1. STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a resistant structure (specially in “design against fracture” and “design against fatigue”) is the consideration of stress concentrations Stress concentration at geometrical notches are always present in a real structure Notches introduce inhomogeneous stress distribution with a stress concentration at the root of the notch Stress concentration factor : K t describes the severity of the notch and depends on the geometry of the notch configuration (shape factor of the notch) K t is referred as the theoretical stress concentration factor: it is based in the assumption of linear elastic material behavior

2. Common examples of stress concentration (a)Gear teeth (b)Shaft keyway (c)Bolt threads (d)Shaft shoulder (e)Riveted or bolted joint (f)Welded joint

3. DEFINITIONS: For the previous example: therefore: following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York (1974) In general K t is the preferred factor to indicate stress concentration

4. THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheet S S r  r0r0 Along the edge of the circular hole: compression at  = 0  for   = 0 ?

5. Circular Hole: STRESS PROFILES We are interested in evaluating: Situation for compressive remote stress (–S): presence of tensile stress! Gradient of Stress in the direction normal to the edge of the hole at the location of s peak : strong gradient Gradient of Stress along the edge of the hole at the location of s peak : much less stronger as along the normal Volume of material subjected to high Stress around the root of the notch: larger for larger notches! (significant to understand notch size effects on fatigue)

6. S S Slow decrease of the stress along the edge compared with the decrease from the edge at the location of  peak

7. Geometrically similar specimens have the same K t but different stress gradients Larger specimens have larger volumes and larger notch surface areas of highly stressed material Important !!!: Notch Size Effects in Fatigue

8. The elliptical hole in an infinite sheet a / b1/313  / a 911/9 KtKt 1.6737 S S use large radii to reduce stress concentration !

9. EFFECT OF NOTCH GEOMETRY ON K t

10. Stress Concentration for an elliptical hole under biaxial loading:  : biaxiality ratio * For the case of a thin walled pressure vessel under pressure  = 0.5 and for the case of a circular hole (a = b): lower than 3 S for uniaxial loading * Same case but elliptical hole with b/a = 2: lower than 3 S for uniaxial loading (actually,   = 1.5 S along the edge of the hole) compare with the square hole (dashed line) with rounded corners with r 10% of hole width: K t = 4.04 for  = 0.5

11. Stress Concentration for a circular hole in a plate under pure shear: Fatigue cracks growing from holes in a shaft subjected to cyclic torsion !

12. Stress concentration factors for a shaft with a grove subjected to: axial load torsion

13. Pin loaded hole:

14. Edge notches and Corrosion Pits Corosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm. Equivalent shape gives very high K t values

15. SUPERPOSITION OF NOTCHES: If a relativelly small notch is added to the root of the main notch  superposition of notches: This overestimates K t because the small notch is not embedded in an homogeneous stress field of magnitude K t1

16. Further Examples: Cross section of a fatigue crack at a sharp corner Lug with small lubrication hole to the lug hole