A ball spline shaft carrying a transverse (radial) load deflects according to standard beam-bending equations, with the governing case selected from a matrix of support conditions and load types [S3].
Deflection at mid-span and the local slope at the load point are calculated using the shaft's geometric moment of inertia I, span l, and the manufacturer's reference modulus of longitudinal elasticity E = 2.06×10^5 N/mm² [S3].
Why the support configuration drives the equation choice
THK's selection catalog distinguishes four primary support/loading combinations, each with a dedicated pair of equations for δmax (maximum deflection) and slope angle (i1 at the load point, i2 at the support point) [S3]. For a "both ends free, concentrated load P at midspan" case, the maximum deflection is δmax = P·l³/(48·E·I), while a "both ends free, uniform load p" case follows δmax = 5·p·l⁴/(384·E·I) [S3]. The "one end fixed" (cantilever) configurations appear in Table 2, with the load and slope terms P·l³/(3·E·I) and P·l²/(2·E·I) referenced for verification [S3]. The Table3–Table6 series supplies pre-computed Z (section modulus) and I (second moment of area) values per model, which is the practical way most designers consume the formulas [S3].
Stress checks that have to run in parallel with the deflection calc
Deflection is not the only check. THK specifies a permissible bending stress of σ = 98 N/mm² and a permissible torsional shear stress of τa = 49 N/mm² for the spline shaft material, derived from yield and a working safety factor [S3]. When bending and torsion act simultaneously, two equivalent moments are computed: Me = √(M² + (α·T)²) for bending, and Te = √(T² + (β·M)²) for torsion, and the larger required shaft diameter of the two is selected [S3][S4]. SACOM's catalog (SLT/SLF series) uses the same dual-diameter approach with equivalent bending moment M_eq and equivalent torsion T_eq for shaft sizing [S4]. A separate rigidity limit caps torsional twist at 1° per 4 m of shaft length (0.25°/m), using G = 7.9×10^4 N/mm² and the polar moment of inertia I_p [S3].
Geometry inputs that actually move the numbers
The four geometric inputs that drive both life and deflection are nominal shaft diameter, ball circle diameter d_p, contact angle (typically 30° in Gothic-arch groove geometry), and the number of load-carrying rows [S2][S4]. PMI/SACOM publishes, for its SLT/SLF line at nominal Ø16/20/25 mm, the root diameter Ød = 15/19/23.9 mm, major diameter ØD0 = 16/20/25 mm, and ball center-to-center diameter Øpd = 17.8/22.2/27.9 mm, with linear mass 1.56/2.44/3.82 kg/m [S4]. NB's catalog spans 4 mm to 100 mm shaft diameters in cylindrical (SSP/SSPM) and flange (SSPF/SSPT) nut forms, with stainless option in SUS440C-equivalent [S1]. Misumi's overview keeps a single baseline: H7 housing bore fit, plastic-retainer components rated below 80°C ambient, and grease lubrication at ship-out [S5].
Comparison: ball spline vs linear bushing vs linear guide under the same load
Misumi rates the three linear-motion families on the same axes, and the comparison is direct: ball spline = Good on radial load and rotational torque; linear bushing = Good radial, Poor torque; linear guide = Excellent radial, Poor torque [S5]. The same table puts cost at Good/Excellent/Acceptable and mass at Excellent/Excellent/Acceptable for spline/bushing/guide respectively [S5]. Thomson's support note reinforces the torque-first bias: ball splines can accept radial load, but only torque ratings are shown in the catalog because radial capacity is governed by the specific load-row geometry [S8]. The practical rule from NB and Misumi is therefore: pick ball spline when torque transmission is a primary requirement, then verify radial deflection as a stiffness check rather than as a load-rating gate [S1][S5].
Selecting the right support case in real machines
Most factory-automation installations use a "both ends free, single concentrated load" model because the shaft is supported by two end bearing blocks and loaded by a single nut positioned at the working offset [S3]. Pick-and-place and tool-changer modules (the canonical ball-spline applications) generally fall into this case with a working span l between supports and a nut force P at the working position [S5]. Where the spline shaft is a cantilevered Z-axis, the "one end fixed" equations apply, and the same beam formula gives a stiffer penalty (P·l³/(3·E·I) tip deflection vs P·l³/(48·E·I) simply supported) [S3]. For combined rotary ball spline (stroke + rotation) and ball-screw-spline hybrids, NB documents use the same deflection matrix but with an additional rotational resonance check tied to the dangerous-speed table [S1].
When the catalog numbers say stop
Radial-load handling on a ball spline has explicit, citable ceilings that designers routinely miss. Misumi's overview caps the operating environment below 80°C because plastic retainers are used; the same page gives rotational clearance bands of -2 to +1 µm (No. 6) widening to -25 to +30 µm (No. 30), and radial runout of the spline nut of 32 to 102 µm over 200 to 1150 mm of supported length [S5]. Thomson flags the catalog bias directly: radial ratings are not published because they depend on the load-row geometry, so sizing must fall back to the manufacturer table or to a measured test [S8]. Life modification factors for spline shafts also narrow the envelope: fH (hardness) 0.5-1.0, fT (temperature) 0.9-1.0, fC (contact, multiple nuts) 0.72-1.0, fW (shock/vibration) 1.0-2.5, all combined into a single f in the L = (f·C/P)³ life equation at 50 km basis [S2]. For related selection logic, see this ball bearing C vs C0 sizing rules breakdown and the ball retainer vs full-complement guide-block decision which both feed into the same stiffness-vs-life trade.
Limits, failure modes, and the resonance trap
The three failure modes the deflection calc is meant to flag are: (1) permanent bending of the shaft when σ exceeds 98 N/mm² under the combined Me/T loading, (2) torsional twist beyond 1° per 4 m of length, and (3) resonance when a rotating spline shaft crosses its first natural frequency [S3]. NB catalogues each rotary spline family (SPR, SPB, SPBR, SPBF) with a maximum rotational speed, and the dangerous-speed calculation in THK's selection guide is the formal way to set that ceiling [S1][S3]. A practical engineering read: compute I from the actual cross-section (root diameter is the lower bound, not nominal ØD0), and use the manufacturer's I table whenever a published model is in play [S3][S4].
Worked quick-reference: inputs and outputs to capture
Every deflection check should record, in one place, the support case label (Table 1 or Table 2 row), the equation used, the I value (mm⁴), E = 2.06×10^5 N/mm², the resulting δmax (mm) and the slope angle i1 at the load point [S3]. If torsion is also present, capture G = 7.9×10^4 N/mm², I_p, and the per-meter twist (°/m) versus the 0.25°/m ceiling [S3]. For a sanity check on the radial rating itself, the conversion formula for simultaneous torque and radial loading uses the equivalent radial load P_E = (T / (i · d_p · cos α)) in the L = (f·C/P_E)³ life equation, where i is the number of load rows, d_p is the ball circle diameter (mm), and cos α is the contact-angle term, values that must come from the manufacturer because they vary by series [S2]. The next verification node for designers is the rotational dangerous-speed table in the same catalog (THK B-section, NB B-10), and the second trackable signal is the deflection angle i2 at the support point, which controls angular alignment of the driven load [S1][S3].
Detailed specification references: ball spline, electronic load, and load cell.