Questions: Stress Concentration and Stress Singularities

Question 1

An engineer is choosing between a 5 mm diameter hole and a 25 mm diameter hole in a large steel plate under uniform tension. Which statement best describes how the stress concentration factor Kt compares between the two holes?

• A. The 25 mm hole has a higher Kt because it interrupts more of the cross-section
• B. The 5 mm hole has a higher Kt because stress crowds more tightly around a smaller feature
• C. Both holes have the same Kt = 3, because Kt depends on shape, not absolute size
• D. The 25 mm hole has a lower Kt because it creates a more gradual stress gradient

Answer: (C)

For a circular hole in a wide plate, Kt = 3 regardless of hole diameter — this is one of the most important results in stress concentration theory. What matters is the geometric shape (circle), not the absolute size. The misconception in options A and B is confusing net area reduction (a different effect on nominal stress) with the stress concentration factor itself, which is purely a function of geometry ratios.

Question 2

A rotating shaft component fails by fatigue cracking at a fillet, despite the nominal applied stress being well below the material's fatigue limit. What is the most likely explanation?

• A. Fatigue failure requires the nominal stress to exceed yield, so this must be a manufacturing defect
• B. The locally amplified stress at the geometric discontinuity exceeds the fatigue limit even though the nominal stress does not
• C. Fatigue is controlled by average cross-sectional stress, so stress concentration is irrelevant to initiation
• D. The fillet reduces cross-sectional area, lowering nominal stress and accelerating fatigue

Answer: (B)

Stress concentration is precisely why fatigue failures occur at stresses far below the nominal fatigue limit. The fillet creates a local stress amplification Kt times the nominal stress; even when the nominal stress is safe, the locally amplified stress at the fillet root may exceed the fatigue limit, initiating microcracks. This is why fatigue failures almost always begin at geometric features — holes, keyways, thread roots, surface scratches — rather than in smooth, uniform regions.

Question 3

Doubling the diameter of a circular hole in a large plate under uniform tension does not change the stress concentration factor Kt at the hole edge.

• T. True
• F. False

Answer: True

Kt for a circular hole in a wide plate equals 3 regardless of hole size, as long as the hole is small relative to the plate width. Kt depends on geometric shape and ratios, not absolute dimensions. This is a striking and often counterintuitive result: a 1 mm hole and a 100 mm hole in an otherwise identical plate have exactly the same peak-stress amplification factor.

Question 4

A sharp crack in an elastic material has an extremely high but finite stress concentration factor Kt, because the tip radius approaches — but seldom quite reaches — zero.

• T. True
• F. False

Answer: False

For a crack with zero tip radius, Kt is formally infinite — this is a stress singularity, not merely a very high Kt. The elastic stress field diverges as σ ∝ K/√r as the distance r from the crack tip approaches zero. The concept of Kt no longer applies; instead, linear elastic fracture mechanics (LEFM) uses the stress intensity factor K to characterize the crack tip field. The transition from Kt (for notches with finite radius) to K (for cracks) marks the boundary between stress concentration and fracture mechanics.

Question 5

Why do fatigue failures almost always initiate at geometric features like holes, fillets, or surface scratches, even when the nominal applied stress is well below the material's fatigue limit?

Model answer: Geometric discontinuities amplify local stress by a factor Kt above the nominal stress. Even if the nominal stress is below the fatigue limit, the locally amplified stress at the feature can exceed the fatigue limit, initiating microcracks that grow with each load cycle until sudden fracture occurs.

This is the central engineering significance of stress concentration: the fatigue limit is a material property measured on smooth specimens with uniform stress, but real components always have notches, holes, and surface irregularities. The local stress at these features equals Kt × σ_nominal, and for a fillet with Kt = 2.5 and a nominal stress of 60% of the fatigue limit, the local stress would be 150% of the fatigue limit — a guaranteed fatigue failure site. Designing for fatigue resistance means minimizing Kt through generous fillet radii, smooth surface finishes, and avoiding abrupt geometry changes.