Curved Beam in Bending
The stress resulting from an applied bending moment is derived from the fact that the resisting moment is simple the integral over the whole section of the moment arm from the neutral axis (y) multiplied by σdA (= dF).
Moment equilibrium is achieved if
The curved beam flexure formula is in reasonable agreement for beams with a ratio of curvature to beam depth of rc/h of > 5 (rectangular section).
As the beam curvature/depth radius increases the difference between the maximum stress calculated by curved beam formula and the normal beam formula reduces. If the ratio is about 8 then a maximum stress error of only about 5% results from using the straight beam formulae.
The above equations are valid for pure bending. In the more normal cases of e.g. crane hooks, the bending moment is due to forces acting on one side of the section under consideration. The bending moment, in this case has to be taken about the centroidal axis , not the neutral axis and the additional tensile or compressive stresses have to be considered to obtain the resultant stresses on the section.