Shaft Deflection Calculator
Calculates the midpoint or tip deflection of a rotating or static shaft under a transverse load, for both simply supported and cantilever configurations. Used by mechanical engineers to verify shaft stiffness and prevent bearing misalignment.
About this calculator
Shaft deflection under a concentrated load depends on the support conditions, load magnitude, shaft geometry, and material stiffness. The deflection formula is δ = (F × L³) / (k × E × I), where F is the applied load (N), L is the shaft length (mm), E is the modulus of elasticity (MPa), and I is the second moment of area for a solid circular cross-section: I = π × d⁴ / 64. The support factor k equals 48 for a simply supported beam (midpoint load) and 3 for a cantilever (tip load), which is reflected in the supportType field. A larger diameter has a fourth-power effect on stiffness — doubling the diameter reduces deflection by a factor of 16. Keeping deflection below 0.001× shaft length is a common design rule of thumb.
How to use
Consider a simply supported steel shaft (E = 200,000 MPa) with L = 500 mm and d = 30 mm, carrying a central load of 1,000 N. First compute I = π × 30⁴ / 64 = π × 810,000 / 64 ≈ 39,761 mm⁴. For a simply supported beam, supportType k = 48. Deflection δ = (1000 × 500³) / (48 × 200,000 × 39,761) = 125,000,000,000 / 381,705,600,000 ≈ 0.327 mm. This is well within acceptable limits for most machinery, confirming the shaft design is adequately stiff for this loading scenario.
Frequently asked questions
What is an acceptable shaft deflection limit for rotating machinery?
A commonly applied guideline is that shaft deflection should not exceed 0.001 times the shaft span (1 mm per metre) between bearings. For precision equipment such as machine tools or turbines, tighter limits of 0.0005× span may be required. Excessive deflection causes bearing edge loading, premature fatigue, vibration, and seal wear. Always cross-check with the bearing manufacturer's maximum misalignment tolerance, which is often the binding constraint.
How does the modulus of elasticity affect shaft deflection calculations?
The modulus of elasticity (Young's modulus) directly determines how much a shaft resists bending — a higher E means less deflection for the same load and geometry. Steel has E ≈ 200 GPa, aluminium ≈ 69 GPa, and titanium ≈ 116 GPa. Switching from steel to aluminium would increase deflection by roughly 2.9× for the same cross-section. This is why steel is overwhelmingly preferred for rotating shafts in power transmission applications despite being heavier than alternatives.
What is the difference between a simply supported shaft and a cantilever shaft in deflection analysis?
A simply supported shaft is constrained at both ends (e.g., between two bearings) and is loaded at or near the midpoint; its stiffness coefficient k = 48. A cantilever shaft is fixed at one end and free at the other (e.g., an overhanging spindle), and the stiffness coefficient drops to k = 3, meaning it deflects 16× more than a simply supported shaft of the same dimensions under the same load. This dramatic difference explains why overhanging loads on shafts should be minimised and why bearing placement relative to the load is critical in drivetrain design.