Shafts, Keys, Splines and Couplings

Introduction

Shafts and their associated connection elements transmit mechanical power between rotating components. A successful design must do more than prevent immediate fracture: it must control deflection and twist, avoid harmful resonance, permit assembly and maintenance, accommodate alignment errors, and remain reliable under fluctuating service loads. Keys, splines, pins, retaining devices and couplings form the mechanical interfaces through which torque, axial force and motion are transferred.

Shaft design is therefore an integrated problem involving strength, fatigue, stiffness, dynamics, manufacturing and assembly. The nominal torque may be calculated from power and speed, but local stress concentrations, surface finish, corrosion, misalignment and start-stop cycles often govern the final design.

Learning Objectives

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

1. Determine shaft torque from power and rotational speed.
2. Analyze shafts subjected to combined bending, torsion and axial loading.
3. Apply static and fatigue criteria to preliminary shaft sizing.
4. Check angular deflection, torsional twist and critical speed.
5. Select and design keys, splines, pins and retaining devices.
6. Compare rigid and flexible couplings and select an appropriate type.
7. Incorporate alignment, assembly, lubrication, inspection and safety requirements into a component design.

1. Shaft Functions and Loading

A shaft is a rotating member that transmits torque and supports mounted elements such as gears, pulleys, sprockets, flywheels, fans and rotors. An axle primarily supports rotating members and normally transmits little or no torque, although the terminology varies by industry.

Typical shaft loads include:

• torsion from transmitted power;
• transverse bending from gear, belt, chain and bearing reactions;
• axial tension or compression from helical gears, thrust mechanisms and preloaded assemblies;
• fluctuating loads caused by starting, stopping, reversing, impact or variable process resistance;
• local contact and stress concentration at shoulders, keyways, grooves and splines.

For a rotating shaft under a stationary transverse force, a material point generally experiences a cyclic bending stress. Torsional stress may be steady or fluctuating depending on the drive system. This distinction is central to fatigue design.

The transmitted power relation is

where P is power in watts, T is torque in newton-metres, ω is angular speed in radians per second, and n is speed in revolutions per minute. Consequently,

For power in kilowatts and torque in newton-metres, a convenient form is T = 9550P_kW/n. These equations assume steady rotational power and consistent units. They do not include transient torque amplification; a service or shock factor may be applied separately.

A torque diagram and a bending-moment diagram should be prepared before sizing a shaft. The critical section is not necessarily where the torque is greatest; it is often a shoulder or keyway where combined stress concentration is severe.

2. Stresses in a Circular Shaft

For a solid circular shaft of diameter d subjected to torque T, the maximum nominal torsional shear stress is

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

These expressions assume elastic, homogeneous, isotropic material and Saint-Venant torsion away from abrupt discontinuities.

The nominal bending stress in a solid circular shaft is

where M is the bending moment. With an axial force F_a, the nominal direct stress is

For a hollow shaft, replace the area and second moment expressions appropriately. The exact normal stress at a notch must include a stress-concentration factor. In fatigue analysis, the fatigue concentration factor K_f is generally more appropriate than the theoretical elastic factor K_t because material notch sensitivity reduces the fully effective concentration.

Combined static loading

At a surface point where bending and axial stresses have the same sign, the normal stress is approximately

The maximum and minimum principal stresses for combined normal stress σ and torsional shear stress τ are

The maximum-distortion-energy, or von Mises, equivalent stress is

For ductile materials under static loading, a preliminary design requires σ_VM ≤ S_y/n_s, where S_y is yield strength and n_s is the desired factor of safety. This criterion is suitable for yielding prediction under multiaxial static stress, but it does not by itself address fatigue, brittle fracture or excessive deflection.

ASME-type shaft sizing relation

A commonly used preliminary relation for a solid shaft is

where K_b and K_t are application factors for bending and torsion, and τ_allow is an allowable equivalent shear stress. This relation is useful for preliminary design, particularly when bending and torque are represented by maximum values. It should not replace a detailed fatigue and stiffness assessment. Different standards and organizations define service factors differently; their definitions must not be mixed.

3. Fatigue Design of Shafts

Many shafts fail by fatigue even when their maximum nominal stress is below the yield strength. A rotating shaft with a constant transverse load experiences fully reversed bending at a surface point, while transmitted torque may remain approximately steady. Thus the bending component is usually alternating and the torsional component often has a mean value.

For each stress component, define

and similarly for shear stress τ. Surface finish, size, reliability, temperature, corrosion and notch effects modify the laboratory endurance limit. The corrected endurance strength should be based on appropriate material data and design standards rather than an unqualified textbook value.

A commonly applied linear Goodman relation for normal stress is

where S_e is corrected endurance strength, S_ut is ultimate tensile strength and n_f is fatigue safety factor. For combined bending and torsion, a practical preliminary approach is to form alternating and mean equivalent stresses using the distortion-energy concept:

and then apply the Goodman relation to σ′_a and σ′_m. This is an engineering approximation. For severe notch effects, nonproportional loading, yielding or complex duty cycles, a more appropriate multiaxial fatigue method is required.

A typical design sequence is:

1. Calculate torque and bearing or gear reactions.
2. Construct shear-force and bending-moment diagrams.
3. Identify shoulders, keyways, grooves and other discontinuities.
4. Resolve stresses into alternating and mean components.
5. Apply fatigue concentration factors and Marin-type modifying factors.
6. Evaluate a fatigue criterion and static overload condition.
7. Increase diameter or improve geometry until strength and reliability requirements are met.

A generous fillet radius, good surface finish, suitable heat treatment and avoidance of abrupt diameter changes often improve fatigue life more effectively than simply increasing the nominal diameter.

4. Shaft Stiffness

A shaft can be strong enough yet unusable because it bends excessively, twists too much or misaligns connected equipment.

For a uniform shaft in torsion, the angle of twist is

where θ is twist in radians, L is shaft length, J is the polar second moment of area and G is shear modulus. For a solid circular shaft, J = πd⁴/32; for a hollow shaft, J = π(d_o⁴-d_i⁴)/32. For stepped shafts, twist contributions are summed:

This assumes elastic torsion, constant torque within each segment and appropriate support conditions.

For lateral deflection, beam theory is used. The second moment of area of a solid circular section is

The deflection depends on loading and bearing support conditions. For example, a simply supported beam with a central point load W has central deflection

where E is Young’s modulus and L is span. This formula is not applicable to an overhung shaft or a shaft with distributed gear loads unless the support and loading model match the stated case. In practical design, bearing stiffness, housing flexibility, gear mesh requirements and seal limits may be as important as shaft stiffness.

5. Critical Speed and Rotor Dynamics

Every rotating shaft has natural frequencies. When operating speed approaches a lateral natural frequency, resonance can produce large vibration, rapid fatigue damage and loss of control. The corresponding rotational speed is called a critical speed or whirling speed.

For a simplified single-disc rotor on a massless shaft, an approximate natural frequency is

where k is lateral shaft stiffness and m is concentrated rotor mass. The critical speed in revolutions per minute is

This model is only a first estimate. Real shafts have distributed mass, several discs, bearing flexibility, gyroscopic effects and damping. A shaft should not be operated continuously near a critical speed unless the rotor-dynamic design specifically permits it. Start-up and coast-down rates must also be considered because a machine may pass through a critical speed even when its steady operating speed is below it.

The design target is commonly either a rigid rotor operating well below the first critical speed or a flexible rotor operating above it with appropriate balancing, damping and control. The required separation margin depends on applicable standards and machine type.

6. Keys and Keyways

A key is a removable element placed between a shaft and hub to transmit torque and prevent relative rotation. Parallel sunk keys are common in general machinery; taper keys provide wedging but may introduce eccentricity and assembly difficulties. Woodruff keys are useful where a short key length and self-alignment are advantageous.

A rectangular key of width b, height h and length L transmitting torque T has approximate tangential force

The average shear stress in the key is

If the key bears against the shaft and hub over approximately half its height, the average bearing stress is

The key must satisfy both shear and bearing criteria using allowable stresses or material-specific design limits. In reality, load distribution is nonuniform, especially near the ends, so a design should not assume perfect uniform contact. The keyway reduces shaft area and creates a stress concentration; the shaft must therefore be checked with the keyway included.

Key selection also depends on hub length, available standards, assembly method, reversing load and required axial location. A key is not normally intended to transmit axial thrust unless specifically designed for that function.

7. Splines

Splines are multiple integral or attached teeth on a shaft that engage matching grooves in a hub. Compared with a single key, splines distribute torque among several teeth, provide better concentricity and can permit axial sliding. Straight-sided and involute splines are widely used; involute forms generally provide improved centering and manufacturability.

Spline capacity is limited by tooth shear, tooth bearing pressure, shaft strength, hub strength, wear and fretting. A simplified average-bearing estimate is

where p_b is allowable bearing pressure, A_m is the effective projected bearing area and r_m is mean spline radius. Because only part of the theoretical tooth area may carry load, an effective engagement or load-sharing factor is required. Standard spline systems should be designed using the relevant dimensional and rating standard rather than relying solely on this simplified relation.

Splines are preferred for sliding couplings, high torque relative to shaft diameter and applications requiring accurate torque transmission. They cost more than keys and require control of backlash, lubrication, surface hardness and contamination.

8. Pins and Retaining Devices

Pins may locate parts, transmit shear, provide overload protection or secure an assembly. Dowel pins are primarily locating elements and should not be treated as general-purpose torque-transmitting members without analysis. Shear pins are intentionally designed as sacrificial elements. Taper pins can provide secure positioning but require suitable reaming and removal access.

For a pin carrying direct shear force F, the nominal shear stress is

where A_s is the resisting shear area. For a single-shear cylindrical pin, A_s = πd_p²/4; for double shear, the idealized area is twice this value. Bearing stress in the connected member must also be checked, as must net-section tension and tear-out. These simplified equations assume uniform load distribution and should be modified for eccentric loading, bending of the pin or clearance-induced impact.

Retaining rings, circlips, shoulders, locknuts, tab washers, castellated nuts and threaded retainers prevent axial movement. Selection depends on axial load, rotational speed, groove strength, repeated assembly, temperature and access. A retaining ring groove reduces shaft cross-section and introduces a stress concentration, so fatigue-critical shafts require a groove assessment. Locking devices must not be used as substitutes for correct fits, shoulders or positive load paths.

9. Couplings

A coupling connects two shafts. It may transmit torque while allowing some axial, angular or parallel misalignment. Couplings are classified as rigid or flexible.

Rigid couplings

Sleeve, clamp and flange couplings are suitable where shafts are accurately aligned and supported. They provide little tolerance for misalignment. If used across misaligned shafts, they impose bending moments and bearing reactions, potentially causing seal, bearing and shaft failures. Flange bolts may be loaded in shear, tension or friction, and the flange and key must be checked as a complete torque path.

Flexible couplings

Flexible couplings accommodate limited misalignment and may also damp torsional vibration. Common types include elastomeric jaw, tyre, grid, gear, disc and diaphragm couplings. Their selection requires torque, speed, bore, temperature, environmental exposure, start-up severity, axial movement and allowable misalignment.

Flexible does not mean unlimited misalignment. Angular and parallel misalignment should be measured and corrected rather than intentionally assigned to the coupling. Excessive misalignment produces heat, wear and alternating forces. Coupling guards are required wherever rotating parts present an entanglement hazard.

A practical selection process is:

1. Determine nominal torque from power and speed.
2. Apply a service factor based on driver, driven machine and duty cycle.
3. Check peak and reversing torque.
4. Check bore, key or spline interface and maximum speed.
5. Verify allowable angular, parallel and axial misalignment.
6. Check temperature, chemicals, lubrication and maintenance requirements.
7. Confirm balancing, guarding and installation space.

The design torque may be represented as T_design = K_sT_nom, where K_s is a documented service factor. The value must come from the coupling manufacturer or an applicable standard; it should not be selected arbitrarily.

10. Alignment and Assembly

Shaft systems fail frequently because of installation errors rather than inadequate nominal strength. Alignment includes offset, angular and axial conditions. Soft foot, pipe strain, bearing-seat errors and thermal growth can change alignment after assembly.

Best practice includes:

• establish datum surfaces and tolerances before machining;
• use calibrated dial indicators or laser alignment equipment;
• inspect runout of shafts, hubs and coupling faces;
• control fits according to load and required removability;
• provide shoulders and end features that create positive axial location;
• avoid forcing hubs onto shafts with uncontrolled impact;
• use correct heating, hydraulic fitting or approved installation tools;
• verify key fit, burr removal and full seating;
• lubricate splines and gear couplings as specified;
• balance rotors after machining and assembly when required;
• document torque values, clearances and inspection results.

A key should fit its side faces without excessive looseness while allowing the hub to assemble without damaging the shaft. A spline should be checked for tooth contact, backlash, axial movement and lubricant compatibility. Coupling bolts should be tightened in the specified sequence and to the specified torque.

11. Worked Example: Preliminary Design of a Solid Shaft

An illustrative shaft transmits 15 kW at 1,200 rpm. At the critical section, the bending moment is 180 N·m and the torque is 119.4 N·m. Ignore axial load for this preliminary static calculation. Use K_b = 1.5, K_t = 1.2 and an allowable equivalent shear stress of 45 MPa.

First, the power relation gives

Using the preliminary sizing relation,

The combined term is

Therefore,

and

A nominal diameter of 35 mm might be selected for the next design iteration. This is not a final approval: the 35 mm shaft must be checked for keyway stress concentration, fatigue, deflection, twist, bearing fits, shoulder fillets and critical speed. If a keyway substantially reduces fatigue strength, a larger diameter or alternative spline connection may be required.

12. Applied Case: Conveyor Drive Shaft

Background

This illustrative case concerns a conveyor driven by a 22 kW motor through a gearbox. The output shaft carries a sprocket and is connected to the gearbox by a flexible coupling. The conveyor experiences frequent starts and occasional jam clearing.

Problem

An initial design based only on nominal torque produced repeated failures near a keyway. Inspection found a sharp shoulder, inadequate coupling alignment and a starting torque substantially greater than the steady running torque.

Analysis

The design team calculated steady torque from power and speed, then obtained the bending reactions from chain tension and bearing spacing. A duty analysis identified high alternating bending caused by the rotating chain pull and intermittent torsional peaks during starts. The keyway was treated as a fatigue notch, and the existing small fillet was found to amplify local stress. Shaft twist also caused sprocket misalignment.

Approach and solution

The revised design used a larger shaft diameter with a generous ground fillet, improved bearing spacing and a shorter overhung sprocket arrangement. The key was lengthened only after confirming hub length and bearing capacity; the shaft was checked separately because increasing key length does not eliminate the shaft keyway notch. A rated flexible coupling with a documented start-up service factor was selected. Laser alignment was specified, and a coupling guard and inspection schedule were added.

Outcome

The revised shaft met the preliminary static and fatigue criteria, reduced angular twist, and provided greater tolerance to thermal and assembly variation. Commissioning measurements confirmed acceptable vibration and coupling temperature. The case demonstrates that reliability resulted from coordinated changes to geometry, duty assessment, alignment and maintenance—not from diameter increase alone.

Learning points

1. Nominal running torque may be inadequate for sizing a cyclic-duty drive.
2. Keyways and shoulders must be included in fatigue analysis.
3. Coupling flexibility cannot compensate for poor alignment.
4. Shaft stiffness and assembly quality can govern performance even when strength is adequate.

13. Limitations, Standards and Professional Practice

Design equations in this chapter are idealized models. Actual design should use verified material properties, manufacturing tolerances, applicable shaft, spline, coupling and guarding standards, and supplier data. Special attention is required for corrosive environments, hydrogen embrittlement, low-temperature service, high-speed rotors, plastic shafts and welded or additively manufactured components.

Computer-aided design and finite-element analysis are valuable for identifying stress concentrations and predicting deflection, but results depend on realistic boundary conditions, mesh quality, contact assumptions and material data. Digital models should be validated against hand calculations, inspection and, where necessary, prototype testing. Sensor-based vibration monitoring and data analytics can detect imbalance, misalignment and bearing deterioration, but automated alarms require engineering thresholds and human review.

Safety practice requires guarding of rotating shafts and couplings, lockout/tagout during maintenance, controlled stored-energy release and documented inspection. Engineers must not conceal overload history, bypass guards or specify a component outside its rated speed or torque. Traceability of materials, heat treatment, inspection and design assumptions is part of professional responsibility.

Chapter Summary

Shaft design integrates torque transmission, bending, axial loading, fatigue, stiffness, dynamics and manufacturability. Torque follows from power and speed, but service factors and transient loads must be considered. Circular-shaft stress equations provide nominal values; shoulders, keyways, grooves and surface condition require concentration and fatigue assessment. Von Mises stress is useful for ductile static loading, while Goodman-type methods provide a preliminary treatment of fluctuating stresses.

Torsional twist, lateral deflection and critical speed can govern a design even when strength is satisfactory. Keys transmit torque economically but weaken the shaft locally. Splines provide distributed torque transmission and sliding capability but require careful control of tooth loading, wear and lubrication. Pins and retaining devices must be designed for their actual function and load path. Rigid couplings require accurate alignment, whereas flexible couplings accommodate limited misalignment and often damp vibration; neither eliminates the need for proper installation.

A professional design process combines analytical calculations, standards, supplier ratings, realistic duty cycles, alignment control, guarding, inspection and maintenance planning.

Key Terms

Shaft: Rotating member used to transmit torque and support machine elements.

Torque: Twisting moment transmitted by a shaft.

Bending moment: Internal moment produced by transverse loading.

Key: Removable element that transmits torque between a shaft and hub.

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

Spline: Multiple external teeth on a shaft engaging matching internal hub grooves.

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

Fatigue: Progressive failure under repeated or fluctuating loading.

Endurance strength: Stress amplitude associated with a specified fatigue-life criterion.

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

Torsional stiffness: Resistance to angular twist under torque.

Lateral stiffness: Resistance to transverse deflection.

Critical speed: Rotational speed near a rotor natural frequency.

Coupling: Device connecting two shafts to transmit torque.

Rigid coupling: Coupling that provides negligible intentional misalignment capacity.

Flexible coupling: Coupling designed to accommodate limited misalignment or vibration.

Runout: Radial or axial deviation of a rotating surface from its ideal datum.

Retaining device: Component preventing unintended axial movement.

Misalignment: Offset, angular or axial deviation between connected shafts.

Review Questions

Short Answer Questions

1. Distinguish a shaft from an axle.
2. Why is a keyway important in shaft fatigue design?
3. State the power–torque–speed relationship.
4. What functions can a flexible coupling perform?
5. Why must critical speed be considered in high-speed shafts?

Long Answer Questions

1. Explain a complete procedure for designing a shaft under combined bending and torsion.
2. Compare keys and splines in terms of load capacity, alignment, assembly and cost.
3. Explain how surface finish, size, fillets and keyways influence fatigue strength.
4. Discuss the differences between rigid and flexible couplings and their alignment requirements.
5. Describe how torsional stiffness, lateral stiffness and critical speed affect machine performance.

Analytical / Application Questions

1. A shaft transmits 18 kW at 900 rpm and experiences a bending moment of 250 N·m. Develop a preliminary diameter calculation using stated service factors and then identify the additional fatigue and stiffness checks required.
2. A keyed shaft fails at the shoulder adjacent to the hub during repeated start-up. Diagnose at least four possible causes and propose a revised design and inspection plan.
3. Select a coupling for a motor-driven conveyor with frequent starts, moderate angular misalignment and a dusty environment. State the required data, selection criteria, safety provisions and commissioning checks.

Multiple Choice Questions with Answers

Question 1

A shaft transmits 12 kW at 1,200 rpm. What is its nominal torque, approximately?

A. 9.55 N·m

B. 95.5 N·m

C. 955 N·m

D. 9,550 N·m

Correct Answer: B. 95.5 N·m

Question 2

Which stress state is represented by σ_VM = √(σ² + 3τ²) for a shaft?

A. Pure axial loading only

B. Combined normal and torsional loading for a ductile static criterion

C. Buckling under compression only

D. Thermal expansion without mechanical load

Correct Answer: B. Combined normal and torsional loading for a ductile static criterion

Question 3

For a solid circular shaft, the nominal bending stress is proportional to which quantity?

A. M/d

B. M/d²

C. M/d³

D. M/d⁴

Correct Answer: C. M/d³

Question 4

What is the primary purpose of a key?

A. To provide shaft lubrication

B. To transmit torque between a shaft and hub

C. To increase critical speed automatically

D. To eliminate all shaft misalignment

Correct Answer: B. To transmit torque between a shaft and hub

Question 5

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

A. A generous smooth fillet

B. A polished uniform diameter

C. A sharp shoulder or keyway

D. A larger bearing span

Correct Answer: C. A sharp shoulder or keyway

Question 6

The angle of twist of a uniform elastic shaft is given by which relation?

A. θ = TL/(JG)

B. θ = JG/(TL)

C. θ = TLG

D. θ = J/(TแL)

Correct Answer: A. θ = TL/(JG)

Question 7

What is a principal advantage of splines over a single key?

A. They always cost less to manufacture

B. They distribute torque among multiple teeth

C. They remove the need for lubrication

D. They prevent every form of misalignment

Correct Answer: B. They distribute torque among multiple teeth

Question 8

A rigid coupling is most appropriate when the connected shafts are:

A. Intentionally offset and unsupported

B. Accurately aligned and properly supported

C. Required to slide axially during operation

D. Subject to unlimited angular misalignment

Correct Answer: B. Accurately aligned and properly supported

Question 9

What does a critical speed represent?

A. The speed at which torque becomes zero

B. A speed near a rotor natural frequency

C. The maximum speed of a stationary bearing

D. The speed at which a key automatically shears

Correct Answer: B. A speed near a rotor natural frequency

Question 10

Which action is best practice before commissioning a coupled shaft system?

A. Remove the coupling guard for visibility

B. Ignore soft foot if nominal torque is low

C. Verify alignment, fastener tightening and allowable operating conditions

D. Operate continuously at the first resonance

Correct Answer: C. Verify alignment, fastener tightening and allowable operating conditions