Miscellaneous strength topics : frequently asked questions

Question 1
Doug,   In thick curved beams I can't understand why you assume that the stress resultant force,   F, acts through the centre of curvature, O, rather than in the cross-section where the stresses are. I've looked at different texts and they have   F acting at the centroidal axis.

Answer
The stress resultant comprises the moment   M in addition to   F. You can move   F transverse to its line of action without affecting the resultant provided   M is altered accordingly (by adding or subtracting   F ∗ distance) - this is basic Statics.
Some texts adopt the centroidal axis as you've discovered, others adopt the neutral axis (which is not the same as the centroidal for curved beams). I've yet to find an elementary text which considers practical bending in its own right, rather than "pure bending then add direct stresses to account for forces".
In a bid to avoid the confusion caused by the centroidal/neutral question and the just- about- inevitable need to extend pure bending results to include forces in practice, I took the bull by the horns with the results you have before you.
You don't have to do it the Wright way - provided you do it a right way you'll get the same stress distribution.