1. Chapter 7 Fatigue Failure Resulting from Variable Loading
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department

2. LECTURE 26

3. 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

4. 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.

5. 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

6. 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

7. In Fig. 7-27, the tensile side of Fig
In Fig. 7-27, the tensile side of Fig 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:
The parabola of Gerber (1874),
The Goodman (1890) (straight) line,
The Soderberg (1930) (straight) line.

9. 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.

10. 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 has the yield strength Syt and the tensile strength plotted along it.

11. The criteria of failure are diagrammed in Fig.7-27:
The Soderberg,
The modified
Goodman
The Gerber
The ASME-elliptic
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

13. FAILURE CRITERIA (mean stress)
1- Modified Goodman Theory (Germany, 1899)
Factor of Safety
For infinite life Failure Occurs When: B
n = OA/OB
Load Line slope

14. FAILURE CRITERIA (mean stress)
2- The Soderberg Theory (USA, 1933)
Factor of Safety
For infinite life Failure Occurs When: B C D E F
n = OC/OB
For finite life fatigue strength Sf = sa replaces Se

15. FAILURE CRITERIA (mean stress)
3- The Gerber Theory (Germany, 1874)
Factor of Safety B C D E F
n = OF/OB
Failure Occurs When:
For finite life σa replaces Se

16. FAILURE CRITERIA (mean stress)
4- The ASME Elliptic
Failure Occurs When:
Factor of Safety B C D E F
n = OE/OB

17. FAILURE CRITERIA (mean stress)
Factor of Safety B C D E F
n = OE/OB
Failure Occurs When
4- The ASME Elliptic
For finite life sa replaces Se

18. 5- The Langer (1st Cycle) Yield Line
FAILURE CRITERIA
5- The Langer (1st Cycle) Yield Line
Failure Occurs When B C D E F
Factor of Safety
n = OD/OB

19. Criteria Equations
(7-43)
(7-44)
(7-45)
(7-46)
(7-47)

21. The stresses nσa and nσm can replace Sa and Sm, where n is the design factor or factor of safety. Then, Eqs. (7-43) to (7-46) become:
(7-48)
(7-49)
(7-50)
(7-51)

22. The failure criteria are used in conjunction with a load line,
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
The failure criteria are used in conjunction with a load line,
Principal intersections are tabulated in Tables 7-9 to Formal expressions for fatigue factor of safety are given in the lower panel of Tables 7-9 to 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.
(7 *)

23. The first column gives the intersecting equations and the second column the intersection coordinates.
There are two ways to proceed with a typical analysis:
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.
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.

24. Intersection of the Static and Fatigue Criteria
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.

25. 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.

26. 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.

27. Special Cases of Fluctuating Stresses
Case 1: sm fixed
Case 2: sa fixed

28. Case 3: sa / sm fixed
Case 4: both vary arbitrarily

29. EXAMPLE 7-11 (Textbook)
Solution
(7-18)
(7-4), p. 329

30. EXAMPLE 7-11 (Textbook) (7-25), p. 331 (7-8), (7-17), p. 325, p. 328
(7-10)

31. (7-*)
(7-28)
(7-10)

32. Figure 7-28
Principal points A, B, C, and Don the designer’s diagram drawn for Gerber, Langer and load line.

33. (7-28)
7-10

34. 7-29
7-11

35. Figure 7-29
Principal points A, B, C, and Don the designer’s diagram drawn for ASME Elliptic, Langer and load lines.

36. 7-11