Design for fluctuating loads

Stress Concentration: Causes and
Remedies
• Stress concentration is defined as the
localization of high stresses due to
irregularities present in the component and
abrupt changes in cross-section of the
component.
• Irregularities may be oil-holes, grooves,
keyways, splines, screw threads and
shoulders.
© Dr. V.R Deulgaonkar, 2018

• To consider the effect of stress concentration
and find out localized stress, a factor known as
stress concentration factor is used, denoted by
Kt
• Kt = {Highest Value of actual stress near
discontinuity} / {Nominal stress obtained by
elementary equations for minimum c/s}
• Kt = max/0 = max/ 0 ; where max ,max are
localized stresses and 0 , 0 are stresses
determined by elementary equations.

Causes of stress concentration
1) Variation in properties of materials: it is due
to internal cracks, flaws, cavities in welds, air
holes in steel components, non-metallic or
foreign inclusions. These variations act as
discontinuities and cause stress
concentration.
2) Load application: in applications as meshing
teeth in gears, balls and races in ball
bearings, rail and the wheel, crane hook and

the chain, a concentrated load is applied over a
very small area resulting in stress
concentration.
3) Abrupt changes in section: Steps are provided
on the shaft in order to mount gears,
sprockets, pulleys and ball bearings leads to
change in c/s and causes stress concentration.
4) Discontinuities in the component: features of
machine components as oil holes, oil grooves,
keyways, splines, screw threads, result in
discontinuities in c/s, and causes stress
concentration in vicinity of them.

5) Machining Scratches: Machining scratches,
stamp mark, inspection mark are surface
irregularities which cause stress
concentration.

Remedies for reduction of stress
concentration
1. Additional Notches and holes in tension
member: Severity of stress concentration is
reduced by use of multiple notches, drilling
additional holes, removal of undesired
material. Method of removal of undesired
material is called principle of minimization of
material.

2) Fillet Radius, Undercutting and Notch for
Member in Bending: A transmission shaft has
shoulders for ball bearings, gears or pulleys to
be mounted. The shoulders generates change
in c/s resulting in stress concentration. the
stress concentration in such cases is reduced
by three ways as,
a) Providing a fillet radius : Results in gradual
transition of change in c/s. In practice fillet
radius dimension is limited by assembly
considerations.

b) Undercut: Fillet radius can be increased by
this operation.
c) Additional Notch: Presence of notch results in
stress concentration, but cutting an additional
notch reduce stress concentration.

3) Drilling additional holes for shaft: In this two
symmetrical holes are drilled on the sides of
the keyway, which press the flow lines and
minimize their bending in the vicinity of
keyway.

4) Reduction of Stress concentration in threaded
members: For a threaded component the
force flow line is bent as it passes from shank
portion to threaded portion, which results in
stress concentration in transition plane.
A small undercut between the shank and the
threaded portion reduces the bending of force
flow line and stress concentration.
Ideally when shank dia. = core dia. There is no
stress concentration.

Fluctuating Stresses
1. The external forces acting on the m/c
component are generally assumed static.
2. In many engineering applications, the
components are subjected to forces which
vary with respect to time. The stresses
induced due to such forces are known as
fluctuating stresses.
3. 80% failures of mechanical components are
due to ‘fatigue failure’ which results from
fluctuating stresses.

4. Stress pattern is variable and irregular in
practical situations as in case of vibrations.
5. Simple models of stress-time relationship are
used for the purpose of design analysis.
6. Popularly sine curve is used for stress-time
relationship.

Mathematical Models for Cyclic Stresses
There are basically three types of mathematical
models to demonstrate cyclic stresses viz.
1. Fluctuating or alternating stresses
2. Repeated Stresses
3. Reversed Stresses

1. Fluctuating or alternating stresses:
a) These stresses varies in a sinusoidal
manner with respect to time.
b) It has some mean value and amplitude
value.
c) It fluctuates between two limits Maximum
and Minimum.
d) The stress can be tensile or compressive or
partly tensile or partly compressive.

2. Repeated Stresses: (min = 0)
a) Varies in a sinusoidal manner with respect
to time, but the variation is from zero to some
maximum value.
b) Minimum stress is zero in this, and hence
the amplitude stress and mean stress are
equal.

3. Reversed Stress: (m = 0)
a) It varies in sinusoidal manner with respect
to time, but it has zero mean stress.
b) In this half portion of the cycle consists of
tensile and the remaining half consists of
compressive stress.
c) In this there is complete reversal from
tension to compression between these two
halves and so mean stress is zero.

Referring to figure above,
max = Maximum Stress
min = Minimum Stress
m = Mean Stress
a = Stress amplitude
We have
m = (max + min)
a = (max - min )

Fatigue Failures
“Fatigue failure is defined as the time delayed
fracture under cyclic loading”.
1. Under fluctuating stresses, the failure of
material occurs at a lower magnitude of
ultimate or sometimes yield strength.
2. Resistance of material to stress magnitudes
decreases with the increase in number of
cycles, and this is the root cause of fatigue
failure.

e.g. bending and unbending of wire
Components such as transmission shafts,
connecting rods, gears, vehicle suspension
springs and ball bearing are commonly
subjected to fatigue failures.
3. Phenomenon of fatigue failure begins with a
crack at some point in the material. This crack
generally occurs at discontinuity regions as
keyways, oil-holes, screw threads. Scratches
on surface, stamp mark, inspection mark
which represent irregularity regions. Defects
in materials as blow holes etc.

4. Such regions are subjected to crack, which
spreads due to fluctuating stresses until the
cross-section of the component is so reduced
that the remaining portion is subjected to
sudden fracture.
5. There are two distinct areas of fatigue failure
as region indicating slow growth of crack with
fine fibrous appearance and region of sudden
fracture with a coarse granular appearance.
6. Fatigue failure is sudden and total.

• Design process is complex as it involves factors
such as number of cycles, mean stress, stress
amplitude, stress concentration, residual
stresses corrosion and creep.

Endurance Limit
Fatigue or endurance limit of a material is
defined as maximum amplitude of completely
reversed stress that the standard specimen
can sustain for an unlimited number of cycles
without fatigue failure.
106 cycles is considered as sufficient number of
cycles to define the endurance limit as the test
cannot be conducted for infinite number of
cycles

The term fatigue life is commonly used with
endurance limit.
Fatigue life is defined as the number of stress
cycles that a standard specimen can complete
during the test before the appearance of first
fatigue crack.
In laboratory, the endurance limit is determined
by means of a rotating beam machine
developed by R.R. Moore.

S-N Curves
A rotating beam m/c diagram is shown in figure
below. Specimen acts as rotating beam
subjected to bending moment. Hence it is
subjected to completely reversed stress cycle.
1. The stress amplitudes can be varied by
changing the bending moment by addition or
removal of weights.
2. With the aid of electric motor the specimen
is rotated.

Figure: Rotating beam fatigue testing Machine

3. Using a revolution counter, the number of
revolutions before the appearance of first
fatigue crack are recorded.
4. In each test, two readings are noted viz.
Stress amplitude (Sf) and number of cycles (N).
5. These readings are used to as two co-
ordinates for plotting a point on S-N diagram.
6. This point is called failure point.
7. The results of such tests are plotted using S-N
curves.

8. S-N curve is the graphical representation of
stress amplitude (Sf) versus the number of
stress cycles (N) before the fatigue failure on a
log-log graph paper.
9. Each test on fatigue testing m/c gives one
failure point on the S-N diagram.
10. These points are scattered in figure and
average curve is drawn through them.
11. It is also called Wohler diagram after
German Engineer, August Wohler who
presented this method in 1870.

Figure: S-N Curve for Steels

NOTCH SENSITIVITY (q)
“It is defined as the susceptibility of a material to
succumb to damaging effects of stress raising
notches in fatigue loading”.
Notch sensitivity factor (q) is defined as the
ratio of increase of actual stress over nominal
stress to increase of theoretical stress over
nominal stress.
Mathematically 0 = Nominal stress obtained by
elementary equations.
Kf*0= Actual Stress & Kt* 0 = Theoretical Stress

• Increase of actual stress over nominal stress
= (Kf*0 - 0 )
• Increase of theoretical stress over nominal
stress = (Kt*0 - 0 )
q = (Kf*0 - 0 ) / (Kt*0 - 0 )
q = (Kf - 1 ) / (Kt - 1 )
The above equation is rearranged as ;
Kf = 1 + q(Kt - 1 ) ----(A)
Following are the observations are from above
equation

• When the material has no sensitivity to
notches
q = 0 & Kf = 1
When the material is fully sensitive to
notches
q = 1 & Kf = Kt
the magnitude of q varies from 0 to 1.
In case of doubt designer should use q = 1 &
Kf = Kt and the design will be on safe side.

Figure: Notch Sensitivity chart
for reversed bending and reversed axial stresses© Dr. V.R Deulgaonkar, 2018

• Fatigue stress concentration factor (Kf ) ; it is
applicable to actual materials and depends
upon the grain size of the material.
• Theoretical stress concentration factor (Kt ) ; it
is applicable to ideal materials which are
homogenous, isotropic and elastic.
• Kf = { Endurance limit of the notch free
specimen} / { Endurance limit of the notched
specimen}

Endurance strength modifying
factors
Notations used regarding endurance limit are
(S’e) and (Se); Where
S’e = Endurance limit stress of a rotating beam
specimen subjected to reversed bending stress
(MPa),
Se = Endurance limit stress of a particular
mechanical component subjected to reversed
bending stress (MPa)

Relations between endurance limit and the
ultimate tensile strength of the material
S’e = 0.5Sut ; For Steels
S’e = 0.4Sut ; For Cast Iron and Cast Steels
S’e = 0.4Sut ; For Wrought aluminium alloys
S’e = 0.3Sut ; For Cast Aluminium
All the above relationships are based on 50%
reliability.

There is difference between (S’e) and (Se) due to
the fact that there are standard specifications
and working conditions for rotating beam
specimen, while the actual components have
different specifications and work under
different conditions.
Different modifying factors are used to account
for this difference and these factors are called
derating factors.
The derating factors reduce the endurance limit
of the rotating beam specimen to suit the
actual component.

The relationship between (S’e) and (Se) is given
as
Se = Ka Kb Kc Kd S’e
Where Ka = Surface finish factor
Kb = Size factor
Kc = reliability factor
Kd = modifying factor to account for the
stress concentration.

Surface Finish Factor :
Surface finish factor takes into account the
poor surface finish and geometric
irregularities on the surface which serve as
stress raisers and cause stress concentration.
It also considers the reduction in endurance
limit due to the variation in surface finish
between the specimen and the actual
component.
For steel components Noll and Lipson devised
an equation as Ka = a(Sut)b ;[If Ka>1,use Ka=1]

Values of a and b are given in table below
For grey cast iron components Ka = 1
Size Factor : The size factor takes into account
the reduction in endurance limit due to
increase in size of the component.
Surface Finish a b
Ground Machined 1.58 -0.085
Cold Drawn 4.51 -0.265
Hot Rolled 57.7 -0.718
Forged 272 -0.995

• Values of size factor are given in table below;
Reliability Factor : depends upon the reliability
that is used in the design of component. For
50% reliability the reliability factor is one. To
ensure that 50% of the parts will survive, the
stress amplitudes on the components should
be lower than the tabulated value of
endurance limit.
Diameter (d) mm Kb
d ⩽ 7.5 1.00
7.5 < d ⩽ 50 0.85
d > 50 0.75

• Reliability factor is used for achieving this
reduction. Reliability factors based on a
standard deviation of 8% are given in table
below.
Reliability R (%) Kc
50 1.000
90 0.897
95 0.868
99 0.814
99.9 0.753
99.99 0.702
99.999 0.659

• Modifying factor to account for stress
Concentration: It is defined as Kd = 1/Kf
Sse = Endurance limit of a component subjected
to fluctuating torsional shear stresses
According to maximum shear stress theory;
Sse = 0.5Se
According to distortion energy theory;
Sse = 0.577Se
For axial loading (Se)a = 0.8Se

Reversed Stresses -Design for Finite
and Infinite life
• Design problems for completely reversed
stresses are divided into two groups as design
for finite and design for infinite life.
Design for Infinite life
• When the component is to be designed for
infinite life, the endurance limit becomes the
criterion of failure. {The amplitude stress
induced in such components} < {endurance
limit in-order to withstand infinite number of
cycles}. © Dr. V.R Deulgaonkar, 2018

• Such components are designed with the help
of following equations;
a = Se /(fs) & a = Sse /(fs)
Where (a) & (a) are stress amplitudes in the
component and Se and Sse are corrected
endurance limits in reversed bending and
torsion respectively.

Design for finite life
• For such case S-N curve is used. A
representative curve is shown in fig below,
which is valid for steels.
• It consists of a straight line AB drawn from
(0.9Sut) at 103 cycles to (Se) at 106 cycles on a
log-log paper.
• Design process for such problems is as under
– Locate point A with co-ordinates [3, log10(0.9Sut)]
since log10(103) = 3

– Locate point B with co-ordinates [6, log10(Se)]
since log10(106) = 6
– Line joining A & B is used as a criterion of failure
for finite-life problems.
– Depending upon the life N of the component,
draw a vertical line passing through log10(N) on
the abscissa. This line intersects AB at point F.
– Draw a line FE parallel to the abscissa. The
ordinate at point E i.e. log10(Sf) , gives the fatigue
strength corresponding to N cycles.
• The value of (Sf) obtained by above process is
used for design calculations.

Figure: S-N Curve

Cumulative Damage in Fatigue
(Miner’s Equation)
• In certain applications, the mechanical
component is subjected to different stress
levels for different parts of the work cycle.
• The life of such component is determined by
Miner’s equation.
suppose a component is subjected to
completely reversed stresses (1) for (n1)
cycles, (2) for (n2) cycles and so on. Let N1 be
the number of stress cycles before fatigue
failure, only if the alternating stress (1) is
acting. © Dr. V.R Deulgaonkar, 2018

• One stress cycle will consume (1/N1) of fatigue
life and since there are n1 such cycles at this
stress level , the proportionate damage of
fatigue life will be (n1/N1). Similarly, the
proportionate damage at stress level (2) will
be (n2 / N2 )and so on. Adding these, we get
(n1/N1) + (n2 / N2 ) + ……. + (nx /Nx ) = 1
Above equation is known as Miner’s equation.

• At times the number of cycles n1 , n2 ,…….. At
stress levels 1 , 2 ,…….. are unknown.
Suppose that 1, 2,…… are proportions of the
total life that will be consumed by the stress
levels 1 , 2 ,…….. etc. Let N be the total life of
the component, then n1 = 1N n2 = 2N
• Using these in Miner’s equation
(1 /N1) + (2 /N2) +…… (x /Nx) = (1 /N)
Also 1 + 2 + 3 + ……+ x = 1

Soderberg, Gerber and Goodman
Lines
• When a component is subjected to fluctuating
stress there is mean stress (m) as well as
stress amplitude (a).
• Mean stress component has effect on fatigue
failure when it occurs in combination with
alternating component.

Figure: Soderberg and Goodman lines

• Fatigue diagram for general case is shown in
figure above; In this diagram
1. Mean Stress (m) is plotted on abscissa
2. Stress amplitude (a) is plotted on ordinate
3. Magnitude of force acting on the component
determines the values of mean stress and
stress amplitudes.
4. When the stress amplitude (a) is zero, the
load is purely static and the criterion of
failure becomes Syt or Sut and these limits are
plotted on abscissa.

5. When the mean stress (m) is completely
zero, the stress is completely reversing and
the criterion of failure is endurance limit Se
plotted on ordinate.
6. When the component is subjected to both
mean stress (m) and stress amplitude (a),
the actual failure occurs at different failure
points as shown in figure.
7. There exists a border, which divides safe
region from unsafe region for various
combinations of (a) & (m).

8. Various criterions are proposed to for
construction of borderline dividing safe zone
and failure zone. They include
a) Gerber Line : It is a parabolic curve joining
Se on the ordinate to Sut on abscissa.
b) Soderberg Line: It is a straight line joining Se
on ordinate to Syt on abscissa.
c) Goodman Line: It is a straight line joining Se
on ordinate to Sut on abscissa.

Discussions……
• Gerber parabola fits the failure points of test
data in the best possible way.
• Goodman line fits beneath the scatter of this
data.
• Gerber parabola & Goodman line intersect at
(Se) on ordinate to (Sut) on abscissa.
• From design considerations Goodman line is
more safe as it lies completely inside the
Gerber parabola and inside the failure points.

• Soderberg line is more conservative failure
criterion and there is no need to consider
even yielding in this case.
• Equation of Soderberg line is given by
(m /Syt) + (a /Se) = 1
• Equation of Goodman line is given by
(m /Sut) + (a /Se) = 1
• Goodman line is widely used as the criterion
of fatigue failure when the component is
subjected to both m & a

• Goodman Line is safe as it is completely inside
the failure points of test data.
• Equation of straight line is simple as compared
with equation of parabolic curve.
• A scale diagram is not required; a rough sketch
is enough for construction of fatigue diagram.

Modified Goodman Diagrams
• Components subjected to fluctuating stresses
are designed by constructing modified
Goodman Diagram. From design
considerations problems are classified into
two categories:
1. Components subjected to fluctuating axial or
bending stresses;
2. Components subjected to fluctuation
torsional shear stresses.

1. Components Subjected to Fluctuating Axial or
Bending Stresses
Figure: Modified Goodman diagram for axial and bending stresses

• Goodman line is modified by combining
fatigue failure with yielding failure.
• In this, the yield strength (Syt) is plotted on
both axes, and a yield line CD at 450 to
abscissa is constructed to join these points to
define failure by yielding.
• Line AF is constructed to Join Se on ordinate
with Sut on abscissa i.e. Goodman Line.
• Point of intersection of lines AF &CD is ‘B’.
• Area OABC represents the region of safety for
components subjected to fluctuating stresses.

• Region OABC is called Modified Goodman
Diagram.
• In this AB is the portion of Goodman line and
BC is the portion of yield line.
• Line OE with slope tan is constructed for
problem solving process as
tan = a / m ;
We know that a/m = (Pa/A) / (Pm/A)
So tan = Pa / Pm

• Magnitudes of Pa & Pm are determined by
maximum and minimum force acting on the
component. Similarly we have
tan = (Mb)a / (Mb)m
Lines AB and OE intersect at point ‘X’ which
indicates the dividing line between safe region
and the region of failure.
Co-ordinates of point X(Sm ,Sa) represent limiting
values of stresses used to calculate the
dimensions of the component.
Permissible stresses are calculated as a = (Sa)/fs
m = (Sm)/fs

2.Components Subjected to fluctuating
torsional shear stresses
Figure: Modified Goodman Diagram for Torsional Shear Stresses

• In this torsional mean stress is plotted on
abscissa while the torsional stress amplitude
on ordinate.
• The torsional yield strength Ssy is plotted on
abscissa and the yield line is constructed
which is inclined at 450 to abscissa.
• Upto a certain point, torsional mean stress has
no effect on torsional endurance limit, so a
line is drawn through Sse on ordinate and is
parallel to the abscissa.

• Point of intersection of this line with the yield
line is B. Area OABC represents region of
safety.
• It is not necessary to construct fatigue
diagram for fluctuating torsional shear
stresses as line AB is parallel to X-axis.
• A fatigue failure is indicated if, a = Sse and a
static failure is indicated if max =a + m = Ssy
• Permissible stresses are a = Sse/fs and
max = Ssy /fs

Fatigue Design Under Combined
Stresses
• With the increase in complexity of application,
the component may be subjected to two
dimensional stresses or to combined bending
and torsional moments.
• The 2D stresses may possess
•
+
Mean Stresses Alternating Stresses

• Such problems which involve combination of
stresses are solved by Distortion-energy
theory of failure.
• General Equation of distortion energy failure is
2 = ½[(x - y)2 + (y - z)2 + (z - x)2 + 6(2
xy +
2
yz + 2
zx)] -------(1)
where x , y , z & xy yz zx are normal
stresses in X, Y, Z directions & shear stresses
in their respective planes.
 = Stress equivalent to 3D stresses.

• For 2D stresses, we know that Z component
does not exist; hence equation 1 becomes
 = (2
x - x y + 2
y)1/2 ---------- (2)
Further the mean and alternating components
of x are xm & xa respectively and for y
are ym & ya
m = (2
xm - xm ym + 2
ym)1/2 and
a = (2
xa - xa ya + 2
ya)1/2
The two stresses m & a obtained by above
equations are used in modified Goodman
diagram to design the component.

• For the case combined bending and torsional
moments, there is a normal stress x
accompanied by torsional shear stress xy
• hence using y = z = yz = zx = 0 in (1)
•  = (2
x + 32
xy )1/2 similarly the mean and
alternating components are
• m = (2
xm + 32
xym)1/2
• a = (2
xa + 32
xya )1/2
• The two stresses m & a obtained by above
equations are used in modified Goodman
diagram to design the component.

Design for fluctuating loads

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