
Parts to AvoidFatigue Failure
 The variation in the size of the component should be as gradual
as possible.
 The holes, notches and other stress raisers should be avoided.
 The proper stress de-concentrators such as fillets and notches
should be provided wherever necessary.
 The parts should be protected from corrosive atmosphere.
 A smooth finish of outer surface of the component increases the
fatigue life.
 The material with high fatigue strength should be selected.
 The residual compressive stresses over the parts surface
increases its fatigue strength.

MODULE 5_Deisign for Fatigue Loads (1).pdf

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