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 nonlinear (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:
sigma = M ci/ (A e ri) and sigma = M co/ (A e ro)
The difficult part is evaluating(or rn). e = R - rn Values that can be used for common sections are given below: