Stress Concentration Factor: Equations and FEA Example

Introduction

A drilled hole can make a component fail locally even when its average stress looks safe. The stress concentration factor explains this difference and connects strength-of-materials equations with finite element analysis. You will learn how to calculate it, verify a circular-hole model, and interpret its relevance to defects in metal additive manufacturing.

Stress concentration factor and nominal stress

A theoretical stress concentration factor measures the elastic amplification of stress near a geometric discontinuity. Engineers define it as Kt = σmax / σnom, where σmax is the relevant peak local stress and σnom is a specified reference stress. Because both stresses have identical units, Kt is dimensionless.

Holes, shaft shoulders, keyways and fillets redirect load through the surrounding material. Sharp transitions generally produce steeper stress gradients than smooth transitions, so a larger fillet radius often reduces the peak. For a given linear-elastic model, geometry and loading determine the factor; increasing material strength does not remove the concentration.

Always identify whether nominal stress uses the gross or net section. For a plate of width W, thickness t and central hole diameter d, these references are σgross = P / (Wt) and σnet = P / ((W − d)t). A factor tabulated against one reference cannot be multiplied by the other.

Stress concentration factor: circular-hole worked example

The classic benchmark concerns stress concentration around a hole in an effectively infinite, isotropic, linear-elastic plate under uniaxial tension. Kirsch’s solution gives the hole-edge hoop stress as σθθ = σ∞(1 − 2 cos(2θ)), where θ is measured from the loading direction. The circular-hole solution assumes an empty, traction-free hole, not a bolt transmitting bearing load.

Consider an illustrative remote tensile stress of σ∞ = 40 MPa. At θ = 90°, cos(180°) = −1, so σθθ = 40(1 + 2) = 120 MPa and Kt = 120 / 40 = 3. At θ = 0°, the hoop stress is −40 MPa: the hole edge experiences local compression there despite remote tension.

The result Kt = 3 is exact for this idealised infinite-plate problem, not every perforated component. Finite width, nearby holes, contact, anisotropy and different loading change the solution. MIT’s stress-concentration laboratory notes distinguish gross-section and net-section factors explicitly.

Finite element analysis and porosity in metal 3D printing

In ANSYS Mechanical or another structural solver, begin with a thin plate and a plane-stress representation where appropriate. Apply remote tension, leave the hole boundary unloaded, and constrain only the rigid-body motions without suppressing the intended deformation. Place outer boundaries sufficiently far away and check that enlarging the plate changes the peak negligibly.

Refine the mesh around the curved hole and compare successive peak hoop stresses, rather than accepting a smooth colour plot as validation. Keep element formulation and stress averaging consistent during this mesh-convergence study. The analytical target of 120 MPa belongs to the stated idealisation, not to an actual simulation performed here.

Porosity in metal 3D printing makes this topic timely. A study published on 7 November 2026 by Park, Nejadseyfi and Langelaar combines analytical pore stress fields with one finite element analysis per optimisation iteration to assess random defects efficiently. Its two-dimensional numerical demonstrations support a research method, not universal certification of printed parts.

Notch sensitivity, common mistakes and exam tips

Do not confuse Kt with the fatigue notch factor Kf or the crack-tip stress intensity factor KI. A common fatigue relationship is Kf = 1 + q(Kt − 1), where q represents notch sensitivity. For illustrative values q = 0.7 and Kt = 3, Kf = 2.4; selecting q requires suitable material and notch data.

Do not multiply an FEA peak by Kt again: a correctly resolved geometry already contains that amplification. Check whether local yielding invalidates a purely elastic prediction, and distinguish a smooth-hole peak from a sharp-corner mathematical singularity. In exams, state assumptions, define the nominal section, retain consistent units and explain why the chosen stress component answers the question.

Conclusion

The stress concentration factor translates a nominal load calculation into a local elastic stress estimate, provided its geometry and reference stress match the problem. Combine analytical benchmarks, mesh convergence and appropriate fatigue or plasticity models before making design decisions. Explore more mechanical engineering topics on Mechtics and share your stress-analysis questions.