Finite Deflections of Curved Beams

A theory was developed to predict load-deflection curves for curved beams under loading conditions that produce finite strains. The theory assumed that the curved beam could be approximated by a number of truncated pie-shaped segments with circular arcs for the inner and outer boundaries. The cross section of each segment was assumed to have a plane of symmetry; the centroidal normal force and bending moment acting on the segment were assumed to lie in the plane of symmetry and to remain constant over the included angle of the segment. Using known tension and compression engineering stress-strain diagrams, internal values of centroidal normal force and moment could be calculated for assumed deformations of the segment. An iterative procedure was used to obtain the deformations which would make the internal values of centroidal normal force and moment equal to the applied values. The deformed shapes of the pie-shaped segments were reassembled to give the deformed shape of the curved beam. Good agreement was found between theory and experiment for nine mild steel curved beams and two commercially available crane hooks.