Shaft Bearing Reactions / Bending Moment Calculator

Static-equilibrium equations

Model the shaft centerline as a simply supported beam from the left bearing center A at x = 0 to the right bearing center B at x = L. Both point loads act downward in the same plane.

At any position x between the supports, the calculator uses the loads to the left of the section:

Worked two-load example

Use the default inputs: L = 1,000 mm, P₁ = 2,500 N at x₁ = 350 mm, and P₂ = 1,500 N at x₂ = 750 mm.

1. Moment equilibrium about A gives R_B = 2,000 N.
2. Force equilibrium gives R_A = 2,000 N.
3. At the first load, M(x₁) = 2,000 × 350 = 700,000 N·mm = 700 N·m.
4. At the second load, M(x₂) = 2,000 × 750 − 2,500 × 400 = 500 N·m.
5. The maximum bending moment is therefore 700 N·m at x = 350 mm, and the maximum internal shear-force magnitude is 2,000 N.

The reactions sum to the applied load, and the bending moment returns to zero at the right simple support. These checks verify internal equilibrium; they do not verify whether the entered loads represent the actual machine.

Why maximum moment can be a region

Between point loads, shear is constant and bending moment changes linearly. If shear is zero between two load positions, the bending moment remains constant across that interval. A symmetric pair of equal loads is a common example. The calculator therefore reports a maximum-moment region start, end, and length instead of always claiming one unique critical coordinate.

Establish the shaft free-body diagram first

Enter forces only after resolving the actual gear, belt, chain, coupling, rotor, or process loads into one bending plane. The load position is the force line of action measured from the left bearing reaction center, not necessarily a component face, shaft shoulder, housing edge, or bearing width edge.

Loads in perpendicular planes must be analyzed separately. If the applicable method permits, combine the resulting bending-moment components at a section as a vector magnitude before transferring the controlling moment to the Shaft Combined Bending and Torsion Calculator. Do not simply add perpendicular force magnitudes as though they act in the same plane.

Bearing reactions are not complete bearing selection loads

The calculated reactions are ideal single-plane transverse support forces. A real bearing arrangement can include radial loads in two planes, axial thrust, preload, moments, internal clearance, housing and shaft deflection, unequal load sharing, thermal effects, and arrangement-specific constraints. Use the bearing manufacturer’s method and verified arrangement geometry before passing a reaction into a bearing rating calculation.

Engineering scope and limitations

This calculator covers a statically determinate shaft between two ideal simple supports with up to two co-directional point loads acting strictly inside the bearing span. It excludes:

• overhung loads, supports outside the entered span, more than two point loads, distributed loads, shaft or component self-weight, and applied couples;
• opposite-direction loads, signed reactions, loads in multiple bending planes, axial forces, torque, gyroscopic loads, and transient or impact loading;
• bearing width, contact-center migration, support stiffness, housing flexibility, preload, clearance, internal bearing load distribution, and misalignment;
• shaft diameter, section changes, stress, stress concentration, deflection, slope, yielding, fatigue, critical speed, vibration, damping, and resonance;
• gear, belt, chain, coupling, or process-force derivation; catalog bearing selection; static or dynamic bearing rating; lubrication; contamination; and life prediction;
• tolerances, manufacturing feasibility, stock selection, inspection, safety factors, standards compliance, and engineering approval.

Use a complete machine free-body diagram, verified operating and design loads, the governing shaft and bearing design method, and qualified engineering review before releasing a drawing or selecting components.