Chapter 5
Design against Fluctuating Load
授課教師:尤春風
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 limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it without permission.
5.1 Stress concentration
Elementary equations:
P t A
Mb y b I
Mt r J
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 ( K t ) is defined as Kt
highest value of actual stress near discontinuity nominal stresses obtained by elementaryequations for minimal cross - section
Fig. 5.1 Stress Concentration
5.2 Stress concentration factor
The nominal stress is given by
P 0 ( w d )t where t is the plate thickness.
Fig. 5.2
Stress Concentration Factor (Rectangular Plate with Transverse Hole in Tension or Compression)
The nominal stress is given by P 0 dt
where t is the plate thickness.
Fig. 5.3 Stress Concentration Factor (Flat Plate with Shoulder Fillet in Tension or Compression)
The nominal stress is given by 0
4
P
d2
where d is the diameter on the small end.
Fig. 5.4
Stress Concentration Factor (Round haft with Shoulder Fillet in Tension)
The nominal stress is given by Mb y 0 I
where d is the diameter on the smaller end.
d4
d I and y 64 2
Fig. 5.5
Stress Concentration Factor (Round Shaft with Shoulder Fillet in Bending)
The nominal stress is given by Mt r 0 J
where d is the diameter on the smaller end.
d4
d J and r 32 2
Fig. 5.6 Stress Concentration Factor (Round Shaft with Fillet in Torsion)
a K t 1 2( ) b 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 Fig. 5.7 Stress Concentration due to Elliptical Hole
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
Flow analogy Fluid mechanics
Solid mechanics
F dA
volume flow q u dA
force
velocity
stress
Flow pattern intensity
Stress concentration factor
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
5.4 Fluctuating 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
Fig. 5.15 Types of Cyclic Stresses
Alternating stress a
max min
2 max min Mean stress m 2 Stress range r max min max Stress ratio R min Amplitude ratio A a m
5.5 Fatigue failure
Fig. 5.16 Shear and Fatigue Failure of Wire (a) Shearing of Wire (b) Bending of Wire (c) Unbending of Wire
Fatigue in railroad axles (I) • Early railroad cars moved on wheels rigidly attached (shrunk) to a solid axle. The bearings were mounted outside the wheels (Fig. a). • The corresponding free-body 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 (II) • 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 (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.
Chapter 5.6 endurance limit
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. 106 cycles is considered as a sufficient number of cycles to
define the endurance limit.
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
S-N curve The S-N curve is the graphical representation of the stress amplitude St versus the number of the stress cycles (N) before the fatigue failure on a log-log graph paper.
St
Fig. 5.20 S-N Curve for Steels
5.7 Low cycle and high cycle fatigue
Fig. 5.21 Low 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
Fatigue stress concentration factor K f is defined as
endurance limit of the notch free specimen Kf endurance limit of the notchedspecimen 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
increae of actual stress over nominal stress q increae of theoretical stress over nominal stress
K f 1 q ( Kt 1)
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 Se and Se' is as follow: where
Se K a Kb K c K d S
' e
K a = surface finish factor K b = size factor
Kc Kd
= reliability factor
Se'
= endurance limit stress of a rotating beam specimen subjected to
= modifying factor to account for stress concentration
reversed beam stress
Se
= endurance limit stress of a particular mechanical component subjected to reversed beam stress
There is an approximate relationship between the endurance limit and the ultimate tensile strength of the material. For steel,
Se' 0.5Sut
For cast iron and cast steel,
S 0.4Sut
For wrought aluminum alloys,
Se' 0.4Sut
For cast aluminum alloys,
Se' 0.3Sut
' e
Surface Finish Factor
Fig. 5.24 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. Table 5.2 Size factor
diameter d (mm) d 7.5 7.5 < d 50 d > 50
Kb
1.0 0.85 0.75
Effective diameter
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.
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 .
d 2 (0.95d ) 2 A95 [ ] 0.0766d 2 4 The effective diameter of any non-circular cross section is given by
A9 5 de 0.0766
A9 5
= portion of cross-sectional area of the non-cylindrical part that is stresses between 95 % and 100 % of the maximum stress d e = effective diameter of non-cylindrical part.
For non-rotating solid shaft, the effective diameter d 0.37d e
For rectangular cross-section the effective diameter d 0.808 bh e
Fig. 5.26 Area above 95% of Maximum Stress
Reliability factor The standard deviation of endurance limit test is 8 % of the mean value. The reliability factor is 1.0 for 50 % reliability.
Reliability R (%) 50
Kc
90
0.897
To ensure insure that more than 50 % of 95 the part will survive, the stress amplitude 99 on the component should be lower than 99.9 the tabulated value of endurance limit.
1.00
0.868 0.814 0.753
99.99
0.702
99.999
0.659
Reliability may be defined as the probability that a machine part will perform its intended function without failure for its prescribed design lifetime.
1 n mean xi n i 1
standard deviation p( x)
1 n 2 ( x ) i n 1 i 1
1 ( x )2 exp[ ] 2 2 2
- x
where p(x) is the probability density function, is the mean value of the quantity, and the standard deviation.
Modifying factor to account for stress concentration The endurance limit is reduced due to stress concentration. To apply the effect of stress concentration, the designer can reduce the endurance limit by K d .
1 Kd Kf
The endurance limit Sse of a component subjected to the fluctuating shear stresses is obtained from the endurance limit in reversed bending ( Se ) using theories of failure.
From the maximum shear theory, From the distortion energy theory,
Sse 0.5Se Sse 0.577Se
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.
Se a , ( fs )
S se a ( fs )
where a , a are stress amplitude in the component and Se , S se are corrected endurance limit in reversed bending and reversing torsion respectively.
(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 (0.9Sut ) at 103cycles to S e at 106 cycles on a log-log paper.
Fig. 5.27 S-N Curve
At 1000 cycles: Bending: S Axial bending:
m
0.9Su t
S m 0.75Su t
5.12 Soderberg and Goodman lines
• When stress amplitude a is zero, the load is purely static and criterion of failure is Syt or Sut . These limits are plotted on the abscissa. • When the mean stress ( m ) is zero, the stress is completely reversing and the criterion of failure is endurance limit S e , that is plotted on the ordinate.
(1) Gerber line A parabolic curve joining S on the ordinate to S on the abscissa. e
yt
(2) Soderberg line A straight line joining S e on the ordinate to
S yt on the abscissa.
The equation of Soderberg line is given as: m a S yt
Se
1
(3) Goodman line A straight line joining Se on the ordinate to Sut on the abscissa The equation of Goodman line is given as
m S ut
a Se
1
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
Modified Goodman Diagram Goodman line is modified by combining fatigue failure with fatigue by yielding. The yield strength S yt is plotted on both the axes-abscissa and ordinate, and a yield line C D 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. A B is the portion of Goodman line and B C is portion of
yield line.
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.
a tan m The point of intersection of lines A B and O E is X. The point X indicates the dividing line between the safe region and the region of failure.
The coordinates of point X ( Sm , Sa ) represent the limiting values of stress, that are used to calculate the dimensions of component.
Fig. 5.40 Modified Goodman Diagram for Axial and Bending Stresses
The modified Goodman diagram for fluctuating torsional shear stress is shown in Fig. 5.41.
Fig. 5.41 Modified Goodman Diagram for Torsional Shear Stresses
Example 5.13 A transmission shaft of cold drawn steel 27Mn2 (
Sut 540N/mm2 ,
and S yt 400 N/mm2) 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:
S 0.5S ut 0.5(500) 25k 0 N/mm ' e
2
K a 0.79 assuming 7.5 d 50 mm, K b 0.85 for 90% reliablity, K c 0.897
S e K a K b K c S e' 0.79 0.85 0.897 250 150.58 N/mm 2 using the distortion energy theory, Sse 0.577S e 0.577 150.58 86.88 N/mm 2 Ssy 0.577S y t 0.577 300 173.1 N/mm 2 1 1 ( M t ) m [(M t ) max ( M t ) min ] [400 100] 150N m 2 2 1 1 ( M t ) a [(M t ) max ( M t ) min ] [400 100] 250N m 2 2 ( M t ) a 250 tan 1.67 ( M t ) m 150
59.04
The modified Goodman diagram is shown in Fig. 5.44. The ordinate of point X is
Ssa or 86.88 N/mm2 .
Fig. 5.44
S sa 86.88 N/mm . 2
a
S sa 86.88 2 43.44 N/mm ( fs ) 2
16 250 103 43.44 3 d d 30.83 mm
5.14 Gerber line
Sa Sm 2 Gerber equation is given as: S ( S ) 1 e ut
Gerber curve takes mean path through failure points. It is more accurate than Goodman or Soderberg line.
Fig. 5.52 Gerber line
5.15 Fatigue design under combined stresses
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. The general equation of distortion energy theory is as follows.
[( x y ) 2 ( y z ) 2 ( z x ) 2 6( xy2 yz2 zx2 )] 2
where x , y , z are normal stresses in X, Y, Z directions and xy , yz , zx are shear stresses in their respective planes.
In case of combined bending and torsional moments, there is a normal stress x accompanied by the torsional shear stress xy.
y z yz zx 0
3 2 x
2 xy
The mean and alternating component of respectively. The mean and alternating component of respectively.
x are xm and xa
xy are xm and xa
The Two stresses m , a are used in the modified Goodman diagram to design the component.
m 3 2 xm
a 3 2 xa
2 xym
2 xya
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 (Sut 540N/mm , and S yt 400 2
N/mm 2 ). The corrected endurance limit of the shaft is
200 N/mm 2. Determine the diameter of the shaft using a factor of safety of 2.
Solution
1 1 ( M b ) m [(M b ) max ( M b ) min ] [600 200] 400N m 2 2 1 1 ( M b ) a [(M b ) max ( M b ) min ] [600 200] 200N m 2 2 1 1 ( M t ) m [(M t ) max ( M t ) min ] [200 70] 135N m 2 2 1 1 ( M t ) a [(M t ) max ( M t ) min ] [200 70] 65N m 2 2 3 32( M b ) m 32(4001000) 4074.37 10 2 xm N/mm d3 d3 d3 32( M b ) a 32(2001000) 2037.18 103 2 xa N/mm d3 d3 d3
16( M t ) m 16(1351000) 687.55 10 2 xym N/mm d3 d3 d3 16( M t ) a 16(65 1000) 331.04 103 2 xya N/mm 3 3 3 d d d 3
3 3 4074 . 37 10 687 . 55 10 2 2 m x2m 3 x2y m ( ) 3 ( ) d3 d3 4244.84 103 2 ( ) N/mm d3
2037.18 10 2 331.04 10 2 a 3 ( ) 3( ) 3 3 d d 3 2116.3310 2 ( ) N/mm 3 d 3
2 xa
2 xya
3
a 2116.33 tan 0.4986 m 4244.84 26.5 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
Sa Sm 1 200 540 Sa tan 0.4986 Sm S a 114.76 N/mm 2 S m 230.16 N/mm 2 Sa 114.76 2 a 57.38 N/mm ( fs ) 2 2116.33 10 57.38 3 d d 33.29 mm 3
Exercise chapter 05 1.
Prob. 5.10
page 201
