Fluctuating Stress
Introduction
Some machine parts are
subjected to static loading. Since many of the machine parts (such as axles,
shafts, crankshafts, connecting rods, springs, pinion etc.) are subjected to
variable or alternating loads (also known as fluctuating or fatigue loads),
therefore we shall discuss, in this chapter, the variable or alternating stresses..
For example in below figure the
fiber on the surface of a rotating shaft subjected to a bending load, undergoes
both tension and compression for each revolution of the shaft.
Any fiber on the shaft is
therefore subjected to fluctuating stresses. Machine elements subjected to
fluctuating stresses usually fail at stress levels much below their ultimate
strength and in many cases below the yield point of the material too. These
failures occur due to very large number of stress cycle and are known as
fatigue failure. These failures usually begin with a small crack which may
develop at the points of discontinuity, an existing subsurface crack or surface
faults. Once a crack is developed it propagates with the increase in stress
cycle finally leading to failure of the component by fracture. There are mainly
two characteristics of this kind of failures:
(a) Progressive development of crack.
(b) Sudden fracture without any
warning since yielding is practically absent. Fatigue failures are influenced
by
(i) Nature and magnitude of the stress cycle.
(ii) Endurance limit.
(iii) Stress concentration.
(iv) Surface
characteristics.
These factors are therefore
interdependent. For example, by grinding and polishing, case hardening or coating
a surface, the endurance limit may be improved. For machined steel endurance
limit is approximately half the ultimate tensile stress.
source: Youtube
Stress cycle
A typical
stress cycle is shown in below figure using standard specimen. The maximum, minimum, mean and
variable stresses are indicated. The mean and variable stresses are given by
In order to find the mean
stress for completely reversed cycle the mean stress is zero. The stress verses
time diagram for fluctuating stress having values σmin and σmax
is shown in fluctuating stress Fig. The
variable stress, in general, may be considered as a combination of steady (or
mean or average) stress and a completely reversed stress component σv.
The following relations are derived from fluctuating stress fig:
Endurance limit
Figure shows the rotating beam arrangement along with
the specimen.
Fig. A
typical rotating beam arrangement.
source: youtube
The loading is such that
there is a constant bending moment over the specimen length and the bending
stress is greatest at the center where the section is smallest. The arrangement
gives pure bending and avoids transverse shear since bending
moment is constant over the length. Large number of tests with varying bending
loads are carried out to find the number of cycles to fail. A typical plot of
reversed stress (S) against number of cycles to fail (N) is shown in below
figure. The zone below 103 cycles
is considered as low cycle fatigue, zone
between 103
and 106 cycles is high cycle
fatigue with finite life and beyond 106 cycles, the zone is considered to be high cycle
fatigue with infinite life.
Fig. A schematic plot of reversed stress (S)
against number of cycles to fail(N) for steel.
Effect of Loading on Endurance Limit—Load Factor
The endurance
limit (σe) of a material as determined by the rotating beam method is for
reversed bending load. There are many machine members which are subjected to loads
other than reversed bending loads. Thus the endurance limit will also be
different for different types of loading. The endurance limit depending upon
the type of loading may be modified as discussed below:
Let
Kb = Load correction factor for the reversed or rotating bending load. Its
value is usually taken as unity.
Ka = Load correction factor for the reversed axial load. Its value may be
taken as 0.8.
Ks = Load correction factor for the reversed torsional or shear load. Its
value may be taken as 0.55 for ductile materials and 0.8 for brittle materials.
∴
Endurance limit for reversed bending load, σeb = σe.Kb
= σe ...( Kb = 1)
Endurance limit
for reversed axial load, σea = σe.Ka
and endurance
limit for reversed torsional or shear load, τe = σe.Ks
Effect of
Surface Finish on Endurance Limit—Surface Finish Factor
When a machine
member is subjected to variable loads, the endurance limit of the material for
that member depends upon the surface conditions. When the surface finish factor
is known, then the endurance limit for the material of the machine member may
be obtained by multiplying the endurance limit and the surface finish factor.
We see that for a mirror polished material, the surface finish factor is unity.
In other words, the endurance limit for mirror polished material is maximum and
it goes on reducing due to surface condition.
Let Ksur
= Surface finish factor.
∴
Endurance limit,
σe1
= σeb.Ksur = σe.Kb.Ksur
= σe.Ksur
...( Kb = 1)
...(For reversed bending load)
= σea.Ksur
= σe.Ka.Ksur ...(For
reversed axial load)
= τe.Ksur = σe.Ks.Ksur ...(For
reversed torsional or shear load)
Note : The surface finish factor
for non-ferrous metals may be taken as unity.
Effect of
Size on Endurance Limit—Size Factor
A little
consideration will show that if the size of the standard specimen is increased,
then the endurance limit of the material will decrease. This is due to the fact
that a longer specimen will have more defects than a smaller one.
Let Ksz
= Size factor.
∴
Endurance limit,
σe2
= σe1 × Ksz ...(Considering surface finish
factor also)
= σeb.Ksur.Ksz
= σe.Kb.Ksur.Ksz
= σe.Ksur.Ksz (Kb = 1)
= σea.Ksur.Ksz
= σe.Ka.Ksur.Ksz ...(For reversed axial
load)
= τe.Ksur.Ksz
= σe.Ks.Ksur.Ksz
... (For reversed
torsional or shear load)
Effect of
Miscellaneous Factors on Endurance Limit
In addition to
the surface finish factor (Ksur), size factor (Ksz)
and load factors Kb, Ka and Ks,
there are many other factors such as reliability factor (Kr),
temperature factor (Kt), impact factor (Ki)
etc. which has effect on the endurance limit of a material. Considering all
these factors, the endurance limit may be determined by using the following
expressions :
1. For the reversed bending
load, endurance limit,
σ'e
= σeb.Ksur.Ksz.Kr.Kt.Ki
2. For the reversed axial
load, endurance limit,
σ'e
= σea.Ksur.Ksz.Kr.Kt.Ki
3. For the reversed torsional
or shear load, endurance limit,
σ'e
= τe .Ksur.Ksz.Kr.Kt.Ki
In solving problems,
if the value of any of the above factors is not known, it may be taken as
unity.
Relation
Between Endurance Limit and Ultimate Tensile Strength
For steel, σe
= 0.5 σu
;
For cast steel,
σe = 0.4 σu ;
For cast iron,
σe = 0.35 σu ;
For non-ferrous
metals and alloys, σe = 0.3 σu
Factor of
Safety for Fatigue Loading
When a
component is subjected to fatigue loading, the endurance limit is the criterion
for failure. Therefore, the factor of safety should be based on endurance
limit. Mathematically,
Stress Concentration:
Whenever a
machine component changes the shape of its cross-section, the simple stress
distribution no longer holds good and the neighbourhood of the discontinuity is
different. This irregularity in the stress distribution caused by abrupt
changes of form is called stress concentration. It occurs for all kinds of
stresses in the presence of fillets, notches, holes, keyways, splines, surface
roughness or scratches etc. In order to understand fully the idea of stress
concentration, consider a member with different cross-section under a tensile
load as shown in below Fig. A little consideration will show that the nominal
stress in the right and left hand sides will be uniform but in the region where
the cross-section is changing, a re-distribution of the force within the member
must take place. The material near the edges is stressed considerably higher
than the average value. The maximum stress occurs at some point on the fillet
and is directed parallel to the boundary at that point.
Theoretical or Form Stress Concentration Factor:
The stress
concentration is based on either the photo-elastic analysis using a circular
polariscope or theoretical or finite element analysis method. The theoretical
or form stress concentration factor is defined as the ratio of the maximum
stress in a member (at a notch or a fillet) to the nominal stress at the same
section based upon net area. Mathematically, theoretical or form stress concentration
factor,
The value of Kt
depends upon the material and geometry of the part.
Imp. Note:
1. In static loading, stress concentration in ductile
materials is not so serious as in brittle materials.
2. In cyclic
loading, stress concentration in ductile materials is always serious because
the ductility of the material is not effective in relieving the concentration
of stress caused by cracks, flaws, surface roughness, or any sharp
discontinuity in the geometrical form of the member.
3. In brittle
materials, cracks may appear at these local concentrations of stress which will
increase the stress over the rest of the section. It is, therefore, necessary
that in designing parts of brittle materials subjected to both static load as
well as fluctuating load.
Stress Concentration due to Holes and
Notches:
Consider a
plate with transverse elliptical hole and subjected to a tensile load as shown in
Fig. (a). We see from the stress-distribution that the stress at the
point away from the hole is practically uniform and the maximum stress will be
induced at the edge of the hole. The maximum stress is given by
and the theoretical stress
concentration factor,
When a/b
is large, the ellipse approaches a crack transverse to the load and the
value of Kt becomes very large. When a/b is
small, the ellipse approaches a longitudinal slit [as shown in Fig. (b)]
and the increase in stress is small. When the hole is circular as shown in Fig.
(c), then a/b = 1 and the maximum stress is three times
the nominal value.
Methods of Reducing
Stress Concentration:
We have already
discussed that, whenever there is a change in cross-section, such as shoulders,
holes, notches or keyways and where there is an interference fit between a hub or
bearing race and a shaft, then stress concentration results. The presence of
stress concentration cannot be totally eliminated but it may be reduced to some
extent. A device or concept that is useful in assisting a design engineer to
visualize the presence of stress concentration and how it may be mitigated is
that of stress flow lines, as shown in Fig.
The mitigation of stress concentration means that the stress flow lines
shall maintain their spacing as far as possible.
Some examples:
Fatigue
Stress Concentration Factor:
When a machine member is subjected to cyclic or fatigue loading, the
value of fatigue stress concentration factor shall be applied instead of
theoretical stress concentration factor. Since the determination of fatigue
stress concentration factor is not an easy task, therefore from experimental
tests it is defined as Fatigue stress concentration factor,
Notch
Sensitivity:
The notch sensitivity
‘q’ for fatigue loading can now be defined in terms of Kf and the
theoretical stress concentration factor Kt and this is given by
Fatigue strength
formulations:
Fatigue
strength experiments have been carried out over a wide range of stress
variations in both tension and compression below figure shows a schematic
diagram of experimental plots of variable stress against mean stress and
Gerber, Goodman and Soderberg lines. But the following are important from the
subject point of view:
1. Goodman method 2. Soderberg method.
1 comments:
commentsI need to find the endurance limit of tool steel when R = 0.2 and R = -0.2. At R = 0, the endurance limit is 60,000 psi. How do I do this?
Reply