Stress Concentrations in Circular Shafts

When torques are applied via flange couplings or gears connected by keys in keyways, the stress distribution near the torque application point deviates from the elastic torsion formula predictions. This occurs because the load is no longer applied uniformly at the shaft ends by rigid plates, but instead concentrated through the coupling or gear mechanism, creating localized stress variations.

Sudden shifts in shaft diameter cause irregular stress concentration distribution, particularly around joint areas. These stress concentrations can be mitigated by incorporating a fillet at the diameter transition. The stress concentration factor, which depends on shaft diameter ratios and fillet size, quantifies the maximum shearing stress at the fillet location.

The stress concentration factor expresses the maximum shearing stress value at a fillet in terms of a dimensionless multiplier. This factor depends on the ratios of shaft diameter and fillet size and can be precomputed and stored for practical use in design calculations, enabling engineers to predict peak stresses without repeated analysis.

The elastic torsion formula analysis method remains valid as long as the maximum stress value does not exceed the material's elastic limit. Once plastic deformations occur, the analysis becomes invalid because plastic deformation results in lower maximum stress values than elastic predictions, requiring different analytical approaches.

Fillets smooth the transition between different shaft diameters, reducing the abruptness of geometric changes that cause stress concentrations. By gradually transitioning the diameter rather than creating a sharp corner, fillets distribute stress over a larger area, lowering the peak stress value and improving shaft durability and reliability.

When plastic deformations occur in a shaft, the maximum stress values become lower than those predicted by elastic analysis. This occurs because plastic deformation redistributes stress and absorbs energy through permanent material deformation, preventing stress from reaching the theoretical elastic limit predicted by the torsion formula.

The stress concentration factor depends directly on shaft diameter ratios and fillet size. As the ratio between the larger and smaller shaft diameters increases, or as fillet size changes, the concentration factor value changes accordingly. These relationships are precomputed and tabulated for practical engineering applications.