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.

针对多模式分布荷载作用下拱的面内非线性弹性稳定性问题，提出了一种数值和近似的解析方法。本文包含六个关键部分。首先，通过线性弹性弯矩和拱端水平反作用力的近似推导，得出了承受多模式分布载荷的拱索轴线的近似解析解。其次，同时采用射击法和二等分法求解面内非线性弹性平衡方程的数值解。第三，根据从第一部分获得的缆索拱形轴的近似解析解，得出平面内非线性弹性平衡的近似解析解，简化了笛卡尔坐标系中的应变位移表达式和虚拟工作原理。第四，提出了一个关键参数，将近似解析解转换为悬链和抛物线拱的对应方程。第五，分析了多模式分布载荷作用下拱的面内非线性弹性对称和非对称屈曲。最后，基于所获得的解析解，推导了承受多模式分布载荷的拱的面内非线性弹性屈曲行为。以跨距为均匀分布，拱形为均匀分布的多模式分布荷载为例，验证了该方法的有效性。