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### The Shaft Design Book

Machine Design Applications, Equations and Calculators

Structural Deflection Equations and Stress

The Shaft Design Book (Design Charts and Calculations for Torsional Properties of Non-Circular Shafts)

Robert I. Isakower

94 Pages

SCIENTIFIC & ENGINEERING APPLICATIONS DIVISION

MANAGEMENT INFORMATION SYSTEMS DIRECTORATE

The Shaft Design Book (Design Charts for Torsional Properties of Non-Circular Shafts)

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Abstract

Design charts and tables have been developed for the elastic torsional stress analyses of free prismatic shafts, splines and spring bars with virtually all commonly encountered cross sections. Circular shafts with rectangular and circular keyways, external splines, and milled flats along with rectangular and X-shaped torsion bars are presented. A computer program was developed at the U.S. Array ARRADCOM, Dover, N.J. site which provides a finite difference solution to the governing (POISSON's) partial differential equation which defines the stress functions for solid and hollow shafts with generalized contours. Using the stress function solution for the various shapes and Prandtl's membrane analogy, the author is able to produce dimensionless design charts (and tables) for transmitted torque and maximum shearing stress. The design data have been normalized for a unit dimension of the cross section (radius or length) and are provided in this report for solid shapes. The eleven solid shapes presented, along with the classical circular cross section solution, provides the means for analyzing 144 combinations of hollow shafts with various outer and inner contours.

Hollow shafts may be analyzed by using the computer program directly or by using the solid shape charts in this report and the orinciples of superposition based on the concept of parallel shafts. The SHAFt Torsion utility program (SHAFT) used for the generation of the data in this handbook is a spin-off of the famous Computer Language for Your Differential Equations (CLYDE) code and employs the same basic mathematical model along with an improved algorithm for maximum stress. The format of the stress charts differs slightly from those of the first report in this series (Technical Report ARMID-TR-78001). Stress/torque ratio curves are presented as being more intuitively recognizable than those of stress.

The elastic stress analysis of uniformly circular shafts in torsion is a familiar and straightforward concept to design engineers. As the bar is twisted, plane sections remain plane, radii remain straight, and each section rotates about the longitudinal axis. The shear stress at any point is proportional to the distance from the center, and the stress vector lies in the plane of the circular section and is perpendicular to the radius to the point, with the maximum stress tangent to the outer face of the bar. (Another shearing stress of equal magnitude acts at the same point in the longitudinal direction.) The torsional stiffness is a function of material property, angle of twist, and the polar moment of inertia of the circular cross-section.

TOC

- The Torsion Problem 1
- Design Charts and Tables
- Accuracy of the Computerized Solution 54
- Parallel Shaft Concept 55
- Bibliography 64
- Appendixes

A Mathematical Model Used in the CLYDE 65

Computer Program

B Extension of Model to Hollow Shafts 78

Tables

Tables

1 Element nomenclature 5

2 Split shaft, volume factor (v) 7

3 Split shaft, slope factor (dO/ds) 9

4 Single keyway shaft, volume factor (V) 11

5 Single keyway shaft, slope factor (dO/ds) 13

6 Two keyway shaft, volume factor (V) 15

7 Two keyway shaft, slope factor (d<5/ds) 17

8 Four keyway shaft, volume factor (V) 19

9 Four keyway shaft, slope factor (dO/ds) 21

10 Single square keyway with inner fillets 23

11 Single spline shaft, volume factor (V) 25

12 Single spline shaft, slope factor (dO/ds) 27

13 Two spline shaft, volume factor (V) 29

14 Two spline shaft, slope factor (dO/ds) 31

15 Four spline shaft, volume factor (V) 33

16 Four spline shaft, slope factor (dΦ/ds) 35

17 Square keyways and external splines volume factor (V), 37

18 Square keyways and external splines, slope factor 39

19 Milled shaft, volume factor (V) 41

20 Milled shaft, slope factor (dO/ds) 43

21 Rectangular shaft 45

22 Pinned shaft, volume factor (V) 47

23 Pinned shaft, slope factor (dO/ds) 49

24 Cross shaft, volume factor (V) 51

25 Cross shaft, slope factor (dO/ds) 53

Figures

1 Membrane analogy 3

2 Split shaft, torque 6

3 Split shaft, stress 8

4 Single keyway shaft, torque . 10

5 Single keyway shaft, stress 12

6 Two keyway shaft, torque 14

7 Two keyway shaft, stress 16

8 Four keyway shaft, torque 18

9 Four keyway shaft, stress 20

10 Single square keyway with inner fillets 22

11 Single spline shaft, torque 24

12 Single spline shaft, stress 26

13 Two spline shaft, torque

14 Two spline shaft, stress

15 Four spline shaft, torque

16 Four spline shaft, stress

17 Square keyways and external splines, torque

18 Square keyways and external splines, stress

19 Milled shaft, torque

20 Milled shaft, stress

21 Rectangular shaft

22 Pinned shaft, torque

23 Pinned shaft, stress

24 Cross shaft, torque

25 Cross shaft, stress

26 Parallel shaft concept

27 Milled shaft with central hole 57

28 Circular shaft with four inner splines

29 Superposition for two spline shaft

30 Superposition A for four spline shaft

31 Superposition B for four spline shaft

32 Illustrative design application 63