Shafts and Axles

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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 31
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Estimating Stress Concentration

• The stress analysis process for fatigue is highly
  dependent on stress concentrations.
• Stress concentrations for shoulders and keyways
  are dependent on size specifications that are not
  known the first time through the process.
• Fortunately, since these elements are usually of
  standard proportions, it is possible to estimate
  the stress concentration factors for initial
  design of the shaft. These stress concentrations
  will be fine-tuned in successive iterations, once
  the details are known.

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Estimating Stress Concentration

• Shoulders for bearing and gear support should
  match the catalog recommendation for the specific
  bearing or gear.
• A look through bearing catalogs shows that a
  typical bearing calls for the ratio of D/d to be
  between 1.2 and 1.5.
• For a first approximation, assume D/d 1.5 can be
  assumed.
• Fillet radius at the shoulder needs to be sized
  to avoid interference with the fillet radius of
  the mating component. There is a significant
  variation in typical bearings in the ratio of
  fillet radius r/d versus bore diameter, with
  typically ranging from around 0.02 to 0.06.

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Estimating Stress Concentration

• Figures A-15-8 and A-15-9 show that the stress
  concentrations for bending and torsion increase
  significantly in this range. For example, with
  D/d 1.5 for bending
• In most cases the shear and bending moment
  diagrams show that bending moments are quite low
  near the bearings, since the bending moments from
  the ground reaction forces are small.

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Estimating Stress Concentration

• In cases where the shoulder at the bearing is
  found to be critical, the designer should plan to
  select a bearing with generous fillet radius, or
  consider providing for a larger fillet radius on
  the shaft by relieving it into the base of the
  shoulder as shown in Fig. 7-9a.
• This effectively creates ahead
• zone in the shoulder area that
• does not carry the bending
• stresses, as shown by the stress
• flow lines.

Fig. 7-9a.
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Estimating Stress Concentration

• A shoulder relief groove as shown in Fig. 7-9b
  can accomplish a similar purpose. Another option
  is to cut a large-radius relief groove into the
  small diameter of the shaft, as shown in Fig.
  7-9c.

Fig. 7-9b.
Fig. 7-9c.
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Figure 7-9 Techniques for reducing stress
concentration at a shoulder supporting a bearing
with a sharp radius. (a) Large radius undercut
into the shoulder. (b) Large radius relief groove
into the back of the shoulder. (c) Large radius
relief groove into the small diameter.
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• This has the disadvantage of reducing the
  cross-sectional area, but is often used in cases
  where it is useful to provide a relief groove
  before the shoulder to prevent the grinding or
  turning operation from having to go all the way
  to the shoulder.
• For the standard shoulder fillet, for estimating
  Kt values for the first iteration, an r/d ratio
  should be selected so Kt values can be obtained.
  For the worst end of the spectrum, with r/d
  0.02 and D/d 1.5, Kt values from the stress
  concentration charts for shoulders indicate 2.7
  for bending, 2.2 for torsion, and 3.0 for axial.

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• A keyway will produce a stress concentration near
  a critical point where the load transmitting
  component is located. The stress concentration in
  an end-milled keyseat is a function of the ratio
  of the radius r at the bottom of the groove and
  the shaft diameter d. For early stages of the
  design process, it is possible to estimate the
  stress concentration for keyways regardless of
  the actual shaft dimensions by assuming a typical
  ratio of r/d 0.02. This gives Kt 2.2 for
  bending and Kts 3.0 for torsion, assuming the
  key is in place.

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• A keyway will produce a stress concentration near
  a critical point where the load transmitting
  component is located. The stress concentration in
  an end-milled keyseat is a function of the ratio
  of the radius r at the bottom of the groove and
  the shaft diameter d. For early stages of the
  design process, it is possible to estimate the
  stress concentration for keyways regardless of
  the actual shaft dimensions by assuming a typical
  ratio of r/d 0.02. This gives Kt 2.2 for
  bending and Kts 3.0 for torsion, assuming the
  key is in place.

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Table 7-1 First iteration estimates for stress
concentration factors Kt Warning These factors
are only estimates for use when actual dimensions
are not yet determined. Do not use these once
actual dimensions are available.
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Fatigue Analysis of Shafts

• The fatigue strength will be determined using
• Distortion-Energy-Gerber
• 2 Distortion-Energy-Elliptic

Rotating Shaft under stationary bending and
torsional moments
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Fatigue Analysis of Shafts
Safety Factor
Gerber
ASME-Elliptic
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Problem 18-10
A geared industrial roll shown in the figure is
driven at 300 rev/min by a force F acting on a
3-in-diameter pitch circle as shown. The roll
exerts a normal force of 30 lbf/in of roll length
on the material being pulled through. The
material passes under the roll. The coefficient
of friction is 0.40. Develop the moment and shear
diagrams for the shaft modeling the roll force as
a concentrated force at the center of the roll,
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Problem 18-10
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Problem 18-10
We have a design task of identifying bending
moment and torsion diagrams which are preliminary
to an industrial roller shaft design.
Gear
Roller
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This approach over-estimates the bending moment
at C, torque at C but not at A.
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Problem 18-11

• Using a 1035 hot rolled steel, estimate the
  necessary diameter at the locations of peak
  bending moment using a design factor of 2. These
  are likely to be fillets at both ends of the
  right hand bearing seat, where the bending moment
  is slightly less than the local extreme.
• Estimating the fatigue stress-concentration
  factor as 2, and using a design factor of 2, what
  is the approximate necessary diameter of the
  bearing seat using the DE-elliptic fatigue
  failure criterion in Problem 18-10?

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Problem 18-11
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Problem 18-11
From static Analysis
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Problem 18-11
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Hollow Shafts

• As an example equation 18-21 is modified to take
  into account the hollow shaft case
• where di and do are respectively the inner and
  outer diameters of the shaft.
• With this, one can consider that the
  stress-strength analysis is completed. You have
  obtained the minimum diameter at the critical
  section that can withstand the applied loads.

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Shaft Design

• One approach is (See Lab Handbook)
• Selecting a material (usually steel)
• Drawing a free body diagram of the shaft
• Performing static equilibrium analysis and
• Locating the critical area
• Performing static stress analysis to find a
  starting diameter size, d.
• Using the value of d in calculating the
  endurance limit (a trial diameter can also be
  used)
• Estimating the critical value of the diameter, d,
  using DE-Gerber or DE-ASME-elliptic methods
• Repeat step 6 if d different from d.
• Building the rest of the shaft by considering the
  machine parts to be mounted on the shaft
  (bearings, gears, pulleys, )
• Performing deflection analysis
• Performing Dynamic analysis

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Shaft Materials

• Shafts are usually made of ductile materials.
• Small shafts with diameters less than 3.5 in (90
  mm) are usually made of Cold Drawn carbon steel
  (AISI 1018-1050).
• Larger diameter shafts are machined from Hot
  Rolled steel.
• Heat treated steels are also used when higher
  strengths are necessary.

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Shaft
N
Y
Design?
Find critical diameter, d
find safety Factor, n
N
Y
N
Y
Shaft rotating?
Shaft rotating?
Static Analysis
Static Analysis
Eq. 6-42 or 6-44
Eq. 6-43 or 6-45
Static Analysis
Fatigue Analysis
Eq. 6-44 or 6-46
d
d'
Fatigue Analysis
n
Reversed
Y
N
bending steady
torque?
Reversed
Y
N
bending steady
torque?
Eq. 18-17 or 18-22
Eq. 18-14 or 18-20
n
Eq. 18-16 or 18-21
Eq. 18-13 or 18-19
d
N
N
d. NE. d
Y
d Critical shaft
diameter
Complete shaft
geometry perform
Deformation analysis
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Geometric Constraints

• Unlike stress, which is a function of local
  geometry and load, deflection is a function of
  the geometry everywhere. Thus, The task of
  deflection and rigidity analyses can be started
  only when the entire geometry of the shaft is
  determined.
• However the approach described in section 18-2,
  which is based on bearing slope constraints as
  limiting, may be used first assuming a uniform
  diameter shaft and using equations 18-1 and 18-2
  to find the diameters at the bearings.

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Stress Concentrations and Shaft Geometry
Shaft shoulders are used to position and provide
necessary thrust supports for elements such as
bearings, gears, pulleys, Provisions must be
made for torque-transfer elements such as keys,
splines, pins The theoretical stress
concentration factors for shoulders, grooves and
transverse holes can be obtained from appendix
A15. Others are

Kt 2.0 for profile key
seats Kt 1.6 for sled runner
keyseats
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Shaft Geometry
To determine the entire geometry of the shaft one
has to rely on existing models. Some of these
models are given in figures 18-1 through 18-8 of
the Textbook. More shaft configurations can be
found in the FAG handbooks of the Design of
Rolling Bearing Mountings.
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Geometric Constraints Shaft Deflection and Slopes

• The transverse deflection of the elastic curve of
  the shaft can be determined by any one of the
  methods studied in Chapter 5.
• The superposition method, which utilizes Appendix
  A-9, is recommended. For complex shaft geometry
  the numerical integration or computer program may
  be used.

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Geometric Constraints Shaft Deflection and Slopes

• The slope at ball bearings should be limited at
  0.25 deg the slope at roller bearings and long
  journal bearings should be a lot less. For
  details on acceptable slopes refer to FAG and SKF
  catalogs.
• For machinery shafting, the deflection should be
  no greater than 0.001 in/ft (0.075 mm/m) of shaft
  length between bearing supports.
• For shafts mounting good quality spur gears, the
  deflection at the gear mesh should not exceed
  0.005 in. (0.125 mm) or F/200 (F is the gear face
  width in inches) and the slope should be limited
  0.0286 deg.
• For shafts mounting good quality bevel gears, the
  deflection at the gear mesh should not exceed
  0.003 in. (0.076 mm).

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