Design against fluctuating load

DESIGN AGAINST FLUCTUATING LOAD

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.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

• 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
Or 𝐾𝑡 =
𝜎𝑚𝑎𝑥
𝜎0
=
𝜏𝑚𝑎𝑥
𝜏0
;
where 𝜎𝑚𝑎𝑥, 𝜏𝑚𝑎𝑥 are localized stresses and 𝜎0 , 𝜏0 are stresses
determined by elementary equations at the discontinuities.

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.

Charts for Stress concentration Factor
Fig.: Rectangular plate with
transverse hole
in tension or compression
Fig.: Flat plate with
shoulder fillet
in tension or compression
Fig.: Round shaft with
shoulder fillet in tension

shoulder fillet in torsion
shoulder fillet in bending

• It is impossible to completely eliminate the effect of stress
concentration, it can only be reduced to some extent.
• It is achieved by providing specific geometry shape to the
component.
• In order to know what happens at the abrupt changes of c/s
or at discontinuity, understanding the ‘flow analogy’ is useful.
There is a similarity between velocity distribution in fluid flow
in a channel and stress distribution in an axially loaded plate.
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,

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
• The external forces acting on the m/c component are
generally assumed to be static.
• 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.
• 80% failures of mechanical components are due to ‘fatigue
failure’ which results from fluctuating stresses.

• Stress pattern is variable and irregular in practical situations
as in case of vibrations.
• Simple models of stress-time relationship are used for the
purpose of design analysis.
• 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: (𝜎𝑚𝑖𝑛 = 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: (𝜎𝑚 = 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,
𝜎𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑆𝑡𝑟𝑒𝑠𝑠
𝜎𝑚𝑖𝑛 = 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑆𝑡𝑟𝑒𝑠𝑠
𝜎𝑚 = 𝑀𝑒𝑎𝑛 𝑆𝑡𝑟𝑒𝑠𝑠
𝜎𝑎 = 𝑆𝑡𝑟𝑒𝑠𝑠 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
We have
𝜎𝑚 =
𝜎𝑚𝑎𝑥 + 𝜎𝑚𝑖𝑛
2
𝜎𝑎 =
𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛
2

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 main characteristic 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.

Fig.: 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.

Fig.: S-N curve for steel

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.

NOTCH SENSITIVITY (𝑞)
“The extent to which the sensitivity of a material to fracture is increased by
the presence of a surface inhomogeneity such as a notch, a sudden change in
section, a crack, or a scratch”.
• 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, 𝜎0 = Nominal stress obtained by elementary
equations.
𝐾𝑓 × 𝜎0 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 & 𝐾𝑡 × 𝜎0 = 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠

• Increase of actual stress over nominal stress = (𝐾𝑓 × 𝜎0 − 𝜎0)
• Increase of theoretical stress over nominal stress = 𝐾𝑡 × 𝜎0 − 𝜎0
𝑞 =
(𝐾𝑓×𝜎0−𝜎0)
(𝐾𝑡×𝜎0−𝜎0)
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.

Fig.: Notch sensitivity charts (for reversed
bending and reversed axial stresses)
Fig.: Notch sensitivity charts (for
reversed torsional shear stresses)

• 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 Limit-Approximate Estimation
Notations used regarding endurance limit are (𝑆𝑒
′
) and (𝑆𝑒);
Where
𝑆𝑒
′
= 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
𝑆𝑒
′
= 0.5 𝑆𝑢𝑡 ; For Steels
𝑆𝑒
′
= 0.4 𝑆𝑢𝑡 ; For Cast Iron and Cast Steels
𝑆𝑒
′
= 0.4 𝑆𝑢𝑡 ; For Wrought Aluminium alloys
𝑆𝑒
′
= 0.3 𝑆𝑢𝑡; For Cast Aluminium
All the above relationships are based on 50% reliability.

• There is difference between (𝑆𝑒
′
) 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 (𝑆𝑒
′
) and (𝑆𝑒) is given as
𝑆𝑒 = 𝐾𝑎𝐾𝑏𝐾𝑐𝐾𝑑𝑆𝑒
′
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.
• Following figure shows the surface finish factor for steel
components.

Fig.: Surface finish factor for steels

• 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 .
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

• Size Factor: The size factor takes into account the
reduction in endurance limit due to increase in size
of the component.
• Values of size factor are given in table below;
Diameter (d) mm 𝑲𝒃
d ⩽ 7.5 1.00
7.5 < d ⩽ 50 0.85
d > 50 0.75

• 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.
• Reliability factor is used for achieving this
reduction. Reliability factors based on a
standard deviation of 8% are given in table.
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}.

• Such components are designed with the help of
following equations;
𝜎𝑎 =
𝑆𝑒
(𝑓𝑠)
& 𝜏𝑎 =
𝑆𝑠𝑒
(𝑓𝑠)
• Where (𝜎𝑎) & (𝜏𝑎) 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 (𝑆𝑢𝑡) 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 𝑙𝑜𝑔10𝑁 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.
𝑙𝑜𝑔10𝑆𝑓, gives the fatigue strength corresponding to N cycles.
❖ The value of (𝑆𝑓) obtained by above process is used for design
calculations.

Fig.: 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 (𝜎1) for (𝑛1) cycles, (𝜎2) for (𝑛2) cycles and so on. Let 𝑁1
be the number of stress cycles before fatigue failure, only if the
alternating stress (𝜎1) is acting.

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 (𝜎2) 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 𝜎1,
𝜎2,…….. are unknown. Suppose that 𝛼1, 𝛼2…… are proportions of
the total life that will be consumed by the stress levels 𝜎1, 𝜎2 ……..
etc.
• Let N be the total life of the component, then
𝑛1= 𝛼1𝑁 𝑛2= 𝛼2𝑁
• Using these in Miner’s equation
𝛼1
𝑁1
+
𝛼2
𝑁2
+……
𝛼𝑥
𝑁𝑥
=
1
𝑁
Also 𝛼1 + 𝛼2 + 𝛼3 … . . +𝛼𝑥 = 1

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

Fatigue diagram for general case is shown in figure above; In this
diagram
1. Mean Stress (𝜎𝑚) is plotted on abscissa
2. Stress amplitude (𝜎𝑎) 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 (𝜎𝑎) 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 (𝜎𝑚) 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 (𝜎𝑚)
and stress amplitude (𝜎𝑎), 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 (𝜎𝑎) & (𝜎𝑚).

8. Various criterions are proposed 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,
𝜎𝑚
𝑆𝑦𝑡
+
𝜎𝑎
𝑆𝑒
= 1
• Equation of Goodman line is given by,
𝜎𝑚
𝑆𝑢𝑡
+
𝜎𝑎
𝑆𝑒
= 1
• Goodman line is widely used as the criterion of fatigue failure when the
component is subjected to both 𝜎𝑚 & 𝜎𝑎.
• 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
Fig.: 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 𝜃 is constructed for problem solving
process as
tan 𝜃 =
𝜎𝑎
𝜎𝑚
;
We know that
𝜎𝑎
𝜎𝑚
=
𝑃𝑎/𝐴
𝑃𝑚/𝐴
So tan 𝜃 =
𝑃𝑎
𝑃𝑚

• Magnitudes of 𝑃𝑎 & 𝑃𝑚 are determined by maximum and
minimum force acting on the component. Similarly we have
tan 𝜃 =
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
𝜎𝑎 =
𝑆𝑎
𝑓𝑠
𝜎𝑚 =
𝑆𝑚
𝑓𝑠

2.Components Subjected to fluctuating torsional shear
stresses
Fig.: 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.
• Up to 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, 𝜏𝑎 = 𝑆𝑠𝑒 and a static failure is
indicated if 𝜏𝑚𝑎𝑥 = 𝜏𝑎 + 𝜏𝑚 = 𝑆𝑠𝑦
• Permissible stresses are 𝜏𝑎 =
𝑆𝑠𝑒
𝑓𝑠
and 𝜏𝑚𝑎𝑥 =
𝑆𝑠𝑦
𝑓𝑠

Design against fluctuating load

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