Ansi Agma6001 d97

AhW/AGMA 6001-097 (Revision of ANWAGMA 6001 -C88)

AMERICAN NATIONAL STANDARD

Design and Selection of Component. for Enclosed Gear Drives

AGMA STANDARD -t =s= Reproduced By GLOBAL -

= - ENGINEERING DOCUMENTS -- w Z With The Permission Of AGMA -- g - Under Royalty Agreement

American Design and Selection of Components for Enclosed Gear Drives Nationa, A$!/fGMA 6001-D97

evrsron of ANWAGMA 6001 -C88]

Standard Approval of an American National Standard requires verification by ANSI that the requirements for due process, consensus, and other criteria for approval have been met by the standards developer.

Consensus is established when, in the judgment of the ANSI Board of Standards Review, substantial agreement has been reached by directly and materially affected interests. Substantial agreement means much more than a simple majority, but not necessarily una- nimity. Consensus requires that all views and objections be considered, and that a concerted effort be made toward their resolution. The use of American National Standards is completely voluntary; their existence does not in any respect preclude anyone, whether he has approved the standards or not, from manufacturing, marketing, purchasing, or using products, processes, or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard. Moreover, no person shall have the right or authorii to issue an interpretation of an American National Standard in the name of the American National Standards Institute. Requests for interpretation of this standard should be addressed to the American Gear Manufacturers Association.

CAUTION NOTICE: AGMA technical publications are subject to constant improvement, revision, or withdrawal as dictated by experience. Any person who refers to any AGMA Technical Publication should be sure that the publication is the latest available from the Association on the subject matter.

jTables and other self-supporting sections may be quoted or extracted in their entirety. Credit lines should read: Extracted from AGMA 6001 -D97, Design and Se/e&ion of Com- ponents forEnclosed Gear Drives, with permission of the publisher, American Gear Manu- facturers Association, 1500 King Street, Suite 201, Alexandria, Virginia, 22314.1

Approved August 7,1997

ABSTRACT This standard outlines the basic practices for the design and selection of components, other than gearing, for use in commercial and industrial enclosed gear drives.

Published by

American Gear Manufacturers Association 1500 King Street, Suite 201, Alexandria, Virginia 22314

Copyright 0 1997 by American Gear Manufacturers Association All rights reserved.

No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without prior written permission of the publisher.

Printed in the United States of America

ISBN: l-55589-883-9

ii

AMERICAN NATIONAL STANDARD ANSIIAGMA 6001-D97

Contents Page

Foreword ............................................................... iv 1 scope .............................................................. 1 2 Definitionsandsymbols ............................................... 1 3 Designconditions .................................................... 1 4 Shafts .............................................................. 4 5 Keys .............................................................. 19 6 Bearings ........................................................... 20 7 Housings .......................................................... 21 8 Threaded fasteners. ................................................. 22 9 M iscellaneous components ........................................... 22

Tables

1 Symbolsusedinequations ............................................ 2 2 Modifying factor for stress concentration, 4 - typical values for keyways in

solid round steel shafts .............................................. 14

Figures

1 Design criteria ....................................................... 5 2 Cyclicloading ....................................................... 7 3 Stress convention showing orbiting element. ............................. 7 4 Surface finish factor, & ............................................... IO 5 Sizefactor,kb ....................................................... 11 6 Reliability factor, k, .................................................. 11 7 Notch sensitivity-steel, 4 ............................................ 12 8 Theoretical stress concentration factor in bending for a circular shaft with a

square shoulder, & (nominal stress is calculated at diameter 4) ........... 13 9 Theoretical stress concentration factor in bending for a circular shaft with

a u-notch, & (nominal stress is calculated at diameter 4) ................ 13 10 Theoretical stress concentration factor in bending for a circular shaft with a

radial hole, & (based on full section without considering hole) ............. 14 11 Torsional deformation ................................................ 15 12 Bending deflection intermediate concentrated load ....................... 16 13 Bending deflection overhung concentrated load ......................... 17 14 Bending deflection intermediate concentrated moment ................... 17 15 Bending deflection overhung concentrated moment ...................... 18 16 Axial deformation ................................................... 18 17 Average shaft and hub radius ......................................... 19 18 Variation of coefficient of friction versus the bearing parameter ............. 21

Annexes

A Allowable stresses for typical key and keyway materials .................. 25 B Allowable stresses for typical threaded fasteners ........................ 27 C Interference fit torque capacity ........................................ 29 D Previous method - shaft design ....................................... 31 E Sample problems - transmission shaft design ........................... 33 F Sample problems - deflection ......................................... 37 G References. ........................................................ 41

. . . III

ANSVAGMA 6091 -D97 AMERICAN NATIONAL STANDARD

Foreword

jThe foreword, footnotes, and annexes, if any, in this document are provided for informational purposes only and are not to be construed as a part of ANSVAGMA Standard 6001-097, Design and Selection of Components for Enclosed Gear Drives.] AGMA 260.02 was approved by the AGMA membership on February 1,1973, and issued in January of 1974. It consolidated with m inor revision, information contained in the following superseded AGMA Standards:

AGMA 255.02 (November 1964), Bolting (Allowable Tensile Stress) for Gear Drives;

AGMA 260.01 (March 1953), Shafting -Allowable Torsional and Sending Stresses; ,

AGMA 260.02 also incorporated allowable stresses for keys;

AGMA 265.01 Bearings -Allowable Loads and Speeds.

The purpose of AGMA 6001 -C88, as a replacement for AGMA 260.02, was to establish a common base for the design and selection of components for the different types of commercial and industrial gear drives.

AGMA 6001 -C88 was expanded to include a generalized shaft stress equation which included hollow shafting, m iscellaneous components, housings, and keyway stress calculations. All design considerations were revised to allow for 200 percent peak load for helical, spiral bevel, spur and herringbone gearing, and 300 percent peak load for wormgearing. The bearing section was updated to include consideration of life adjustment factors, bearing lives other than 5000 hours and reliability levels other than Ll 0.

During the preparation of AGMA 6001 -C88, a considerable amount of time was spent on the shaft design section in an effort to include the most recent theories on shaft stresses and material characteristics. The standard included the existing practice for shaft design, and for reference purposes, appendix C included a description of, and excerpts from, ANSI/ASME B106.1 M , Design of ‘liansmssjon Shafting, published in 1985.

AGMA 6001 -C88 was approved by the membership in May 1988 and approved as an American National Standard on June 24, 1988.

This revision, AGMA 6001 -D97, has been expanded to include more recent theories on shaft design and analysis. Also, equations for shaft deformation were added. AGMA 6001 -D97 was approved by the membership in October 1996 and approved as an American National Standard on August 7,1997.

Suggestions for improvement of this standard will be welcome. They should be sent to the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314. -

iv

AMERICAN NATlONAL STANDARD ANSI/AGMA 6001 -D97

PERSONNEL of the AGMA Component Design Committee

Chairman: D. McCarthy ......................... Dorris Company Vice Chairman: D. Cressman .................... Philadelphia Mixers Corporation

ACTIVE MEMBERS

R. Errichello ................................... GEARTECH J.B. Hagaman .................................. Cone Drive Operations, Inc. R. Holzman .................................... Milwaukee Gear Company, Inc. J. Lisiecki ...................................... The Falk Corporation D.R. McVie ................................... Gear Engineers, Inc. K. Newton ..................................... Rockwell Automation/Dodge W.F. Schierenbeck .............................. Xtek, Inc. R.G.Smith ..................................... Philadelphia Gear Corporation R. Tameja ..................................... Peerless-Winsmith, Inc. F.C. Uherek .................................... Flender Corporation J.J. Vielhauer .................................. The Cincinnati Gear Company

D.Behlke......................................TwinDisc,Inc. R.E. Brown .................................... Caterpillar, Inc. R.Z. Johnston .................................. University of Maine S. Miller ....................................... The Cincinnati Gear Company C. Mischke .................................... Iowa State University A.E. Phillips .................................... Rockwell Automation/Dodge A. Williston ..................................... Dorris Company

V

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AMERICAN NATIONAL STANDARD ANSIIAGMA 6901 -D97

American National Standard -

Design and Selection of Components for Enclosed Gear Drives

1 Scope

This standard provides an acceptable practice for the design and selection of components for enclosed gear drives. Fundamental equations provide for the proper sizing of shafts, keys, and fasteners based on stated allowable stresses. Other components are discussed in a manner to provide an awareness of their function or specific requirements. This standard applies to the following types of commercial and industrial enclosed gear drives, individually or in combination: spur, helical, herringbone, bevel and worm.

1 .l Exceptions

The equations in this standard are not applicable when gear drives are subjected to vibratory conditions where there ,may be unpredictable fatigue failure.

The procedure for design or selection of the specific gear components is varied and complex and is beyond the scope of this standard. Designers must refer to the specific. rating or enclosed drive standards for this aspect of drive design.

1.2 Intended use

The equations and values presented provide a general approach to design. Deviations from the methods and values stated in this standard may be made when justified by experience, testing, or more specific analysis. It is intended for use by experienced gear designers capable of selecting reasonable values based on their knowledge of the performance of similar designs and the effect of such items as lubrication, deflection, manufacturing toler-

ances, metallurgy, residual stresses, and system dynamics. It is not intended for use by the engineering public at large.

2 Definitions and symbols

The symbols and definitions used in this standard may differ from those in other AGMA standards. The user should not assume that familiar symbols can be used without a careful study of the applicable section(s) and equation(s).

2.1 Definitions

The terms used, wherever applicable, conform to the following standards:

AGMA 904-C96, Metrk Usage

ANSI Y10.3-1968, Letter Symbols for Quantities Used in Mechanics of Solids

ANSIIAGMA 1012-F90, Gear Nomenclature, Definitions of Terms with Symbols

2.2 Symbols

The symbols used in this standard are shown in table 1.

SI units of measure are shown in parentheses in table 1 and in the text. Where equations require a different format or constant for use with SI units, a second expression is shown after the first, indented, in smallertype, and with “M” included in the equation number.

Example:

%e = Wf FP

0.785

%e = WfFP

0.785(0 - 0.9382P)2

. ..(70)

. ..(70M)

The second expression uses SI units.

3 Design conditions

This standard should be used in conjunction with appropriate current AGMA standards. When the

1

ANSI/AGMA 6001 -D97

operating conditions are known, each component of the drive shall be designed to meet those conditions. When operating conditions are not known, all load carrying components of the drive shall be designed to support the stated mechanical rating of the drive for continuous duty based on a unity service factor (1 .O). External loads must be considered as acting in

Symbol A A, 4 4s a B C

D de 4 4 E F

FP

fif fiP F ya G HB h Z J JIZ K Kt 4, k

: k

2 9

tr L, M m N

directions and rotations producing the most unfavorable stresses unless more specific information is available. Due allowances must be made for peak loads.

For enclosed drives designed to operate under specific conditions such as load, speed, duty cycle and life, components may be selected accordingly.

Table 1 - Symbols used in equations

Coefficient Term

Compressive area of key in keyway Shear area Cross sectional area Distance from support to concentrated load Coefficient Coefficient Fastener nominal diameter Shaft diameter adjacent to section being analyzed Shaft inside diameter Shaft outside diameter Modulus of elasticity Concentrated load Peak load factor Fatigue safety factor Peak load safety factor Allowable stress to yield strength factor Modulus of rigidity Brine11 hardness number Radial step Second area moment of cross section Second polar moment of area Second polar moment of area of nth section of shaft Constant Theoretical stress concentration factor in bending Fastener torque coefficient Fatigue strength modification factor Surface finish factor Size factor Reliability factor Temperature factor Lie factor Modifying factor for stress concentration M iscellaneous effects factor Length of shaft Length of the nth section of shaft Bending moment Coefficient Number of stress cycles

Units

in2 (mm2) in2 (mm? in2 (mm? in (mm)

in (mm) in (mm) in (mm) in (mm)

lb/in2 (N/mm? lb (NJ

b/is (N/mm2) HB

in (mm) in4 (mm4) in4 (mm4) in4 (mm4)

in (mm) in (mm)

lb in (Nm)

First referenced

Fig 4 Eq 66 Eq 69 Eq 63 Eq 46 Fig 4 Eq 37 Eq 70 Fig a Eq6 Eq6 Eq 46 Eq 46 Eq5 Eq 1 Eq2 Eq5 Eq 41 Eq 30 Fig 8 Eq 46 Eq 41 EqM ma Eq 38 Eq 71 Eqa Eq 35 Eq= Eq 35 Eq 35 Eq 35 Eq 35 Eq 35 Eq 41 Eq 44 Eq 7 Eq 37 Eq 37

(continued

2

AMERICAN NATIONAL STANDARD ANSI/AGMA 6001 -D97

Table 1 (continued) First

Symbol Term Units referenced Nfo Permissible number of momentary peak load cycles Eq 39 It Fastener threads per inch in-f Eq 70 P Fastener thread pitch (mm) Eq 70M px Axial force lb N Eq 10 4 Notch sensitivity Eq 38 R Reliability (survival rate) Eq 36 % Surface finish lrin (w) Fig 4 r Notch radius, fillet radius, hole radius in (mm) Fig 7 *a Average radius along the key length in (mm) Eq 69 kc Average radius at compressive load area in (mm) Eq 68 &a Allowable compressive stress lb/k? (N/mm? Eq 66

; Sf at 1 O6 stress cycles lb/is (N/mm? Eq 37 Modified fatigue strength (endurance limit) lb/is (N/mm? Eq 3

% Basic fatigue strength (endurance limit) of polished, unnotched lb/in2 (N/mm2) Eq 32 test specimen in reverse bending

SP Fastener proof load stress lb/is (N/mm? Eq 72 s sa Allowable shear stress lb/is (N/mm9 Eq 67

z Ultimate tensile strength lb/k? (N/mm? Eq 30 Tensile yield strength lb/in2 (N/mm2) Eq 3

s, Calculated compressive stress lb/in2 (N/mm? Eq 68 %k Calculated key shear stress lb/in2 (N/mm2) Eq 69 St, Calculated tensile stress in fastener lb/in2 (N/mm2) Eq 70 T Torque lb in (Nm) Eq 6 7 Fastener torque lb in (Nm) Eq 71 V Transverse shear force lb 0’0 Eq 8 Wf Applied tensile load on fastener lb 0’4 Eq 70 Y@ Fastener tensile preload lb V’J) Eq 71 x Distance from support to cross section in (mm) Eq 46 Y Deflection of shaft in (mm) Eq 46 6 Elongation in (mm) Eq 63 8 Angular position of shaft element radians Fig 3 06 Shaft slope radians Eq 47 ec Critical stress angle radians Eq 28 e, Angle of twist radians Eq 41 v Poisson’s ratio 4 9 oa Alternating component of stress lb/in2 (N/mm2) Eq 12 0, Alternating axial normal stress lb/k? (N/mm2) Eq 13 % Alternating radial normal stress lb/in2 (N/mm? Eq 13 0a.z Alternating tangential normal stress lb/in2 (N/mm3 Eq 13 ‘=b Axial normal stress due to bending lb/$ (N/mm2) Eq 7 0, Mean component of stress lb/in2 (N/mm2) Eq 11 omax Maximum stress Ib/ir? (N/mm2) Fig 2 0 min Minimum stress lb/r? (N/mm? Fig 2 %x Mean axial normal stress lb/is (N/mm9 Eq 14 GY Mean radial normal stress lb/k? (N/mm? Eq 14 %a Mean tangential normal stress lb/is (N/mm9 Eq 14 OP Axial normal stress due to tension or compression lb/r? (N/mm? Eq 10

(continued)

3

ANSIIAGMA 9001 -D97 AMERICAN NATIONAL STANDARD

Table 1 hmcluded~

Symbol Term 0, Total axial normal stress % Total radial normal stress 0, Total tangential normal stress 6 Von Mises alternating stress 4, Von Mises mean stress Q Equivalent uniaxial stress under peak loading Got al Von Mises total stress ht Alternating torsional shear stress b Alternating radial shear stress bYZ Alternating axial shear stress ha Alternating tangential shear stress kuy Mean radial shear stress hYZ Mean axial shear stress hrp Mean tangential shear stress 4 max Maximum torsional shear stress 4 min Minimum torsional shear stress h Mean torsional shear stress kY Total radial shear stress Cryz Total axial shear stress Gta Total tangential shear stress b Torsional shear stress r, Shear stress due to shear force [test 1 Test shear stress to find 9,

Units lb/in’ (NlmmL) lb/it? (N/mm? lb/in* (N/mm9 lb/in* (N/mm*) lb/is (N/mm? lb/k? (N/mm? lb/in* (N/mm*) lb/k? (N/mm)* Ib/i$ (N/mm? lb/in* (N/mm2) lb/k? (N/mm? lb/in* (N/mm? lb/in* (N/mm7 lb/k? (N/mm? lb/in* (N/mm*) lb/in* (N/mm*) lb/r? (N/mm? lb/in* (N/mm*) lb/is (N/mm*) lb/in* (N/mm? lb/in* (N/mm? lb/ins (N/mm? lb/in* (N/mm*)

First referenced

Eq 15 Eq 15 Eq 15 Eq3 Eq3 Eq 39 Eq5 Eq 20 Eq 13 Eq 13 Eq 13 Eq 14 Eq 14 Eq 14 Eq 19 Eq 19 Eq 19 Eq 15 Eq 15 Eq 15 Eq6 Eq8 Eq 27

It is recommended that the cumulative fatigue damage criteria proposed by Miner (Miner’s Rule) be employed to evaluate the effects of variable loading on the life of components.

3.1 Load spectrum analysis 3.3 System analysis

This standard assumes that within the operating speed range, the system of connected rotating parts is compatible and free from critical speeds and torsional or other types of vibrations, no matter how induced.

3.2 Momentary peak loads

This standard is based on an allowable momentary mechanical peak load rating of 200 percent (2.0 x mechanical rating) for spur, helical, herringbone, and bevel gear drives and an allowable momentary mechanical peak load rating of 300 percent (3.0 x mechanical rating) for worm gear drives. Frequency and duration of peak loads must be considered when designing and selecting components. If the frequency or duration of peak loading becomes significant (greater than 100 cycles during the design life), the designer should consider a cumulative fatigue analysis such as Miner’s Rule.

The gear drive designer or manufacturer is not responsible for the system analysis unless this provision is clearly identified by contractual agreement.

4 Shafts

This section covers a stress analysis procedure applicable to cylindrical steel shafts used in conjunction with and as a part of enclosed gear drives. This analysis may or may not be applicable to other materials.

Shaft stress is but one consideration in the design of shafting. A shaft must have proper radial, axial, and

AMERICAN NATIONAL STANDARD ANSl/AGMA 6001 -D97

torsional stiffness to lim it deflections to acceptable levels and to avoid unwanted vibratory motion.

4.1 Design criteria

Shafts must pass two stress analysis tests to be considered adequately designed. First, they must be designed to resist fatigue failure due to cyclic loading over their intended life. All operating loads, including momentary peak loads, must be considered, and a M iner’s Rule analysis (see IS0 TR 10495) may be required to properly account for the different stress states. See 4.6.8.

considerably greater than unity shall be used. If the consequence of failure is m inimal, a safety factor closer to unity may be used.

4.2 Fatigue safety factor

The elliptic equation has been selected for analysis of the fatigue failure mode. It is found in references [3], [lo] and [15].* This equation is:

. ..(3)

Fsp 1.0 where

. ..(l)

Fsf is fatigue safety factor.

Second, they must be designed to withstand momentary peak loads without distress or permanent deformation.

Fsp a 1.0 where

. ..(2)

where

on is Von M ises alternating stress, lb/in2 (N/mm2);

!f is modified fatigue strength, lb/in2 (N/mm?; & is Von M ises mean stress, lb/in2 (N/mm?; J; is tensile yield strength, lb/in2 (N/mm2); Fsf is fatigue safety factor.

This equation can be rewritten to solve for the fatigue safety factor.

FSI, is peak load safety factor.

The requirement for both of these criieria is to compare the stress condition for the given shaft geometry and loading to properties of the shaft material. Each stress condition applied to the shaft must be checked with equation 1 and equation 2 such thatFsfz 1 .O andFs, z 1 .O. See figure 1. Where nominal or estimated material properties are used, a factor of safety, Fsf, greater than 1.0 is recommended.

Fsf = [is)‘+l( 2 )] 0.5

&l . . .

SY

(4)

In selecting a value for safety factor, the consequence of failure shall be considered. If the consequence of failure is high, a safety factor

For the design to be considered acceptable for fatigue condition, the resulting fatigue safety factor, Fsf, must be equal to or greater than 1 .O.

4.3 Peak load safety factor

The following peak load analysis equation is used to solve for the peak load safety factor:

Stress curve associated

Mean stress Figure 1 - Design criteria

* Numbers in brackets throughout the text, [I, refer to publications listed in annex G.

5

ANSl/AGMA 6001497 AMERICAN NATIONAL STANDARD

where

Fsp is peak load safety factor;

Fya is allowable stress to yield strength factor;

J; is tensile yield strength, lb/in2 (Nhm2);

FP is peak load factor;

?&d is Von Mises total stress, lb/in2 (N/mm2). CAUTION: Equation 5 is based on a ductile material. For purposes of this standard, a material is considered ductile if the tensile elongation of the core material is at least 10%. For nonductile materials, the effects of stress concentration should be considered. See 4.5.1.

If i&d includes stresses which are not a function of load, such as stress resulting from the weight of components or stress resulting from shrink fit of components, Fsp may be conservative. Consider- ations may be given to only applying I$ to those stresses of Got& that are load related.

For the design to be considered acceptable for peak load condition, the resulting peak load safety factor, Fsp, must be equal to or greater than 1 .O.

The safety factors are to be chosen based on experience and engineering judgement.

4.3.1 Allowable stress to yield strength factor, Fya The allowable stress to yield strength factor is to provide conservatism over the stress resulting from expected peak load conditions and variations in the tensile yield strength. Values between 0.66 and 0.80 have traditionally been employed for this variable. Unless otherwise agreed upon, a value of 0.75 is recommended.

4.3.2 Peak load factor, Fp

The peak load factor accounts for momentary peak loads over the unity service factor load. In the absence of other known conditions, the following values for the peak load factor are to be used:

For spur, helical, herringbone, and bevel gear drives, Fp= 2.0.

For wormgear drives, Fp= 3.0.

4.4 Calculated stresses

There are four major types of loading applied to shafting that constitute the simplified case. These result in torque (7), bending moments (M), shear forces (V), and axial tension or compression (f”). The equations converting these forces to stresses are given in equations 6 to 10. Positive forces and stresses are in tension and negative forces and stresses are in compression. All forces and stresses

must be those which would be present when the drive is loaded to its unity (1 .O) service factor rating. For torque:

z* = 16 Td,

II d: ( - d;) t = 16OOOTd,

t +? - 4)

. ..(6)

. ..(6M)

‘is is torsional shear stress, lb/in2 (N/mm2); T is torque, lb in (Nm); 4 is shaft outside diameter, in (mm); 4 is shaft inside diameter, in (mm).

For bending moment:

(3b = 32Md,

JC (di - d;)

Ob = 32OOOMd,

. ..(7)

,..(7M)

where

ab is axial normal stress due to bending, lb/in2 (N/mm2);

M is bending moment, lb in (Nm). For shear force:

t is shear stress due to shear force, lb/in2 (N/mm2);

V is transverse shear force, lb (N). and

K= (1 + 244 + 2.24

(1 + v)(d$ + d;) . ..(9)

where

V is the material’s Poisson’s ratio.

For a solid steel shaft, wherev = 0.3, K= 1.23 and for a thin walled hollow steel shaft, K approaches 2.0.

For axial tension or compression:

ap = nj;: d?)

. .

where

4 is axial normal stress due to tension or compression, lb/in2 (N/mm?;

6

AMERICAN NATIONAL STANDARD ANSIIAGMA 6001 -D97

PX is axial force, lb (N).

All of these stresses can have alternating and mean components. See figure 2.

Therefore: 0, = 0.5 (amax + omin) = mean component of stress . ..(n) 0~ = 0.5 (Qma - Qmin) = alternating component of stress . ..(12)

(Equations 11 and 12 are generalized for a normal stress, but also apply to shear stresses.)

The case of completely reversed stress, where 4 = omax and G = 0, occurs for the axial normal bending stress, ob, and the shear stress, q, (due to transverse shear force) when the shaft is rotating and is subjected to a constant direction load. This is a common loading condition.

The location of the maximum and m inimum stress intensities due to these loads varies as the stress element being analyzed orbits with the rotating shaft. See figure 3. In this figure, the loads are fixed and only the shaft element under analysis is rotating with

the shaft. If all loads are treated as positive values and are therefore additive, the critical locations to be examined are at positions A (3 = 0) and B (3 = JC/~) and at positions 8 between them. Depending on the relationships between the four stresses, any position of 8 between 0 and Ic/2 may be the location of maximum stress.

Tension (+I

Compression

Figure 2 - Cyclic loading

Element at position A

Figure 3 - Stress convention shoying orbiting element

7

ANSI/AGMA 6991 -D97 AMERICAN NATIONAL STANDARD

Where stress is not uniformly distributed around the periphery of the shaft, the stress intensity of an element orbiting between A and B will vary approximately in a sine or cosine pattern as appropriate.

Most shaft stress analyses are concerned only with position A, as the bending stress, 4, is typically much larger than the transverse shear stress, T+, which produces the bending.

stress at position B or 6, between A and 6, can become important in a short shaft section with a high shear force as may be found adjacent to a bearing.

4.4.1 The aeneral case

k is mean tangential normal stress, lb/in2 (N/mm9 ;

* is mean radial shear stress, lb/in2 (N/mm?; % is mean axial shear stress, lb/in2 (N/mm?; b is mean tangential shear stress, lb/l$

(N/mm2). Von Mises total stress:

Gal = wwfx - QyY + (sy - %I2 + @&! - &>21 + 3[T&? + T& + l&2]}o-5 . ..(15)

where

&,d is Von Mises total stress, lb/in2 (N/mm?; arx is total axial normal stress, lb/in2 (N/mm?;

For a complete 3-dimensional fatigue analysis, the Von Mises stresses will be used. Refer again to figure 3. These are given in equations 13 through 15, where the total or maximum stress is the sum of the alternating and mean stresses (9jc = a, + & + . ..).

% is total radial normal stress, lb/k? (N/mm?; o, is total tangential normal stress, lb/in2

@J/mm% hr is total radial shear stress, lb/in2 (N/mm?; -I+ is total axial shear stress, lb/in2 (N/mm2);

Von Mises alternating stress: 0, = {OS[(cs~ - c&y)2 + (oq - c&)2

+ (oa - oax)2] + 3[&$ + ‘hyz2 + l&2]}oe5 . ..(13)

where

Gl is Von Mises alternating stress, lb/r? (N/mm2);

o, is alternating axial normal stress, lb/in2 (N/mm2);

o+ is alternating radial normal stress, lb/in2 (N/mm2);

o,, is alternating tangential normal stress, lb/k? (N/mm2);

‘hxy is alternating radial shear stress, lb/in2 (N/mm2) ;

hYz is alternating axial shear stress, lb/is (N/mm2);

k= is alternating tangential shear stress, Ib/ir? (N/mm?.

Von Mises mean stress:

4, = ww%x - %zyj2 + @tIy - %Kj2 + @m - %x)2] + 3[kxy2 + Tmyz2 + k2])o-5

. ..(14)

where

4, is Von Mises mean stress, Iblin2 (N/mm2);

h is mean axial normal stress, lb/in2 (N/mm?;

% is mean radial normal stress, lb/in2 (N/mm?;

b is total tangential shear stress, lb/k? @J/mm?.

All unused terms in the above equations are set equal to zero. The results of equations 13 through 15 are used in equations 3 through 5.

4.4.2 The simplified case There are stresses which are not included in these simplified case formulas but to which, nonetheless, the designer must be alert and appropriately ad- dress if they are encountered. These include, but are not limited to, hoop stresses due to press fitted parts, pressure on hollow shafts, thermal stresses, stresses due to unbalance and centrifugal forces, and residual stresses, any of which can become significant. For the simplified case, all stresses on the free (outer) surface of the element are zero (i.e., no interference fitted elements), as are shear stress in the radial direction and the tangential stress. However, experience has shown that the effect of the interference fit can be modeled using the simplified case and the modifying factor for stress concentration, 4, as discussed in 4.6.6. Keeping in mind sign convention: At position 6:

&=($‘,cd+t+ . ..(16) &=7ysine+~ . ..(17) q=I&q.p+=O . ..(18)

After determining the forces and stresses on the shaft at the critical sections, determine the amount of each stress that is alternating and the amount that is mean, per equations 11 and 12. For many cases a

AhtERICAN NAIONAL STANDARD ANSI/AGMA 6001 -D97

and q are completely alternating, whereas 4 is entirely mean. The torsional shear stress, ‘4, often has both a mean (T,,J and an alternating component hzt) -

‘I;nr = O-5 (% max + ‘ir min)

hr = 0.5 (3 max - ‘4 min)

If such is the case, at position 6,

. ..(19)

. ..(20)

Mean stresses:

%Lx=qo Gzr=T?u

Alternating stresses:

. ..(21)

. ..(22)

o,=ob~s6 (maxat0=0) . ..(23)

h=qsin8+Gr (maxatt3=$) . ..(24)

The mean components of stress have m as the first subscript (h, b) while the alternating components have a as the first subscript (oar, b).

For standard catalogued gear drives that are not subjected to torque reversals, alternating torques in the range of 25% to 50% are suggested. If no data exists, a worst case 50% alternating torque (that is between zero and the maximum torque, where m t = st) should be used. A value lower than 25% may be used if experience shows that reduced torque fluctuations exist.

If frequent torque reversals are present, consider setting ht = T,,, and k = 0 (100% alternating torque).

Von M ises alternating stress for the simplified case: 0, = pa,2 + 3&]0*5 . ..(25)

Von M ises mean stress for the simplified case: 4, = 1~2 + 3b2p5 . ..(26)

Calculate equations 23 through 25 for 8, depending on Ttst and ht:

02 b

- 3t$ &St = 3%

If % r c Gstt

8, = sin-’ 2 ( 1

.,

,..(27)

If Tzr 2 %?st, 8, = zradians

where

ec is the critical stress angle, deg.

and use the components o, and & for the maximum value of 5, to calculate Von M ises total stress for the simplified case:

iota1 = [(c& + oar)2 + 3(& + T&J2]“*5 . ..(29)

This represents the maximum value of stress at any point of the shaft surface at that location for the general case. The results of equations 25,26 and 29 are used in equations 3 through 5. 4.5 Mechanical properties of steel

Mechanical property values, determined from test, of the specified grade of steel in its processed and heat treated condition should be used whenever available. 4.51 Estimated properties of steel

It is important to note that the estimated material properties may deviate significantly from actual test values. Thus, an appropriate factor of safety should be used. In the absence of test data, the following properties of forged or rolled steel shafting material may be used. For through hardened steel shafts, the ultimate tensile strength can be based on the Brine11 hardness of the shaft, near the outer surface, at the diameter under consideration. For steel shafts that are case hardened by processes such as carburizing and nitriding, the ultimate tensile strength is based on the Brinell hardness of the shaft underneath the hardened case unless a detailed analysis or experience indicates that a different tensile strength should be used. See sample problem 3 in annex E. The yield strength and modified fatigue strength can be calculated based on the ultimate tensile strength. The following equations have been developed by curve fitting representative test results from references [3], 1181 and [19]. Ultimate tensile strength:

S, = 500 HB . ..(30) St4 = 3.45 HB . ..(30M)

where S, is ultimate tensile strength, lb/in2 (N/mm2); HB is Brinell hardness number, HB. NOTE: The value of 500 in equation 30 is a typical value. In reference [Xl, 450 is mentioned as a m inimum value.

Tensile yield strength: J; = 0.94 s, - 12 500

4 = 0.94 S,, - 86.2 where

. ..(31) . ..(31 M)

J; is tensile yield strength, lb/k? (N/mm?. Basic fatigue strength:

sf, = 0.5 s, if S, S- 200 000 lb/in2 . ..(32) qe = 0.5 Lc& if& s 1380 N/mm2 . ..(32M)

NOTE: The value of 0.5 in equation 32 is an average value. The values can range from 0.4 to 0.6 depending upon heat treatment. See [15] for detailed information.

9

ANSYAGMA 6001 -D97 AMERICAN NATIONAL STANDARD

sf, = 100 000 lb/in2 if S, > 200 000 lb/ins . ..(33) sf, = 690 N/mm2 ifs, 5 1380 N/mm2 . ..(33M)

where

4.6.1 Surface finish factor, k

% is basic fatigue strength of polished, unnotched test specimen in reverse bending, lb/in2 (N/mm?.

The surface finish factor accounts for the difference between the actual shaft and a highly polished test specimen. Values for & are given in figure 4.

4.6.2 Sii factor, &

4.6 Modified fatigue strength

Since the fatigue strength is largely influenced by physical conditions, environmental conditions, and application conditions as well as material conditions, the basic fatigue strength must be modified.

Sf=k+e . ..(34) where

The size factor accounts for the increased likelihood of encountering a fatigue initiating defect as the shaft diameter increases. Values for b are given in figure 5.

4.6.3 Reliability factor, k

?f is modified fatigue strength, lb/r? (N/mm2);

k is fatigue strength modification factor.

The fatigue strength modification factor, k, is the product of the fatigue factors (k through 4). The fatigue safety factor, Fsf, is greatly affected by the fatigue strength modification factor. Therefore, care should be exercised in determining the values of the fatigue factors [3].

k=kJdclrdk$~ . ..(35)

The reliability factor accounts for variation or scatter in fatigue test data for samples of a given material. The reliability factor is determined by the desired level of reliability and the dispersion of the test data. Unless otherwise agreed upon, a value of k = 0.817 (for 99% reliability) should be used.

The formulas and data in figure 6 are for a normal distribution where the standard deviation is 8% of the mean.

NOTE: For high levels of reliability (greater than 0.99) the reliability factor is very sensitive to the failure distribution assumed. The equation shown in figure 6 may only provide an approximation of the actual reliability factor for these levels.

0.8

lu” 0.6

finish; &

4

Surface

60 80 160 180 240 1000 lb/k?

4bO 5bCl 690 760 860 9bO 1600 11100 1200 l;OO 1400 1 ;OO 1600 1700 N/mm2 Ultimate tensile strenoth. S

Figure 4 - Surface finish factor, & [lo]

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AMERICAtj NATIONAL STANDARD ANSI/AGMA 6001 -D97

& 0.8

&.r 0 8 s 8 cn 0.7

0.6 i ' ' _ 0 2 4 6 8 10 12 14 16 18 20 in I I I,, I I I I I I I, I I I I, I I I 0 50 100 150 200 250 300 350 400 450 500mm

Shaft diameter, (I;,

Figure 5 - Size factor, k6 [lo]

Reliability, R, is the fraction of a group of apparently limits of steels. However, between normal operating identical parts that exceed a given life. temperatures ranging from about -20°F (-29°C) to

R number of pieces exceeding a given life 250°F (121 “C), the fatigue strength characteristics

= total number of pieces subjected to loading of most steels are essentially unchanged. For this

. ..(36) temperature range, a temperature factor b = 1.0 may be used.

4.6.4 Temperature factor, & CAUTION: Consideration must be given to the loss of hardness and strength of some materials due to the

Extreme operating temperatures affect the fatigue tempering effect at high temperatures.

1 .oo 0.98 0.96 0.94 0.92

42 0.90 6 0.88 g 0.86 2 0.84 9 0.82 8 0.80

I I i

0.9 Reliability, R 0.99 0.999

For normal distribution where the standard deviation is 8% of the distribution mean.

Figure 6 - Reliability factor, k [3] [lo]

11

ANSYAGMA 6001 -D97 AMERICAN NATIONAL STANDARD

For applications outside this temperature range, the fatigue properties should be determined by actual tests [3]..

4.6.5 Life factor, k

shoulder, or other discontinuity where the effective stresses have been ampllfied. The effect of stress concentration on the fatigue strength of the shaft is represented by the modifying factor for stress concentration, kf.

Lie factor, &, is taken as unity (1.0) at lo6 stress cycles. At greater than 1 O6 stress cycles, k should be taken as unity, but in fact may be continually decreasing with increasing number of stress cycles at an unknown (lesser) rate. k, is greater than unity between 1 Os and 1 O6 stress cycles and in this range may be calculated as follows:

kf=l+q;K,-lj . ..(38)

where

4 is notch sensitjvity; & is theoretical stress concentration factor in

bending.

A single fatigue stress concentration factor, that in bending (I&), is utilized as representative of any stress condition. For many applications, only a small error will result from this assumption because the other stress concentration factors are very close to that in bending. However, if a different stress concentration factor (other #an bending) is required and it is significantly different than I$, a modification to this analysis may be necessary.

lOCN-m ke= s e

. . . 8 (37)

where m is ‘/a loglc IO.8 SJSe];

C is log10 ((0.8 &J2/Se];

Se is sf at lo6 stress cycles (where 4 = 1 .O);

N is number of stress cycles (between 1 Os and 10s).

Below 1 Os stress cycles, the value of & obtained at 1 O3 cycles should be used.

4.6.6 Modifying factor for stress concentration, k/ Experience has shown that a shaft fatigue failure almost always occurs at a notch, hole, keyway,

0.8

Q- $ 0.7 ‘Z ‘B 5 z 0.6 v

2 0.5

0.4

Notch sensitivity, q, accounts for the phenomenon that low strength steels are less sensitive to fatigue at notches than are high strength steels. Values for q are shown in figure 7 for ductile (i.e., elongation 2 10%) through hardened steel shafts. The theoretical stress concentration factor in bending, Ir;, as shown in figures 8 through 10 are taken from reference 13. These values for & represent some of the more common cases.

tl-l- su . 4- = 260030 [ 1 (O086r-0'3861

for S, in lb/in2

(0300r4-386) & . 4- = 1791.4 [ 1 for S, in N/mm2

(r = shaft radius = 0.16 in (4.1 mm) maximum)

r = n inches (n mm) I I I I I I I I I I I I

60 80 100 120 140 160 180 200 220 240 260 1000 lb/in2 I I I I I I I I I I I I I I I

.

400 500 600 700 800 800 1000 1100 1200 1300 1400 1500 1800 1700 1800 N/mm2 Ultimate tensile strength, S,

Figure 7 - Notch sensitivity - steel, q [lo]

12

AMERICAN NATIONAL STANDARD ANSVAGMA 6001 -D97

5.0-yiwere ^ ” = Kl+Ke ,) +K# +K,(@ ~ I lot 0.25 s h/r 5 2.0 I for 2.0 s h/r I 20.0

Kl -

-3.790 + 0.958 0.847 + l-716,@- 0.506h/r

i& j-O.790 + 0.417,/h/r-0.2 :46h/r 1 -

1.0 I I I I I 0.0 0.1 012 0.3 0.4 015 O!S

w4 Figure 8 - Theoretical stress concentration factor in bending for a circular shaft with a square

shoulder, 4 (nominal stress is calculated at diameter 4,)[13]

2.5

2.0

1.5

1.0 1 I I 0.0 0.1 I 0.2 0.3 I

0.4 I

0.5 -_ d.6

Figure 9 - Theoretical stress concentration factor in bending for a circular shaft with a u-notch, 4 (nominal stress is calculated at diameter 4) [13]

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ANSl/AGMA 6991 -D97 AMERICAN NATlONAL STANDARD

3.8

3.8

3.2

2.8

2.8

2.4

4.0 II

- where For di/do I 0.9 and 2/d, I 0.3

KI 3.ooo - K2 -6.69~1.620di/do + 4.432(di/doj2

- 0.7 - 0.6 - 0.5 ; 0.4

solid

Figure 10 - Theoretical stress concentration factor in bending for a circular shaft with a radial hole,. & (based on full section without considering hole) [13]

Table 2 gives typical values for 4 for standard keyways in solid round steel shafts. Changes in keyway proportions, corner radii, size of shaft, and fit with mating members can greatly alter the values of +. Often press-fit assembly of hub and shaft is used with or without provisions of a key. Fatigue stress modifying factors for interference fits vary widely but are often quoted as being in the general range of 0.50. Combinations of interference fit and keyway give values of 4 typically in the range of 0.4 to 0.33.

Table 2 - Modifying factor for stress concentration, 9 -Iypicai values for keyways

in solid round steel shafts’) m

Profiled Sled-runner keyway keyway bending bending

steel StreSS stress Annealed (less than 200 HB) 0.63 0.77 Quenched and drawn (over 200 HB) 0.50 0.63 NOlE l) Nominal stresses should be based on the section modulus for the shaft section with the keyway effect ignored.

Experimental verification is preferred for super- position of stress concentration factors. Without verification, the smaller values should be used. One reason is that the possibility of a fatigue failure originating in the region of an interference fit is often aggravated by fretting corrosion.

4.6.7 M iscellaneous effects factor, kg

Since fatigue failures nearly always occur at or near the surface of the shaft where the stresses are greatest, surface condition strongly influences fatigue life. A number of factors affecting the fatigue lim it have values not readily found in design texts. Some of these factors are:

- heat treatment (such as case hardening and decarbonization);

- residual stresses (such as cold rolling, peening, and welding);

- corrosion (such as stress corrosion cracking, fretting corrosion); - plating or surface coating.

Although only lim ited quantitative data have been published for these factors, they should be considered and accounted for if applicable. Some of these factors can have a considerable effect on the shaft endurance characteristics. In the absence of

14


published data, it is advisable to conduct fatigue tests that closely simulate the shaft condition and its operating environment. Use published data or test data when available. If none of the above conditions or other m iscellaneous effects contribute to the endurance of the shaft being analyzed, kg may be set to 1.0. If any of these conditions reduce the endurance strength, consider setting kg to less than 1.0; if any of these conditions increase the endurance strength, kg may be greater than 1 .O.

4.6.8 Permissible number of peak load cycles

If the number of momentary peak load (I$ x the unity service factor load) cycles are significant, they can become the dominant factor in the stress analysis. If a M iner’s Rule analysis has not been performed including these loads, the permissible number of momentary peak load cycles, NfO, to avoid excessive fatigue is determined as follows:

If 1.0 s Fg c Fp, then

loc ( 1 I/m

$3 = F

2 l/2 E m [ 01 --

i; sy

. ..(39)

. ..(40)

where

Nfo

00

cm

is permissible number of momentary peak load cycles;

is equivalent uniaxial stress under peak loading, lb/in2 (N/mm2);

are as defined in 4.6.5.

If Fg z Fp, the permissible number of peak load cycles analysis is not necessary.

If Nfo is less than or equal to the number of momentary peak load cycles of the application, redesign with lower calculated stresses necessary.

a is

If Fs~ < 1.0, the design is unacceptable. See 4.2.

4.7 Deformation

This section is intended to give an overview of deformation of steel shafts within enclosed gear drives. Deformation of bearings, housings, and other components is beyond the scope of this standard. For a more general treatment of deforma-

ANSI/AGYA 6001 -D97

tion, a mechanical design text or similar reference should be consulted.

Deformation is the deviation of a shaft from its original or ideal shape. All shafts deform when they are subjected to stress. For a particular application, the deformation may be so small that it cannot be measured with usual techniques, or it may be large enough to be observed with the unaided eye.

Deformation can cause m isalignment of components mounted on the shaft, reduce seal performance, and alter bearing and gear tooth load contact patterns. All shafts should be designed so that shaft deformation is within suitable lim its for reliable operation of gears and other components that are affected by shaft deformation.

Only homogeneous isotropic cylindrical shafts subjected to elastic strains will be considered here. Design conditions such as varying material properties, inelastic strains, complex loading conditions, or non-cylindrical shapes may require a more detailed analysis. Finite element analysis and physical testing may be appropriate alternative methods of obtaining deformation information.

While the equations presented in this section cannot predict exact deformations, calculated deformations can often indicate the suitability of a shaft for its application.

4.7.1 Torsion

Torsional deformation is measured as the angle of shaft twist.

Figure 11 is a sketch of a cylindrical shaft subjected to pure torsion. The angle of twist for this ideal shaft can be calculated from equation 41.

Figure 11 - Torsional deformation

15

ANSI/AGMA 6001 -D97 ANlERICAN NATIONAL STANDARD

et = $$ . ..(41)

1000 TL 0, = GJ . ..(41 M)

where

e, is angle of twist, rad;

T is torque, lb in (Nm);

L is length, in (mm);

G is modulus of rigidity, lb/in* (N/mm?;

J is second polar moment of area, in4 (mm4).

For steel:

GE 11.5 x 1 O6 lb/in* (79 300 N/mm*)

, ,..(42)

where

4 is shaft outside diameter, in (mm);

4 is shaft inside diameter, in (mm).

For solid steel shafts equation 41 can be simplified to:

0, = 8.86 x 10-7TL d4 0

e t

=0X284 TL 4

. ..(43M)

The angle of twist between shaft ends or specific cross sections can be calculated by adding the twist angles of the intervening shaft lengths that have constant properties.

8, -TILI ; T2L2 ; . . . +Wn

Gl Jl G2J2 Gn Jn . ..(44)

‘t = “O” TILl T2L2 TnLn -+-+...++ G,J, G2J2 Gn Jn I

. ..(44M) For a homogeneous shaft which consists of n different diameters and is subjected to a constant torque:

(45) . . .

where

& is the length of the cth section, in (mm);

J.. is the second polar moment of area of the nth section, in4 (mm4).

4.7.2 Bending

Bending deflection is deformation which is measured perpendicular to the axis of the shaft. Slope, which is the rate of change of deflection, can also be an important design criterion.

Several common simply supported shaft loading cases are shown. These loading cases assume a uniform cylindrical shaft reacting to concentrated forces and moments. Many shafts can be evaluated as some combination of these loading cases. Shaft loads and reactions can often be separated in order to simplify calculation of the deformations which can then be added vectorially to give the total deformation.

Additional loading cases with distributed loads or different supporting arrangements can be found in several mechanical design texts or similar references.

Cases involving shafts with several diameters or indeterminate bearing arrangements should not be analyzed with the methods shown here. Finite element analysis and numerical integration are other methods for estimating deflection and slope and can be applied to a wide variety of cases.

4.7.2.1 Intermediate concentrated load

Figure 12 is a diagram of a simply supported shaft with a concentrated load applied between the supports. Forces and reactions are assumed to act at right angles to the shaft. Zero deflection at the reactions (bearings) is assumed.

Y

* L .

I x * -a F

d---T X

eb

Figure 12 - Bending deflection intermediate concentrated load

The equations for deflection and slope for a simply supported shaft subjected to an intermediate concentrated load are different depending upon the relative position of the load to the shaft section of interest.

16


Forxcaand(L-a)>a:

Y= F(x)(L - a)(x2 - 2uL + P2) . ..(46)

0, = F(L - a)(3x2 - 2QL + 02)

6EL5 . ..(47)

The equations for deflection and slope for a simply supported shaft subjected to an concentrated overhung load are different depending upon the relative position of the section of interest to the supports.

Ifx<L

Forx>aand(L-a)>a:

y _ F(a)(L - x)(x2 - 2Lx + 02) 6EIL

8, = F(a)@ - 3x2 - zL2 - a2) 6ELL

. ..(46)

. ..(49)

where

Y

F

L

u

x

E

66

I

is deflection of shaft at x, in (mm), (positive upward as shown in figures 12 - 15);

is concentrated load, lb (N), (positive downward as shown in figures 12 - 15);

is length of shaft between supports, in (mm);

is distance from support to concentrated load, in (mm);

is distance from support to cross section of interest, in (mm);

is shaft material modulus of elasticity, lb/in2 (N/mm2);

is shaft slope atx,

is second area moment of cross section, in4 (mm4).

I=%(& - d;) . . 4.7.2.2 Overhung concentrated load

Figure 13 is a diagram of a simply supported shaft subjected to a concentrated overhung load. Forces and reactions are assumed to act at right angles to the shaft. Zero deflection at the reactions (bearings) is assumed.

a a

X

F

Figure 13 - Bending deflection overhung concentrated load

y = F@>(x) (L2 - x2) 6ELc

. ..(51)

. . ..(52)

Ifx>L

F(x - L) (x - L)2 - u(3x - L) Y

I = . ..(53)

F C 3(x - L)2 - a(6x - 4L) 8, = 1 . ..(54)

6EI

4.7.2.3 Intermediate concentrated moment

Figure 14 is a diagram of a simply supported shaft wlth a concentrated moment applied between the supports. The reactions are assumed to act at right angles to the shaft. Zero deflection at the reactions (bearings) is assumed.

:rl Figure 14 - Bending deflection intermediate

concentrated moment

The equations for deflection and slope for a simply supported shaft subjected to an intermediate concentrated moment are diierent depending upon the relative position of the load to the shaft section of interest.

Ifxsa

Y = M(x)(6d - 2L2 - 3a2 - x2) . ..(55)

Y= lOOOM(x)(6uL - 2L2 - 39 - x") ...t55MI

e _ M(6uL - 2L2 - 3a2 - 3x2) b- 6EIL

. ..(56)

8 -loooM(6aL- b-

- 3a2 - %x2) . . . (56M)

17

ANSIIAGMA 6001~D97 AMERICAN NATIONAL STANDARD

lfx>L

Y = - M(L - x)(L - 3x) . ..(61)

6El - lOOOM(L -x)(L -3x)

Y= . ..(61 M)

8, = -M(6x- :

6El . ..(62)

8, = -1OOOAq~ - 4L)

6El 4.7.3 Axial

. ..(62M)

Elongation is axial deformation and is measured parallel to the axis of the shaft. A positive elongation is the result of a tensile shaft stress and a negative elongation is the result of a compressive shaft stress. Figure 16 is a sketch of a shaft subjected to an axial force. Elongation of this ideal shaft can be calculated from the following equation.

Ifx>a

Y = M(3a2L - 3a2x - 2L% + 3Lx2 - x3) . ..(57)

1000M(3& - 3a2, - 2L2x + 3Lx2 - x3) Y =

. ..(57M)

8, = M(ti - 3a2 - 2L2 - 3x2) . ..(56)

8, = 1oooM(6Lx - 3a2 -

6EIL. 2L2 - 3x2) . ..(56M)

where

M is concentrated moment, lb in (Nm).

4.7.2.4 Overhung concentrated moment

Figure 15 is a diagram of a simply supported shaft subjected to an overhung concentrated moment. Reactions are assumed to act at right angles to the shaft. Zero deflection at the reactions (bearings) is assumed.

Y

a *

I MTJ

Figure 15 - Bending deflection overhung concentrated moment

The equations for deflection and slope for a simply supported shaft subjected to an overhung concentrated moment are different depending upon the relative position of the section of interest to the shaft supports.

IfxeL

Y= M(x)(L2 - x2) . ..(59)

Y= lOOOM(x)(L2 - 9) . ..(59M)

M(L2 - 3x2) ‘b= (jjg~

. ..(60)

8, = lOOm4(L2 - 52) . ..(60M)

6 p,L =- AC&

. . .

where 6 is elongation, in (mm); px is axial force, lb (N); L is length, in (mm); & is cross sectional area, in* (mm*).

w

Figure 16 - Axial deformation

For cylindrical shafts:

,..(64)

For steel E & 30 x 1 O6 lb/in* (207 666 N/mm*). For solid cylindrical steel shafts equation 63 can be simplified to:

a 424 x lo-8P,L = 4

. ..(65)

6 6.15 x lo-@& =

4 . . w(65M)

16

AMERICAN NATIONAL STANDARD ANSI/AGYA 6001-097

4.7.4 Permissible deflection

The amount of deflection permissible in a shaft is dependent, to a great extent, upon the particular condition of operation encountered; hence, no general rules can be given. However, the shaft must be stiff enough to lim it the deflection of key power elements such as gears and pulleys. The allowable slope of the shaft through the bearings should not exceed the allowable m isalignment lim its of the bearings. Moreover, the judgment and the experience of the designer must be relied upon to determine the extent to which either lateral or torsional deflection may be permitted.

s, = 0.5 s,,

where . ..(67)

S so is allowable shear stress, lb/in* (N/mm*).

See annex A, table A.1 for typical values of S,, and S co’

5.4 Compressive stress calculation

The compressive stress in a key or keyway resulting from the transmitted torque may be calculated using equation 68. The calculated compressive stress, Q,, should not exceed the allowable compressive stress, S,, .

TIip s, = - r&b

..(68)

5 Keys SC = 1000 TFP

rac& .I

where

s,

T

Gc

4

FP

..(68M)

Keys are detachable components which, when assembled into keyways, provide a positive means for transmitting torque between the shaft and hub. Four common types of keys are square, rectangular, tapered, and Woodruff.

is calculated compressive stress, lb/in* (N/mm*);

5.1 Sizes and tolerances

Standard key and keyway sizes, tolerances and fits may be obtained from ANSI 817.1, Keys and Keyseats, ANSI 817.2, WoodrM Keys and Key- seats, or AGMA 9002-A86, Bores and Keyways for Flexible Couplings (inch Series).

is shaft torque, lb in (Nm); is average radius at compressive load area, in (mm) (see figure 17); is compressive area of key in contact with shaft or hub, in* (mm*); is peak load factor = 2.0for spur, helical, herringbone and bevel gear drives = 3.0 for worm gear drives.

5.2 Allowable compressive stress

The allowable key, shaft and hub compressive stresses are commonly based on 70 percent of the component material yield strength.

s, = 0.7 J; . ..(66) where

NOTE: Each component shall be evaluated based on the allowable stress for its material and hardness.

s is allowable -compressive stress, lb/in* Co (N/mm*);

s; is tensile yield strength of key, shaft or hub at the keyway section, lb/in* (N/mm*).

See annex A, table A. 1 for typical values of S,, and J;.

NOTE: The allowable stress for surface hardened components shall be based on the core hardness ofthe material except when a detailed analysis justifies a higher allowable stress.

Figure 17 - Average shaft and hub radius

5.5 Shear stress calculation

5.3 Allowable shear stress

The allowable key shear stress shall be based on 50 percent of the allowable key compressive stress.

The shear stress in a key resulting from the transmitted torque may be calculated using equation 69. The calculated key shear stress, %k, should not exoeed the allowable shear stress, S,, .

19


TFP ‘sk = z

where

. ..(69) of the bearing system and bearing lubrication system is critical to proper functioning of gear drives.

6.1 Roller and ball bearing selection criteria

%k is calculated key shear stress lb/in2 (N/mm21 ;

AS is shear area, in2 (mm2);

ra is average shaft radius along the key length, in (mm).

5.6 Keyed interference fit

When an interference fit is used in conjunction with a key, the torque to be transmitted by the key may be reduced by the minimum torque capacity of the interference fit. The minimum torque capacity of the interference fit is the minimum capacity at the most unfavorable conditions of speed, temperature, dimensional tolerances and sliding coefficient of friction. For reversing loads, either a tight fit key is to be used or the restraining torque of the interference fit must exceed the peak torque applied to the joint. Due to the irregular shape created by the keyway, the calculation of this torque capacity is beyond the scope of this standard and must be established by experiment.

5.7 Keyless interference fit

Roller and ball bearings generally are selected according to LlO life calculated by the methods of the bearing manufacturer and adjusted for the factors given below. A minimum calculated life of 5000 hours LlO, without consideration of adjustment factors, has historically been the standard for gear drives. The LlO life is the length of time that 90 percent of a group of apparently identical bearings will equal or exceed before a subsurface originated fatigue spall reaches a predetermined size at the surface. The L10 lie is associated with a 90 percent reliability level.

A bearing’s dynamic load rating is the load that the bearing will carry for a specified number of revolu- tions at a specified reliability level. Bearing life shall be calculated according to the load supported by the bearing resulting from both internal and external loads. Life in hours shall be calculated using the rotational speed of the bearing. In addition to dynamic load rating, other selection criteria such as static load rating, minimum load requirements, and speed limitations should be considered.

When an interference fit is used without a key, the restraining torque resulting from the interference fit shall exceed the peak torque at the joint.

Annex C presents one method for calculation of torque due to interference fit.

CAUTION: When calculating the torque capacity of an interference fit, consideration should be given to:

6.1 .l Reliability

Reliabilii levels other than 90 percent may be calculated. For more specific analysis, consult the bearing manufacturer.

6.1.2 Life adjustment factors

A bearing’s rating is given for operation under a

- thermal effects of dissimilar materials; specified set of operating conditions. The lie should be adjusted for the following factors:

- centrifugal effects;

- hoop stresses:

- residual stresses;

- dimensional tolerances; - coefficient of friction as determined by surface finish, material, and lubrication.

6 Bearings

Bearings are required to support shafting in accurate alignment while supporting shaft loads and allowing shafts to rotate or translate, or both. Proper selection

Lubrication. A bearing’s dynamic rating is established for specific conditions of lubricant viscosity, speed and temperature which affect lubrication. Adjustment should be made for actual lubricant viscosity, speed and temperature as specified by the gear drive designer.

Load zone. A bearing’s dynamic rating is generally based on a nominal amount of internal clearance which generates a load zone (loaded arc) within the bearing of between 150” and 180”. Adjustment should be made for the actual load zone under the loads imposed on the bearing.

Alignment. Adjustment should be made for misalignment magnitudes which may reduce bear-

20

AMERICAN NATIONAL STANDARD ANSIJAGMA 6001 -D97

ing life. The misalignment may be due to size, form or position tolerances; shaft or housing deflection under load; or a combination of the above.

Hydrodynamic A *

Bearing material. A bearing’s dynamic rating is based upon hardened, good quality bearing steel. If premium steels are used, the life may be adjusted due to the reduction of impurities in the steel.

Usable life. A bearing’s dynamic rating is based upon a laboratory spall size which normally will not hinder the performance of a gear drive. A spall size which is considered detrimental to the performance of the gear drive may be a usable life criterion.

5 -5; 0 E ‘ii E ‘Ei E 8 0

NOTE: The life calculation methods used by beating manufacturers are based upon subsurface originated fatigue damage which leads to spalls. Other types of bearing damage which may reduce bearing lie include, but are not limited to, surface originated spalling due to bruises from contaminants in the lubricant, plastic yielding or brinelling due to overload, retainer damage and scoring or scuffing due to lack of lubrication.

Consult the bearing manufacturer for specific life adjustment factors.

1 I I Lubricant

Beering viscositv x Rubbing

SPeed parameter Projected area pressure

Figure 18 - Variation of coefficient of friction versus the bearing parameter

6.2 Sleeve bearing selection criteria Operating temperature. In general, for tin based babbii, metal temperature does not exceed 260°F (127°C).

Sleeve bearings operate in three basic regimes:

- boundary lubrication;

- mixed film lubrication;

- full film hydrodynamic lubrication.

Figure 18 illustrates the typical variation of the coefficient of friction as a function of the bearing parameter for the three lubrication regimes.

NOTE: The criteria above may be exceeded when sophisticated analyses are performed on bearing materials, construction, clearance, eccentricity ratio, shaft slope, length to diameter ratio, lubricant viscosity, and operating temperature.

Hydrodynamic bearings subjected to infrequent momentary peak loads shall be designed such that the yield point of the bearing materials is not exceeded.

6.2.1 Boundary and mixed film regimes For specific information refer to the bearing manufacturer, or [4].

In the boundary and mixed film lubrication regimes, a pressure-velocity criterion for the specific material of the bearing should be checked.

7 Housings

6.2.2 Hydrodynamic regime

In the hydrodynamic regime, the main design criteria are:

Unit loading. In general, bearings are operated at less than 750 lb/in* (5.2 N/mm? unit loading.

The housing provides accurate alignment of the gears and bearings with sufficient strength and stiffness to maintain alignment under maximum internal and external loading. The housing also provides some, but not necessarily all, of the following:

Oil film thickness. In general, oil film thickness is greater than 0.0008 in (0.02 mm).

- reservoir for retention of lubricant; - exclusion of contaminants; - drain and fill locations;

21

ANSI/AGMA 6001 -D97

- inspection covers;

- oil level indicators; - surface area for heat dissipation;

- feet or mounting pads;

- motor mounting surface;

- Wing lugs.

8 Threaded fasteners

The purpose of threaded fasteners is to clamp two or more joint members together. The fasteners shall be of sufficient tensile strength and quantity to withstand the maximum internal and external design loads and prevent movement between the joint members. Fasteners may also be subjected to shear loading. This condition requires additional analysis and is beyond the scope of this standard.

8.1 Tensile stress

The forces to be considered are those developed by the mechanical rating of the gear drive in addition to the external loads. The equation for calculating tensile stress is as follows:

%e = WfFP

0.785(0 - qr ’ ..(70)

%e = WfFP

0.785(0 - 0.9382P)2 . ..( 70M)

where

St,

Wf

5

D

n

P

is calculated tensile stress, lb/in2 (N/mm2);

is applied tensile load on fastener, lb (N);

is peak load factor = 2.0for spur; helical, herringbone and bevel gear drives = 3.0 for worm gear drives;

is fastener nominal diameter, in (mm);

is fastener threads per inch, in-‘;

is fastener thread pitch, m m .

The calculated tensile stress should not exceed the allowable stress. The allowable stress is 80% of the fastener tensile preload stress in order to ensure joint integrity. The fastener tensile preload stress is 75% of the proof load stress in order to avoid fastener breakage. Allowable stress values for

typical threaded fasteners for tensile preloads of 75% of proof load are shown in annex 6. Other values may be used based on testing or experience.

8.2 Torque

The following equation may be used to estimate torque for inducing a given tensile preload 151:

Tf=&cD Wfi . ..(71) Ktc 5 w*

Tf= looo . ..(71 M)

where

?T Ktc

D

43

is fastener torque, lb in (Nm); is fastener torque coefficient; = 0.2 for nonplated steel fasteners. In applications where lubricants such as thread locking compounds, greases, oils and waxes are applied to the fasteners, Kt, may be assumed to be as low as 0.12 [5]; is nominal diameter of threaded fastener, in (mm) ; is fastener tensile preload, lb (N). Common practice is to torque the fastener to provide a tensile preload of 75 percent of the proof load.

Wb = 0.75Sp[0.,85(D - @ I . ..(72)

Wfi = 0.75 4,L[0.785 (D - 0.9382 PI;1

where . ..(72M)

S, is proof load stress, lb/n2 (N/mrr?). CAUTION: The preload shall be adequate to prevent movement between the joint members with due consideration given to deformation of the members.

8.3 Engagement length

The thread engagement in tapped holes shall be of sufficient length that the shear strength in the internal and external threads is greater than the tensile preload in the fastener.

9 Miscellaneous components

The following sections discuss frequently used components of industrial enclosed gear drives. A specific gear drive may not require all of the components listed and may utilize other components which are not included.

9.1 Shims

Shims are generally used to position gears, bearings, or other components. When also used as

22

AMERICAN NATIONAL STANDARD ANSIIAGYA 6001 -D97

gaskets, shims shall provide proper sealing. The shim material must withstand the minimum ambient and maximum operating temperatures at the shim. The material shall remain dimensionally stable at initial assembly and when exposed to temperature extremes for the life of the drive. The shim material shall be compatible with the gear drive lubricant and outside atmosphere, including contaminants.

CAUTION: The total shim pack thickness should not exceed the shim manufacturer’s recommendation.

9.2 Gaskets

Gaskets are used to retain lubricant and to exclude contaminants. They should not be used to position components. The gasket material shall be compatible with the minimum ambient and maximum operating temperatures at the gasket.

Gasket material is generally intended to compress at assembly but shall remain dimensionally stable when subjected to temperature extremes. The gasket material shall be compatible with gear drive lubricant and outside atmosphere, including contaminants.

9.3 Oil seals

Oil seals are used with rotating shafts to retain the lubricant and to exclude contaminants. Considera- tions concerning oil seal selection include but are not limited to hardness, material, pressure at the seal, shaft finish, roundness, concentricity, bearing endplay, speed, minimum ambient and maximum operating temperatures at the oil seal, lubricant, outside environment and expected life.

CAUTION: Oil seals should be selected in accordance with the seal manufacturer’s recommendations. Con- tact type seals should be considered wearable and replaceable items over the life of the gear drive.

9.4 Breathers

Breathers maintain pressure balance between the inside of the drive and the external atmosphere, while excluding environmental contaminants. The location should be such that oil leakage through the breather is prevented. When the pressure differen- tial is less than the acceptable component sealing limits, a breather may not be required.

9.5 Expansion chambers

Expansion chambers are devices that maintain pressure balance between the inside of a gear drive

and the external atmosphere while isolating the two Environmental conditions and some applications may preclude the use of breathers, in which case expansion chambers may be used.

9.6 Oil level indicators

Oil level indicators are used to identify the proper oil level with the gear drive mounted in a specified position. The manufacturer shall specify under which condition the oil level is to be checked, static or operating.

Typical oil level indicators include pipe plugs, sight gauges, standpipes and dipsticks.

CAUTION: A pressure buildup inside a gear drive will cause a false reading on a vented oil gauge.

9.7 Bearing retainers

Bearing retainers are devices other than the housing that maintain the axial or radial positions of the bearings. All retainers shall be designed to locate and maintain dimensional stability for the bearings and gears in accordance with the bearing and gear manufacturers’ specifications. Bearing retainers include but are not limited to locknuts, keeper plates, end caps, cartridges or carriers and snap rings.

9.8 Grease retainers

Grease retainers are generally located between the bearing cavity and oil sump to retain grease in the bearings.

9.9 Dowels and pins

Many different types of dowels and pins are used to provide positive location or to prevent movement between two or more parts under load.

9.9.1 Dowels and pins used for positive location

These devices are generally used to return parts to the exact position required if disassembly is necessary. Care should be taken to assure that the required holes are the proper size.

9.9.2 Dowels and pins used to prevent movement

These devices shall be selected based on the maximum design loads. Generally two or more of these devices are used. Care should be taken to assure that the required holes are the proper size.

CAUTION: In the above cases, the dowel and pin manufacturers recommendations for fit and strength should be followed.

23

ANSl/AGMA6901-D97

9.10 Spacers

Spacers are generally used to position bearings, gears, and other components. The spacer construction and material shall be of sufficient strength, stiffness and size to provide proper support for adjacent components under maximum internal and external design loads. Spacers shall withstand required assembly forces.

9.11 Seal retainers Seal retainers are generally used to position the seal in proper relationship to a shaft, or to lock a split type seal in place.

9.12 Fastener locking devices

Locking devices may be provided to lock fasteners in place. Typical locking devices include lo&washers, self-locking fasteners, locking compounds, locking tabs, and lock wiring.

24

AMERICAN NATIONAL STANDARD ANSI/AGMA 6001-097

Annex A (informative)

Allowable stresses for typical key and keyway materials rhe foreword, footnotes, and annexes, if any, are provided for informational purposes only and should not be construed as a part of ANSl lAGMA 6001 -D97, Design and Selection of Components for Enclosed Gear Drives.]

A.1 Purpose The purpose of this annex is to provide reference information required for key and keyway calculations for typical materials used in enclosed gear drives. Refer to 5.2 and 5.3 of ANWAGMA 6001-D97 to determine allowable stresses for materials and hardnesses not listed in this annex.

A2 Allowable stress

The allowable stress values in the following table are based on 70 percent of the material yield point. Other values may be used based on testing or experience.

Table A.1 - Allowable stresses for typical key and keyway materials

Hardness Yield strength, S, Allowable stress, lb/in* (N/mm9

Key or hub material HB lb/in* (Nlmn?) Shear, S, Compressive, S, Source AISI 1018 126 54 000 18900 37 800 1) Cold drawn ’ (370) (130) (260) AISI 1045 179 77 000 26 950 53 900 1) Cold drawn (530) (185) (370) AISI 4140 320 110 000 38 500 77 000 1) Heat treated W ’) (265) WO) Cast iron 160 - 190 85 0002) -- 59 500 3)

Class 30 (590) (415) Ductile iron 187 -255 80 0002) -- 56 000 3)

80-55-06 (550) (385) Heat treated forged bronze 162 48 000 -- 33 600 4)

(copper alloy no. 90673) (330) (230) Nickel-tin bronze (WE 65) 102 30 000 -- 21 000 4)

(copper alloy no. C90700 (205) (145) centrifugal cast) . Aluminum bronze (copper 195 alloy no. C95499 heat treated)

NOTES:

54 000 -- 37 800 4)

(370) (260)

l) SAE - Society of Automotive Engineers *) Compressive yield strength 3, MH - Machinery’s Handbook 4, CDA - Copper Development Association

25

(This page is intentionally left blank.)

26


Annex B (informative)

Allowable stresses for typical threaded fasteners

[The foreword, footnotes, and annexes, if any, are provided for informational purposes only and should not be construed as a part of ANSVAGMA 6001 -D97, Design end Selection of Components for Enclosed Gear Drives.]

B.l Purpose

The purpose of this annex is to provide reference information required for fastener calculations.

Table B.l - Allowable tensile stress for typical inch threaded fasteners

Grade Proof load Allowable designation Products Nominal diameter, in stress, lb/h?

Tensile preioadl) stress, IbfiG stress) lb/in2

SAEl Bolts, Screws, l/4 through l-1/2 33 0003) 24 750 19 800

Studs ___- sAE2 Bolts, l/4 through 3/44) 55 0003) 41 250 33 000

Screws, Studs Over 3/4 to 1 -l/2 33 000 24 750 19800

ASTM A-449 Bolts Over 1 -l/2 to 3 55 000 41 250 33 000

sAE4 Studs l/4 through 1 -l/2 65 000 48 750 39 000

- sAE5 Bolts, l/4 through 1 85 000 63 750 51000 Screws,

Studs Over 1 to 1 -l/2 74 000 55 500 44 400

WE 75) Bolts, l/4 through 1 -l/2 105 000 78 750 63 000 Screws

ASTM A-354 Grade BC

Bolts Over l/4 to 2-l/2 105 000 78 750 63 000

SAE8

Bolts

Bolts, Screws,

Studs

Over 2-l /2 to 4 95 000 71 250 57 000

l/4 through l-1/2 120 000 90 000 72 000

SAE 8.1

SAE 8.2

Studs

Bolts, Screws

l/4 through l-1/2 120 000 90 000 72 000

l/4 through 1 120 000 90 000 72 000

NOTES: l) The fastener tensile preload stress values in the table are based on torquing the fastener to produce atensile preload of 75% of its proof load to avoid fastener breakage. Other values of percentage of proof load may be used, based on testing or experience. 2, The allowable stress values in the table are based on 60% of the fastener tensile preload stress in order to ensure joint integrity. 3, Proof load test: Requirements in these grades apply only to stress relieved products. 4) Grade SAE 2 requirements for sizes l/4 through 3/4 inch apply only to bolts and screws 6 inches and shorter in length, and to studs of all lengths. For bolts and screws longer than 6 inches, Grade SAE 1 requirements shall apply. 5) Grade SAE 7 bolts and screws are roll threaded after heat treatment. Reference: Fastener Standards, Cleveland, Ohio: Industrial Fasteners Institute, 1966, Sixth Edition.

27

ANSIIAGMA 6001~097 AMERICAN NATlONAL STANDARD

Table B.2 - Allowable tensile stress for typical metric threaded fasteners

Proof load Tensile pre- IS0 property Nominal diameter, stress, load’) stress,

A;;;F;e

class mm N/mm* N/m& N/mm*’

4.6 5.0 through 36 225 169 135

4.8 1.6 through 16 310 232 186

5.8 5.0 through 24 380 285 228

8.8 16.0 through 36 600 450 360

9.8 1.6 through 16 650 488 390

10.9 5.0 through 36 830 622 498

12.9 1.6 through 36 970 728 582

NOTES: 1) The fastener tensile preload stress values in the table are based on torquing the fastener to produce a tensile preload of 75% of its proof load to avoid fastener breakage. Other values of percentage of proof load may be used, based on testing or experience. 2) The allowable stress values in the table are based on 80% of the fastener tensile preload stress in orderto ensure joint integrity. Reference: SAE Handbook, Warrendale, Pennsylvania: Society of Automotive Engineers, Inc., 1986.

28

AMERICAN NATlONAL STANDARD ANSI/AGMA 6901 -D97

Annex C (informative)

Interference fit torque capacity

[The foreword, footnotes, and annexes, if any, are provided for informational purposes only and should not be construed as a part of ANSIIAGMA 6001 -D97, Design and Selection of Components fix Enclosed Gear Drives.]

C.l Purpose

The purpose of this annex is to provide a typical calculation method for determining the amount of torque that can be transmitted by an interference fit between cylindrical surfaces without discontinuities.

C.2 Calculations

PC = 6

B B*+A* Ei(B’-&) + 4?;2, - $ + k

I . ..(C.l)

For similar metals E = Ei = E, and TV = b

E6 (c* - B*)(B2 - A*) PC = 283 (C* - A*)

. ..(C.2)

s, = p, . .

s,= Bpc B*+A* [ 1 B* -A* . .

,.(C.3)

.(C.4)

T = xB2J’Jf 2

. ..(C.S)

T = ~B*pcL.f 2mcl

., ..(C.5M)

where

A is inside diameter - inner member, in (mm); B is nominal outside diameter - inner

member, in (mm); c is outside diameter - outer member, in

(mm);

E

43

f L

PC &

s, T

6

f-4 PO

is modulus of elasticity, lb/r? (N/mm*); is modulus of elasticity - outer member, lb/in* (N/mm?; is modulus of elasticity - inner member, lb/$ (N/mm?; is coefficient of friction; is fit length, in (mm); is interface pressure, lb/in9 (N/mm*); is tangential stress at the interface-inner member, lb/n* (N/mm?; is tangential stress at the interface-outer member, lb/in* (N/mm?; is torque capacity resulting from interference fit, lb in (Nm); is diametral interference, in (mm); is Poisson’s ratio - inner member; is Poisson’s ratio - outer member.

Inner member Outer member

Figure C.l - Calculation terminology

29



30

AMERICAN NATIONAL STANDARD ANSVAGYA 6001 a97

125-

16 loo-

i2 I ?ij 12 a, 75 9 3 g 50-

0

a

25- 4

o- 9. , 160 200 240 280 320 360 400 440

Brine11 hardness I I I 1 I I , I

80 100 120 140 160 180 200 220 Tensile strength, 1000 lb/in2

J I 1 I t I 600 800 1000 1200 1400

Tensile strength, N/mr$

Annex D (informative)

Previous method - shaft design

pheforeword, footnotes, and annexes, if any, are provided for informational purposes only end should not be construed as a pert of ANSIIAGMA 6001 -D97, Design and Selection of Components for Encked Gear Drives.]

D.l Purpose

The purpose of this annex is to include the previous shaft design section of ANSVAGMA 6001-C88 (formerly AGMA 260) for reference purposes.

0.2 Shafting

The general equations for torsional and bending stress are shown in equations D.l and D.2. While the allowable stresses shown in figure D.l do not separately consider the effects of such things as shaft size, surface finish, operating temperature, corrosion, residual stresses and reliability, this method was the accepted practice for shaft design that has been included in AGMA standards and used successfully for many years.

02.1 Shaft stress calculation

Nominal shaft stresses are calculated as follows. The applicability of equations D.l and D.2 to the design of thin wall shafts where the ratio (&a) > 0.9 has not been established.

16Td, s, = . ..(D.l)

R(d$ - d;) 16 OOOTd,

.% = 3+,$ - d!) . ..(D.lM)

32Md, Sb =

n(d;t - df) . ..(D.2)

320OOMd,

Sb = zc(d$ - d?) . ..(D.2M)

Figure D.l - Allowable stress for steel shafts

31

ANSl/AGMA 6001 -D97 AMERICAN NATIONAL STANDARD

where

s, is calculated torsional shear stress, lb/i&’ <N/mm?;

T is shaft torque, lb in (Nm);

do is shaft outside diameter, in (mm); 4 is shaft inside diameter, in (mm); .Q is calculated bending stress, lb/k? (N/mm2); M is bending moment, lb in (Nm).

For solid shafting, equations D.l and D.2 simplify to:

16 T ss = - scd3,

16OOOT s, = - xd;

(D.3)

32M ‘b = 31;d3

0

32000M Sb = - xds

D.2.2 Allowable stress

For steel shafts the calculated stress due to bending and the calculated stress due to torsion shall not exceed the values shown in figure D.1. These stresses may exist simultaneously.

The allowable stress for steel shafts that are hardened by processes such as case carburizing or nitriding should be based on the core hardness of the material unless a detailed analysis or experience indicates that a different allowable stress be used.

D.2.3 Stress concentration

Shaft stresses concentrate near a change in the shaft or where a load is applied to the shaft. Typical stress concentrators include but are not limited to

key joints, shoulders, grooves, splines and interference fits. Notch sensitivity accounts for different materials reacting differently to the same theoretical stress concentration. The allowable stresses shown in figure D.l provide for stress concentrations, including notch sensitivity, up to 3.0. When the actual stress concentration is greater than 3.0 a detailed analysis is required.

D.2.4 Specific life

When designing a shaft for a specific number of rotating cycles, the allowable bending stresses of figure D.l may be multiplied by the factors in table D.l.

Table D.l - Allowable stress multipliers for shafting

Cycles up to 1000 cycles Over 1000 to 10 000 cycles Over 10 000 to 100 000 cycles Over 100 000 to 1 million cycles Over 1 million cycles

D.2.5 Deflection

Factor 2.4 1.8 1.4 1.1 1.0

Deflection (lateral, torsional and axial) is a function of the loading on the shaft, modulus of elasticity and the size of the shaft, and is independent of the hardness. Damage to bearings, gear teeth or other components may occur if deflection is excessive.

D.2.6 Peak loads

Bending and torsional stresses resulting from peak loads must be analyzed to assure that the mechanical properties of the shaft material are not exceeded.

32


Annex E (informative)

Sample problems - transmission shaft design t’Theforeword, footnotes, and annexes, if any, are provided for informational purposes only and should not be construed as a part of ANSIIAGMA 6001-097, Design end Selection of Components for Encbsed Gear Drives.]

E.l Purpose

The purpose of this annex is to provide examples of the application of clause 4, Shafting. The equations of this section will be applied to practical shafting problems.

E.2 Sample problem number 1

An AlSl 4140 steel helical pinion shaft, through hardened to 360 HB is subject to 10 000 peak load cycles @ 200% operating load and has the following loads imposed on it at the shafts critical section:

T

M

V

PX

= 1565 lb in (50% alternating); = 328 in lb (100% alternating - shaft is rotating); = 678 lb (100% alternating - shaft is rotating); = 424 lb (constant).

The sectional properties of the shaft at this point are:

4 = 0.94 in;

4 = 0.0 in;

4 = 1.25 in (as shown in figure 8); r = 0.12 in (as shown in figure 8).

The critical section is at a radius adjacent to a shoulder.

The stresses are:

z, = l6 (1565) = 9596 lb/in2 Jr (o.94)3

For a 50% alternating torque,

Tzr = hnt = ; = 4798 lb/in2

ab = = = 4022 lb/b’ JE (o.94)3

z, = 4 (678) (l=) = 1201 lb/i,,2 ?c (o.94)2

OP = -%%fk = 611 lb/i& = h x (o.94)2

=a.x =4022cos8

ha = 1201 sin 8 + 4798

&St = (4022)2 - 3(1201)2

3(1201) = 3289 < 'hr

Since%, is greater than%,, then the max stress is at 8, = 90”. Therefore o, = 0 and h = 5999 lb/in’.

Ga = {(0)2 + 3(5999)2}05 = 10 391 lb/in2

4, = {(611>2 + 3(4798)2}05 = 8333 lb/in’

i& = {(611+ 0>2 + 3(4798 + 5999)2}o*5 = 18 711 lb/in2

&I =500(360HEi)=180000 lb/in2

J; = 0.94(180 000) - 12 500 = 156 700 lb/in2

sf, = 05(180 000) = 90 000 lb/in2

Listed below are the fatigue lim it modification factors for this example:

kl = 30.3(180000)-"~315 = 0.670

NOTE: Shaft is machined to a 125 & surface finish.

&I = 0.869(0.94)"-"'=0.874

kc = 0.512(ln 1/O.99)o-11+ 0.508 = 0.817 (99% reliability)

k;i = 1.0

k = 1.0 (lo6 cycles)

!f = 1111 + 0.93 (1.6 - l)] = 0.64

where

Ir; = 1.60 (fkom figure 8)

% = 1.0

k = 0.670 (0.874) (0.817) (1.0) (1.0) (0.64) (1.0) = 0.306

s/ = 0.306 (90 000) = 27 557 lb/in2

The resulting safety factors for fatigue failure analysis and the peak load failure analysis are:

Fti = l/{(lO 391/27 557)2 + (8333/156 700)2)“.5 = 2.63

Fsp = 0.75(156700)=3 14

2(18711) - N$, is not calculated as F @ > Fp (Fp = 2 for helical gears).

33

ANSVAGMA 6091497 AMERICAN NATIONAL STANDARD

E.3 Sample problem number 2 omal = {[O + 6487 cos (31.8”)]’ + 3[6.577

An AlSl 4140 steel worm gear shaft throuah hardened to 360 HB is subject to 10 000 peak load cycles @ 300% operating load and has the following loads imposed on it at the shafts critical section:

T = 152 751 in lb (25% alternating);

M = 56 498 in lb (100% alternating - shaft is rotating);

V = 28 000 lb (100% alternating - shaft is rotating);

px = 0.

The sectional properties of the shaft at this point are:

43 = 4.46 in;

4 = 0.0 in.

This critical section has a keyway and a gear pressed onto it with an interference fit (9 taken as 0.33).

The stresses are:

+ 2204 sin (31.8”) + 2192]2}o.5 = 18 061 lb/in’

The calculation of the fatigue (endurance) lim it of the shaft is as follows:

&I = 500 (360 HB) = 180 000 lb/in2

J; = 0.94(180 000) - 12 500 = 156 700 lb/in2

sfe = 0.5 (180 000) = 90 000 lb/in2

Listed below are the fatigue lim it modification factors for this example:

kn = 30.3(180 OOO)-“.315 = 0.670 NOTE: Shaft is machined to a 125 & surface finish.

kl = O.869(4.46)4-o97 = 0.752

kc = 0.512fln 1/O.99)o-11 + 0508 = 0.817 (99% reliability)

b = 1.0

k = 1.0 (lo6 cycles)

67 = 0.33

kg = 1.0

tr = 16 = 8769 lb/in2 ax (4.46)3

k = 0.670 (0.752) (0.817) (1.0) (1.0) (0.33) (1.0) = 0.136

For a 25% alternating torque (torque is fluctuating between the maximum value and one-half maximum value):

% t = %I4 = 2192 lb/in2

tnr = 3~J4 = 6577 lb/in2 = h

9 = 0.136 (90 000) = 12 240 lb/in’

The resulting safety factors for the fatigue failure analysis and the peak load failure analysis are as follows:

Gf =1/{(8008/12 240)2+(11 392/156 700)2}o.5 = 1.52

ob = 32 = 6487 lb/in2 rc (4.46)3

F SP = 0.75 (156 700) = 2 17

3(18061) .

4 (a ‘O”) (lez) = 2204 lb/in2 F = JC (4.46)2

OP =o=c&

% =6487cosCI ’

ha = 2204 sin 0 + 2192

Tlesf = tH87>2 - 3(2204)2 = 4160, ‘ht 3 (2204)

As 1.0 5 Fgs Fp, calculate the permissible number of peak load cycles.

0, = 8008 w

= 24 617 lb/in’

c = lqIo[ o.8)l$,w,]2] = 6.229

Therefore, m = l/3

0.8, \&“V “VU, !\ I1 Qtl Mn\l = o 356g

0, = sin-1 2192 ( 1

loslo[( 12240 .

4160 = 0.555 radians = 31.8” 1

Gl = {[6487 ~0s (31.S0)12 + 3[2204 sin (31.8”) Nfi = = 140 329 cycles

+ 2192]2}o.5 = 8008 lb/in2 I

4, = {02 + 3 (6577)2}o.5 = 11392 lb/in2 As NfO is greater than the 10 000 peak load cycles expected in service, this is acceptable.

34

ANSI/AGMA 6001 -D97

E.4 Sample problem number 3

A carburized and hardened AISI 9310 steel helical pinion shaft has a core hardness of 300 HB and a 63 & surface finish at the shafts critical section. Only 100 000 cycles are required during service, and no peak loading is present. The loading is:

T = 4000 lb in (25% alternating); M = 3000 in lb (100% alternating - shaft is

rotating); V = 1000 lb (100% alternating - shaft is

rotating);

px = 0. The sectional properties are:

4 = 1.25 in;

4 = 0.0 in;

4 = 1.75 in (as shown in figure 8); r = 0.06 in (as shown in figure 8).

The critical section is at a radius adjacent to a shoulder.

The stresses are:

z, = 16 (4000)

x (1.25Q = 10 430 lb/in2

For a 25% alternating torque;

ht = >/4 = 2607 lb/in2

14nt = 3~J4 = 7823 lb/in2 = &

=b _ 32 (3000)

x (1.2q3 = 15 646 lb/in2

% = 4 CIOoo) (1-Z) = 1002 lb/in2 Jr, (1.25)’

OP =o=c&

ax =15646cos8

b = 1002 sin 8 + 2607

ttest = (15 646)2 - 3(1002)2 = 80 434 > %I 3 (1002)

Therefore,

qc

Set 8, = 0.

_a. 0, = 15 646 lb/in2 and z, = 2607 lb/in2

G2 = ((15 646y + 3(26O7)2}o-5 = 16 285 lb/in’

s, = {(0>2 + 3(7823)2}o.5 = 13 550 lb/in’

GOtal = ((0 + 15 646)2 + 3(7823 + 2607)2}o.5 = 23 899 lb/in2

For a surface hardened part, it is usually conservative to analyze it at its core hardness. For this example, 300 HB is arbitrarily chosen. This particular part has a surface hardness of greater than 600 HB and also some residual compressive stress at the surface, whose beneficial effect is beyond the scope of this standard.

The calculation of the fatigue (endurance) lim it at the 300 HB core hardness is as follows:

44 = 500 (300) = 150 000 lb/in2

J; = 0.94 (150 000) - 12 500 = 128 500 lb/in2

s/, = 0.5 (150 000) = 75 000 lb/in2

k =%kbkkdk~ckg

k = 14.2 (150 000)4-244 = 0.775

4 = 0.869 (1.25)-“.097 = 0.850

k = 0.817 @ 99% reliability

ki = 1.0 @ 100°F

4 = 1/(1+0.87(2.14-l)} = 0.503

kg = 1.0

To calculated,, set k = 1.0, k = 0.271; Se = k sfe = 20304 lb/in2.

For this example, the partwill only be run for 100 000 cycles. Therefore, k will be calculated at 1 O5 cycles.

m = l/3 logI 0.8 [ (gg)] = 0.257

C = log10 [(0.8 Su)2/Se] = 5.851

ke = (lo5-851)(loo -o-257) = l . 81 000 20 304

@ 1o5 cyc,es

sf = 20 304 (1.81) = 36 750 lb/in2

Thus,

Fif = 1/{(16 285/36 750)2 + (13.550/128 500)2}“.5 = 2.20

F sp = 0.75 (128 500) = 2 o2

2(23 899) -

35

ANSIfAGrnA 6091-097 AMERICAN NATIONAL STANDARD


36

AMERICAN NATIONAL STANDARD ANSIIAGUA 6001 -D97

Annex F (informative)

Sample problems - deflection

[The foreword, footnotes, and annexes, if any, are provided for informational purpases only and should not be construed as a part of ANSIIAGMA 6001 -D97, Design and Selection of Components for Enclosed Gear Drives.]

F.1 Purpose

The purpose of this annex is to illustrate the method of numerical integration in the determination of shaft deflection and slope. A sample shaft will be presented broken into various nodes. Then deflection and slope about its neutral axis will be determined at the various nodes. The main objective of this example is to calculate the slope of the shaft through the journal bearing so that the bearing can be analyzed for sufficient clearance.

F.2 Sample problem number 1

Determine the slope and the deflection of the shaft at a point A as shown in figure F.l.

Solution: Refer to 4.7.2.1. The various parameters are:

L is length of the shaft = 90 in;

x is distance where deflection and slope are required = 30 in;

a is distance from support to the concentrated load = 70 in;

I is moment of inertia = (sc/64)& = 0.7854 in4; E is Young’s modulus = 30 x lo6 lb/in2;

Y is deflection of shaft at x, in;

@b is shaft slope at x, rad;

Shaft Diameter = 2 in

30 in

R2 Figure F.l

80 lb Shaft Diameter = 2 in L 60 in l

70in_I“An

R2 Figure F.2.

F is load = 120 lb. Since x < a, use equation 46 for determining the deflection and equation 47 for slope.

F(x)@ - a)x2 - 2aL + a2 Y= 6EIL

. ..(46)

0, = F(L - a) 3x2 - U + a2

6EIL . ..(47)

Substituting in the above equations, we get:

120(30)(90 - 70) 302 - 2(90)(70) + 702 Y= 6(30) lo6 (O-7854)(90) . ..(F.l)

= -0.0385 in (minus sign indicates deflection is downward)

8, = 120(90 - 70)[3(302) - 2(70)(90) + 7021

6(30)( 106)(0.7854)(90) . ..(F.2)

= -0.0009 rad (minus sign indicates slope is pointed downward as x increases)


Determine the slope and the deflection of the shaft at a point A as shown in figure F.2.


L is length of the between supports = 90 in; X is distance where deflection and slope are

required = 70 in;

129 lb

37

ANSI/AGMA 6001 -D97

I is moment of inertia = (x&4)& = 0.7854 in4;

E is Young’s modulus = 30 x 1 OS lb/in2;

Y is deflection of shaft at 4 in;

eb ‘is shaft slope at x, rad;

F is load = 80 lb.

Since x > O, use equation 48 for determining the defiection and equation 49 for slope.

Y= F(u)(L - x)(x* + a* - 2Lx)

6EIL . ..(48)

8, = F(a)(6Lx - 3x2 - 2L2 - a*)

6EIL . ..(49)


80(30)(90 - Y

70)[70* + 302 - 2(90)(70)] =

6 (30) ( 106) (0.7854)(90) . ..(F.3)

= -0.0256 in (minus sign indicates deflection is downward)

6, = 80(30)[6(90)(70) - 3(702) - 2(90*) - 30*]

W)( 10Q)@78WP0~ = 0.0011 rad

. . . (F 4)


Determine the slope and the deflection of the shaft at a point A as shown in figure F.3.

a is distance from support to the concentrated load = 30 in.


L is length of shaft between supports = 90 in;

X is distance where deflection and slope are required = 30 in;

0 is distance from support to the concentrated load = 20 in;

I is moment of!nertia = @/84)& = 0.7854 in4;

E is Young’s modulus = 30 x 1 O6 Ib/in2;

Y is deflection of shaft at x, in;

eb is shaft slope at 4 rad;

F is load = 120 lb. Shaft Diameter = 2 in

k-3Oin-4


Since x < L, use equation 51 for determining the deflection and equation 52 for slope.

. ..(51)

8, = F(a)(L* - 3x2) . ..(52)


Y = 120(20)(30)(90* - 302) = o 0407 in

6 (30) ( 106)(0.7854)(!JO) ’ . ..(E5)

8, = 120(20)[9~* - 3Po*)] = o oolo rad 6(30)(106)(0.7854)(90) ’ . ..(F.6)


The steel pinion shaft presented above in figure F.4 will be analyzed. The slope of the shaft about its neutral axis at point A and F will be determined for further bearing analysis.

The shear and moment diagrams for this problem are illustrated in figure F.5.

The deflection and slope of the simply supported. beam will be calculated at the points indicated in the figure. Numerical integration will be used to determine these values [14].

The following two successive integrals will be used: x

9 = ii+ . ..(F.7)

0 X

v= w I . ..(F.8)

0

The slope is calculated from: X

. . . (F.9)

Second integration yields deflection: y=l#+c]x+c* . ..(F.lO)

120 lb

c

Rl R2 Figure F.3

38


A B C D 1.75 in 9298 lb

I

E F

‘lr AV

t 3.5 in ’

IL t _ 5.0 in 5.38 in 2.5 in

Dia X Dia - - - - Dia Dia

-L ,.A

- 5.0 in I-

i.Oin+ ,Iin-4 ,+ 1.31 in

Figure F.4 - Shaft geometry and loading

6859 +

ob”,

Figure F.5 - Shear and moment diagrams

Where Cl and C2 are determined by the values of x and q at the supports where deflection is zero:

(F.ll)

. . ..(F.12)

Rewriting equation F.7 using the trapezoidal rule yields:

+i+2 = % + 4[ @gi+, + (g)i] (xi+2 - Xi)

. ..(F.13)

r

X

2439

-X

Applying Simpson’s rule, equation F.8 produces:

Wi+d=Vi + 1f6(&+4+4&+2 + (pi)&+4 -4) . ..(F.14)

Table F.l presents the tabulated values for the equations presented, applied to the example to determine the deflection and slope of the shaft. Therefore, the slope of the shaft through the center of bearing A is -0.000 28 radians and through the center of bearing F is 0.000 16 radians.

39

ANSIIAGMA 6001 -D97

Table F.l

Station

A

B

C

D

E

F

E =301

X

(in) 0

0.875

1.75

3.375

5.00

7.50

10.00

13.875

17.75

18.41

19.08

i 0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21

(steel s

0 3.50 3.50 3.50 3.50 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.38 5.38 5.38 5.38 2.50 2.50 2.50 2.50

0 SW

I (in4)

0 7.366 7.366 7.366 7.366 31.42 31.42 31.42 31.42 31.42 31.42 31.42 31.42 41.12 41.12 41.12 41.12 1.918 1.918 1.918 1.918

0

(inyb) M /El 0 0 0 0

6002 27.16 6002 27.16

12003 54.32 12003 12.73 23149 24.56 23149 24.56 34 295 36.38 34 295 36.38 28198 29.92 28198 29.92 22100 23.45 22100 17.92 12649 10.25 12649 10.25

3195 2.59 3195 55.53 1585 27.55 1585 27.55

0 0 0 0

Values for MIEI, c$ and 3 at-e x 1 O6 c2 = 0

Y (in)

0

&O -0.00028

11.88

47.53 27.72 -0.00037 -0.00018

77.83

127.3 291.1 -0.00085 -0.000 10

210.2

276.9 1329 -0.000 96 0.000 05

331.5

356.4 3860 0.000 13

383.8

392.8 4359 19

-0.00020

0 0.000 16

40

, Annex G (informative)

Q

References

rhe foreword, footnotes, and annexes, if any, are provided for informational purposes only and should not be construed as a part of ANSIIAGMA 6001 -D97, Design and Selection of Components for Enclosed Gear Drives.]

c -

The following documents are either referenced in the text or included for additional information.

1. AGMA 904-C96, Metric Usage.

2. ANSIIAGMA 9002-A86, BoresAnd Keyways for Flexible Couplings.

3. ANSI/ASME B106.1M-1985, Design Df Trans- mission Shafting (second printing). 4. Cast Bronze Bearihg Design Manual. Evanston, IL: Cast Bronze Bearing Institute. 5. Fastenerstandards. Cleveland, Ohio: Industrial Fasteners Institute, 1988. Sixth Edition. 6. Hopkins, Bruce R. Design And Analysis Cf ShaftsAnd Beams. New York: McGraw Hill Book Company, 1970. 7. Juvinall, Robert C. Stress, Strain And Strength. New York: McGraw Hill Book Company, 1967. 8. Metric Fastener Standards. Cleveland, Ohio: In- dustrial Fasteners Institute, 1983, Second Edition. 9. Peterson, R. E. Stress Concentration Factors. New York: John Wiley and Sons, 1974. 10. Shigley, Joseph E. and Mitchell, Larry D. Me- chanical Engineering Design. New York: McGraw Hill Book Company, 1983, Fourth Edition.

11. SAE J 429 January, 1980.

12. Wellauer, Edward J., Design of Shafting For Gear Drives, AGMA Technical Paper P246.01, Oc- tober 1966.

13. Roark, Raymond J. and Young, Warren C., Formulas for Stress and Strain New York: McGraw Hill Book Company, 1975, Fifth Ediiion.

14. Shigley, Joseph E. and Mischke, Charles R., Standard Handbookof Machine Design. New York: McGraw-Hill Book Company, 1986.

15. Shigley, Joseph E. and Mischke, Charles R., Mechanical Engineering Design, New York: McGraw-Hill Book Company, 1989, Fiih Edition.

16. Sullivan, J. L., “Fatigue Life Under Combined Stress,” Machine Design, January 25, 1979.

17. ANSIIAGMA 2001 -C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth.

18. NASA, Reference 1123, Design of Power Transmitting Shafts, S. Lowenthal, 1984.

19. Bethlehem Steel Corp., Modem Steels and Their Properties, Seventh Edition, 1972.

41

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PUBLISHED BY AMERICAN GEAR MANUFACTURERS ASSOCIATION 1500 KING STREET, ALEXANDRIA, VIRGINIA 22314

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Ansi Agma6001 d97