Chapter 18

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

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## Chapter 18

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Chapter 18

Shafts and Axles

Dr. A. Aziz Bazoune

King Fahd University of Petroleum & Minerals

Mechanical Engineering Department

Chapter Outline

18-1 Introduction ……….92218-2 Geometric Constraints ……….92718-3 Strength Constraints ……….93318-4 Strength Constraints – Additional Methods ……….940

18-5 Shaft Materials ……….94418-6 Hollow Shafts ……….94418-7 Critical Speeds (Omitted) ……….94518-8 Shaft Design ……….950

LECTURE 30

18-1 Introduction ……….92218-2 Geometric Constraints ……….92718-3 Strength Constraints ……….933

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

CH-18 LEC 30 Slide 4

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

CH-18 LEC 30 Slide 5

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

CH-18 LEC 30 Slide 6

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)

CH-18 LEC 30 Slide 7

Fatigue Analysis of shafts

where

(18-15)

CH-18 LEC 30 Slide 8

Fatigue Analysis of shafts

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)

CH-18 LEC 30 Slide 9

Fatigue Analysis of shafts

(18-18)

CH-18 LEC 30 Slide 10

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

CH-18 LEC 30 Slide 11

CH-18 LEC 30 Slide 11

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)

CH-18 LEC 30 Slide 12

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)

CH-18 LEC 30 Slide 14

Shaft Diameter Equation for the DE-Elliptic Criterion

• At a shoulderFigs. A-15-8 and A-15-9 provide information about Kt and Kts.
• For a hole in a solid shaft, Figs. A-15-10 and A-15-11 provide about Kt and Kts .
• For a hole in a solid shaft, use Table A-16
• For grooves use Figs. A-15-14 and A-15-15

CH-18 LEC 30 Slide 15

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.

CH-18 LEC 30 Slide 16

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)

CH-18 LEC 30 Slide 17

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)

CH-18 LEC 30 Slide 18

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-rangestresses 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.

CH-18 LEC 30 Slide 19

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.

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.

CH-18 LEC 30 Slide 23

Example

Solution

7-20

7-20

CH-18 LEC 30 Slide 25

Figure 7-20

Figure 7-21

CH-18 LEC 30 Slide 26

18-22

Solderberg

18-14

18-24

CH-18 LEC 30 Slide 28