Stress concentration and Saint-Venant's principle

Stress concentration and stress concentration factor

Stress concentration is the localised rise of stress near a sudden change in cross-section or geometry (holes, notches, grooves, keyways, fillets, sharp corners, threads), where the flow of internal force is disturbed. The stress at such a place is much higher than the average stress calculated from $P/A$.

The stress concentration factor is

$$K_t = \frac{\sigma_{max}}{\sigma_{nom}}$$

where $\sigma_{max}$ is the peak stress at the discontinuity and $\sigma_{nom}$ is the nominal stress based on the net area. $K_t \ge 1$ and depends on geometry (not on the material, as long as it is elastic). For a small circular hole in a wide plate in tension $K_t = 3$.

 Stress flow lines crowd near a hole:
  ─────┐   ┌─────
  ─────┤ ○ ├─────   σ_max at the edge of the hole
  ─────┘   └─────

Effects are serious for brittle materials and for fatigue loading, whereas ductile materials under static loading yield locally and redistribute stress.

Saint-Venant's principle

It states that the stress and strain produced at points sufficiently far from the place of application of a load (or from a discontinuity) depend only on the resultant of the load, not on the exact way it is distributed; the differences between two statically equivalent loadings become negligible at a distance about equal to the largest cross-sectional dimension of the member.

With regard to stress concentration, it tells us that the disturbance is local: high stresses exist only in the neighbourhood of the hole, notch or point load, and at a distance of about one width from it the stress becomes uniform ($P/A$). So the formula $\sigma = P/A$ may be used away from these places and the concentration need be considered only locally.

Why a factor of safety is used

The factor of safety ($FOS = \dfrac{\text{ultimate or yield stress}}{\text{permissible (working) stress}}$) is used because of:

1. Uncertainty in the actual loads (overloading, impact, dynamic effects).
2. Variation in material properties, defects and non-uniformity.
3. Approximations in analysis and stress concentrations, workmanship and fabrication errors.
4. Deterioration with time (corrosion, fatigue, wear).
5. The need for safety of life and property, and to keep stresses in the elastic range.

Typical values: 1.5 to 2 for steel (yield basis), 3 to 4 for brittle materials (ultimate basis), higher for dynamic loads.