Chapter 9

Shaft Design

Transmission shafts transmit torque from one location to anotherSpindles are short shaftsAxles are non-rotating shafts

Figure 9.1 is an example of a shaft with several features. It is a shaft for a Caterpillar tractor transmission1.

Figure 9.1: Example of a typical shaft design

1From Frederick E. Giesecke, Technical Drawing, Chapter 13.

1

9.1 Shaft Loads

• Torsion due to transmitted torque• Bending from transverse loads (gears, sprockets, pulleys/sheaves)∗

o * a pulley and a sheave are essentially the same thingSteady or FluctuatingSteady transverse-bending load ♦ fully reversing bending stress (fatigue failure)

9.2 Attachments and Stress Concentrations

Steps and shoulders are used to locate attachment (gears, sheaves, sprockets)Keys, snap rings, cross pins (shear pins), tapered pinsUse generous radii to reduce stress concentrationsClamp collarsSplit collarPress fits and shrink fitsBearings may be located by the use of snap rings, but only one bearing is fixedIssues - axial location, disassembly, and element phasing (e.g., alignment of gear teeth for timing)

MACHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000

F I G U R E 9 - 2

Various Methods to Attach Elements to Shafts

shaft

snap ring

sprocket gear

clampcollar

bearing

press fit

key

sheave

taper pin

bearing

stepstep

step

press fit

step

hub hub

frame frame

axialclearance

Figure 9.2: Example of a shaft with various attachments and details

9.3 Shaft Materials

• Steel (low to medium-carbon steel)• Cast iron• Bronze or stainless steel• Case hardened steel

2

9.3.1 Shaft Power

Power is the time rate of change of energy (work).work = Force * distance or Torque * angle, so Power = Torque * angular velocity

Pwr = Torq ∗ ω (9.1)

9.4 Shaft Loading, Approaches to Analysis

Most general form - A fluctuating torque and a fluctuating moment, in combination.If there are axial loads, they should be “taken to ground” as close to the load as possible.Given knowledge of the moments and the torques (i.e., mean and alternating components) Use the “DesignSteps for Fluctuating Stresses” in Section 6.11 in combination with the multiaxial-stress issues addressed inSection 6.12.

9.5 Shaft Stresses

Bending Stress

σalt = kfMac

I(9.2)

σmean = kfmMmc

I(9.3)

Torsional Shear Stress

τalt = kfsTar

J(9.4)

τmean = kfsmTmr

J(9.5)

9.5.1 Shaft Failure in Combined Loading

9.6 Shaft Design

9.6.1 General Considerations

1. To minimize both deflections and stresses, the shaft length should be kept as short as possible andoverhangs minimized.

2. A cantilever beam will have a larger deflection than a simply supported (straddle mounted) one for thesame length, load, and cross section, so straddle mounting should be used unless a cantilever shaft isdictated by design constraints. (Figure 9-2 shows a situation in which an overhung section is requiredfor serviceability.)

3. A hollow shaft has a better stiffness/mass ratio (specific stiffness) and higher natural frequencies thana comparably stiff or strong solid shaft, but will be more expensive and larger in diameter.

4. Try to locate stress-raisers away from regions of large bending moment if possible and minimize theireffects with generous radii and relief.

5. General low carbon steel is just as good as higher strength steels (since deflection is typical the designlimiting issue).

6. Deflections at gears carried on the shaft should not exceed about 0.005 inches and the relative slopebetween the gears axes should be less than about 0.03 degrees.

3

MACHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton, Prentice-Hall 2000

σ τa

e

a

esS S

+

=2 2

1σ τa

e

m

ysS S

+

=

2 2

1

τm

ysS

τa

esS

σa eS σa eS

from ref. 2

from ref. 3

from ref. 3

(a) Combined stress fatigue-test data for reversed bending combined with static torsion (from ref. 4)

(b) Combined stress fatigue-test data for reversed bending combined with reversed torsion (from ref. 5)

F I G U R E 9 - 3

Results of Fatigue Tests of Steel Specimens Subjected to Combined Bending and Torsion (From Design of Transmission Shafting, American Society of Mechanical Engineers, New York, ANSI/ASME Standard B106.1M-1985, with permission)

Figure 9.3: Shaft failure in combined loading

4

7. If plain (sleeve) bearings are to be used, the shaft deflection across the bearing length should be lessthan the oil-film thickness in the bearing.

8. If non-self-aligning rolling element bearings are used, the shaft’s slope at the bearings should be keptto less than about 0.04 degrees.

9. If axial thrust loads are present, they should be taken to ground through a single thrust bearing perload direction. Do not split axial loads between thrust bearings as thermal expansion of the shaft canoverload the bearings.

10. The first natural frequency of the shaft should be at least three times the highest forcing frequencyexpected in service, and preferably much more. (A factor of ten times or more is preferred, but this isoften difficult to achieve).

Designing for Fully Reversed Bending and Steady Torsion

ASME Method (ANSI/ASME Standard for Design of Transmission Shafting B106.1M-1985.Uses the elliptical curve of Figure 9-3.Equations 9.5e and 9.6a,b.9.6 can be applied only for• constant torque• fully reversed moment.• No axial load

d = 3

√√√√32SafetyFactor

π

√(kf

Ma

Sf)2 +

34(Tm

Sy)2 (9.6)

More general loading cases require Equation 9.8.See Example 9..

9.6.2 Shaft Deflection

Deflection is often the more demanding constraint. Many shafts are well within specification for stress butwould exhibit too much deflection to be appropriate.

9.6.3 Keys and Keyways

P9-03.pdf Shaft Design for Problems 9-6, 9-9. 9-11, and 9-12

F I G U R E P 9 - 3

P

bbearings are self-aligningso act as simple supports

a

l

T

dgear

Figure 9.4: Shaft with overhung gear

Example -Homework Problem 9-2

5

9.6.4 Splines

9.6.5 Interference Fits

Components can be attached to a shaft without a key or spline by using an interference fit.There are two methods used to assemble these components:

• press fit

• shrink (and/or expansion) fit

The amount of interference is importantThe analysis of interference follows from the equations for pressure on thick-walled cylinders.A rule of thumb that is used is one to two thousands of diametral interference per unit of shaft diameter,e.g., a shaft of two inch diameter would have 0.004 inches of interference with an attached gear hub.Machinists use a simplified approach to this – 1/1000 of interference for each inch of diameter.

However, there is a formal approachStandards have been developed for these fits.

Metric Preferred Metric Limits and Fits — ANSI B4.2-1978.

US Customary Preferred Limits and Fits for Cylindrical Parts —ANSI B4.1-1967

9.7 Terms related to Fits and Tolerances ANSI B4.2-1978

9.7.1 Definitions

D – basic size of the holed – basic size of the shaftδu – upper deviationδl – lower deviationδF – Fundamental deviation∆D – tolerance grade for the hole∆d – tolerance grade for the shaftTolerance – the difference between the maximum and minimum size limits of the dimensions of a partNatural tolerance – a tolerance equal to ± three standard deviations from the meanClearance – amount of space between an internal and external memberInterference – the amount of overlap between an internal and external memberInternational Tolerance Grade Numbers (IT) – designate groups of tolerances such that the tolerances for aparticular IT number have the same relative level of accuracy, i.e., IT 9Smaller numbers mean tighter tolerances, IT 6 through IT 11 are used for preferred fits.For a 32 mm hole we might use 32H7

• The H establishes the fundamental deviation and the number 7 defines a tolerance grade of IT7. Thegrade number specifies a tolerance zone.

For the mating shaft we might have 32g6

9.7.2 Table of Tolerance Grades 2

Lower and Upper Deviations

• For shaft letter codes c, d, f, g, and h2Shigley Table E-11, page 1188.

6

Table 9.1: International Tolerance Grades

Basic SizesAll values in mm Tolerance Grades

A < d ≤ B IT6 IT7 IT8 IT9 IT10 IT110-3 0.006 0.010 0.014 0.025 0.040 0.0603-6 0.008 0.012 0.018 0.030 0.048 0.075

6-10 0.009 0.015 0.022 0.036 0.058 0.09010-18 0.011 0.018 0.027 0.043 0.070 0.11018-30 0.013 0.021 0.033 0.052 0.084 0.13030-50 0.016 0.025 0.039 0.062 0.100 0.16050-80 0.019 0.030 0.046 0.074 0.120 0.190

80-120 0.022 0.035 0.054 0.087 0.140 0.220120-180 0.025 0.040 0.063 0.100 0.160 0.250180-250 0.029 0.046 0.072 0.115 0.185 0.290250-315 0.032 0.052 0.081 0.130 0.210 0.320315-400 0.036 0.057 0.089 0.140 0.230 0.360

– Upper deviation = fundamental deviation

– Lower deviation = upper deviation – tolerance grade

• For shaft letter codes k, n, p ,s, and u

– Lower deviation = fundamental deviation

– Upper deviation = lower deviation + tolerance grade

• Hole letter code is H

– Lower deviation = 0

– Upper deviation = tolerance grade

9.7.3 Fundamental Deviations for Shafts – Metric Series

These are related to the tolerance grades. See the table below. Capital letters always refer to the hole (orbore) and lowercase letters are used for the shaft.

9.7.4 Fit Types

Table 9.3 provides a linguistic description for commonly used references to fit types.

9.8 Flywheel Design

One of the biggest issues with regard to flywheels is balancing. Because they are, by intention, devices withlarge inertias, balancing them to remove eccentric loading and thus lower the loading on bearings and othercomponents is very important.Flywheels develop large stresses at their inter hub connection due to dynamic forces caused by the spinning.These stresses can lead to failure. Careful design is required to avoid catastrophic failure.

9.9 Critical Speeds

There are three types of vibration that are encountered with shafts:

7

Table 9.2: Fundamental Deviations for Shafts Metric Seriesbasic

dimension Clearance Transition InterferenceA < d ≤ B Upper Deviation Letter Lower-Deviation Letter

c d f g h k n p s u0-3 -0.060 -0.020 -0.006 -0.002 0 0 +0.004 +0.006 +0.014 +0.0183-6 -0.070 -0.030 -0.010 -0.004 0 +0.001 +0.008 +0.012 +0.019 +0.023

6-10 -0.080 -0.040 -0.013 -0.005 0 +0.001 +0.010 +0.015 +0.023 +0.02810-14 -0.095 -0.050 -0.016 -0.006 0 +0.001 +0.012 +0.018 +0.028 +0.03314-18 -0.095 -0.050 -0.016 -0.006 0 +0.001 +0.012 +0.018 +0.028 +0.03318-24 -0.110 -0.065 -0.020 -0.007 0 +0.002 +0.015 +0.022 +0.035 +0.04124-30 -0.110 -0.065 -0.020 -0.007 0 +0.002 +0.015 +0.022 +0.035 +0.04830-40 -0.120 -0.080 -0.025 -0.009 0 +0.002 +0.017 +0.026 +0.043 +0.06040-50 -0.130 -0.080 -0.025 -0.009 0 +0.002 +0.017 +0.026 +0.043 +0.07050-65 -0.140 -0.100 -0.030 -0.010 0 +0.002 +0.020 +0.032 +0.053 +0.08765-80 -0.150 -0.100 -0.030 -0.010 0 +0.002 +0.020 +0.032 +0.059 +0.102

80-100 -0.170 -0.120 -0.030 -0.012 0 +0.003 +0.023 +0.037 +0.071 +0.124100-120 -0.180 -0.120 -0.036 -0.012 0 +0.003 +0.023 +0.037 +0.079 +0.144120-140 -0.200 -0.145 -0.043 -0.014 0 +0.003 +0.027 +0.043 +0.092 +0.170140-160 -0.210 -0.145 -0.043 -0.014 0 +0.003 +0.027 +0.043 +0.100 +0.190160-180 -0.230 -0.145 -0.043 -0.014 0 +0.003 +0.027 +0.043 +0.108 +0.210180-200 -0.240 -0.170 -0.050 -0.015 0 +0.004 +0.031 +0.050 +0.122 +0.236200-225 -0.260 -0.170 -0.050 -0.015 0 +0.004 +0.031 +0.050 +0.130 +0.258225-250 -0.280 -0.170 -0.050 -0.015 0 +0.004 +0.031 +0.050 +0.140 +0.284250-280 -0.300 -0.190 -0.056 -0.017 0 +0.004 +0.034 +0.056 +0.158 +0.315280-315 -0.330 -0.190 -0.056 -0.017 0 +0.004 +0.034 +0.056 +0.170 +0.350315-355 -0.360 -0.210 -0.062 -0.018 0 +0.004 +0.037 +0.062 +0.190 +0.390355-400 -0.400 -0.210 -0.062 -0.018 0 +0.004 +0.037 +0.062 +0.208 +0.435

8

Table 9.3: Fit Types and their descriptionType of fit Reference Description SymbolClearance Loose running fit For wide commercial tolerances or al-

lowances on external membersH11/c11

Free running fit Not for use where accuracy is essential,but good for large temperature varia-tions, high running speeds, or heavyjournal pressures

H9/d9

Close running fit For running on accurate machinesand for accurate location at moderatespeeds and journal pressures

H8/f8

Sliding fit Where parts are not intended to runfreely, but must move and turn freelyand locate accurately

H7/g6

Locational clearance fit(snug fit)

Provides snug fit for location of station-ary parts, but can be freely assembledand disassembled

H7/h6

Transition Locational transitional fit For accurate location, a compromise be-tween clearance and interference

H7/k6

Locational transitional fit(wringing fit)

For more accurate location wheregreater interference is permissible

H7/n6

Interference Locational transitional fit(tight fit)

For parts requiring rigidity and align-ment with prime accuracy of locationbut without special bore pressure re-quirements

H7/p6

Medium Drive Fit For ordinary steel parts or shrink fitson light sections, the tightest fit usablewith cast iron

H7/s6

Force Fit Suitable for parts which can be highlystressed or for shrink fits where theheavy pressing forces required are im-practical

H7/u6

• Lateral vibration

• Shaft whirl

• Torsional vibration

9.10 Couplings

Many applications require us to connect one shaft to another axially. This is done with the use of couplings.Note that the possibility of getting the two shafts perfectly aligned (linearly and angularly) is essentiallyzero, so couplings are typically designed to accomodate some misalignment. Couplings come in many shapes,sizes, and degrees of misalignment. One type of coupling you might be familiar with is the universal joint,see Figure 9.5. A recent inovation used with front wheel drive is the CV (constant velocity) joint.Another type used widely for connections to electric motors is a flexible coupling, see Figure 9.6.

9.11 Summary

While shafting can be purchased as a stock item, most applications require some customization of thelayout and dimensioning to accommodate the attachment of components and bearings. Almost all shafts are

9

Figure 9.5: Typical automotive universal joint

Figure 9.6: Small flexible couplings

designed for high cycle fatigue (HCF), and are made of steel, since it has an fatigue limit. One is cautionedto applied the shaft diameter design equations presented in Norton (Equation 9.6 & Equation 9.8) properlysince specific requirements must be met to apply these equations.Many other factors come into play during the shaft design process. These may include:

• keyways and keys

• splines

• couplings

• shaft vibrations and balancing

• flywheels

10