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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Dr. A. Aziz Bazoune King Fahd University of Petroleum & Minerals Mechanical Engineering Department CH-18 LEC 30 Slide 1

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I 18-1 Introduction ……….922 18-2Geometric Constraints ……….927 18-3Strength Constraints ……….933 18-4Strength Constraints – Additional Methods ……….940 18-5Shaft Materials ……….944 18-6Hollow Shafts ……….944 18-7Critical Speeds (Omitted) ……….945 18-8Shaft Design ……….950 18-1 Introduction ……….922 18-2Geometric Constraints ……….927 18-3Strength Constraints ……….933 18-4Strength Constraints – Additional Methods ……….940 18-5Shaft Materials ……….944 18-6Hollow Shafts ……….944 18-7Critical Speeds (Omitted) ……….945 18-8Shaft Design ……….950 CH-18 LEC 30 Slide 2

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I 18-1 Introduction ……….922 18-2Geometric Constraints ……….927 18-3Strength Constraints ……….933 CH-18 LEC 30 Slide 3

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 4 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 Fatigue Analysis of shafts M m : Midrange bending moment, σ m : Midrange bending stress M a : alternating bending moment,σ a : alternating bending stress T m : Midrange torque,τ m : Midrange shear stress T a : alternating torque, τ m : Midrange shear stress K f : fatigue stress concentration factor for bending K fs : fatigue stress concentration factor for torsion CH-18 LEC 30 Slide 4

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 5 For solid shaft with round cross section, appropriate geometry terms can be introduced for C, I and J resulting in Fatigue Analysis of shafts M m : Midrange bending moment, σ m : Midrange bending stress M a : alternating bending moment,σ a : alternating bending stress T m : Midrange torque,τ m : Midrange shear stress T a : alternating torque, τ m : Midrange shear stress K f : fatigue stress concentration factor for bending K fs : fatigue stress concentration factor for torsion CH-18 LEC 30 Slide 5

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I 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 The Gerber fatigue failure criterion (18-12) where A and B are defined by the radicals in Eq. (8-12) as CH-18 LEC 30 Slide 6

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Fatigue Analysis of shafts (18-13) The critical shaft diameter is given by or, solving for 1/n, the factor of safety is given by (18-14) CH-18 LEC 30 Slide 7

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Fatigue Analysis of shafts (18-15) where CH-18 LEC 30 Slide 8

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Particular Case Critical Shaft Diameter CH-18 LEC 30 Slide 9 Fatigue Analysis of shafts (18-16) Safety Factor (18-17) 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 M m = 0 and T a = 0, which simply drops out some of the terms. CH-18 LEC 30 Slide 9

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 10 (18-18) Fatigue Analysis of shafts CH-18 LEC 30 Slide 10

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 11 Shaft Diameter Equation for the DE-Elliptic Criterion Remember The Elliptic fatigue-failure criterion is defined by (18-12) where A and B are defined by CH-18 LEC 30 Slide 11

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 12 Shaft Diameter Equation for the DE-Elliptic Criterion Substituting for A and B gives expressions for d, 1/n and r: (18-19) Critical Shaft Diameter Safety Factor (18-20) CH-18 LEC 30 Slide 12

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 13 Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 13

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 14 Particular Case Critical Shaft Diameter (18-21) Safety Factor (18-22) 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 M m = 0 and T a = 0, which simply drops out some of the terms. CH-18 LEC 30 Slide 14

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 15 At a shoulder Figs. A-15-8 and A-15-9 provide information about K t and K ts. For a hole in a solid shaft, Figs. A-15-10 and A-15-11 provide about K t and K ts. 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

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 16 The value of slope at which the load line intersects the junction of the failure curves is designated r crit. It tells whether the threat is from fatigue or first cycle yielding If r > r crit, the threat is from fatigue If r < r crit, the threat is from first cycle yielding. CH-18 LEC 30 Slide 16

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 17 For the Gerber-Langer intersection the strength components S a and S m are given in Table 7-10 as (18-23) CH-18 LEC 30 Slide 17

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I Shaft Diameter Equation for the DE-Elliptic Criterion CH-18 LEC 30 Slide 18 For the DE-Elliptic-Langer intersection the strength components S a and S m are given by (18-24) CH-18 LEC 30 Slide 18

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 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. CH-18 LEC 30 Slide 19

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 20

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 21

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I 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. CH-18 LEC 30 Slide 22

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 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 σ a and σ m. (σ a + σ m ) will always be greater than or equal to σ’ max, and will therefore be conservative. CH-18 LEC 30 Slide 23

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 24

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 25 Example Solution 7- 20 CH-18 LEC 30 Slide 25

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 26 Figure 7-20 Figure 7-21 CH-18 LEC 30 Slide 26

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 27

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 28 18- 22 18- 14 18- 24 Sold erbe rg CH-18 LEC 30 Slide 28

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 29

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 30

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 31

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 32

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 33

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 34

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 35

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Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts ME 307 Machine Design I ME 307 Machine Design I CH-18 LEC 30 Slide 36

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