Variable Stresses in Machine Parts
n
185
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 K
sur
= Surface finish factor.
∃ Endurance limit,
!
e1
= !
eb
.K
sur
= !
e
.K
b
.K
sur
= !
e
.K
sur
(
∵
K
b
= 1)
(For reversed bending load)
= !
ea
.K
sur
= !
e
.K
a
.K
sur
(For reversed axial load)
= %
e
.K
sur
= !
e
.K
s
.K
sur
(For reversed torsional or shear load)
Note : The surface finish factor for non-ferrous metals may be taken as unity.
6.66.6
6.66.6
6.6
EfEf
EfEf
Ef
fect of Size on Endurance Limit—Size Ffect of Size on Endurance Limit—Size F
fect of Size on Endurance Limit—Size Ffect of Size on Endurance Limit—Size F
fect of Size on Endurance Limit—Size F
actoractor
actoractor
actor
A little consideration will show that if the size of the standard specimen as shown in Fig. 6.2 (a)
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 K
sz
= Size factor.
∃ Endurance limit,
!
e2
= !
e1
× K
sz
(Considering surface finish factor also)
= !
eb
.K
sur
.K
sz
= !
e
.K
b
.K
sur
.K
sz
= !
e
.K
sur
.K
sz
(
∵
K
b
= 1)
= !
ea
.K
sur
.K
sz
= !
e
.K
a
.K
sur
.K
sz
(For reversed axial load)
= %
e
.K
sur
.K
sz
= !
e
.K
s
.K
sur.
K
sz
(For reversed torsional or shear load)
Notes: 1. The value of size factor is taken as unity for the standard specimen having nominal diameter of
7.657 mm.
2. When the nominal diameter of the specimen is more than 7.657 mm but less than 50 mm, the value of
size factor may be taken as 0.85.
3. When the nominal diameter of the specimen is more than 50 mm, then the value of size factor may be
taken as 0.75.
6.76.7
6.76.7
6.7
EfEf
EfEf
Ef
fect of Miscellaneous Ffect of Miscellaneous F
fect of Miscellaneous Ffect of Miscellaneous F
fect of Miscellaneous F
actoractor
actoractor
actor
s ons on
s ons on
s on
Endurance LimitEndurance Limit
Endurance LimitEndurance Limit
Endurance Limit
In addition to the surface finish factor (K
sur
),
size factor (K
sz
) and load factors K
b
, K
a
and K
s
, there
are many other factors such as reliability factor (K
r
),
temperature factor (K
t
), impact factor (K
i
) etc. which
has effect on the endurance limit of a material. Con-
sidering all these factors, the endurance limit may be
determined by using the following expressions :
1. For the reversed bending load, endurance
limit,
!'
e
= !
eb
.K
sur
.K
sz
.K
r
.K
t
.K
i
2. For the reversed axial load, endurance limit,
!'
e
= !
ea
.K
sur
.K
sz
.K
r
.K
t
.K
i
3. For the reversed torsional or shear load,
endurance limit,
!'
e
= %
e
.K
sur
.K
sz
.K
r
.K
t
.K
i
In solving problems, if the value of any of the
above factors is not known, it may be taken as unity.
In addition to shear, tensile, compressive and
torsional stresses, temperature can add its own
stress (Ref. Chapter 4)
Note : This picture is given as additional information
and is not a direct example of the current chapter.
186
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A Textbook of Machine Design
6.86.8
6.86.8
6.8
RelaRela
RelaRela
Rela
tion Betwtion Betw
tion Betwtion Betw
tion Betw
een Endurance Limit and Ultimaeen Endurance Limit and Ultima
een Endurance Limit and Ultimaeen Endurance Limit and Ultima
een Endurance Limit and Ultima
te te
te te
te
TT
TT
T
ensile Strensile Str
ensile Strensile Str
ensile Str
engthength
engthength
ength
It has been found experimentally that endurance limit (!
e
) of a material subjected to fatigue
loading is a function of ultimate tensile strength (!
u
). Fig. 6.4 shows the endurance limit of steel
corresponding to ultimate tensile strength for different surface conditions. Following are some
empirical relations commonly used in practice :
Fig. 6.4. Endurance limit of steel corresponding to ultimate tensile strength.
For steel, !
e
= 0.5 !
u
;
For cast steel, !
e
= 0.4 !
u
;
For cast iron, !
e
= 0.35 !
u
;
For non-ferrous metals and alloys, !
e
= 0.3 !
u
6.96.9
6.96.9
6.9
Factor of Safety for Fatigue LoadingFactor of Safety for Fatigue Loading
Factor of Safety for Fatigue LoadingFactor of Safety for Fatigue Loading
Factor of Safety for Fatigue Loading
When a component is subjected to fatigue loading, the endurance limit is the criterion for faliure.
Therefore, the factor of safety should be based on endurance limit. Mathematically,
Factor of safety (F. S.) =
Endurance limit stress
Design or working stress
e
d
!
&
!
Note: For steel, !
e
= 0.8 to 0.9 !
y
where !
e
= Endurance limit stress for completely reversed stress cycle, and
!
y
= Yield point stress.
Example 6.1. Determine the design stress for a piston rod where the load is completely
reversed. The surface of the rod is ground and
the surface finish factor is 0.9. There is no stress
concentration. The load is predictable and the
factor of safety is 2.
Solution. Given : K
sur
= 0.9 ; F.S. = 2
The piston rod is subjected to reversed
axial loading. We know that for reversed axial
loading, the load correction factor (K
a
) is 0.8.
Piston rod
Variable Stresses in Machine Parts
n
187
Fig. 6.5. Stress concentration.
If !
e
is the endurance limit for reversed bending load, then endurance limit for reversed axial
load,
!
ea
= !
e
× K
a
× K
sur
= !
e
× 0.8 × 0.9 = 0.72 !
e
We know that design stress,
!
d
=
0.72
0.36
2
ea e
e
FS
!!
&&!
Ans.
6.106.10
6.106.10
6.10
StrStr
StrStr
Str
ess Concentraess Concentra
ess Concentraess Concentra
ess Concentra
tiontion
tiontion
tion
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
Fig. 6.5. 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.
6.116.11
6.116.11
6.11
TheorTheor
TheorTheor
Theor
etical or Foretical or For
etical or Foretical or For
etical or For
m Strm Str
m Strm Str
m Str
ess Concentraess Concentra
ess Concentraess Concentra
ess Concentra
tion Ftion F
tion Ftion F
tion F
actoractor
actoractor
actor
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,
K
t
=
Maximum stress
Nominal stress
The value of K
t
depends upon the material and geometry of the part.
Notes: 1. In static loading, stress concentration in ductile materials is not so serious as in brittle materials,
because in ductile materials local deformation or yielding takes place which reduces the concentration. 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 such as castings, care should be
taken. In order to avoid failure due to stress concentration, fillets at the changes of section must be provided.
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. If the stress at any point in a member is above the
endurance limit of the material, a crack may develop under the action of repeated load and the crack will lead to
failure of the member.
6.126.12
6.126.12
6.12
StrStr
StrStr
Str
ess Concentraess Concentra
ess Concentraess Concentra
ess Concentra
tion due to Holes and Notchestion due to Holes and Notches
tion due to Holes and Notchestion due to Holes and Notches
tion due to Holes and Notches
Consider a plate with transverse elliptical hole and subjected to a tensile load as shown in Fig.
6.6 (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
!
max
=
2
1
∋(
!
∀
)∗
+,
a
b
188
n
A Textbook of Machine Design
and the theoretical stress concentration factor,
K
t
=
2
1
max
a
r
!
∋(
&∀
)∗
!+ ,
When a/b is large, the ellipse approaches a crack transverse to the load and the value of K
t
becomes very large. When a/b is small, the ellipse approaches a longitudinal slit [as shown in Fig. 6.6
(b)] and the increase in stress is small. When the hole is circular as shown in Fig. 6.6 (c), then a/b = 1
and the maximum stress is three times the nominal value.
Fig. 6.6. Stress concentration due to holes.
The stress concentration in the notched tension member, as
shown in Fig. 6.7, is influenced by the depth a of the notch and radius
r at the bottom of the notch. The maximum stress, which applies to
members having notches that are small in comparison with the width
of the plate, may be obtained by the following equation,
!
max
=
2
1
∋(
!∀
)∗
+,
a
r
6.136.13
6.136.13
6.13
Methods of Reducing StrMethods of Reducing Str
Methods of Reducing StrMethods of Reducing Str
Methods of Reducing Str
ess Concentraess Concentra
ess Concentraess Concentra
ess Concentra
tiontion
tiontion
tion
We have already discussed in Art 6.10 that whenever there is a
change in cross-section, such as shoulders, holes, notches or keyways and where there is an interfer-
ence fit between a hub or bearing race and a shaft, then stress concentration results. The presence of
stress concentration can not 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
Fig. 6.7. Stress concentration
due to notches.
Crankshaft
Variable Stresses in Machine Parts
n
189
and how it may be mitigated is that of stress flow lines, as shown in Fig. 6.8. The mitigation of stress
concentration means that the stress flow lines shall maintain their spacing as far as possible.
Fig. 6.8
In Fig. 6.8 (a) we see that stress lines tend to bunch up and cut very close to the sharp re-entrant
corner. In order to improve the situation, fillets may be provided, as shown in Fig. 6.8 (b) and (c) to
give more equally spaced flow lines.
Figs. 6.9 to 6.11 show the several ways of reducing the stress concentration in shafts and other
cylindrical members with shoulders, holes and threads respectively. It may be noted that it is not
practicable to use large radius fillets as in case of ball and roller bearing mountings. In such cases,
notches may be cut as shown in Fig. 6.8 (d) and Fig. 6.9 (b) and (c).
Fig. 6.9. Methods of reducing stress concentration in cylindrical members with shoulders.
Fig. 6.10. Methods of reducing stress concentration in cylindrical members with holes.
Fig. 6.11. Methods of reducing stress concentration in cylindrical members with holes.
The stress concentration effects of a press fit may be reduced by making more gradual transition
from the rigid to the more flexible shaft. The various ways of reducing stress concentration for such
cases are shown in Fig. 6.12 (a), (b) and (c).
190
n
A Textbook of Machine Design
6.146.14
6.146.14
6.14
FF
FF
F
actoractor
actoractor
actor
s to be Considers to be Consider
s to be Considers to be Consider
s to be Consider
ed while Designing Machine Ped while Designing Machine P
ed while Designing Machine Ped while Designing Machine P
ed while Designing Machine P
arar
arar
ar
ts to ts to
ts to ts to
ts to
AA
AA
A
vv
vv
v
oidoid
oidoid
oid
FF
FF
F
aa
aa
a
tigue Ftigue F
tigue Ftigue F
tigue F
ailurailur
ailurailur
ailur
ee
ee
e
The following factors should be considered while designing machine parts to avoid fatigue failure:
1. The variation in the size of the component should be as gradual as possible.
2. The holes, notches and other stress raisers should be avoided.
3. The proper stress de-concentrators such as fillets and notches should be provided
wherever necessary.
Fig. 6.12. Methods of reducing stress concentration of a press fit.
4. The parts should be protected from corrosive atmosphere.
5. A smooth finish of outer surface of the component increases the fatigue life.
6. The material with high fatigue strength should be selected.
7. The residual compressive stresses over the parts surface increases its fatigue strength.
6.156.15
6.156.15
6.15
StrStr
StrStr
Str
ess Concentraess Concentra
ess Concentraess Concentra
ess Concentra
tion Ftion F
tion Ftion F
tion F
actor factor f
actor factor f
actor f
or or
or or
or
VV
VV
V
arar
arar
ar
ious Machine Memberious Machine Member
ious Machine Memberious Machine Member
ious Machine Member
ss
ss
s
The following tables show the theoretical stress concentration factor for various types of
members.
TT
TT
T
aa
aa
a
ble 6.1.ble 6.1.
ble 6.1.ble 6.1.
ble 6.1.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) f) f
) f) f
) f
or a plaor a pla
or a plaor a pla
or a pla
te with holete with hole
te with holete with hole
te with hole
(of diameter (of diameter
(of diameter (of diameter
(of diameter
dd
dd
d
) in tension.) in tension.
) in tension.) in tension.
) in tension.
d
b
0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
K
t
2.83 2.69 2.59 2.50 2.43 2.37 2.32 2.26 2.22 2.17 2.13
Fig. for Table 6.1 Fig. for Table 6.2
TT
TT
T
aa
aa
a
ble 6.2.ble 6.2.
ble 6.2.ble 6.2.
ble 6.2.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) f) f
) f) f
) f
or a shaftor a shaft
or a shaftor a shaft
or a shaft
with transverse hole (of diameter with transverse hole (of diameter
with transverse hole (of diameter with transverse hole (of diameter
with transverse hole (of diameter
dd
dd
d
) in bending.) in bending.
) in bending.) in bending.
) in bending.
d
D
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
K
t
2.70 2.52 2.33 2.26 2.20 2.11 2.03 1.96 1.92 1.90
Variable Stresses in Machine Parts
n
191
TT
TT
T
aa
aa
a
ble 6.3.ble 6.3.
ble 6.3.ble 6.3.
ble 6.3.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) f) f
) f) f
) f
or steppedor stepped
or steppedor stepped
or stepped
shaft with a shoulder fillet (of radius shaft with a shoulder fillet (of radius
shaft with a shoulder fillet (of radius shaft with a shoulder fillet (of radius
shaft with a shoulder fillet (of radius
rr
rr
r
) in tension.) in tension.
) in tension.) in tension.
) in tension.
Theoretical stress concentration factor (K
t
)
D
d
r/d
0.08 0.10 0.12 0.16 0.18 0.20 0.22 0.24 0.28 0.30
1.01 1.27 1.24 1.21 1.17 1.16 1.15 1.15 1.14 1.13 1.13
1.02 1.38 1.34 1.30 1.26 1.24 1.23 1.22 1.21 1.19 1.19
1.05 1.53 1.46 1.42 1.36 1.34 1.32 1.30 1.28 1.26 1.25
1.10 1.65 1.56 1.50 1.43 1.39 1.37 1.34 1.33 1.30 1.28
1.15 1.73 1.63 1.56 1.46 1.43 1.40 1.37 1.35 1.32 1.31
1.20 1.82 1.68 1.62 1.51 1.47 1.44 1.41 1.38 1.35 1.34
1.50 2.03 1.84 1.80 1.66 1.60 1.56 1.53 1.50 1.46 1.44
2.00 2.14 1.94 1.89 1.74 1.68 1.64 1.59 1.56 1.50 1.47
TT
TT
T
aa
aa
a
ble 6.4.ble 6.4.
ble 6.4.ble 6.4.
ble 6.4.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) f) f
) f) f
) f
or a steppedor a stepped
or a steppedor a stepped
or a stepped
shaft with a shoulder fillet (of radius shaft with a shoulder fillet (of radius
shaft with a shoulder fillet (of radius shaft with a shoulder fillet (of radius
shaft with a shoulder fillet (of radius
rr
rr
r
) in bending.) in bending.
) in bending.) in bending.
) in bending.
Theoretical stress concentration factor (K
t
)
D
d
r/d
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
1.01 1.85 1.61 1.42 1.36 1.32 1.24 1.20 1.17 1.15 1.14
1.02 1.97 1.72 1.50 1.44 1.40 1.32 1.27 1.23 1.21 1.20
1.05 2.20 1.88 1.60 1.53 1.48 1.40 1.34 1.30 1.27 1.25
1.10 2.36 1.99 1.66 1.58 1.53 1.44 1.38 1.33 1.28 1.27
1.20 2.52 2.10 1.72 1.62 1.56 1.46 1.39 1.34 1.29 1.28
1.50 2.75 2.20 1.78 1.68 1.60 1.50 1.42 1.36 1.31 1.29
2.00 2.86 2.32 1.87 1.74 1.64 1.53 1.43 1.37 1.32 1.30
3.00 3.00 2.45 1.95 1.80 1.69 1.56 1.46 1.38 1.34 1.32
6.00 3.04 2.58 2.04 1.87 1.76 1.60 1.49 1.41 1.35 1.33
192
n
A Textbook of Machine Design
TT
TT
T
aa
aa
a
ble 6.5.ble 6.5.
ble 6.5.ble 6.5.
ble 6.5.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) f) f
) f) f
) f
or a stepped shaftor a stepped shaft
or a stepped shaftor a stepped shaft
or a stepped shaft
with a shoulder fillet (of radius with a shoulder fillet (of radius
with a shoulder fillet (of radius with a shoulder fillet (of radius
with a shoulder fillet (of radius
rr
rr
r
) in torsion.) in torsion.
) in torsion.) in torsion.
) in torsion.
Theoretical stress concentration factor (K
t
)
D
d
r/d
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
1.09 1.54 1.32 1.19 1.16 1.15 1.12 1.11 1.10 1.09 1.09
1.20 1.98 1.67 1.40 1.33 1.28 1.22 1.18 1.15 1.13 1.13
1.33 2.14 1.79 1.48 1.41 1.35 1.28 1.22 1.19 1.17 1.16
2.00 2.27 1.84 1.53 1.46 1.40 1.32 1.26 1.22 1.19 1.18
TT
TT
T
aa
aa
a
ble 6.6.ble 6.6.
ble 6.6.ble 6.6.
ble 6.6.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
))
))
)
ff
ff
f
or a gror a gr
or a gror a gr
or a gr
oooo
oooo
oo
vv
vv
v
ed shaft in tension.ed shaft in tension.
ed shaft in tension.ed shaft in tension.
ed shaft in tension.
Theoretical stress concentration (K
t
)
D
d
r/d
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
1.01 1.98 1.71 1.47 1.42 1.38 1.33 1.28 1.25 1.23 1.22
1.02 2.30 1.94 1.66 1.59 1.54 1.45 1.40 1.36 1.33 1.31
1.03 2.60 2.14 1.77 1.69 1.63 1.53 1.46 1.41 1.37 1.36
1.05 2.85 2.36 1.94 1.81 1.73 1.61 1.54 1.47 1.43 1.41
1.10 2.70 2.16 2.01 1.90 1.75 1.70 1.57 1.50 1.47
1.20 2.90 2.36 2.17 2.04 1.86 1.74 1.64 1.56 1.54
1.30 2.46 2.26 2.11 1.91 1.77 1.67 1.59 1.56
1.50 2.54 2.33 2.16 1.94 1.79 1.69 1.61 1.57
2.00 2.61 2.38 2.22 1.98 1.83 1.72 1.63 1.59
−. 2.69 2.44 2.26 2.03 1.86 1.74 1.65 1.61
Variable Stresses in Machine Parts
n
193
TT
TT
T
aa
aa
a
ble 6.7.ble 6.7.
ble 6.7.ble 6.7.
ble 6.7.
TheorTheor
TheorTheor
Theor
etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
actor (
KK
KK
K
tt
tt
t
) of) of
) of) of
) of
a gra gr
a gra gr
a gr
oooo
oooo
oo
vv
vv
v
ed shaft in bending.ed shaft in bending.
ed shaft in bending.ed shaft in bending.
ed shaft in bending.
Theoretical stress concentration factor (K
t
)
D
d
r/d
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
1.01 1.74 1.68 1.47 1.41 1.38 1.32 1.27 1.23 1.22 1.20
1.02 2.28 1.89 1.64 1.53 1.48 1.40 1.34 1.30 1.26 1.25
1.03 2.46 2.04 1.68 1.61 1.55 1.47 1.40 1.35 1.31 1.28
1.05 2.75 2.22 1.80 1.70 1.63 1.53 1.46 1.40 1.35 1.33
1.12 3.20 2.50 1.97 1.83 1.75 1.62 1.52 1.45 1.38 1.34
1.30 3.40 2.70 2.04 1.91 1.82 1.67 1.57 1.48 1.42 1.38
1.50 3.48 2.74 2.11 1.95 1.84 1.69 1.58 1.49 1.43 1.40
2.00 3.55 2.78 2.14 1.97 1.86 1.71 1.59 1.55 1.44 1.41
. /012 3045 3067 6084 6044 6076 6012 6056 6095 6093
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T
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ble 6.8.ble 6.8.
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etical stretical str
etical stretical str
etical str
ess concentraess concentra
ess concentraess concentra
ess concentra
tion ftion f
tion ftion f
tion f
actor (actor (
actor (actor (
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eded
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shaft in torsion.shaft in torsion.
shaft in torsion.shaft in torsion.
shaft in torsion.
Theoretical stress concentration factor (K
ts
)
D
d
r/d
0.02 0.04 0.08 0.10 0.12 0.16 0.20 0.24 0.28 0.30
1.01 1.50 1.03 1.22 1.20 1.18 1.16 1.13 1.12 1.12 1.12
1.02 1.62 1.45 1.31 1.27 1.23 1.20 1.18 1.16 1.15 1.16
1.05 1.88 1.61 1.40 1.35 1.32 1.26 1.22 1.20 1.18 1.17
1.10 2.05 1.73 1.47 1.41 1.37 1.31 1.26 1.24 1.21 1.20
1.20 2.26 1.83 1.53 1.46 1.41 1.34 1.27 1.25 1.22 1.21
1.30 2.32 1.89 1.55 1.48 1.43 1.35 1.30 1.26 — —
2.00 2.40 1.93 1.58 1.50 1.45 1.36 1.31 1.26 — —
. 2.50 1.96 1.60 1.51 1.46 1.38 1.32 1.27 1.24 1.23
194
n
A Textbook of Machine Design
Stepped shaft
Example 6.2. Find the maximum
stress induced in the following cases
taking stress concentration into
account:
1. A rectangular plate 60 mm ×
10 mm with a hole 12 diameter as
shown in Fig. 6.13 (a) and subjected
to a tensile load of 12 kN.
2. A stepped shaft as shown in
Fig. 6.13 (b) and carrying a tensile
load of 12 kN.
Fig. 6.13
Solution. Case 1. Given : b = 60 mm ; t = 10 mm ; d = 12 mm ; W = 12 kN = 12 × 10
3
N
We know that cross-sectional area of the plate,
A =(b – d) t = (60 – 12) 10 = 480 mm
2
∃ Nominal stress =
3
2
12 10
25 N / mm 25 MPa
480
W
A
:
&& &
Ratio of diameter of hole to width of plate,
12
0.2
60
d
b
&&
From Table 6.1, we find that for d / b = 0.2, theoretical stress concentration factor,
K
t
= 2.5
∃ Maximum stress = K
t
× Nominal stress = 2.5 × 25 = 62.5 MPa
Ans.
Case 2.
Given : D = 50 mm ; d = 25 mm ; r = 5 mm ; W = 12 kN = 12 × 10
3
N
We know that cross-sectional area for the stepped shaft,
A =
22 2
(25) 491 mm
44
d
;;
:& &
∃ Nominal stress =
3
2
12 10
24.4 N/ mm 24.4 MPa
491
W
A
:
&& &
Ratio of maximum diameter to minimum diameter,
D/d = 50/25 = 2
Ratio of radius of fillet to minimum diameter,
r/d = 5/25 = 0.2
From Table 6.3, we find that for D/d = 2 and r/d = 0.2, theoretical stress concentration factor,
K
t
= 1.64.
∃ Maximum stress = K
t
× Nominal stress = 1.64 × 24.4 = 40 MPa
Ans.
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