Transcript LEC 26 CH-07 - KFUPM Open Courseware

ME 307
Machine
Design I
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 25
Slide 2
ME 307
Machine
Design I
Modified Goodman
Diagram
 It has midrange stress plotted along the
abscissa
and all other components of
stress plotted on the ordinate, with
tension in the positive direction.
 The endurance limit, fatigue strength,
or
finite-life
strength
whichever
applies, is plotted on the ordinate above
and below the origin.
 The midrange line is a 45o line from the
origin to the tensile strength of the
part.
Figure 7-24
Modified Goodman diagram showing all the strengths
and the limiting values of all the stress components for
a particular midrange stress
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 25
Slide 3
ME 307
Machine
Design I
Plot of Fatigue Failures for Midrange Stresses in both Tensile
and Compressive Regions.
Figure 7-25
Plot of fatigue failures
for midrange stresses
in both tensile and
compressive regions.
Normalizing the data
by using the ratio of
steady strength
components to tensile
strength Sm/Sut,
steady strength
component to
compressive strength
Sm/Suc, and strength
amplitude component
to endurance limit
Sa/S’e enables a plot
of experimental
results for a variety of
steels.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 25
Slide 4
ME 307
Machine
Design I
Master Fatigue Diagram.
Figure 7-26
Master fatigue
diagram for AISI
4340 steel with
Sut = 158
Sy = 147 kpsi.
The stress
component at A
are
σmin = 20,
σ max = 120,
σ m = 70,
σ o = 50
all in kpsi
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 25
Slide 5
ME 307
Machine
Design I
Fluctuating Stresses
Mean Stress Effect (R  -1)
2. Representing mean
stress effect using
modified Goodman
Diagram
S is for strength
Failure data for Sm in tension and in compression
COMPRESSIVE mean stresses are BENEFICIAL (or have no effect) in fatigue
TENSILE mean stresses are DETRIMENTAL for fatigue behavior
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 6
ME 307
Machine
Design I
 In Fig. 7-27, the tensile side of Fig. 7-25 has been redrawn in terms of
strengths, instead of strength ratios, with the same modified Goodman criterion
together with four additional criteria of failure.
 Such diagrams are often constructed for analysis and design purposes; they are
easy to use and the results can be scaled off directly.
 The early viewpoint expressed on a diagram was that there existed a locus (sa,
sm) diagram was that there existed a locus which divided safe from unsafe
combinations of
(sa, sm) .
 Ensuing proposals included:
1. The parabola of Gerber (1874),
2. The Goodman (1890) (straight) line,
3. The Soderberg (1930) (straight) line.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 7
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 8
ME 307
Machine
Design I

As more data were generated it became clear that a fatigue criterion,
rather than being a “fence”, was more like a zone or band wherein the
probability of failure could be estimated. We include the failure criterion
of Goodman because
It is a straight line and the algebra is linear and easy.
It is easily graphed, every time for every problem.
It reveals subtleties of insight into fatigue problems.
Answers can be scaled from the diagrams as a check on the
algebra.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 9
ME 307
Machine
Design I

Either the fatigue limit Se or the finite-life strength Sf is plotted on the
ordinate of Fig. 7-27.

These values will have already been corrected using the Marin factors of
Eq.(7-17).

Note that the yield strength is plotted on the ordinate too.

This serves as a reminder that first-cycle yielding rather than fatigue
might be the criterion of failure.

The midrange-stress axis of Fig. 7-27 has the yield strength Syt and the
tensile strength plotted along it.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 10
ME 307
Machine
Design I


The criteria of failure are diagrammed in Fig.7-27:
1.
The Soderberg,
2.
The modified
3.
Goodman
4.
The Gerber
5.
The ASME-elliptic
6.
Yielding
The diagram shows that only the Soderberg criterion guards against any yielding,
but is biased low.

Considering the modified Goodman line as a criterion, point A represents a
limiting point with an alternating strength Sa and midrange strength Sm . The slope
of the load line shown is defined as
Dr. A. Aziz Bazoune
.
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 11
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 12
ME 307
Machine
Design I
1-
FAILURE CRITERIA (mean stress)
Modified Goodman Theory (Germany, 1899)
Factor of Safety
For infinite life Failure Occurs When:
a m 1
 
Se Su n
Sa Sm
 1
Se Su
n = OA/OB
Load Line slope
Sa
r
Sm
Dr. A. Aziz Bazoune
B
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 13
ME 307
Machine
Design I
2-
FAILURE CRITERIA (mean stress)
The Soderberg Theory (USA, 1933)
Factor of Safety
For infinite life Failure Occurs When:
a m 1
 
Se S y n
Sa S m
 1
Se S y
n = OC/OB
F
D
For finite life fatigue
B
E
C
strength Sf = a
replaces Se
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 14
FAILURE CRITERIA (mean stress)
ME 307
Machine
Design I
3-
The Gerber Theory (Germany, 1874)
Failure Occurs When:
Sa
Se
2
 Sm 
   1
 Su 
Factor of Safety
n = OF/OB
F
B
D E
C
Factor of Safety
2
n a  n m 

 1
Se  Su 
For finite life σa replaces Se
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 15
FAILURE CRITERIA (mean stress)
ME 307
Machine
Design I
4-
The ASME Elliptic
Failure Occurs When:
2
 Sa   S m 
     1
 Se   S y 
2
Factor of Safety
2
2
 n a   n m 

 
 1
 Se   Sy 
F
B
Dr. A. Aziz Bazoune
C
D E
n = OE/OB
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 16
FAILURE CRITERIA (mean stress)
ME 307
Machine
Design I
4-
The ASME Elliptic
Failure Occurs When
2
 Sa   S m 
     1
 Se   S y 
2
Factor of Safety
2
 n a   n m 
 1

  
 Se   S y 
2
F
B
For finite life sa
Dr. A. Aziz Bazoune
D E
C
replaces Se
n = OE/OB
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
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Slide 17
FAILURE CRITERIA
ME 307
Machine
Design I
5-
The Langer (1st Cycle) Yield Line
Failure Occurs When
Sa S m

1
S yt S yt
Factor of Safety
a m 1
 
S yt S yt n
F
D
B
n = OD/OB
Dr. A. Aziz Bazoune
E
C
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
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Slide 18
ME 307
Machine
Design I
Criteria Equations
(7-43)
(7-44)
(7-45)
(7-46)
(7-47)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 19
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 20
 The stresses nσa and nσm can replace Sa and Sm, where n is the design
ME 307
Machine
Design I
factor or factor of safety. Then, Eqs. (7-43) to (7-46) become:
(7-48)
(7-49)
(7-50)
(7-51)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 21
ME 307
Machine
Design I
We will emphasize the Gerber and ASME-elliptic for fatigue failure
criterion and the Langer for first-cycle yielding. However, conservative
designers often use the modified Goodman criterion. The design
equation for the Langer first -cycle-yielding is
(7 *)
The failure criteria are used in conjunction with a load line,
Principal intersections are tabulated in Tables 7-9 to 7-11. Formal
expressions for fatigue factor of safety are given in the lower panel of
Tables 7-9 to 7-11. The first row of each table corresponds to the
fatigue criterion, the second row is the static Langer criterion, and
the third row corresponds to the intersection of the static and fatigue
criteria.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 22
ME 307
Machine
Design I
 The first column gives the intersecting equations and the second
column the intersection coordinates.
 There are two ways to proceed with a typical analysis:
1. One method is to assume that fatigue occurs first and use one
of Eqs. (7-48) to (7-51) to determine n or size, depending on
the task. Most often fatigue is the governing failure mode.
Then follow with a static check. If static failure governs then
the analysis is repeated using Langer Static yield equation.
2. Alternatively, one could use the tables. Determine the load
line and establish which criterion the load line intersects first
and use the corresponding equations in the tables.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 23
ME 307
Machine
Design I
Fatigue
Criterion
Static
Langer
Criterion
Intersection of
the Static and
Fatigue Criteria
TABLE (7-9)
Amplitude and Steady
Coordinates of Strength and
Important Intersections in
First Quadrant for Modified
Goodman and Langer Failure
Criteria.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 24
ME 307
Machine
Design I
Gerber
Langer
Intersection
of Gerber and
Langer
TABLE (7-10)
Amplitude and Steady
Coordinates of Strength and
Important Intersections in
First Quadrant for Gerber
and Langer Failure Criteria.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 25
ME 307
Machine
Design I
ASME
Elliptic
Langer
Intersection of
ASME Elliptic
and Langer
TABLE (7-11)
Amplitude and Steady
Coordinates of Strength
and Important
Intersections in First
Quadrant for ASME Elliptic
and Langer Failure
Criteria.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 26
Special Cases of Fluctuating
Stresses
ME 307
Machine
Design I
•Case 1: m fixed
n
Sa
a
•Case 2: a fixed
n
Dr. A. Aziz Bazoune
Sm
m
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
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Slide 27
ME 307
Machine
Design I
•Case 3: a / m fixed
n
Sa
a

Sm
m
•Case 4: both vary arbitrarily
1 a m


n Se Sut
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 28
ME 307
Machine
Design I
EXAMPLE 7-11 (Textbook)
Solution
(7-18)
Dr. A. Aziz Bazoune
(7-4), p. 329
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 29
ME 307
Machine
Design I
EXAMPLE 7-11 (Textbook)
(7-25), p. 331
(7-8), (7-17), p. 325, p. 328
(7-10)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 30
ME 307
Machine
Design I
(7-*)
(7-28)
(7-10)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 31
ME 307
Machine
Design I
Figure 7-28
Principal points
A, B, C, and Don
the designer’s
diagram drawn
for Gerber,
Langer and load
line.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 32
ME 307
Machine
Design I
(7-28)
Dr. A. Aziz Bazoune
7-10
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
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Slide 33
ME 307
Machine
Design I
7-11
7-29
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
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Slide 34
ME 307
Machine
Design I
Figure 7-29
Principal points
A, B, C, and Don
the designer’s
diagram drawn
for ASME
Elliptic, Langer
and load lines.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 35
ME 307
Machine
Design I
7-11
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 26
Slide 36