1. Shafts and Shaft Components
Lecture 3
Ali Vatankhah
Chapter 7
Shafts and Shaft Components

2. Chapter Outline

3. SHAFT
The term shaft usually refers to a component of circular cross-section that rotates and transmits power from a driving device, such as a motor or engine, through a machine.
Shafts can carry gears, pulleys and sprockets to transmit rotary motion and power via mating gears, belts and chains.
A shaft may simply connect to another via a coupling.
A shaft can be stationary and support a rotating member such as the short shafts which support the non-driven wheels of automobiles, often referred to as spindles.
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4. SHAFT
A shaft is a rotating member usually of circular cross-section (solid or hollow), which transmits power and rotational motion.
Machine elements such as gears, pulleys (sheaves), flywheels, clutches, and sprockets are mounted on the shaft and are used to transmit power from the driving device (motor or engine) through a machine.
Press fit, keys, dowel, pins and splines are used to attach these machine elements on the shaft.
The shaft rotates on rolling contact bearings or bush bearings.
Various types of retaining rings, thrust bearings, grooves and steps in the shaft are used to take up axial loads and locate the rotating elements.
Couplings are used to transmit power from drive shaft (e.g., motor) to the driven shaft (e.g. gearbox, wheels).
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5. SHAFT Shaft design considerations include:
1. Size and spacing of components (as on a general assembly drawing), tolerances
2. Material selection and material treatments
3. Deflection and rigidity
bending deflection
torsional deflection
slope at bearings
shear deflection
4. Stress and strength
static strength
fatigue
reliability
5. Frequency response
6. Manufacturing constraints
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6. Shaft Design
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7. Shaft- A rotating member used to transmit power.
Shaft Types
Shaft- A rotating member used to transmit power.
Axle- A stationary member used as support for rotating elements such as wheels, idler gears, etc.
Spindle- A short shaft or axle (e.g., head-stock spindle of a lathe).
Stub shaft- A shaft that is integral with a motor, engine or prime mover and is of a size, shape, and projection as to permit easy connection to other shafts
Line shaft- A shaft connected to a prime mover and used to transmit power to one or several machines
Jackshaft- (Sometimes called countershaft). A short shaft that connects a prime mover with a line shaft or a machine
Flexible shaft- A connector which permits transmission of motion between two members whose axes are at an angle with each other
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8. Hollow Versus Solid Shafts
Shaft Design
Most shafts are round but they can come in many different shapes including square and octagonal. Keys and notches can also result in some unique shapes.
Hollow Versus Solid Shafts
Hollow shafts are lighter than solid shafts of comparable strength but are more expensive to manufacture. Thusly hollow shafts are primarily only used when weight is critical. For example the propeller shafts on rear wheel drive cars require lightweight shafts in order to handle speeds within the operating range of the vehicle.
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9. Shaft Design
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10. Shaft Design
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11. Shaft Design
Material Selection
Geometric Layout
Stress and strength
Static strength
Fatigue strength
Deflection and rigidity
Bending deflection
Torsional deflection
Slope at bearings and shaft-supported elements
Shear deflection due to transverse loading of short shafts
Vibration due to natural frequency

12. Shaft Materials
Deflection primarily controlled by geometry, not material
Stress controlled by geometry, not material
Strength controlled by material property

13. Shaft Materials
Shafts are commonly made from low carbon, CD or HR steel, such as ANSI 1020–1050 steels.
Fatigue properties don’t usually benefit much from high alloy content and heat treatment.
Surface hardening usually only used when the shaft is being used as a bearing surface.

14. Shaft Materials
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15. Shaft Materials
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16. Shaft Materials Hot Rolled
Hot rolling is a mill process which involves rolling the steel at a high temperature (typically at a temperature over 1700° F), which is above the steel’s recrystallization temperature. When steel is above the recrystallization temperature, it can be shaped and formed easily, and the steel can be made in much larger sizes. Hot rolled steel is typically cheaper than cold rolled steel due to the fact that it is often manufactured without any delays in the process, and therefore the reheating of the steel is not required (as it is with cold rolled). When the steel cools off it will shrink slightly thus giving less control on the size and shape of the finished product when compared to cold rolled.
Uses: Hot rolled products like hot rolled steel bars are used in the welding and construction trades to make railroad tracks and I-beams, for example. Hot rolled steel is used in situations where precise shapes and tolerances are not required.

17. Shaft Materials Cold Rolled
Cold rolled steel is essentially hot rolled steel that has had further processing. The steel is processed further in cold reduction mills, where the material is cooled (at room temperature) followed by annealing and/or tempers rolling. This process will produce steel with closer dimensional tolerances and a wider range of surface finishes. The term Cold Rolled is mistakenly used on all products, when actually the product name refers to the rolling of flat rolled sheet and coil products.
When referring to bar products, the term used is “cold finishing”, which usually consists of cold drawing and/or turning, grinding and polishing. This process results in higher yield points and has four main advantages:

18. Shaft Materials Cold Rolled
Cold drawing increases the yield and tensile strengths, often eliminating further costly thermal treatments.
Turning gets rid of surface imperfections.
Grinding narrows the original size tolerance range.
Polishing improves surface finish.
All cold products provide a superior surface finish, and are superior in tolerance, concentricity, and straightness when compared to hot rolled.
Cold finished bars are typically harder to work with than hot rolled due to the increased carbon content. However, this cannot be said about cold rolled sheet and hot rolled sheet. With these two products, the cold rolled product has low carbon content and it is typically annealed, making it softer than hot rolled sheet.
Uses: Any project where tolerances, surface condition, concentricity, and straightness are the major factors

19. Shaft Materials
Cold drawn steel typical for d < 3 in.
HR steel common for larger sizes. Should be machined all over.
Low production quantities
Lathe machining is typical
Minimum material removal may be design goal
High production quantities
Forming or casting is common
Minimum material may be design goal

20. Issues to consider for shaft layout Axial layout of components
Supporting axial loads
Providing for torque transmission
Assembly and Disassembly
Fig. 7-1

21. Axial Layout of Components
Fig. 7-2
Choose a shaft configuration to support and locate the two gears and two bearings.
(b) Solution uses an integral pinion, three shaft shoulders, key and keyway, and sleeve. The housing locates the bearings on their outer rings and receives the thrust loads.
(c) Choose fanshaft configuration.
(d) Solution uses sleeve bearings, a straight-through shaft, locating collars, and setscrews for collars, fan pulley, and fan itself. The fan housing supports the sleeve bearings.

22. Supporting Axial Loads
Axial loads must be supported through a bearing to the frame.
Generally best for only one bearing to carry axial load to shoulder
Allows greater tolerances and prevents binding
Fig. 7-3
Fig. 7-4

23. Providing for Torque Transmission
Common means of transferring torque to shaft
Keys
Splines
Setscrews
Pins
Press or shrink fits
Tapered fits
Keys are one of the most effective
Slip fit of component onto shaft for easy assembly
Positive angular orientation of component
Can design key to be weakest link to fail in case of overload

24. Assembly and Disassembly
Arrangement showing bearing
inner rings press-fitted to shaft
while outer rings float in the
housing. The axial clearance
should be sufficient only to
allow for machinery vibrations.
Note the labyrinth seal on the
right.
Fig. 7-5
Similar to the arrangement of
Fig. 7–5 except that the outer
bearing rings are preloaded.
Fig. 7-6

25. Assembly and Disassembly
In this arrangement the inner ring of the left-hand bearing is locked to the shaft between a nut and a shaft shoulder. The locknut and washer are AFBMA standard. The snap ring in the outer race is used to positively locate the shaft assembly in the axial direction.
Note the floating right-hand bearing and the grinding runout grooves in the shaft.
Fig. 7-7
This arrangement is similar to
Fig. 7–7 in that the left-hand bearing positions the entire shaft assembly.
In this case the inner ring is secured to
the shaft using a snap ring.
Note the use of a shield to prevent dirt generated from within the machine from entering the bearing
Fig. 7-8
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26. SHAFTS: General Design Rules
Keep shafts short, with bearings close to the applied loads. This will reduce deflections and bending moments, and increases critical speeds.
Place necessary stress raisers away from highly stressed shaft regions if possible. If unavoidable, use generous radii and good surface finishes. Consider local surface-strengthening processes (shot-peening or cold-rolling).
Use inexpensive steels for deflection-critical shafts because all steels have essentially the same modulus of elasticity.
Early in the design of any given shaft, an estimate is usually made of whether strength or deflection will be the critical factor. A preliminary design is based on that criterion; then, the remaining factor is checked.
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27. SHAFTS: Rules of Deflection
Deflections should not cause mating gear teeth to separate more than about .005 in. They should also not cause the relative slope of the gear axes to change more than .03 deg.
The shaft deflection across a plain bearing must be small compared to the oil film thickness.
The shaft angular deflection at a ball or roller bearing should generally not exceed .04 deg. unless the bearing is self-aligning.
Rule Of Thumb: Restrict the torsional deflection to 1° for every 20 diameters of length, sometimes less.
Rule Of Thumb: In bending the deflection should be limited to .01 in. per foot of length between supports.
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28. Shaft design procedure for a shaft experiencing constant loading
Determine the shaft rotational speed.
Determine the power or torque to be transmitted by the shaft.
Determine the dimensions of the power transmitting devices and other components mounted on the shaft and specify locations for each device.
Specify the locations of the bearings to support the shaft.
Propose a general form or scheme for the shaft geometry, considering how each component will be located axially and how power transmission will take place.
Determine the magnitude of the torques throughout the shaft.
Determine the forces exerted on the shaft.
Produce shearing force and bending moment diagrams so that the distribution of bending moments in the shaft can be determined.
Select a material for the shaft and specify any heat treatments etc.
Determine an appropriate design stress, taking into account the type of loading (whether smooth, shock, repeated, reversed).
Analyse all the critical points on the shaft and determine the minimum acceptable diameter at each point to ensure safe design.
Specify the final dimensions of the shaft. This is best achieved using a detailed manufacturing drawing, and the drawing should include all the information required to ensure the desired quality. Typically this will include material specifications, dimensions and tolerances (bilateral, runout, datums etc.;), surface finishes, material treatments and inspection procedures.
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29. Shaft design procedure (Simple Steps)
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30. Static Loading
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31. Static Loading
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32. ASME Design for transmission shaft
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33. Shigley’s Mechanical Engineering Design

34. General principles that should be observed in shaft design
Keep shafts as short as possible with the bearings close to applied loads. This will reduce shaft deflection and bending moments and increase critical speeds.
If possible, locate stress raisers away from highly stressed regions of the shaft. Use generous fillet radii and smooth surface finishes, and consider using local surface strengthening processes such cold rolling.
If weight is critical use hollow shafts.
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35. General principles that should be observed in shaft design
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36. Shaft Design for Stress
Stresses are only evaluated at critical locations
Critical locations are usually
On the outer surface
Where the bending moment is large
Where the torque is present
Where stress concentrations exist

37. Shaft Stresses
Standard stress equations can be customized for shafts for convenience
Axial loads are generally small and constant, so will be ignored in this section
Standard alternating and midrange stresses

38. Shaft Stresses
Customized for round shafts

39. Shaft Stresses
Combine stresses into von Mises stresses

40. Shaft Stresses
Substitute von Mises stresses into failure criteria equation. For example, using modified Goodman line,
Solving for d is convenient for design purposes

41. Shaft Stresses
Similar approach can be taken with any of the fatigue failure criteria
Equations are referred to by referencing both the Distortion Energy method of combining stresses and the fatigue failure locus name. For example, DE-Goodman, DE-Gerber, etc.
In analysis situation, can either use these customized equations for factor of safety, or can use standard approach from Ch. 6.
In design situation, customized equations for d are much more convenient.

42. Shaft Stresses
DE-Gerber
where
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43. Shaft Stresses
DE-ASME Elliptic
DE-Soderberg

44. Shaft Stresses for Rotating Shaft
For rotating shaft with steady bending and torsion
Bending stress is completely reversed, since a stress element on the surface cycles from equal tension to compression during each rotation
Torsional stress is steady
Previous equations simplify with Mm and Ta equal to 0

45. Checking for Yielding in Shafts
Always necessary to consider static failure, even in fatigue situation
Soderberg criteria inherently guards against yielding
ASME-Elliptic criteria takes yielding into account, but is not entirely conservative
Gerber and modified Goodman criteria require specific check for yielding

46. Checking for Yielding in Shafts
Use von Mises maximum stress to check for yielding,
Alternate simple check is to obtain conservative estimate of s'max by summing s'a and s'm

47. Example 7-1

48. Example 7-1

49. Example 7-1

50. Example 7-1

51. Example 7-1
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52. Estimating Stress Concentrations
Stress analysis for shafts is highly dependent on stress concentrations.
Stress concentrations depend on size specifications, which are not known the first time through a design process.
Standard shaft elements such as shoulders and keys have standard proportions, making it possible to estimate stress concentrations factors before determining actual sizes.

53. Estimating Stress Concentrations

54. Reducing Stress Concentration at Shoulder Fillet
Bearings often require relatively sharp fillet radius at shoulder
If such a shoulder is the location of the critical stress, some manufacturing techniques are available to reduce the stress concentration
Large radius undercut into shoulder
Large radius relief groove into back of shoulder
Large radius relief groove into small diameter of shaft
Fig. 7-9

55. Example 7-2

56. Example 7-2
Fig. 7-10

57. Example 7-2

58. Example 7-2

59. Example 7-2

60. Example 7-2

61. Example 7-2

62. Example 7-2

63. Example 7-2

64. Example 7-2

65. Example 7-2
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66. Example 7-2

67. Example 7-2

68. Example 7-2

69. Deflection Considerations
Deflection analysis at a single point of interest requires complete geometry information for the entire shaft.
For this reason, a common approach is to size critical locations for stress, then fill in reasonable size estimates for other locations, then perform deflection analysis.
Deflection of the shaft, both linear and angular, should be checked at gears and bearings.

70. Deflection Considerations
Allowable deflections at components will depend on the component manufacturer’s specifications.
Typical ranges are given in Table 7–2

71. Deflection Considerations
Deflection analysis is straightforward, but lengthy and tedious to carry out manually.
Each point of interest requires entirely new deflection analysis.
Consequently, shaft deflection analysis is almost always done with the assistance of software.
Options include specialized shaft software, general beam deflection software, and finite element analysis software.

72. Example 7-3
Fig. 7-10

73. Example 7-3

74. Example 7-3
Fig. 7-11

75. Example 7-3

76. Example 7-3

77. Adjusting Diameters for Allowable Deflections
If any deflection is larger than allowed, since I is proportional to d4, a new diameter can be found from
Similarly, for slopes,
Determine the largest dnew/dold ratio, then multiply all diameters by this ratio.
The tight constraint will be just right, and the others will be loose.

78. Example 7-4

79. Angular Deflection of Shafts
For stepped shaft with individual cylinder length li and torque Ti, the angular deflection can be estimated from
For constant torque throughout homogeneous material
Experimental evidence shows that these equations slightly underestimate the angular deflection.
Torsional stiffness of a stepped shaft is

80. Critical Speeds for Shafts
A shaft with mass has a critical speed at which its deflections become unstable.
Components attached to the shaft have an even lower critical speed than the shaft.
Designers should ensure that the lowest critical speed is at least twice the operating speed.

81. Critical Speeds for Shafts
For a simply supported shaft of uniform diameter, the first critical speed is
For an ensemble of attachments, Rayleigh’s method for lumped masses gives

82. Critical Speeds for Shafts
Eq. (7–23) can be applied to the shaft itself by partitioning the shaft into segments.
Fig. 7–12

83. Critical Speeds for Shafts
An influence coefficient is the transverse deflection at location I due to a unit load at location j.
From Table A–9–6 for a simply supported beam with a single unit load
Fig. 7–13

84. Critical Speeds for Shafts
Taking for example a simply supported shaft with three loads, the deflections corresponding to the location of each load is
If the forces are due only to centrifugal force due to the shaft mass,
Rearranging,

85. Critical Speeds for Shafts
Non-trivial solutions to this set of simultaneous equations will exist when its determinant equals zero.
Expanding the determinant,
Eq. (7–27) can be written in terms of its three roots as

86. Critical Speeds for Shafts
Comparing Eqs. (7–27) and (7–28),
Define wii as the critical speed if mi is acting alone. From Eq. (7–29),
Thus, Eq. (7–29) can be rewritten as

87. Critical Speeds for Shafts
Note that
The first critical speed can be approximated from Eq. (7–30) as
Extending this idea to an n-body shaft, we obtain Dunkerley’s equation,

88. Critical Speeds for Shafts
Since Dunkerley’s equation has loads appearing in the equation, it follows that if each load could be placed at some convenient location transformed into an equivalent load, then the critical speed of an array of loads could be found by summing the equivalent loads, all placed at a single convenient location.
For the load at station 1, placed at the center of the span, the equivalent load is found from

89. Example 7-5
Fig. 7–14

90. Example 7-5

91. Example 7-5
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92. Example 7-5

93. Example 7-5
Fig. 7–14

94. Example 7-5

95. Example 7-5

96. Setscrews resist axial and rotational motion
They apply a compressive force to create friction
The tip of the set screw may also provide a slight penetration
Various tips are available
Fig. 7–15

97. Setscrews
Resistance to axial motion of collar or hub relative to shaft is called holding power
Typical values listed in Table 7–4 apply to axial and torsional resistance
Typical factors of safety are 1.5 to 2.0 for static, and 4 to 8 for dynamic loads
Length should be about half the shaft diameter

98. Used to secure rotating elements and to transmit torque
Keys and Pins
Used to secure rotating elements and to transmit torque
Fig. 7–16

99. Taper pins are sized by diameter at large end Small end diameter is
Tapered Pins
Taper pins are sized by diameter at large end
Small end diameter is
Table 7–5 shows some standard sizes in inches
Table 7–5

100. Keys come in standard square and rectangular sizes
Shaft diameter determines key size
Table 7–6

101. Keys
Failure of keys is by either direct shear or bearing stress
Key length is designed to provide desired factor of safety
Factor of safety should not be excessive, so the inexpensive key is the weak link
Key length is limited to hub length
Key length should not exceed 1.5 times shaft diameter to avoid problems from twisting
Multiple keys may be used to carry greater torque, typically oriented 90º from one another
Stock key material is typically low carbon cold-rolled steel, with dimensions slightly under the nominal dimensions to easily fit end-milled keyway
A setscrew is sometimes used with a key for axial positioning, and to minimize rotational backlash

102. Head makes removal easy Projection of head may be hazardous
Gib-head Key
Gib-head key is tapered so that when firmly driven it prevents axial motion
Head makes removal easy
Projection of head may be hazardous
Fig. 7–17
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103. Woodruff keys have deeper penetration
Useful for smaller shafts to prevent key from rolling
When used near a shoulder, the keyway stress concentration interferes less with shoulder than square keyway
Fig. 7–17

104. Woodruff Key
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105. Woodruff Key

106. Stress Concentration Factors for Keys
For keyseats cut by standard end-mill cutters, with a ratio of r/d = 0.02, Peterson’s charts give
Kt = 2.14 for bending
Kt = 2.62 for torsion without the key in place
Kt = 3.0 for torsion with the key in place
Keeping the end of the keyseat at least a distance of d/10 from the shoulder fillet will prevent the two stress concentrations from combining.

107. Example 7-6
Fig. 7–19

108. Example 7-6

109. Example 7-6

110. Retaining Rings
Retaining rings are often used instead of a shoulder to provide axial positioning
Fig. 7–18

111. This requires sharp radius in bottom of groove.
Retaining Rings
Retaining ring must seat well in bottom of groove to support axial loads against the sides of the groove.
This requires sharp radius in bottom of groove.
Stress concentrations for flat-bottomed grooves are available in Table A–15–16 and A–15–17.
Typical stress concentration factors are high, around 5 for bending and axial, and 3 for torsion
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112. Limits and Fits
Shaft diameters need to be sized to “fit” the shaft components (e.g. gears, bearings, etc.)
Need ease of assembly
Need minimum slop
May need to transmit torque through press fit
Shaft design requires only nominal shaft diameters
Precise dimensions, including tolerances, are necessary to finalize design

113. Tolerances
Close tolerances generally increase cost
Require additional processing steps
Require additional inspection
Require machines with lower production rates
Fig. 1–2

114. Standards for Limits and Fits
Two standards for limits and fits in use in United States
U.S. Customary (1967)
Metric (1978)
Metric version will be presented, with a set of inch conversions

115. Nomenclature for Cylindrical Fit
Upper case letters refer to hole
Lower case letters refer to shaft
Basic size is the nominal diameter and is same for both parts, D=d
Tolerance is the difference between maximum and minimum size
Deviation is the difference between a size and the basic size
Fig. 7–20

116. Tolerance Grade Number
Tolerance is the difference between maximum and minimum size
International tolerance grade numbers designate groups of tolerances such that the tolerances for a particular IT number have the same relative level of accuracy but vary depending on the basic size
IT grades range from IT0 to IT16, but only IT6 to IT11 are generally needed
Specifications for IT grades are listed in Table A–11 for metric series and A–13 for inch series

117. Tolerance Grades – Metric Series
Table A–11

118. Tolerance Grades – Inch Series
Table A–13

119. Deviations
Deviation is the algebraic difference between a size and the basic size
Upper deviation is the algebraic difference between the maximum limit and the basic size
Lower deviation is the algebraic difference between the minimum limit and the basic size
Fundamental deviation is either the upper or lower deviation that is closer to the basic size
Letter codes are used to designate a similar level of clearance or interference for different basic sizes

120. Fundamental Deviation Letter Codes
Shafts with clearance fits
Letter codes c, d, f, g, and h
Upper deviation = fundamental deviation
Lower deviation = upper deviation – tolerance grade
Shafts with transition or interference fits
Letter codes k, n, p, s, and u
Lower deviation = fundamental deviation
Upper deviation = lower deviation + tolerance grade
Hole
The standard is a hole based standard, so letter code H is always used for the hole
Lower deviation = 0 (Therefore Dmin = 0)
Upper deviation = tolerance grade
Fundamental deviations for letter codes are shown in Table A–12 for metric series and A–14 for inch series

121. Fundamental Deviations – Metric series
Table A–12

122. Fundamental Deviations – Inch series
Table A–14

123. Specification of Fit
A particular fit is specified by giving the basic size followed by letter code and IT grades for hole and shaft.
For example, a sliding fit of a nominally 32 mm diameter shaft and hub would be specified as 32H7/g6
This indicates
32 mm basic size
hole with IT grade of 7 (look up tolerance DD in Table A–11)
shaft with fundamental deviation specified by letter code g (look up fundamental deviation dF in Table A–12)
shaft with IT grade of 6 (look up tolerance Dd in Table A–11)
Appropriate letter codes and IT grades for common fits are given in Table 7–9

124. Description of Preferred Fits (Clearance)
Table 7–9

125. Description of Preferred Fits (Transition & Interference)
Table 7–9

126. Procedure to Size for Specified Fit
Select description of desired fit from Table 7–9.
Obtain letter codes and IT grades from symbol for desired fit from Table 7–9
Use Table A–11 (metric) or A–13 (inch) with IT grade numbers to obtain DD for hole and Dd for shaft
Use Table A–12 (metric) or A–14 (inch) with shaft letter code to obtain dF for shaft
For hole
For shafts with clearance fits c, d, f, g, and h
For shafts with interference fits k, n, p, s, and u

127. Example 7-7

128. Example 7-8

129. Stress in Interference Fits
Interference fit generates pressure at interface
Need to ensure stresses are acceptable
Treat shaft as cylinder with uniform external pressure
Treat hub as hollow cylinder with uniform internal pressure
These situations were developed in Ch. 3 and will be adapted here

130. Stress in Interference Fits
The pressure at the interface, from Eq. (3–56) converted into terms of diameters,
If both members are of the same material,
d is diametral interference
Taking into account the tolerances,

131. Stress in Interference Fits
From Eqs. (3–58) and (3–59), with radii converted to diameters, the tangential stresses at the interface are
The radial stresses at the interface are simply
The tangential and radial stresses are orthogonal and can be combined using a failure theory

132. Torque Transmission from Interference Fit
Estimate the torque that can be transmitted through interference fit by friction analysis at interface
Use the minimum interference to determine the minimum pressure to find the maximum torque that the joint should be expected to transmit.