This page provides engineering formulas and worked examples for estimating the strength of keys and splines used to transmit torque between shafts and hubs. The calculations consider shaft shear stress together with key and spline shear and compressive (bearing) stresses.
The methods are useful for preliminary mechanical design and for understanding the load path through a keyed or splined shaft connection. Service factors are included to account for operating conditions, fatigue, wear and load distribution.
The following notes provide methods for calculating reasonable estimates of the stresses resulting from an applied torque. The calculations allow the operating conditions to be included through a service factor Ks.
Depending on the connection being considered, the service factor may include:
The wear factor is primarily relevant to splines and other connections where relative axial movement occurs.
The resulting design stress should be compared with an appropriate material design strength, allowing for the required factor of safety.
The original RoyMech calculations were also compared with external spline calculation methods. Different spline design methods can use different definitions of allowable stress, service factors and fixed or flexible connections. Care is therefore required when comparing numerical results between methods.
Note: Tables of the factors and design strengths used by the original RoyMech method are provided on the Spline Factors page.
Before checking the key or spline, it is good practice to check the torque capacity of the shaft itself. Where a keyway or spline removes material from the shaft, the reduced shaft section should also be considered.
Note: Tables of the various factors and design strengths are provided on the Spline Factors page.
|
Note: Tables of the various factors and design strengths are provided on the Spline Factors page.
A key transmitting torque should normally be checked for both shear and compressive (bearing) stress. The reduced shaft section produced by the keyway should also be checked.
To determine the torque capacity of the key:
|
It is reasonable for an initial calculation to use x = h/2. The geometry shown below provides a more representative value of x. 
|
The bearing stress in the hub should also be considered where the hub material has a significantly lower design strength than the shaft or key material.
Based on shear
Based on compressive strength
Note: Tables of the various factors and design strengths are provided on the Spline Factors page.
For a straight-sided spline, torque is transmitted through several spline faces around the circumference. The calculation below estimates the compressive stress on the working faces.
|
Based on compressive strength
Note: Tables of the various factors and design strengths are provided on the Spline Factors page.
For an involute spline the torque is distributed through a number of engaging teeth. The following approximate method checks tooth shear and compressive stress together with the strength of the reduced shaft section.
For the 30° spline geometry used in the original RoyMech method, h = 0,9m for the flat-root form and h = m for the fillet-root form. The applicable spline standard should be checked for the geometry being used. |
Three example calculations are provided below using the same transmitted power and rotational speed. The key and splines are assumed to be close-fit or guided connections.
The original RoyMech example uses the factors and material design strengths from the Spline Factors page. The purpose of the example is to show the calculation method and to compare the resulting stresses for a key, a straight-sided spline and an involute spline.
A transmitted power of 10 kW at 1500 rev/min is used. A nominal shaft diameter of 25 mm is then used as the basis for the three connection examples.
|
|
Assume a basic shaft diameter Do = 25 mm. The torsional shear stress in the shaft is:
τ = 16 T Ks / (π Do3)
τ = 16 × 63,66 × 1000 × 2 / (π × 253) = 41,5 MPa
Factor of safety = Ss / τ = 200 / 41,5 ≈ 4,8
Select an 8 mm × 7 mm key, 70 mm long.
Le = effective length of key =
70 - (2 × 4) = 62 mm
x = 2,94 mm
x1 = 3,26 mm
Di = reduced shaft diameter =
25 - 4 = 21 mm
τ = 16 T Ks / (π Di3)
= 16 × 63,66 × 1000 × 2 / (π × 213) = 70,0 MPa
Factor of safety ≈ 200 / 70,0 = 2,86
For this example the compressive stress between the key and the mating surface is the more restrictive of the key checks.
σc = T Ks / [Le (D/2) x]
= 63,66 × 1000 × 2 × 2 / (62 × 25 × 2,94) = 55,8 MPa
Factor of safety ≈ 130 / 55,8 = 2,33
Original comparison calculation: approximately 58,5 MPa
Select a straight-sided spline designated 25 - 6 × 21 × 25. For this example:
τ = 16 T Ks / (π Di3)
= 16 × 63,66 × 1000 × 2 / (π × 213) = 70,0 MPa
Factor of safety ≈ 200 / 70,0 = 2,86
σc = T Ks / (Le n r y)
= 63,66 × 1000 × 2 / (25 × 6 × 11,5 × 1,4) = 52,7 MPa
Factor of safety ≈ 130 / 52,7 = 2,47
Original comparison calculation: 52,7 MPa
Select an involute spline with module m = 1 and 24 teeth. The example geometry is:
τ = 16 T Ks / (π Die3)
= 16 × 63,66 × 1000 × 2 / (π × 22,53) = 56,9 MPa
Factor of safety ≈ 200 / 56,9 = 3,51
Original comparison calculation: 56,9 MPa
τ = 4 T Ks / (L z t D)
= 4 × 63,66 × 1000 × 2 / (10 × 24 × 1,5708 × 24) ≈ 56,3 MPa
Factor of safety ≈ 200 / 56,3 = 3,55
σc = 2 T Ks / (L z h D)
= 2 × 63,66 × 1000 × 2 / (10 × 24 × 0,9 × 24) ≈ 49,1 MPa
Factor of safety ≈ 130 / 49,1 = 2,65
Original comparison calculation: approximately 46,3 MPa
The example demonstrates why a shaft-to-hub connection should not be assessed only from the nominal shaft diameter. Cutting a keyway or spline changes the minimum shaft section, while the connection itself introduces shear and bearing stresses that must also be checked.
For each design the engineer should therefore consider the complete load path: shaft, reduced shaft section, key or spline teeth, hub contact surfaces and the effects of the actual operating duty.
The simplified calculations are useful for initial sizing and for comparing alternative arrangements. Detailed spline design additionally requires consideration of actual tooth load distribution, dimensional tolerances, alignment, manufacturing accuracy, fatigue, wear and the requirements of the applicable spline standard.