Glossary term
Stress Concentration
The local amplification of stress that occurs near geometric discontinuities such as holes, notches, and fillets.
Definition
phenomenonStress concentration is the local amplification of stress that occurs in the vicinity of geometric discontinuities — holes, notches, fillets, grooves, threads, keyways, or weld toes — where the smooth flow of internal forces is disrupted.
In a uniformly loaded body with no geometric irregularities, stress distributes smoothly across any cross-section. When a discontinuity is introduced, the load-carrying area is locally reduced and the force flow lines are forced to deviate and crowd together around the irregularity, producing a peak stress that can be several times higher than the nominal stress in the undisturbed cross-section. Stress concentrations are critical in fatigue design because cracks almost always initiate at points of elevated local stress, even when the nominal stress is well below the fatigue limit.
In any structural component carrying a load, the internal forces must be transmitted continuously from one cross-section to the next. In a prismatic bar with a smooth, uniform cross-section, this transmission is uniform — the stress is the same at every point of a given cross-section. When a geometric irregularity is introduced — a hole drilled through the bar, a shoulder where the diameter changes, a groove cut for a retaining ring, a weld bead with an irregular toe — the load-carrying area is locally reduced and the internal force-flow lines must deviate around the obstruction. This crowding of force lines produces a local peak stress that can be many times higher than the stress in the undisturbed region.
The stress concentration factor
The severity of a stress concentration is quantified by the theoretical stress concentration factor K_t, defined as the ratio of the maximum local stress \sigma_\text{max} to the nominal stress \sigma_\text{nom} calculated for the net cross-section:
K_t is a dimensionless geometric factor that depends on the shape and size of the discontinuity, the geometry of the surrounding section, and the type of loading (tension, bending, torsion), but not on the material — it is a purely elastic, geometric quantity. Values of K_t are tabulated in reference handbooks (Peterson’s Stress Concentration Factors is the standard reference) and can also be determined by finite element analysis or experimental methods such as photoelasticity.
For a circular hole in a wide plate under uniaxial tension, the classical result from elasticity theory gives K_t = 3 at the edge of the hole. For a sharp notch or a crack-like defect, K_t can be much higher — theoretically unbounded for a perfectly sharp crack, which is why fracture mechanics uses the stress intensity factor rather than K_t in that limit.
Fatigue strength reduction factor
In fatigue design, the theoretical stress concentration factor K_t overestimates the reduction in fatigue strength for notched components, because ductile materials can locally yield and redistribute stress at the notch root. The fatigue strength reduction factor K_f accounts for this by introducing the notch sensitivity index q:
where q ranges from 0 (material insensitive to notches — full redistribution) to 1 (material fully sensitive — K_f = K_t). Notch sensitivity depends on material strength, grain size, and the notch root radius. High-strength steels and aluminium alloys have higher notch sensitivity than low-strength steels. The fatigue strength of a notched component is:
where S_e is the endurance limit of the smooth specimen.
Design strategies
The fundamental strategy against stress concentrations is to smooth the load path — to allow forces to flow gradually without abrupt direction changes. Practical measures include: using generous fillet radii at section transitions rather than sharp corners; avoiding unnecessary holes or cutouts in highly stressed regions; specifying smooth surface finish at stress raisers; using compressive residual stresses (shot peening, surface rolling) at notch roots to offset the locally elevated tensile stress; and placing unavoidable discontinuities (bolt holes, keyways) in regions of lower nominal stress.
In fatigue-critical components — aircraft structural joints, crankshafts, connecting rods, pressure vessel nozzles — stress concentration management is as important as material selection and load analysis. A component made from a high-strength material but with a poor notch geometry can have a shorter fatigue life than one made from a lower-strength material with a well-designed geometry.
Common mistakes
A common mistake is refining a finite element mesh until a very sharp notch gives a large local stress and then treating that number as a directly usable fatigue stress without considering notch radius, material notch sensitivity, plasticity, and surface condition. Another is improving material strength while leaving a poor load path unchanged. A strong stress-concentration review states nominal stress definition, geometry, loading mode, root radius, source of K_t, fatigue reduction factor, surface condition, residual stress, and inspection requirement.