Design of shafts, keys, bearings and springs

In short

Shafts carry torque and bending between prime movers and driven machines; keys lock hubs to shafts; bearings support the shaft and locate it; springs store energy and apply controlled force. These four are the "machine elements" core of machine design — every gearbox, pump, conveyor, motor mounting and vehicle suspension is a combination of them. Design means sizing each element for strength, rigidity and life against standard allowable stresses, catalogue ratings and Indian standards (IS 1570 steels, IS 2048 keys, ISO 281/IS rolling-bearing ratings, IS 4454 and IS 7906 springs). This block is heavily examined in GATE and in university Design of Machine Elements papers, and it is the usual first technical round at Kirloskar, Thermax, TVS, Tata Motors, Mahindra, L&T, Godrej, BHEL, SKF, Schaeffler and Rane.

Explained

Shafts: types, loads and the design approach

• A shaft is a rotating member transmitting power; an axle is a stationary member supporting a rotating part; a spindle is a short shaft on a machine tool; line, counter and jack shafts are classes of transmission shafting.
• Loads: pure torsion (couplings), bending (self weight, gear/belt/chain pull), axial (helical and worm gears, thrust), and almost always combined. Rotation makes bending stress fully reversed → fatigue governs real shafts.
• Design is for strength (stress limit) and for rigidity (deflection and twist limits). The larger of the two diameters is chosen, then rounded up to a standard size.
• Materials: plain carbon steel 40C8, 45C8, 50C4 (IS 1570) for general shafting; hot-rolled mild steel for low duty; alloy steels 40Ni2Cr1Mo28, 16Mn5Cr4 / 20MnCr5 (case hardened) for gear and machine-tool spindles; EN8/EN9 in trade usage.
• ASME code allowable shear: the smaller of 30% of yield or 18% of ultimate strength; for commercial-grade shafting take τ ≈ 55 N/mm². Reduce by 25% (to ≈ 40 N/mm²) wherever a keyway exists. Allowable bending ≈ 2τ.
• Shock and fatigue factors Kb (bending) and Kt (torsion) multiply the moments: stationary shaft with gradual load 1.0/1.0; rotating shaft gradual load 1.5/1.0; minor shock 1.5–2.0 / 1.0–1.5; heavy shock 2.0–3.0 / 1.5–3.0.
• Stress raisers: keyways, shoulders, press fits, retaining-ring grooves, oil holes. Use generous fillets (r ≥ 0.1d), grind undercuts, avoid a keyway at the point of maximum bending moment.
• Hollow shafts give the same torque for 25–35% less mass (material near the axis carries little shear stress) and are used in propeller, axle and machine-tool spindle applications; cost and manufacturing limit their general use.
• Design steps: draw the shaft layout → mark bearing/gear/pulley positions → find tangential, radial and axial forces → draw bending moment diagrams in two planes and combine (M = √(Mh² + Mv²)) → find torque at each section → size by equivalent twisting/bending moment at the critical section → check twist, deflection and critical speed → fix standard diameter and fillets.

Rigidity, critical speed and standard sizing

• Lateral deflection limits: about 0.001 L for general shafting, and far tighter at gears (typically ≤ 0.05 mm or ≤ 0.01 × module) to keep tooth contact; slope at a bearing ≤ 0.0008–0.001 rad (plain) or as the bearing's misalignment rating allows.
• Torsional deflection limits: about 0.25°–3° per metre, or the workshop rule of 1° in a length of 20 diameters for power shafting; much tighter for machine-tool feed and indexing shafts.
• Critical (whirling) speed occurs when running speed equals the shaft's natural bending frequency; operate below 0.75 Nc or above 1.25 Nc. Mass of mounted rotors, not the shaft alone, usually decides it.
• Standard diameters: round up to preferred sizes (commonly 1 mm steps up to 50 mm, 2 mm up to 100 mm, then 5 mm); bearing bores force 5 mm multiples above 20 mm.

Keys, keyways and splines

• A key is a wedge-shaped or prismatic element in a keyway that transmits torque between shaft and hub (gear, pulley, coupling, sprocket) and prevents relative rotation.
• Types: sunk keys — rectangular/square parallel key (most common), taper key with 1:100 taper, gib-head key (removable), feather key (allows axial sliding, screwed to shaft), Woodruff key (semicircular, self-aligning, for tapered shafts and light duty); saddle keys — flat and hollow, transmit by friction only, light duty; tangent/Kennedy keys — pairs fitted at right angles for heavy reversing duty; round/pin keys; splines — multiple integral keys (straight-sided per IS 3665 or involute) for high torque with axial movement, as in gearbox and propeller shafts.
• Standard proportions: w = d/4 and t = d/6 as a design rule; in practice take w × t from IS 2048 (parallel keys and keyways) for the shaft diameter band — e.g. 16 × 10 for 50–58 mm, 18 × 11 for 58–65 mm shafts.
• Two failure modes: shear across the key width on the shaft–hub interface, and crushing (bearing) on the key side faces. With t < w and σc = 2τ, crushing usually needs the longer key, so it governs.
• Key length is limited by hub width (usually l ≤ 1.5d); if one key is too short, use two keys 90°–180° apart (take 75–80% of each key's capacity) or a spline.
• The keyway weakens the shaft: torsional strength factor e = 1 − 0.2(w/d) − 1.1(h/d), h = t/2 — typically 0.80–0.85. Do not apply this and the ASME 25% reduction together; one allowance is enough.
• Keys are deliberately made slightly weaker than the shaft so that the cheap element fails first; a shear pin is the same idea taken further.

Bearings: types, rating and selection

• Sliding (plain/journal) bearings carry load over a lubricated film area; rolling-contact (anti-friction) bearings carry it on balls or rollers. Rolling bearings have low starting friction, need little lubricant and are standardised; plain bearings are cheaper, quieter, better in dirt and shock, and take very large loads in small space.
• Rolling bearing types: deep groove ball (radial + moderate thrust, cheapest, most used), angular contact ball (combined load, mounted in pairs, preloaded), self-aligning ball and spherical roller (misalignment), cylindrical roller (heavy radial, no thrust), taper roller (heavy radial + thrust — wheel hubs), needle roller (small radial space), thrust ball/roller (axial only).
• Designation (e.g. 6205): first digit = type (6 deep groove ball, 7 angular contact, 1 self-aligning ball, 2 spherical roller, 3 taper roller series 302xx/303xx, N cylindrical roller); second = dimension (diameter) series (2 light, 3 medium, 4 heavy); last two × 5 = bore in mm (05 → 25 mm). Suffixes: ZZ/2Z shields, 2RS seals, C3 clearance.
• Basic dynamic load rating C = constant radial load a group of identical bearings can carry for 10⁶ revolutions with 90% survival. Basic static rating C0 = load causing 0.0001 D permanent deformation at the most loaded contact; check C0 for slow or oscillating bearings.
• Rating life L10 is the life reached or exceeded by 90% of a population; median life ≈ 5 × L10. Reliability above 90% is handled by a reliability factor (ISO 281 life-modification factor a1, with a_iso covering lubrication and contamination).
• Equivalent load P = (X·V·Fr + Y·Fa) × Ks — X, Y from the catalogue table depending on Fa/Fr versus e and on Fa/C0; V = 1 for a rotating inner ring, 1.2 for a rotating outer ring; Ks = service/shock factor (1.0 gearbox steady, 1.5–3 crushers, rolling mills).
• Selection procedure: fix shaft diameter → decide load direction and misalignment → choose type → compute P → fix required life in hours (ball bearings in industrial gearboxes 20,000–30,000 h; automobile 1,500–5,000 h; machine-tool spindles 20,000 h+) → compute required C → pick the smallest bearing in the catalogue with C ≥ required → check speed limit, C0 and sealing.
• Mounting: one bearing is axially fixed (locating) and the other floats to allow thermal expansion; the ring that rotates relative to the load gets the interference fit (usually k6/m6 shaft, H7 housing); locate with shoulders, lock nuts and lock washers, circlips or end plates.
• Journal bearing design uses bearing pressure p = W/(L·d) on the projected area (typical 1–8 N/mm² depending on duty), l/d ratio 0.8–1.5, radial clearance c ≈ 0.001d, and the bearing modulus ZN/p; design at about three times the modulus at minimum friction so a transient dip does not drop the bearing into boundary lubrication. Heat generated (μWV) must equal heat dissipated by the housing or oil flow.
• Failure modes: rolling-contact fatigue spalling (the only one the L10 equation predicts), brinelling from shock during assembly, false brinelling from vibration at rest, fretting, contamination, electrical pitting and lubricant failure — in service, lubrication and cleanliness decide life far more often than the rating does.

Springs: types, design and practice

• Functions: absorb shock (suspension, buffers), store and release energy (clocks, toys, starters), apply force (clutch, brake, valve, relief valve), measure force (spring balance), isolate vibration.
• Types: helical compression, extension/tension and torsion springs; conical and volute; leaf (laminated) springs; disc/Belleville washers; spiral (clock) springs; rubber and air springs.
• Terminology: wire diameter d, mean coil diameter D, spring index C = D/d (keep 4–12; 6–9 is best for coiling), active coils n, total coils nt, solid length = nt·d, free length, pitch, stiffness/spring rate k (N/mm), and spring surge (a compression wave along the coils when excitation approaches the spring's natural frequency — a valve-spring problem).
• End styles (compression): plain (nt = n), plain and ground (nt = n + 1), squared and squared and ground (nt = n + 2); squared and ground is standard for axial seating.
• Stress: the axial load F produces a torque T = F·D/2 on the wire, so the main stress is torsional shear, increased by direct shear and by coil curvature; both are packed into the Wahl factor K. Use the stress-concentration-free form only for purely static loading.
• Materials (IS 4454): Part 1 patented cold-drawn unalloyed steel wire, grades 1–4 (most cold-coiled springs); Part 2 oil-hardened and tempered wire (SW, VW, HS); Part 3 stainless; chrome-vanadium and chrome-silicon for high temperature and valve springs; phosphor bronze and beryllium copper for corrosion and electrical contacts; Inconel for hot service. Wire strength rises sharply as diameter falls, so always read allowable stress against the chosen d.
• Allowable shear stress ≈ 0.4–0.5 of the wire's ultimate tensile strength for static duty and about 0.3 Sut for severe fatigue duty; shot peening and presetting (scragging) raise fatigue life substantially. Design and calculation practice follows IS 7906 (helical compression, extension, hot-coiled and torsion springs in separate parts); automobile leaf springs follow IS 1135.
• Checks: solid-length clash (keep clearance ≈ 15% of working deflection, or 1 mm per active coil), buckling (a free-standing spring with Lf/D above about 5 needs a guide rod or bore), surge (natural frequency at least 15–20 times the forcing frequency), set and relaxation at temperature.
• Extension springs fail at the end hooks (combined bending and shear) and usually carry initial tension; torsion springs are loaded in bending, so bending stress and angular deflection formulas apply.
• Leaf springs: a stack of graduated leaves approximates a triangular plate of uniform stress; the master leaf carries the eye and the extra load. Nipping — giving the graduated leaves more initial camber than the full-length leaves — equalises stress and removes the overload on the master leaf. Rebound clips, centre bolt and U-bolts complete the assembly.
• Series and parallel: springs in series (same load, deflections add) give a softer rate; in parallel / concentric nested springs (deflection shared, loads add) give a stiffer unit in short space — nested springs are wound in opposite hands to avoid interlocking.

Key points

• Equivalent twisting moment — Te = √((KbM)² + (KtT)²); size solid shafts from it with the torsional formula.
• Equivalent bending moment — Me = ½[KbM + Te]; the larger of the two diameters governs.
• ASME allowable shear — ≈ 55 N/mm² for commercial shafting, cut 25% to ≈ 40 N/mm² when a keyway is present.
• Strength vs rigidity — short, heavily loaded shafts fail on stress; long, precise shafts are decided by twist and deflection.
• Hollow shaft — same torque for about 30% less mass because stress varies linearly with radius.
• Key failure — shear across width w, crushing on the flank of thickness t; with σc = 2τ, crushing needs a key 1.5× longer, so it governs.
• Equal-strength key length — l = 1.571 d (shear) or 2.356 d (crushing) when the key must match the full shaft torque with w = d/4, t = d/6.
• Keyway strength factor — e = 1 − 0.2(w/d) − 1.1(h/d), usually 0.80–0.85 of plain shaft torsional strength.
• Dynamic rating C — load for 10⁶ revolutions at 90% reliability; C0 is the static (indentation) rating.
• L10 life — (C/P)³ million revolutions for ball bearings, (C/P)^10/3 for roller bearings; median life ≈ 5 × L10.
• Equivalent load — P = (X·V·Fr + Y·Fa)·Ks; without axial load and with a rotating inner ring, P = Fr·Ks.
• Journal bearing — designed on projected pressure W/(Ld), l/d ≈ 1, c ≈ 0.001d, and kept at about 3× the minimum-friction bearing modulus.
• Spring index C = D/d — below 4 is hard to coil, above 12 buckles and tangles; 6–9 is the practical band.
• Wahl factor K — combines direct shear and curvature effects; always include it for fatigue loading.
• Stiffness k = Gd⁴/(8D³n) — stiffness rises as the fourth power of wire diameter and falls with the cube of coil diameter.
• Series/parallel — series softens (1/k = Σ1/ki), parallel/nested stiffens (k = Σki).
• Nipping — pre-camber of graduated leaves that equalises leaf-spring stresses and relieves the master leaf.
• Spring surge — resonance of the coils; keep the spring's natural frequency 15–20 times the excitation frequency.

Formulas and examples

Example 1 — Shaft under combined bending and torsion. A shaft transmits 20 kW at 300 rpm; bending moment 800 N·m; rotating shaft with minor shock, Kb = 1.5, Kt = 1.0; τ = 40 N/mm² (keyway present), σb = 80 N/mm².

• T = 9550 × 20/300 = 636.7 N·m.
• Te = √((1.5 × 800)² + 636.7²) = √(1,440,000 + 405,300) = 1358.4 N·m = 1,358,400 N·mm.
• d³ = 16Te/(πτ) = 16 × 1,358,400/(π × 40) = 172,930 → d = 55.7 mm.
• Me = ½(1200 + 1358.4) = 1279.2 N·m → d³ = 32 × 1,279,200/(π × 80) = 162,870 → d = 54.6 mm.
• Shear governs; select d = 60 mm (next convenient bearing-bore size).
• Rigidity check, 1 m between bearings, twist limit 1° = 0.01745 rad: J = TL/(Gθ) = 636,700 × 1000/(80,000 × 0.01745) = 455,900 mm⁴ → d⁴ = 32 × 455,900/π = 4.644 × 10⁶ → d = 46.4 mm. Strength governs, so 60 mm is safe on both counts.

Example 2 — Hollow shaft for the same duty, di/do = 0.6, same Te and τ.

• do³ = 16Te/(πτ(1 − 0.6⁴)) = 172,930/0.8704 = 198,680 → do = 58.4 mm, di = 35 mm.

• Same torque capacity for 30% less mass, at a 5% larger envelope.

Example 3 — Key for the 60 mm shaft, τ = 40 N/mm², σc = 80 N/mm², T = 636,700 N·mm.

• Rule-of-thumb size w = d/4 = 15 mm, t = d/6 = 10 mm; IS 2048 standard for a 60 mm shaft is 18 × 11 mm — use the standard.
• Tangential force F = 2T/d = 2 × 636,700/60 = 21,220 N.
• Shear: l = F/(wτ) = 21,220/(18 × 40) = 29.5 mm.
• Crushing: l = 2F/(tσc) = 42,440/(11 × 80) = 48.2 mm → take l = 50 mm, well within a typical 70–90 mm hub.
• Keyway factor e = 1 − 0.2(18/60) − 1.1(5.5/60) = 1 − 0.060 − 0.101 = 0.839 — a plain-shaft calculation would need d increased by (1/0.839)^1/3 = 1.060; here the ASME 25% reduction already covers it.

Example 4 — Deep groove ball bearing selection. Fr = 3000 N, Fa = 1500 N, 1200 rpm, required life 10,000 h, X = 0.56 and Y = 1.5 from the catalogue table, service factor Ks = 1.2, rotating inner ring (V = 1).

• P = (0.56 × 3000 + 1.5 × 1500) × 1.2 = (1680 + 2250) × 1.2 = 3930 × 1.2 = 4716 N.
• L10 = 60 × 1200 × 10,000/10⁶ = 720 million revolutions.
• C = P × 720^1/3 = 4716 × 8.963 = 42,270 N ≈ 42.3 kN.
• Choose the smallest catalogue bearing with C ≥ 42.3 kN — a 6308/6309-size deep groove ball bearing in common catalogues (exact C values differ slightly between makers, so always read the maker's table).
• Check back with C = 42.3 kN: L10 = (42.3/4.716)³ = 8.97³ = 721 million rev → Lh = 721 × 10⁶/(60 × 1200) = 10,020 h ✓. The same duty on a roller bearing (a = 10/3) would need C = 4716 × 720^0.3 = 33.9 kN — roller bearings use the larger life exponent, so they need a smaller rating for the same life.

Example 5 — Helical compression spring. Max load 500 N, deflection 25 mm, C = 5, allowable τ = 420 N/mm², G = 81,370 N/mm², squared and ground ends.

• K = (4 × 5 − 1)/(4 × 5 − 4) + 0.615/5 = 1.1875 + 0.123 = 1.3105.
• d² = K·8FC/(πτ) = 1.3105 × 8 × 500 × 5/(π × 420) = 26,210/1319.5 = 19.86 → d = 4.46 mm → take d = 5 mm (standard wire), so D = 5 × 5 = 25 mm.
• Actual stress: τ = 1.3105 × 8 × 500 × 5/(π × 25) = 26,210/78.54 = 333.8 N/mm² < 420 ✓.
• Active coils: n = δGd/(8FC³) = 25 × 81,370 × 5/(8 × 500 × 125) = 10,171,250/500,000 = 20.3 → n = 21.
• k = Gd⁴/(8D³n) = 81,370 × 625/(8 × 15,625 × 21) = 50,856,250/2,625,000 = 19.37 N/mm → δ at 500 N = 25.8 mm ✓.
• nt = 21 + 2 = 23; solid length = 23 × 5 = 115 mm; clearance ≈ 15% of 25.8 ≈ 4 mm → free length = 115 + 25.8 + 4 ≈ 145 mm; pitch = 145/22 = 6.6 mm.
• Energy stored U = ½ × 500 × 25.8 = 6450 N·mm = 6.45 J.
• Buckling check: Lf/D = 145/25 = 5.8, above the usual free-standing guideline of about 5 — so guide the spring over a rod or inside a bore, or redesign with a larger C and shorter length.

Where it is used

• Gearbox and drive manufacture (Elecon, Shanthi Gears, Bonfiglioli India, Greaves): shaft sizing by equivalent moment, bearing life for 20,000–30,000 h, keys and splines on every gear seat.
• Automotive (Tata Motors, Mahindra, Ashok Leyland, Rane, Gabriel): valve springs, clutch and brake springs, leaf springs to IS 1135 for LCV/HCV rear suspension, taper roller hub bearings, axle shafts.
• Pumps, compressors and fans (Kirloskar Brothers, KSB, Thermax, Elgi): rotor shafts checked for whirling speed, mechanical-seal springs, deep groove and angular contact bearing sets.
• Spring and bearing industry (Jamna Auto, Stumpp Schuele & Somappa, NBC/NEI, SKF India, Schaeffler India, Timken India): wire drawing to IS 4454, shot peening, hot coiling, catalogue life computation and application engineering roles.
• Railways and heavy engineering (RDSO specifications, BHEL, L&T, Jindal/Tata Steel mills): coil and leaf springs for wagons and bogies, mill roll necks on plain bearings, crane and hoist shafts under heavy shock factors.
• Job roles: design engineer (DME/GD&T, CAD/CAE), application engineer at bearing and spring makers, maintenance and reliability engineer (bearing failure analysis, vibration), quality engineer on shaft and spring inspection.

Interview questions

• Q: Why is the ASME allowable shear reduced by 25% when the shaft has a keyway? A: The keyway is a sharp stress raiser and removes material, cutting torsional strength to roughly 80–85% of the plain shaft; the 25% reduction is a simple blanket allowance for it.
• Q: Which usually governs a key's length — shear or crushing? A: Crushing, because the standard thickness is less than the width and the crushing allowable is about twice the shear allowable, making the required length about 1.5 times the shear length.
• Q: Why is a hollow shaft stronger per unit mass? A: Torsional shear stress varies linearly with radius, so material near the axis contributes little; removing it loses little strength but a lot of weight.
• Q: A shaft passes the stress check but the gears whine and wear unevenly. What would you check? A: Rigidity — lateral deflection and slope at the gear and bearings; shafts for gears are often deflection-governed, not stress-governed.
• Q: What exactly does L10 life mean? A: The life in revolutions or hours that 90% of an identical bearing population will reach or exceed under a given load; 10% are expected to have failed by then, and the median life is about five times L10.
• Q: If the load on a ball bearing is doubled, what happens to its life? A: It falls to one-eighth, because life varies as the inverse cube of the equivalent load.
• Q: Why are angular contact bearings mounted in pairs and preloaded? A: A single bearing takes thrust in one direction only; a preloaded pair takes it both ways, removes clearance and raises stiffness and running accuracy.
• Q: What is the Wahl factor and when may it be ignored? A: A multiplier combining direct shear and coil curvature with the torsional shear in spring wire; for purely static loading the simpler shear-stress factor suffices, but it must be included for fatigue duty.
• Q: Why is the spring index kept between about 4 and 12? A: Below 4 the wire cracks during coiling and the curvature stress is severe; above 12 the spring is slender, buckles easily and tangles in handling.
• Q: What is spring surge and how is it avoided? A: A travelling compression wave when the excitation frequency approaches the spring's natural frequency, causing coil clash and fatigue failure; it is avoided by keeping the natural frequency 15–20 times the forcing frequency, using conical or variable-pitch springs or a damper.
• Q: What is nipping in a leaf spring and why do it? A: Giving the graduated leaves a larger initial camber than the full-length leaves, so that on assembly the pre-stress equalises the bending stress across all leaves and relieves the overloaded master leaf.
• Q: Two identical springs are fitted first in series, then in parallel. Compare the rates. A: In series the combined rate is half that of one spring; in parallel it is double — series softens, parallel stiffens.

Common mistakes

• Mixing units: feeding torque in N·m into formulas expecting N·mm, or kW into the N·m torque relation without the 9550 factor.
• Applying both the 25% ASME keyway reduction and the keyway strength factor e, so the shaft is oversized twice.
• Forgetting the shock/fatigue factors Kb and Kt, or using stationary-shaft values for a rotating shaft.
• Sizing a shaft on strength alone and never checking twist, deflection at the gear or critical speed.
• Using the wrong life exponent (3 for ball, 10/3 for roller) or forgetting the 60 N factor when converting revolutions to hours.
• Taking the catalogue Fr as P when a thrust load exists, instead of computing P = XVFr + YFa with the correct X and Y for Fa/Fr versus e.
• Using the computed exact wire diameter instead of the next standard wire size, and then not recomputing stress, coils and stiffness.
• Omitting the solid-length clearance and buckling check, so the spring bottoms out or bows under load.

Related topics

Machine design: failure theories and fatigue · Bearings, lubrication and tribology · Gear design · Clutches and brakes · Bolted, riveted and welded joints · Strength of materials: torsion and bending of beams