Shafts, Keys, Splines and Couplings

Introduction and context

Shafts and axles are fundamental machine elements used to transmit power, support rotating components, and maintain accurate relative motion. A shaft normally transmits torque and may also carry bending loads, whereas an axle primarily supports rotating elements and is commonly subjected to bending with little or no transmitted torque. Keys, splines, and couplings provide the mechanical interfaces through which torque is transferred between a shaft and mounted components or between two shafts.

A successful design must satisfy more than static strength. It must control angular twist, lateral deflection, fatigue damage, vibration, critical speed, wear, assembly requirements, and manufacturing constraints. The designer must also consider stress concentrations at shoulders, keyways, retaining grooves, threads, and fits. In industrial equipment, failures of shafts and couplings can cause secondary damage, production losses, and safety hazards.

Learning objectives

After studying this chapter, the student should be able to:

• distinguish shafts, axles, and spindles;
• calculate shaft stresses due to torque, bending, and combined loading;
• select a suitable shaft diameter using strength and rigidity criteria;
• apply fatigue design concepts and stress-concentration factors;
• evaluate shaft deflection, angle of twist, and critical speed;
• design and check keys and understand spline selection;
• select rigid and flexible couplings for service conditions;
• identify retaining-device, vibration, lubrication, and maintenance requirements.

1. Functional classification and design requirements

A shaft is a rotating member designed to transmit torque. It may simultaneously experience bending from gears, pulleys, chains, or overhung loads. An axle supports rotating parts but generally does not transmit torque; railway axles and trailer axles are examples. A spindle is usually a short, accurately manufactured rotating shaft, such as a machine-tool spindle.

Shaft design is governed by several requirements:

1. Strength: stresses must remain below allowable values under steady, fluctuating, and transient loading.
2. Rigidity: excessive torsional twist or lateral deflection can cause poor gear alignment, seal failure, and inaccurate motion.
3. Fatigue resistance: repeated stress cycles can initiate cracks, particularly at geometric discontinuities.
4. Dynamic stability: operating speed should be sufficiently separated from critical speeds.
5. Manufacturability: diameter steps, fillets, keyways, fits, heat treatment, and inspection must be practical.
6. Reliability and maintainability: bearings, seals, couplings, and retaining devices must be accessible and secure.

Common shaft materials include medium-carbon steels, alloy steels, stainless steels, and, in special applications, titanium or nickel alloys. Material selection depends on strength, toughness, corrosion resistance, heat treatment, cost, and compatibility with the environment.

2. Torque, power, and torsional stress

For steady rotational power, torque and angular speed are related by

where P is power in watts, T is torque in newton-metres, ω is angular velocity in radians per second, and N is speed in revolutions per minute. This equation assumes negligible transmission loss. If efficiency is η, the required input power is P_in = P_out/η.

For a circular shaft under pure torsion, the torsion equation is

where J is the polar second moment of area, τ is shear stress at radius r, G is shear modulus, θ is twist in radians, and L is shaft length. The equation assumes a homogeneous, prismatic, circular shaft, elastic behavior, and Saint-Venant torsion.

For a solid circular shaft of diameter d,

and the maximum nominal torsional stress is

For a hollow shaft with outside diameter d_o and inside diameter d_i,

The hollow form can provide lower mass for a specified torsional rigidity, although it may be less economical to manufacture.

The angle of twist of a uniform shaft is

For stepped or nonuniform shafts, the total twist is the algebraic sum:

The twist must be expressed in radians. Degrees may be obtained by multiplying radians by 180/π.

3. Bending and combined loading

A shaft carrying gears, pulleys, sprockets, or flywheels is subjected to transverse forces. The resulting bending moment varies along the shaft. At a section, the nominal bending stress in a solid circular shaft is

where M is bending moment. In a hollow shaft,

The maximum bending stress occurs at the outer surface. Transverse shear stress due to a shear force is often much smaller than bending stress in slender rotating shafts, but it should be considered in short, heavily loaded members.

When bending and torsion act together, the surface is subjected to normal stress σ and shear stress τ. A widely used distortion-energy approach defines an equivalent von Mises stress:

For static or slowly varying loading, a preliminary diameter may be obtained by requiring σ_VM ≤ S_y/n, where S_y is yield strength and n is the design factor. This method assumes ductile material, elastic behavior, and a suitable interpretation of the combined stress state.

A commonly used shaft-design form combines bending moment and torque through an equivalent twisting moment:

and then

Here K_b and K_t are design factors accounting for shock, fatigue, stress concentration, or service conditions, and τ_allow is the selected allowable shear stress. These factors must not be inserted indiscriminately; they should reflect the chosen design standard and loading model.

Worked example: preliminary shaft diameter

A solid steel shaft transmits 12 kW at 600 rpm. It experiences a bending moment of 450 N·m. Use K_b = 1.5, K_t = 1.2, and an allowable shear stress of 45 MPa. Find a preliminary diameter.

First, calculate torque:

T = 9550P_kW/N = 9550(12)/600 = 191 N·m

The equivalent twisting moment is

T_e = √[(1.5×450)² + (1.2×191)²] = 701 N·m

Therefore,

d = [16(701×10³)/(π×45)]^(1/3) = 43.0 mm

A practical design would select the next suitable standard diameter, then recheck keyway weakening, fatigue, deflection, fits, and manufacturing requirements. The result is only a preliminary strength diameter.

4. Fatigue design of shafts

Rotating bending is often fully reversed at a material point: a surface element alternately experiences tensile and compressive stress as the shaft rotates. Torque may be steady, fluctuating, or reversed. Fatigue analysis separates stress into alternating and mean components:

Similar definitions apply to shear stress. A modified Goodman relation is frequently used for preliminary design:

where S_e is the corrected endurance limit, S_ut is ultimate tensile strength, and n is the design factor. This relation is an empirical linear interaction and should not be treated as universal. The endurance limit must be corrected for surface finish, size, reliability, temperature, loading mode, and other effects.

Stress concentrations are important at keyways, shoulders, grooves, and threads. The theoretical concentration factor K_t describes elastic stress amplification, while the fatigue concentration factor K_f is commonly expressed as

where q is notch sensitivity, with 0 ≤ q ≤ 1 for the usual engineering model. A design should use generous fillet radii, avoid abrupt diameter changes, improve surface finish, and locate keyways away from maximum bending regions where possible.

5. Rigidity, deflection, and critical speed

Strength alone is insufficient. Excessive lateral deflection can produce gear misalignment and uneven bearing loading. For a beam segment, the relationship between bending moment and curvature is

where y is transverse deflection, E is Young’s modulus, and I is the second moment of area. Boundary conditions and load distribution determine the complete deflection curve. For a solid circular section, I = πd⁴/64.

Torsional rigidity is controlled by GJ. A design may specify a maximum angle of twist per unit length, such as radians per metre, according to the machine’s precision and operating requirements. Gear shafts generally require tighter alignment and deflection limits than simple line shafts.

A rotating shaft with attached masses has natural frequencies. When operating speed approaches a natural frequency, resonance may occur. The corresponding speed is called a critical speed or whirling speed. For a simplified single-disc rotor with lateral stiffness k and mass m,

and the corresponding speed in revolutions per minute is

This simplified model neglects shaft distributed mass, gyroscopic effects, damping, bearing flexibility, and multiple modes. Real rotor design normally uses beam models or finite-element rotor-dynamic analysis. The operating range should avoid resonance or pass through it rapidly with controlled acceleration when permitted by the design.

Balancing is essential for high-speed shafts. Static unbalance produces a centrifugal force approximately equal to F = meω², where m is unbalanced mass and e is eccentricity. Alignment, bearing condition, coupling runout, and thermal growth also influence vibration.

6. Keys and keyways

A key is a removable element placed between a shaft and hub to transmit torque and prevent relative rotation. Common forms include parallel keys, sunk keys, taper keys, gib-head keys, and woodruff keys. A keyway reduces shaft strength because it removes material and creates a stress concentration.

For a rectangular sunk key of width w, height h, and length L, torque produces tangential force

assuming the effective radius is approximately d/2. The nominal shear stress in the key is

If crushing is based on the projected bearing area approximately (h/2)L, the compressive stress is

These equations assume uniform load distribution and neglect stress concentration, friction, and end effects. In practice, standard key dimensions and allowable stresses are taken from recognized standards and manufacturers’ data.

The required key length is therefore checked from both modes:

and

The larger value governs. The hub length should also be sufficient to prevent excessive bearing pressure and rocking.

Keys should be fitted carefully. Excessive clearance causes impact and fretting; excessive interference can distort the hub or make assembly difficult. Keyways should be inspected for burrs, cracks, and poor surface finish. For high-cycle or high-speed service, a splined connection may be preferable.

7. Splines

Splines are multiple integral teeth formed on a shaft and matching grooves in a hub. Compared with a single key, splines distribute torque over many teeth, provide better concentricity, and can permit axial sliding in applications such as vehicle transmissions and machine-tool feeds.

Spline selection considers torque, speed, tooth form, number of teeth, pressure angle, engagement length, axial movement, lubrication, wear, and manufacturing method. Failure modes include tooth shear, tooth bending, surface pitting, fretting, wear, and hub bursting.

A simplified average bearing-pressure check may be expressed as

where d_m is mean spline diameter, L_e is effective engagement length, and h_e is effective tooth height. The exact coefficient depends on the definition of tooth geometry and the number of loaded teeth; therefore, standard spline calculations should use the applicable ISO, ANSI, DIN, or manufacturer procedure. The assumption that every tooth carries equal load is conservative only in some contexts because manufacturing tolerance and elastic deformation cause unequal load sharing.

8. Retaining devices and shaft details

Axial retention may be provided by shoulders, retaining rings, locknuts, end plates, spacers, collars, circlips, or interference fits. Selection depends on axial load, speed, assembly frequency, temperature, and serviceability.

A shoulder must be large enough to carry the axial force without excessive contact stress. Its transition should include a fillet, but the fillet must be compatible with the bearing chamfer or gear hub relief. Locknuts should be secured by tab washers, thread deformation, split pins, or other approved methods where vibration is present. Retaining rings require correctly machined grooves and must be rated for the direction and magnitude of the load.

Interference fits transmit torque through friction. A fit must be checked for contact pressure, hoop stress in the hub, assembly method, thermal expansion, and fretting risk. A retaining device should never be treated as an alternative to correcting an undersized shaft or an inadequate torque interface.

9. Couplings

A coupling connects two shafts to transmit torque. Couplings may be rigid or flexible.

Rigid couplings

Rigid couplings maintain a fixed angular and radial relationship between shafts. Sleeve, clamp, and flange couplings are common. They are appropriate where shafts are accurately aligned and there is little relative movement. Their advantages include simplicity, high torsional stiffness, and accurate torque transmission. Their limitation is that they transmit forces caused by misalignment directly to bearings and shafts.

Flange couplings may use fitted bolts or ordinary bolts. Fitted bolts transmit shear directly through the shank; friction-grip joints depend on bolt preload and interface friction. The coupling flange and bolts must be checked for shear, bearing, bending, and fatigue.

Flexible couplings

Flexible couplings accommodate angular, parallel, and axial misalignment while transmitting torque. Examples include elastomeric jaw, tyre, grid, gear, disc, diaphragm, Oldham, and universal-joint couplings. They may also damp torsional vibration and isolate shock loads.

Selection requires the following information:

• transmitted power and speed;
• starting, stopping, reversing, and overload torque;
• shaft diameters and available space;
• maximum angular, parallel, and axial misalignment;
• torsional stiffness and damping requirements;
• temperature, chemicals, dust, and lubrication;
• balancing grade and maintenance access.

A service factor is often applied to nominal torque:

where K_s reflects driver type, driven-machine characteristics, starts per hour, shock, and duty cycle. The selected coupling must have a rated torque at least equal to T_design, and its maximum speed and bore capacity must also be adequate. Catalog ratings must be interpreted according to the manufacturer’s stated conditions; they are not interchangeable between coupling types.

Flexible couplings do not eliminate the need for alignment. Misalignment consumes coupling life, increases bearing load, and may generate heat. Alignment should be performed using dial indicators or laser systems, with allowance for thermal growth where relevant.

Worked example: key sizing

A 50 mm diameter shaft transmits 1,000 N·m through a rectangular key 14 mm wide and 9 mm high. The allowable key shear stress is 60 MPa and allowable crushing stress is 120 MPa. Determine the minimum length from the two criteria.

From shear,

L_s = 2T/(d w τ_allow) = 2(1,000,000)/(50×14×60) = 47.6 mm

From crushing,

L_c = 4T/(d h σ_c,allow) = 4(1,000,000)/(50×9×120) = 74.1 mm

Crushing governs, so a practical standard length must be at least 74.1 mm, subject to hub geometry and the applicable design standard. The shaft keyway must then be included in the shaft fatigue and stress-concentration assessment.

Applied case: illustrative conveyor drive

Background

Consider an illustrative belt-conveyor drive transmitting 30 kW at 300 rpm. The motor and gearbox are connected through a flexible coupling, and the gearbox output shaft carries a sprocket and bearing-supported overhung load.

Problem

A preliminary solid shaft based only on torsional strength is relatively small. During trial operation, vibration increases near operating speed, the keyway shows fretting, and the elastomeric coupling fails prematurely.

Analysis

The design team first calculates nominal torque from T = 9550(30)/300 = 955 N·m. The overhung sprocket force creates bending, so a combined bending-torsion analysis is required. The keyway is located close to the maximum bending moment, increasing fatigue stress. Shaft deflection causes angular misalignment at the coupling. The operating speed is also close to a rotor natural frequency, amplifying vibration.

Approach and solution

The shaft diameter is increased after applying combined-loading and fatigue criteria. A larger fillet radius and improved surface finish are introduced, while the keyway is moved away from the peak bending region where the layout permits. A standard spline is considered for the high-cycle connection. The coupling is replaced by a torsionally suitable flexible coupling with a higher service factor and adequate misalignment capacity. Laser alignment is performed, and a rotor-dynamic check verifies separation from the operating-speed range. The sprocket is moved closer to its bearing to reduce bending moment.

Outcome

The revised design reduces shaft stress, deflection, fretting, and coupling temperature. Vibration is reduced by improved alignment, balancing, and increased separation from critical speed. The example demonstrates that shaft reliability is a system problem: increasing diameter alone may not correct misalignment, resonance, or an unsuitable coupling.

Learning points

• Torsional strength is only one design criterion.
• Keyways and shoulders must be included in fatigue analysis.
• Coupling selection requires torque, speed, misalignment, environment, and dynamic data.
• Bearing placement strongly affects bending moment.
• Vibration diagnosis should consider unbalance, alignment, looseness, resonance, and coupling behavior.

Challenges, standards, and contemporary practice

The principal design challenge is the interaction among strength, rigidity, fatigue, dynamics, and manufacturability. A shaft that is strong enough may be too flexible; a very stiff shaft may be heavy and costly. Corrosion, fretting, lubricant contamination, thermal gradients, and poor assembly can reduce life substantially.

Designers should use applicable standards for dimensions, fits, keys, splines, bearings, balancing, and couplings. Traceable material certificates, controlled heat treatment, nondestructive inspection, and documented torque procedures improve reliability. Safety-critical shafts may require ultrasonic, magnetic-particle, dye-penetrant, or eddy-current inspection depending on material and defect sensitivity.

Computer-aided design and finite-element analysis are valuable for stress, deformation, and modal evaluation, but results depend on boundary conditions, mesh quality, material data, and contact assumptions. AI-assisted design optimization can explore diameter, fillet, material, and mass alternatives, but it does not replace engineering judgment, standards compliance, physical validation, or responsibility for safety. Digital vibration monitoring and condition-based maintenance can identify imbalance, misalignment, bearing deterioration, and coupling wear before catastrophic failure.

Ethically, the designer must not reduce dimensions or omit inspection merely to meet cost or weight targets. All safety factors, assumptions, and deviations from standards should be recorded. Guards must be provided around rotating shafts and couplings, and lockout procedures must be established before maintenance.

Best-practice design workflow

1. Define power, speed, duty cycle, loads, environment, life, and allowable deflection.
2. Draw free-body diagrams and obtain bearing reactions, shear forces, and bending moments.
3. Determine torque from power and speed, including transient and service factors.
4. Select a material and preliminary diameter using combined strength criteria.
5. Check fatigue with corrected endurance data and stress concentrations.
6. Check angle of twist, lateral deflection, bearing reactions, and critical speeds.
7. Design the key, spline, fit, shoulder, groove, or other torque interface.
8. Select a coupling using rated torque, speed, misalignment, environment, and torsional behavior.
9. Specify manufacturing tolerances, surface finish, fillets, balancing, inspection, and retention.
10. Validate the design through calculation, simulation, prototype testing, and monitoring.

Chapter Summary

Shafts transmit torque and commonly carry bending, while axles primarily support rotating components. Shaft design must satisfy strength, fatigue, torsional rigidity, lateral rigidity, dynamic stability, manufacturing, and maintenance requirements. The torsion equation provides the basis for calculating shear stress and angle of twist in circular shafts; combined bending and torsion require an equivalent-stress or equivalent-torque approach. Fatigue design must account for mean and alternating stresses, corrected endurance strength, and stress concentrations at keyways, shoulders, grooves, and threads.

Deflection and critical-speed checks are essential because a shaft can be statically strong yet dynamically unreliable. Keys transmit torque but weaken the shaft and must be checked for shear and crushing. Splines distribute load among multiple teeth and are appropriate for high-cycle or sliding connections, but their design should follow recognized standards. Rigid couplings require accurate alignment, whereas flexible couplings accommodate specified misalignment and may provide damping. Reliable design integrates shaft geometry, retaining devices, bearings, couplings, alignment, balancing, inspection, and safe maintenance.

Key Terms

Shaft: Rotating machine member that transmits torque.

Axle: Member that supports rotating parts and usually does not transmit torque.

Spindle: Short, accurately manufactured rotating shaft.

Torsion: Loading that produces twisting about a member’s longitudinal axis.

Polar second moment of area: Geometric property governing circular-section torsional resistance.

Angle of twist: Relative angular rotation produced by torque.

Bending moment: Internal moment caused by transverse loading.

Equivalent von Mises stress: Distortion-energy measure used to assess combined ductile-material stress.

Fatigue: Progressive damage caused by repeated or fluctuating stress.

Stress concentration: Local increase in stress caused by geometric discontinuity.

Key: Removable element transmitting torque between shaft and hub.

Keyway: Groove machined in a shaft or hub to receive a key.

Spline: Multiple-tooth shaft-and-hub torque connection.

Retaining ring: Ring fitted in a groove to resist axial movement.

Rigid coupling: Coupling that does not intentionally accommodate shaft misalignment.

Flexible coupling: Coupling that transmits torque while accommodating specified misalignment.

Critical speed: Rotational speed near a system natural frequency.

Unbalance: Unequal mass distribution causing centrifugal vibration.

Service factor: Multiplier accounting for duty, shock, and operating conditions.

Fretting: Surface damage caused by small-amplitude oscillatory relative motion.

Review Questions

Short Answer Questions

1. Distinguish a shaft from an axle and a spindle.
2. State the assumptions behind the circular-shaft torsion equation.
3. Why does a keyway reduce shaft fatigue strength?
4. What is the purpose of a flexible coupling?
5. Define critical speed and explain why it matters.

Long Answer Questions

1. Explain a complete procedure for designing a shaft subjected to bending and torsion.
2. Derive the torsional stress and angle-of-twist relations for a solid circular shaft.
3. Discuss fatigue design of shafts, including mean stress, alternating stress, surface finish, size, and stress concentration.
4. Compare keys and splines with respect to load capacity, alignment, manufacturability, and applications.
5. Compare rigid and flexible couplings and describe the information required for coupling selection.

Analytical / Application Questions

1. A shaft transmits 20 kW at 750 rpm and experiences a bending moment of 300 N·m. Explain what additional information is required before selecting its final diameter, and identify the governing checks.
2. A keyed connection repeatedly develops fretting near the hub. Analyze possible causes and recommend corrective actions involving fit, alignment, load distribution, surface condition, and retention.
3. A machine operates close to a measured critical speed. Propose a systematic engineering plan involving measurement, balancing, stiffness changes, damping, operating-speed selection, and validation.

Multiple Choice Questions with Answers

Question 1

Which quantity is directly related to transmitted power and rotational speed by T = 9550P_kW/N?

A. Bending moment in N·m

B. Torque in N·m

C. Angle of twist in radians

D. Bearing pressure in MPa

Correct Answer: B. Torque in N·m

Question 2

For a solid circular shaft, the nominal maximum torsional stress is proportional to:

A. T/d

B. T/d²

C. T/d³

D. T/d⁴

Correct Answer: C. T/d³

Question 3

A shaft transmits 12 kW at 600 rpm. Its torque is approximately:

A. 19.1 N·m

B. 191 N·m

C. 955 N·m

D. 5,730 N·m

Correct Answer: B. 191 N·m

Question 4

In the key design example, which failure mode determined the required key length?

A. Tensile rupture

B. Buckling

C. Shear

D. Crushing

Correct Answer: D. Crushing

Question 5

Which feature most directly increases fatigue stress concentration in a shaft?

A. A generous fillet

B. A smooth polished surface

C. An abrupt shoulder and keyway

D. A gradual diameter transition

Correct Answer: C. An abrupt shoulder and keyway

Question 6

A rigid coupling is most appropriate when:

A. Large misalignment is expected

B. Shafts are accurately aligned

C. High damping is required

D. Axial movement must always be unrestricted

Correct Answer: B. Shafts are accurately aligned

Question 7

The simplified natural frequency of a single-mass rotor supported by stiffness k is:

A. ω_n = k/m

B. ω_n = m/k

C. ω_n = √(k/m)

D. ω_n = √(m/k)

Correct Answer: C. ω_n = √(k/m)

Question 8

Which connection generally provides multiple load-sharing teeth and may permit axial sliding?

A. Spline

B. Plain washer

C. Taper pin only

D. Retaining ring

Correct Answer: A. Spline

Question 9

Why is shaft deflection checked in addition to shaft strength?

A. Deflection determines material density

B. Deflection can cause gear misalignment and bearing overload

C. Deflection eliminates all fatigue stresses

D. Deflection measures key hardness

Correct Answer: B. Deflection can cause gear misalignment and bearing overload

Question 10

A coupling service factor is primarily used to account for:

A. Shaft color

B. Driver and driven-machine duty and shock

C. Ambient lighting

D. The number of keyways only

Correct Answer: B. Driver and driven-machine duty and shock