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

2. Chapter Outline
7-1 Introduction to Fatigue in Metals     Approach to Fatigue Failure in Analysis and Design    Fatigue-Life Methods     The Stress-Life Method     The Strain-Life Method     The Linear-Elastic Fracture Mechanics Method    The Endurance Limit     Fatigue Strength     Endurance Limit Modifying Factors    Stress Concentration and Notch Sensitivity     Characterizing Fluctuating Stresses     Fatigue Failure Criteria for Fluctuating Stress     Torsional Fatigue Strength under Fluctuating Stresses     Combinations of Loading Modes    Varying, Fluctuating Stresses; Cumulative Fatigue Damage     Surface Fatigue Strength     Stochastic Analysis    373

3. LECTURE-21 7-7 The Endurance Limit
7-8 Fatigue Strength   Endurance Limit Modifying Factors

4. 7-7 The Endurance Limit   
A quick method of estimating endurance limits is needed:
for preliminary and prototype design
for some failure analysis
Experimental results for
rotating-beam tests
simple tension tests
of specimens taken from
the same bar are shown in
Figure 7.18.
Figure 7-18
Graph of endurance limits versus tensile strengths from actual test results for a large number of wrought irons and steels.

5. Figure 7-18
Graph of endurance limits versus tensile strengths from actual test results for a large number of wrought irons and steels. Ratios of S’e/Sut of 0.60, 0.50, and 0.40 are shown by the solid and dashed lines. Note also the horizontal dashed line for of S’e=107 kpsi. Points shown having a tensile strength greater than 214 kpsi have a mean endurance limit of S’e=107 kpsi and a standard deviation of 13.5 kpsi.

6. For steels, the relationship between the tensile strength and the endurance limit is given by
(7-8)
where
: is the minimum tensile strength. The prime mark on in this equation refers to the rotating-beam specimen itself.
The unprimed symbol is for the endurance limit of any particular machine element subjected to any kind of loading.

7. The endurance limits for various classes of cast irons, polished or machined, are given in Table A-24. Aluminum alloys do not have an endurance limit. The fatigue strengths of some aluminum alloys at 5(108 ) cycles of reversed stress are given in Table A-24.

8. 7-7 Fatigue Strength Region of low cycle fatigue:
The fatigue strength is only slightly smaller than the tensile
strength
Region of high Cycle Fatigue
The purpose of this section is to develop methods of approximation of the S-N diagram in the high-cycle region, when information may be as sparse as the results of a simple tension test. Experience has shown high-cycle fatigue data are rectified by a logarithmic transform to both stress and cycles-to-failure.

9. 7-8 Fatigue Strength   
In the region of high cycle fatigue, the equation relating the fatigue strength to the number of cycles to failure may be given by the empirical curve fit equation:
where is the number of cycles to failure and a and b are given by
(7-12)
(7-13)
(7-14)
where is found from Figure 7-19.

11. If a completely reversed stress is given, setting in Eq
If a completely reversed stress is given, setting in Eq. (7-12), the number of cycles-to-failure can be expressed as
Low-cycle fatigue is often defined (see Fig. 7-10) as failure that occurs in a range of cycles.
(7-15)

14. Example SOLUTION: Given a 1050 HR steel, estimate
a) The rotating-beam endurance limit at 106.
b) The endurance strength of a polished rotating beam specimen corresponding to 104 cycles to failure.
c) The expected life of a polished rotating-beam specimen under a completely reversed stress of 55 kpsi.
SOLUTION:
a) From Table A-20,
From Eq. (7-8)
b) From Fig. (7-19) for

15. Example (Cont.’d) From Eq. (7-13) and (7-14) == Thus Eq. (7-12) is:
for cycles to failure, the above equation becomes

16. Example (Cont.’d) c) From Eq. (7-15), with
Keep in mind that these are only estimates.

17. 7-9 Endurance Limit Modifying Factors
The rotating-beam specimen used in the laboratory to determine endurance limits is prepared very carefully and tested under closely controlled conditions. It is unrealistic to expect the endurance limit of a mechanical or structural member to match the values obtained in the laboratory. Some differences include
Material: composition, basis of failure, variability
Manufacturing: method, heat treatment, fretting corrosion, surface condition, stress concentration
Environment: corrosion, temperature, stress state, relaxation times
Design: size, shape, life, stress state, stress concentration, speed, fretting, galling

18. Marin’s Equation   
Marin identified factors that quantified the effects of surface condition, size, loading, temperature, and miscellaneous items. Marin’s Equations is therefore written as:
(7-17)
Endurance limit at the critical location of a machine part in geometry and condition of use
rotary-beam test specimen endurance limit

19. When endurance tests of parts are not available, estimations are made by applying Marin factors to the endurance limit.

20. Table 7-4 Parameters for Marin surface modification
(7-18)
where is the minimum tensile strength and and are to be found in Table 7-4.
Table 7-4 Parameters for Marin surface modification
factor, Eq. (7-18)

21. The size factor for bending and torsion may be given by:
(7-19)
For axial loading there is no size effect, so
(7-20)

22. QUESTION:
What to do with Eq.(7-19) if a round bar in bending is not rotating or when a non-circular cross-section is used?
ANSWER:
Use effective dimension where
(7-23)
as the effective size of a round corresponding to a non-rotating solid or hollow round. Table 7-5 provides areas of common structural shapes undergoing non-rotating bending

23. Table 7-5
Areas of common non-rotating structural shapes

24. Average values for the load factor are given by
(7-25)