In-plane non-linear elastic stability of parabolic arches with different rise-to-span ratios

Investigations of in-plane nonlinear elastic buckling and postbuckling of parabolic shallow arches have been reported in some literature, where a basic assumption that curve arc differential being equaled to horizontal line differential was used to facilitate the derivation. However, this assumption can be accepted only when the rise to -span ratio of an arch is quite small, and when the rise-to-span ratio of an arch is greater than 1/12.5, the analytical predictions for buckling and postbuckling responses of the arch based on the assumption have significant deviations from the corresponding finite element results. It is known that parabolic arches having a rise to -span ratio from 1/8-1/4 are often used in the engineering practice, especially in the arch bridges. This paper aims to improve accuracy of nonlinear analyses for parabolic arches with different rise-to-span ratios. A more precise strain expression that includes an approximate curve arc differential item in the Cartesian coordinate system is derived in this paper. Based on the strain expressions and principle of virtual work, nonlinear differential equilibrium equations are derived and more accurate analytical solutions for the buckling and post buckling behavior are derived. Parabolic arches with the rise-to-span ratio ranging from 1/12.5-1/3 are taken as examples to investigate the effectiveness of the proposed method for prediction of the buckling and postbuckling behavior by comparisons with the predictions of a finite element method. The comparisons show that the solutions are in good agreements with the finite element results and that the new strain expression can lead to more accurate closed-form analytical solutions for the buckling and postbuckling behavior of parabolic arches with different rise-to-span ratios. It is found that the rise-to-span ratio has a significant influence on the in-plane nonlinear buckling behavior of parabolic arches.