Splines and Serrations for Shaft-Hub Connections: Failure Mode 4

Failure Mode 4: Bursting Stress on Internal Splines

Internal splines can fail by bursting — the sleeve splitting apart radially. This is actually three stresses combined:

Stress 1 — Radial load component:

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

Where tw = wall thickness of the internal spline sleeve (outside diameter minus major diameter, divided by 2), and L = full spline length.

Stress 2 — Centrifugal tensile stress:

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

Where Doi = outside diameter of the spline sleeve.

Stress 3 — Beam loading (tooth bending):

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

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

Combined bursting stress:

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

Allowable Tensile Stresses:

Estimating Spline Size: The Diameter-Torque Relationship

Before diving into detailed calculations, experienced engineers use empirically derived curves to estimate the required spline pitch diameter for a given torque. These curves represent different application categories:

Spline Length: The Overlooked Variable

For fixed splines:

• A length of one-third the pitch diameter gives the spline the same shear strength as the shaft (assuming uniform tooth loading)
• But since spacing errors cause only half the teeth to be fully loaded, the practical length for balanced strength is two-thirds the pitch diameter
• If weight is not critical, this may be increased to equal the full pitch diameter

For flexible splines:

• Long lengths do not contribute to load-carrying capacity when misalignment must be accommodated
• Maximum effective length should be determined from empirical charts based on pitch diameter and misalignment severity

For fixed splines without helix modification, the effective length Le should never exceed:

$$L_e \leq \frac{5000 \times D^{3.5}}{T}$$

Crowned Splines — Mastering Misalignment

When Straight Teeth Aren't Enough

Remember the practitioner from the opening story? His spline failed because the shaft alignment was slightly off — within spec for the couplings on either side, but enough to create devastating end-loading on straight spline teeth.

Straight-toothed flexible splines can handle misalignments of less than 1 degree. Beyond that, wear accelerates exponentially and tooth-end loading becomes the primary failure mode.

Crowned splines solve this by modifying the tooth profile along its length. Instead of parallel flanks running straight from one end to the other, a crowned tooth has a barrel-shaped profile that distributes contact across a curved surface.

The Geometry of a Crowned Spline

        ┌──────────────────────────────────────┐
        │      CROWNED vs. STRAIGHT TOOTH      │
        │                                      │
        │  STRAIGHT:         CROWNED:          │
        │  ┌────────┐       ╭────────╮         │
        │  │████████│       │██████  │         │
        │  │████████│       │████████│         │
        │  │████████│       │████████│         │
        │  │████████│       │██████  │         │
        │  └────────┘       ╰────────╯         │
        │                                      │
        │  Uniform width    Barrel-shaped       │
        │  along length     profile             │
        │                                      │
        │  End-loads under  Contact centered    │
        │  misalignment     under misalignment  │
        └──────────────────────────────────────┘

The key dimensions of a crowned spline are:

• r₁ = Radius of the crown (in the axial direction)
• r₂ = Radius of curvature of the crowned tooth
• D = Pitch diameter
• F = Face width
• A = Crown height (relief at the ends of the teeth)

Design Rules for Crowned Splines

Crown height A should always be somewhat greater than:

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

Where θ is the maximum expected misalignment angle.

Radius of curvature r₂:

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

Crown radius r₁:

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

Where φ is the pressure angle of the spline.

Compressive stress on crowned spline teeth:

$$S_c = 2290\sqrt{\frac{2T}{D \times N \times h \times r_2}}$$

This stress must be less than the values in the crowned spline column of the compressive stress table (which allows four times the stress of straight splines, as shown earlier).

The Trade-Off

Crowned splines with precise alignment have considerably less capacity than straight splines of the same size operating with precise alignment. The crown reduces the contact area.

But when large misalignments exist — as they inevitably do in real machinery — the crowned spline has greater capacity because it distributes the load smoothly instead of concentrating it at the tooth ends.

American Standard tooth forms may be used for crowned external members so that they may be mated with straight internal members of standard form. This means you don't need special internal members — only the external spline requires the crown modification.

The Silent Killer — Fretting Damage to Splines

What Is Fretting?

Fretting is wear that occurs when cyclic loading — such as vibration — causes two surfaces in intimate contact to undergo small oscillatory motions with respect to each other.

During fretting, high points (asperities) of the mating surfaces adhere to each other, and small particles are pulled out, leaving minute, shallow pits and powdery debris. In steel parts exposed to air, this metallic debris oxidizes rapidly and forms a red, rustlike powder or sludge — hence the designation "fretting corrosion."

Why "Fretting Corrosion" Is a Misleading Name

Fretting is mechanical in origin. It has been observed in materials that don't oxidize at all — gold, platinum, and nonmetallics. The corrosion (oxidation) that accompanies fretting of steel parts is a secondary factor, not the primary damage mechanism.

The primary mechanism is adhesive wear at micro-scale: surfaces stick, tear apart, generate debris, and repeat millions of times.

Where Fretting Occurs in Splined Systems

Fretting can occur wherever cyclic loading creates relative micro-motion between contacting surfaces. In the context of splines and related connections:

• Splined joints (the most common site)
• Interference fits
• Bolted, keyed, pinned, and riveted joints
• Between wires in wire rope and flexible shafts
• Friction clamps
• Small-amplitude oscillation bearings

Why Fretting Is Devastating

Fretting is dangerous for three reasons:

1. It destroys close fits. The material removal changes the clearances and load distribution of the spline.
2. Debris clogs moving parts. The oxide powder acts as an abrasive, accelerating further damage.
3. Fatigue failure is accelerated. Stress levels required to initiate fatigue in fretted parts are much lower than for undamaged material. The micro-pits act as crack initiation sites.

Countermeasures Against Fretting

Vibration and cyclic loading are the root causes. If these cannot be eliminated:

• Greater clamping force may reduce movement — but if not effective, may actually worsen the damage
• Lubrication may delay the onset of damage
• Hard plating or surface hardening may be effective, not by reducing fretting, but by increasing the fatigue strength of the material to resist crack initiation
• Plating with soft, lubricious materials (such as silver or molybdenum disulfide coatings) onto contacting surfaces is effective — until the plating wears through
• Converting from flexible to fixed spline design (with proper piloting) eliminates the rocking motion that causes axial fretting

Inspection — The Difference Between Professional Work and Guesswork

Inspection with Gages

Spline gages are used for routine production inspection. The standard defines several types:

GO and NOT GO Gages

GO gages inspect maximum material conditions (maximum external dimensions, minimum internal dimensions). They control the minimum looseness or maximum interference. A product is acceptable if the GO gage enters or goes on.

NOT GO gages inspect minimum material conditions (minimum external dimensions, maximum internal dimensions). They control the maximum looseness or minimum interference. A product is acceptable only if the NOT GO gage does NOT enter or go on.

Critical limitation: A NOT GO gage can inspect only one dimension. Attempting to simultaneously check multiple dimensions with a single NOT GO gage can produce false acceptances — the gage might fail to enter even though individual dimensions are out of tolerance, because their combined relationship happens to create interference.

Analytical Inspection: Measurement with Pins

When gages aren't available (prototype work, short runs, or supplementary inspection), individual dimensions can be measured using precision pins.

For internal splines — measurement between pins:

1. Calculate the involute of the pressure angle at pin center
2. Look up the corresponding pressure angle φi from involute function tables
3. Compute the measurement Mi:
   • Even number of teeth: Mi = Db × sec(φi) − di
   • Odd number of teeth: Mi = Db × cos(90°/N) × sec(φi) − di

Where di = 1.7280/P for 30° and 37.5° pressure angle splines, or 1.9200/P for 45° pressure angle splines.

For external splines — measurement over pins:

1. Calculate the involute of the pressure angle at pin center
2. Look up φe and sec(φe) from involute function tables
3. Compute the measurement Me:
   • Even number of teeth: Me = Db × sec(φe) + de
   • Odd number of teeth: Me = Db × cos(90°/N) × sec(φe) + de

Where de = 1.9200/P for all external splines.

Remember: Pin measurements determine actual dimensions — they do not directly determine fit between mating parts. Pin data must be combined with variation analysis to evaluate effective dimensions.

When Analytical Inspection Is Required

Analytical inspection (measuring individual dimensions and variations) is required in several situations:

• Supplementing gage inspection where NOT GO composite gages are used in place of NOT GO sector gages and variations must be controlled
• Evaluating rejected parts to determine whether gage rejection was marginal or definitive
• Prototype and short-run parts where spline gages aren't justified
• Controlling individual variations to prevent any one error from consuming too much of the total tolerance budget

The Metric World — ISO/ANSI Metric Module Splines

A Hard Metric Standard

ANSI B92.2M-1980 (R1989) is the American version of the ISO involute spline standard (ISO 4156). This is explicitly not a "soft metric" conversion of any previous inch-based standard.

Critical distinction: Splines made to the metric module standard are not intended for use with components made to the B92.1 (inch) standard or other previous standards. The two systems are fundamentally different and must not be mixed.

Metric Module Nomenclature

Where the inch standard uses diametral pitch (P), the metric standard uses module (m) — the metric equivalent of the reciprocal of diametral pitch.

Tolerance Classes and Fit Classes

The metric module standard defines:

• Tolerance Classes 4, 5, 6, and 7 (same numbering as the inch standard, but calculated differently)
• Fit Classes d, e, f, and h for the external spline, where the internal spline is always Class H (no tooth thickness modification)

Tolerance Formulas

The total tolerance for metric module splines is calculated using tolerance units:

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

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

The total tolerance (T + λ) for each class:

Basic Rack Profiles

The metric module standard defines four basic rack profiles for different pressure angle and root configurations:

1. 30° Flat Root Spline — The most common general-purpose profile
2. 30° Fillet Root Spline — For heavy-load applications at standard pressure angle
3. 37.5° Fillet Root Spline — For specialized medium-tooth applications
4. 45° Fillet Root Spline — For fine-pitch, high tooth-count applications

The British Standards — A Parallel Evolution

BS 2059:1953 — Straight-Sided Splines and Serrations

The British Standard for straight-sided splines was prepared on the hole basis — the hole is the constant member, and fits are obtained by varying the shaft size. This is the opposite philosophy from many American practices where the shaft is the constant member.

Part 1 covers 6 splines only (regardless of shaft diameter) in two depths:

• Shallow
• Deep

The splines are bottom fitting with top clearance — contact occurs at the minor diameter.

Three grades of fit:

Part 2 covers straight-sided 90° serrations with nominal diameters from 0.25 to 6.0 inches, also in three fit grades:

BS 3550:1963 — Involute Splines

This standard is complementary to BS 2059 and aligns with the American ANSI B5.15-1960 standard. The basic dimensions are identical for major diameter fit and side fit configurations.

Coverage:

• Side fit, flat root: 2.5/5.0 to 32/64 pitch, 6 to 60 splines
• Major diameter, flat root: 3.0/6.0 to 16/32 pitch, 6 to 60 splines
• Side fit, fillet root: 2.5/5.0 to 48/96 pitch, 6 to 60 splines

BS 6186 — Metric Module Involute Splines

BS 6186, Part 1:1981 is identical to sections 1 and 2 of ISO 4156 and with ANSI/ASME B92.2M-1980 (R1989). This represents the convergence of international standards for metric involute splines — a rare instance of genuine harmonization across standards bodies.

Manufacturing — Where Design Meets Reality

How External Splines Are Made

External splines can be produced by five primary methods, each with distinct characteristics that affect quality, cost, and applicability:

Key manufacturing insight: Cold-formed (rolled) external splines are usually of the fillet root design. A chamfer cannot be provided by the cold-forming process, so when major diameter fit is required, the corner clearance must be provided on the internal spline instead.

How Internal Splines Are Made

Internal splines are produced primarily by two methods:

Broaching is the dominant method for internal splines in production environments. The SAE standard spline fitting dimensions apply specifically to soft broached holes — the tolerances are designed to be readily maintained by standard broaching methods.

Deburring Splines

After machining, spline teeth inevitably have burrs that must be removed. The standard industrial approach uses wire-fill radial brushes.

The setup for brushing spline bores differs from brushing gears: brushes are located off-center rather than on the centerline. This off-center positioning ensures the brush wire tips contact the tooth flanks at the correct angle for effective burr removal without damaging the involute profile.

Polygon-Type Shaft Connections — The Alternative

When Splines Aren't the Answer

Involute and straight-sided splines dominate shaft connection design, but they're not the only option. Polygon-type connections — so called because they resemble regular polygons with curved sides — offer an alternative approach for both fixed and sliding connections.

Three-Sided vs. Four-Sided Designs

German DIN Standards 32711 and 32712 define metric polygon connections in two variants:

Strength Calculations for Polygon Connections

The strength analysis of polygon connections involves several specialized formulas:

Section modulus (bending):

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

Polar section modulus (torsion):

• Three-sided: ZP = 0.196 × DM⁴/DA

Where DM = D1 + 2e (mean diameter), DA = outer diameter, DI = inner diameter, and e = eccentricity defining the polygon profile.

Dimensional note: DM = D1 + 2e. Pressure angle Bmax ≈ 344e/DM degrees for three sides, and ≈ 299e/DM degrees for four sides.

Drawing Data and Specification — Communicating Your Design

Standard Drawing Data Format

The ANSI standard provides a specific, tabulated format for specifying splines on engineering drawings. Following this format prevents misunderstandings between design and manufacturing.

Internal Involute Spline Data (Example: 30° Flat Root Side Fit):

┌─────────────────────────────────────┐
│ INTERNAL INVOLUTE SPLINE DATA       │
│ Flat Root Side Fit                  │
├─────────────────────────────────────┤
│ Number of Teeth         xx          │
│ Pitch                   xx/xx       │
│ Pressure Angle          30°         │
│ Base Diameter            x.xxxxxx REF│
│ Pitch Diameter           x.xxxxxx REF│
│ Major Diameter           x.xxx max   │
│ Form Diameter            x.xxx       │
│ Minor Diameter           x.xxx/x.xxx │
│ Circular Space Width:               │
│   Max Actual             x.xxxx      │
│   Min Effective          x.xxxx      │
├─────────────────────────────────────┤
│ Optional:                           │
│ Max Measurement Between  x.xxxx REF │
│   Pins                              │
│ Pin Diameter             x.xxxx      │
└─────────────────────────────────────┘

External Involute Spline Data (Example: 30° Flat Root Side Fit):

┌─────────────────────────────────────┐
│ EXTERNAL INVOLUTE SPLINE DATA       │
│ Flat Root Side Fit                  │
├─────────────────────────────────────┤
│ Number of Teeth         xx          │
│ Pitch                   xx/xx       │
│ Pressure Angle          30°         │
│ Base Diameter            x.xxxxxx REF│
│ Pitch Diameter           x.xxxxxx REF│
│ Major Diameter           x.xxx/x.xxx │
│ Form Diameter            x.xxx       │
│ Minor Diameter           x.xxx min   │
│ Circular Tooth Thickness:           │
│   Max Effective          x.xxxx      │
│   Min Actual             x.xxxx      │
├─────────────────────────────────────┤
│ Optional:                           │
│ Min Measurement Over     x.xxxx REF │
│   Pins                              │
│ Pin Diameter             x.xxxx      │
└─────────────────────────────────────┘

Important: Dimensions marked REF (reference) should not be used as criteria for part acceptance or rejection. They exist for engineering and manufacturing guidance.

Design note on side fit splines: The internal spline major diameter is shown as a maximum dimension only, and the external spline minor diameter as a minimum dimension only. The minimum internal major diameter and maximum external minor diameter must clear the specified form diameter and thus need no additional control.

Interchangeability — The Hidden Compatibility Matrix

Cross-Standard Compatibility

One of the most complex aspects of spline engineering is understanding which standards are compatible with which. The ANSI B92.1-1970 standard provides explicit compatibility data with older standards:

External splines made to ANSI B92.1-1970 will mate with older internal splines as follows:

Internal splines made to ANSI B92.1-1970 will mate with older external splines as follows:

Key exceptions to be aware of:

• (A): The external major diameter, unless chamfered or reduced, may interfere with the internal form diameter
• (B): For 15 teeth or fewer, the minor diameter of the internal spline (unless chamfered) will interfere with the external form diameter
• (C): For 9 teeth or fewer, the minor diameter of the internal spline (unless chamfered) will interfere with the external form diameter
• (D): The internal minor diameter (unless chamfered) will interfere with the external form diameter
• (E): Same as (D)
• (F): For 10 teeth or fewer, the minimum chamfer on the major diameter of the external spline may not clear the internal form diameter
• (G): Depending on the pitch of the spline, the minimum chamfer on the major diameter may not clear the internal form diameter

Bottom line: If you're replacing or mating with splines made to older standards, you must check the specific exception conditions for your tooth count and configuration. "It should be compatible" is not good enough — verify mathematically.

The Complete Design Workflow — Putting It All Together

Step-by-Step: From Torque Requirement to Finished Specification

Here is the complete workflow that transforms a torque requirement into a fully specified, inspectable spline design. This is the process that separates the the practitioner-before-the-failure from the engineer-who-never-fails.

Step 1: Define the Load

• Determine transmitted torque (T)
• Classify the power source and load type → select Ka
• Determine if the spline is fixed or flexible
• Estimate the required life in torque cycles (for Kf) and/or revolutions (for Kw)

Step 2: Estimate Initial Size

• Use the diameter-torque relationship curves to estimate pitch diameter
• Select appropriate curve (A through E) based on application category

Step 3: Select Spline Configuration

• Choose pressure angle (30° unless specific conditions favor 37.5° or 45°)
• Choose flat root or fillet root based on load severity
• Choose side fit or major diameter fit based on centering requirements
• Select number of teeth (even numbers, 10–30 typical)

Step 4: Determine Fit Type and Tolerance Class

• Fixed permanent assembly → tighter fit, lower tolerance class
• Sliding under load → looser fit, appropriate clearance
• Consider manufacturing capabilities → match tolerance class to achievable accuracy

Step 5: Calculate All Four Stress Modes

• Torsional shear under root → Formula (1) or (2)
• Shear at pitch diameter → Formula (3)
• Compressive stress on tooth flanks → Formula (4) or (5)
• Bursting stress on internal member → Formulas (6), (7), (8), combined

Step 6: Verify Against Allowable Stresses

• Compare each calculated stress to the material-appropriate allowable value
• All four must pass — the design is only as strong as its weakest failure mode

Step 7: Determine Spline Length

• Fixed: minimum 2/3 pitch diameter for balanced strength
• Flexible: use maximum effective length charts
• Check effective length limit: Le ≤ 5000 × D^3.5 / T

Step 8: Specify Drawing Data

• Follow ANSI tabulated format
• Include all required dimensions
• Mark REF dimensions appropriately
• Specify tolerance class and fit type

Step 9: Define Inspection Requirements

• Specify GO/NOT GO gage requirements
• Define pin measurement procedures and dimensions
• State any analytical inspection requirements

Step 10: Consider Failure Prevention

• Evaluate fretting risk → specify surface treatment if needed
• Assess misalignment → specify crowning if greater than 1°
• Evaluate fatigue life → specify fillet root if marginal
• Define maintenance/inspection intervals

Epilogue: What the practitioner Learned

Six months after the failure, the practitioner was back. Not with a replacement of the same design, but with a fundamentally re-engineered spline coupling that incorporated every lesson from the investigation:

• He'd switched from flat root to fillet root to reduce stress concentration
• He'd specified crowned external teeth because the site survey revealed 2.3° of angular misalignment between the motor and gearbox
• He'd applied the correct application factors — Ka = 2.4 for the combination of motor (uniform) and conveyor (intermittent shock), where his original design had used Ka = 1.0
• He'd checked all four failure modes instead of relying on a single shear calculation
• He'd specified surface treatment to resist the fretting that had accelerated the failure
• He'd doubled the inspection requirements, including pin measurements on every part

The redesigned coupling ran for seven years without a single issue. The mining operation's chief engineer eventually told the practitioner it was the most reliable coupling in their entire fleet.

The lesson isn't that splines are difficult. The lesson is that splines reward thoroughness.

Every formula in this guide exists because someone, somewhere, learned a lesson the hard way. The application factors came from field failures. The allowable stress tables were calibrated against broken parts. The fretting damage section reflects decades of mysterious premature failures that were eventually traced to micro-motions invisible to the naked eye.

You now have all of this accumulated wisdom in one place.

Your Next Step

You've just read the most comprehensive single-source guide to spline engineering available outside a university library. But reading is not mastering.

Here is your challenge:

Take the next spline connection you encounter — whether in a design you're creating, a machine you're maintaining, or a system you're analyzing — and run it through the complete design workflow in Part Fifteen.

Calculate all four failure modes. Check the interchangeability table if older components are involved. Evaluate the fretting risk. Question whether the fit type and tolerance class are truly appropriate.

If you do this once, thoroughly, you will understand splines at a level that most engineers never reach.

And you'll never be the person standing in a maintenance bay at 3:00 AM, staring at the pieces of a connection that everyone assumed was "just a standard part."

What's the most challenging spline application you've faced? What failure mode surprised you? Share your experience — every engineer's lesson becomes another engineer's advantage.

Reference Standards:

• ANSI B92.1-1970 (R1993) — American National Standard Involute Splines
• ANSI B92.2M-1980 (R1989) — Metric Module Involute Splines
• ISO 4156 — Straight Cylindrical Involute Splines
• BS 2059:1953 — Straight-Sided Splines and Serrations
• BS 3550:1963 — Involute Splines
• BS 6186:1981 — Involute Splines, Metric Module
• SAE Standard Splined Fittings (4, 6, 10, and 16-Spline)
• DIN 32711 / 32712 — Polygon Shaft Connections

The Complete Engineering Guide to Shaft-Hub Power Transmission

The Gearbox That Destroyed a Production Line

the practitioner didn't hear the failure. He felt it.

A deep, gut-level shudder ran through the floor of the assembly plant at 2:47 AM. By the time he reached the CNC transfer line, the damage was already done. A splined coupling between the main drive motor and the gearbox had stripped — not catastrophically, but insidiously. The external spline teeth on the drive shaft had worn past their compressive limits over six months of operation, and the resulting misalignment had cascaded through the entire drivetrain.

Total cost: 72 hours of downtime, a destroyed gearbox, and a replacement shaft flown in from overseas.

The root cause? The original designer had specified a straight-sided 4-spline fitting where an involute fillet root spline was needed. The application involved intermittent shock loads from an internal combustion engine driving hydraulic actuators. The straight-sided spline lacked the self-centering action needed under dynamic loads, and the stress concentrations at the flat root accelerated fatigue failure.

the practitioner's story isn't unique. Spline failures account for a disproportionate share of drivetrain downtime across automotive, aerospace, agricultural, and industrial machinery — not because splines are inherently weak, but because they are consistently misunderstood, under-specified, or poorly matched to their application.

This guide changes that.

You are about to learn everything that separates a novice from a spline expert — from the fundamental difference between straight-sided and involute profiles, through ANSI/ISO standard dimensional systems, to the stress analysis formulas that predict whether your spline will survive 10 million torque cycles or fail at 10,000.