Splines and Serrations for Shaft-Hub Connections: Load Distribution Factors (Km) for Flexible Splines

Load Distribution Factors (Km) for Flexible Splines

For fixed splines, Km = 1. For flexible splines, select based on misalignment and face width:

Fatigue-Life Factors (Kf)

A torque cycle consists of one start and one stop — not the number of revolutions.

Wear-Life Factors (Kw) for Flexible Splines

Unlike fatigue-life factors, wear-life factors are based on the total number of revolutions since each revolution of a flexible spline results in a complete cycle of rocking motion.

Stress Formula 1: Shear Stress Under Roots of External Teeth

For a solid shaft:

$$S_s = \frac{16 \times T \times K_a}{\pi \times D_{re}^3 \times K_f}$$

For a hollow shaft:

$$S_s = \frac{16 \times T \times D_{re} \times K_a}{\pi \times (D_{re}^4 - D_h^4) \times K_f}$$

Where:

• Ss = Torsional shear stress (psi)
• T = Transmitted torque (lb-in)
• Dre = Minor diameter of external spline (root diameter)
• Dh = Inside diameter of hollow shaft
• Ka = Application factor
• Kf = Fatigue-life factor

Stress Formula 2: Shear Stress at Pitch Diameter

$$S_s = \frac{4 \times T \times K_a \times K_m}{D \times N \times L_e \times t \times K_f}$$

The factor of 4 assumes that only half the teeth carry the load due to spacing errors. For poor manufacturing accuracy, change this factor to 6.

Stress Formula 3: Compressive Stress on Tooth Sides

For flexible splines:

$$S_c = \frac{2 \times T \times K_m \times K_a}{D \times N \times L_e \times h \times K_w}$$

For fixed splines:

$$S_c = \frac{2 \times T \times K_m \times K_a}{9 \times D \times N \times L_e \times h \times K_f}$$

Where h = depth of engagement:

• Flat root splines: h ≈ 0.9/P
• Fillet root splines: h ≈ 1/P

Allowable Shear Stresses

Allowable Compressive Stresses

Note the dramatic difference between straight and crowned spline allowable compressive stress. Crowned splines permit 4× higher compressive stress because they eliminate end-loading of the teeth.

Allowable Tensile Stresses

Bursting Stresses: The Silent Killer of Internal Splines

Internal splines can fail catastrophically by bursting — the sleeve literally splits apart. Three types of tensile stress contribute:

. Radial Load Tensile Stress (S₁)

$$S_1 = \frac{T \times \tan\phi}{\pi \times D \times t_w \times L}$$

Where:

• tw = Wall thickness of internal spline = (OD of spline sleeve − spline major diameter) / 2
• L = Full length of spline
• φ = Pressure angle

. Centrifugal Tensile Stress (S₂)

$$S_2 = \frac{1.656 \times (rpm)^2 \times (D_{oi}^2 + 0.212 \times D_{ri}^2)}{1{,}000{,}000}$$

Where:

• Doi = Outside diameter of spline sleeve
• Dri = Major diameter of internal spline

. Beam Loading Tensile Stress (S₃)

$$S_3 = \frac{4 \times T}{D^2 \times L_e \times Y}$$

Where Y is the Lewis form factor obtained from a tooth layout. For internal splines of 30° pressure angle, Y ≈ 1.5 is a satisfactory estimate. The factor of 4 assumes only half the teeth carry load.

Total Bursting Stress

$$S_t = \frac{K_a \times K_m \times (S_1 + S_3) + S_2}{K_f}$$

This total must be less than the allowable tensile stress from the table above.

Crowned Splines for Large Misalignments

When shafts cannot be precisely aligned — misalignments up to 5 degrees — crowned splines become the only viable solution.

What Makes Crowned Splines Different

A crowned spline has a barrel-shaped profile on the external teeth. The crown radius r₁ and the radius of curvature of the crowned tooth r₂ are related by:

$$r_1 = r_2 \times \tan\phi$$

Where φ is the pressure angle of the spline.

Design rule for crown height (A):

$$A > \frac{F}{2} \times \tan(\text{misalignment angle})$$

Where F is the face width.

Approximate radius of curvature:

$$r_2 = \frac{F^2}{8A}$$

Compressive Stress for Crowned Splines

The compressive stress calculation for crowned splines must account for the contact pattern, and the computed value should be less than the allowable compressive stress (which is 4× higher for crowned splines than for straight splines).

Trade-Off: Crowned vs. Straight Under Precise Alignment

Crowned splines have considerably less capacity than straight splines of the same size when both operate with precise alignment. However, when large misalignments exist, the crowned spline has greater capacity because it prevents the devastating end-loading that destroys straight spline teeth.

Standard tooth forms may be used for crowned external members so they can mate with straight internal members of standard form.

Fretting Damage: The Invisible Destroyer

the practitioner's spline didn't fail from a single overload. It failed from fretting — a slow, insidious wear mechanism that is the number one cause of spline degradation in service.

What Fretting Is

Fretting is wear that occurs when cyclic loading causes two surfaces in intimate contact to undergo small oscillatory motions relative to each other. During fretting:

1. High points (asperities) of the mating surfaces adhere to each other
2. Small particles are pulled out, leaving minute, shallow pits
3. A powdery debris accumulates
4. In steel parts exposed to air, this debris oxidizes rapidly, forming a red, rust-like powder

This oxidized debris is the origin of the term "fretting corrosion," though fretting is mechanical in origin, not chemical. It has been observed in gold, platinum, and non-metallic materials — materials that don't oxidize.

Why Fretting Is So Dangerous

• Destroys close fits — the debris accumulates and changes the dimensional relationship between mating parts
• Clogs moving parts — debris migration causes secondary failures
• Accelerates fatigue failure — stress levels required to initiate fatigue in fretted parts are much lower than for undamaged material

Where Fretting Occurs

Fretting sites include interference fits, splined joints, bolted joints, keyed joints, pinned and riveted joints, between wires in wire rope, flexible shafts and tubes, between leaves in leaf springs, friction clamps, small-amplitude bearings, and electrical contacts.

Countermeasures

Inspection Methods for Involute Splines

Analytical Inspection (Measurement with Pins)

Analytical inspection — direct measurement of individual dimensions — is required when:

• Supplementing gage inspection (e.g., when NOT GO composite gages replace sector gages)
• Evaluating parts rejected by gages
• Inspecting prototype parts or short production runs
• Controlling individual variations that might assume too great a portion of the overall tolerance

Pin Measurement Formulas — Internal Splines

Step 1: Find involute of pressure angle at pin center:

$$\text{inv } \phi_i = \frac{s}{D} + \text{inv } \phi_D - \frac{d_i}{D_b}$$

Step 2: Look up φi in involute function tables, find sec φi.

Step 3: Compute measurement between pins:

For even number of teeth: $M_i = D_b \times \sec\phi_i - d_i$

For odd number of teeth: $M_i = (D_b \times \cos\frac{90°}{N}) \times \sec\phi_i - d_i$

Where:

• di = 1.7280/P for 30° and 37.5° pressure angle splines
• di = 1.9200/P for 45° pressure angle splines

Pin Measurement Formulas — External Splines

Step 1: Find involute of pressure angle at pin center:

$$\text{inv } \phi_e = \frac{t}{D} + \text{inv } \phi_D + \frac{d_e}{D_b} - \frac{\pi}{N}$$

Step 2: Look up φe and sec φe.

Step 3: Compute measurement over pins:

For even number of teeth: $M_e = D_b \times \sec\phi_e + d_e$

For odd number of teeth: $M_e = (D_b \times \cos\frac{90°}{N}) \times \sec\phi_e + d_e$

Where de = 1.9200/P for all external splines.

Worked Example: Pin Measurement Calculation

Given: Internal spline, 30° pressure angle, tolerance class 4, 3/6 diametral pitch, 20 teeth.

Finding maximum actual space width (s):

• Minimum effective space width: sv = π/(2×3) = 0.52360
• λ = 0.0027 × 0.71 = 0.00192
• m = 0.00176 × 0.71 = 0.00125
• s = 0.52360 + 0.00192 + 0.00125 = 0.52677

Computing pin measurement:

• D = N/P = 20/3 = 6.66666
• inv 30° = 0.053751
• di = 1.7280/3 = 0.57600
• Db = D × cos 30° = 6.66666 × 0.86603 = 5.77353

Step 1: inv φi = 0.52677/6.66666 + 0.053751 − 0.57600/5.77353 = 0.03300

Step 2: φi = 25°46.18′, sec φi = 1.11044

Step 3: Mi = 5.77353 × 1.11044 − 0.57600 = 5.8352 inches

Metric Module Involute Splines (ANSI B92.2M-1980, R1989)

The metric module standard is the American National Standards Institute version of the ISO 4156 international standard. This is a "hard" metric system — not a soft conversion from inch-based standards.

Critical warning: Splines made to this metric standard are NOT intended for use with components made to the B92.1 or other inch-based standards. A "soft" conversion (multiplying inch dimensions by 25.4) does not produce compatible metric module splines. For example, a 10 diametral pitch hob calculates to a 2.54 module hob — a module that does not exist in the metric standard.

Features Retained from the Inch Standard

• 30°, 37.5°, and 45° pressure angles
• Flat root and fillet root side fits
• Four tolerance classes (4, 5, 6, and 7)
• Tables for a single class of fit
• The effective fit concept

Major Differences from the Inch Standard

• Modules from 0.25 through 10 mm replace diametral pitch
• Dimensions in millimeters instead of inches
• "Basic rack" concept replaces the previous dimensional system
• Major diameter fit removed — only side fit configurations
• ISO symbols replace previous notation
• Three defined clearance fits can be calculated

Standard Modules

The standard modules in the metric system are: 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.5, 3, 4, 5, 6, 8, and 10 mm.

All modules from 0.5 to 10 apply to 30° and 37.5° splines. For 45° fillet root splines, only 0.25 to 2.5 module applies.

Metric Module Dimension Formulas

Where m = module (mm), Z = number of teeth, αD = standard pressure angle.

Metric Module Major Diameter Formulas

Metric Module Fit Classes

Four classes of side fit are provided:

The tooth thickness modifications h, f, e, and d are fundamental deviations selected from ISO R286 ("ISO System of Limits and Fits"). They are applied to the external spline by shifting the tooth thickness total tolerance below the basic tooth thickness.

Tooth Thickness Modification (es) for Selected Fit Classes

Metric Module Effective Variation (λ)

The effective variation in the metric system is calculated as:

$$\lambda = 0.6 \times \sqrt{F_p^2 + f_f^2 + F_\beta^2} \quad \text{(millimeters)}$$

Where:

Where g = length of spline in millimeters.

Total Tolerance Formulas (T + λ) — Metric Module

Where tolerance units are:

$$i^* = 0.001(0.45\sqrt[3]{D} + 0.001D) \quad \text{for } D \leq 500 \text{ mm}$$

$$i^* = 0.001(0.004D + 2.1) \quad \text{for } D > 500 \text{ mm}$$

$$i^{**} = 0.001(0.45\sqrt[3]{S_{bsc}} + 0.001 \times S_{bsc})$$

Reduction of External Spline Diameters for Fit Classes (es/tan αD)

This table is essential for calculating external spline major and minor diameters across fit classes:

British Standard Straight Splines (BS 2059:1953)

For engineers working with British equipment or international projects requiring compliance with BS standards, understanding BS 2059 is essential.

Straight-Sided Splines

BS 2059 Part 1 covers 6 splines only, regardless of shaft diameter, with two depths termed shallow and deep. The splines are bottom-fitting with top clearance.

Design basis: Prepared on the hole basis — the hole is the constant member, and different fits are obtained by varying the shaft size.

Three Grades of Fit

All fits allow clearance on the sides (widths), but in Fit 1, the minor diameters of hole and shaft may be identical.

° Serrations

Covers serrations with nominal diameters from 0.25 to 6.0 inches with three fit grades:

Related British Standards

• BS 3550:1963 — "Involute Splines" — complementary to BS 2059, with basic dimensions matching ANSI/ASME B5.15-1960 for major diameter fit and side fit
• BS 6186, Part 1:1981 — "Involute Splines, Metric Module, Side Fit" — identical with ISO 4156 and ANSI/ASME B92.2M-1980

Polygon-Type Shaft Connections: The Alternative

Beyond involute and straight-sided splines, polygon-type connections offer a third option for fixed and sliding shaft-hub connections. Named for their resemblance to regular polygons with curved sides, they are standardized in German DIN Standards 32711 (three-sided) and 32712 (four-sided).

Choosing Between Three-Sided and Four-Sided Designs

Strength Formulas for Polygon Connections

Section modulus (bending):

• Three sides: Z = 0.098 × DM⁴ / DA
• Four sides: Z = 0.15 × DI³

Polar section modulus (torsion):

• Three sides: ZP = 0.196 × DM⁴ / DA

Where DM = D₁ + 2e, DA is the envelope diameter, and e is the eccentricity parameter.

Drawing Data and Specification

Proper communication of spline requirements on engineering drawings prevents manufacturing errors. The ANSI standard recommends a tabulated format for spline specifications, which eliminates the need for graphic illustration of the spline teeth.

Required Drawing Data (ANSI B92.1-1970, R1993)

The following data must appear on every spline drawing:

1. Number of teeth
2. Pitch (diametral pitch / stub pitch)
3. Pressure angle
4. Base diameter (reference)
5. Pitch diameter (reference)
6. Major diameter (with tolerances)
7. Minor diameter (with tolerances)
8. Circular space width or tooth thickness (effective and actual limits)
9. Form diameter
10. Tolerance class
11. Fit type (side fit or major diameter fit)
12. Root form (flat root or fillet root)

Professional tip: Reference dimensions (noted "REF") should never be used as criteria for part acceptance or rejection. They are provided for engineering and manufacturing purposes only.

Improvement method and result

Six months after the catastrophic failure, the practitioner stood in front of the same CNC transfer line — now running at full capacity with zero spline-related downtime.

What changed?

He replaced the 4-spline straight-sided coupling with a 30° involute fillet root spline designed to the following specifications:

• Tooth count: 20 (even number for measurement compatibility)
• Pitch: 6/12 (balancing tooth strength with manufacturing ease)
• Fit type: Side fit, Class 5 tolerance
• Root type: Fillet root (for the stress concentration reduction required by shock loading)
• Application factor: Ka = 2.4 (IC engine driving intermittent shock loads)
• Material: Case-hardened steel (RC 58–63) for both members
• Crowned external spline to accommodate the 0.5° shaft misalignment measured at the coupling

The results were transformative:

• Torque capacity increased 340% over the original straight-sided design
• Self-centering action eliminated the vibration that had been damaging downstream bearings
• Crowned teeth prevented the fretting damage that had been the original failure mode
• Fillet root design raised the fatigue life from an estimated 100,000 cycles to over 10,000,000

The total cost of the redesign — including new spline tooling, machining, and installation — was less than one-quarter of the cost of the single failure event it prevented.

Your Next Step

You now have the complete engineering knowledge base for spline design, specification, and analysis. The question is: what will you do with it?

Here are three actions depending on where you stand:

If you are designing a new spline connection:

• Start with the diameter-torque estimation charts to get in the right ballpark
• Select involute over straight-sided unless you have a compelling reason not to
• Always use even tooth numbers
• Apply the correct application factor — this is where most designs go wrong
• Specify fillet root for any application involving shock loads or high cycle counts

If you are troubleshooting a spline failure:

• Check for fretting damage first — it is the most common failure mode
• Recalculate the application factor using the actual power source and load characteristics
• Verify the misalignment — if it exceeds 1°, crowned splines are mandatory
• Inspect the root geometry — flat roots under heavy loads are a red flag

If you are specifying replacement splines:

• Consult the interchangeability tables before assuming old and new standards are compatible
• Verify the measurement system — inch-based and metric module splines are not interchangeable
• Match the tolerance class to the actual application requirements, not to what was previously specified

The most expensive spline in any machine is the one that fails.

Design it right the first time.

What spline challenge are you currently facing? Whether it is a new design, a failure analysis, or a standards compliance question, the formulas and data tables in this guide give you the foundation to solve it. Bookmark this page — it is a reference you will return to for decades.