Shaft Geometric Stress-Concentration Calculator

Content

Estimate theoretical elastic stress-concentration factors for stepped, cross-drilled, and circumferentially grooved shafts, or screen a bending diameter.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Geometry and load inputs

Ratios are dimensionless positive magnitudes. Select only a listed D/d curve for shoulder and groove modes.

Screening result

Primary result1.37
Shaft discontinuity geometryA stepped shaft with shoulder radius, a transverse hole, and a circumferential groove, labeled with large diameter D, small diameter d, and radius r.Ddrholegroove
Idealized discontinuities. Use the ratio definition shown for the selected mode; the correlations report theoretical elastic factors.

How to use

  1. Select one of seven stress-concentration modes or the Bending diameter mode.
  2. For a shoulder or groove, select the listed large-to-small diameter ratio D/d and enter the fillet or groove-radius ratio r/d. Groove correlations require 0 < r/d ≤ 0.3.
  3. For a transverse hole, enter hole-to-shaft ratio d/D. Kt,A is the shaft surface at the hole; Kt,B is below the shaft surface inside the hole.
  4. For diameter sizing, enter bending moment in N·m, yield strength in MPa, endurance ratio, d/D, and an independently selected bending Kt.
  5. Select Calculate, or edit a field for an immediate update. Reset restores the shoulder-torsion example.
  6. Interpret Kt as a local theoretical elastic stress multiplier. The sizing result reports reduced diameter d and large diameter D in mm.
  7. Correct warnings before interpretation and verify ratio applicability, material response, fatigue-notch sensitivity, surface finish, residual stress, and loading independently.
  8. Do not use these curve fits for plastic notch response, crack-like defects, combined multiaxial fatigue, noncircular shafts, keyways, spline roots, or final safety/code qualification.

Worked example

1. Select Shoulder · torsion, D/d = 1.20, and r/d = 0.10. 2. Calculate. The coefficients are A = 0.83425 and b = −0.21649, giving Kt = 1.37. 3. An elastic nominal torsional stress of 50 MPa would screen to about 68.5 MPa locally before fatigue-notch or plasticity corrections.

Equations used

Shoulder and groove modes: Kt = A(r/d)^b. A and b come from the selected D/d correlation table for the chosen tension, bending, or torsion geometry. Kt is rounded to two decimals after evaluation.

Transverse hole in torsion, with x=d/D: Kt,A = 3.92150 − 24.435x + 234.06x² − 1200.5x³ + 3059.5x⁴ − 3042.4x⁵; Kt,B = 3.9702 − 9.292x + 27.159x² + 30.231x³ − 393.19x⁴ + 650.39x⁵ + 15.451x⁶.

Transverse hole in bending: Kt = 1.58990 − 0.63550 log10(d/D).

Stle

Bending sizing: σe = (σe/Rₚ₀.₂)Rₚ₀.₂; d = [32MKt/(πσe)]^(1/3); D = d/(d/D). M uses N·m, stresses use Pa, and diameters are reported in mm to one decimal.

D is the larger shaft diameter, d is the smaller shaft diameter except in hole modes where d is hole diameter, r is fillet/groove radius, M is positive bending-moment magnitude, Rₚ₀.₂ is yield strength, σe is allowable endurance stress, and Kt is dimensionless. No load direction is inferred.

Theory and method

Geometric discontinuities disturb nominal stress flow and create local elastic maxima. These curve fits convert dimensionless geometry ratios into theoretical concentration factors for isolated idealized shapes. They do not include material notch sensitivity; a fatigue-notch factor Kf is generally lower than Kt and requires material and size data.

Assumptions and limits: homogeneous isotropic linear elasticity, smooth axisymmetric shoulder/groove geometry or a clean circular transverse hole, nominal loading matching the selected mode, and ratios represented by the fitted curves. Groove r/d is limited to 0.3. Hole formulas are polynomial/logarithmic correlations and should not be extrapolated casually toward zero or unity. Verify with current handbook charts or finite-element analysis for critical designs.

FunctionalProduct

Reference: A. van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, geometric stress-concentration and shaft-sizing calculation collection. Consult a current stress-concentration handbook and applicable shaft/fatigue standard for final design.

Continue in TriboSolver to refine the geometry, local stress field, fatigue loading, and material response.

Drive-Shaft Fatigue Strength Calculator supports nominal combined-stress and fatigue screening after selecting an appropriate geometric factor.

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