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
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 stress distribution due to the moment only can be found by balancing the
externally applied moment against the internal resisting moment. The result is:
The difficult part is evaluating e (or rn). e = R - rn
Values that can be used for common sections are given below:
||0.5 d(bi + bo)
||0.5 (ri + ro)
||ri + (d/3)(bi + 2 bo)/(bi + bo)
||0.5 (ri + ro)
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.
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.
Return to module introduction
David J Grieve, 16th August 2004.