Mechanical Engineering Calculators: Simply Supported Shaft Deflection / Slope Calculator

Shaft deflection equations

The calculator models the shaft centerline as a uniform Euler-Bernoulli beam from the left bearing center at x = 0 to the right bearing center at x = L. Each entered force is a positive downward point-load magnitude acting in the same bending plane.

For a solid or concentric hollow circular section, the area second moment for bending is:

For one point load P at x = a, let b = L − a. The downward deflection contribution at a position x is:

The domain engine adds the two load contributions by linear superposition. It differentiates the same piecewise functions to obtain slope θ = dδ/dx, then solves θ(x) = 0 for the maximum-deflection coordinate. Deflection is positive downward; with ordinary downward loading, the reported left slope is positive and the right slope is negative under this sign convention.

Worked two-load example

Use the defaults: L = 1,000 mm, P₁ = 2,500 N at x₁ = 350 mm, P₂ = 1,500 N at x₂ = 750 mm, solid diameter D = 40 mm, and E = 200 GPa.

1. The solid circular section gives I = 125,663.706 mm⁴.
2. Static equilibrium gives R_A = 2,000 N and R_B = 2,000 N.
3. Superposition gives δ(x₁) = 2.425455 mm and δ(x₂) = 1.881676 mm.
4. Solving the total zero-slope condition gives δ_max = 2.676878 mm at x = 489.047979 mm.
5. The support slopes are θ_A = +0.490141° and θ_B = −0.478743°.
6. The maximum bending moment shared with the reaction model is 700 N·m.

As a separate verification case, one centered load produces the classical δ_max = PL³/(48EI) result and equal-magnitude opposite support slopes. Automated tests also verify unit equivalence, load superposition, span scaling, the diameter fourth-power relationship, hollow sections, boundaries, warnings, and invalid inputs.

How to establish the inputs

Use bearing reaction centers—not housing faces or shaft shoulders—as the endpoints of L. Measure each force line of action from the left reaction center. Derive actual gear, belt, chain, rotor, coupling, process, and weight loads before using this tool. If only one point load is required, set the other load to zero and keep its unused position within the support span.

Enter the actual uniform section that resists bending. A shoulder, bore change, keyway, cross-hole, or assembled component may make the uniform-section idealization inappropriate even when the nominal diameter appears constant. Enter a verified elastic modulus rather than treating the default value as automatic material selection.

Interpreting deflection and slope

The result is an elastic model response, not an allowable value. Acceptable deflection and slope depend on the machine: gear mesh alignment, bearing misalignment capability, seal behavior, air gap, coupling alignment, process accuracy, clearance, vibration, and governing design criteria can impose different limits.

The blue SVG curve is deliberately exaggerated. Its coordinates come from 11 calculation-engine samples, but the visual scale is normalized to make the shape readable. Use the numeric outputs for engineering work.

Loads in perpendicular bending planes must be solved separately. A scalar deflection from one plane cannot simply be added to a perpendicular scalar result. Establish vector direction and the applicable combination method before evaluating total displacement, slope, stress, or clearance.

Small-deflection model review

Euler-Bernoulli theory assumes small displacement and neglects transverse shear deformation. This calculator raises a warning when the calculated δ_max/L exceeds 1%. That threshold is deliberately labeled a model-review trigger: it is neither a universal serviceability limit nor a pass/fail rule. Short, deep, low-shear-modulus, compliant-support, or locally flexible systems may need a more detailed method even below that trigger.

Engineering scope and limitations

This calculator covers one straight, uniform, solid or concentric hollow circular shaft between two ideal simple supports, with up to two co-directional point loads strictly inside the bearing span. It excludes:

• overhung segments, cantilevers, more than two point loads, distributed loads, shaft self-weight, applied couples, and signed or reversing loads;
• stepped diameters, tapers, shoulders, grooves, keyways, cross-holes, splines, local contact deformation, and stress concentrations;
• bearing width, support translation or rotation stiffness, housing compliance, preload, clearance, contact-center migration, and nonlinear load sharing;
• Timoshenko shear deformation, large displacement, plasticity, residual stress, creep, temperature-dependent properties, and material selection;
• axial force, torque, perpendicular-plane combination, gyroscopic effects, transient loads, shock, vibration, damping, critical speed, and resonance;
• allowable-deflection selection, gear or seal alignment approval, static strength, yielding, fatigue, buckling, tolerances, manufacturing feasibility, inspection, safety factors, standards compliance, and engineering approval.

Use a complete machine free-body diagram, verified geometry and material data, governing acceptance criteria, and qualified engineering review before releasing a shaft drawing or selecting bearings and connected components.