# Flexure formula insufficient for model

This is just a though experiment that I had.

Two hooks (the hooks are just extrusions of the sketches shown in the pictures below, with both hooks being extruded by the same amount). If we use the flexure formula to calculate the stress at point A in hook 1 and hook 2 then we get the same bending stress despite the two hooks have different shapes. This is because they have the same Moment applied to them, the same MMOI for the cross-section and are the same distance away from their respective neutral axes. How do I include the effect of the curved shape on hook 1 in my calculation for the bending stress?

• Without knowing the diameter of hook1, how do you calculate the moment?
– r13
Commented Sep 26, 2022 at 18:55

It may seem counterintuitive but --barring plasticity effects-- the stresses on point A should be the same in both cases. (this is probably the most difficult concept to grasp in statics 101).

I.e. Unless you have deformations that range into the plasticity region of the material (and other non linear effects) the result should remain the same.

If you are interested in the maximum stresses in general (e.g. for a cross-section of the hook, then obviously the shape matters.

• Along with the bending moment stress there should also be a direct normal stress. Is this stress just 300/Area at the cross-section at A? This seems highly unintuitive aswell because if this is the case then: regardless of how wacky of the shape the hook has, as long as the cross section at A is connected to the point of application of force then the direct normal stress will be the same. Commented Sep 26, 2022 at 11:15
• the bending moment induced stress has two components. The normal and the shear. Usually the shear stress is neglected (at least for long and thin beams). The shape of the normal bending stress is like this, while the normal stress due to axial forces is more uniform.
– NMech
Commented Sep 26, 2022 at 19:03
• But the bottom line is that, yes, for a concentrated load, in a scenario like yours, in the first order analysis (large deformations ignored) the reaction forces (and therefore the stresses) on the cross-section of the hook are independent of the shape.
– NMech
Commented Sep 26, 2022 at 19:05