Stresses in a Crane Hook

1 Introduction
A crane hook is a curved beam and the simple theory of straight, shallow beam bending does not apply. The stress distribution across the depth of such a beam, subject to pure bending, is non linear (actually hyperbolic) and the position of the neutral surface is displaced from the centroidal surface towards the centre of curvature. A rigorous solution is quite complex and some simplifying assumptions are made in practice.
The diagrams below show the nomenclature used:
In the expressions below
h = the depth of the beam.
A = the beam cross section area.
co = distance from the neutral surface to outer fibre.
ci = distance from the neutral surface to inner fibre.

2 Suggested Simple Approach
The stress distribution due to the moment only can be found by balancing the externally applied moment against the internal resisting moment. The result is:

sigma = M y/(A e(rn - y)) The critical stresses occur at the inner and outer surfaces: sigmai = M ci/(A e ri) and sigmao = M co/(A e ro)
The difficult part is evaluating e (or rn). e = R - rn Values that can be used for common sections are given below:
  Rectangle Trapezium Circle
Area b d 0.5 d(bi + bo) 3.14159 d2/4
R 0.5 (ri + ro) ri + (d/3)(bi + 2 bo)/(bi + bo) 0.5 (ri + ro)
rn d/ln(ro/ri)

The equations for the stress, sigma, are for pure bending and for a crane hook the bending moment is due to a force acting on one side of the cross section. In this case the bending moment is calculated about the centroidal axis, not the neutral axis. Also additional tensile and / or compressive stresses must be added to the bending stresses, given by the two equations above, to obtain the total stresses acting on the section.

The most highly stressed points in a typical crane hook area E and B, see diagram below:

sigmaE = F R ci/(A e ri) + F/A (tension) and
sigmaB = - F R co/(A e ro) + F/A

The cross section of many crane hooks can be approximated to that of a trapezium giving rise to only a small error so the data given above can be used. Alternatively a more rigorous approach can be used, see reference (i) below.

3 Reference
i) 'Advanced Mechanics of Materials', by A P Boresi and O M Sidebottam, John Wiley and Sons, 4th Ed. 1985, ISBN: 0-471-84323-7.

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David J Grieve, 16th August 2004.