Multi-Bearing Shaft Support Load Distribution Calculator

Loads and supports

Results update automatically. All positions use one axial origin; positive moment adds to the force moment.

Radial force PN Force position x_Pmm Applied moment MN·mm

Support data

Enter each support position and effective coefficient C for R=Cδ^3/2.

#	Position x (mm)	C (N/mm^3/2)

Support reactions

The largest reaction is highlighted; the table is the canonical reaction result.

Maximum support reaction—

#	x (mm)	R (N)	δ (mm)

Convergence diagnostics

Equations used

For support i, x_i is axial position in millimetres, C_i is the entered effective support coefficient in N/mm^3/2, δ₀ is shaft translation at the support-span midpoint x₀, and θ is small rigid-body tilt in radians. Compression and reaction are positive.

δ_i = max[δ₀ + θ(x_i − x₀), 0]

R_i = C_i δ_i^3/2

The maximum enforces unilateral contact: a support carries no tensile reaction. The nonlinear solver finds δ₀ and θ that satisfy static equilibrium.

ΣR_i = P

ΣR_ix_i = Px_P + M

P is radial force in newtons, x_P is its position in millimetres, and M is the applied free moment in N·mm. Positive M shifts the combined resultant toward increasing x: x_R=x_P+M/P.

r_F = ΣR_i − P [N]

r_M = ΣR_ix_i − (Px_P + M) [N·mm]

r_δ = max_i|δ_i − max[δ₀ + θ(x_i − x₀), 0]| [mm]

These are the reported force, moment, and compatibility residuals. The compatibility residual is the largest absolute departure from the unilateral rigid-shaft displacement law.

L = x_max − x_min; u_i = (x_i − x₀)/L; q = θL

r_u = ΣR_iu_i − [P(x_P − x₀) + M]/L

ε = √[(r_F/P)² + (r_u/P)²]

L is support span in mm, u_i is the dimensionless normalized support coordinate, q is tilt scaled to mm, r_u is the normalized-coordinate moment residual in N, and ε is the dimensionless convergence norm. Damped Newton iteration uses dR_i/dδ_i=(3/2)C_i√δ_i. It accepts ε≤10⁻¹² during iteration, allows a final round-off check at ε≤10⁻¹⁰, and otherwise stops with an error after at most 80 iterations.

Assumptions and limits

The shaft is rigid in one radial plane; support compression is an affine function of axial position.
Supports are compressive-only and follow the entered Hertz 3/2-power law. Coefficients must already represent the relevant bearing, housing, fit and operating condition.
Loads are static. Clearance, preload, shaft bending, housing flexibility, bearing moment, internal rolling-element loads, dynamics and fatigue life are excluded.
The combined resultant must lie within the support span. Do not use this mode when shaft bending or changing contact geometry materially affects load sharing.

How to use

Use this rigid-shaft mode only for a static radial load system with at least three supports in one plane.
Enter the compressive radial force P in N and its axial position x_P in mm.
Enter any free applied moment M in N·mm; positive moment adds to Px_P.
Enter each support position x_i in mm and positive effective coefficient C_i in N/mm^3/2.
Use Add support for four or more supports. Remove extra rows only when at least three remain.
Results calculate automatically after every valid edit; there is no separate Calculate button.
Use Reset example to restore the three-support published validation case.
Read reactions and displacements by support, then confirm force and moment residuals are near zero and the solver says Converged.
Correct any red warning before using outputs. A resultant outside the support span has no compressive-only static solution in this model.
Do not use these results for flexible-shaft systems, transient loads, bearing life, contact stress, preload, clearance, bearing moments, or catalog selection.

Reference

Harris, T. A. and Kotzalas, M. N. (2007), Advanced Concepts of Bearing Technology: Rolling Bearing Analysis, Fifth Edition, Volume II, Chapter 9, Example 9.3. doi:10.1201/9781420006582. The published reactions are rounded; the solver retains full precision.