CALCULATION OF BENDING STRESS AND COMPARISON WITH ESTIMATED FAILURE STRESS
The bending stress applied to a round bar, as a function of deflection, is a straightforward solid mechanics calculation. The first step is to use geometric relationships to find the radius of curvature of the bolt as a function of depth of the yoke (essentially the same as the subtended chord of the deflected bolt) and the deflection of the bolt in the deformed yoke holes. This analysis assumes that the deflected bolt forms a circular arc. The diagram below shows the geometry and defines the displacement d of the bolt, the chord c (or depth of the yoke = 40 mm) and the radius of curvature R of the bent bolt.
The required relationship between bolt deflection d and radius of curvature of the bolt R given by:
The radius of curvature R can be related to surface bending stress via the equation:
where E is the modulus of elasticity (210 GPa), r is the radius of the bolt (5.5 mm) and R is the radius of curvature of the bolt under load, defined in the diagram.  Combining the above two equations gives:
This equation gives the stress induced at the surface of bolt under a bending load which causes a deflection d in the bolt.  This can be compared with the stress to cause fracture obtained from a suitable stress intensity equation. The case of a semi-elliptic crack in a round bar, subject to bending, has been solved by several workers and two recent publications are referenced below.
Summarising the results for the present case indicates that an approximate relationship between stress intensity factor and applied stress is given by

where a is the crack depth and Y is a geometry correction factor that depends on crack aspect ratio (depth/surface length) and the ratio of crack depth to bolt diameter.  For the likely variation in these parameters for the cracked shear bolt, the geometry correction factor could vary between 0.95 and 1.12.  At fracture, the value of K equals the fracture toughness KC which is likely to lie between 70 MPa m and 110 MPa m.

The applets below allow calculation of the bending stress induced by deflection (perhaps in a range of 0.1-0.5 mm) and the fracture stress in the presence of a fatigue crack.  Examine the effects on the stress calculations of varying the parameters KC, Y, a and d over their possible ranges.  In such an approximate analysis, the values of stress obtained may seem high in comparison with the tensile strength, but does this approach provide support for the possibility that the shear bolt may have failed first, followed shortly afterwards by the tractor coupling bolts as the aircraft veered outwards past the tractor?

References
  1. Yan-Shin Shih and Jien-Jong Chen (2002), The stress intensity factor study of an elliptical cracked shaft, Nuclear Engineering and Design, Vol. 214 pp.137-145.
  2. M da Fonte and M de Freitas (1999), Stress intensity factors for semi-elliptical surface cracks in round bars under bending and torsion, International Journal of Fatigue, Vol. 21 pp.457-463.

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