LECTURE # 23

Chapter 7Fatigue Failure Resulting from Variable Loading Dr. A. Aziz Bazoune

King Fahd University of Petroleum & MineralsMechanical Engineering DepartmentDr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 25 Slide 2LECTURE 26Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I 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.

CH-07 LEC 25 Slide 3

Figure 7-24Modified Goodman diagram showing all the strengths and the limiting values of all the stress components for a particular midrange stressModified Goodman DiagramDr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 25 Slide 4Plot of Fatigue Failures for Midrange Stresses in both Tensile and Compressive Regions.Figure 7-25Plot 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/Se enables a plot of experimental results for a variety of steels.

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 25 Slide 5Master Fatigue Diagram.Figure 7-26

Master fatigue diagram for AISI 4340 steel with Sut = 158 Sy = 147 kpsi.

The stress component at A are

min = 20, max = 120, m = 70, o = 50

all in kpsi

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 6Fluctuating StressesFailure data for Sm in tension and in compressionCOMPRESSIVE mean stresses are BENEFICIAL (or have no effect) in fatigue TENSILE mean stresses are DETRIMENTAL for fatigue behavior

S is for strength2.Representing mean stress effect using modified Goodman Diagram

Mean Stress Effect (R -1)Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 7In 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:The parabola of Gerber (1874), The Goodman (1890) (straight) line, The Soderberg (1930) (straight) line.Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 8

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 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 becauseIt 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 LoadingME 307 Machine Design I CH-07 LEC 26 Slide 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. 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 LoadingME 307 Machine Design I CH-07 LEC 26 Slide 11The criteria of failure are diagrammed in Fig.7-27:The Soderberg,The modifiedGoodmanThe GerberThe ASME-ellipticYieldingThe 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 LoadingME 307 Machine Design I CH-07 LEC 26 Slide 12

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 13

BFAILURE CRITERIA (mean stress)1-Modified Goodman Theory (Germany, 1899)

Factor of SafetyLoad Line slope

n = OA/OBFor infinite life Failure Occurs When:Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I

BCDEFn = OC/OBCH-07 LEC 26 Slide 14FAILURE CRITERIA (mean stress)2- The Soderberg Theory (USA, 1933)

Factor of SafetyFor infinite life Failure Occurs When:For finite life fatigue strength Sf = sa replaces Se

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 15Factor of Safety

BCDEF

n = OF/OBFailure Occurs When: Factor of SafetyFor finite life a replaces SeFAILURE CRITERIA (mean stress)3- The Gerber Theory (Germany, 1874) Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 16Factor of Safety

BCDEF

n = OE/OBFAILURE CRITERIA (mean stress)4- The ASME Elliptic Failure Occurs When: Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 17Factor of Safety

BCDEF

n = OE/OBFailure Occurs WhenFor finite life sa replaces SeFAILURE CRITERIA (mean stress)

4- The ASME Elliptic Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 185- The Langer (1st Cycle) Yield LineFAILURE CRITERIAFailure Occurs WhenFactor of Safety

n = OD/OB

BCDEF

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 19

(7-43) (7-44) (7-45) (7-46) (7-47) Criteria EquationsDr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 20

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 21The stresses na and nm 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) Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 22We 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 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.

(7 *) Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 23The 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.Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 24

TABLE (7-9)

Amplitude and Steady Coordinates of Strength and Important Intersections in First Quadrant for Modified Goodman and Langer Failure Criteria. FatigueCriterionStatic Langer CriterionIntersection of the Static and Fatigue CriteriaDr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 25TABLE (7-10)

Amplitude and Steady Coordinates of Strength and Important Intersections in First Quadrant for Gerber and Langer Failure Criteria. GerberLanger Intersection of Gerber and Langer

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 26TABLE (7-11)

Amplitude and Steady Coordinates of Strength and Important Intersections in First Quadrant for ASME Elliptic and Langer Failure Criteria. ASME EllipticLanger Intersection of ASME Elliptic and Langer

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 27Special Cases of Fluctuating StressesCase 1: sm fixed

Case 2: sa fixed

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 28Case 3: sa / sm fixed

Case 4: both vary arbitrarily

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I

CH-07 LEC 26 Slide 29

EXAMPLE 7-11 (Textbook)

Solution(7-18) (7-4), p. 329 Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 30EXAMPLE 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 LoadingME 307 Machine Design I CH-07 LEC 26 Slide 31

(7-*)(7-28)(7-10)

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 32

Figure 7-28

Principal points A, B, C, and Don the designers diagram drawn for Gerber, Langer and load line.Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 33

(7-28)7-10Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 34

7-297-11

Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 35

Figure 7-29

Principal points A, B, C, and Don the designers diagram drawn for ASME Elliptic, Langer and load lines.Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I CH-07 LEC 26 Slide 36

7-11Dr. A. Aziz Bazoune Chapter 7: Fatigue Failure Resulting from Variable LoadingME 307 Machine Design I