Effects of approximations on non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches

• Approximation leads to some mistakes of mechanical concept and errors of solution.
• Approximation leads to incorrect prediction for buckling of arches with f/L < 0.08.
• Approximation (dy/dz)2≪1(dy/dz)2≪1 is valid only for parabolic arches with f/L < 0.08.
• Caution is needed when using the analytical solutions based on the approximation.

In the non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches, to overcome the difficulty in deriving an accurate expression for the non-linear normal strain, an approximation assumption that the derivative of the vertical coordinate with respect to the horizontal coordinate satisfies (dy/dz)1≪1(dy/dz)1≪1 has been adopted in many investigations. The merit of the assumption is that it leads to the same differential equations of equilibrium and the same solutions as those for shallow circular arches. However, the accuracy of the assumption and the limitation of the analytical solutions have not been examined and because of the approximation, the analytical solutions may lead to significant errors for the buckling loads of shallow parabolic arches in some cases. This paper investigates the effects of the approximation assumption on the accuracy of in-plane buckling and postbuckling analyses of pin-ended and fixed shallow parabolic arches by comparing the analytical solutions with their finite element counterparts. It is found that the analytical solutions based on the assumption have some limitations because the assumption holds approximately only for extremely shallow parabolic arches, but is not valid for most shallow parabolic arches. The analytical solutions for the buckling loads based on the assumption are larger than the corresponding finite element results for parabolic arches with a rise-to-span ratio greater than 0.08, the error of the analytical solution increases with an increase of the rise-to-span ratio of the arch, and the sources for the errors are identified and discussed. Hence, caution should be exercised when using the analytical solutions to predict the buckling load of shallow parabolic arches, particularly of those with a rise-to-span ratio greater than 0.08.