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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

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

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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.


  • 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


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


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