BIZOMAT MAKEOS · Gears
Spline Joint Design (Engagement Length and Stresses)
How long must this involute spline engage to carry the torque, and are the shaft and teeth within their allowable stresses?
Design of a metric-module involute spline joint (30° flat-root proportions): pitch, minor diameter, tooth thickness and working depth from module and tooth count, root shear stress of the shaft, the engagement length needed for tooth shear at the pitch line and for tooth compression (fixed or flexible spline), and the stresses at the length chosen, by the same method as the spline torque capacity tool.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Torque | N·m | Torque the joint transmits. |
| Module | mm | Module of the spline (pitch diameter ÷ teeth). |
| Number of teeth | Number of spline teeth. | |
| Engagement length | mm | Length over which the teeth of shaft and hub engage. |
| Bore of a hollow shaft (0 = solid) | mm | Bore diameter if the shaft is hollow. |
| Spline type | choice | A fixed spline does not slide; a flexible spline slides or accepts misalignment and is limited by wear. |
| Spacing accuracy | choice | Accuracy of tooth spacing, which sets how many teeth share the load. |
| Application factor | Application factor for shock in the drive. | |
| Load distribution factor | Load distribution factor for misalignment along the length. | |
| Fatigue life factor | Fatigue life factor for the number of torque cycles. | |
| Wear life factor (flexible splines) | Wear life factor for flexible splines. | |
| Allowable shear stress | MPa | Allowable shear stress of the spline material (your design value). |
| Allowable compressive stress | MPa | Allowable compressive stress on the tooth flanks (your design value). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Torque
- What it is
- The torque the spline joint transmits between shaft and hub in normal running, which loads the teeth and twists the shaft at its minor diameter.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 10000000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Every stress and the engagement length needed rise in direct proportion to torque.
- Get it right
- Enter the steady working torque; shock is covered by the application factor, so do not add it twice.
What this means: Module
- What it is
- The size of the spline teeth in the metric system: pitch diameter divided by number of teeth. It sets pitch diameter, tooth thickness and working depth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.25 mm to 20 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger module enlarges every dimension, so root shear and the length needed fall.
- Get it right
- Metric module is not diametral pitch; a spline specified in teeth per inch needs converting first.
What this means: Number of teeth
- What it is
- How many teeth the spline has around the shaft. With the module it gives the pitch and minor diameters.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 100. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More teeth raise the pitch and minor diameters, lowering root shear and the length needed.
- Get it right
- Count the teeth on one member; shaft and hub have the same number.
What this means: Engagement length
- What it is
- The axial length over which the shaft and hub teeth actually overlap and share the load.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- It does not change the length needed; a longer engagement lowers tooth shear and compressive stress.
- Get it right
- Use the overlap of the toothed lengths, not the hub length, if chamfers or reliefs shorten it.
What this means: Bore of a hollow shaft (0 = solid)
- What it is
- The diameter of the hole through the shaft if it is hollow; zero means a solid shaft. It weakens the shaft against twisting at the root.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger bore raises the root shear stress; tooth stresses are unaffected.
- Get it right
- Use 0 for a solid shaft. The bore must be smaller than the minor diameter of the teeth.
What this means: Spline type
- What it is
- Whether the spline is fixed, with no sliding, or flexible, sliding axially or taking misalignment. Flexible splines are limited by wear on the flanks.
- Units and format
- Choose one of 2 options from the list: Fixed (no sliding); Flexible (sliding or misaligned).
- Example
- The form opens with “Fixed (no sliding)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Flexible splines use the wear life factor and a much lower compressive capacity, so the length for compression grows.
- Get it right
- Choose flexible if the hub slides in service or the shafts are not held in line, even if the movement is small.
What this means: Spacing accuracy
- What it is
- How accurately the teeth are spaced around the spline, which decides how many teeth actually share the load.
- Units and format
- Choose one of 2 options from the list: Good (load shared by about half the teeth); Ordinary (load shared by about a third of the teeth).
- Example
- The form opens with “Good (load shared by about half the teeth)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Ordinary spacing raises tooth shear by half again compared with good spacing.
- Get it right
- Choose ordinary unless the spline is made and inspected to a close spacing tolerance.
What this means: Application factor
- What it is
- A multiplier for shock and torque peaks from the driving and driven machines, applied to the steady torque.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 4. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Every stress and the length needed rise in direct proportion.
- Get it right
- Take it from the spline standard or your design basis for the machines involved.
What this means: Load distribution factor
- What it is
- A multiplier for load concentrated at one end of the teeth by misalignment and shaft deflection along the engagement length.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 4. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Tooth shear, compressive stress and the length needed rise in proportion; root shear does not.
- Get it right
- It grows with longer engagement and poorer alignment; take it from the spline standard.
What this means: Fatigue life factor
- What it is
- A factor that adjusts the stresses for the number of torque cycles or reversals the spline will see in its life.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller factor raises root shear, tooth shear and fixed-spline compression, so the length needed grows.
- Get it right
- It sits below the stresses in the formulas, so a value under 1 raises them; take it from the spline standard.
What this means: Wear life factor (flexible splines)
- What it is
- A factor for the wear life wanted on the flanks of a flexible spline, depending on the number of revolutions and sliding.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- For a flexible spline, a smaller factor raises compressive stress and the length for compression.
- Get it right
- It is used only when the spline type is flexible; it has no effect on a fixed spline.
What this means: Allowable shear stress
- What it is
- The shear stress you allow in the spline material, used both for twisting at the shaft root and for tooth shear at the pitch line.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher allowable shortens the length for tooth shear and raises the root shear margin.
- Get it right
- Use your design allowable for the material and heat treatment, not its ultimate shear strength.
What this means: Allowable compressive stress
- What it is
- The contact (compressive) stress you allow on the tooth flanks where shaft and hub teeth press together.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The length for compression is inversely proportional to it.
- Get it right
- Use your design value for the flank material and hardness; flexible splines need a value limited by wear.
Method and formulas
- Pitch diameter D = m N; external minor diameter D_re = m (N − 1.35); tooth thickness t = π m ÷ 2; working depth h = m (30° flat-root proportions).
- Root shear S_s = 16 T K_a ÷ (π D_re³ K_f) for a solid shaft (hollow: 16 T D_re K_a ÷ [π (D_re⁴ − D_h⁴) K_f]).
- Tooth shear at the pitch line S = k T K_a K_m ÷ (D N L t K_f), k = 4 (good spacing) or 6; compressive S_c = 2 T K_m K_a ÷ (9 D N L h K_f) fixed, or 2 T K_m K_a ÷ (D N L h K_w) flexible.
- Length needed = the larger of the lengths that bring tooth shear and compression to their allowables.
- Example: 500 N·m on a 20-tooth module 2 spline needs 4.7746 mm of engagement; the root shear stress is 58.884 MPa.
Limitations
- Simplified method; full spline rating uses the standard’s factors and tooth geometry.
- Flat-root proportions; fillet-root splines have a slightly larger minor diameter.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size the engagement length of a splined hub
- Check a splined shaft end for a torque increase
- Compare fixed and flexible (sliding) splines
Who it is for
- Design engineers
- Maintenance engineers
- Power transmission engineers
What you get
- Spline proportions from module and teeth
- Root shear stress
- Engagement length needed and stresses at the length chosen
Interpreting the result
- Engagement beyond about one pitch diameter adds little: misalignment concentrates the load at one end.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2255–2282. Topic reference; no source tables or recommended settings are reproduced.
- ANSI B92.1 Involute Splines and Inspection; ISO 4156 Straight cylindrical involute splines
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Open the Spline Joint Design (Engagement Length and Stresses) calculator