BIZOMAT MAKEOS · Power Transmission
Involute Spline Stresses and Torque Capacity
Are the shear and compressive stresses in this involute spline within the allowables for the torque?
Stresses in a 30° involute spline by the handbook (Dudley) formulas: shear under the roots of the external teeth for a solid or hollow shaft, shear at the pitch diameter, and compressive stress on the tooth sides for fixed or flexible splines, using application, load-distribution, fatigue-life and wear-life factors.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Transmitted torque | N·m | Torque through the spline. |
| Pitch diameter | mm | Pitch diameter. |
| Root diameter of the external spline | mm | Root (minor) diameter of the external spline. |
| Bore of a hollow shaft (0 = solid) | mm | Inside diameter of a hollow shaft. |
| Number of teeth | Number of spline teeth. | |
| Effective length | mm | Effective length carrying load (limited by misalignment; never more than the engaged length). |
| Circular tooth thickness at the pitch diameter | mm | Circular tooth thickness at the pitch diameter. |
| Depth of engagement | mm | Radial depth of tooth engagement (about 0.9 module for flat root, 1 module for fillet root). |
| Spline | choice | A fixed spline is piloted or shrunk on so it cannot rock; a flexible spline rocks and wears. |
| Spacing accuracy | choice | Spacing accuracy of manufacture. |
| Application factor | Application factor for the power source and load. | |
| Load distribution factor (1 for fixed) | Load distribution factor for misalignment (1 for fixed splines). | |
| Fatigue-life factor | Fatigue-life factor for the number of torque cycles. | |
| Wear-life factor (flexible) | Wear-life factor for the revolutions of a flexible spline. | |
| Allowable shear stress | MPa | Allowable shear stress for your material and hardness. |
| Allowable compressive stress | MPa | Allowable compressive stress for your material and spline type. |
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: Transmitted torque
- What it is
- The torque passed through the spline connection from shaft to hub.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 100000000 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
- All three stresses rise in proportion.
- Get it right
- Use the nominal torque here; peaks from the driver and load are covered by the application factor.
What this means: Pitch diameter
- What it is
- The pitch diameter of the spline, the diameter at which tooth thickness and pressure angle are defined: module times number of teeth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shear at the pitch diameter and compressive stress are inversely proportional to it.
What this means: Root diameter of the external spline
- What it is
- The root (minor) diameter of the external spline teeth on the shaft, where shear under the roots is checked. It must be smaller than the pitch diameter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 28 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Root shear in a solid shaft falls with the cube of this diameter.
What this means: Bore of a hollow shaft (0 = solid)
- What it is
- The bore of a hollow splined shaft; zero for a solid shaft. It must be smaller than the root diameter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 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.
What this means: Number of teeth
- What it is
- The number of spline teeth around the shaft.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 200. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 24 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shear at the pitch diameter and compressive stress are inversely proportional to it.
What this means: Effective length
- What it is
- The length of spline that actually carries load: never more than the engaged length, and shorter where misalignment concentrates load at one end.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 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
- Shear at the pitch diameter and compressive stress are inversely proportional to it.
What this means: Circular tooth thickness at the pitch diameter
- What it is
- The circular (arc) thickness of one tooth measured along the pitch circle.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.96 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shear at the pitch diameter is inversely proportional to it.
What this means: Depth of engagement
- What it is
- The radial depth over which the internal and external teeth overlap and bear on each other.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.125 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The compressive stress on the tooth sides is inversely proportional to it.
What this means: Spline
- What it is
- Fixed for a spline that is piloted or press-fitted so the hub cannot rock; flexible for one that can rock or slide and so wears. It selects the compressive stress formula.
- Units and format
- Choose one of 2 options from the list: Fixed; Flexible.
- Example
- The form opens with “Fixed” selected as a worked example. Choose the option that matches your case.
What this means: Spacing accuracy
- What it is
- How evenly the teeth are spaced in manufacture, which decides how many teeth are taken to share the load in the shear formula at the pitch diameter.
- Units and format
- Choose one of 2 options from the list: Normal (half the teeth carry load); Poor (factor 6).
- Example
- The form opens with “Normal (half the teeth carry load)” selected as a worked example. Choose the option that matches your case.
What this means: Application factor
- What it is
- The application factor for torque peaks from the power source and the driven load above the nominal torque.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. 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
- All three stresses rise in proportion.
- Get it right
- Take it from spline design references or your design rules for the actual driver and load.
What this means: Load distribution factor (1 for fixed)
- What it is
- The load distribution factor for uneven load along the teeth caused by misalignment; it is applied to the pitch-line shear and the compressive stress.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. 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.
What this means: Fatigue-life factor
- What it is
- The fatigue-life factor for the number of torque cycles the spline will see; it divides the root and pitch-line shear and the fixed-spline compressive stress.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 3. 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.
What this means: Wear-life factor (flexible)
- What it is
- The wear-life factor for the number of revolutions a flexible spline will run; used only in the flexible-spline compressive stress.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 5. 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.
What this means: Allowable shear stress
- What it is
- The allowable shear stress for the spline material and hardness, compared with the larger of the root and pitch-line shear stresses.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 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.
- Get it right
- Use an allowable working value from material data or spline design references, not the ultimate shear strength.
What this means: Allowable compressive stress
- What it is
- The allowable compressive (bearing) stress on the tooth sides for the material, hardness and spline type.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 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.
Method and formulas
- Root shear, solid: S_s = 16 T K_a ÷ (π D_re³ K_f); hollow: 16 T D_re K_a ÷ [π (D_re⁴ − D_h⁴) K_f].
- Pitch-line shear: S_s = 4 T K_a K_m ÷ (D N L_e t K_f), with 6 in place of 4 for poor spacing accuracy.
- Compression, flexible: S_c = 2 T K_m K_a ÷ (D N L_e h K_w); fixed: 2 T K_m K_a ÷ (9 D N L_e h K_f).
- Example: 500 N·m in a 24-tooth spline gives 139.2 MPa root shear.
Limitations
- 30° involute splines; bursting of internal splines is a separate check.
- Factors and allowables are user inputs from the spline standard and material data.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a spline for a new torque or duty
- Find the effective length a spline needs
- Compare fixed and flexible spline designs
Who it is for
- Design engineers
- Gear designers
- Driveline engineers
What you get
- Shear stress under the roots
- Shear stress at the pitch diameter
- Compressive stress on the teeth
Interpreting the result
- If compression governs, lengthen the engagement or make the spline fixed (piloted).
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2266–2270. Topic reference; no source tables or recommended settings are reproduced.
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Open the Involute Spline Stresses and Torque Capacity calculator