1. Chapter 18 Shafts and Axles Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department

2. Chapter Outline
18-1 Introduction ……… Geometric Constraints ……… Strength Constraints ……… Strength Constraints – Additional Methods ……….940
18-5 Shaft Materials ……… Hollow Shafts ……… Critical Speeds (Omitted) ……… Shaft Design ……….950

3. LECTURE 30
18-1 Introduction ……… Geometric Constraints ……… Strength Constraints ……….933

4. Fatigue Analysis of shafts
Neglecting axial loads because they are comparatively very small at critical locations where bending and torsion dominate. Remember the fluctuating stresses due to bending and torsion are given by
Mm: Midrange bending moment, σm: Midrange bending stress
Ma : alternating bending moment, σa: alternating bending stress
Tm: Midrange torque, τm: Midrange shear stress
Ta : alternating torque, τm: Midrange shear stress
Kf: fatigue stress concentration factor for bending
Kfs: fatigue stress concentration factor for torsion
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5. Fatigue Analysis of shafts
For solid shaft with round cross section, appropriate geometry terms can be introduced for C, I and J resulting in
Mm: Midrange bending moment, σm: Midrange bending stress
Ma : alternating bending moment, σa: alternating bending stress
Tm: Midrange torque, τm: Midrange shear stress
Ta : alternating torque, τm: Midrange shear stress
Kf: fatigue stress concentration factor for bending
Kfs: fatigue stress concentration factor for torsion
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6. Fatigue Analysis of shafts
Combining these stresses in accordance with the DE failure theory the von-Mises stress for rotating round, solid shaft, neglecting axial loads are given by
(18-12)
where A and B are defined by the radicals in Eq. (8-12) as
The Gerber fatigue failure criterion
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7. Fatigue Analysis of shafts
The critical shaft diameter is given by
(18-13)
or, solving for 1/n, the factor of safety is given by
(18-14)
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8. Fatigue Analysis of shafts
where
(18-15)
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9. Fatigue Analysis of shafts Critical Shaft Diameter
Particular Case
For a rotating shaft with constant bending and torsion, the bending stress is completely reversed and the torsion is steady. Previous Equations can be simplified by setting Mm = 0 and Ta = 0, which simply drops out some of the terms.
Critical Shaft Diameter
(18-16)
Safety Factor
(18-17)
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10. Fatigue Analysis of shafts
(18-18)
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11. Shaft Diameter Equation for the DE-Elliptic Criterion
Remember
(18-12)
where A and B are defined by
The Elliptic fatigue-failure criterion is defined by
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12. Critical Shaft Diameter
Shaft Diameter Equation for the DE-Elliptic Criterion
Substituting for A and B gives expressions for d, 1/n and r:
Critical Shaft Diameter
(18-19)
Safety Factor
(18-20)
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13. Shaft Diameter Equation for the DE-Elliptic Criterion
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14. Critical Shaft Diameter
Shaft Diameter Equation for the DE-Elliptic Criterion
Particular Case
For a rotating shaft with constant bending and torsion, the bending stress is completely reversed and the torsion is steady. Previous Equations can be simplified by setting Mm = 0 and Ta = 0, which simply drops out some of the terms.
Critical Shaft Diameter
(18-21)
Safety Factor
(18-22)
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15. Shaft Diameter Equation for the DE-Elliptic Criterion
At a shoulder Figs. A-15-8 and A-15-9 provide information about Kt and Kts.
For a hole in a solid shaft, Figs. A and A provide about Kt and Kts .
For a hole in a solid shaft, use Table A-16
For grooves use Figs. A and A-15-15
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16. Shaft Diameter Equation for the DE-Elliptic Criterion
The value of slope at which the load line intersects the junction of the failure curves is designated rcrit.
It tells whether the threat is from fatigue or first cycle yielding
If r > rcrit, the threat is from fatigue
If r < rcrit, the threat is from first cycle yielding.
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17. Shaft Diameter Equation for the DE-Elliptic Criterion
For the Gerber-Langer intersection the strength components Sa and Sm are given in Table 7-10 as
(18-23)
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18. Shaft Diameter Equation for the DE-Elliptic Criterion
For the DE-Elliptic-Langer intersection the strength components Sa and Sm are given by
(18-24)
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19. Note that in an analysis situation in which the diameter is known and the factor of safety is desired, as an alternative to using the specialized equations above, it is always still valid to calculate the alternating and mid-range stresses using the following Eqs.
and substitute them into the one of the equations for the failure criteria , Eqs. (7-48) to (7-51) and solve directly for n.
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22. In a design situation, however, having the equations pre-solved for diameter is quite helpful.
It is always necessary to consider the possibility of static failure in the first load cycle.
The Soderberg criteria inherently guards against yielding, as can be seen by noting that its failure curve is conservatively within the yield (Langer) line on Fig. 7–27, p. 348.
The ASME Elliptic also takes yielding into account, but is not entirely conservative throughout its range. This is evident by noting that it crosses the yield line in Fig. 7–27.
The Gerber and modified Goodman criteria do not guard against yielding, requiring a separate check for yielding. A von Mises maximum stress is calculated for this purpose.

23. To check for yielding, this von Mises maximum stress is compared to the yield strength, as usual
For a quick, conservative check, an estimate for σ’max can be obtained by simply adding σa and σm . (σa + σm ) will always be greater than or equal to σ’max, and will therefore be conservative.
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25. Example
Solution
7-20
7-20
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26. Figure 7-20
Figure 7-21
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27. CH LEC Slide 27

28. 18-22
Solderberg
18-14
18-24
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30. CH LEC Slide 30