Transferred torque

where:

	P	transferred power [lb ft]
	n	speed [min -1 ]

Minimum shaft diameter

1. shaft inside diameter d h > 0

a)

b) if d min ≤d h -> d min = 1.1 d h [in]

if else, the following formula is used to find the best diameter:

2. shaft inside diameter d h = 0

where:

	d min	minimal shaft diameter [inm]
	d h	shaft inside diameter [in]
	T	torque [lb ft]
	K a	application factor
	K f	fatigue-life factor
	S v	desired safety
	τ Al	Allowable Shear Stress

General calculation

Minimum splines length to transfer the torque

1. Fixed connection:

2. Flexible connection:

where:

	T	torque [lb ft]
	K a	application factor
	K f	wear-life factor
	K w	application factor
	K m	load distribution factor
	S v	desired safety
	d s	middle diameter = (D + d) / 2 [in]
	D	outside diameter of groove section [in]
	d	inside diameter of groove section [in]
	N	number of grooves [-]
	h	height of groove = (D - d) / 2 [in]
	s	chamfer [in]
	h st	connection height h st = h - 2 s [in]
	p Dmin	allowable pressure on supporting surface of shaft or groove [psi]

Allowable pressure

1. Fixed connection::
2. Flexible connection:
where:

	T	torque [lb ft]
	K a	application factor
	K f	wear-life factor
	K w	application factor
	K m	load distribution factor
	S v	desired safety
	d s	middle diameter = (D + d) / 2 [in]
	D	outside diameter of groove section [in]
	d	inside diameter of groove section [in]
	N	number of grooves [-]
	h	height of groove = (D - d) / 2 [in]
	s	chamfer [mm]
	h st	connection height h st = h - 2 s [in]
	l f	active key length [in]

Strength Check

p min ≤ p Ds

p min ≤ p Dh

where:

	p min	minimal calculated h/2 pressure [psi]
	p Ds	allowable pressure in shaft [psi]
	p Dh	allowable pressure in hub [psi]

Shear stress under roots of external teeth

For a transmitted torque T, the torsional shear stress induced in the shaft under the root diameter of an external spline

For a solid shaft

For a hollow shaft

Shear stress at the pitch diameter of teeth

The shear stress at the pitch line of the teeth for a transmitted torque T

Calculated spline length:

L f = min {L, Le} [in]

Compressive stresses on sides of spline teeth

Allowable compressive stresses on splines are much lower than for gear teeth, since nonuniform load distribution and misalignment result in unequal load sharing and end loading of the teeth.

For flexible splines

For fixed splines

where:

Calculated spline length	L f = min {L, Le} [in]
Depth of teeth engagement h h ≅ 0.9 / P [in]	for splines with flat groove bottom
h ≅ 1 / P [in]	for splines with filleted groove bottom

Bursting stresses on splines

Internal splines may burst due to tensile stress from the radial component of the transmitted load, centrifugal tensile stress, tensile stress due to the tangential force at the pitch line causing bending of the teeth.

1. Radial load tensile stress

where: Wall thickness of internal spline	t w = D oi - D ri [in]
2. Centrifugal tensile stress
3. Beam loading tensile stress

where: Calculated spline length

L f = min {L, L e } [in]

4. The total tensile stress tending to burst the rim of the external member

Crowned splines for large misalignments

Crowned splines can accommodate misalignments of up to about 5 degrees. Crowned splines have considerably less capacity than straight splines of the same size if both are operating with precise alignment. However, when large misalignments exist, the crowned spline has greater capacity.

American Standard tooth forms may be used for crowned external members so that they may be mated with straight internal members of Standard form.

Compressive stress on the teeth

where:

Radius of curvature of the crowned tooth	r 2 ≅ F 2 / 8 A [in]
Radius of the crown	r 1 = r 2 tan Φ [in]
Depth of teeth engagement h h ≅ 0.9 / P [in]	for splines with flat groove bottom
h ≅ 1 / P [in]	for splines with filleted groove bottom

Meaning of used variables

T	torque [lbft]
n	speed [min -1 ]
D	pitch diameter [in]
D ri	major (root) diameter of internal spline [in]
D re	root diameter of external spline [in]
D h	inside diameter of a hollow shaft [in]
D oi	outside diameter of grooved hub [in]
N	number of grooves [-].
h	height of groove [in]
L f	calculated spline length [in]
L	active spline length [in]
L e	maximum effective length [in]
t	actual tooth thickness, circular [in]
t w	wall thickness of internal spline = outside diameter of spline sleeve minus major diameter, all divided by two [in]
Φ	stroke angle [deg.]
Y	Lewis form factor obtained from a tooth layout. The Y = 1.5 value is used in the calculation.
F	pressure angle [deg]
S v	desired safety
A	relief of crown at the ends of the teeth [in]
r 1	fillet radius of crown spline [in]
r 2	curvature radius of rounded teeth for crown spline [in]
K a	application factor
K f	wear-life factor
K w	application factor
K m	load distribution factor
K s	factor of tooth side
	K s = 0.5 for splines with common and higher accuracy of mounting (only one half of teeth carries the load)
	K s = 0.3 for splines with lower production and assembly accuracy (only one third of teeth carries the load)