chapter05(design against for fluctuating load)
TRANSCRIPT
PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limiteddistribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using thisPowerPoint slide, you are using it without permission.
Chapter 5
Design against Fluctuating
Load
授課教師:尤春風
I
yM bb
J
rM tA
Pt
Elementary equations:
Stress concentration is defined as the localization of high stresses due
to the irregularities presents in the component and abrupt changes of
the cross section.
Stress concentration factor ( ) is defined as
section -cross minimalfor equations elementaryby obtained stresses nominal
itydiscontinunear stress actual of luehighest vatK
5.1 Stress concentration
tK
Fig. 5.1 Stress Concentration
Fig. 5.2 Stress Concentration Factor (Rectangular Plate with
Transverse Hole in Tension or Compression)
5.2 Stress concentration factor
The nominal stress is given by
where t is the plate thickness.
tdw
P
)(0
Fig. 5.3 Stress Concentration Factor (Flat Plate with Shoulder Fillet in
Tension or Compression)
td
P
0
The nominal stress is given by
where t is the plate thickness.
Fig. 5.4 Stress Concentration Factor (Round haft with Shoulder
Fillet in Tension)
20
4d
P
The nominal stress is given by
where d is the diameter on the
small end.
Fig. 5.5 Stress Concentration Factor (Round Shaft with Shoulder
Fillet in Bending)
I
yM b 0
The nominal stress is given by
where d is the diameter on the
smaller end.
2 and
64
4 dy
dI
Fig. 5.6 Stress Concentration Factor (Round Shaft with Fillet in Torsion)
The nominal stress is given by
where d is the diameter on the
smaller end.
2 and
32
4 dr
dJ
J
rM t 0
Fig. 5.7 Stress Concentration due to Elliptical Hole
)(21b
aKt
a = half width (or semi-axis) of ellipse perpendicular to the direction
of load
b = half width (or semi-axis) of ellipse in the direction of load
Following guidelines are considered for the stress
concentration factor:
(1)Ductile material under the static load
When the stress in the vicinity of the discontinuity
reaches the yield point, there is plastic deformation,
resulting in a redistribution of stresses. This plastic
deformation or yielding is local and restricted to very
small area in the component. There is no perceptible
damage to the part as a whole.
(2) Ductile material under the fluctuating load
When the load is fluctuating, the stresses at the
discontinuity exceed the endurance limit, the
component may fail. Therefore, endurance limit of the
components made of the ductile material is greatly
reduced due to stress concentration.
Following guidelines are considered for the stress
concentration factor:
(3) Brittle material
The effect of stress concentration is more severe in case of
brittle material, due to their instability to plastic
deformation.
5.3 Reduction of stress concentation
Fluid mechanics Solid mechanics
volume flow force
velocity stress
Flow pattern intensity Stress concentration factor
dAF dAuq
Flow analogy
Fig. 5.8 Force Flow Analogy
(a) Force Flow around Sharp Corner
(b) Force Flow around Rounded Corner
Fig. 5.9 Reduction of Stress Concentration due to V-notch
(a) Original Notch (b) Multiple Notches
(c) Drilled Holes (d) Removal of Undesirable Material
Fig. 5.10 Reduction of Stress Concentration due to Abrupt
Change in Cross-section
(a) Original Component (b) Fillet Radius
(C) Undercutting (d) Addition of Notch
Fig. 5.11 Reduction of Stress Concentration in Shaft with Keyway
(a) Original Shaft (b) Drilled Holes (c) Fillet Radius
Fig. 5.12 Reduction of Stress Concentration in Threaded Components
(a) Original Component (b) Undercutting
(c) Reduction in Shank Diameter
Fig. 5.15 Types of Cyclic Stresses
• S-N curve obtained from a rotating beam
test has completely reverse d stress state.
• Many stress histories will not have
completely reverse d stress state
5.4 Fluctuating stresses
Alternating stress
2
minmax
m
2
minmax
a
Mean stress
minmax rStress range
Stress ratio min
max
R
Amplitude ratio
m
aA
Fig. 5.16 Shear and Fatigue Failure of Wire (a) Shearing of Wire
(b) Bending of Wire (c) Unbending of Wire
5.5 Fatigue failure
• Early railroad cars moved on wheels rigidly attached
(shrunk) to a solid axle. The bearings were mounted
outside the wheels (Fig. a).
• The corresponding freebody diagram shows the
bearing supports of the beam shaft with vertical
forces acting at each wheel (Fig. b).
• At any instant, the axles is loaded in bending with
maximum stresses at top and bottom (Fig. c).
Fatigue in railroad axles (I)
• Because of rotation ̧the material at any point undergoes a
complete stress cycle every revolution (Fig. d).
• During operation, stress cycles accumulate rapidly, and
fracture may occur at either of the two bearings.
• Fatigue fracture surfaces often display two distinctly
different zones. The one section, often discolored by
corrosion, usually exhibits a pattern of lines or beach marks
(Fig. f).
Fatigue in railroad axles (II)
Fatigue in railroad axles (III)
• At times, the beach marks are so fine that they are visible
only magnification (such as is possible with an electron at
great microscope). Crack origin and direction of progression
are often indicated by these markings, which thus give a
clue to possible material flaws or inadequate design. The
other zone of the fracture usually has the bright, grainy
appearance of ductile rupture or fracture.
Mechanism of fatigue feature
• Crack initiation
• Crack propagation
• Fracture
Crack-initiation stage
• Some regions of geometric stress concentration in location
of time-varying that contains a tensile component.
• As the stresses at the notch oscillate, local yielding may
occur due to the stress concentration, even though the
nominal stress is below yield strength of the material.
• The localized plastic yielding causes distortion and creates
slip bands along the crystal boundaries of the material.
• As the stress cycles, additional slip bands occur and
coalesce into microscopic cracks.
• Because of their association with shear stress and
slip, microcracks are oriented with the maximum
shear stress. They may grows across several grains.
Crack propagation stage
• The sharp crack creates stress concentrations larger than
those of the original notch, and a plastic zone develops at the
crack tip each time a tensile stress opens the crack, blunting
its tip and reducing the effective stress concentration.
• This process continues as long as the local stress is cycling
from below the tensile yield to above the tensile yield at the
crack tip.
• The crack growth is due to tensile stress and the crack
grows along planes normal to the maximum tensile stress.
Fracture
• The growth of the cracks continues until a critical size is
reached such that one more application of the load brings
about instability and fracture.
Endurance limitChapter 5.6 endurance limit
The fatigue or endurance limit of a material is defined as the
maximum amplitude of completely reversed stress that the
standard specimen can sustain for an unlimited number of
cycles without fatigue failure.
cycles is considered as a sufficient number of cycles to
define the endurance limit.
610
Fig. 5.17 Specimen for Fatigue Test
Fig. 5.18 Rotating Beam
Subjected to Bending moment
(a) Beam,
(b) Stress Cycle at Point A
Fig. 5.19 Rotating Beam Fatigue Testing Machine
Fig. 5.20 S-N Curve for Steels
The S-N curve is the graphical representation of the stress
amplitude versus the number of the stress cycles (N) before
the fatigue failure on a log-log graph paper.
S-N curve
tS
tS
Fig. 5.21 Low and High Cycle Fatigue
5.7 Low cycle and high cycle fatigue
• Low cycle fatigue
Any fatigue failure, when the number of stress cycles are
less than 1000, is called low cycle fatigue.
This case is treated as the static condition and a larger
factor of safety is used and design on the basis of
ultimate strength or yield strength.
• High cycle fatigue
Any fatigue failure, when the number of stress cycles are
more than 1000, is called high cycle fatigue.
Components are designed on the basis of endurance limit
stress. S-N curve, Soderberg, Goodman are used in
design.
5.8 Notch sensitivity
specimen notched theoflimit endurance
specimen freenotch theoflimit endurancefK
)1( 1 tf KqK
Notch sensitivity is defined as the susceptibility of a material
to succumb to the damaging effects of stress raising notches in
fatigue loading.
The notch sensitivity factor is defined as
stress nominalover stress al theoreticof increae
stress nominalover stress actual of increaeq
Fatigue stress concentration factor is defined asfK
Fig. 5.22 Notch Sensitivity Charts (for Reversed Bending
and Reversed Axial Stresses)
Fig. 5.23 Notch Sensitivity Charts (for Reversed Torsional
Shear Stresses)
Chapter 5.9 Endurance limit ---Approximate estimation
The relationship between and is as follow:
where
= surface finish factor
= size factor
= reliability factor
= modifying factor to account for stress concentration
= endurance limit stress of a rotating beam specimen subjected to
reversed beam stress
= endurance limit stress of a particular mechanical component
subjected to reversed beam stress
'
ee SKKKKS dcba
bK
aK
cK
dK
eS
'
eS
'
eS
eS
There is an approximate relationship between the endurance
limit and the ultimate tensile strength of the material.
For steel,
For cast iron and cast steel,
For wrought aluminum alloys,
For cast aluminum alloys,
ute SS 5.0'
ut
'
e 4.0 SS
ut
'
e 4.0 SS
tSS u
'
e 3.0
Fig. 5.24 Surface Finish Factor
Surface Finish Factor
Size factor
The rotating beam specimen is small with 7.5 mm diameter.
The endurance limit reduces with increasing the size of the
component.
For bending and torsion, the value of size factor are given in
Table 5.2.
diameter d (mm)
d 7.5 1.0
7.5 < d 50 0.85
d > 50 0.75
bK
Table 5.2 Size factor
Fig. 5.25
Effective diameter is based on an equivalent circular cross-section.
Kuhuel assumes a volume of material that is stresses to 95 % of
maximum stress or above. As high stress volume.
Effective diameter
0766.0
95Ade
The effective diameter of any non-circular cross section is given by
222
95 0766.0]4
)95.0([ d
ddA
= portion of cross-sectional area of the non-cylindrical part that is
stresses between 95 % and 100 % of the maximum stress
= effective diameter of non-cylindrical part.ed
95A
The effective diameter is obtained by equating the volume of the material
stresses at and above 95 % of the maximum stresses to the equivalent
volume in the rotating beam specimens.
The area stressed above 95 % of the maximum stress is the area of a ring,
having an inside diameter of 0.95 d and outside 1.0 d .
Fig. 5.26 Area above 95% of Maximum Stress
For rectangular cross-section the effective
diameter bhde 808.0
dde 37.0
For non-rotating solid shaft, the effective
diameter
Reliability factor
Reliability R (%)
50 1.00
90 0.897
95 0.868
99 0.814
99.9 0.753
99.99 0.702
99.999 0.659
cK
The standard deviation of endurance limit test is 8 % of the
mean value.
The reliability factor is 1.0 for 50 %
reliability.
To ensure insure that more than 50 % of
the part will survive, the stress amplitude
on the component should be lower than
the tabulated value of endurance limit.
Reliability may be defined as the probability that a machine
part will perform its intended function without failure for its
prescribed design lifetime.
where p(x) is the probability density function, is the mean
value of the quantity, and the standard deviation.
- ]2
)([exp
2
1)(
)(1
1deviation standard
1mean
2
2
1
2
1
xx
xp
xn
xn
n
i
i
n
i
i
Modifying factor to account for stress concentration
f
dK
K1
The endurance limit is reduced due to stress concentration.
To apply the effect of stress concentration, the designer can
reduce the endurance limit by .dK
The endurance limit of a component subjected to the
fluctuating shear stresses is obtained from the endurance
limit in reversed bending ( ) using theories of failure.
From the maximum shear theory,
From the distortion energy theory,
ese 5.0 SS
ese 577.0 SS
seS
eS
5.10 Reversed stresses design for finite and infinite life
(1) Infinite life
Endurance limit is the criterion of failure. The amplitude
of stress should be lower than the endurance limit in order
to withstand the infinite number of cycles.
)( ,
)( fs
S
fs
S sea
ea
where are stress amplitude in the component and
are corrected endurance limit in reversed bending
and reversing torsion respectively.
aa ,
see SS ,
(2) Finite life
When the components is to be designed for finite life,
S-N curve (Fig. 5.27) can be used. It consists of a
straight line AB drawn from at cycles
to at cycles on a log-log paper.
310610
)9.0( utS
eS
Fig. 5.27 S-N Curve
At 1000 cycles:
Bending:
Axial bending:utm
utm
SS
SS
75.0
9.0
5.12 Soderberg and Goodman lines
• When stress amplitude is zero, the load is purely static
and criterion of failure is . These limits are plotted
on the abscissa.
• When the mean stress ( ) is zero, the stress is completely
reversing and the criterion of failure is endurance limit ,
that is plotted on the ordinate.
meS
a
utyt or SS
(1) Gerber line
A parabolic curve joining on the ordinate to on the abscissa.
(2) Soderberg line
A straight line joining on the ordinate to on the abscissa.
The equation of Soderberg line is given as:
(3) Goodman line
A straight line joining on the ordinate to on the abscissa
The equation of Goodman line is given as
eS
eS
ytS
utS
eS ytS
1e
a
yt
m SS
1e
a
ut
m SS
Fig. 5.39 Soderberg and Goodman Lines
Any combination of mean and alternating stress that lies on or
below Goodman line will have infinite life.
Goodman line is widely used in the criterion of fatigue failure
when the component is subjected to mean stress as well as
stress amplitude.
(1) Goodman line is safe from design consideration because it
is completely inside the failure points of test data.
(2) The equation of straight line is simple.
5.13 Modified Goodman Diagram
Goodman line is modified by combining fatigue failure with
fatigue by yielding.
The yield strength is plotted on both the axes-abscissa and
ordinate, and a yield line is constructed to join two points
to define failure by yielding.
The region OABC is called modified Goodman diagram. All
the points inside the modified Goodman diagram should cause
neither fatigue failure and yielding.
is the portion of Goodman line and is portion of
yield line.
DC
ytS
BACB
Modified Goodman Diagram
A line is drawn through on the ordinate and parallel to the
abscissa. The point of intersection of this line and yield line is
B. The area OABC represents the region of safety.
The region OABC is called modified Goodman diagram.
All the points inside the modified Goodman diagram should
neither fatigue failure nor yielding.
The point of intersection of lines is X. The point X
indicates the dividing line between the safe region and the
region of failure.
EOBA and
m
atan
The coordinates of point X ( ) represent the limiting
values of stress, that are used to calculate the dimensions of
component.
am SS ,
Fig. 5.40 Modified Goodman Diagram for Axial and Bending
Stresses
Fig. 5.41 Modified Goodman Diagram for Torsional Shear Stresses
The modified Goodman diagram for fluctuating torsional
shear stress is shown in Fig. 5.41.
Example 5.13
A transmission shaft of cold drawn steel 27Mn2 (
) is subjected to a fluctuating torque which
varies from -100 N-m to 400 N-m. The factor of safety is 2 and
expected reliability is 90 %. Neglecting the stress concentration,
determine the diameter of the shaft.
Solution:
,N/mm540 2
ut S2
yt N/mm 400 and S
897.0,reliablity %90for
85.0,mm 505.7 assuming
79.0
N/mm025)500(5.05.0
c
2
ut
'
K
Kd
K
kSS
b
a
e
04.59
67.1150
250
)(
)(tan
mN250]100400[2
1])()[(
2
1)(
mN150]100400[2
1])()[(
2
1)(
N/mm 1.173300577.0577.0
N/mm 88.8658.150577.0577.0
, theoryenerg distortion theusing
N/mm 58.150
250897.085.079.0
mt
at
mintmaxtat
mintmaxtmt
2
ytsy
2
ese
2
'
M
M
MMM
MMM
SS
SS
y
SKKKS ecbae
The modified Goodman diagram is shown in Fig. 5.44. The ordinate of
point X is .N/mm 88.86or 2
saS
Fig. 5.44
mm 83.30
44.43
1025016
N/mm 44.432
88.86
)(
.N/mm 88.86
3
3
2sa
2
sa
d
d
fs
S
S
a
Fig. 5.52 Gerber
line
5.14 Gerber line
1)( 2
ut
m
e
a S
S
S
SGerber equation is given as:
Gerber curve takes mean path through failure points. It is more
accurate than Goodman or Soderberg line.
5.15 Fatigue design under combined stresses
The general equation of distortion energy theory is as follows.
where are normal stresses in X, Y, Z directions and
are shear stresses in their respective planes.zyx , ,
zxyzxy , ,
The bending moment as well as torsional moment may have
two components – mean and alternating stresses. Such
problems involving combinational of stresses are solved by the
distortion energy theory of failure.
2
)](6)()()[( 222222
zxyzxyxzzyyx
In case of combined bending and torsional moments, there is a
normal stress accompanied by the torsional shear
stress .
0 zxyzzy
22 3 xyx
xyx
xaxm and
The mean and alternating component of are
respectively.
The mean and alternating component of are
respectively.
x
xy
xaxm and
22
m 3 xymxm
22
a 3 xyaxa
The Two stresses are used in the modified Goodman
diagram to design the component.
am ,
Example 5.20
A transmission shaft carries a pulley midway between the two
bearings. The bending moment at the pulley varies from 200
N-m to 600 N-m , as the torsional moment in the shaft varies
from 70 N-m to 200 N-m. The frequency of variation of
bending and torsioal moments are equal to the shaft speed. The
shaft is made of steel fee 400 (
). The corrected endurance limit of the shaft is
200 . Determine the diameter of the shaft using a
factor of safety of 2.
400 and ,N/mm540 yt
2
ut SS
2N/mm
2N/mm
2
3
3
33
abxa
2
3
3
33
mbxm
mintmaxtat
mintmaxtmt
minbmaxbab
minbmaxbmb
N/mm1018.2037
d
)1000200(32
d
)(32
N/mm1037.4074
d
)1000400(32
d
)(32
mN65]70200[2
1])()[(
2
1)(
mN135]70200[2
1])()[(
2
1)(
mN200]200600[2
1])()[(
2
1)(
mN400]200600[2
1])()[(
2
1)(
d
M
d
M
MMM
MMM
MMM
MMM
Solution
2
3
3
2
3
32
3
32
xya
2
xaa
2
3
3
2
3
32
3
32
xym
2
xmm
2
3
3
33
atxya
2
3
3
33
mtxym
N/mm)1033.2116
(
)1004.331
(3)1018.2037
(3
N/mm)1084.4244
(
)1055.687
(3)1037.4074
(3
N/mm1004.331
d
)100065(16
d
)(16
N/mm1055.687
d
)1000135(16
d
)(16
d
dd
d
dd
d
M
d
M
5.26
4986.084.4244
33.2116tan
m
a
The modified Goodman diagram is shown in Fig. 5.56. The
coordinate of point X are obtained by solving the following
two equations simultaneously.
Fig. 5.56
mm 29.33
38.571033.2116
N/mm 38.572
76.114
)(
N/mm 16.230
N/mm 76.114
4986.0tan
1540200
3
3
2a
2
m
2
a
m
a
ma
d
d
fs
S
S
S
S
S
SS
a
Exercise chapter 05
1. Prob. 5.10 page 201