The elastic stability of curved beams

Abstract

Calculations are made of the ranges of validity of three theories used for investigating the instability of curved beams. These include linear and nonlinear (elastica) extensible theories, as well as a nonlinear inextensible one. The specific example chosen for the investigation is that of a curved beam hinged to fixed supports at its ends, and subjected to equal but opposite end moments. The values of the moments at which instability occurs are based on the classical criterion. Of the three theories investigated, the nonlinear extensible one is, of course, the most accurate. However, if the radius of gyration half-span parameter p = 0·025 (see Notation), it is found that the linear theory predicts the buckling moments accurately for beams having a rise span ratio λ 2l less than 0·13. Beyond this value of λ 2l , the nonlinear extensible and inextensible theories predict substantially the same buckling moments. For p = 0·05, the corresponding value of λ 2l is 0·19. Finally, it is observed that buckling cannot occur if p = 0·025, λ 2l < 0·024 and p = 0·05, λ 2l < 0·048 . These values can be obtained either by the linear or nonlinear extensible theories.