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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

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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

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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

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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 ?

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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)

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

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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

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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 !

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EFFECT OF NOTCH GEOMETRY ON K t

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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

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

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

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Pin loaded hole:

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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

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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

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

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