Involute splines are widely used in mechanical engineering to transmit torque between shafts and hubs. They provide multiple engaging teeth around the shaft circumference and can be used for both fixed and sliding shaft-to-hub connections.
This guide covers straight cylindrical involute splines in accordance with ISO 4156, including geometry, modules, pressure angles, tolerances and outline design calculations.
Important note:
The engineering information and numerical data on this page were originally
compiled with reference to the relevant British Standards and Machinery's
Handbook. The detailed tables and equations should be treated as outline
engineering reference information. Detailed design, manufacture and inspection
should be completed using the applicable current standard or another
authoritative technical source.
Involute splines are toothed shaft connections that use an involute tooth profile similar to that used for gears. They transmit torque through multiple engaging tooth flanks and may be used for fixed or sliding connections between shafts and hubs.
The notes and tables below relate to straight cylindrical involute splines, metric module, side fit, in accordance with the ISO 4156 system. The spline system described here is a side-fit system, with centring based primarily on the sides of the spline teeth.
Involute splines provide multiple load-carrying tooth surfaces around the shaft. External splines may be produced by processes including hobbing or gear shaping, while internal splines may be produced by broaching or gear shaping. To control the fit, the effective space width and other dimensional characteristics of the internal and external spline are controlled within specified tolerances.
A simple method of making an initial involute spline selection is to establish a pitch circle diameter (D) and module (m). For example, for an approximate 50 mm outside diameter spline with a module of 2 mm, an initial pitch diameter of 48 mm gives:
z = D / m = 48 / 2 = 24 teeth
This is only an initial geometric selection. Detailed dimensions, tolerances, fit, strength and manufacturing requirements must be established using the applicable standard and engineering requirements.
The fit of the spline is controlled through the dimensional and tolerance requirements of the spline standard. General ISO limits and fits information is available on the RoyMech ISO Limits and Fits pages.
Shaft connections based on involute splines can be used for transmitting
cyclic and shock torsional loads and for both fixed and sliding connections
between shafts and hubs.
The spline profile uses involute teeth with nominal pressure angles of
30°, 37.5° or 45°.
The spline system covered by ISO 4156 is primarily a side-fit system in which
the tooth flanks establish the functional fit.
The tooth root may be flat or filleted.
ISO 4156 uses the following modules.
For pressure angles of 30° and 37.5° the module series is:
0,5; 0,75; 1; 1,25; 1,5; 1,75;
2; 2,5; 3; 4; 5; 6; 8; 10
For a pressure angle of 45° the module series is:
0,25; 0,5; 0,75; 1; 1,25;
1,5; 1,75; 2; 2,5
Male Involute Spline
A simple initial geometric selection can be made by determining the pitch circle diameter (D) and module (m). The number of teeth is:
z = D / m
For example, using a pitch circle diameter of 48 mm and module 2 mm gives:
z = 48 / 2 = 24 teeth
The resulting spline must then be checked against the standard for the required pressure angle, root form, fit class, dimensions, tolerances and inspection requirements.
Involute splines distribute transmitted torque through multiple tooth flanks rather than through a single key. They can therefore provide substantial torque capacity within a compact shaft-to-hub connection and are suitable for both fixed and sliding arrangements.
The actual load distribution between spline teeth is influenced by manufacturing accuracy, alignment, fit, tooth geometry and elastic deformation. Spline capacity and shaft strength should therefore be assessed for the particular application rather than inferred solely from the number of teeth.
Compared with a simple keyed connection, an involute spline is generally more complex to design, manufacture and inspect. The choice between an involute spline, straight-sided spline, key or another shaft-to-hub connection depends on the required torque, alignment, axial movement, manufacturing method, inspection requirements and service conditions.
The current BS ISO 4156 series for straight cylindrical involute splines, metric module, side fit comprises:
Legacy-data note: The engineering data on this RoyMech page was originally prepared with reference to the earlier BS ISO 4156:2005 series. The 2005 editions have been superseded. The current standards should therefore be consulted for new design, manufacture and inspection work.
An example set of designations for a mating spline pair with 32 teeth, 2,5 module, a 30° pressure angle, fillet root and class 5 fit is:
Mating: INT/EXT 32z ° 2,5m ° 30R ° 5H/5f ISO 4156
Internal spline: INT 32z ° 2,5m ° 30R ° 5H ISO 4156
External spline: EXT 32z ° 2,5m ° 30R ° 5f ISO 4156
| Pressure Angle | α | degree (o) |
| Pressure Angle At Pitch Diameter |
αD | degree (o) |
| Number of teeth | z | - |
| Module | m | D / z |
| Pitch (Circular) | p | m.π |
| Basic Tooth thickness Circular |
S | p/2 |
| Basic space width Circular |
E | p/2 |
| Effective space width Circular (minimum) |
EV min | p/2 |
| Pitch diameter | D | m.z |
| Base diameter | Db | m° z ° cos αD |
| Base pitch | pb | m° π ° cos αD |
| Fundamental deviation, external | esv | Tooth modification factor – results from fit (h, f, e, d); see table below |
|
Minimum major diameter, internal 30°, flat root 30°, fillet root 37,5°, fillet root 45°, fillet root |
Dei min |
m° (z + 1,5) m° (z + 1,8) m° (z + 1,4) m° (z + 1,2) |
| Maximum major diameter, internal | Dei max | Dei min + (T + λ) / tan αD |
|
Minimum form diameter, internal 30°, flat root & fillet 37,5°, fillet root 45°, fillet root |
DFi min |
m° (z + 1) + 2.cF m° (z + 0,9) + 2.cF m° (z + 0,8) + 2.cF |
| Minimum minor diameter, internal | Dii min | DFe max + 2° cF |
|
Maximum minor diameter, internal: m =< 0,75 0,75 < m < 2 m > 2 |
Dii max |
Dii min + IT 10 Dii min + IT 11 Dii min + IT 12 |
|
Maximum actual Space width class 4 class 5 class 6 class 7 |
Emax |
EV min + (T + λ) EV min + (T + λ) EV min + (T + λ) EV min + (T + λ) |
| Minimum actual Space Width | Emin | Ev.min + λ |
| Maximum effective Space Width | EV max | Ev.min + TV |
|
Maximum major dia external. 30°, flat root & fillet 37,5°, fillet root 45°, fillet root |
Dee max |
m° (z + 1) + esv / (tan αD) m° (z + 0,9) + esv / (tan αD) m° (z + 0,8) + esv / (tan αD) |
|
Minimum major diameter, external: m =< 0,75 0,75 < m < 2 m => 2 |
Dee min |
Dee max - IT 10 Dee max - IT 11 Dee max - IT 12 |
| Maximum form diameter | DFe max |
|
|
Maximum minor diameter, external 30°, flat root 30°, fillet root 37,5°, fillet root 45°, fillet root |
Die max |
m° (z - 1,5) + esv / (tan αD) m° (z - 1,8) + esv / (tan αD) m° (z - 1,4) + esv / (tan αD) m° (z - 1,2) + esv / (tan αD) |
| Minimum minor, external | Die min | Die max - (T + λ) / tan αD |
| Maximum effective tooth thickness | SV max | S + esV |
|
Minimum actual tooth thickness class 4 class 5 class 6 class 7 |
Smin |
SV max - (T + λ) SV max - (T + λ) SV max - (T + λ) SV max - (T + λ) |
| Maximum actual tooth thickness | Smax | Sv.max - λ |
| Minimum effective tooth thickness | SV min | Sv.max - TV |
| Total tolerance, Space width or tooth thickness |
T + λ | See table below |
| Maximum effective clearance | CV max | Ev max - Sv min |
| Minimum effective clearance | CV min | Ev min - Sv max |
| Form tooth height | hs | See sketch and table for basic rack profile below |
Note: The form circle is the circle used to define the depth of involute
profile control.
For an external spline it is located near and above the minor diameter,
and for an internal spline near and below the major diameter.
| Parameter | Pressure Angle | |||
| 30o | 37,5o | 45o | ||
| Flat root | Fillet root | |||
| Major Space height | 0,75.m | 0,9.m | 0,7.m | 0,6.m |
| Major Tooth height | 0,5.m | 0,5.m | 0,45.m | 0,4.m |
| Form tooth height (hs) | 0,6.m | 0,6.m | 0,55.m | 0,5.m |
| Minor tooth height | 0,75.m | 0,9.m | 0,7.m | 0,6.m |
| Root radius (ρFi) | 0,2.m | 0,4.m | 0,3.m | 0,25.m |
| Root radius (ρFe) | 0,2.m | 0,4.m | 0,3.m | 0,25.m |
| Form radius (cF) | 0,1.m | 0,1.m | 0,1.m | 0,1.m |
Important Note: These values are all reductions and so they are negative (-ve) values.
| Pitch Dia. D (mm) |
Selected fit class | |||
| d | e | f | h | |
| Thickness reduction (esv) (mm) | ||||
| =< 3 | 0,020 | 0,14 | 0,006 | 0 |
| 3 to 6 (inc) | 0,030 | 0,020 | 0,010 | 0 |
| 6 to 10 (inc) | 0,04 | 0,025 | 0,013 | 0 |
| 10 to 18 (inc) | 0,05 | 0,032 | 0,016 | 0 |
| 18 to 30 (inc) | 0,065 | 0,040 | 0,020 | 0 |
| 30 to 50 (inc) | 0,080 | 0,050 | 0,025 | 0 |
| 50 to 80 (inc) | 0,10 | 0,060 | 0,030 | 0 |
| 80 to 120 (inc) | 0,12 | 0,072 | 0,036 | 0 |
The Total Tolerance (T + λ) is the sum of the Machining Tolerance (T) and the Effective variation (λ).
| Spline Tolerance class (T + λ) | Formula for (T + λ) | Formula for i1 & i2 |
| 4 | 10i1 + 40i2 |
i1 = 0,001[0,453.Sqrt (D) +0,001D ] i2 = 0,001[0,453.Sqrt (S +0,001 S ] Note: i1 applies for D < 500mm |
| 5 | 16i1 + 64i2 | |
| 6 | 25i1 + 100i2 | |
| 7 | 40i1 + 160i2 |
The effective variation is the combined effect of the total index variation (Fp), the positive profile variation (ff) and the tooth alignment variation (Fβ). These combined effects are calculated using the following equation.
| Spline Tolerance class |
Fp | ff | Fβ |
| 4 | 0,001.[ 2,5 Sqrt(M.z.β /2 ) + 6,3] | 0,001 [1,6.m (1+0,012z) + 10] | 0,01 [0,8.sqrt(g) + 4] |
| 5 | 0,001.[ 3,55 Sqrt(M.z.β /2 ) + 9] | 0,001 [2,5.m (1+0,012z) + 16] | 0,01 [1,0.sqrt(g) + 5] |
| 6 | 0,001.[ 5 Sqrt(M.z.β /2 ) + 12,5] | 0,001 [4.m (1+0,012z) + 25] | 0,01 [1,25.sqrt(g) + 6,3] |
| 6 | 0,001.[ 7,1 Sqrt(M.z.β /2 ) + 18] | 0,001 [6,3.m (1+0,012z) + 40] | 0,01 [2,0.sqrt(g) + 10] |
| Pitch Dia. D (mm) |
α = 30 | α = 37,5 | α = 45 | α = 30 | α = 37,5 | α = 45 | α = 30 | α = 37,5 | α = 45 | α = All |
| fit class = d | fit class = e | fit class = f | fit class = h | |||||||
| esv / (tan αD) - (mm) | ||||||||||
| =< 3 | 0,035 | 0,026 | 0,020 | 0,024 | 0,018 | 0,014 | 0,010 | 0,008 | 0,006 | 0 |
| 3 to 6 inc. | 0,035 | 0,039 | 0,030 | 0,035 | 0,026 | 0,020 | 0,017 | 0,013 | 0,010 | 0 |
| 6 to 10 inc. | 0,052 | 0,052 | 0,040 | 0,043 | 0,033 | 0,025 | 0,023 | 0,017 | 0,013 | 0 |
| 10 to 18 inc. | 0,069 | 0,065 | 0,050 | 0,055 | 0,042 | 0,032 | 0,028 | 0,021 | 0,016 | 0 |
| 18 to 30 inc. | 0,087 | 0,085 | 0,065 | 0,069 | 0,052 | 0,040 | 0,035 | 0,026 | 0,020 | 0 |
| 30 to 50 inc. | 0,113 | 0,104 | 0,080 | 0,087 | 0,065 | 0,050 | 0,043 | 0,033 | 0,025 | 0 |
| 50 to 80 inc. | 0,139 | 0,130 | 0,100 | 0,104 | 0,078 | 0,060 | 0,052 | 0,039 | 0,030 | 0 |
| 80 to 120 inc. | 0,173 | 0,156 | 0,120 | 0,125 | 0,094 | 0,072 | 0,062 | 0,047 | 0,036 | 0 |
| 180 to 250 inc. | 0,208 | 0189 | 0,145 | 0,147 | 0,111 | 0,085 | 0,074 | 0,056 | 0,043 | 0 |
| 250 to 315 inc. | 0,251 | 0,222 | 0,170 | 0,173 | 0,130 | 0,100 | 0,087 | 0,065 | 0,050 | 0 |
| 315 to 400 inc. | 0,294 | 0,248 | 0,190 | 0,191 | 0,143 | 0,110 | 0,097 | 0,073 | 0,056 | 0 |
ISO 4156 defines the dimensional and tolerance system for the mating internal and external spline. The required fit should be selected according to the functional requirements of the connection and verified using the applicable standard.