A crane hook is used for lifting and moving heavy objects and is often found in industrial applications. Design a crane hook to carry a load F. The hook has a rectangular cross section with width b (minimum 0.2 mm) and height h.
Optimize the crane hook design to minimize the volume of the hook. The hook is manufactured from a complete rectangular wire ring that is clipped and bent to give the final hook shape. The outer radius of the hook is ro and the inner radius is ri with a minimum inner diameter of 3.0 mm. The height is the difference between the outer and inner radius h=ro-ri. The bending moment is M=FR with a force F of 100 N (10.2 kg for a static load on earth). The centroid radius is R and the neutral axis radius is rn.
$$r_n = \frac{h}{\ln\left(r_o/r_i\right)}$$
The difference between the centroid radius and the neutral axis radius is e. The stress at point A is
$$\sigma_A = \frac{M \left(r_o-r_n\right)}{b\;h\;e\;r_o}$$
The stress at point B is
$$\sigma_B = \frac{M \left(r_n-r_i\right)}{b\;h\;e\;r_i}$$
The stress at points A and B should not exceed the yield strength of the steel at 430 N/mm2.
Solution
The optimal solution is:
Optimal Volume: 37.50 mm^2 Optimal outer radius: 3.39 mm Optimal inner radius: 1.50 mm Optimal hook width: 1.29 mm