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.
c_{o} = distance from the neutral surface to outer fibre.
c_{i} = 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(r_{n}  y))
The critical stresses occur at the inner and outer surfaces:
sigma_{i} = M c_{i}/(A e r_{i}) and
sigma_{o} = M c_{o}/(A e r_{o})
The difficult part is evaluating e (or r_{n}). e = R  r_{n}
Values that can be used for common sections are given below:

Rectangle 
Trapezium 
Circle 
Area 
b d 
0.5 d(b_{i} + b_{o}) 
3.14159 d^{2}/4 
R 
0.5 (r_{i} + r_{o}) 
r_{i} + (d/3)(b_{i} + 2 b_{o})/(b_{i} + b_{o}) 
0.5 (r_{i} + r_{o}) 
r_{n} 
d/ln(r_{o}/r_{i}) 


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:
sigma_{E} = F R c_{i}/(A e r_{i}) + F/A (tension) and
sigma_{B} =  F R c_{o}/(A e r_{o}) + 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: 0471843237.
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