In-plane non-linear elastic stability of arches subjected to multi-pattern distributed load. (September 2020) - Hu, Chang-Fu; Li, Zhang; Liu, Zhi-Wei; Chen, Shen-Shen
Abstract: This paper proposes a numerical and an approximate analytical method for the in-plane non-linear elastic stability of arches under the multi-pattern distributed load. There are six key parts in this paper. Firstly, the approximate analytical solution for funicular axis of arches subjected to the multi-pattern distributed load is derived from the approximations of linear elastic bending moment and horizontal reaction force in the arch end. Secondly, the numerical solution of the in-plane non-linear elastic equilibrium equation is solved by using the shooting method and the bisection method simultaneously. Thirdly, the approximate analytical solutions for the in-plane non-linear elastic equilibrium are derived according to the approximate analytical solution for funicular arch axis obtained from the first part, the simplified strain-displacement expression in Cartesian coordinate system and the virtual work principle. Fourthly, a key parameter is proposed to transform the approximate analytical solutions into the corresponding equations of catenary and parabolic arches. Fifthly, the in-plane non-linear elastic symmetric and asymmetric buckling of arches under the multi-pattern distributed load is derived analytically. Lastly, the in-plane non-linear elastic buckling behaviors of arches subjected to multi-pattern distributed load are deduced based on the obtained analytical solutions. The multi-pattern distributed load with the uniformly distributed load along the span and the uniformly distributed load along the arch is selected as example to verify the proposed method. Comparisons with numerical solutions demonstrate that the proposed approximate analytical solutions for the funicular arch axis and linear elastic horizontal reaction force agree well with the results of Runge-Kutta method and the proposed approximate buckling predictions have sufficient accuracy compared with the results of finite element method in different rise-to-span ratios, relative slenderness and arch axis parameters. Highlights: Approximate closed-form solution for funicular axis was proposed. Numerical solution was obtained by shooting method and bisection method. Analytical solutions of in-plane nonlinear elastic stability were derived. Analytical solutions can be transformed into equations of catenary and parabolic arches. Critical slenderness were derived to distinguish different buckling behaviors.