Keys, Splines and Couplings: Method

Keys, Splines, Pins and Couplings: Method and Sources

A hub is fixed to a shaft by a key, a spline, a pin, a set screw, a press fit or a bolted flange, and the torque crosses the joint as a tangential force at the shaft surface or the pitch circle. Every check on this page is that force divided by an area: a shear plane, a crushed face, a bolt shank. The areas come from the geometry of the standard parts; the standards themselves (ANSI B17.1 and B17.2 for keys, ANSI B92.1 and SAE J499 for splines, the AGMA coupling standards) are cited, not reproduced. What the tool ships as tables is what the U.S. Navy and Army machinist manuals print of them, the keyseat factors of a NASA reference publication, and the spline formulas and allowables of the Army helicopter design handbooks. The closed forms are checked by an independent numpy/scipy script (References/Keys-Splines/checks/). The defaults are illustrative, stated on the page, except the involute spline, which is a NASA-tested part.

Parallel keys

Standard sizes. NAVEDTRA 12204 (Machinery Repairman, 1990, p. 3-41) prints Table 3-4, an excerpt of USAS B17.1-1967 Table 1: for shafts of 7/8 to 1 1/4 in a 1/4 in square key (3/16 in deep rectangular), 1 1/4 to 1 3/8 in 5/16, 1 3/8 to 1 3/4 in 3/8, 1 3/4 to 2 1/4 in 1/2, 4 1/2 to 5 1/2 in 1 1/4 and 6 1/2 to 7 1/2 in 1 3/4, with key lengths of 4 to 16 widths; square keys are recommended to 6 1/2 in shafts and rectangular keys above. TC 9-524 (Fundamentals of Machine Tools, 1996, pp. 8-26/8-27 and Appendix A Table 8-5) gives the rules behind the table, width about a quarter of the shaft diameter, flat-key thickness about a sixth, length at least 1 1/2 diameters, keyseat depth half the key, and lists the square and flat keys for every 1/16 in of shaft from 1/2 to 2 in with the shop dimension S, bottom of the keyseat to the opposite side of the shaft. The tool's "standard size" select carries these bands and fills the width and height, which stay editable.

Keyseat geometry. With the key half in the shaft, the keyseat edge sits on the shaft surface at a chord of width w; the surface rises above the chord by y = d/2 − √((d/2)² − (w/2)²), so the depth at the centre line is h/2 + y and the gauged dimension is S = d − h/2 − y. The test reproduces all fifty S values of Table 8-5 to the printed 0.001 in except one: the flat-key row for the 1/2 in shaft prints 0.455 where the geometry (and the pattern of the neighbouring rows) gives 0.445, a misprint in the Army table.

Stresses. The torque T on a shaft of diameter d is carried across the key as the force F = 2T/d at the shaft surface. The key shears along the shaft surface over w × L, τ = 2T/(d w L), and is crushed on the half of its height in the hub and the half in the shaft, σ = F/((h/2) L) = 4T/(d h L) on each side. The shear allowable is 0.577 S_y/n (the distortion-energy shear yield of the key at the design factor); the bearing allowable on each side is S_y/n of the softer of the two parts in contact, so a cast-iron or aluminum hub governs its own side. The required length is the largest of 2T/(d w τ_all) and 4T/(d h σ_all) for the two sides, printed beside the machinist rules (1.5 d; 4 w to 16 w). These are the formulas every machine-design text writes (Shigley Section 7-7, cited); they assume the force uniform along the key, which is why the rules cap the length at 16 widths and why a hub longer than about 1.5 diameters should be checked as two keys.

Taper and gib-head keys have a taper of 1/8 in per foot (NAVEDTRA 12204 p. 3-41): the height falls by L/96 from the large end, so the tool prints both ends and uses the mean height for the bearing. A driven taper key also wedges the hub against the shaft and carries part of the torque by friction; the check does not credit it. Feather keys let the hub slide under torque. With one key the hub bears on the shaft opposite the key with the same force F, so two surfaces rub and the axial force to move the hub is 2 μ F; with two keys 180° apart each carries F/2 and the force is μ F (the two cases Machinery's Handbook states, cited). AMCP 706-201 puts the coefficient of friction of a lubricated steel spline at 0.1 to 0.15, which serves as a guide for keyed hubs.

Woodruff keys

A Woodruff key is a segment of a disc that seats in a semicircular keyseat milled by a cutter of the same diameter and width; it cannot roll in its seat and is easy to remove, which is why NAVEDTRA 12204 recommends it for pulleys. Its number encodes its size: the last two digits give the diameter in eighths of an inch and the leading digits the width in 32nds (NAVEDTRA 12204 p. 3-42 and TC 9-524 p. 8-26 both print the examples 404 = 1/8 × 1/2 in and 1012 = 5/16 × 1 1/2 in; TC 9-524 also 204 = 1/16 × 1/2). TC 9-524 Table 8-4 (Appendix A p. A-30) lists fifteen numbers with their keyway depth in the shaft, the height of the key above the shaft and the shaft diameters they serve; the key height is the sum of the two, and the test checks that this sum is the same for every key of one diameter (0.203 in for the 1/2 in keys up to 0.594 in for 1 3/8 in) and that the height above the shaft is half the width. The table's last row prints the width of key 1011 as "5/10": by its number and its 0.1562 in height above the shaft it is 5/16 in.

The stresses follow the segment geometry. With R the key radius, the chord of the key at a height y above the bottom of the arc is 2√(R² − (y − R)²). The shear plane at the shaft surface is the chord there times the width; the hub bears on the strip of the key face between the shaft surface and the flat top; the shaft bears on the segment below the surface. The strip areas are the closed-form integral of the chord, checked against quadrature. Because a Woodruff seat is deep, it weakens the shaft more than a parallel keyseat; the RP-1123 factors below, which are for parallel keyways, are a floor.

The shaft at the keyseat

NASA RP-1123 (Loewenthal, Design of Power-Transmitting Shafts, 1984, Table V p. 13, from Peterson) gives typical fatigue stress concentration factors of keyways in solid round steel shafts as strength-reduction factors k_e = 1/K_f: for an annealed shaft (under 200 BHN) 0.63 in bending and 0.77 in torsion with a profiled (end-milled) keyway and 0.77 for both with a sled-runner keyway; for a quenched-and-drawn shaft (over 200 BHN) 0.50 and 0.63 profiled, 0.63 and 0.63 sled-runner. The nominal stresses are those of the full shaft section (16T/(πd³) in torsion). The tool prints K_f = 1/k_e and the fatigue-effective torsional stress K_f τ_nom for a cyclic torque; for the shaft's fatigue design itself use the Shaft Design tool, which carries the full RP-1123 method.

For the static torque capacity the tool applies the old ASME code rule that a shaft with a keyseat carries 75 % of the torque of the plain shaft (the "0.75 factor", reported by Machinery's Handbook, cited): T_keyed = 0.75 τ_all π d³/16 with τ_all = 0.577 S_y,shaft/n. The margin is against the entered torque.

Involute splines

Geometry. An involute spline is a gear tooth of half depth, usually at a 30° pressure angle: pitch diameter D = N/P for N teeth of diametral pitch P, circular pitch π/P, basic circular tooth thickness π/(2P), base diameter D cos φ. The major and minor diameters are written (N ± c)/P; the tool's "tooth form" select fills the coefficients with the basic proportions of the involute spline standard (ANSI B92.1, cited only: external major (N + 1)/P at 30°, external minor (N − 1.35)/P flat root or (N − 1.8)/P fillet root, internal minor (N − 1)/P, internal major (N + 1.35)/P or (N + 1.8)/P; 37.5° and 45° forms with 0.9 / 1.6 and 0.8 / 1.2) and they stay editable for a drawing's own values. The engaged depth is half the difference of the external major and the internal minor diameter, 1/P at 30°. Serrations are the fine-pitch 45° form of the same standard and are checked the same way. The default is the spline NASA tested for the Space Shuttle main engine's oxidizer preburner pump (Ku, Walton and Lund 1994, NASA CP-3239): 42 teeth, 20/40 pitch, 30° fillet-root side fit, whose printed pitch diameter 53.3 mm and base diameter 46.1937 mm the geometry reproduces.

Stresses. The Army helicopter handbooks (AMCP 706-201 par. 7-6.2.4 and AMCP 706-202 par. 4-2.2.3.4) write four stresses. The nominal bearing stress S_br = 2T/(D² F) (eq. 7-52 / 4-55) assumes every tooth in contact over the depth D/N; it is not the true bearing pressure, but it is the stress the handbook's allowables are calibrated to. The tooth shear at the pitch line S_s = 2T/(γ D N t_c F) (eq. 4-56) divides the tangential force among the fraction γ of the teeth that actually carry load: 0.25 to 0.45 for gaged commercial splines, 0.45 to 0.75 for gaged and measured ones, 0.75 to 0.95 for analytically measured ones. The torsional shear of the shaft under the spline is 16 T D_re/(π(D_re⁴ − D_i⁴)) on the external minor diameter (eq. 4-57). The bursting stress of the internal member (eq. 4-58) adds the hoop stress of the rim from its own rotation, ρ v² at the rim speed (the handbook's 7.095 × 10⁻⁶ ρ (n D₂)² in lb, in and rpm, which the test reproduces within 0.1 %), to the hoop stress from the radial component of the tooth load, 4 T tan φ/(π F D² ((D₂/D₁)² − 1)). The tool prints the bearing stress with γ and the computed engaged depth beside the nominal value.

Allowables. AMCP 706-201 Table 7-4 (p. 7-80) gives the allowable bearing stress of steel splines at maximum operating power with the nominal formula: loose (floating) assembly 5,000 lb/in² as machined without lubrication, 7,500 ground, 15,000 hardened and ground; clamped assembly 7,500, 10,000 and 20,000; and 500 (loose) or 1,000 lb/in² (clamped) for steel on aluminum. The handbook adds that low L/D, lubrication, materials and precision allow more, and high L/D, flexing members and poor geometry less. The shear stresses are compared with 0.577 S_y/n. AMCP 706-202 fixes the limit torque of a helicopter spline at 1.5 times the maximum continuous torque and the ultimate at 1.5 times the limit, prefers face widths of 0.4 to 1.0 pitch diameters for torque transmission (0.8 to 2.0 for location) and takes 0.4 T/D as a safe design axial force for sliding an oil-lubricated spline (eq. 4-54); AMCP 706-201 prefers L/D of 2 or less and puts the coefficient of friction at 0.1 to 0.15.

SAE straight-sided splines

TC 9-524 Table 8-8 (Appendix A p. A-32) prints the SAE parallel-side series for 3/4 to 6 in shafts: the width is 0.241 D for 4 splines, 0.250 D for 6, 0.156 D for 10 and 0.098 D for 16, and the minor diameter is 0.850 / 0.900 / 0.910 / 0.910 D for a permanent fit, 0.750 / 0.850 / 0.860 / 0.860 D for a hub that slides when not under load, and 0.800 / 0.810 / 0.810 D (6, 10 and 16 splines) for one that slides under load. The depth is h = (D − d)/2 and the mean radius (D + d)/4. The SAE capacity formula T = p N h r_m L takes every spline bearing at the side pressure p; the machine-design texts quote it with p = 1,000 lb/in², a wear limit for sliding fits (cited only, entered here). The tool also prints the bearing with the contact fraction, T/(γ N h r_m L), the shear at the spline base, T/(γ N W r_m L), and the shaft's torsional shear at the minor diameter.

Bolted flange couplings

The coupling is selected and checked at T_s = SF × T, the service factor coming from the coupling standards and catalogues (ANSI/AGMA 9001 and the makers' tables by driver and load; cited only, entered). With n bolts on a circle of diameter D_b sharing the torque equally, each carries F_b = 2T_s/(n D_b) tangentially (the test sums the bolt forces about the axis and recovers T_s); it is in single shear on its shank, F_b/(π d_b²/4), and bears on each flange over d_b t_f; the flange shears around the hub over π D_h t_f at the arm D_h/2, τ = 2T_s/(π D_h² t_f). Before the bolts bear, the clamped faces carry torque by friction, T_fr = μ F_i n (faces) D_b/2; the clamping load per bolt is k S_p A_t from the thread and grade of the Bolt Preload and Torque library (tensile-stress area and proof strength; k = 0.75 reusable, 0.90 permanent) or entered. When T_fr exceeds T_s the joint does not slip and the bolt shear is the condition after slip or of fitted bolts in reamed holes. Each hub's key is checked under T_s with the key formulas above. Shrink and press-fit hubs are the subject of the Interference and Press Fit tool.

Flexible couplings

A flexible coupling absorbs misalignment but reacts it: the catalogues give an angular stiffness (moment per degree), a lateral stiffness (force per unit offset) and an axial stiffness per coupling size. The reactions M_a = k_a θ, F_r = k_r δ and F_x = k_x δ_x load the shafts and bearings; at the bearing nearest the coupling the bending moment is M_a + F_r L_ov. The stiffnesses are entered as plain numbers in the page's units; the catalogue values are the makers' and are not reproduced. AMCP 706-201 par. 7-6.3 lists the coupling failures of helicopter drive lines (fatigue from misalignment, loss of lubricant) as the reason to keep these reactions small.

Cross pins and set screws

A cross pin through hub and shaft is sheared at the two planes where it leaves the shaft; the two forces F = T/d form the couple T, so τ = 4T/(π d_p² d). In the shaft the pin's bearing pressure grows with the distance from the axis: a triangular distribution whose moment equals T peaks at 6T/(d_p d²) at the surface (the test finds the same peak by solving the moment integral). In the hub the pin bears on each side over (D_h − d)/2 at the mean radius (D_h + d)/4, 4T/(d_p(D_h² − d²)). The hole removes a strip from the shaft: the tool subtracts the strip's polar moment (integrated over the strip; the common π d⁴/32 − d_p d³/12 is within 1 %) and prints the shaft shear on the net section without a stress concentration factor; RP-1123 Figure 2 gives K_t for a radial hole. American Standard taper pins (taper 1/4 in per foot, NAVEDTRA 12204 p. 3-42) are listed by number with their large-end diameters from TC 9-524 Table 7-7 (Appendix A p. A-13).

A set screw holds by the friction of its point against the shaft. The makers publish a holding power per screw size at the seating torque (Shigley's Section 7-7 reproduces such a table, from about 50 lb for a No. 0 screw to 1,000 lb for 1/4 in and 7,000 lb for 1 in; cited only, the value is entered); since the screw resists the tangential force 2T/d at the shaft surface, the torque the screws hold is n P_h d/2. Set screws are light-duty fastenings; a flat on the shaft, or a key with the screw over it, is the usual way to keep a hub from working loose.

Torsional stiffness of the assembly

Shaft lengths, couplings, splines and keyed hubs act as torsional springs in series: the twists add, so the compliances add, 1/k = Σ 1/k_i. A shaft element has k = G J/L with J = π(d⁴ − d_i⁴)/32; a coupling its catalogue stiffness; a joint a measured value. The tool prints the series stiffness, the twist under the entered torque and each element's share of it. The softest element dominates, which is why a flexible coupling usually sets the torsional natural frequency of a drive; the series stiffness is the spring to enter in the Multi-DOF Vibration and Critical Speed tool.

Sources and copyright

Everything shipped as a number comes from a U.S. government work: NAVEDTRA 12204 (Navy, 1990), TC 9-524 (Army, 1996), NASA RP-1123 (1984), AMCP 706-201 (1974) and AMCP 706-202 (1976), and NASA CP-3239 (1994). The ANSI, SAE and AGMA standards, Peterson's charts, Shigley and Machinery's Handbook are cited; their tables are entered inputs with the defaults named above. The involute diameter coefficients in the "tooth form" select are the standard's basic proportions as commonly reproduced and are flagged as entered values to be checked against the drawing or the standard. The 0.75 keyseat rule, the SAE 1,000 lb/in² basis and the set-screw holding power are single cited facts.