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Chapter 18
Shafts and Axles
Dr. A. Aziz Bazoune King Fahd University of
Petroleum Minerals Mechanical Engineering
Department
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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) .945 18-8 Shaft Design
.950
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LECTURE 30
18-1 Introduction .92218-2 Geometric
Constraints .92718-3 Strength Constraints
.933
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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, sm Midrange
bending stress Ma alternating bending
moment, sa alternating bending stress Tm
Midrange torque, tm Midrange shear stress Ta
alternating torque, tm Midrange shear stress
Kf fatigue stress concentration factor for
bending Kfs fatigue stress concentration factor
for torsion
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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, sm Midrange
bending stress Ma alternating bending
moment, sa alternating bending stress Tm
Midrange torque, tm Midrange shear stress Ta
alternating torque, tm Midrange shear stress
Kf fatigue stress concentration factor for
bending Kfs fatigue stress concentration factor
for torsion
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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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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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Fatigue Analysis of shafts
where
(18-15)
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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)
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Fatigue Analysis of shafts
(18-18)
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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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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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Shaft Diameter Equation for the DE-Elliptic
Criterion
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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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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-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

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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 gt rcrit, the threat is from fatigue
• If r lt rcrit, the threat is from first cycle
  yielding.

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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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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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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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• 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. 727, 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. 727.
• 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.

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• 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
  smax can be obtained by simply adding sa and sm
  . (sa sm ) will always be greater than or equal
  to smax, and will therefore be conservative.

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Example
Solution
7-20
7-20
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Figure 7-20
Figure 7-21
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18-22
Solderberg
18-14
18-24
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